e . ,
Bel {i~ If cO~) te'fl:,i ·~!;;i\!!) ~'I ~,~i_."!i;~~
,
.
8 March 19.7-7.' 81 :JSD: j'o~'s~ii
..
.'
.
Memo To:
Mr. L. Hochreiter
Copies To:
Messrs. W. Byrd, T. Eidson, G. Grimes, L. Lortz, O .. K. }'1ccaskill" W. Thatcher
~ailboom'to Fuselage'Atta~h~;~t Holes -
Subject:
~Maximum
Enclosure:
Allowal:le Diameter
Table I
Th,~
various helj;C:;:~lf:t-er maintenance manuals do ~ot have a consistent for al16w q bll; hq.le 55 zes for the four tailboom to fuselage attachment holes .. ":Also, the manner in which these allowables are pres~nted is not qonsistent.
ba~seline
For .. field maint~nance p~rposes, the Airframe Structures Group rec._Dm.mends the maximum allo\-iable hole size be set at .. 015 above ·the--blue" prin't minimum hole Cli.arn~ter for these attachment holes. It is suggested that these allowable ho~e sizes be pres2nted in th~ ,.tciilboom' s,ection of the ,various maintenance-manuals and hole allowables for both the tailboom- ana. fus.elage be addressed in' this sec~ion. "
'.
The Airframe Structures Group also recomm~nds that bushing of,' the~e holes by l-iaterial Review.Board <1-IRB) disposition be limited to holes t.rt'at: exceed one hc:!lf (bl)' the .difference between. the maximum hole d~.~meter allowed fbI:' wear 'and the maximum blue print .hole diameter. ''', ~,
'
Tabl.e I present~ the reco~.!I\ended maximum hole diameters for field .f.\a.ir.tenance pu~,p~~~s ai{d J.iRB o.ction. (o/fl1~you .pleas~; .. z!~s:nt the aliowable wear limits from Table I to the Serv~ce Eng"~n:~er1ng Sroup and request that they revise the maintenance manua"ls accordingly.
-,'
-~
,I'
o.
•
.
Baker Group Engineer Airframe structures Se~ior
•
•
•
---TABLE I
T_A_I_LB_O_O_M~,_T_O~F-U-S~EL-A-G-E--A-T-T-A-C-HM_E_.N_T__H_O_L_E_S__- __MA_X_I_~~1U_M_I-A-L-L-O-W-A-BL-E~.-H_O_L~E__S_I_Z~E~__------__---~~~i)
______- r___
Maximum
Maximum Allowab1e Hole Size· -
Model .'
( Re f • )
D/P Hole Size
Allowable Hole Size .(l"!eld Maintenance)
205
205-030-713 2'05-031-801
.501/.506 .376/.381
• 516 • 3 91
206
206-031-003 206-031-004
urr
.376/.378
.391
206-033::-003 206-033-004
Upr & Lwr
.376/.378
.391
.384
212-030-156
Upr L.H. Upr R.H.
.563/ .568· .501/ . . 506
.578 .5).6
.573 .511
Drawing No.
206L
212
I
& Lwr
(MRS
I .. ~
Action)":"~'11 "
, .
'i
!:~:;
(4 lIole,s)
(4 Holes)
212-030-128
.3£1 .386 ------~--------~----r_----------------~-----. -~----~--~/.~~--------~·-·-·--~------------_. 214 214-030-24B Upr L.H. • 7 50/ • 7 5 6 •7 65 • 7 6' 0 214-030-314 Upr R.B. .625/.6jl .640 .635 & R. H.
.376/.3r.~
214-030-315
Lwr L. It. & R. H •
.625!.6~1
.640
.635
209-030-11::
Upr " Lwr (4'Holes)
.5:01/.506
.516
\ .511
,Lwr L. Ii.
I
AII-1G
~
------~-------------+-----------------~.--_--_--~I~--~--------------------+-------------~ 209-031-113 .635 Alt-1J I Up,;- L. n • R. H. &
2'0 9- 0J1- n7G
Lwr Tit H,
&
R. H.
.625/.631 .563/.568
.640 .578
.573
209 .. 031-000
________ r ___________ • _______________~~_~__----·~----·~------------------_r------------_a
209-033-002
All-IS
209-033-114
upr & Lwr
209-033-810
(4 Holes)
~·501/. ,
506
.516
.511
209-033-811
-------~--------~----+----------------~------.------~------------------~--------------~ .635, .~JI-IT 209-031-227 Upr L.H. & R.B. .625/~631 .640 .573 209-032-000 Lwr L H & R.B. .5'53/.568 .578
(. "
----.
\___·__·____ . . .___ . -,.---...!..--------~----J....------I
-----
t
\.
~'
. • 51l,~ :" • 3 8 6·... '~:,:_:j: f'~ "'!:. · ' -''''':1,):,' .384 ,,'
·0
lNTER-OFFJCE MEMO
9- March 1978 8l:GRA: jo-705
Memo To:
Production Airframe Stress
Copies To:
Messrs.
o. o. n..
Subject:
Baker, P. Bauer, L. Lortz, K. McCaskill, J. McGuigan, Pos te: r, r:. Sc r.oq Q!1, H.. !~COHl;1
MINIMUM Thickness of Aluminum Airframe Parts
The purpose of this memo is to establish the procedure to use when specifying thicknesses and conducting analyses on Aluminum Airframe parts. Airframe Design prefers that we sepcify the MINIHUH thickness required for structural consideration. Therefore, the procedure to use for analysis of all Aluminum Airframe parts with the exception of basic extrusion (shapes), drawn tubing, and sheet stock, shall be the same as defined for chern milled Aluminum parts in BHT IOH 8l:GR~:jo-693 dated 3 February 1978 and repeated below. Analyses of basic extrusion (shapes), drawn tubing and sheet stock should be conducted using NOMINAL dimensions .. Calculate the required thickness for struct~ral consideration based on standard analysis procedures. Specify the MINIHU!t1 thickness to be the required thickness minus the tolerance shown bela,'; : Thickness Range
I
{
.012 .037 .046 .097 .141 .173 .204 .250
to to to to to to to to
.036 .045 .096 .140 .172 .203 .249 .320
Tolerance
.002 .003 .004 .005 .008 .010. .011 .011
Note: If it is possible to hold htcr tolerances than shown above, use NOMINAL thickness in the analysis. The above table is based on the standard tolerances specified for 36 to 48 inch wide aluminum sheet stock shown in ANSI H35.2 1975. It is standard practice to use nominal sheet stock thicknesses for analyses with the above tolerances. Therefore, the same tolerance is acceptable for analysis of other aluminum parts.
Page Ttvo
9 March 1978 81:GRA:jo-70S
.l'-\.s an example t If you calculate treqd = .035 in., you should specify tmin = .035 - .002 = .033 in. The tmax will be based on Design and Manufacturing considerations. The final analysis will be based on tmin + tolerance from the ~bove table. For our example, the analysis would be based on tmin + ttol = .033 + .OOL= .035 in. It is in our best interest to try to influence Design and Manufacturing to maintain as tight a tolerance as possible in order to minimize weight.
G. R. A sm~ller, Jr. Group F.nqinccr Production Airframe Stress
Bell Helicopter i i :):1 i it.J: I INTER-OFFICE MEMO
19 June 1985 81 : .:'1EG : q 1 v- 21 5
~'1E~10
T. Attridge, C. Baskin, R. Battles, J. Braswell,
TO:
M. Ernest, W. Koch, L. Lortz, M. Lufkin, R. Murray. P. Patel, G. Perry, J. Reynolds, E. Schellhase, W. Sundland, K. Tessnow, R. Wardlaw C:JPIES 70:
R..';lsmi l1er, R. Barrett, F. House, A. Fountain, L. Lynch, E. Rose1e"r, O. Sims, H. Thomas
Diagram for Metallurgical :est Locations on Forging/ Cas tin 9 0r avii n9s
~UBJEC-;-:
~ffective
immediately, all new forging and designated casting drawing3
shall include a diagram which defines locations for metallurgical tests. ~ocat~ons
•
for grain flow, tensile bars, and microstructure Evaluations, 5 ha 11 be s hOltm i n th e d i a gram .
a sap p1i cab 1e ,
r all forgings, a ll?rototype Forging Test Diagram ~ shall be required. 7his diagram will be similar to the x-ray diagram required for most castings. 1\'/0 or :nree reduced size vielt/s of the ;::a(t ''''; 11 be reQuired :0 s~ow grain flow, ~ensile bar, and microstructure test locations. il
or primary Ir~·:AP" part shall have Since these castings ~lready require an x-ray diagram, tensile bar and microstr~cture test ~:Jcatisns c,)n be snovm on the viettls of the dia"9ram.
'=3st~ngs
=n
one snd
','/hich make a critical
":<_?AY lHin
FOU~JDRY
1t~~AC'
CONTROL TEST DIAGRAW'.
tQl ~aterials Group will coordinate with Stress, Desi;n Group, ~he ~etallurgical Laboratory to determine and designate the be tested. The Metals Grou8 ~ill ~ark ~he test print for the Design Group to incorporate on the drawing prior to submitting to the Check GrouD.
~prcJr,ate ~reas !o ~ocations on a check ~ngineering
~his information on the engineering drawing will eliminate oroblems encoun[ered over the past years.
o
Metallurgical test locations designated with design and stress input will insure that critical or highly stressed areas are adequately evaluated. This v-1i1l eliminate requesting additional tests after reviewing pro~otype forging or foundry control casting reports.
... 10 ,June 1985 3 1 : 2·1EG : Cj 1 ~J - 21 S
?age 2 .~
o
Test locatiol1s on ~~.e cra'.-/ina \.Ji 11 ::ssure :i1at -~rts~(e ella 1ua ted t:le same ·..:nen ;Jroduced by ':';:0 v::!ldors ::r '.'ihen vendors are changed.
o
Cha~~es
for destructive ~2StS by vendors can be compared and cantroiled at time or '~;'.lOte. Charges. (:7etalL1rgiC31 f:\/a1uations vary considerably from vendor to vendor aeoending on the number of tests they perform. When the same tests are soecified for each vendor, direct comparison of costs can be ~ade.
The requirements for the forging and casting teSt diagram ·. ii11 be incorporated into the DRM at next revision.
7)1.~b!~ .1. t. llreene Group Enqineer
~i:r Director Vehicle Design
•
j-1etal i·~aterials
.~..:_", .. ;;.J.
,
. BELL HELIC8PTER COMPANY Inter-Office Heme
•
5 December 1974 81:LL: 5w-3013 Memo to:
Airframe Desi&n Gr.up
Copies to:
Messrs. R. Alsmiller. O~ Baker, D. Braswell, B. DeLorme, T. Eidson, E. Fischer, D. Higby, L. Hochreiter. R. Lynn, O. McCaskill. J. McGuigan. D. Poster
Subject:
STRUCTL~AL
NUTPLATE POLICY IN AIRFRAME DESIGN
It has recently been established that the airframe drawings have an inconsistency in types of nutplates/hole sizes in both basic structure and removable panels. This memo is written to clarify the airfram~ g·roups position pertaining to structural applications and standardization of nutplates. For all future designs the following parameters will be used for hole sizes and nutplate selection except where structural requirements dictate a closer tolerance hole to guarantee int~grity of the airframe design. The use of reduced spacing nutplates and regular fixed nutplates will be used only with the approval of airframe supervision.
For 3/16 inch threaded fasteners using nutplates:
a.
In the base structure (frame caps, doublers, etc.), use a .193/.198 diameter hole and a floating type nutplate. Do not attach nutplates to materials less than .032 thickness for structural application.
b.
In the removable pariel/door/structure, u~a .203/.208 diameter hole. A larger hole might be authorized if stress levels permit.
For 1/4 inch threaded fasteners using nutplates:
a. -,
/t
. h.
In the base structure, use a .256/.262 diameter hole and a floating, type nutplate. Do not attach nutplates to materials less than .040 inch thickness for structural applications. In the removable panel/door/structure, usa a .264/.270 diameter hole. A larger hole eight be authorized if stress levels permit.
For 5/16
a.
h. ~
inc~
threaded fasteners using nutplates:
In the base structure, usc a .316/.322 diameter hole and a floating type nutplatc. Do not attach nutplates to materials less than .050 inch thickness for structural applications. In the removable panel/door/structure, use a .327/.333 diameter hole.
.. A'larger hole might be authorized if stress levels permit.
Page 2
S December 1974 81:LL:sw-J013
For 3/8 inch threaded fasteners using nutplates:
•
a.
In the base structure. use a .378/.384 diameter hole and a floating type nutplate. Do not attach nutplates to materials less than .050 inch thickness for structural applications.
b.
In the removable panel/door/structure, use a .391/397 diameter hole. A larger hole might be authorized if stress levels permit.
Bell Hesicopterii~:;ii!·J:J DIVISIOn 01
It:.tlrotl Inc
INTER-OFFICE MEMO
3,.February 1978 Bl:GRA:jo-693
-
.
Memo To:
Production Airframe Stress Group
o.
Copies To:
Messrs.
Subject:
Minimum Thickness on Chern r4illed Aluminum Sheet or Wrought Alloys
Baker, P. Bauer, L. Lortz, O. K. McCaskill, J. McGuigan, D. Poster, E. Scroggs,.R. Scoma
Due to the increased use of chern milling for weight and cost reduction, it is necessary to clarify the Structures Group position on the chern milled thickness to use in analyses and reports. The following procedure should be used when specifying thicknesses and conducting analyses on sheet or wrought aluminum alloys.
-. ,--~
Calculate the required thickness for structural consideration based on standard analysis procedures. Specify the minimum . thickness to be the re~uired thickness minus the tolerance shown below: Thickness Ransr e .012 .037 .046 ·.097 .141 .173 .204 .250
to to to to to to to to
.036 .045 .09'6 .140 .172 .203 .249 .320
Tolerance .002 .003 .004 .005 .008 .010 .011 .013
Note: If it is possible to hold tight.er tolerances than shown above, it is permissible to use nominal thickness. The above table is based on the standard tolerances specified for 36 to 48 inch wide aluminum sheet stock shown in ANSl H35.'2 ~ 1975. It is standard practice to use nominal sheet stock thicknesses for analyses with the above tolerances. Therefore, the same tolerance is acceptable for analysis of chern milled aluminum parts.
I
3 February 1978 Bl:GRA:jo-693
Page Two
~
As an example,
-
If you ~alculate treqd = .035 in., you should specify tmin = .035 - .002 = .0)3 in. The tmax will be based on Design and Manufacturing considerations. The final analysis will be based on trnin + tolerance from the above table. For our example, the analysis would be based on tmin + ttol = .033 +" .002 = .035 in. It is in our best interest to try to influence Design and Manufacturing to maintain as tight a tolerance as possible in order to minimize weight.
~(aL~~j
G. R. Alsmiller, J~ Group Engineer Production Airframe Stress
Jell
~1ay
1988
MEMO TO:
ALL
1Oth
STRUCTURES
HeticD~ler II ~ , , ;. •
AND
DESIGN
(INCLUDING
LIAISON)
PERSONNEL,
G.M. GRITZKA, R. FEWS FROM:
A. WATERHOUSE
SUBJECT:
USE
OF 7075-173 FOR PRIMARY AND CRITICAL PARTS
A copy of the attached memo must be inserted in the rel evant pl ace(s)
of all copies of Structures and Design manuals, i.e. the Airframe Design Manual, Rotor Systems Design Manual, Landing Gear Design Manual t Structural DeSign Manual, Fatigue Design Handbook, Rotor Stress Group Manual, and any other des; gn or anal ysi s gui del ines used.
A. Waterhouse
-
Chief of Structures \
Attachment:
AW/dd/19/031
rOM F. Wagner to Vehicle Design Managers, "Shot-Peening Requirements for Primary and Critical Parts made from 7075-T73 11 , 27 April 1984.
.. Bel.' Helicopter' j j:; itt.] :t OMIIOI'IOf f.-,ron Inc,
e,
INTER·OFFICE MEMO
April 27,.1984
Memo
S
to:
Messrs. B. Alapic,
s.
Rober~st
o.
Baker, W. Cresap, R. Duppstadt, E. Roseler
Copies to:
Messrs. C. Davis, M. Glass, R. Lynn, G. Rodriguez, S. Viswanathan
Subj ect:
Shot-Peening Requirements for Primary and Critical Parts made from 707S-T73
'
I
\,
The following normal design procedure is to be implemented on all future designed 707S-T73 parts as a result of a recently completed fatigue committee study: All primary and critical parts made from 707S-I73 will require BPS 4409 shot-peening.
•
Since this is a structural shot-peening as compared to Mil Spec peening, those parts which require fatigue testing will be tested in the peened condition. Field repairs of damage will only be authorized in locations on the subject parts which would not require a peening of the repaired area. You are requested to inform your designers and insert a copy of this memo into the DeSign Manual of each appropriate Design Group.
Director Vehicle Design
r""'c:. _ _ _ _..........- - - _
..
,
.II
'OATt - -_ _- - - - - ftfro;'tT _ _ _ _ __
IotOO[L
.-t:VIS(O _ _ _ _- - - - -
THItEl\DJ::n CONNECTIONS The {ollo\-,.'ing rules are sta.ted !~r the case of threaded· CO:1~~zctions betv.rcen 1'\\'0 tubes subjected to axial tensile {orcc. P. For bQlt-andnut asscn1blics, the same rul:s apply, by substituting Di = O.
~hc.
InvestiGate
(1)
follovling stresses:
Tensile stress in outer tube at root section of thread
p
ft .'
(2) Ten.sile stress ,in outer tube at relic! groove
(3)
Tensile stress in inner tube at root scctiO:1 of thread
." .
D~J , .... ~
,
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.
10400tl.
R(VaStO _ _ _ _ _ _- -_ __
-e
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....
'
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TI-IREJ\DED CONNECTIONS Conttd
"
(4) 'rensilc stress in inner tube at relief groove
:
.. ...
(5) Shear stres s ac ros s threads. along 2. cylindrical surface having a diameter c,:,.ual to the lninixnum pitch' diameter •
.
F--" LJf\H_OJ\DED '. -TOO-fH HEiGriT ..- -'TOADED TOOTH HEIGHT
. ,I
Unloaded Tooth Height = 1~1inimun) ma.jor diamete:- of i~ner tt:be'min'·.!.s ma)drnum minor dia.metc r of outer tube. (- ~ - :'-
Loa.ded Tooth H clGM ;
Unlo~de~ too~h heiGht mim:..s!j ~::-:·.:.s 6 1 '.
~..
0
.
. . (6) Bearing stress on the surface of contact bctv,:~cr.. i~ne= n.:1d outer threads.
,
p
where:
L-lcnC!h ,of cn:;."q;cmcnt f.j,=~la~tic c.~c;c~sc i~ diarnctcro! inner tt.:oc
IJ. () = e 1.1
5
lie inc,. c :l. ~ c i n Gi ~L mel c r o!
0
ul e r t l\ be
•
... DATE: _ _ _ _ _Ft~vl1[O
_ _ __
RErORT ________________
_ _ _ _ _ _ _ _ __
MOOfL,
RCVlStD _ _ _ _ _ _ _ __
.----.-
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.. _.;.._
~
______________
.
....
i
Tf-lREi\DED CONNECTIONS Contto. •
(7) 11:oop compressi.on
•
..
stre~s
in inner tube
.. The corresponding elastic decreas'c in ,dia,meter &...
'.'
or inner tube is
•
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-. :
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. ...
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(8) Hoop tension str'ess in outer tube '.
...
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..
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.
i:;~';:-~::ll ~h:-:::a-:' ?:1:1
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18
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..
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• 03969 .. 04410.. .04962 .05670 .06107
.06615
.' . = ..
",
. d
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.. O,r....,.,,,... Z .~
• 0 171;0 • 02030 ....... .. 02436 .02706 .03045 • Oj~SO .03747 .04059
.072.17·
.04429
.07939
.04871 .05·113 .060£:9 .06959 .08 119 .09;-i3
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-'~==============~-.-~~-~~.-~======================~
STANDARD DESIGN BEND RADII& ~I"£aL\L AN 0 CONDITION 1-_-r-_~_-r-_-""_'"""l¥-fiiF~;;:IA;::.:L:.......;.r~J.rc:;:;;';::":';""':'::"""".:.rS~S--'_--'_ _r----'r--_"'-_"'-_""--_-I FOR.'£O AT ROOM T£.MP .012 .. 016 .. 020 .025 .032\.. ~~ .. OS~ .063}.071 .. 080 .090 .100 .. 125 .160 .190\-250
202 .... 0
.03.03 .06 .06 .0& 1.':9 .06 06 06 .09 .12 !.l6 l 50S !-O ;-:--_ _ _ _. t...Q.~_ ._Q.~~'_ 03 06 I 06 u50sr.:Hj4.1 ,,03 03 .06 .06 .06 +.09 .02_..Jl.~_ .02 .03 ... OJ"I :0:" M 50S ~-H~4· I 606 ,_--=0~_ _ _ _-i~03~+--=O~3......r...:0~3-+.~O:-:=3:--t~.06 i.06 N'606:'-T( l .06.06 ..... 06 -,-09 _.._12 I 16 .06 .06 .06 .06 .06 1.09 707i-T{ I .09 .09 .09 .l2 .16 .25 A~I..::!tl
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rAZ:i~a~o
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:31
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.. 62 .• 15 1..09 .1.2
.09 1.16
25 .25
.:5 .. 28
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• .LO .31 .09 .L6 .09 09 31 .16 .37
.US .37
Jig
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I-50 !.1S r.81 \1.00
.25 .37 .12 .25 .l2 12 .l7 .25 .50
.25 .SO .16 .25
• .Ja I .... ~ .90 .2S .. 25 ..J8.38 ..56
.Jl .:"S
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.. 31
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I!, CO?!E'_;.:..A_ _
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con A
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~ rrYPJ:
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.09 .18 .06 .09 i.~_6 06 ,18 09 .25
.e32 .. 0l.4(] .osa .06':' .080 .100 .. 126 .. 177 .200 .225 .250 .3t2
r'I...:O~~..:::.8-T.:..;:O=6~z..7.J.i.0_B_Or.l::.:0_0;....p:._..!!!..1: .:~S: .-;-:•.!.:L.:.60:::...;-:.:..2I,i;.O.;;.O+.:..:2:..:S:.::2~.3:.,:l:.,::9-;.::.'J~6=-oo.;.l..;.•. ;.!.to;.:s:.;lr,. .;:.~. : . 50=+-=.:.;:s;.;:.6.:.2.+-1_-;-1__! _~
costa 0721 090 .OS~ ~0721_090
II!. COMP 0
112 .lu!i .180
11.S 1.253
LJ5S
.t1.2 .l!.l4 .1801..225 1.2831.355
.LaOO .. tcsol.soo .6251 .40ol.~so .5001.6l5
*FOR NON-SnucnJRA.t. USE ONLY
TITANIUM CONDITION
COMMON
DESIGNATION
TYPE I, COMP A TYPE I, COMP B TYPE I, COMP C
~ ~~
COMMERCIALLY PURE
TYPE II, COMP A Ti-SA1- 2. SSn TI'PE II!, COMP C Ti .. 6Al-4V TYPE Illy COMP D Ti-6Al-4V ELI
~ C
1.1.1
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SHEET 1 REVISED. BELL DESIGN STANDARD
COOE 'DENT .1'40. 91""
Drawinq Number.-
BEND RADII-STANDARD DESIGN
160001 2 2 SHEET
OF
.'
Bell Helicopter' i
j:, i.t·;:)
STRUCTURAL INFORMATION MEMO NO TBD
July 26th, 1993
•
MEMO TO:
BHTI/BHTC AIRFRAME STRUCTURES GROUPS
COPIES TO:
G.R. ALSMtLLER, K.M. STEVENSON
SUBJECT:
MINIMUM THICKNESS FOR STRUCTURAL STATIC ANAL VSIS OF AIRFRAME pARTS.
The purpose of this memo is to establish the policy with respect to the thickness to be used in structural static analysis and reports. Analyses of all airframe forgings, castings, machinings and chern-mill parts shall be conducted using minimum drawing thicknesses. Analyses of stock material (such as extrusion. drawn tubing, sheet. plate, bar, etc) should be conducted using nominal thicknesses.
Airframe Structures BHTI
GUkd
•
Technology BHTC
-
~ m, )''-f/ ~~
lh-/d-#'1J~ standard tolerances/sheet and plate
61,1l.
TABLE 7.7 Thickness TolerancesID ",UOYS 20.... 202~, 2036, 2124, 2219. 300", SQ.S2. S083, 5086, .51 SA. 5252, 525", S~. $4S6. .5652. 606'. 7075. 7G79, 717.. ....ND BRAZING SHEET NOS. 11. 12. 21. 22. 21. AND 24.. NOTE: AlSO APPLICABLE TO THE ALLOYS USTED WHEN SUPPLIED AS ALct.AO. SPECIFIED WIDTH-i ...
Sprc.FIED THICKNESS
In.
0_ .s.c
Up , thru 18
thru
O"er 60 thru
60
66
: Ow.r
66
i
thru : 72
Ov.r : O_r 72 i 78 thru thru 78 ! U
i
Over . Ow« 84 iI 90 thrv thN
90
O".r
O"er
I
Owr
96 112 I 1"" thru i thrv 1 thnt 132 II 1" 156
I
96
lOver 156
I thru 168
I I
TOURANCI-in, p'us ..... nUn••
,001
.0015 .00.4 .005 .00S
t:
!
'e
.006
.0035 .00.4 .00,( .004 .005 .00"'5 ; .0045 : .OOS
.006 .007 .007
.006 .006 .007
.006 .006 .007
.001
.007
.010 .012
.012
I .OOS
.007 .il09 .013
! .007
: .009
j
! .013
i .013
0.321-0."38
.019 .025 .030
.019
.035 .0.40
.025 ; .030 .035 .0.40
.0.(5 .052
2.251.2.750
.0-'5 .052 .060. 1 .07S·
2.751 ·l.ooo
.060 .075
.090
.090
3.00\·-4.000
.110 .125 .135
.110 .IlS
TABLE 7.9
.005 .005
! .135
.011
.019 .025
iJm
.005
I
! .008 i .OO~ I .011
' .010 .Otl
i .OT5 .013
I
j
.013
1.025
.016
.018
.018
.020
!
J
I
. .006
.006
: .Q06
.001 .007
I .007 .007 j.Ol . .
.020
.023 .025
.0'23 .030 .OJ7 .0.45 .051
.Od
052
.045 .052 .060 .075 .090
.0.(5 .057 .060 .075 .090
.045 .052
I .075 I .090
.052 .060 .075 .090
.110 .125 .135
.110
1 .110
.110
.no
.1.40
.t25 .1l.5
.125
.125 .135
.150
1..060
I
i
I
I .125 .135
Width Tolerances
I .020
.013 .015 .017 .018 .020
i .0.(5
I
: .018
.016 .016 .020 1 .020
I .008
t .0.40
i
I
.012 .012
I .008
.007
030 1. I .OU
.035 .040'
.009 .011
.008 .008
. .030 .035 .0.40
1
.007 Jla1 .008 .012 .012 .016 .016
.025 .030 .035 .0.(0
0030
....
.006
J
: .007
.012 : .018
.016
i .018
.016 ; .017
.011
j
.017
.Q1B
.018 .020
; .022 1.02",
!.020 !
.019
I
.013
.. I
.IlS
.0lO I
.035 .040
.060 .07S
.090
.060 .070 .080 .100 .120
.160
: .Q2.5
I .030 t .037 i .045
.025 .030 .031 .0",5
1·052
.052
i .060 i .070 i .OBO
.070
.060 .080
.•100 ; .120
! .120
i .ISO .140
! .160
•• 100
!
.026
! .028
i
..
I I
I
I
1.042'
: .053
j .o~5 l.cu~
: .057
! .055 ! .055 ! .065 I .070 : .065 : .065 ; .075 . .o8Q
; .071 t .088 : .D98
! .025
i .030
I .026
I .035 .O~S
.075 .08a .100 .125
..033 : .035 I
.0~5
: .075 I .088 : .100 : .12.5 .150
.150
I
.UQ
.160
i
.150
.. 160 1.170
.160
I
I .Q23
i
1.020
i .019
i .025
.ou
.OU .016
1
..
.
~ ......
i
j I
.006
.008 i .010
S.OOI·iI.OOO
.-
: .004
.006
.00.45 ! .0045 .006 I .006
".001·5.000
!
I .006
I
1.376.1.625 1.626·1.875 1.876·'2.250
•••• :
.00
0.126·0. t 40 0.1-41.0.112 0.173·0.203 0.20"·0.249 0:250·0.320 0.439·0.625 0.626.0.875 0.816·1.125 1.126·1.375
I
j
1.035 ! .038 j
.0.43 .054
.oU
I.059 l.O9O
; .067
.t08
1 ....
i
.:60
I
SHEARED FLAT SHEET AND PLATE SPEC1flED WIDtH-id.
SPECIFIED tHICKNESS
,
Up ,hru
in.
Ow... 6 thnI 24
Ow., 24 thru 60
0".., 96 thru 132
0'11." &0 ,h,u 96
TCLERANClt.;;-ea.
\ ~
0.006-0.124 o. I 25·0.249 0.2~O.0 .• 99
TABLE 7.10
+~
=~ =~Z
=~
=¥lz =~2 +o/t,
='h6 =-¥.J2
=lh
+-Ma
+~~
Length Tolerances
SHU.IEO FU.T SHEET AND PUTE SPECtflEO UNGtH-in.
SPECIFIED THICKNlSS
h"
Up thru
30
O"er 3Q tlu" 60
0".,
6Q
th,u 120
ay.,
120
thn. HO
OVM
240
thr" 360
0.,., 360 tl"" 480
0 ... 1' £80
th,,, 600
I
Ower 600
,h,,, 720
TOLERANCES:1) -i ...
e
Q.006·0. t 2-' 0.125-0.24.9 0.2!i0·0.499
:: Yi, ::t~2 +~
For all numbertd footnotes, see pa9~ 120
:!:~z
=¥'!Z +~
::~ :t~
+*.
=%2 =~2 +~
=*6 =~2 +*6
!:~2
=~
+~
=%2
-=Yte
~lYt5
+* 119
(e
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.,
.!:-:::: 0.25 0.50 0.75 1.00
0.15
0.20
0.40
0.50
************~***************** 3.0'+ 2.31 *********~*******~ 2.94 2.43 1.68 *':~¥*::= ~~¥-:.'! 2.4e 3.12 1.93 1.2.7 *****~ 2.50 1.57 2.06 1.00 1 .. 74 2.47 0.19 1.29 0.64 1.4'1 1.09 2..10 1.95 1.30 O.9~ 0.52 3.11 0.43 2.99 1.74 1.14 0.80 2.7S 1.57 1.00 0.69 0.35 2.64 1.43 O.8S 0.60 0.28 2.48 1.30 0.80 0.52 0.23 2.32 0.72 0.45 0.19 l.1Ci 2.18 1.0B 0.64 0.39 0.34 2.06 1.OC 0.57 1.95 0.92 0.52 0.30
2.61 1.73 1.21 0.89
2.12 1.29 0.85
*
1.2:3 1.50 1.75 2.00
0.10
0.30
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•
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2.75 3.00 3.25 3.50 3.75 4.00
T a.ble E('(OY"
0'"
RbjP
Pllllt's
O.6a 0.52.
0.40 0.31 0.23
O.e4 0.47 0.33 0.25 0.18 0.13
I
0.18
,
If
t1 i«~
{
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tftf
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S/b= 0.20
0.10
0.15
0.20
0.30
0.40
0.50
a./b==O. 2. 5
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Revision F
"# l1-1--S
·STRUCTURAL DESIGN MANUAL VOLUME I
•
RESTRICTED DISCLOSURE NOnCE THE DRAWINGS. SPECIFICATIONS, DESCRIPTIONS. AND OTHER TECHNICAL DATA ATTACHED HERETO ARE PROPRIETARY AND CONfiDENTIAL TO BEll HELICOPTER TEXTRON INC. AND CO~STITUTE TRADE SECRETS FOR PURPOSES OF THE TRADE SECRET AND FREEDOM OF INFORMATION ACTS. NO DISCLOSURE TO OTHERS, EITHER IN THE UNITED STATES OR ABROAD, OR REPRODUCTION OF ANY PART OF THE INFORMATION SUPPLIED IS TO BE MADE, AND NO MANUFACTURE. SALE. OR USE OF ANY INVENTION OR DISCOVERY DISCLOSED HfREIN SHALL 8E MADE. EXCEPT BY WRITTEN AUTHORIZATION OF SELL HELICOPTER TEXTRON 'NC. THIS NOTiCE Will NOT OPERA TE TO NULLIFY OR LIMIT RIGHTS GRANTED 8Y CONTRACT. THE DATA SUBJECT TO THIS RESTRICTION IS CONTAINED IN All SHEETS AND IS DISCLOSED TO PERSONNEL OF 8ElL HELICOPTER TEXTRON INC. FOR THE PURPOSES(S) OF INTERNAL USE AND DISTRIBUTION ONLY.
Bell Helicopter' i i:j i .t•]:I
Bell Helicopter iii:; i it·]~J DlIiIslOO 01 Te~tron Inc.
INTER-OFFICE MEMO
27 October 1977 S1:GLJ:jo-666
Memo To:
Holders of Structural Design Manual
Subject:
Revision A
Changes to the Structural Design Manual made by Revision A are listed below. Please remove superseded pages and add revised pages and new pages to your copy_ Revised pages: 2-1, 2-5, 2-6, 3-18, 3-19, 3-27, 3-57, 4-13, 4-17, 6-1, 6-2, 6-44, 6-52, 6-58, 6-63 thru 6-67, 6-80, 6-81, 6-82, 7-50, 8-11, 9-9, 9-24, 10-4, 10-20, 10-22 thru 10-26, 10-37, 10-39 thru 10-42, 10-44 thru 10-47, 10-49, 10-50, 10-52, 10-57, 10-58, 10-71, 10-73, 11-9, 11-10, 11-11, 11-73, 11-74, 12-93, 12-94, 13-5, 13-7, 13-57, 14-10. New pages: 10-20a, 10-20b, 13-6a.
dq~
Experi~
G. L . Jor
Approved:
Airframe Stress
Bell Helicopter' itt i tI.]: I DIIIISIon of Texlron Inc
INTER~OFFICE
MEMO
.12 May 1978 8l:GLJ:jo-726
Memo To:
Holders of Structural Design Manual
Subject:
Revision B
Changes to the Structural Design Manual made by Revision Bare listed below. Please remove superseded pages and add revised pages and new pages to ,your copy. Insert this memo preceding the Table of Contents in Volume I. Volume I revised pages: Title Page, v, vi, V111, 2-2, 3-4, 3-11, 3-57, 3-79, 4-16, 4-22, 4-23, 4-34, 4-46, 6-11, 6-15, 6-18, 6-24, 6-32 thru 6-36, 6-38, 6-41, 6-61, 6-62, 6-64 thru 6-67, 6-80, 6-82, 6-87, 6-88, 8-5, 9-18, 9-19, 9-34, 9-38. Volume I new pages: 6-65a, 6-65b, 6-67a, 6-67b, 6-89 thru 6-94. Volume II revised pages: Title Page, v, vi, viii, 10-7, 10-22, 10-58, 10-70, 10-71, 10-75, 11-74, 12-8.
an
G. L.~ Experimental Airframe Stress
Approved:
~l CGiligan; ~~ '~~ o::L' . /
M. J.
Chief of Structures Technology
lleU UeUco,::~ :tcr ij-:);.j Il\)~~ ~] ()IIISlon ollelrronll'lC,
INTER-OFFICE MEMO
9 October 1979 Sl: ELB: jo-908
Memo To:
Holders of Structural Design Manual
Subject:
Revision C
Please ihsert the following revised pages in your copy of the Structural Design Manual and insert this memo preceding the Table o·f Contents in Volume I as a summary of the changes of Revision C .
,
.Additional corrections and suggested inclusions may be submitted to the undersigned for incorporation in the next revision. Volume I changes: Revised Title Page, vi, 2-1, 2-2, 2-3, 2-4, 2-5, 2~6, 3-2, 3-64, 4-24, 5-2, 6-7, 6-8', 6-54, 7-4, ~-5, 8-5, 8-6, 9-26, 9-58
Added Pages 6-93, 6-94, 6-95/6-96 Changed Page number 6-93, 6-94 to 6-97, 6-98 Volume I I changes: ' Revised Title Page, vi, 10-15, 11-18, 11-25, 11-30, 11-31, 1,1-:--j2,",.rl:':~'3~··11--75, ~ ~1~~~,~~-t-~12-3;·-·13:'·21~ '13"':23, 13-47
E. L. Brach Experimental Airframe Stress
I"
APPROVED: . I
\,-'
Te.c~nol<)gy .
,
;";,",,
.... :;i:.:_ •••
.
,..:..
Bell Helicopter i i ~:, i hI.]: I A Sub~lrt.a'y 01
Ie."
nn h\~
INTER·OFFICE MEMO j.
2 September 1983 8l:DL:wc-1475 MEMO TO:
Holders of Structural Design Manual
SUBJECT:
REVISION D
Please insert the following pages in your copy of the Structural Design Manual and insert this memo preceding the Table of Contents in Volume I as a summary of the changes of Revision D. Volume I changes: Revised Title Page, iii, 2-1, 2-4 thru 2-12. Added Pages: 2-13, 2-14, 2-15.
Volume II changes: Revised Title Page, page iii.
D. Lindsay Airframe Structures
Approved:
Manager of Structural Analysis
Bell Helicopter' i ~:, i
hI.': I
j,
INTER·OFFICE MEMO
17 April 1985 81:DL:db-1475
MEMO TO:
Holders of Structural Design Manual
SUBJECT:
REVISION E
Please insert the following pages in your copy of the Structural Design Manual apd insert this memo preceding the Table of Contents in Volume I as a summary of the changes of Revision E. Volume I changes: Revised Title Page, 3-17, 4-11, 4-24, 5-18, 6-13, 6-39, 6-52, 6-56, "6-59, 6-64, 6-73, 6-82, 6-68, 6-69, 7-19, 7-27, 7-31, 8-3, 8-15, 9-52, 9-113. Volume II changes: Revised Title Page, 10-19, 10-21, 10-24, 10-26, 10-28, 10-31, 10-35, 10-39, 10-40, 10-41, 10-45, 10-47, 10-70," 10-71, 11-82, 11-101, 11-112, 11-113, 12-20, 12-.21.
D. L1ndsay ( Airframe Structures
Approved:
Materials
Bell Helicopter' i
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INTER-OFFICE MEMO
81:JCS:jw-2110 22 October 1987
(
MEMO TO:
Holders of Structural Design Manual
SUBJECT:
REVISION F
Please insert the following pages in your copy of the Structural Design Manual· and insert this memo preceding the Table of Contents in Volume I as a summary of the changes of Revision F. Volume I
changes:
Title Page, Table of Contents, Section 1, Section 2, to 3-83, 4-9, 4-11, 6-6, 6-58, 7-9, 7-11, 7-71.
3-54,
3-77
Volume II changes: Title Page, Table of Contents, 10-26, 10-41, 10-47, 11-28, 13-7.
J.
. S
Structural Methods
Approved:
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•
STRUCTURAL DESIGN MANUAL Revision F
TABLE OF CONTENTS VOLUME I
REFERENCES [NTRODUCTION
xiii xv
SECTION 1 - PROCEDURES General Design Coverage Critical Parts Flight Safety Parts Check List for Drawing Review ,Envelope, Source Control~ and SpeCification Control Drawings Stress Analysis Structures Report Introductory Data Body of the Report General Information Structural Information Memos Purpose Preparation Published SIM·s
1.1 1.2 1.2.1 1.2.2 1.2.3 1.2.4 1.3 1.. 4
•
1.4.1 1.4.2 1.4.3 1.5 1.5.1 1.5.2 1.5.3
1-1 1-1
1-1 1-2 1-2 1 5 1-6 1-8 1-8 1-9
1-10 1-11
1-11 1-11
1-11
SECTION 2 - COMPUTER PROGRAMS 2.1 2.2 2.3 2.3.1 2.3.2 2.4 2 . 4.1 2.4.2 2.4.2 . 1 2.4.2 . 2 2.4.2 .. 3 2.4.3 2.4.4
2.5 2.5 . 1 2.5.2 2.5.3
General Computer Facilities Finite Element and Supplementary Programs Finite Element Programs Finite Element Preprocessors and Postprocessors Approved Structural Analysis Methodology (ASAM) The LODAM System The CASA System CASA System Features CASA System Architecture CASA Programs Description The CPS2TSO System Other Programs Miscellaneous Programs Load Programs Unrelated to Finite Element Analysis DynamiC Structures Analysis Fatigue Evaluation
2 1 2-1 2-1 2-1 2-2 2-5 2-5 2-5 2-6 2-7 2-9 2-12 2-16 2-17 2-17
2-17 2-17
SECTION 3 - GENERAL 3.1 3.1.1
Properties of Areas Areas ~nd Centroids
3-1 3-1 i;;
STRUCTURAL DESIGN MANUA.L TABLE OF CONTENTS (Continued)
SECTION 3 - GENERAL (Continued) 3. 1.2 3.1.3 3.1.4
3.1 . 5 3. 1.6 3. 1.7
3.2 3.3 3.4 3.5 3.6 3. 7 3.8 3.9 3.9.1 3.9.2 3.10
3.11 3.11.1 3.11.2 3.12 3. 12. 1 3.12.2 3.12.3 3.12.4 3.12.5 3.12.6 3.13 3 . 13.1 3.13.2
3.13.3 3.13.4 3.14
Moments of Inertia Polar Moment of Inertia Product of Inertia Moments of Inertia About Inclined Axes Principal Axes Radius of Gyration Mohr's Circle for Moments of Inertia Mass Moments of Inertia Section Properties of Shapes Bend Radi i Hardness Conversions Graphical Integration by Scomeano Method Conversion Factors The International System of Units Basic SI Units Symbols and Notation Weights Shear Centers Shear Centers of Open Sections Shear Center of Closed Cells Strain Gages The Wire Strain Gage The Foil Strain Gage The Weldable Strain Gage The Strain Gage Rosette Strain Gage Temperature Compensation Stress Determination from Strain Measurements Acoustics and Vibrations Uniform Beams Rectangular Plates Columns Stress and Strain in Vibrating Plates Bell Process Standards
3-2 3-3 3-5 3-6 3-7 3-7 3-8 3-9 3-10
3-36 3-36 3-36 3-45 3-45 3-45 3-45 3-52 3-52 3-53 3-68 3-70
•
3 70
3-71 3-71 3-72 3-73 3-73 3-76 3-79 3-81 3-81 3-84 3-85
)
SECTION 4 - INTERACTION 4.1 4.2 4.3
4.4 4.4.1 4.4.2 4.5
4.S.1 iv
Material Failures Theories of Failures Determination of Principal Stresses Interaction of Stresses Procedure for M.S . For Two Loads Acting Procedure for M.S. for Three loads Acting Compact Structures Biaxial Stress Interaction Relationships
4-1 4-2 4-3 4-5 4-11 4-11 4-12 4-12
•
STRUCTURAL DESIGN MANUAL Revision F TABLE OF CONTENTS (Continued)
SECTION 4 - INTERACTION (Continued)
)
4.5.1.1 4.5.1.2 4.5.1.3 4.5.2 4.5.3 4.5.4 4.5.5 4.5.6
Max Shear Stress Theory Interaction Equations Octahedral Stress Theory Interaction Equations M.S. Determination Uniaxial Stress InteractiOn Relationships Thick Walled Tubular Structures Unstiffened Panels Unstiffened Cylindrical Shells Stiffened Structures
4-13 4-13
4-17 4-19
4-19 4-19 4-20 4-20
SECTION 5 - MATERIALS
)
5.1 5.1.1 5.1.2 5. 1.3 5.2 5.2.1 5.2.2 5.2.3 5.3 5.3.1 5.3.2 5.3.3 5.3.4 5.3.5 5.4 5.4.1 5.4.2 5.4.3 5.5 5.6 5.7 5.. 7.1 5.7.2
Genera 1 Material Properties Selection of Design Allowables Structural Design Criteria Material Forms Extruded, Rolled and Drawn Forms Forged Forms Cast Forms Aluminum A11 oys Basic Aluminum Temper Designations Aluminum Alloy Processing Fracture Toughness of Aluminum Alloys Resistance to Stress-Corrosion of Aluminum Alloys Mechanical Properties of Aluminum Alloys Steel Alloys Basic Heat Treatments of Steel Fracture Toughness of Steel Alloys Mechanical Properties of Steel Alloys Magnesium Alloys Titanium Alloys Stress-Strain Curves Typical Stress-Strain Diagram Ramberg-Osgood Method of Stress-Strain Diagrams
5-1 5-1 5-1 5-3 5-3 5-3 5-5 5-6 5-8
5-8 5-9 5-13 5-13
5-19 5-21 5-22 5-23 5-24 5-24 5-28 5-29 5-29 5-31
SECTION 6 - FASTENERS AND JOINTS 6.1 6.2 6.2.1 6.2.2 6.2.2.1 6.2.2.2 6.2.2.3
General Mechanical Fasteners Joint Geometry Mechanical Fastener Allowables Protruding Head Solid Rivets Flush Head Solid Rivets Solid Rivets in Tension
6-1 6-1 6-2 6-3 6-3 6-3 6-6
v
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DESlG:N· MANUAL
~.'
TABLE OF CONTENTS (Continued)
SECTION 6 - FASTENERS AND JOINTS (Continued) 6.2.2.4 6.2.2 . 5 6.2.2.6 6.2.2.7 6.2.2.8 6.3 6.3.1 6.3.2 6 .. 3.3 6.3.4
Threaded Fasteners Blind Rivets Swaged Collar Fasteners Lodcbo 1ts Torque Values for Threaded Fasteners Metallurgical Joints Fusion Welding - Arc and Gas Flash and Pressure Welding Spot and Seam Welding Effect of Spot Welds on Parent Metal 6.3.5 Welding of Castings 6.4 Mechanical Joints 6.4.1 Joint Load Analysis 6.4.1.1 One-Dimensional Compatibility 6.4.1.2 Constant Bay Properties - Rigid Sheets 6.4.1.3 Constant Bay Properties - Rigid Attachments 6 .. 4.1.4 The Influence of Slop 6.4.1.5 _ Two-Dimensional Compatibility 6.4.2 Joint Load Distribution - Semi Graphical Method 6.4.2.1 Fastener Pattern Center of Resistance 6.4.2.2 Load Determination 6.4.3 Attachment Flexibility 6.4.3.1 Method I - Generalized Test Data 6.4.3.2 Method II - Bearing Criteria 6.4.4 Lug Design 6.4.4.1 Nomenclature 6~4.4.2 Analysis of Lugs with Axial Loads 6.4.4.3 Analysis of Lugs with Transverse Loads 6.4.4.4 Analysis of Lugs with Oblique Loads 6 .. 4.4.5 Analysis of Pins 6.4.4.6 Lugs with Eccentrically Located Hole 6.4.4.7 Lubrication Holes in Lugs 6.4.5 Stresses Due to Press Fit Bushings 6.4.6 Stresses Due to Clamping of Lugs 6.4.7 Single Shear Lug Analysis 6.4.8 Socket Analysis 6.4.9 Tension Fittings Tension Clips 6.4.10 Cables and Pulleys 6 .. 5
6-6 6-10 6-10 6-11 6-11 6-15 6-15 6-15 6-15 6-15 6-18 6-19 6-19 6-19 6-41 6-42 6-42 6-48 6-50 6-50 6-51 6-52 6-52 6-54 6-55 6-56 6-57 6-58 6-60 6-60 6-71 6-71 6-74 6-79 6-79 6-82 6-87 6-93 6-96
)
SECTION 7 - PLATES AND MEMBRANES 7"1
7.2 vi
Introduction to Plates Nomenclature for Analy~is of Plates
7-1 7-1
•
STRUCTURAL DESIGN MANUAL Revision F TABLE OF CONTENTS (Continued)
SECTION
)
7. 3 7.3.1 7.3.2 7.3.3
7.4 7.4.1 7.4.2 7.5 7.6 7.7 7.8 7 .. 8 . 1 7.8.2 7.9 7.10 7.11 7.12 7.13 7.14 7 . 15
7 - PLATES AND MEMBRANES (Continued)
Axial Compression of Flat Plates Buckling of Unstiffened Flat Plates ;n Axial Compression Buckling of Stiffened Flat Plates 'in Axial Compression (rippling Failure of Flat Stiffened Plates in Compression Bending of Flat Plates Unstiffened Flat Plates, In-Plane Bending Unstiffened Flat Plates, Transverse Bending Shear Buckling of Flat Plates Axial Compression of Curved Plates Shear Loading of Curved Plates Plates Under Combined Loadings Flat Plates Under Combined Loadings Curved Plates Under Combined Loadings Triangular Flat Plates Buckling of Oblique Plates Introduction to Membranes Nomenclature for Membranes Circular Membranes Long Rectangular Membranes Short Rectangular Membranes
7-2 7-3 7-15 7-27 7-34 7-34 7-37 7-50 7-50 7-58 7-58 7-58 7-58 7-70 7-70 7-74 7-74 7-74 7-76 7-79
SECTION 8 - TORSION Torsion of Solid Sections Torsion of Thin-Walled Closed Sections Torsion of Thin-Walled Open Sections Multicell Closed Beams in Torsion Plas·tic Torsion Allowable Stresses
8.1 8.2 8.3 8.4 8.5 8 .. 6
)
8-1 8-2 8-8 8-11 8-14 8-15
SECTION 9 - BENDING 9.1 9.2 9.2.1 9.2.2 9 .. 2.3 9.3 9.3.1 9.3 .. 2 9.3.3 9.3 .. 4 9.3.5 9.3.6 9 . 3.7
General Simple Beams Shear, Moment and Deflection Stress Analysis of Symmetrical Sections Stress Analysis of Unsymmetrical Sections Strain Energy Methods Castiglianols Theorem Structural Deformation Using Strain Energy Deflection by the Dummy Load Method Analysis of Redundant Structures Analysis of Redundant Built-Up Sheet Metal Structures Analysis of Structures with Elastic Supports Analysjs of Structures with Free Motion
9-1 9-1 9-1
9-23 9-24 9-25 9-25 9-25 9-27 9-36 9-40 9-46 9-46
vii
S,TRUCTURAL DESIGN MANU,AL TABLE OF CONTENTS (Continued)
SECTION 9 - BENDING (Continued) 9.4 9.5
9.5.1 9 .. 5.2 9 .. 6 9.6 .. 1 9.6.2 9.6.3 9.6 . 4 9.6.5 9.6.6 9.7
9.8 9.9
Continuous Beams by Three-Moment Equation Lateral Buckling of Beams lateral Buckling of Deep Rectangular Beams lateral Buckling of Deep I Beams Plastic Analysis of Beams Bending About a~ Axis of Symmetry Bending in a Plane of Symmetry Complex Bending Evaluation of Intercept Stress, fo Plastic Bending Modulus, Fb Application of Plastic Bending Curved Beam Correction Factors for Use in Straight Beam Formula Bolt-Spacer Combinations Subjected to Bending Standard Bending Shapes
9-49 9-51 9-52 9-54 9-58 9-58 9-105 9-105 9-107 9-113 9-113
9-116 9-118 9-118
)
viii ~.--~.- .. -
.~
T- . ,
(',;,1' \
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'~~~~.~ STRuc-rURAl DESIGN MANUAL
Revision F
TABLE OF CONTENTS VOLU~1E
II
SECTION 10 - BUCKLING 10.1
')
10.1.1 10.1.2 10.1.3 10.2 10.2.L 10.2.2 10.3 10.4 10.5 10.5.1 10.5.2 10.5.3 10.5.4 10.5.5 10.5.6 10.5.7 10.5.8 10.6 10.7 10.8 10.9
Shear Resistant Beams Introduction Unstiffened Shear Resistant Beams Stiffened Shear Resistant Beams Shear Web Reinforcement for Round Holes Doubler Reinforcement 45° Flange Reinforcement -Shear Beams with Beads Shear Buckling Incomplete Diagonal Tension Effective Area of Uprights Moment of Inertia of Uprights Effective Column Length Discussion of the End Panel of a Beam Analysis of a Flat Tension Field Beam Uprights Analysis of a Flat Tension Field Beam Uprights Analysis of a Flat Tension Field Beam Uprights and Access Holes Analysis of a Tension Field Beam with Inter-Rivet Buckling Compressive Crippling Effective Skin Width Joggled Angles
10-1 10-1 10-1 10-5 10-13 10-13 10-17 10-17 10-20
10-20 10-21 10-21 10-22 10-22
with Single 10-22
with Double with 5i ng1 e" Curved Panels
10-37 10-42 10-49 10-60
10-69 10-73 10-75
SECTION 11 - COLUMNS AND BEAM COLUMNS
)
Simple Columns Long Elastic Columns Short Columns 11.1.2 11.1.3 Columns with Varying Cross Section 11.1.4 Column Data for Both Long and Short Columns 11.2 Beam Columns 11.2.1 Beam Columns with Axial Compression Loads 11.2.2 Beam Columns with Axial Tension Loads 11.2.3 Multi-Span Columns and Beam Columns 11.2.4 Control Rod Design 11.2.5 Beam Columns by the Three-Moment Procedure 11.3 Torsional Instability of Columns 11.3.1 Centrally Loaded Columns 11.3.1.1 Two Axes of Symmetry 11. 1 11.1.1
11-1 11-3 11-3 11-18 11-33 11-72 11-72 11-79 11-79 11-100 11-100 11-101 11-101 11-101
ix
TABLE OF CONTENTS (Continued)
SECTION 11 - COLUMNS AND BEAM COLUMNS (Continued) 11.3.1.2 11.3.1.3 11.3.2
General Cross Section Cross Sections with One Axis of Symmetry Eccentrically Loaded Columns
11-102 11-102 11-109
)
SECTION 12 - FRAMES AND RINGS 12.1 12.2 12.2.1 12.2.2 12.2.3 12.3 12.4 12.4.1 12.4.2 12.4.3 12.4.4
General Analysis of Frames by Moment Distribution Sign Convent ion . Moment Distribution Procedure Sample Problem Formulas for Simple Frames Analysis of Rings Analysis of Rigid Rings with In-Plane Loading Analysis of Rigid Rings with Out-of-Plane Loading Analysis of Frame Reinforced Cylindrical Shells Frame Analysis by the Dummy Load Method
12-1 12-1
12-1 12"-2 12-5 12-7 12-22 12-22 12-53 12-53 12-91
SECTION 13 - SANDWICH ANALYSIS 13.1 Materials 13.1.1 Facing Materials Core Materials 13.1.2 13.1.3 Adhesives Methods of Analysis 13.2 13.2.1 Wrinkling of Facings Under Edgewise Load Continuous Core 13 . 2.1.1 13.2.1.2 Honeycomb Core 13.2.2 Dimpling of Facings Under Edgewise Load Flat Rectangular Panels with Edgewise Compression 13.2.3 13 . 2.4 Flat Rectangular Panels Under Edgewise Shear Flat Panels Under Uniformly Distributed Normal Load 13.2.5 Sandwich Cylinders Under Torsion 13.2.6 Sandwich Cylinders Under Axial Compression 13.2.7 Cylinders Under Uniform External Pressure 13.2.8 Beams 13.2.9 Attachment Details 13.3 13.3.1 Edge Design Doublers and Inserts 13.3.2 Attachment Fittings 13.3.3
13-1 13-3 13-4 13-7 13-15 13-16 13-17 13-17 13-17 13-21 13-43 13-48 13-62 13-65 13-77 13-80 13-101 13-101 13-103 1 103
SECTION 14 - SPRINGS 14.1 x
Abbreviations and Symbols
14-1
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STRUCTURAL DESIGN MANUAL Revision F TABLE OF CONTENTS (Continued)
SECTION 14 - SPRINGS (Continued)
)
14.2 14.2.1 14.2.2 14.2.3 14.2.4 14.2.5 14.2.6 14.2.7 14.2 . 8 14 . 2.9 14.2 . 10 14.3 14.3.1 14.3.2 14 . 3 .. 3 14.3.4 14.4 14.4.1 14.4.2 14 .. 4.3 14.4 . 4 14.4 . 5 14.4 . 6 14.5 14.5 . 1 14.6 14.7 14.7.1 14.7.2 14.7.3 14 .. 7. 4 14.8 14.8 .. 1 14.8.2 14.8.3 14.8.4 14.8.5 14.8.6
Compression Springs Design Formulas Buckling Helix Direction Natural Frequency, Vibration and Surge Impact Spring Nests Spring Index (Old) Stress Correction Factors for Curvature Keystone Effect Design Guidelines for Compression Springs Extension springs Design Formulas End Design Initial Tension (Preload) Design Guidelines for Extension Springs Torsion Springs Design Formulas End Design Change in Diameter and Length Helix of Torsion Springs Torsional Moment Estimation Design Guidelines for Torsion Springs Coned Disc (Belleville) Springs Design Formulas Flat Springs Material Properties Fatigue Strength Other Materials Elevated Temperature Operation Exact Fatigue Calculation Spring Manufacture Stress Relieving Cold Set to Solid Gri nd i ng Shot Peening Protective Coatings Hydrogen Embrittlement
SECTION 15 15 . 1 15.1.1 15.2
14-2 14-4 14-4 14-7 14-7 14-8 14-8 14-8 14-8 14-9 14-9 14-10 14-10 14-10 14-12 14-13 14-15 14-15 14-15 14-17 14-18 14-18 14-18 14-20 14-20 14-22 14-22 14-22 14-26 14-26 14-27 14-28 14-28 14-30 14-30 14-30 14-31 14-31
THERMAL STRESS ANALYSIS
Strength of Materials Solution General Stresses and Strains Uniform Heating
15-1 15-2 15-3 xi
ST.RUCT.URAlDESIGN MANUAL TABLE OF CONTENTS (Concluded)
SECTION 15 - THERMAL STRESS ANALYSIS (Continued) 15.2.1 15.2.2 15 . 2.3 15.2.4 15.2.5 15.2.6 15.3 15.3.1 15 . 3.2 15.4 15.4.1 15.4.2 15.5 15.6 15.6.1 15.6.2 15.6.3 15.7 15.7.1 15.7.2 15.8
Bar Restrained Against Lengthwise Expansion Restrained Bar with a Cap at One End Partial Restraint Two Bars at Different Temperatures Three Bars at Different Temperatures General Equations for Bars at Different Temperatures Non-Uniform Temperatures Uniform Thickness Varying Thickness Linear Temperature Variations Restrained Rectangular Beam. Uniform Face Temperatures Pin-Ended Beams Combined Mechanical and Thermal Stresses Flat Plates Plate of General Shape Square Plates Flat Plates with Uniform Heating Temperature Effects on Joints Preload Effects Due to Temperature Thermally Induced Loads in Material Thermal Buckling
15-3 15-3 15-4 15-4 15-5 15-5 15-5 15-6 15 6
15-6 15-7 15-7 15-8
15-8 15-9 15-9 15-9 1"5-12 15-12 15-13 15-15
)
xii
STRUCTURAL DESIGN MANUAL REFERENCES 1. MIL-HDBK-5, Metallic Materials and Elements for Aerospace Vehicle Struetures, Department of Defense, 1974. 2. Stress Analysis Manual, Air Force Flight Dynamics Laboratory, Wright-patterson Air Force Base, 1969. 3. Astronautic Structures Manual, Structures and Propulsion Laboratory, Marshall Space Flight Center, 1975. 4. Griffel, W., Handbook of Formulas for Stress and Strain, Frederick Ungar Publishing Company, 1966. 5. Roark, R. J., Formulas for Stress and Strain, McGraw-Hill Book Company, 1965. 6. Baker, E': H., Capelli, A. P., Kovalevsky, L., Rish, F. L., and Verette, R. M., Shell Analysis Manual, North American Aviation, Inc., SID-66-398, 1966. 7. Reissner, E., On the Theory of Thin Elastic Shells, H. Reissner Anniversary Volume, 1949. 8. Clarke, R. A., On the Theory of Thin" Elastic Toroidal Shells, Journal of Mathematics and Physics, Volume 29, 1950. 9. Ramberg, W. and Osgood, W., Description of Stress-Strain Curves by Three Parameters, NACA TN 902, 1943. 10. Switzky, H., Forray, M., and Newman, M., Thermo-Structural Analysis Manual, Technical Report Number WADD-TR-60-5l7, 1962.
)
11. Gerard, G. and Becker, H., Handbook of Structural Stability, Part I - Buckling of Flat Plates, NACA TN 3781, 1957. 12. Becker, H., Handbook of Structural Stability, Part II - Buckling of Composite Elements, NACA TN 3782, 1957. 13. MIL-STD-29A, Drawing Reiuirements for Mechanical Springs, U. Govt. printing Office, 962.
s.
14. MIL-HDBK-23, Structural Sandwich Composites, Department of Defense, 1968. 15. Peery, D. J., Aircraft Structures, McGraw-Hill Book Company, 1950. 16. Bruhn, E. F., Analysis and Design of Flight Vehicle Structures, Tri-State Offset Company, 1965.
xiii
STRUCTURAL DESIGN MANUAL
eli REFERENCES (Continued) 17. Niles, A. S. and Newell, J. S., Airplane Structures, John Wiley and Sons, 1943. 18. Sechler, E. E. and Dunn, L. G. , Airplane Structural Analysis and Design, John Wiley and Sons, 1942.
)
xiv
INTRODUCTION The purpose of the Bell Helicopter Structural Design Manual is to provide a source of structural analysis methods, material data, procedures and policies applicable to structural design.
The data contained herein are largely condensations of
in~
formation obtained from the Federal Government, universities, textbooks, technical publications and Bell reports and memoranda.
As much as possible the sources
are noted and shown in a list of references at the beginning of the manual.
This manual is intended to provide the structural designer with necessary design information in the form of equations, curves, tables and step-by-step procedures. Derivations are purposely omitted to make the manual easier to use. and suggestions regarding this manual are encouraged.
Corrments
They should be directed
to the Chief of Structural Technology, Bell Helicopter Textron.
)
-
.
xv /xvi
STRUCTURAL DESIGN MANUAL Revision F SECTION 1 PROCEDURES 1.1 General The Structures Technology Section of Engineering has the primary responsibilityof insuring structural integrity of all Bell Helicopter products at minimum weight and cost. To this end, the.following procedures are outlined. )
1.2 Design Coverage The Structures Engineer must follow a design from its inception. His requirements and suggestions must be submitted to the Design Engineer as the design progresses. The Structures Engineer must frequently review the work of the designer and if possible be ready to sign the drawing when it is comp1ete. I n most cases, no changes shou 1d have to be made to a draw; ng ,after it has been submitted to the Structures Group for signature. Structures Engineers are ,responsible for obtaining concurrence, and signatures where required, of Materials Technology and Fatigue Groups. Components should be substantiated by using established and recognized stress analysis methods or by comparison to existing test data. If, for specific problems design support test data are needed, Structures Engineers should initiate test requests. 1.2.1 Critical Parts All parts of the helicopter structure which are subjected to significant oscillatory loads, or which are of prime importance to safety of flight, will come under the heading of critical parts and will be treated as follows: - All critical parts will be evaluated for static design loading, fatigue, fretting. corrosion, residual stresses~ stress corrosion, corrosion fatigue and other environmental cond;tions~
)
- Analysis should include primary and secondary modes of failures as well as the effects of bearing friction or stiffness under load. 'All such analysis should be kept in a drawing check notebook. -
any significant design changes, alternate parts or materials will be reviewed with special care. The requirement for reQualification tests will be a Joint decision of the Structures Group Engineer and Project Engineer (and D.E.R. where applicable). In all tests conducted, the objectives of the test will be stated prior to the test. where possible, the method of interpretation of test results will be defined prior to the test. Test results that are unexpected or unexplained will be pursued to a solution.
1-1
STRUCTURAL DESIGN ·MANUAL Revision F - Flight and laboratory test requests should be accompanied by a brief writeup of expected results where possihle and Lab and Flight Test personnel will be requested to alert affected groups of any unexpected results. - All Structures personnel will work with the Fatigue Group, Materials Technology Group, Laboratory personnel and the appropriate design group in the design of new parts to reduce stress concentrations and stress corrosion susceptibility and to improve fretting characteristics. -
In all new designs, maximum use will be made of service experience from similar parts of structural. configurations used on previous models.
1.2.2 Flight Safety Parts A flight safety part ;s defined as any part, assemblj or installation whose failure, malfunction or absence could cause loss or seiious damage to the aircraft and/or serious injury or death to the occupants or group support personnel. A part is considered a FSP if item 1.2.2.1 is affirmative and anyone of item 1.2.2.2 is affirmative. 1.2.2.1 Primary failure or malfunction affects the safe operation of the aircraft. 1.2.2.2 a. The part has a predicted or demonstrated finite life. b. A 10% reduction in laboratory working strength would result in an unlimited life becoming a finite life. c. Loss of function could occur due to improper assembly or·operation. d. Fabrication of the part involves a manufacturing process which, if improperly performed, has high probability of changing material properties significantly impacting life. Flight Safety Parts are further governed by BHTI Requirements Specification 299-947-478. Each FSP must show at least one critical characteristic. 1.2.3 Check List for Drawing Review The following check list ;s a guide for drawing review: General Considerations Are critical net sections okay? - Take into 1-2
account~area
reductions for stretched formed parts.
)
STRUCTURAL DESIGN MANUAL Revision F Be aware of clamp-up stresses and associated stress corrosion problems. -
Account for all local discontinuities and centroid shifts. Consider all secondary effects such as tension field loads. Avoid abrupt changes in area. Use no one-rivet shear clips. Check for column stability on all compression members. Stress relieve and heat treat after welding.
-
Specify machine finishes. Design for thermal stresses and strains. Reduce allowables for temperature., Check for no yield at limit load as well as no failure at ultimate since some materials have low yield/ultimate relationship. For material allowables. use "A" values with all statically determinate structures. Use "B" values with redundant structures and structures designed for crash conditions, if specifically approved by the procuring agency. Investigate stiffness requirements. Check strain compatibility of joined structures.
-
)
Check structural deflection.
- When necessary to use dissimilar metals, design for their use, especially in joints. Design for wear, abrasion and fretting. -
Design for corrosion and stress corrosion due to residual stresses.
-
Account for friction. Check for surface treatments such as paint, chrome, hard anodizing, cadmium, etc. Proof load parts per specification requirements (such as hoists, control systems, hydraulics, etc.).
- Make the structure as light as possible. 1-3
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STRUCTURAL DESIGN· MANUAL
Revision F
- Verify that all processes necessary to the strength of the structure are properly called out.
•
Fasteners Verify that bolt grip lengths are adequate to prevent threads in bearing . Avoid using rivets in primary tension applications. Make joint critical in sheet bearing rather than
sh~ar
of fasteners.
)
Specify torque requirements on all bolts. Do not use bolts less than ! inch diameter in load carrying capacity without specific approval of Structures Group Engineer. - Avoid using blind fasteners in engine inlets or by the tail rotor side of a fin. 00 not use nutplates with reduced rivet spacing.
Avoid using rivets less than 1/8, inch in diameter in structural applications. - Avoid mixing bolts and rivets in shear jOints. Avoid using tension and shear fasteners as load sharing attachments in joint design.
•
Examine hole tolerances and fits. - Avoid using screws in primary tension applications with repeated loads. - NAS quality bolts shall be installed in the movable portion of all control system joints. - Commercial applications require dual locking devices on threaded fasteners or analysis to show fail safe with one fastener missing. Composite Structures - Verify the use of appropriate adhesive and supporting .BPS. - Complete and check the destructive test diagram for sandwich construction. - Assure that appropriate extensions or cutoffs are included for destructive test of laminates. Verify that the structure carries the appropriate classification. 1-4
•
STRUCTURAL DESIGN MANUAL Revision F Identify critical areas on the destructive test diagram. -
~erify
that processes called out will produce the desired structure.
- Verify that the part or assembly carries the appropriate classification. See the DRM, Section 2H-13 for definitions. Castings Verify that the casting carries the appropriate classification. - Review for approval of weld repair and the associated reduction in strength. (Reference SIM No.9) -
Verify that x-ray standards are appropriate for the casting classification. Identify critical areas on the x-ray diagram.
1.2.4 Envelope, Source Control,
a~d
Specification Control Drawings
All Structures personnel who have occasion to sign Envelope, Source Control, or Specification Control Drawings shall, as a minimum, establish that the drawing adequately defines the following: - Configuration - Mounting and mating dimensions -
Dimensional limitations (interferences)
-
Performance (loads, environment, life, etc.)
- Weight limitations - Reliability requirements Interchangeability requirements -
Test requirements
- Verification requirements (analysis or test) - Material limitations (example, no castings allowed, etc.) -
Casting classification if allowed (also casting factor)
-
Primary Part designation
-
Reference to applicable specification. 1- 5
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STRUCTURAL DESIGN MANUAL
:-. i';7
Revision
Also, if special inspections and tests such as x-rays and static tests are required, the Project Engineer should be alerted so that plans can be made to procure parts for the required tests. "Approved Sources of Supplyll or IISuggested Sources of Supply" shall be approved by Structures only if the proposed vendor item meets all structural requirements. This may mean vendors must submit stress analyses of their design or test data as a part of their proposal. On all Primary Parts or other items with significant structural requirements, the Structures Engineer shall retain a copy of the approved design, vendor stress analysis and test data and file this information in the proper drawing check notebook. 1.3 Stress Analysis A stress analysis of each component of the helicopter structure ;s required. The analysis should be in the following format:
- at
x 11 paper (BHTI stress pad)
- Analyst1s name (no initials) and date at the top of each page -
Part number of the part being analyzed at the top right hand corner of each page
-
Number all pages (1 of 10, etc.).
Each analysis must contain the following information: -
Sketch of ' the part being analyzed. Should be to scale but must contain enough dimensions to derive loads or calculate critical sections.
-
Free body diagram. Must contain all loads for a particular Must be in static equilibrium in all views.
condition~
- All the loads cases necessary to determine the critical cases for the part. Various sections of the part may be designed by different loads conditions. Be careful to identify loads as limit or ultimate. - Material of which the part ;s made. -
Material allowable stresses and source. the procuring agency.
The source must be approved by
- Step by step procedure for the stress analysis of all failure modes. If equations being used are not widely recognizable standard stress equations (i.e., PlAt Me/I, Vqjlt, etc.) the source of the equation must be stated .
1-6
)
STRUCTURAL DESIGN MANUAL Revision F - Margin of safety calculation. Any information which was used in the design of the part must be included. - Specify applicable factors (casting, fitting, etc.). - Assumptions that you made in order to "idealize" a structure, unless they would be clearly understood by another person. Calculations that are superseded by a redesign must be clearly marked "void ll or lIobsolete U and referenced to the new calculations. The stress notes must be kept up to date by the Structures Engineer until the time the engineering drawing is approved by the Structures Group. At this time the stress notes are filed in a master file from which they will not be removed. This file is maintained in each project area by the Lead Structures Eng i neer. Drawing Tolerances for Section Properties and Stress Analysis For analyses of basic extrusion (shapes), drawn tubing and sheet stock, nominal dimensions will be used. Nominal dimensions are defined as those from which th~ tolerance is added or subtracted. If a dimension is given as an upper and lower limit, the nominal dimension is the mean value. For aluminum airframe parts other than those specified above where it is desirable to specify a min-max thickness, the following procedure will be used: - Calculate lne required thickness for structural consideration based on standard analysis methods. Specify the minimum thickness to be the required thickness minus the tolerance shown below:
)
Thickness Range
Tolerance
to to to to to to t(! to
.002 .003 .004 .005 .008 .010 .011 .013
.012 .037 •046 .097
.141 .173 .204 . 250
.036 ~O45
.096 .. 14iJ
.172 . 20" .~49
.320
NOTE:
Examl2le If t (required) .120 in~ Specify: t (min) = .120-.005 = .115 in . t (max) may be .125, .130, 132 ••. depending on the material thickness tolerance •
If it ;s possible to hold tighter tolerances than shown above, use nominal thickness in the analysis. The Structures Engineer should seek the closest tolerance possible commensurate with cost considerations in order to minimize weight.
1-7
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STRUCTURAL DESIGN MANUAL
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Revision F
1.4 Structures Report The following informatipn is submitted as a format for structures reports. It ;s necessarily a general outline but should be adhered to in order to produce a document that has clearly defined subject matter, is accurate in technical aspects and is clearly readable as well as reproducible. 1.4.1
Introductory Data
Preceding the technical content or body of the report is: a. Title Page - Must contain all authors, group and project approval, OER approval if commercial, contract number if military and revision status. b. Revision Status Listing (for a multi-volume report) c. Revision Summary Page d. Proprietary Rights Notice e. A page of individual author signatures (when too numerous for the Title Page) f. Table of Contents - must contain all the major section headings and basic divisions within each section. Each topic will have the appropriate page number. If the report is multi-volume, each volume will contain a full Table of Contents for all volumes. g. References - (See General Information) h. list of Figures (not included in reports having less than 200 pages) i. List of Tables (not included in reports having less than 200 pages)
j. Definitions and Symbols k. Three-View Drawing (basic design criteria, loads and fuselage reports) 1. Basic Lines Data (loads report and airframe analysis)
)
m. Sign Convention n. Table of Minimum Margins of Safety"- On a part with multiple margins, both high and low, no margin greater than .25 will be ~hown. Minimum margins of safety include only the minimums on each part analyzed- and generally speaking, margins greater than 1.0 are not shown. Discretion must be used and margins greater than 1.0 may be shown on critical parts or system such as controls if all margins are greater than 1.0.
1-8
•
STRUCTURAL DESIGN MANUAL Revision F o. Introduction - The report introduction should always refer to the Basic Design Criteria Report as a source of helicopter data, horsepower, C.G. vs G. Wt.~ load conditions, load factors, etc. 1.4.2 Body of the Report
)
The body of the report ;s the structural analysis and should be broken down into major c'omponents, parts or system and conform to the outl ine as shown in the Table of Contents. Pages will be numbered sequentially within major sections; i.e., 1.001 ~ •• , 2.001 • . . , etc. Page numbers for all pages preceding the body of the report shall be small Roman numerals, as i, iv, xii .. • •
Stress analysis of components, details or systems: d.
Should be written on Bell stress pad ( originals should always be used in a report and copies filed in drawing checks or elsewhere).
b. A discussion should precede the analysis of each major component, part or system. This should include a description of the system or structures being analyzed, a reference to the loading conditions or design criteria, a summary of the methods of analysis used, the assumptions made and should often include statements regarding parts that are not analyzed as "being not critical. 11 load configurations or conditions not analyzed should also be noted as "not critical." c. Page headings will include nomenclature and part number. d. A sketch and/or geometry of the part or component being analyzed with the critical loan condition and load direction shown. Reactions shall be shown ar~ lne part or component will be statically balanced. When possl~le~ the axis shown should conform to the helicopter sign convention in the preface of the report, location of the part should be identified in d positive manner such as W.l.ls, B.l.ts or STA's and when not clear IIUp!!, fwd etc. should be noted. Should the analysis to follow include several details, sections, or panels, etc., these should be lettered for identification. II
II ,
e. Identification of material(s) and properties. Note the allowable stresses and/or loads and a leference to the source by page number or show the computed a l l ...... able. f. Special
fa:~0rs accounting for stress concentrations, cast materials, fatigue considerations, etc. should be identified together with their source. These factors will appear not in loads but in margin of safety calculations. .
g. The analysis is generally written for ultimate loads unless it ;s necessary to show compliance with limit load or fatigue criteria.
1-9
STRUCTURAL DESIGN MANUAL h. A margin of safety or fatigue life calculation concludes the analysis. The correct ~argin should be shown. If a part is loaded in tension and bending. stress ratios for tension and bending modulus should be calculated ang a margin of safety based on the sum of the stress ratios shown. Show ~he margin calculation at the extreme right hand side of the page. " i.
When the analysis is extensive and it is deemed advisable to summarize results in a table, shown the tabulated results with reference to pages from which the results were obtained.
This table of IISummary of Results" should precede the analysis. " The entire picture is, therefore, shown in the "D;scussionn~ the "Geometry and Loads and the "Summary of Results"". ll
,
1.4.3 General Information a. Since our reports are reproduced, write with a soft lead for maximum reproducibility. b. Extensive usage of flag notes and footnotes is not advised. c. Adequately reference what you put down; i.e., do not assume the reader knows where this information came from. Many of our readers are in foreign countries. When making the reference, give page number as well as the title. " d. Use of the Tenses - Computations appearing in the report should be referred to in the present tense. The past tense is used when referring to work not appearing in the report~ but which was done as a prerequisite to data in the report. Do not slip into the present imperative or past or future tenses. Use the third perso"n throughout. e. Be neat; do not overcrowd the page. f. Avoid the usage of 11 x 17 pages, if practical.
Proper planning will
minimize this . g. A "List of References u , nproprietary Rights Notice", a IIRevision li page and a "Distribution List ll will be in a1' volumes. h. Coordinate with the project office and/or contractual data to determine who is on the distribution list. i.
References should list Bell reports first, generally beginning with the nBasic Design Criteria", other reports, textbooks such as npeery", "Bruhn u , IlTimoshenko"" etc., MIL-HBDKs, NACA Technical notes and vendor data, i.e., honeycomb data, bearing catalogs, etc., in the order shown.
j. Discussions, references, general data, table of contents, etc., on all
reports shall be typed. 1-10
STRUCTURAL DESIGN MANUAL Revision F 1.5 Structural Information Memos 1.5.1
Purpose
Structural Information Memos (SIM) are distributed by the Director of Structures Technology to make new and unique structural design information available to members of the Structures groups and appropriate design groups. 1.5.2 Preparation Each new SIM submitted for approval shall include a cover memo addressed to the Director of Structures Technology, giving a brief synopsis of the Material. The memo shall be signed by the originator and approved by the SIM coordinator. The originator of each SIM shall be responsible for establishing the credibility and accuracy of his information and for preparing the SIM for distribution. Each SIM shall "stand on its own" and be thoroughly checked and referenced. Format of the material is left to the discretion of the originator. 1.5.3 Published SIM's Previously published SIM's are included in the following pages. The signatures have been removed, but the originals the SIM's are available from structures technology.
of
)
1-11
STRUCTURAL DESIGN MANUAL STRUCTURES INFORMATION MEMO NO.1 June 15, 1972
SUBJECT:
PROCEgURE FOR STRUCTURES INFORMATION MEMO (SIM)
REFERENCES:
(a) (b)
As required As required
ENCLOSURES:
(a) (b)
SIM Index SIM Distribution
This memo is written to establish a procedure for making new and unique structural design information available to members of the Structures groups and appropriate design groups. Much useful information is either generated or collected by members of the Structures groups during the norma1 performance of their duties. This information ;s usually available to a limited number of persons and ;s often filed away and forgotten. In order to prevent valuable information from becoming useless and forgotten, the Structures Information Memo ;s hereby established as the vehicle for conveying this information. The Methods and Materials Structures Group Engineer will be the coordinator for all SIM's and will assist in determining what information is valuable enough to publish. He will retain all originals~ will assign SIM index number, and update the index and distribution list as required. Each SIM shall include a cover memo, addressed to the Chief of Structural Design, giving a brief synopsis of the material. The memo shall be signed by the originator and approved by the SIM coordinator. The originator of each SIM shall be responsible for establishing the credibility and accuracy of his information and for preparing the SIM for distribution. Each SIM shall "stand on its ownl! and be thoroughly checked and referenced. Format of the material is left to the discretion of the originator, however, it should be remembered that all SIMas will be considered for incorporation in a Structures Manual to be issued at a later date. Similar significant structural information originating in any design group will be welcomed and handled in the same manner. Any additions or deletions to the distribution list should be directed to the SIM coordinator.
1-12
STRUCTURAL DESIGN MANUAL Revision F STRUCTURES INFORMATION MEMO NO.2 August 27, 1973 SUBJECT: REPORT FORMAT FOR STRUCTURAL ANALYSIS
)
The following information is submitted in an effort to clarify questions regarding the above subject by persons involving in writing stress Analysis reports. It is necessarily a general outline but should be adhered to in. order to produce a document that has clearly defined subject matter, ·is accurate in technical aspects and is clearly readable as well as reproducible. This is our product. 1.
Preceding the technical content or body of the report is: a. Title Page
b. Proprietary Rights Notice c. Revi s ion Status L; sti ng (when a report has mu 1t ipl e va lume·s) d. A r~vision page .(blank in a new report except for report number and volume number in place of page number) e. A page of individual author signatures (when too numerous to be listed on the Title Page). f. Table of Contents (the first numbered page) g. References (see General Information) h. List of Figures (not included in reports having less than 200 pages) i. List of Tab 1es ( I I II II II " II II II) j. Definitions and Symbols k. Three-View Drawing (basic design criteria, loads and fuselage reports) 1. Basic Lines Data (loads report and airframe analysis) m. Sign Convention n. Table of Minimum Margins of Safety On a part with multiple margins, both high and low, no margin greater than .25 will be shown. Minimum margins of safety include only the minimums on each part analyzed and generally speaking, margins greater than 1.0 are not shown. Discretion must be used and margins greater than 1.0 may be shown on critical parts .or system such as controls if a11 marg; ns are gr·eater than 1.0. o. I nt roduct i on The report introduction should always refer to the Basic Design Criteria Report as a source of helicopter data, horsepower, C.G. vs G. Wt. load conditions, load factors. etc. \I
)
2.
Body of the report: The body of the report ;s the structural analysis and should be broken down into major components, parts or systems and conform to the outline as shown in the Table of Contents. Pages will be numbered sequentially within major sections; i.e., 1.001 . . . , 2.001 . . • , etc. Page numbers 1-13
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STRUCTURAL .D.ESIG" MANUA_L
/
".' / -:f .
Revision F
for all pages preceding the body of the report shall be small roman numerals, as i, iv, xii • . . 3.
Stress analysis of components, details or systems: a. Should be written on Bell stress pad (see example on page 1-12; originals should always be used in a report and copies filed in drawing checks or elsewhere). b. A discussion should precede the analysis of each major component, part or system. This should include a description of the system or structure being analyzed, a refetence to ~he loading conditions or design criteria. a summary of the methods of analysis used, the assumptions made and should often include statements regarding parts that are not analyzed as "being not critical~. Load configurations or conditions ~ot analyzed should also be noted as "not critical". . c. Page headings will include nomenclature and part number. on page 1-16).
(See example
d. A sketch and/or geometry of the part or component being analyzed with the critical load condition and load direction shown. Reactions shall be shown and the part or component will be statically balariced. When possible the axis shown should conform t~ the helicopter sign convention in the preface of the report~ location of the part should be identified in a positive manner such as W.L.'s, B.l.IS or STA's and when not clear "UpU, ufwd u , etc. should be noted. Should the analysis to follow include several details, sections, or ~an~ls, etc., these should be lettered for identification. ' e. Identification of material(s) and properties. Note the allowable stresses and/or loads and a reference to the source by page number or show the computed allowable. f. Special factors accounting for stress concentrations, cast materials, fatigue considerations 9 etc. should be identified together with their source. These factors will appear not in loads but in margin of safety calculations. g. The analysis is generally written for ultimate loads unless it is necessary to show compliance with limit load or fatigue criteria. h. A margin of safety or fatigue life calculation concludes the analysiS. Show the margin calculation at the extreme right hand side of the page. No negative margins of safety may be shown; however, zero margins are acceptable. i. When the analysis is extensive and it is deemed advisable to summarize
results in a table, show the tabulated results with reference to pages from which the results were obtained. This table of IISummary of Resultsil should precede the analysis. The entire picture is therefore
.
1-14
)
s-rRUCTURAL DESIGN MANUAL Revision F shown in the IIDiscussion lt , the "Geometry and Loads Results lt
ll ,
and the "Summary of
•
General Information
4.
a. Since our reports are reproduced, write with a soft lead, such as "F" grade for maximum reproducibility. b. Extensive usage of flag notes and footnotes is not advised.
)
c. Adequately reference what you put down; i.e., donlt assume the reader knows where this information came from. Many of our readers are in foreign countries. d. Be neat. dont' overcrowd the page.
e. Avoid the usage of 11 x 17 pages, if practical. minimize this.
Proper planning will
f. For multi-volume reports, a complete table of contents will be shown for
all volumes in Volume I. g. A table of contents for that volume will be shown in volumes other than Volume I. h. A "List of References ll "Proprietary Rights Notice't" a "Revision u page and a UDistribution Listll will be in all volumes. t
i.
Co-ordinate with the project office and/or contractual data to determine who is on t~e distribution list.
j. References should list Bell reports first, generally beginning with the
"Basic Design Criteria,U other reports, textbooks such as IIPeery,1J "Bruhn," uTimoshenko," etc., MIL-HOBK's, NACA Technical notes and vendor data ~.e. Hexcel data, bearing catalogs, etc. in the order shown. Note: Structures manuals from other companies are not legitimate references.
~
i.
Oiscussions, references, general data, table of contents, etc. on all reports shall be typed.
1 15
STRUCTURAL DESIGN MANUAL Your Marne Here ey ____________________
~
.ell Helicopter' i =t:t i
.t., :I
MODEL
PAGE _ _ __
Leave Blank on Com.
CM EC KE 0 _ _ _ _ _ _ _--1
RPT
Add dwg. number part being
Ianalyzed TITLE HERE
in this box.
Leave space for
revision letter later
I
~TAIL PART, NOMENCLATURE HERE State
geometry~
loads. detan and location or reference Sect. A., Pg. ___
where this is shown.
See 13.
Compute the actual ultimate stress level. gene1al1y from limit loadS, referencing a report or page number for the loads. [t
may be necessary to compute section
loads on the section being
propert1es~
analyzed or to determine and show a static balance prior to computation of the stress level. Compute an allowable or state a reference for the allowables being used. State limit loads and yield a110wables when these are used to prove structure is non-yielding at limit load.
State the margin of safety. This is the pyrpose and conclusion of the analysis.
Be sure to include
fitting factors, casting factors in the M.S. and so state 9 i.e. t lIusing l.15
fitting factor u •
State which formula is used such as.
M.S.
L
1 t
+ R.) (Factor)
-1
= +.xx
~ Conftne the analysiS within these limits and thereby preserve the neatness of the report •
1-16
= (R
Vl,e Of disclosure of aata 01'1 this polqe 1\ ~Ul)lecl to the reUrlctlon on tl'le ,We Olloqe. Cl
•
STRUCTURAL DESIGN MANUAL Revision F STRUCTURES INFORMATION MEMO NO.3 August 28, 1973 SUBJECT:
LOAD SHEETS FOR STATIC TEST OF CASTINGS
This memo is written to standardize the data furnished on the load Sheets prepared for the Mechanical laboratory and .used by them in the static tests of castings. The following Sheets.
informat;on~
as a minimum. should be included on all Load
1. Title, part no., and name, thus: CASTING LOAD SHEET Part No. 204-XXX-XXX-X, Bellcrank, Cyclic Control 2. Indicate loads as 1l1imit" or Uultimate
ll
•
Ultimate preferred.
3. Draw sketches of part showing the external load application, direction and magnitude, and the reactions (usually designed by , and ). Sufficient views shall be used to completely define the critical loading condition. Each view shall show the reactions necessary to place the part, with its applied external load(s), in a state of static equilibrium. The loads and reactions shall be the same as those used in the structural analysis to insure that the part will be tested in the same manner as it was analyzed. Where moment vectors ( ) are used a note shall be included to indicate whether the right or left hand rule is applicable. t
4. When available the report number from which loads and reactions were obtained shall be referenced. thus: ) .7
Ref. Report 205-XXX-XXX 5. A note shall give a brief description of the loading condition! thus: Loading condition:
8G Forward Crash
6. A note shall indicate the casting factor, thus: Loads inc1ude a 1.33 casting factor Where the casting factor is unity, so indicate, thus: Casting factor of 1.00 is applicable
1-17
STRUCTURAL DESIGN MANUAL 7. Any other special information necessary to assure that the casting will be tested as It was analyzed. 8. All Load Sheets shall be prepared on stress pad paper. An example Load was omitted.
Sh~et
is attached.
Note that the rule for the moment vectors
\
1-18
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•
STRUCTURAL DESIGN MANUAL Revision F BY
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1-19
STRUCTURAL DESIGN MANUAL STRUCTURES INFORMATION MEMO NO.4 27 February 1974
SUBJECT:
BOLTS IN MOVEABLE CONTROL SYSTEM JOINTS
In order to avoid the possibility of installing an understrength bolt and to provide increase resistance to repeated loads. the following policy shall be implemented on the Model 409, Model D306~ Model 301; the production series of the Model 214 and Model 206L; and all future designs.
)
quality bolts shall be installed in the movable portion of all control system joints.
- NAS
•
1-20
•
STRUCTURAL DESIGN MANUAL Revision F
STRUCTURES INFORMATION MEMO NO.5 12 March 1974
SUBJECT:
)
JUMP TAKEOFF LOADS
Recently~ it has come to my attention that we are not addressing the rotor tilt for the jump takeoff conditions in a consistent manner. In order to provide a uniform approach, the following procedure shall be followed:
- Assume the helicopter has landed on a slope of specified magnitude ;n any direction (normally 6°) and executes a vertical takeoff at maximum load factor for this condition. The rotor tilt will be that which is necessary to execute this maneuver.
)
1-21
STRUC.TU.RAL DESIGN MANUAL STRUCTURES INFORMATION MEMO NO.6 2 August 1974
SUBJECT:
DETERMINATION OF FAILURE MODES
ENCLOSURE:
Suggested Form for Recording Failure Modes
Beginning with the Model 222, and effective for all future design activity, the Airframe and Dynamic Structures Groups will establish and maintain a notebook which shows the first and second predicted failure modes for all structural elements. The maintenance of these notebooks will be the responsibility of the lead structures engineer for each project.
)
The determination of these failure modes will consider static and dynamic loads along with other contributing factors, such as temperature, corrosion, and fabrication effects. The primary control will be maintained at the subassembly level (i.e., engine mount, bulkhead, main beam, etc.). Primary and secondary failure modes for static and fatigue loading will be determined for each subassembly. For those elements which are subjected to static or fatigue testing, the results of those tests will be entered in the notebook. In addition, any service problems encountered in the production cycle of the element will be entered. A suggested form for these records is enclosed. To aid the designer in his determination of these failure modes, the structural design groups will supply the deSigner with the critical loads for the structural element under consideration. These will be supplied in the form of a sketch or free body of the element with the applied loads and reactions. These loads will be updated as the mathematical model is refined during the design process. The establishment and maintenance of these records can mean much in establishing the rationale for a particular design, tracking its performance and guiding similar designs in the future. Your cooperation in implementing the procedure is essential to its success.
)
1-22
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1-23
STRU,CTURAL DESIGN MANUAL STRUCTURES INFORMATION MEMO NO.7 8 August 1974
SUBJECT: STRUCTURES APPROVAL OF ENVELOPE, SOURCE CONTROL, AND SPECIFICATION CONTROL DRAWINGS It has recently come to my attention that some of the subject type drawings do not always contain adequate information to allow us to properly validate the item to the government or the FAA. For example, castings may be purchased from a Source Control Drawing without proper inspection or test requirements being fully met within the company_ The Source Control Drawing may make no reference to x-ray requirements, static test requirements or any other special inspections required on castings. Therefore, all Structural Design personnel who have occasion to sign Envelope, Source Control, or Specification Control Drawings shall, as a minimum, establish that the drawing adequately defines the following: o
o
o o o
o o o o o o o
o
Configuration Mounting and mating dimensions Dimensional limitations (interferences) Performance (loads, environment, life, etc.) Weight limitations Reliability requirements Interchangeability requirements Test requirements Verification requirements (analysis or test) Material limitations (example, no castings allowed, etc.) Casting classification if allowed (also casting factor) Primary Part designation Reference to applicable specification
Also, if special inspections and tests such as x-rays and static tests are required, Project should be alerted so that plans can be made to procure parts for the required tests. UApproved Sources of Supply" or "Suggested Sources of Supplyll shall not be approved by Structures until we are completely satisfied that the proposed vendor item does meet all structural requirements. This may mean vendors must submit stress analyses of their deSign or test data as a part of their proposal. On all Primary Parts or other items with significant structural requirements, the Structures Engineer shall retain a copy of the approved design, vendor stress analysis and test data and file this information in the proper Drawing Check Notebook.
,)
Revision F It is hoped that other design groups will use this or some other check list for processing these type drawings.
)
1-25
STRUCTURAL DESIGN MANUAL STRUCTURES INFORMATION MEMO NO.8 26 February 1975
SUBJECT:
EDGE DlSTANCE REQUIREMENTS FOR NAS 1738 AND HAS 1739 BLIND RIVET INSTALLATION
As stated in MIL-HDBK-5B, paragraph 8.1.4, Blind Fasteners, liThe strength values were established from test data and are applicable to joints having values of elD equal to or greater than 2.0. Where ejD values less than 2.0 are used, tests to SUbstantiate yield and ultimate strengths must be made. 1I On page 1-11 of MIL-HOBK-5B, e is defined as the distance from a hole centerline to the edge of the sheet and 0 is the hole diameter.
The ultimate and yield strength values for NAS 1738 locked spindle blind rivets are based on a hole diameter of 0.144 for a 1/8 rivet, 0.177 for a 5/32 rivet, and 0.2055 for a 3/16 rivet, reference MIL-HDBK-5B, Table 8.1.4.1.2{d). The shank diameter for the HAS 1738 and HAS 1739 rivets are 0.140 for a 1/8 rivet, 0.173 for a 5/32 rivet, and 0.201 for a 3/16 rivet. Loft and sometimes Engineering Design will dimension edge distances and parts for the NAS 1738 blind rivet based on two times the 5/32 value (.31) rather than two times the 0.177 MIL-HDBK-58 value (.36), for example. This practice results in a rivet edge distance of less than 2.0; therefore, the MI HOBK-S8 strength values in Table 8.1.4.1.2(1) for HAS 1738B rivets are not applicable. In conclusion, to ensure the correct edge distance is used when planned patterns of NAS 1738 and HAS 1739 rivets are installed, Structures Group recommends that the correct edge distance dimension be specified on the face of the drawing for rivet patterns rather than using the drawing note that states rivet elO is equal to two times the rivet shank diameter. Also, special attention must be given to skin overlaps, and bulkhead and stiffener flange dimensioning. The edge distance for the countersunk NAS 1739 rivet of 2.5 times the rivet shank diameter is valid because MIL-HDBK-58 "values for the HAS 1739 rivet are based on two times the hole diameter. The table below summarizes the recommended minimum nominal edge distance values for NAS 1738 and HAS 1739 blind spindle locked rivets. Rivet . Size 1/8 5/32 3/16
1-26
EDGE DISTANCE NAS 1738 NAS 1739 .29 .36 .41
.32 .39 . 47
STRUCTURAL DESIGN MANUAL Revision F
STRUCTURES INFORMATION MEMO NO.9 4
)
May 1976
SUBJECT:
MECHANICAL PROPERTIES REDUCTION FACTORS FOR CASTINGS WITH FOUNDRY WELD REPAIR
REFERENCES:
a)
BHC Report 599-233-909, liThe Effect of Weld Repair on the Static and Fatigue Strengths of Various Cast Alloys"
b)
BPS FW 4470 - In Process Welding of Castings
c)
ASM Technical Report W6-6.3, IIStatic and Fatigue Properties of Repair Welded Aluminum and Magnesium Premium Quality Castings"
Future casting drawings should have a note that permits the in process welding of castings per BPS 4470. To allow for this weld repair, parts should be analyzed using the following reductions in allowables. Reduction Factors for Foundry Weld Repair Material
Ultimate Tensile
Yield Tens i le
Elongation
Endurance Limit
·356-T6
10%
5%
0
10%
A356-T6
10%
13%
0
10%
AZ91-T6
25%
22%
50%
10%
ZE41-T5
10%
0
50%
10%
17-4 PH
0
2%
30%
10%
In those circumstances where the part cannot be sized to allow for weld repair throughout the part, a weld map should be provided on the drawing to indicate those areas which may receive weld repair. If the entire part is so critical that no weld repair can be permitted and the part cannot be redesigned, the drawing and all analysis should be clearly marked "No Weld Repair Allowed". All 201 Aluminum Alloy castings shall be marked IINo Weld Repair Allowed ll
•
1-27
STRUCTURAL DESIGN. MANUAL STRUCTURES INfORMATION MEMO NO. 10 9 March 1978
SUBJECT: DESIGN CRITERIA FOR DOORS AND HATCHES Unless otherwise specified in a Detail Specification or Structural Design Criteria Report, the Structural Design Criteria presented herein should be used on new designs for the following: 1. Access doors 2. Hinged or sliding canopies 3. Sliding doors 4. Passenger doors 5. Crew doors 6. Cargo compartment doors 7. Emergency doors 8. Escape hatches All loads associated with the use and operation of doors and hatches terminated in the latches and hinges and their attachment to the airframe. The sources of these loads are: 1. 2. 3. 4. 5. 1.
Open canopy during approach or taxi operation Gusts Outward push from personnel Air loads Rough handling Open canopy during approach or tax; operation
If a sliding or hinged canopy is used, it should be designed to withstand an air load from tax; operations of up to 60 kt. 2.
Gusts
All doors that are subject to damage by ground gusts and wind loads from other helicopters being run up or taxied nearby or flown close overhead, should be provided with a means to absorb the energy resulting from a 40 kt ground gust occurring during opening or closing. Doors and access doors or panels that have a positive hold-open feature should be capable of withstanding gust loads to 65 kt when the door or panel is in the open position and unattended. 3.
Outward push from personnel
Due to possible inadvertent loading by personnel, passenger doors should be capable of withstanding an outward load of 200 lb. without opening. Also, 1-28
STRUCTURAL DESIGN MANUAL Revision F doors between occupied compartments shall be capable of withstanding a load of 200 lb. in either direction without opening. These loads are assumed to be applied upon a 10 sq. in. area at any point on the surface of the door. Yielding and excessive deflections are permitted but the door 'must not open. 4.
Air loads
The air loads on doors and hatches for helicopters probably are minimal when compared to the many personnel-oriented loads. The air loads, however, should be investigated, including the application of the appropriate gust criteria. All doors should be capable of withstanding air loads up to VD in the closed position. All sliding cargo and passenger doors should be capable of withstanding air loads up to 120 knots in the full open position and up to 80 knots in any partially open position. 5.
Rough handling
All doors and hatches that are likely to recelve rough handling during their lifetime should be capable of withstanding loads they are expected to receive in operation. Passenger and crew doors should withstand a 150 pound load applied downward at the most critical location without permanent deformation. All other doors that are unlikely to be stepped on or used as a handhold or which are marked with a IINO STEP" or liND HANDHOLD II decal should withstand a 50 pound load parallel to the hinge pin axes and a 50 pound load perpendicular to the surface without permanent deformation.
)
1-29
STRUCTURAL DESIGN MANUAL STRUCTURES INFORMATION MEMO NO. 11 16 January 1980 SUBJECT:
EMERGENCY FLOAT KIT LOADS
In addition to the existing design conditions, emergency float kit loads must developed for the following conditions:
be
)
1. Floats in the water at 0.8 bag buoyancy and combined with salt water drag for 20 knots forward speed. These loads will be treated as limit loads. These loads will be applied at angles corresponding to the righting moments, but not to exceed 20°. 2. For skid mounted floats;
a)
A computer drop will be done in a tail down attitude for limit sink speed. Skids will be checked for a positive M.S. at yield.
b)
Crosstubes will not yield with the helicopter in the water, floats inflated and no rotor lift.
)
1-30
•
STRUCTURAL DESIGN MANUAL Revision F
STRUCTURES INFORMATION MEMO NO. 12 7 August 1981
. SUBJECT: 7050-T73 RIVETS IN LIEU OF 2024-T31 (ICE BOX) RIVETS
)
7050-T73 rivets will be utilized in lieu of 2024-T31 "00 11 (ice bOX) rivets as of 20 July 1981. The 7050 rivets can be stored at room temperature, thereby eliminating numerous problems that exist with the 2024 rivets. The following policy will be implemented. 1.
Manufacturing will utilize the 7050-T73 rivets to supersede the MS 2024600 and MS 204700D rivets (Reference, the SUPER SESSION LIST, BHT Standard 170-001, Revision "G II ) , effective the target date of 20 July 1981.
2.
The 100 flush and protruding head 7050 aluminum alloy rivets are delineated in BHT Standards 110-174 and 110-175. respectively.
3.
All new drawings initiated after this date will callout the 7050 rivets for 3/16 and 1/4 inch diameters. Approval for other diameters MUST be obtained from applicable Structures and DeSign Group Engineers prior to utilization.
4.
The 7050 rivets "will not" be utilized to replace IIADI! rivets (generally used in 5/32 inch diameters and smaller) at this time (not cost effective).
5.
The driven shear strengths for both the 7050 and 2024 rivets are established for an Fsu = 41 ksi. Until MIL-HDBK-5 allowables are available, 2024-T31 MIL-HDBK-5 data in the 3/16 and 1/4 inch diameters, for both protruding and 100 flush heads, are acceptable for 7050-T73 installations and should be so identified for report referencing. It;s antiCipated that 7050-T73 MIL-HDBK-5 allowables will be available during the 1981-1982 time frame.
0
0
)
1-31
STRUCTURAL DESIGN MANUAL STRUCTURES INFORMATION MEMO NO. 13 31 August 1981 SUBJECT:
FITTING FACTORS, THEIR DEFINITION AND APPLICABILITY
Reference:
FAR 29.623, 29.619
A fitting factor is a 1.15 load factor~ applied to limit loads, and is in addition to the 1.50 factor of safety. It accounts for uncertanties such as deterioration in service manufacturing process variables and unaccountability in the inspection processes. For design considerations, a fitting shall be defined as part(s) used in a primary structural load path whose principal function is to provide a load path through the joint of one member to another. The connecting means is generally a single fastener. A fitting factor is applicable to the fitting, the fastener bearing on the joined members, as well as the attachments joining the fitting(s) to the structure. It is particularly considered when failure of such fitting should not allow load redistribution in a manner that would provide continued safe flight and that load redistribution cannot be verified by analysis or test. Obviously then, a fitting factor is applied to non-redundant connecting members in primary load path applications the failure of which may affect safety of the aircraft and its occupants. It is applied until the load is distributed into the surrounding back-up structure to which the fitting ;s attached. A fitting factor is not applicable to: a)
Crash load factors that are the only design condition and/or crash load factors that exceed limit load factors x 1.5 x 1.15.
b)
A continuous riveted joint(s) ;n basic structure when section properties remain consistent throughout the joint and the joint consists of approved practices and methods such as splices of main beam caps riveted door post caps to bulkheads, riveted skin splice doublers, continuous riveted skins to longerons, continuous riveted structure such as bulkheads to beams or intercostals, or frames, etc.
c)
An integral fitting beyond the point where section properties become typica1 of the part. Example, integrally fabricated lug on a forging, or machining.
d)
Welded joints.
1-32
)
STRUCTURAL DESIGN MANUAL Revision F
)
e)
To a member when a larger load factor is used such as a larger special bearing factor, a 1.25 casting factor, a 1.33 fatigue factor, a 1.33 retention factor of seats and safety belts.
f)
Systems or structure when they are verified by limit or ultimate load tests. The fixed control system is an example of this exception.
g)
Bonded inserts and/or fittings in sandwich panels.
h)
A fitting in redundant connecting members.
1-33
STRUCT·U·RAL :DESIGN'MANUAL
•
STRUCTURES INFORMATION MEMO NO. 14 25 January 1982
SUBJECT:
STRUCTURAL APPROVAL POLICY
Reference:
Structures Information Memo No.7 - "Structures Approval of Envelope. Source Control and Specification Control Drawings"
Structures Group approval of any drawing is defined as structural approval of all parts called out on that drawing regardless of whether or not they are Bell designed parts. It is therefore the responsibility of the Structures Engineer who signs a drawing to satisfy himself that all components of that drawing, including vendor part numbers, standard parts and speCification controlled items, meet Bellis structural requirements for that particular installation. For components that are defined by Bell Procurement Specification, Specification Control Drawing or Source Control Drawing, the guidelines of SIM No.7, as amplified here, are to be followed. The Structures Engineer must be assured that the controlling Bell speCification or drawing contains adequate requirements for vendor stress analysis and/or structural test proposal and results report to assure that strength requirements are met. Provision should be made for FAA conformity and for Bell witness of testing, if required. In the case of a product defined entirely by vendor1s drawings and procured by their part number, the Structures Engineer must notify the Project
Engineer in writing of the extent of structural sUbstantiation by analysis or testing required from the vendor. Provision should be made for FAA conformity and for Bell witness of testing, if required. It must be made clear that drawing approval ;s contingent upon successful completion of analysis or testing and submittal of these data for structures approval. If Bell testing is indicated, EWAs and schedules must be written to establish these tests.
1-34
)
~\ STRUCTURAL DESIGN MANUAL
,,~.,..
Revision F STRUCTURES INFORMATION MEMO NO. 15
(This memo is not published)
)
1-35
STR.U·CTURAL .DESIGN MANUAL STRUCTURES INFORMATION MEMO NO. 16 22 February 1983
SUBJECT:
PROPERTIES FOR CRES 17-4 PH CASTINGS
Reference:
(a)
MIL-HDBK-5 51st Meeting Agenda, Item 79-21, "Design Allowables (Derived Properties) for 17-4PH (HIOOO) Castings t " April 1981, Pg. 2-166 Attached t
MIL-HOBK-5 does not currently contain shear and bearing properties for 17-4PH castings. Properties to be used for analysis of 17-4PH castings shall be as shown in Table I. This table includes properties for the two recommended and most often used tensile strength ranges at BHTI. Properties for 17-4PH castings in the tensile range of 150-170 KSI per Reference (a) have been derived for and approved by the Mll-HDBK-5 committee. These properties are scheduled for inclusion in MIl-HDBK-5, Revision D. The properties in Table I were established or derived as follows: (1)
Tensile Ultimate - Minimum tensile strengths and tensile strength ranges are those most commonly used at SHTI.
(2)
Tensile Yield - Tensile yield strengths were established by comparing yield to ultimate ratios obtained from a large quantity of tensile test results. Test data from separately cast test bars and bars cut from castings were evaluated. These data came primarily from lot certifications submitted to BHTI by casting suppliers. Tensile yield strengths shall be 20 KSI lower than the ultimate tensile for tensile strengths equal to or less than 155 KSI minimum and 25 KSI for tensile strengths greater than 155 KSI minimum.
•
(3) Tensile Range of 155-175 KSI - Properties for compressive yield, shear and bearing for the tensile range of 155-175 KSI in Table I were derived by direct ratio of Reference (a) properties with ultimate tensile strengths as reference. (4)
Tensile Range of 170-200 KSI - Properties for compressive yield, shear and bearing for the tensile range of 170--200 KSI in Table I were derived by direct ratio of the 155-175 KSI tensile range properties, as shown in Table I, plus a conservative 5% reduction with ultimate tensile strength as reference.
Properties for tensile ranges not shown in Table I shall be derived in accordance with Items (2) and (4) except properties for the tensile range of 150-170 KSI shall be as identified in the attached Table 2.6.9.0(j) of MILHOBK-50. Properties shown in Table I and the procedures herein shall also be applicable to 15-5PH castings. 1-36
~ ~
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STRUCTURAL DESIGN MANUAL Revision F
.J
Applicable casting factors or minimum margins of safety shall be maintained per appropriate Design Criteria. Note the design approach to minimize or eliminate static test of castings due to cost and schedule impacts.
TABLE I *PROPERTIES FOR CRES 17-4PH CASTINGS )
Tensile Range 155 - 175 KSI Ftu Fty Fcy
-
Fsu
-
Fbru(1.5) Fbru(2.0) Fbry(1.5} Fbry(2.0)
-
e
155 135 136 101 262 340 195 229 8%
Tensile Range 170 - 200 KSI
Ftu Fty Fey Fsu Fbru (1.5) Fbru(2.0) Fbry(1.5) Fbry(2.0) e
-
170 145 141 105 272 354
-
203
-
239
6%
*Some property values may be higher than those previously derived and shown in MIL-HDBK-SC for wrought products.
)
1-37
. '\.~ tf!f0 . /r.4
+\--:
t
-
.
\
"~., STRUCTU.RA·L DESIGN MANUA.L Revision F MIL .. HDBK-5D
t June 1983
TABLE 2.6.9.0U}. Design and Physical Properties of 11-4PH Stainless Steel Casting
Specification
II"
••
It;
..
"
•
"
"
AMS 5355
AMS 5398
Investment casting
Sand casling
.......
Form ............. , ....... Condition .............•...
H900
H9.25
Hlooo
H 1100
H925
Thickness. in ...............
· ..
. ..
. ..
. ..
..
Basis .....................
Sa
Sa
Sa
Sa
Sa
180
180
160
ISO
150 130
130 120
180 150
Mechanical properties: FIIJ . ksi . " ................... " ......
F", ksi .......... " ................... F(\. ksi .. . .. . .. . .. " .... " ........ Fsu. ksi F bru • b ksi: '"
•
..
"
.......
1\
.....
if.
•
.,.
"
...
(ej D= 1.5) ........•.... (e/ D=2.0) .....•.......
Fh,p bleSt: (e/D=l.S) ............. (e/ D=2.0) ............. e, percent .............. "' ........ " .. RAt percent ......... " ........... 3
E. 10 ksi
...
""
......
t
..
.,
« . . . . .,
Ec. 10) ksi .. ................... ., .. G, 10) ksi ............... }J
...
;II
..........
"
...
II
..
&
.,
.........
·..
· ..
... . ..
132 98
. .. .. .
.. . · ..
. .. . ..
254
. .. · ..
.. . . ..
6
6 IS
15
. .. . ..
.. .
. ..
. ..
329
. ..
. ..
189
.. .
222'
.. .
. ..
8
8 15
6 12
20
28.5 30.0
12.7 0.27
p hysical properties: J
w.lbiin. C. K, and a ....•...••.•• ................
.,
.,
..
;II
III
0.282 (H900) See Figure 2.6.9.0
'For separatel~ cast bars. Pro~rt.e$ of test specimens machined from caslings shall be as agreed upon by purcha!l~ and suppllef DIkanng values are "dry pm" values per Scclion 1.4.7.1
(1)
This Table is sCheaule for publication in the futu.re MIl-HDBK-5,
Revision 0 ..
1-38
•
STRUCTURAL DESIGN MANUAL Revision F
STRUCTURES INFORMATION MEMO NO. 17 24 January 1984
SUBJECT:
LATERAL LOAD CRITERIA FOR COLLECTIVE CONTROL
To preclude inadvertent damage from handling. the following additional criteria will be met on all future collective control systems: 170 pound limit load applied separately in a horizontal plane inboard or outboard at the center of the collective handgrip.
)
1 39
STRUCTURAL DESIGN MANUAL Revision F SECTION 2
COMPUTER PROGRAMS 2.1 GENERAL
\
This section presents the computer aided analyses available to the structure engineers at Bell Helicopter. It provides a brief description of the computer program systems and associated computer programs. More detailed information can be obtained from relevant documentation available in the Structural Methods Group. 2.2 Computer Facilities The major computer facilities used by structure engineers are the IBM mainframe computing system, which utilizes an IBM 3081 and an IBM 3090 computers, and the TSO (Time Sharing Option) operating system. TSO video terminals. supporting text and graphics, are located in all engineering departments. CADAM scopes, the special purpose graphics terminals, are also available for access to the Computer Aided Design package CADAM. Depending on the setup of the computer programs, a computer job can be tun in an interactive mode or a batch mode. The resulting printout and plotting wil', by default, be sent to the computing center for process, but can also be routed to one of the local printers if preferred. Permanent datasets are stored and managed by the PANVALET data base management system. Magnetic tapes can be used to store large or inactive data, or for exchange of information with outside sources (approval required). A number of VAX super mini-computers and IBM PC's are available for engineering use. They are usually installed for the purpose of running certain special purpose programs. 2.3 Finite Element and Supplementary Programs
)
The finite element method has become one of the most important techniques for determining structural loads and performing stress analysis for sophisticated aircraft structures. The two major and very time-consuming steps in the finite element analYSis are the model generation and output interpretation. Many finite element preprocessors and postprocessors have been developed for these purposes. The preprocessors are used for automatic finite element mesh generation. The postprocessors are used for checking and plotting the model, as well as sorting, plotting, and processing the results for a variety of purposes. This section will introduce the finite element programs and their preprocessors and postprocessors available at Bell. 2.3.1 Finite Element Programs NASTRAN - General Purpose Finite Element Analysis I
2-1
";~~
d/-jt-\1
~B.II ;
",*LH!oP"I!.R
-
/
.STRUCTURAL DESIGN MANUAL
NASTRAN was originally developed by NASA. However~ many enhanced versions of NASTRAN have been developed thereafter. Bell has two versions: MSC/NASTRAN and COSMIC/NASTRAN. The MSC/NASTRAN is developed and maintained by MacNeal-Schwendler Corporation (MSC). It is a large-scale general purpose digital computer program which solves a wide variety of engineering problems, including static and dynamic structural analyses, acoustics, etc., by the finite element method. MSC/NASTRAN has made many additions and changes to the original NASTRAN; such as additional elements, composite laminate analysis, improved dynamic analysis methods, etc. The COSMIC/NASTRAN ;s the NASTRAN version currently maintained by the COSMIC. It is more like the original NASTRAN and does not have many features of MSC/NASTRAN. ANSYS - Engineering Analysis System The ANSYS software is developed and maintained by Swanson Analysis Systems, Inc. It is a large-scale general purpose finite element program •. Analysis capabilities include static and dynamic; elastic, plastic, creep a-nd swelling; buckli"ng; small and large def1ections; and other engineering analyses. SECR02 - Static Finite Element Program (RPT 599-272-001) This program utilizes the stiffness approach to perform a static finite element analysis of a structure. Contained in the program ;s a fracture mechanics element that calculates the stress intensity factor of a crack. 2.3.2 Finite'Element Preprocessors and Postorocessors PAT RAN - Finite Element Preprocessor and Postprocessor The PATRAN software is developed and maintained by POA Engineering. PATRAN has extensive geometric modeling and graphic capabilities. Its capabilities are numerous but include the finite element preprocessing and postprocessing; such as creating a finite element model and presenting the output graphically. PATRAN consists of many processing modules; including Conceptual Solid Modeling Advanced Geometry Modeling, Finite Element Modeling, Linear Statics and Dynamics Analysis, Composite Materials Design and Analysis, Results Evaluation, X-V Plotting, Engineering Animation, etc. PATRAN itself does not contain a finite element solver. PATRAN data have to be translated to and from finite element programs (including NASTRAN and ANSYS) through interface programs . t
2-2
•
STRUCTURAL DESIGN MANUAL Revision F ANASYS
Preprocessor (PREP7) and Postprocessor
The ANSYS program has a preprocessor (PREP7) and a postprocessor of its own. The preprocessor contains mesh generation capability and geometric plotting. The postprocessor routines will plot distorted geometries, stress contours, safety factor contours, temperature contours, mode shapes, time history graphs, and stress-strain curves. The postprocessor also has the routines for algebraic modification, differentiation, integration of calculated results, and other functions. )
CADAM - Computer Aided Design Package The CADAM software is primarily a computer aided design tool. However, its CADAM/FEM module is capable of creating and editing a 3-D finite element model. CADAM and NASTRAN data can be translated through the interface programs. The most important advantage of this approach is that the finite element model of a structure can be created directly from design drawings so that accurate modeling and time savings can be obtained. GIGN01 (old program NASPLOT) - NASTRAN Model Plotter This program is deck directly. element numbers program must be
a NASTRAN postprocessor and will process NASTRAN data It can plot the finite ele~ent model grid points and for a specified subsections and arbitrary views. The executed on a graphics terminal.
SECR64 - Model Generation and AnalysiS of Cutouts in Composite Materials (RPT 599-162-919) This program generates a NASTRAN finite element model of a rectangular orthotropic plate containing popular cutout shapes in.order to calculate the strains and margins of safety for each lamina. SECR64 - Model Generator for Rectangular Plates with Circular Patches (RPT 599-162-922)
)
This program generates a NASTRAN finite element model of a flat composite rectangular panel containing a circular hole with a flat circular composite patch. SESN04 - Finite Element Model Data Generator for Shells (RPT 599-162-912) This program generates a NASTRAN finite element model for any shell type structure which can be mathematically defined. SESN05 - Finite Element Model Data Generator for Rectangular Plates, Solids, and Laminated Plates (RPT 599-162-913)
2-3
STRUCTURAL DESIGN MANUAL This program generates a NASTRAN finite element model for a rectangular plate, doubler stack, or a laminated plate. SESN06 - Finite Element Data Generator for Pin-Loaded lugs (RPT 599-162-917) This program generates a NASTRAN finite element model for ~ pin-loaded lug. The hole in the lug can be either concentric or eccentric. The lug can be modeled with a bushing and a pin. SE1703 - Finite Element Data Generator for Truss Tailboom This program generates d NASTRAN finite element model of a truss tailboom. Arbitrary station input and number of longerons are allowed. SDSSOl - Unit Inertia Forces and Weight Generator (RPT 299-099-252) This program uses a WAVES weights file and a NASTRAN grid deck to obtain an inertia representation of the helicopter. The option is given to have either weights or equivalent unit inertia forces. SiSS12 - NASTRAN Critical Loading Selector (RPT 299-099-252) This program is used with SESBIO and compares the forces and stresses calculated during the execution of SESBIO. It selects the critical load and loading condition for each member in the finite element model. The critical loads selected are printed in report format and saved on-tape.
• '
SESN09 - Shear and Bending Moment Diagrams Generated from NASTRAN Force Input (RPT 299-099-252) This program uses the inertia loads and corresponding applied loads for a NASTRAN airframe finite element model to create the shear forces and bending moment diagram associated with _each given design load condition and gross weight configuration. The program generates the diagram as Calcomp plots and tabular tables. SESN13 - Element Grouping Program This program groups the NASTRAN output for a particular area of the structure regardless of the element numbering sequence. It;s particularly useful for bulkheads, skins, stringers, tailboom, etc. SESSOl - Automatic NASTRAN Element Sizing This program will generate a NASTRAN model with more representative element size early in a project design phase; A NASTRAN model with unit element areas is run against a set of design conditions. Selected elements are sized for these internal loads. New NASTRAN property cards are punched for rod, bar and shear panel elements.
2-4
)
STRUCTURAL DESIGN MANUAL Revision F SESBIO - Basic NASTRAN Airframe Output Option {RPT 299-099-252} This program transforms NASTRAN output into a form more suitable for structural analysis. Shear flows, end loads and moments are generated. 2.4 Approved Structural Analysis Methodology (ASAM)
)
The computer programs approved for structural analysis at Bell are collected in the ASAM (Approved Structural Analysis Methodology) system. The documentation for the system overview and individual programs are on file in the Structural .Methods Group. These documents are also on line of the computer and can be obtained by request thru a self instructive procedure on a TSO terminal. The ASAM software currently contains three computer systems; which are lOOAM, CASA, and CPS2TSO in addition to separate individual programs categorized under the title IIProgram u • The ASAM system is not static; it is constantly being revised and expanded. The user should consult the active system for the latest system capabilities. ASAM ;s a TSO based system. The user can access the TSO and entering the command IIASAM"; this will bring menu to the screen. ASAM is a menu directed system, option the system will display successive menus, for the desired program or function is reached.
system by logging onto the primary analysis i.e. by selecting a menu option selection, until
It ;s suggested that before using the system the user accesses ASAM to request for the documentations for system overview and to get familiar with its menus and structure. In addition, he should request the .documentation for the program of his interest prior to attempting the analysis. The documentation of the program includes an explanation of the theory/methodology, technical references, and major analytical equations used, if practical. It also contains user instructions for proper execution of the program, along with example problems with their associated input and output data. The ASAM documentation can be referenced in the Bell IS official reports to meet government agency's or customer's requirements. )
2.4.1
The lOOAM System
The lODAM is a system for structural loads development. This system is currently under development and will be available for use at a future date. 2.4.2 The CASA System The CASA (Computer Aided Stress Analysis) system contains a group of matured analysis programs. It provides the structures engineer with a consolidated stress information system. The primary goal of CASA is to increase the efficiency of engineers by reducing the manhours required to perform structural analysis and to produce reports. The description of the system and its features. as given next, are general in nature.
2-5
STRUCTURAL DESIGN MANUAL 2.4.2.1 CASA System Features. follows: 1.
The Primary CASA system features are as
A computer stored stress information system accessed through TSO The CASA System collects the stress disCipline into a computer stored stress information system. It uses computing facilities to provide the computations, the utility linkag-es, and the high speeds required to process the information efficiently. It eliminates time consuming data manipulations which were previously done by hand. It accomplishes these tasks within the CASA System itself and also allows access to external programs and systems such as NASTRAN, WAVES~ and CADAM.
2.
Automated Modular Stress Analysis Programs The CAS A application progra~s are a collection of automated stress analysis programs with the following features: - They are structured to be used by both entry level and experienced stress analysts. - They all are interfaced bY.the user through interactive menus and prompting on TSO terminals. - The inputs to the programs require the minimum amount of manual calculations. - The programs create an input dataset when they are run. This dataset can be edited and used to run the program without the user having to answer each prompt on an individual basis. - The programs provide the user with options as to the format of the output data. He may choose either temporary output format or the report format (Form 8441). - The output from .the calculations performed by the programs are presented on standard formats that are concise and easily understood. - All application programs provide the user with an automated .system for the permanent saving of input datasets on the CASA system database. - The flexibility and applicability of the programs are maintained by speCifying general solutions of basic theory such as tension, compression, and shear. When combined with appropriate utilities and interface, the analysis of airframe fuselage sections, tailbooms, bulkheads, and mechanical systems can be attained.
3.
CASA has self contained tutorial and documentation functions. The CAS A system provides an up to date tutorial and set of documentation for each application program. The tutorial function presents the user
2-6
STRUCTURAL DESIGN MANUAL Revision F with an introduction, a set of program capabilities and limitations, and a detailed step by step set of user instructions on how to execute the program. The documentation function provides the user with a printed copy of the methodology/description of each option and includes the necessary graphics. 4.
Report ready formatted output. Where possible, the CAS A programs perform analysis to the compilation of margin of safety summaries on an output report format. This format is the standard stress pad format (including heading, border, proprietary note, etc.). Where graphics are required space is left on the pages. This report ready format (8.5 inches by 11.0 inches with holes punched) is ready for immediate cataloging and inclusion into stress reports. Each application program presents its output in a standardized, concise, and readable form.
5.
Secured and protected data and programs. The CASA system provides the user with an automated system to permanently identify and save all input datasets that have been used to produce final report analysis. Once an analysis is designated as a final report analysis the security function automatically labels it with all the information required to save it on a permanent database. This dataset can be recalled and used to rerun the analysis in the future to obtain the same results. The CASA programs in all revisions are also saved. In addition, all current programs are protected from unauthorized changes to the source code. The final report datasets that are saved on the database are protected from any alteration to their contents. The database automatically names its datasets to prevent duplication or overwriting.
6 ..
)
Uses structural design manual methods.
The application programs utilize BHT approved methodologies and/or structural deSign manual (SDM) methods. SDM variable identification ;s maintained, where possible~ for ease of reference and identification. 7.
Produces reports and documentation suitable for submittal to regulating and procuring agencies. The documentation and analysis produced by the CASA system for both reports and methodology has been formatted and produced in a manner so that it is suitable for submittal to the FAA and Military authorities. The methodology for all CASA programs can be found in BHT Report No. 299-099-252, Volume IV.
2.4.2.2 CASA System Architecture. The CASA System architecture (see Figure 1) gives the CASA user an overview of the capabilities of the present (Phase IV) and future systems. Each item on the figure can be associated with software that is used by the CASA System in performing its functions. The 2-7
STRUCTURAL DESIGN, MANUAL
COMPUTER AIDED CASA
S
TRESS ANAL YSIS
EXECUTIVE MODULE
I DATABASE/ EXT . INTERFACES ~TA
I
I NAS'TRAN
•r:::t3M
MSC NASl'RN>I
esrros
I
•CADAH t€5tt
TUT~IAL
I
ANALVSIS
OUTPUT
I
TEJ4'OftAA\'
ANALYSIS
•tCAD
I
ANALYSES
MENUS
SON ANAl... YSES LUG ~ PIN ANAlYSIS
TUTORIAl.
LOAD CASE SELECTION
•I:AM
•C-61
DATA PREPARATIct4
FOR DIAS. TEN.
...
WAVES
.. FUT1..AE r:::.APMILITY
1
.
DATABASE DOCUMENTATIDN
I
I F'lNAL
REPCRTS
I CONTROL. LINtCAGE NASTRAN LOADS LOADS TAPE COPYI SAVE UTILITY THREE OIMENSIONAl fASTENER LOADS
DATA PREPARATION • DIAG_TEN. ANAL
UNSYMETRICAl aEND EXTERNAL DAHSET
DIAB. TEN. USIfi"G
EXTERNAL DATASET
UNSVMETRICAL BEND t DUG TEN
LOAD SELECTION FeR AXIAL L St€AR ELEH
STEP COLUMN 8UCK FOR CONTROL TUBES
AX1AL ELEM. TEN. L c:otP. ALLOWAISLES
REPORT OF OATA&ASE OATASETS
AXlAL £lEM. MARQINS Cf!' SAFETY
DATABASE STORAGE FOR INPUT OATASETS
Sl£AR ELEM. AlL.OWABLES
SECTION ANAL.YSIS PROGRAM
(WEB)
I=~ TATICH
ANALYSES
FOR DIAG. T04.
StEAR ELEM. MARGINS a: SAFETY
C:ONPOSITE JOI NT
CONTRa.. LINKAGE QEOMETRY PROCESSOR
COMPOSITE BEAM ANAYLSIS
Figure 1. 2-8
USER
INTERNAL INTERFACES
•NlN4
[)A.TASETS
TSO MENUS
APPlICATION PROGRAMS
SPECS
e
I
INTERACTIVE CONTROL PR(M>llNG I
I
NIL-SC
•M5IC:
I
EXlERNAL
1 N1ERF AlIS
I
DOCUMENTATION/ TUTORIAL MODULE
ANALYSIS MODULE
MODULE
ARCHIVAL
)
I
ANAL.YSIS
Phase IV CASA System Architecture
)
STRUCTURAL DESIGN MANUAL Revision F executive module provides the user with interactive control and interfaces to the three primary modules: the analysis module, the documentation/tutorial module, and the database/external interface module. The executive software transforms CASA into a modern stress information system that will allow future expansion to be done in a timely and cost effective manner. New application programs and functions can be plugged into the system and use the existing software in many of their options. 2.4.2.3 CASA Programs Description )
CV004P(A) - Structural Design Manual Lug and Pin Analysis This program performs a stress analysis for a lug or lug and bolt/pin combination according to the procedures and methodology presented in section 6 of this manual. CV005P - Load Case Selection for Diagonal Tension Analysis This utility program is used to copy selected load cases from an Airframe Options Loads Tape to a temporary file. The file will be used as the NASTRAN input loads for a diagonal tension analysis.
•
CV006P - Data Preparation for Diagonal Tension Analysis This program prepares an input dataset for CASA program CV008P (Diagonal Tension). It prepares the geometriC data, material properties, and the loads from a NASTRAN Airframe Options Tape for input into the Diagonal Tension program. CV007P - Data Preparation with Diagonal Tension Analysis 'This program prepares an input dataset for CASA program CV008P (Diagonal Tension). It prepares the geometric data, material properties, and the loads from a NASTRAN Airframe Options Tape and performs a diagonal tension analysis of the structure.
)
CV008P - Diagonal Tension This program uses the methods outlined in NACA TN2661 to analyze a flat or curved panel that has developed incomplete diagonal tension. The program is set up to analyze a multi-bay structure with single or multiple loading conditions. It does not require NASTRAN generated loads. CV009P - Load Case Selection for Axial and Shear Element Summaries This program is used to create a critical loads dataset from an Airframe Options Tape. These loads are used by programs CVOllP and CV013P (Axial and Shear Elements Summary) to calculate margins of safety for the structure.
2-9
STR,UCTURAL DESIGN MANUAL CVOIOP - Axial Elements Tension and Compression Allowables This program produces the axial element allowable loads for tension, compression, crippling. compressive yield, and interrivet buckling for airframe structures. It processes the axial element IDs, geometry, and material allowables for use by program CVOIIP (Axial Elements Al10wables and Margins of Safety). CVOIIP - Axial Elements Allowables and Margins of Safety Summary This program produces an axial elements' margins of safety summary for airframe structures. It processes the axial element allowables from program CVOI0P and combines them with the NASTRAN loads from program CV009P to produce the margins of safety. CV012P - Standard, Lightened Hole, and Beaded Shear Web Allowables This program produces the shear element allowable loads for airframe structure. It processes the shear element IDs, geometry, and material allowables for use by program CV013P (Shear Elements Allowables and Margins of Safety). CV013P - Shear Elements Allowables and Margins of Safety Summary This program produces a shear elements margins of safety summary for airframe structures. It processes the shear element allowables from program CV012P and combines them with the NASTRAN loads from program CV009P to produce the margins of safety. l
CV014P - Control Linkage Geometry Processor This program is used as a geometry processor to analyze the kinematics of a control linkage. It produces the necessary NASTRAN bulk data to represent the movement of the control linkage. The linkages are modeled as cranks, idlers, slides, actuators, rods, jackshafts, and mixers by using modeling operators. CV015P - Control Linkage Geometry - NASTRAN Loads - Airframe Options This program is used as a geometry processor to analyze the kinematics of a control linkage. It produces the necessary NASTRAN bulk data to represent the movement of the control linkage. The linkage ;s modeled as cranks, idlers, slides, actuators, rods, jackshafts. and mixers by using modeling operators. In addition, it calculates the internal loads and reactions in the control system due to specified input loading conditions using COSMIC/NASTRAN as a solver. CV017P - 3-D Fastener Pattern loads This program calculates the three dimensional loads on a fastener pattern using rigid body mechanics. It allows weighing of the fastener . 2-10
)
STRUCTURAL DESIGN MANUAL Revision F areas in proportion to the stiffness that the individual fastener and its backup structure provides in a given direction. CVOIBP - Section Properties and Unsymmetrical Bending Analysis This program calculates the section properties, unsymmetrical bending stresses, element loads, and shear flows for a single or multiple cell torque box. It is applicable to standard stiffener/skin and sandwich constructions. CV019P{A} - Load Case Selection for Unsymmetrical Bending and Diagonal Tension Analysis This program ;s used to copy selected load cases from a Shears and Moments History Tape to a TSO dataset. This dataset contains th~ , NASTRAN loads which will be used by program CV019(B) to prepare a load input dataset for the program CV019P(C) (Unsymmetr;c Bending Analysis). CV019P(B) - Data Preparation for Unsymmetrical Bending Analysis This program prepares an input dataset for program CV019P(C) (Unsymmetrical Bending Analysis) using input dataset produced by program CV019P(A) and the geometry data of the structure. CV019P(C) - Section Properties and Unsymmetrical Bending Analysis This program calculates the section properties, unsymmetrical bending stresses, element loads, and shear flows for a single or multiple cell torque box. It is applicable to standard stiffener/skin and sandwich constructions. It should be used when a NASTRAN model is available for the structure and when a diagonal tension analysis follows the unsymmetrical bending analysis. This program requires a different input dataset from program CV018P and produces different output. CV019P{D) - Oat a Preparation for Diagonal Tension Analysis
)
This program produces an input dataset for program CV019P(E) (Diagonal Tension Analysis). It combines the geometry inputs with the loads output from program CV019P(C) to produce the desired input dataset. CV019P(E) - Diagonal Tension Analysis This program uses the methods outlined in NACA TN2661 to analyze a flat or curved panel that has developed incomplete diagonal tension. The program ;s set up to analyze a multi-bay structure with single or multiple loading conditions. It accounts for the increase or decrease in the web buckling allowable due to the compression or tension stresses in the web.
2-11
STRUCTURAL DESIGN MANUAL CV020P - Step Column Buckling for Control Tubes This program calculates the critical buckling loads for a stepped and pinned end control tube or column. The program includes empirical factors to adjust the analysis for control tubes. These factors can be omitted for columns that are not control tubes. The program only checks for buckling and not for local instability. It uses the Tangent Modulus Theory and allows the user to define his own materials. CV023P - Section Analysis This program can be used to perform a section analysis on cross section made from metallic and certain composite materials. It calculates the section properties, the resultant loads on the section, the resu1ting stresses/strains of general polygonal sections. It is based on beam theory as opposed to plate theory.
)
CS024P - Bolted Joint Stress Analysis This program computes stress distributions on a lamina or laminate basis for unloaded or loaded (bolt bearing) holes in isotropic or ansitrop;c materials and predicts failure based on lamina properties and user selected failure criterion. The program should be used for preliminary design only. The margins of safety it calculates are approximate because of the many assumptions made.
~
CS025P - Composite Beam Analysis This program computes section properties and stresses for a beam of composite materials with irregular cross-sections using a finite element method. Section properties include cross-sectional area, centroids, moments of inertia, location of the shear center, shear coefficients, torsional constants, and warping constant. The stresses include the normal stress due to axial force and bending moments and shear stress due to transverse shear forces and twisting moment~ 2.4.3 The CPS2TSO System Most programs, developed at Bell or acquired from outside sources, are not normally installed into CASA directly. After being checked out and approved for use, they are normally included in the CPS2TSO system. The CPS2TSO system does not have all the features of CASA. Each CPS2TSO program has documentation, but does not produce the results on the final report form. A CPS2TSO program will be transformed to CASA when it is fully checked for its reliability and accuracy. The programs currently contained in the CPS2TSO system are described in the following: CPSOIP (old program AIRFACTORS) - Airload Distribution on the Fuselage The external loadings needed for the structural analysis of airframe structure include the distributed loads representing the air load 2-12
)
STRUCTURAL DESIGN MANUAL Revision F pressure distribution. These airload distributions for various maneuvers are normally developed form wind tunnel data. The C-81 program determines the contributing inertia force of each mass/weight item of the helicopter and the applied airloads necessary to balance a given maneuver. These airloads are applied at the C-81 model reference point. Program SD5001, or similar programs, calculates a unit airload distribution for a unit force and/or movement at a reference point. . Then CPSOIP ;s used to determine the scale factors to apply the unit load datasets produced by programs like S05001, to produce loads that are equivalent to the applied airloads from C~81. CPS02P (old program DL3301) - Crack Growth Program This program analytically calculates the crack growth rates and crack sizes for flat plates and cylinders under axial loading. CPS03P (old program NFSN05) - Inertia Scale Factors For Point Loads This program calculates an inertia scale factor for each of the six degrees of freedom for each airframe loading condition. In addition, the forces and moments about the helicopter center of gravity are computed. These scale factors and the appropriate inertia forces are combined on a NASTRAN load card. CPS04P (old program SCAV17) - Composite Laminate Analysis This program performs a linear point stress analysis for a composite laminate. It predicts the initial and progressive failures using one of the selected failure theories. Applied loads include ;nplane forces and moments, transverse shear, and thermal effects. A library for commonly used materials has been built into the program in addition to an option that allows the user to input materials. The output contains laminate and ply-by-ply strains and stresses, and margins of safety. It can also produce NASTRAN composite plate data cards, failure envelope plots, and carpet plots for laminate properties.
)
CPS05P (old program SFCR02) - Section Properties and Unsymmetrical Bending Analysis This program determines the section properties, unsymmetrical bending stresses, element loads and shear flows for a single cell torque box. It accounts for sandwich materials, monocoque construction, local stiffening of element. and allows for a safety factor to be used in determining the loads. CPS07P (old program SD5001) - Representation of Forward and Aft Pressure Distributions by Panel Point Loads Once the airload distribution acting on a helicopter has been determined, the task of representing this distribution by means of concentrated loads at selected panel points remains. This program 2-13
STRUCTURAL DESIGN MANUAL distributes the airloads to the panel points for various loading conditions at a reference point, which ;s usually the center of gravity location. CPS08P (old program 5E5002/5E1713) - Engine Loads Program This program calculates the forces and moments at the engine center of gravity due to flight maneuvers and gyroscopic coupling of the rotating components. It uses the engine mass properties and aircraft accelerations to calculate the engine loads. The program can be used for single or twin engine aircraft. CPSIOP (old program SLHW02) - Dynamic Landing Gear Analysis This program develops the loads in a crosstube landing gear for vertical and inclined plane landings. 80th conventional crosstube attachments and the three point attachment with stops are allowed. The user has the option to have the program develop the load deflection curve or input it himself. The rotor lift, gross weight, and e.G. may be varied to analyze more than one landing load case at a time. After the loads are developed, a stress analysis i.s given that can be used to help size the crosstube. CPSllP (old program PB280l) - Plastic Bending Analysis This program develops the load deflection characteristics and the internal plastic bending stresses of a cantilevered or a symmetrically loaded simply supported beam with a tubular cross section. Arbitrary end loading is allowed with either large deflection or small deflection theory. CP512P (old program SP5009) - Pylon Loads Computer Programs This program calculates the forces on a conventional transmission pylon system. The forces and moments about the lift links are calculated, these moments are used to determine which if any, pylon mounts are bottomed. Based on the stop geometry, the loads on the mounts are calculated. The program can be executed in a mode that accounts for the deflections of the pylon system in calculating the moments about the lift link; or the moments can be calculated assuming the deflections are negligible. t
CPSI3P (old program ST4101) - Tension Beam Analysis (for Blade Fatigue Specimens and Masts) This program serves as an aid in statically determining an initial set of loads for a blade fatigue specimen. For a given axial load the tip moment and shear are determined so that the bending moment distribution in the specimen approximates a flight loads moment distribution as closely as a least squares curve fit will permit.
2-14
)
STRUCTURAL DESIGN MANUAL Revision F CPS14P (old program TW3301) - Torsional Analysis of Flexures with Rectangular & I Sections This program performs an analysis for flexures having rectangular or I cross sections that vary arbitrarily with spanwise distance. The stiffening effects due to centrifugal force, elongation of the outer fibers, and warping restraint due to end fixity are considered and the maximum shear stress at each section is calculated. CPS15P (old program NLAMOl) - Simultaneous Linear Equations (High Accuracy Solution) This program solves a set of simultaneous linear equations using IMSL routine LEQT2F, which ;s the linear equation solution ful1-storage-modehigh accuracy solution routine. The program solves the equations to the desired accuracy and informs the user of the accuracy achieved. CPS16P (old program LD2112) - One Dimensional Joint Analysis This program calculates the distribution of an axial load thru a one dimensional fastener pattern. It is applicable to a mechanical joint composed of two similar or dissimilar elastic materials. CPS17P (old program SRANOl) - Frame Energy Solution This program calculates the internal shears and moments in a frame or bulkhead when a set of balanced shear flows and/or concentrated loads are applied. The solution ;s based upon the assumption that the elastic energy causing deformation of the frame produces consistent deformation at any paint. CPS18P (old program OB1) - Composite Tube Analysis
)
This program performs an analysis for composite or metal tubes and columns. It calculates the ultimate buckling loads and the natural frequencies for the tube or column assuming pin ends. It allows each segment of the tube to have its own material properties and for this reason it is preferred for composites. It does not check for local failures this should be done independently. t
t
CPS19P (old program SDAN08) - Design Loads (with a Panel Point Weight Distribution) This program combines a set of external loads and balancing load factors with the panel point unit shears and moments to produce airframe design loads. CPS23P (old program SESE01) - NASTRAN Model Description Report Generator This program selects and sorts structural member data from a NASTRAN Bulk Data Dec~. The selected structure is associated with its defining 2-15
STRUCTURAL DESIGN MANUAL grids section properties, and material data. A complete definition of this data is produced on a report format. The structural members may be selected by the following means: t
Members totally in a region defined by a rectangular parallelpiped Members which are completely defines by an input grid list Members whose element IDs are contained in a list CPS24P (Old program SP2803) - Pylon Support Loads (A Laminate Analysis) This program calculates the loads ;n the pylon supports and the resulting applied airframe loads. The rotor loads and airframe accelerations are used to determine the axial bi-pod loads, the loads applied the airframe attachments, the mount, and the stop. CPS25P (old program SESB13) - Shear and Moments Envelope Program This program plots shear and moment envelopes using specified cases selected from a SESN09 Shears and Moments History Tape. The enve10pe is a plot of the maximum and minimum values at each station location. This program is similar to program SESB13 but it allows two additional features; a plot heading and a figure deSignation for the plot. CPS26P - Composite Laminate Buckling Delamination Analysis This program is designed to assess the capability of a laminate to resist near surface delamination growth. The delamination is assumed to exist in a laminate. The strain energy release rate and related buckling and threshold strains are then calculated and plotted against the delamination sizes so that a more delamination resistant laminate can be designed. CPS27P - The Crippling Strength of Compression Members This program determines the crippling strength of a compression member with a simple or complex cross section based on a set of empirical design curves. The user may select from a number of different design curves obtained from Bell Design Manuals and U.S. Government Reports. In addition, the program can develop crippling design curves from the users own experimental data. CPS28P - Determination of Aircraft Tiedown Loads This program is developed to analyze a tiedown configuration and to determine the load distribution in the tiedown cables. The significant features of the analytical model include using a rigid body fuselage with a flexible tailboom and landing gear structures~ and using onedirectional load carrying springs for cables. This methodology is
2-16
)
STRUCTURAL DESIGN MANUAL Revision F generalized to be applicable for structures of any configuration possessing these characteristics. 2.4.4 Other Programs Individual Programs not appropriate or unable to be incorporated in the above systems will be included in this category. Possible examples are those programs of other disciplines or programs without source code and those that are unable to be transformed to a form compatible with any of the ASAM systems. This part of ASAM ;s currently under development. 2.5 Miscellaneous Programs Other miscellaneous programs used by structure engineers are described in the following: 2.5.1
Load Programs Unrelated to Finite Element Analysis
SOAN02 - Unit Panel Point Shears and Moments (RPT
299-099~252)
This program uses panel pOint weight distribution from computer program SDCSOI to create unit shear and moment data. SDAN39 - Shear and Moment Plotting Program (RPT 299-099-252) This program plots the panel point shear and moment data created by SDAN08. SDCSOl - Panel Point Unit Weight Distribution (RPT 299-099-252) Unit panel paint weight distributions are created using 50CS01. The program inputs a weights tape and outputs weight distributions to be used as input to SDAN02. 2.5.2 Dynamic Structures AnalYSis caCR02 - Rotor Blade Section Mass and Stiffness Properties (RPT 299-099-749) Airfoil coordinates are extracted from tape storage and used in calculating mass and stiffness properties for various rotor blade cross sections. CSCR02 - Stress Maps for Rotor Blade Cross Sections (RPT 299-099-749) Bending plus axial stresses are plotted for steady and oscillatory loads along the outer surface of rotor blade cross sections using airfoil coordinates from a tape.
2-17
STRUCTURAL DESIGN MANUAL 2.5.3 Fatigue Evaluation FFAM06 - Flight Test Data Generator This program converts measured flight loads in scalor units from a Xerox 530 GOC tape into engineering units. This data can be saved on a file tape for use with other programs. Output includes span plots, load vs. airspeed plots, and tab listings of mean and oscillatory loads depending on the program option chosen. FFSTOI - Loads-Airspeed Comparison Plot Program This program allows "Loads vs. Airspeed" plots from different flights, or different models~ to be plotted together for comparison. Loads are taken from FECSOl or FECS09 file tapes. FECR22 - Flight Data Organization Program This program produces a listing of sorted loads for a given Item Code, using FEeS01 file tape as input. The user can request_o?cillatory loads and/or maximum and minimum peak loads. Each gross weight and C.G. ;s sorted separately and the condition number and altitude "Line" is ~iven with each column. DLCR04 - Fatigue Life Calculation This program reads loads from FEeSOl file tape, computes a stress set, and determines the fatigue life of a component. ,Histogram plots. with various selection options which apply to th~ plots only, can also be generated from this program. Part II of the program is used to create; update. delete and list frequency of occurrence spectra used for calculation of the fatigue lire. DLCR2l - Cycle Counted Fatigue Life Calculati6n This program calculates the fatigue life of a component by considering the damage caused by each rotor revolution. This;s used with loads from FFCR03 file tape. This program may also be used as a cycle count program, producing a cycle count listing and override cards for program OLCR04. DACR62 - Harmonic Fatigue Damage The purpose of this program is to obtain a summation of damage caused by various harmonics (up to 9/Rev) within a rotor cycle for a g-;ven record. The program reads a time history tape digitized continuously at high rate.
, ,2-18
) "
"~.
STRUCTURAL DESIGN MANUAL Revision F DLCSIO - Span Plot-Regression This program reads FEeSO! file tape and creates span plots for main rotor and tail rotor blades. A curvilinear regression up to a fourth order is performed after interrogating the data. Output includes plots and punched cards for loads at a given span station. FLASH - Fatigue Life Analysis System for Helicopter
)
This is a computer software system that was developed to speed up fatigue life evaluation for helicopter certification. The system uses the computer to manipulate the data and perform analysis in the fatigue life evaluation process which consists of measuring the fatigue loads in flight. comparing these loads to fatigue test data, and then making computations of the expected fatigue life.
)
2-19
STRUCTURAL DESIGN MANUAL SECTION 3 GENERAL
3.0
GENERAL
This section, for the most part, deals with sections. Properties of various types of sections are shown. In addition, conversion factors, graphical integration, Rockwell hardness, strain gages, etc., are described.
3.1
PROPERTIES OF AREAS
In structural analysis, certain properties of areas are needed, such as location of the centroid; first moment of the area and the second moment or moment of inertia of the area with respect to an axis either perpendicular to the area or lying in the plane of the area and the product of inertia of the area with respect to a set of perpendicular axes lying in the area. These properties are defined in this section.
3.1.1
•
Areas and Centroids
The area of a generalized shape, as shown in Figure 3.1, is the sum of all of the incremental areas, dA.
x ~----------------------------~x
)
FIGURE 3.1
A
A'1
GENERALIZED AREA
3.1
The centroid of an area is that point in the plane of the area about any axis through which the moment of the area is zero; it coincides with the center of gravity of a body with the same shape having an infinitely thin homogeneous thickness. Equation 3.2 and 3.3 define the centroids of an area:
3-1
STRUCTURAL DESIGN MANUAL Revision C x =
fxdA
3.2
A
2:
D
1=1 fYdA Y= A =
YiAj
~n
i=l
3.1.2
_I
3.3
i\i
Moments of Inertia
The moment of inertia is the second moment of area. The moment of inertia of an element of area such as dA in Figure 3.1 with respect to a given axis is defined as the product of the area and the square of the distance from the axis to the element. It is shown mathematically as:
3.4 The sum of the moments of inertia of all the elements in a generalized area is defined as the moment of inertia of the area, that is, 3.5
3.6
The subscripts x and y indicate the axis about which the moment of inertia is taken. The moment of inertia of any area about any axis is equal to the moment of inertia of the area about an axis through the centroid of the area and parallel to the given axis plus the product of the area and the square of the distance between the two parallel axes; that is
3.7 1y = 10
3-2
+
_2
(x) A
3.8
•
STRUCTURAL DESIGN MANUAL
•
These equations can accomplish transformation of moment of inertia of plane areas only between parallel axes and one of the two parallel axes must pass through the centroid as shown in Figure 3.2.
x
,
8
y
)
x
~
X
FIGURE 3.2
PARALLEL AXIS TRANSFORMATION
The term 10 in equations 3.7 and 3.8 is the moment of inertia of the area about its own centroidal axis. Figure 3.4 shows an example of a typical moment of inertia calculation by the tabular method. 3.1.3
Polar Moment of Inertia
The polar moment of inertia is the moment of inertia of an area about an axis perpendicular to the plane of the area. The elementary area, dA, in Figure 3.3 lying in the plane xy has y
-
X
r
/
,)
..t
ET Y
~
X
Z
FIGURE 3.3
POLAR MOMENT OF INERTIA
3-3
IAY/ A =
(..)
I
+:-
y
.0833/.1204 = .6919 in.
~AX/A- .0007/.1204
= .00.58 in. Ix"""~Ay2+IIox_y2IA =.0631 + .0036 .. .1204 ('6919)2 ~
= .0091
H
G)
c
.~
w •
~
t1j
~
y'
m~ g~ ~ ,....
tl:!, ~' "'\ ..~It--,.·
in.4
I9=IAX2+~IOy-X'2I,A- .0072 + .0022 - .1204 (,0058)2 = .0094 in.
i= .
',,~CD
4
Xy-I,AXY+I.loxy-XYI,A= .0007 + 0 - (.0058)(.. 6919)(.1204) = .0002 in.4
/"< J1C -Iy- ) ::.
an 2{3 = - 2 I xv
tot
3
en
.75
......
;:ID
- 2 ( .0002) / G. 0091-.0094) :: 1.3333
= n .....
xl
I:!j
0
~
Y
~
I
c:
t=.040
3: 0
X
~ ~
ELE
I:%j ~
1
2
8
3
;:.
4
H
A
Y
y2
AY
Ay2.
lox
X
X2
AX
AX2
lOY
AXY
CJ ~
H'
Z
:.:. r-
z 8 0
::a
.028 .036 .028 .0284
.92 .45 .92
.545
.846 .203 .846 .297
.0258 .0162 .0258 .0155
.0237
.0073 .0237 .0084
.0024 .0012
-.35 -.02 .366 .036
.1225
.0004 .1340 .0013
-.0098 -.0007 .0102 .0010
.0034 .0000 .0038 .0000
.0011 .0011
n
~
-.0090 -.0003 .0094 .0006
en
-
Ci)
Z
31:
t""
n
:.:.
c: ~
Z
8
c: ::a:r-
H
0 Z
.1204
.0833
.0631
.0007
.0072
.0022
STRUCTURAL DESIGN MANUAL
e) Ix =
fy 2dA,
(Equation 3.5)
Iy =
fx 2dA,
(Equation 3.6)
The moment of inertia about the z axis is 3.9 but r
2
x
2
+
y
2
3.10
then 3.11
The polar moment of inertia of an area is therefore equal to the sum of the moment of inertia of the area about two mutually perpendicular axes. Thus I
3.1.4
z
IX
+
Iy = Ip
3.12
Product of Inertia
The elementary area, dA, in Figure 3.1 is located at a distance x from the y axis and y from the x axis. The product of the area multiplied by the coordinate distances is then, xydA and is called the product of inertia. This term is a mathematical property that is dependent upon the area itself and its loc'ation relative to two mutually perpendicular axes. The value of the product of inertia for the entire area is
)
3.13
The subscripts of I serve to define the pair of axes of reference. Unlike the moments of inertia, products of inertia involve the first powers of the coordinate distance and such products may be positive, negative or zero. An example is shoWn in Figure 3.4. Products of inertia can be transferred between axes. Thus the product of inertia of an area about any pair of mutually perpendicular axes is equal to the sum of the products of inertia of that area about a parallel mutually perpendicular pair through the centroid plus the product of the distances between the axis times the area as shown in Figure 3.5.
3-5
STRUCTURAL DESIGN MANUAL 3.14
Ixy y
x
X
y
)
~-----------------L----X
FIGURE 3.5 3.1.5
PRODUCT OF INERTIA
Moments of Inertia About Inclined Axes
Unsymmetrical sections are quite common and it is often necessary to find the moments of inertia about an inclined axis. The general procedure is to first find the moment of inertia about some set of rectangular axes through the centroid and transfer to a second set of axes also through the centroid making an angle 9 with the first set of axes using the following relationships and Figure 3.6.
~ ~
3.15
3.16 Iuv
= 1/2(Ix-Iy )
sin29 + Ixy cos29
3.17
,.)
~~---------T----~----X
FIGURE 3.6 MOMENTS OF INERTIA ABOUT INCLINED AXES
3-6
STRUCTURAL DESIGN MANUAL The angle axes. 3.1.6
)
e
is posi.tive when produced by a counterclockwise motion from the original
Principal Axes
It is often necessary to determine the maximum and minimum values of moments of inertia. These occur about an axis passing through the centroid of the section. The axes are called the principal axes and1~he orientation is such that the moment of inertia is either greater than or less than for any other axis passing through the centroid. The principal axes location makes an angle with the original axes of tan29 = 2Ixy Iy.I x There are two values are two values of ~ original x axis, the other axis, inclined be minimum.
3.18 of 29 differing by 180°, having the same tangent. Then there differing by 90°. About one axis at the angle Q from the value of the moment of inertia will be maximum and about the at 90 0 to the first, the value of the moment of inertia will
Substitution of the values of Q into equations 3.15 and 3.16 produces the principal moments of inertia. 3.19 3.20
)
The sum of the principal moments of inertia, like the sum of any two moments of inertia about mutually perpendicular axes, is the polar moment of inertia. It should be noted that the product of inertia. of an area about a pair of axes which are principal axes of inertia is zero. The product of inertia about a pair of axes, one of which is an axis of summetry, is also equal to zero. Thus the axes of sym~ metry are principal axes of inertia. 3.1.7
Radius of Gyration
The radius of gyration of an area with respect to a given inertia axis may be defined as a distance to the point where the area would be concentrated in order to produce the same amount of inertia. Thus P =JI/A
3.21
3-7
STRUCTURAL DESIGN MANUAL The subscript for I determines the inertia axis for the respective radius of gyration. 3.2
MOHR·S CIRCLE FOR MOMENTS OF INERTIA
The equations for the moments and products of inertia about inclined axes may be found using a semigraphic solution which aids in visualizing the relationship between the moments of inertia about various axes. The method is called Mohr's circle. Using Figure 3.7, the following procedure describes Mohr's circle for inertia.
Ix
.....-----'9"1-? cos2
Iu; Iv - - -____
f
f3
I -I
I xy = .J:L:Y sin2J3 2
o
U
Ix
.....- - - - - - - I u FIGURE 3.7
MOHR'S CIRCLE FOR MOMENTS OF INERTIA
( 1)
Calculate Ix, Iy and Ixy for the section. The x and y axes can be at any orientation but it should be located so that Ix is greater than Iy.
(2)
Draw a set of rectangular axes. Label the horizontal axis Ix and the vertical axis Ixy. This is shown in Figure 3.7.
(3)
Moments of inertia (second moments of areas) are always positive, but products of inertia can be positive or negative. Positive moments of inertia are plotted to the right of the origin. Positive products of inertia (Ixy) a~e plotted above the Ix axis and negative values below.
(4)
Layoff the distance OAI along the Ix axis equal to Ix and A'A parallel to the Ixy axis equal to Ixy. Label the point (Ix, Ixy) as point A.
(5)
In a similar manner locate point B by making OBI equal to Iy and B'B equal to -Ixy , the value of Ixy with the algebraic sign reversed.
3-8
STRUCTURAL DESIGN MANUAL (6)
Draw the line AB intersecting the Ix axis at C and draw a circle of diameter This circle is Mohr's circle for moments of inertia and each point on the circle represents Iu and Iuv for any orientation of the u and v axes. The abscissa represents Iu and the ordinate Iuv.
AB.
The following relationships can be developed using Figure 3.7. Maximum moment of inertia:
') Iup
= 1/2 (Ix+Iy) + J1Xy2 + 1/4
(Ix-1y)2
3.22
Minimum moment of inertia: 3.23 These are the moments of inertia about the principal axes. The angle of the principal axis with respect to the reference axis is tan2j3
2Ixy
Ix-Iy
3.24
The sign of P is taken as positive for counterclockwise movement from the reference x axis (Ix in Figure 3.7). The moment and product of inertia (Iu, IV, Iuv) may be determined at any angle 29 from the principal axes. In the cross section the angle is 9 while in Mohrts circle it is plotted as 29. From Figure 3.7 the values can be derived to be
)
3.25
3.3
Iv = Ix sin 29 + Iy cog 29 + Ixy sin29
3.26
Iuv = Ixy cos29 + (Ix;Iy ) sin29
3.27 .
MASS MOMENTS OF INERTIA
The inertia resistance to rota~ional acceleration is that property of a body which is co~only known as mass moment of inertia. If a body of mass m is allowed to rotate 'about an axis at an angular acceleration a, an element of this mass dm, will have a component of acceleration tangent to the circular path of ra, with the
3-9
STRUCTURAL DESIGN MANUAL tangential force on the element being radm. Since the distance to the element is r, the resulting moment of the force equals r 2adm. Integrating the elements of the body gives
This expression is known as the mass moment of inertia of the body, where a is dropped because it is constant for a given rigid body.
)
If the body is of constant mass density, the differential, dm, may be replaced with pdV, since dm = pdV, and the following expression results 3.29 The units of mass moment of inertia are commonly expressed as lb-ft-sec 2 or slug-ft 2 •
3.4
SECTION PROPERTIES OF SHAPES
Tables 3.1 through 3.5 show section properties of various sections. sents the properties of standard tubings.
Table 3.6 pre-
~
~
)
3-10-
Revision B'
RECTANGLE
r-X'
)
I
X
r.L ---i-- ~i
A
=
i
=. b/2
bd
Y
1 2 -:.t
Y P, - J
d/2
db!
=F
It_I
'2-2
3
bd
P3"3
12 - 2
= 12'
I
= bd
=
bd
2
-6-
= 0.2887 b :.
O.2887d
:
O.5773d
3
3
k--b~
3-3 II-I _ X
RECTANGLE
= bd Y = bd
d b2
-a-
A
~ b 2 t d2
L;_t
b3 d 3 6 (h 2+ d 2)
11_ 1_
b 2 d2
.,
•
,
Y -6v'b 2 td 2 =
P. 1-'
RECTANGLE
)
A _
d2 )
= bd
Y=
I
bd
,J S (b 2...
bSIN8
2 2
'-1
+ d COS8
::: bd (b SIN
2
COS! 8)
12
!.t.:L = bd ( b 2 SIN 2 6 (b SIN
j
p
8 +d
2
_
Ib 2
1-' - v-
8 + d 2 COS 2g )
8 + d case)
SIN!
8 + d 2 cos 2 8 12
.
TABLE 3.1 PROPERTIES OF COMMON SECTIONS 3-11
STRUCTURAL DESIGN MANUAL HOLLOW RECTANGLE 3
y t 2
~ bd 3 - ac 3
12 ( bd
P. 2-2
=
- ac)
11_2. .-_2_ _
V(i)d - oc )
HOLLOW SQUARE
SQUARE
2
= d Y = O.5d Y:s = I. 414d _ d4 12 _ 2 - T A
d4
13 - 3 = T2 =
2
c
-
= Y = 1._. = 13 -3 z: X
12 "2= 1.. -..
2
O.5d
d4
=
-
12
c4
4d 4 - 3d 2 c2 - c· 12
II_I
1,-, _ d~ j -S
".-t = O.2887d TABLE 3.1 (CONT·D) PROPERTIES OF COMMON SECTIONS 3-12
A = d
p._ 1 = '3-3 =0.289 .j'd
P 2-2
=p .-4
=
2
.. C 2
(1..:,.1__.:;.. .__
Vd Z
_
c2
"8~
STRUCTURAL DESIGN MANUAL
TRAPEZOID (0 + b) d 'A= - - -
2
x= ~
=
d3
(
b2 + 4 a b + a 2) 36 (a + b)
--=:..--..::..~~..=..:;;..~~
, .. I
I
[C _ b~a]
d(b+2o) 3(o+b}
V= I
~
+
=
2-2
d 5 (b + 30) 12
p...~ Sra+b)
~_~ d
2
./2 (b + 4 ab +
2
0
)
-v'=: (0+:b )
OBLIQUE TRIANGLE bd Z
-1 1,,1
Y ="12
)
I = bd 5 2-2 12
.
a
1'.."3
b:
2
I .... , V.
=
P. . .=
.236 d
bd
24
Pz ..i .408 d ~.; .707d
p'_=:..236 . / b2 + e2 _ be
I = bd (b2+ e 2 - be ) 4 ... 4
36
TABLE 3.1 (CONTtD) PROPERTIES OF COMMON SEtTIONS 3-13
STRUC,JURAL DESIGN MANUAL OBTUSE
If
TRI.ANGLE 4~
2
bd h : 12 Y 2 It:.L = bd 24
A=M
2 -_ y .d3
)',
y= b +2c 3
~_I= .236d
a = bd b+c
I._f
bd! 36
I2 ..~
bd! 12
)
' ....4. 236 ~ bZ+ c 2 + be l 2 '6 ..~. 408 ~ b +c +:3 be
S
I=~ I-a 4 .
I = ~(b2+bc+CI) ••• 36 l
I8-6 :J!.L(b +3bc+3c 12
CJRCLE A
.".
2
•
= 40 = . 78540
Z '
::. 3.14ISR,
2
----DaR y-x--2 ..
4
II_I=~..i :4 D =.049090 =.7854R I2_i 14.:
1,_.:: y
~
X
s..a 4
-I
TABLE 3.1 (CONT'D) PROPERTIES OF COMMON SECTIONS 3-14
R4
Ia-3=.098'80".O.7854RS
P =p ::. JL I .. ,
17'
4
.'
)
STRUCTURAL DESIGN MANUAL SEMI- CtRClE
A = ~ R2
= 1.5708 R2
'1'= Mi 31T = 0 • 4244R Y=R
3r-
[:!L - ..§..] :: 0.1 097 R4
1 :: R4
0
'''',
8 971' I 1:.3927 R"
4
P.1- .:: .2643 R
2 .. 2
4
= . 3927 R 4-4
1
I
Ii..
2-
= 1.9637 R4
2= • 50
R
1.a...: 1.118 R
a"3
01
QUARTER CIRCLE A:: .7854 Ri
Y=j =.4244 R I 2
I
CIRCULAR
SECTOR
1:.0549 R4
1..1 , I:
2-2
=1 a... J
1 :: .1964 R" 4-4
A :: R2
(I
y: ~ [R S~N a ]
,Y:
R SINa 4
a -SINa COSo) 4 I : O.25R (a -SIN a COSo) + R " 2~2 2 I ... ,': O.25R
(
4
)
I
=O.25R 4 (tJ _
3-3
16 SIN a
941
"s I N2 a
+ SI~2 q )
I...lo.25R 4 (a + SINa COSo) p.>.=O.50R.JI- SINo oCOSG a"- =.J IRia 2
a
IN RADIANS
Z.. 2
p. = 0 50R.) I + .SINaCOSa _ 16SiN~(!
. 5 ..S
.
,a
9412
'.>.= O.50RJ 1+ SING oCOSa
TABLE 3.1 (CONT'D) PROPERTIES OF COMMON SECTIONS 3-15
STRUCTURAL DESIGN MANUAL CIRCULAR SEGMENT A : O. 50 R 2 (2 a -
- _ x -
y:
SIN 2 a)
3
4 R SIN a 3(2a - SIN 2a)
RSIN a 3 2 = 25 A R [I - ~ ( S IN a COS a 3 a-51 N a COS
I·. o.
1
1 = I._t O.50R _ 2 2
I 2
4
)] Q
2
(2a - SIN 2-a)(SIN a ) ,
'.
) 6
:025AR2[1+2SINaCOSa ]_4RSINQ a - SINa COSa 9A
3-' .
I 4-4
~O.25AR2[1+ .
a IN RADIANS
2 SINJaCOSa ] a - 5 IN a COS a
1_ 2 5 IN
3
a CO S a 3 ( a - S ~ N a C OS a )
-JIIIIIIIIII"!'..----. 21 _ 3
R2
(
2 CI
-
3
SIN a )
CIRC,ULAR COMPLEMENT A X
= O.2146R 2 = Y = 0.22 34R
P.= 0.3159 R 11_.= 12 - 2 = 0.007545 R4
Is_,= 0.01198 R 4 1..
_.=
O.003J06R 4
15 _ 5= 16 - 6 =0.0182 R4
'1_'= '2-2 :
0.187 R
, 5. .; "-6.= 0 .• 292 R
-t
TABLE 3.1 (CONTtD) PROPERTIES OF COMMON SECTIONS 3-16
)
.)
STRUCTURAL DESIGN MANUAL Revision E
OBLIQUE FILLET :5
~i2 4
A
=
(TANS - 8 ) R 2
Y
::
[1[TAN
p
=
[SEC8 -
cose
2
I-I
6 COSS 651 N 8
4-4
SINt9TAN29 3(TAN8-8)
JR
4
2
2
(If 2 SIN 8 )- 8 _ S I N STAN 8 ] R4 4 9 (TAN S -8 >. 2
38- SIN 8 CoS8t3 - 2 SIN 8) _SlN
4
4
STAN 8 ] R4 9(TAN9-8l
12
'59 _ 8 - SIN 8 CO 5 9 ] R"
s :: [SEC! 8-COS S _
I
a]R
8-
=
... [TA N B( TAN 28 + SIN S )_ 12 - 2 6"
:: [ SIN
2 SINt9 TAN 3(TAN8 -8)
i
.. [SIN 8 TAN8 + SINe II .. 13'
J
8] R
SINt9 TA N 3(TAN8-8)
4 2
S+SINS COSS _ SIN STAN'*8 JR 4 4 9(TAN9-8)
HOLLOW CIRCLE A::
3.1416 (R2- r
2
)'
'i:: y = R 4 I, .. , = 1.-.= 0.7854 (R - r ::
P."I
=o.
...!.:! 1
Y
= P4-4 = Pa-s
)
4
12 - 2 = 13 - 3
P2-2
.
•
..
2
O. 7854 (5 R - 4 R r 2
5 (R t r 2 15R t r2 d
=v
2
2
- r"
4
)
1/2
)
2
4-_ r4) (R _ = 1 4 -4 = ___7_8_5_4_ _ _
x
R
TABLE 3.1 (CONTtn) PROPERTIES OF COMMON SECTIONS 3-17
~ ,,~,,~ ~\-'\'1
STRUCTURAL DESIGN MANUAL
Revision A
ECCENTRIC HOLLOW CIRCLE 2
A : ". (R 2
y
=
-
r2 )
e r2
R2 - r2
IJ _I = . .78 54 ( R4 - r 4 4
12 .2: . 7 e5 4 (R - r
4
)-
)
HOLLOW SEMI- CIRCLE
A = 1. 5 708 (R 2_ r 2 x
=R
y = 2
I
I-I
0.4244 (Rt
la-a: 0 . 3927(R 1.- 4 = o.
t
TABLE 3.1 (CONT'D) PROPERTIES OF COMMON SECTIONS 3-18
r2
R+r)
::; 0.3927 (R4_ r 4 ) -1.5708(R 2 _r2)y
I 2 ... 2 = 0.3927 (R4- r .
3
)
4
4
4
)
r ) 22' 2 3927 (R - r: ) ( 5 R + -
'...,....
!
/
",'I" :
~
I
I
,
•
""
'.,
\
'\
'
\ \
" " "it~ I ,'\' Bell j
,
\~
. """
\ ,...... ':..l.-"
I
"
.... ,.
STRUCTURAL DESIGN MANUAL
."*
Revision A
HOLLOW CIRCULAR SECTOR 4
:5
)
TABLE J. I (CONT'O) PROPERTIES OF COMMON SECTIONS 3-19
STRUCTURAL D'ESIGN MANUAL PARABOLA
= 4/3 =b
A 5
2
•
X
Y =
ab
0.40£1
a:
I, ,= 0',09143 b II = O.2667ob Is
r----I--------LL ----1.....-
2
i
-I, Y
HALF PARABOLA
~i
b- ,
:: 0, 3048 a ~ b
1""4 = . 5 7 14 a! b P, _I = . 2 619 a 'z.2= .4472b
=
A
15 - 5 = 1.6 ab
3
'.-4 ::.65460 PO-5
'~~
,t
"I
= 1.095 b
2/3 ab
y = 0.400 x = O.375b 1•. = 0.C4571 a' b S 12 :: 0.039580 b I, = 0.15238 aSb I.. = O. 1333 D b 3 3
y
l a - 5 =,.28570 b
Pa... =. 2619 0 P2 .. i = .2437b
' ....... = .4472b Pa- e = .65460
'COMPLEMENT OF HALF
PARABOLA
= 1/3 ab y = 0.700 x = O.75b
A'
I. I2
= 0.01762
o:S b
= 0.0125 ob!
.... = .22990
P
,~
TABLE 3.1 (CONT'D) PROPERTIES OF COMMON SECTIONS 3-20
)
STRUCTURAL DESIGN MANUAL PARABOLIC FI LLET IN RIGHT ANGLE
1 t
'1
a
:: 0.3536 t
b A
.: 0.7071 t l :: O.1667t
i I.
=V:
O. SOOt
::
O.05238t 4
II::
-----~'"
....--- t
ELLIPSE
... b/2 A = 3.1416 a b P,-. i :: Q PI -! :: I. U8b b Y :: 0/2 Pz-2 l :: O.7854ab I, = 1.1180 3 12 ': O. 78540 b '4-4
-
-
•
I a-I :: 5/4 ". ab 3
HALF ELLIPSE 2
' '1
•
r-
I
T 2
A X
= t. 5708b
Y
:: 0.42440
=
V
la
b
= O.3927o ' b
.!...L
x-1
TABLE 3.1 (CONT'D) PROPERTIES OF COMMON SECTIONS
3.927 ab
= 3.927a l
I. ::O.I098a'b 12 = O.3927ob'
a
b
I 4-4 = 5/4 ... b'a
=
4
2
)
I
a b
STRUCTURAL DESIGN MANUAL
• QUARTER ELLIPSE .......Ji~
21
..
A y i
= 0.7854ob = 0.4244 a = 0.4244 b = 0.05488 a!
It b 5 I •.:: 0.05488 a b J! 14
= O. 1963 0 3 b a = O. 1963 a b
ELLIPTIC COMPLEMENT I
= 0.2146 ab - y = 0.7766 a i = 0.7766 b A
II :: 0.00754 0' b 3
12 = 0.00754 ab
TABLE 3.1 (CONT'D) PROPERTIES OF COMMON SECTIONS
3-22
)
""8! ..
STRUCTURAL DESIGN MANUAL
HOLLOW ELLIPSE
HOLLOW SEMI- ELLIPSE
..
3
I
) 2
.. A
i y
= = =
1._.=
3
3.1416 (ob - cd)
a b 0.7854 (0 b' - cd')
12 - 2 = 0.7854
(0 b! -
l
cd )+3.14IS (ob-cd)b
2
1,_,= O.7854(Olb - cad ) !
I
I._ 4= 0.7854(0 b- c d 1+ 1r
(0
b - cd)a
I
)
NOTE: -
ABOVE PROCEDURES IMPLY A NON - UNIFORM WALL THICKNESS
TABLE 3.1 (CONT'D) PROPERTIES OF COMMON SECTIONS 3-23
~8
STRUCTURAL DESIGN MANUAL
DOUBLE WEDGE SECTION. ct A : -2_ t
Y
=--r
1._.=
ct 3 48
= .0208 c t
,)
3
I ~-t -- ~4t 2 : .0416 ct 2
81- CONVEX SECTION A :
2 3" ct
t V = .. 2
1._.= 11-1
-:
y
m
3
ct =.038tct
Sct l
.
105 : .0761 ct
S
2
VALUES OF A 'AND'I. ARE BASED ON BI-PARABOLIC SECTION. ERROR IS NEGLIGIBLE WHEN tIc
TABLE 3.1 (CONT'D) PROPERTIES OF COMMON SECTIONS 3-24
<
0.10 •
)
STRUCTURAL DESIGN MANUAL = t( b +
A
ANGLE
e)
-x = t( 2a2(a+ +h) h)+ a 2 - 3
I
-3
+ by
t(h - y) x
3 -
I
y
-Y =
t(b - x)
=
I
3
-
t(b
+
2e)
2(b
+
+
e
2
e)
a(i - t)3
_3
+ hx - e(i -
t)3
3
+ abeh_t_ 4(b + e}
-
xy - -
A = 2htl + at h 2 t + .St 2 a y = 1 2htl + at
CHANNEL
= 2t 1h 3 +
I x
If
I
T SECTION
12
=
I
+ bd 2
2
+ 2ee ) + 2ee
a(h -
3
i) -
2e(b -
i) +
3
dx
3
x X
X
,t
Z SECTION
+ .St 2 a)2 + at
1 2htl
- 3
Y;:I~' i-, 1b~c ',. ,-IdJ-
2 _ (h t
hb 3 _ ea 3
=
~ ::::: h _ ~(dh dh
I-~~L
3
3
A ::::: ae
y
•
Y
at
Iy
A
=
=
ea
3
+
12
bd
3
ted + 2a) y =
Ix
= ~2
[ bd
Iy
= ~2
[deb +
I
=. I x
x, tanZDC.
3
2b -
t
2
- a ( d - 2 t) 3 ]
2
cos oJ.
3
2
a)3 - 2a e - 68b c] - I
' 2
y
sin CIt.
cos 2cL 2 = (dt - t 2 )(b - bt)
Ix - Iy
TABLE 3.1 (CONT'D) PROPERTIES OF COMMON SECTIONS 3-25
STRUCTURAL DESIGN MANUAL -
-x
= bh
A
PARALLELOGRAM I
bh 12
bh
I
2
bh(a
=
I
2
+ b 2) 12
3
= -3-
x.
2
Y
bh 6
= -(2a
y.
e
-Y = -h
+ b
a
3
=-
x
I
=
2
+ 2b 2 + 3ab)
,~, ,..
REGULAR HEXAGON
A = .866h
I I
REGULAR OCTAGON
=
P
y = h/2
a
= .0601h
y
= .2766h
XI_
I
I
=
x
)
-'
Z
= b/2 =
x
I
.1203h
4
I
4
=
Y'
.3969h
4
4
= 2.8286r 2
A
=y =r I = I = x Y
x
x.
=
I . I
NOSE RIB
A
x I I I I
xI
= chord line.
=
p
= =
I
YI.
.6381 r
=
4
3.4667 r
1.2763 r
4
4
= a(b
+ c)(19b 2 + 26b c + 19c 2)
480
x Xt
= .1333(ab
=
y .
y~
+ ac)(b 2 - be + c 2 )
3 .0418 a (b
=
+
4 .2857 a (b
c)
+ c)2
Based on parabolic segment.
TABLE 3.1 (CONTID) - PROPERTIES OF COMMON SECTIONS 3-26
-----
)
2/3a(b + c) .6a y = .375(b - c)
I
P
= I
x
+ I
Y
STRUC1·URAL DESIGN MANUAL Revision A ~---.------------------------------------,
CENTROID
~-
---
t
r
Area
y
t
r
Area
y
)
TABLE 3.2 - AREA, CENTROID & MOMENT OF INERTIA OF 90° BENDS
3-27
STRUCTURAL DESIGN MANUAL y
I ,
x
x
B
M
!
I y A
·1
B
9/16 1/4
5/8
1/4
11/16 1/4
3/4 7/8
1
5/16
1 1/8 5/16
1
3/8 5/16
1 1/2
1 1/0
3/8
2
1/2
·t
TABLE 3.3 - PROPERTIES OF ANGLES 3-28
STRUCTURAL DESIGN MANUAL
,----E
r·
L
,
)
TABLE 3.3 (CONT'n) - PROPERTIES OF ANGLES
3-29
STRUCTURAL DESIGN MANUAL y
I
------~--~~----+------_T--~~---
X
A
1 .
T
A
.020
1/2
.~5
.0}2
.025
,
M
AREA
R
.13Q .141 .l~14 .1 '2 .1 )9 .1 t5 .115 .1 1 .1 18 .1 1 .1 ~
i2 019 2 .02J_ 2 .0.10 2 .026 J 32 .O}3
9/~ .032 .041 .040 1 .O}2 :5 ~ .O3~ ,/8 .01 o l,j .• 040 .~ 1 ., 2 .O~n .0 2 2 ~2 11/i6 .01 0 1 (I . .0<1
,
'/4
0 1
13/16 ·
7/8
!~
&
7,'32 .10 l~
.ll'l
l@
.200
.072
'at> 7,
,}2
.113
1 1
.O-n
L012
.081
.~
},16 .110 cl..ll
M2 1
.1121 ~ .040 1 .0'''11
")
~ ~2
-
.O"U .5. 12 - ,--
271,
.200 .2~
.145 .300
~ .1~1
·J~~h .103
.Q5.1t i~6 .n 1 1/8 .012 if32 ..lli .081 1 4 .166 Jm. 9, 12 .1Bl .102 11 2 .20 ~ 1 1/"+ .040 1 ,
1 1/4 .0..]2
.061
.tt
.'07 .,05 .~ll .~lQ
-
.186 .206
.102 11
2
~.J
.0341
1
.040
1 'I
.020
1
.Q21
Wj
-:lOb .31 2 .134 .3'
.0251 .3 1 .0:51
• ~ • 32
3 16 .166
1 3/8 .Q72
.0020 .211
~l
7, 2 .185 1r4 .206
.~4
.21~
o .~~ 16 .222 ~
.250
.0051 .218 .000, .277 .0070 .274 .oo8lI .273 .0091 .270 .0077 ,"51U .009b .317 .0116 ...ll~ .0128 .313 .0141 .312 .0134 .310 .01Q9 .356
.Q91
9_ 2 .22
.102 11
2 .250
.0";1
2 .l~
~
1 1/2 .0
5
2 16
.16
21 32 .20
.on
1 '"
.OS1
9, 32 .25_2
.2~
.102 11 32 .280 .0 1 5, 32 .150
.00,li}, 16 1 5/8 .0 2 7')2 .011 1 'Il .~ 1 9 '}2
.198 .221 .248 .275
.102 J.J. t~2
. ~o~ & 1 ·3/32 .112 ...QI~ ..l.16 .214 .0 ~ ~ '32 .239
1 3/'" .Oi 1 1" .261 .()i 1 9.',2 .2<17 .102 11 '}2 .341
.,_~
.O}49 .4'5
.1 0
.~6B. 2~ .O}~. 111
.'91 .Q'_6I . ,..2 •()II.2tJ .IIO~
.4~'-
..!,l
.11, .01&11 • ~T. • If] .u I~ • '1 . 2} ..Q ~ • 7" .~ ~ ~:
,"-p
.~i}6.0 1" .\66 ,4'5..Q ~O .:)19 .1
.,
~
• ).1J
18
.05l'b
.·~l'_
,I'.~. ~~_"-
1 '-!!E1 • 1--> 1,,68. 07Qo .' Ll
.1a67 .1t7' IQ< .1Kl .41 .~)Q
~1 a' 9 .0666 .
.Q'I39
I)
.000J ," ). .0911.' .1008.'
.0136 .351 .016Q .356
.Q'U
•0241 .,49 .OUlt • !tOO .O~qo , -_. ------ .. -....391
") ',\2 . HI'
.~
2
3 (16 .24 .072 1 ',\2 .27"'\ ...Q6l..l 4 307
. "\2Q .080Q .1 ~O ."'l~" .0QQ8 :, 57 .' I' .1114 .1 ~
.' ItS .ill1 .t 36 .
.0Ql9'-..2.""2 .''\'' .nB2 ,102 III [12.-..J82.
.t3'!l . I 6~ .1'\27 .1 in
TABLE 3.3 (CONT'D) - PROPERTIES OF ANGLES
3-30
.~I· ,
2
.~4
.,>1 ,31
1 4
.3<:0 .U23l
.356 .02~ .~ .3f2 .(Y2~ .~ .31 .0,1 .,
~
~1
.}2'3 .0185 S}5 .0221 ·'55 .}} .0226 .}51 .'Ill .33-
), 16 .1.5.0 7/32 .161
.0015 .191 .0018 .19' .0022 .19'
.'21t2 • .21l8 • .2~ .
.12'
.5, 32
T
.064
.221 .0041 .23'
_~ 22 .~
·.t>4 ,1 ~
A
.0008 ~f73 .0011 .171 .0013 .17t
.21tl .OO}9 .23t
.242 .249 .O!,-r • 26J. 270
~ ~
.t'l'U
1
Pxx
,
1
.081
Ix..x
.-000'; .1')9 .0006 .158 .0007 ·ill
.2}3 .OO~l .2~> .01 1 ' .228 .. OO~O .257 5/32 .0 ) .2}~ • ~ '-?~
• .. 0
.040
y
1'8
4 2
• 0 r2
..
.20) .0029 .210 .20, .0026 .'23E .212 .00'1 .23i
~l ~ '2 ~, .032 ) }2 .0~'5 LJ 0 l~ .0' ~. 1 5.'32 .0 0 ~ .3~6 .006 2 1. 32 .0'~5
·
..
I
.)
STRUCTURAL DESIGN MANUAL y
---L----x
x---~--
...------1--- -
A
B
T
5/8
1/1!
.032 .040
3/4
1/1+
.032
1/8
1/4
3/8 1/4 1
3/8
1 3/8
)
0~2
.040 .032 .040 .032 .040 .040 .051
.064 .0}2
1/4
.0hO
1/2
.040 .051 .06)+
1 1/8
1 1/4
.040
.040
3/8 _ .051 3/8
.064 .040 .0'51 .06t~
Jl5.1
11/2
1/2
~lt
.072 .0'51 13/4
1/2
.06J~
072 .01)1
2
5/8
.064 .081
.064 2'1/2
,
3/4
3/4
.072 .091
.064 OQl
.102
-
A
y -,
,
AREA!
~.
-
~
.
PXX
Pyy
3/32 .031 .QB- .0002 .0016 .07~ 1/8 .057 .080 .0002 .0016 .073 3/32 .03'5 .06~ f---!--9.9 02 ~5.. .072 .042 liB 071 0028 .OP02 )/12 oo-n 070 039 062 0002 1/8 .047 .060 .0002 .001.1.2 .069 3/32 .O!~7 .105 .0006 .0051 .115 1/8 .057 .111 .0007 .00)9 .113 3/32 .043 .058_ .0002 .0052 .068 .0)2 .063 .0002 .0059 .067 1/8 .062 .103 .0003 .0082 .112 1/8 5. 32 .076 .110 _~0009 .0096 .110 3/16 .09 2 .litf .0011 .0110 .10n 5/32 .047 .05 4 .0002 .0071 .066 1/8 .057 .022 .0002 .0001 .066 1/8 ·Q77 .141 .00H3 .0140 .1,)4 5/32 .095 .1hB .0022 .0167 .1'52 3/16 .116 ~ -.0026 .019h .150 liB .072 .092 .0008 • OlJ.j.I~ .108 5/32 .089 .09d .0010 .0170 .106 3/19 . lOti .1.99_ .0012 .012.0 .104 1/8 .077 .087 .0009 .olB2 .106 '5/32 .095 .093 .0010 .0217 .104 5/16 .116 .100 .0012 .02')2 .102 ')/"32 114 ~2e .002'5 ,0339 .l46 ~/16 002Q ,0')99 14'} 140 13" 7 '32 .1')4 .141 .00)2 .0427 14"3 ') 32 .122.- .lH3 .0026 .04q6 .142 "3 16 .1'56 .12'5 0031 .0')89 141 7 "52 172 1~0 0031 063~ IhO 5/32 .1'53 . ll~7 .00r)1 .c8J..6 .183 3/16 .ltJ8 .1,)4 .0061 .Q978 .181 l/)~ .230 : .16,) .0013· .1147 .179 3/16 .236 ). r7)~ . 0111 .1946, .217 7 )2 .262 I 178 0122 2128 216 9 ')2 • .. 322 ! .190 .0147 .2rj21 .211+ ') 16 .268 : .1'57 .0117 .~0'59 209 Q 1')2 17? ?()(-) III '--6 ~h7 b..onh 11/32 .404 I .180 .0168 .4279 .20h
.226 .?18
R
M
Ixx
Iyy
.267
.Orr
,2~Q
~oB
.299,
.330 ·323
.3 Jn
.330
.385 .356
.346 .386 .377 .427 .419. .hl0 .447 4'38
.42]
.hS'7 .1nB ~467
'>4h 5~4
.,)27
62') IL615
.607
~
.
.731 . .721
.706 .Q~
.
GOl ~5
1 069 nuu 1.030
.1
, TABLE 3.4 - PROPERTIES OF CHANNELS
\
3-3t
STRUCTURAL DESIGN MANUAL
CIRCUMFERENCE RADIUS OF MOMENT OF AREA DIAMETER (IN.) GYRATION (p) (INCHES) r<'RACTION DECIMAL (SQ. INCHES) INERTIA (1) .0469 .5891 6.066 x 10-; .02761 .1875 3/16 .6381 .0508 8.358 x 10. . .03241 .2031 13/64 .6872 .0547 .0001125 .03758 .2188 7/32 .7363 .0586 .0001481 .2344 .04314 15/64 .0625 .7854 .0001918 .2500 .04909 1/4 .8836 .0703 .0003072 .06213 .2813 9/32 .9818 .0781 .0004682 .07670 .3125 5/16 .0860 1.0799 .0006854 .09281 .3438 11/32 .0938 1.1781 .0009710 .1105 .3750 3/8 .1016 1.2763 .001337 .4063 .1296 13/32 .1094 1.3744 .001798 .1503 .4375 7/16 .1172 1.4726 .002370 .1726 .4688 15/32 .1250 1.5708 .003068 .1964 .5000 1/2 .1328 1.6690 .003910 .2217 .5313 17/32 .1406 1.7671 .004914 .2485 .5625 9/16 .1485 1.8653 .006101 .2769 19/32 .5938 .1563 1.9635 .007490 .. 3068 5/8 .6250 .1641 2.0617 .009104 .3382 .6563 21/32 .1719 2.1598 .01097 .3712 .6875 11/16 2.2580 .1797 .01310 .4057 .7188 23/32 2.3562 .1875 .01553 .4418 .7500 3/4 .1953 2.4544 .01829 .7813 .4794 25/32 2.5525 .2031 .02139 .8125 .5185 13/16 .2110 2.6507 .02488 .8438 .5591 27/32 .2188 2 .. 7489 .02878 .8750 .6013 7/8 .2246 2.8471 .03311 .6450 .9063 29/32 .2344 2.9452 .03792 .6903 .9375 15/16 .2422 3.0434 .04323 .7371 .9688 31/32 .2500 3.1416 .7854 .04908 1.0000 1 .. 2656 3.3380 .8866 .06256 1.0625 1 1/16 3.5343 .2813 .07863 .9940 1.1250 1 1/8 .2969 3.7307 .. 09761 1.1875 1.1075 1 3/16 .3125 3.9270 .1198 1.2500 1.2272 1 1/4 .3281 4.1234 .1457 1.3125 1.3530 1 5/16 .3438 4.3197 .1755 1.3750 1.4849 1 3/8 4.5161 .3594 .2096 1.4375 1.6230 1 7/16 .. 3750 4.7124 .2485 1.5000 1.7671 1 1/2 .3906 .2926 4.9088 1.5625 1 .. 9175 1 9/16 .4063 5.1051 .3423 1.6250 2.0739 1 5/8 .4219 .3980 5.3015 1.6875 2.2365 1 11/16 .4375 5.4978 .4603 1.7500 2.4053 1 3/4 5.6942 .4531 .5298 1.8125 2.5802 1 3/16 .4688 .6067 5.8905 1.8750 2.7612 1 7/8 .4844 6.0869 .6917 1.9375 2~9483 1 15/16 6.2832 .5000 .7854 2 2.0000 3.1416 TABLE 3.5 - PROPERTIES OF CIRCLES 3-32
)
STRUCTURAL DESIGN MANUAL
)
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DIAMETER (IN.) FRACTION DECIMAL 2 1/16 2.0625 2 1/8 2.1250 2 3/16 2.1875 2 1/4 2.2500 2 5/16 '2.3125 2 3/8 2.3750 2 7/16 2.4375 2 1/2 2.5000 2 9/16 2.5625 2 5/8 2.6250 2 11/16 2.6875 2 3/4 2.7500 2 13/16 2.8125 2 7/8 2.8750 2 15/16 2.9375 3 3.0000 3 1/16 3.0625 3.1250 3 1/8 3 3/16 3.1875 3 1/4 3.2500 3 5/16 3.3125 3 3/8 3.3750 3 7/16 3.4375 3.5000 3 1/2 3 9/16 3.5625 3.6250 3 5/8 3 11/16 3.6875 3 3/4 3.7500 3 13/16 3.8125 3 7/8 3.8750 3 15/16 3.9375 4 4.0000 4 l/B 4.1250 4 1/4 4.2500 4 3/8 4.3750 4 1/2 4.5000 4 5/8 4.6250 4 3/4 4.7500 4 7/8 4 .. 8750 5 5.0000 5 1/8 5.1250 5 1/4 5.2500 5 3/8 5.3750 5.5000 5 1/2
AREA 3.3410 3.5466 3.7583 3.9761 4.2000 4.4301 4.6664 4.9087 5.1572 5.4119 5.6727 5.9396 6.2126 6.4918 6.7771 7.0686 7.3662 7.6699 7.9798 8.2958 8.6179 8.9462 9.2806 9.6211 9.9678 10.3.21 10.680 11.045 11.416 11.793 12.177 12.566 13.364 14.186 15.033 15.904 16.800 17.721 18.666 19.635 20.629 21.648 22.691 23.758
MOMENT OF CIRCUMFERENCE RADIUS OF GYRATION (p) INERTIA (I) (INCHES) 6.4796 6.6759 6.8723 7.0686 7.2650 7.4613 7.6577 7.8540 8.0504 8.2467 8.4431 8.6394 8.8358 9.0321 9.2285 9.4248 9.6212 9.8175 10.0139 10.2102 10.4066 10.6029 10.7993 10.9956 11.1920 11.3883 11.5847 11.7810 11.9774 12.1737 12.3701 12.5664 12.9591 13.3518 13.7445 14.1372 14.5299 14.9226 15.3153 15.7080 16.1006 16.4933 16.8860 17.2787
.8883 1.0009 1.1240 1.2580 1.4038 1.5618 1.7328 1.9175 2.1165 2.3307 2.5607 2.8074 3.0714 3.3537 3.6549 3.9761 4.3179 4.6813 5.0673 5.4765 5.9101 6.3689 6.8540 7.3662 7.9066 8.4765 9.0764 9.7072 10.371 11.067 11.799 12.556 14.214 16.025 17.992 20.128 22.550 25.124 27.738 30.675 33.852 37.321 40.980 44.915 -
.5156 .5313 .5469 .5625 .5781 .5938 .6094 .6250 .6406 .6563 .6719 .6850 .7031 .7188 .7344 .7500 .7656 .7813 .7969 .8125
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-
TABLE 3.5 (CONT'D) - PROPERTIES OF CIRCLES
3-33
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No.
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2~
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.701ll .9050
.120
~H
ArCfl
w.u
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6.1915
1.3·195 1.1191
1.3268
I.30B4
.8897 1.01798 1.1103 1.47)6 1.16Ul 1.4694 1.5635 1.4608
1.3224 ".2158 4.9473
2.3281
S.6H2 6.lOn 7.5387
3.5476 1.)965
1.9319 VI 484 1.'1378 2.6561 1.4274 2.9683
2.2271
.9898
2.8098 3.1001 3.9627 5.03Y1
1.2"&8 1.4179
5.9166 6./)Hj
1.5625 9.0612 10.4217
2.62H 3.3143
1.4170
1.5682 1.5619 l.SYJ8
1.7612
1.'H91
2.231:10
1.5~67
2.6296 ).OOlB 3.361l
1.5156 ). SO~l
1.5261
4.0272 1.4846 1·6)19 1.4644 1.1054
1.6566
1.}95S L'}B'}l 1.9707 2.507,1
1.6S03 ] .6462 J .il}']) 1.()2'j!
7.00~9
2949'1
1.6145
8.0107
3.3n9 1.60)9
3.76-15 4.6dO} 5.9550
n~
2.6S7S 3.11l9 l.S]4l
;11
4.156$
89738 1O.77M
3.77I'io1 .... :BN
"II
5.H42
12.<1224
.5.2)05 1.S525 1.7187 L76!7 1.714<> 2.1919 1.7H9 2.1~}1 1.7n.. 3.1871 J.7028
~~
a·14
) .2821
}.87SB
l}
1.4619
4.4012
H
,;.
l.KW7
5.4798
1.5934
1.57;8
1.5S03
'J:J
I.. lin 2.834A
ill
2'
1.2g~:i
9,40H9
}.7b)6
1.6~22
h
3.7106
JO.j~07
4.2203
S·071H ,5.H6.5':1
).61:511 1.6610
"
691'10}
8.21')2
II ..
4.6019
12.6'1\3
:l'
~.4'1b7
14.66'17
III
TABLE 3.6 (CONT'D) - PROPERTIES OF ROUND TUBING
1.6-106
3-35
STRUCTURAL DESIGN MANUAL .3.5 The
BEND RADII minim~m
bend radii for sheet materials are given in Tables 3.7 and 3.8.
Table
3.7 shows the minimum radii obtainable by cold forming the sheet while Table 3.8 gives the minimum radii by hot forming the sheet. Figure 3.8 shows the minimum
flange width. Shorter flanges may be obtained by trimming after forming, but this is expensive and shall not be specified unless additional tooling and processing costs can be justified. ~----------------------------------~
w=2R+t ---
FIGURE 3.8-
3.6
-'
STANDARD DESIGN BEND RADII
HARDNESS CONVERSIONS
Table 3.9 presents the conversions for hardness numbers to ultimate tensile strength. In this table the ultimate strength values are in the range 50 to 304 ksi. The corresponding hardness number is given for each of three hardness machines; Vickers, Brinell and three scales of the Rockwell machine • .3. 7
GRAPHICAL INTEGRATION BY SCOMEAN-O METHOD
It is often necessary to integrate curves of unusual or irregular shapes. There is a convenient method of integrating even the most complex shapes. It is called graphical integration and is attributed to Scomeano. For the purposes of this discussion, assume the curve tq be integrated has the form of y=f(x). The curve could be composed of several different shapes; such as, fl(x), for-x=O to nl; f 2 (x), for x=nl to n2; f.3(x) , for x=n2 to n3, and so forth. Figure 3.9 shows a typical problem for which graphical integration will be used. Since the integral of the curve y=f(x) is, equal to the area under the curve, the maximum ordinate of the first integral of f{x) will be the area under the y=f(x) curve. It is necessary to chose a scale for the ordinate of the integral curve so that the maximum value of the ordinate will lie in the working area of the graph. The area under the y=f(x) curve is estimated. In the example in Figure 3.9, the area can eaSily be calculated since the f(x) curves are straight lines. It is 134 and the scale for the first integral is chosen to give easily read divisions and to show the maximum value within the limits of the paper.
3-36
)
I,.
\fUJ
BELL HEUCOPTER
~-
COMPANY
.
DESIGN STANDARD
STANDARD DESIGN BEND RADII
BEND RADIUS ____...,......1
L
THICKNESS
MINIMUM FLANGE WIDTH
R (FROM BEND RADII CHART)
)
W (MINIMUM FLANGE WIDTH) = 2R + T
NOTES:
In
Z
o
&
&
SHORTER FLANGES ~y BE OBTAINED BY TRIMMING AFTER FORMING, BUT THIS OPERATION IS EXPENSIVE AND SHALL NOT BE SPECIFIED UNLESS ADDITIONAL TOOLING AND PROCESSING "COSTS CAN BE JUSTIFIED.
£
BEND RADII LISTED ARE FOR ANGLES BETWEEN 0 0 1 AND 110 INCLUSIVE. RADII FOR LARGER ANGLES MUST ~E COORb~ATED WITH PRODUCTION ENG.
&. ® 4.
TI~IUM
PER MIL-T-9046
THE SUNDARD BEND RADII TOLERANCE IS
> BELL DESIGN
STANDARD
TITLE
BEND RADII-STANDARD DESIGN /O~4-
± .02.
CODE I DENT NO. 97499
Drawinq Nu mber
I",. BELL 'iIJJ HELICOPTER
DESIGN STANDARD
COMPANY
. STANDARD DESIGN BEND RADII ~ MATERIAL THICKNESS MATERIAL A~D CONDITION fOR.."1ED AT ROOM T::MP .012 .016 .020 .025 .032 .040 .050 .063 .0711.080 .090 .100 .l2S .160 .190 .250 .18 .2.) • .l') • .Jl .~I! .:if) .. 06 .00 .Ofj .ulj .12 _J.O I • .1.0 .03 ,.03 1.06 l2024-0 31 .31 .37 .31 • SO .is .. ~O l~~~06 .06 .09 .12 .16 .l8 .25 06 A 12024 ...T( } .03 .03 .03 .03 .06 .06 .06 .06 .09 .09 -'59 .l2 .16 .lS ;.25 .3~ l 5052-0 .03 .03 .06 .06 .06 .09 .09 .12 .16 .16 .18 .25 .25 .31 .38 • .56 ~~S2-H3:" )i 5OS2-H34'* .02 ! ..O~ ~2 .03 .03 .0..3 .06 .06 .oel .09_ .. Oil .12 t 16061 .. 0 06 06 .09 -,,09 03 09 .12 .16 .L8 06 06 03 .03 38 .0-.3 2~_ N 1~061-Tf ) 16 .18 25 31 31 37 31 50 .75 •.88 ~2S06 06 .06 .09 .12
~ 7075-0
.06 .09
707S-T( 1 .
M ~Z31a~Q
-.-
A G ~z3i.... R2q
I
r-' .- -
-
.
300 SERIES 5 ~~RI~S T 300 SERIES E ~O SER-IES f300 SERIES "l30 At.'IL &
~.ARBON Sn.
"'-.-
--
.
~
.09 : .'pg"
.03
AI.'lL lis HARD \.:i HARD 3/4 HARD HARD I.OW
03
--,,06
.09 .. 09
--
I T TYPE I
024 .030 .024 048 054 .054
COMP A COMP B
A N TYPE I.. caMP C t U TYPE II J:OHP A ItA
'~.
1&
l'YPE ..lII. COMP C mE III t OOMP D
l'
!., *FOa ..... -
.03 03 .06 .09 .09 .06
~ORM
I
.06 .12
.06 .09
~Og
.00 .16
, . Jr'- ~la_
- ~.18 .•18 .18 -.25-. .31 .. - _. ---
4130 NORM it 8630
. TlTY!!
.06 .09
.03 -,,06 .. 06 .06
.03 --,,06 .09 .12 .16 .06
.06 06 .Og .16 .19 .06
.09
.09
.12
.09
.12
032 .. 040 .. 040 .OSC .(i32 .040 .064 .080 .072 .090 072 .090
.09 .25
.. 09 .25
.12 .31
.16 .37
.16 • .,37
.. 18 .50
.25
'~i5~
.25
~31~
.37 t .50
.75
.JS-T:~'1
I
I
.50
.50
.62
.7S ;87
• .,37 .50 .62 ,,87- [..00 " LSO-
.31 .75
.50
t' - -_ .
1..00 ;"50 1..75
,
...
~
.12 .19 .22 .09
.06 .09 .16 .22 .22 .09
.09 .09 16 12 .25 1.25 .25 .2S .25 .28 .12 .16
.09 16 .38 .31 •..Jl .16
.38 .19
.12
.16
.19
.. 25
.31
09
064 080 .100 .062 .080 .100 .125 .050 .0,64 .080 .100 .lOO .118 .160 .200 .112 .1"4 .180 225 .112 .144 .180 .225 ~OSO
.21
.17i .213 .177 .319 ,.283 355 .283 .355
.126 .157 .126 .252
.12 19 .38 .38
.12 ... 25 . 50 .47
.. 50
. ., I
-
.22
.50 .25
.28
.75 L.OO .7S ll.:OO.34 .~O
.3l
• .,31
.38
.50
.~7
.50
.50
.62
.200 .225 .250 .312 .. 240 .270 .300 • .,375
.. 200 .225 .250 .~OS .450 .400 .450 • SOO .400 .450 .500 .360
.312 • .562
.625 .625
NON-8mUCTtJRAL USE ONLY
it
COMMON DESIGNATION
TITANIUM CONDITION
~ ~ I
TYPE TYPE TYPE TYPE TYPE TYPE
~
~" til
>
0
COMMERCIALLY I, COMP A PURE COMP B I, C I, COMP 11,- COMP A Ti-5Al-2.5Sn III, COMP C Ti .. 6Al-4V ELI III~ COMP D Ti-6A!-4V
.:
Q. Q.
!-- '"-
'" Z
0
'.
--
'" > ....,
® SHEET
1 REVISED.
~ ~
.
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~a:,1'u ',;'
,
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BELL DESIGN STANDARD -AT'!I:'"
/" J"t'7t'(" t·~ /1 ;:. i£- 'lo
(~ a:;~i~~
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v~:/~-7(.)
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TITLE
CODe IDENT 1'40.97499
Ora winq Number BEND
RADII-STANDARD DESIGN
-~
.-
r
~
-.
.. 06
-
160-001 2 ' 2 SHEET
~F
STRUCTURAL DESIGN MANUAL ~~TERIAL AND CONDITION fOR...-£D AT ROOM TEMP
2024-0
.03
A 2024-T{ )
.06
~o.?2-0
.OJ .OJ
16061-0
.03
H7075-0
.06
l
U 50S2-H34 M i50s2-H3~;'
,
.0£_
N 6061-T( )
Nt
.03 .06
.06$ti
.Ob
.09
.12
.16
.09 .18
.03 .03
.03
.06
.06
.06
.09
l!\Z318-0
A G IAz3i.8-R24
1. 09
.03 ' .03 .03 .03 .06 .06 .09 .09 HARD .09 .09
S ~EB..lf:S ~ HARD T 300 SERIES ':i HARD e 300 SER-IES3Zq HARD
f 4130 j60 SERIES A.'lL &
LOW
CARBON STL 4130 NORM & 8630
.06
!NORM
TITYPE
I. COMP A
T TYPE I. COMP B
A
N mE It COl-1P C f
W TYPE II.
Nt.
cq~ A
TYPE III COHP C rrYPE III, COMP 0
.06
03 ~~-I .06
'. i-8-- .-18
300 SERIES ANL
.06
L..
.06 .06 .09- .09
7075-TC 1
1
MATERIAL THICKNESS
.012 .016 .020 .025 .032 .OLiO .050
•
.09
6 .03
~
.03 .09 1.12 ! .16 .06 1.06 1.09 .12 .16 .25
:09 ' .J!;
~.O80 .T6
_.37
.16 50
.18 .75
.50
.25 .50
.75
.87
. 56 .15
.75
.~f-
.09 .25
I
.18
.25
.2.5
.~f
.6.1 . .75
.37
.31
.18 .37 .09 .18 .. 09 .09 .37 .18
.12 .25 .12
.12
.3t_ ~_7_
,:81 LOa ;.. 50 L 7S
• 18
.25
.31
.. 50
.50
.03 .06 .06
06
,06 09
.12 .06
.03 .06 .09 .12 .16 .. 06
.. 19 .06
.
.06 .09 09 1.12 16 .25' 22 .25 22 .25 .09 .12
.09 1.16 .25 .28 .28 .16
.31 .. 16
.19
.47 .50 .47 1 .50 .22 .25
.09
.09
.12
.l2
.16
.22
.25
.31
.:31
.09
..
6
.
.31 .75 .18 .31
.09 .16 .09 09 .31 .16 .37
18
.25 .31
.25 .50 .16 .25
• 2S, .. , .06 .09 .12 .16 .06 .09 09 06 .25 31 .12 .16 .31 .37
.09 .06 06
.090 .100 .L15 .160
.09 .16 .38
.:31
.12
.12 25 .3s-T • so 19
.38 .38
.31
.~~
.50 .28
~
.02E., .032 .040 .050 .064 .. 080 .100 .126 .177 .200 .225 .250 .312 .030 .0LtO .050 .. 062 .080 .100 _1-25 .157 .213 .2LiO ".270 .300 .375
.024 .032 .040 .050 .048 .064 .OBO .100 .. 05Lt .072 090 .112 .054 072 .090 .112
.064 .080 .100 .126 .128 .160 .200 .252 .144 .tBO .225 .283 .144 .180 .225 .283
.177 .200 .225 .250 .312
.319 .360 .~OSe·562 .355 .400 .450 .625 .JSS .. 4QO .450 .500 .625
COMMON DESIGNATION
TITANIUM CONDITION
COMMERCIALLY I, COMP A PURE I, COMP B I, COMP C II, COMP A Ti-5Al-2.5Sn TYPE III, COMP C Ti-6Al-4v TYPE III, COMP D Ti-~1-4V ELI
TYPE TYPE TYPE TYPE
) Note:
0
1 .. Bend radii listed are for angles between 1 and 110 0 inclusive. Radii for larger angles must be coordinated with production engineering. 2. The standard bend radii tolerance is + 0.015. 3. Reference Bell Design Standard '160 - '002.
TABLE 3.7
~
STANDARD DESIGN BEND RADII (COLD FORMING)
3-31
STRUCTURAL DESIGN MANUAL
~TERLAL
AND CONOITIO~. ~
MATERIAL THiCKNESS .012 .016 .020 .025 .032 .040 .050 .063 .071 .080 .090 .lOO .125 .160
A
L U
'i' 7075-T( )@
3000F
.06
.06
.06
.06
.09
.. 12
.16
.19
.25
.31
.38
.41
• SO
350 F
.06
.06
.06
.06
.06
.09
.09
.t6
.16
.18
.25
.25
25
'r .06
.09
.09
.09
.16
.18
.25
.31
.27
50
.50
.50
.. 62
.69
N
~ /' ~
M ~Z31B .. O @
Q
A G ~Z31B-H24 @ 350°F
riTE I. COM]? A @ qOO· -600° F T
TYPE ~
I
r.
COMP B
400:1 - 600· F
T TYPE I. COHP C
A N
@ LlOO° -600°
F
I ..
.018 .02L, .0.3C .-037 .048 .080 .100 .126 .142 .160 .180 .200 .250 I
.ou~
.024 .,030 .037 .Oil8 .080 .100 .126 .142 .160 .180 .200! .2SC
.018 1.024 .030 .031 .OLl8 .080 .100 .l26 .142 .160! .180 .200 .250
t
U TYPE II. COMP A M S 400 0 -600" F
e I rI
.036 .048 .060 .075 .096 .120 .LSO .189 .248 .. 280 .315 .350 .437 ..
caMP C
.024 .032 .040 .050 .064 .080 .100 .126 .142 .160 .lS0 .200 .250
[rYPE III. COHP 0 9 'tOO O _600 0 'F
.024 .032 .040 .050 .061.1 .080 .100 .126 .142 .160 .180 .200 .. 2.50
ITYll @
./.too ...... &00" F
TITANIUM CONDITION TYPE 1, COMP A TYPE 1, COMP B TYPE 1, COMP C TYPE II. COMP A TYPE III. COMP C D TYPE 1119 COMP ... Note:
COMMON DESIGNATION COMMERCIALLY PURE Ti-: SAl .. 2. 5Sn Ti-6Al-4V Ti-6Al-4V ELI 0
0
1. Bend radii listed are for angles between 1 and 110 inclusive. Radii for larger angles must be coordinated with Production Engineering. 2. The standard bend radii tolerance is + 0.015. 3. Reference Bell Design Standard 160 - 002.
TABLE 3.8 - STANDARD DESIGN BEND RADII (HOT F0RMING)
3-38
STRUCTURAL DESIGN MANUAL t
, I
Tensile strength
ksi
)
'
:I
•
Rockwell
VickersFirth Diamond
Brinell
10mm Btl Ball
A Scale
B Scale
60 kg Hardness Number
Hardness Number
120 deg
100 kg 1/16 in.
150 kg 120 deg
Diamond Cone
Dia Stl Ball
Diamond. Cone
3000 kg
---
50
104
92
52
108
96
54
112
100
--
56
116
104
--
58
120
108
60
125
113
62
129
117
64
135
122
C Scale
58
--
61
--
64
--
66
.
.
----
--
68
--
70
---
72
----
66
139
127
---
68
143
131
--
77.5
70
149
136
---
79
--
80.5
--
82
--
85.5
----
87
--
74 "
76
72
153
140
74
157
145
76
162
150
--
sa
78
167
154
Sf
84.5
80
171
158
52
82
177
162
53
,
--
."
"
~
.
i
! ~
TABLE 3.9 - HARDNESS CONVERSION TABLE 3-39
STRUCTURAL DESIGN MANUAL ~
VickersFirth
Tensile Strength
Diamond
Hardness
ks!
Number
Rockwell
BrineU'
3000 kg 10mm Stl Ball Hardness Number
A Scale
B Scale
C Scale
60 kg
100 kg 1/16 in.
150 kg 120 deg
120 deg Diamond
Cone ---
Dia Stl Ball
.
,.
83
179
165
53.5
87.5
--
85
186
171
54
89
--
87
189
174
55
90
--
89
196
180
56
91
--
91
203
186
,56.5
92.5
--
93
207
190
57
93.5
--
95
211
193
57
94
--
97
215
197
57.5
95
--
99
219
201
57.5
95.5
--
102
227
210
59
97
--
104
235
220
60
98
19
107
240
225
60.5
99
20
110
245
230
61
99.5
21
112
250
235
61.5
100
22
115
255
241
62
101
23
118
261
247
62.5
101. 5
24
120
267
253
63
102
25
"
TABLE 3.9 (CONT'D) - HARDNESS CONVERSION TABLE 3-40
Diamond Cone
"
)
"8 Tensile Strength
kai
)
STRUCTURAL DESIGN MANUAL
VickersFirth Diamond
Brinell 3000 kg, 10mm Stl Ball
Hardness
Hardness Number
Number
Rockwell A Scale
B Scale
C Scale
60 kg 120 deg Diamond Cone
100 kg 1/16 in. Dia Btl Ball
150 kg 120 deg Diamond: Cone
103
26
123
274
259
63.5
126
281
265
64
--
27
129
288
272
64.5
--
28
132
296
279
65
--
29
136
304
286
65.5
--
30
139
312
294
66
--
31
142
321
301
66.5
32
147
330
309
67
150
339
318
67.5
----
155
348
327
68
--
35
160
357
337
68.5
--
36
165
367
347
69
--
37
170
376
357
69.5
--
38
176
386
367
70
181
396
377
70.5
188
406
387
71
194
417
398
71.5
-
-----
33 " 34
39
40 41 42
TABLE 3.9 (CONT'n) - HARDNESS CONVERSION TABLE 3-41
r
I
I
STRUCTURAL DESIGN MANUAL !
VickersStrength Firth Tensile
Diamond
ksi
Hardness Number
Brinell' 3000 kg 10mrn st1 Ball Hardness Number
Rockwell A Scale
B Scale
C Scale
60 kg 120 deg Diamond
100 kg
150 kg 120 deg. Diamond Cone
Cone
1/16 in. Dia Stl Ball
201
428
408
72
--
43
208
440
419
72.5
--
44
215
452
430
73
--
)45
221
465
442
73.5
--
46
231
479
453
74
--
47
237
493
464
75
--
48
246
508
476
75.5
--
49
256
523
488
76
264
539
500
273
556
283
573
76.5
---
51
512
77
--
52
524
77.5
----
53
..
294
592
536
78
~O4
611
548
78.5
--
TABLE 3.9 (CONTln) - HARDNESS CONVERSION TABLE
3-42
50
54 55
)
STRUCTURAL DESIGN MANUAL
e
l
1400
l~
200
1300 1200 1100
)
1000 900 800 160 16 -'-,
';-'_~
__ ' __
l
1--~1 I
-
i
1
700 140 14
;~ ""i - ~ -~ -; ~
,
e)
600 120 12
,
~".
I
'
I
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i
t
i
I!
;--t :
I t ,
'~; ..
~
_ ..
1~ ~.
.~i
~
.-
)
~
~ ~ ~
500 100 10
---
.,
400
80
8
300
60
6
200
40
4
100
20
2
,
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...
r
-
_
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i
,
I
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t
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:
t
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0
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0
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4
6
..
,
,- !
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,
8
10
12
"
14
'
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-. 1-1
i
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16
18
20
~
0
LENGTH
1""1
><
.....
'
~ 'FIGURE 3.9 - GRAPHICAL INTEGRATION
tI.()
3-43
STRUCTURAL DESIGN MANUAL The next step is to locate a "pole. u This is accomplished by dividing any ordinate of the first integral curve by the corresponding ordinate of the y=f(x) curve. For the case in Figure 3.9, select the first integral ordinate of 120. The corresponding y=f(x) ordinate is 12. The pole distance from the origin is then 120/12=10. Now it is necessary to determine from which side the integration is to be made. It is customary to integrate from the left but integration from either side is possible. If from the left, draw a vertical line at x=lO from the origin. If from the right, draw a vertical line at x=IO to the left of the maximum x value of the y=f(x) curve. In the example the integration is from the left and the pole is located at x=10 from the origin.
.)
The next step is to determine the incremental x, (Ax), to be. used. In the example, y=f(x) is a straight line throughout, so~x=l is arbitrarily selected. The smaller theAx, the smoother the integral curve. ~x can be changed as the integration progresses if more accuracy is needed. The first step in the integration is to lay out~x=l from the origin. Locate the median point on the y=f(x) within this ~x increment. For a straight line, this will be the midpoint but if the y=f(x) is a curve within the~x', then the median must be estimated,. This is done by assuming a loea tiona Draw a horizontal line through the point. Check the areas above the horizontal line and below the horizontal bounded by y=f(x) within the Ax increment. When these two areas are equal, the median is the point where the horizontal line crosses the y=f(x) curve. Project the horizontal line to the vertical pole. section and the origin of the y=f(x) curve. Mark the x=l vertical line. Layout another increment between x=l and x=2. Locate the median as before the vertical pole. Determine the slope from this fer this slope to the point previously located at from x=l to x=2.
Draw a line between this intera point where this line crosses of Ax. This time it will extend and project a horizontal line to intersection to the origin. Trans-. x=l. Draw a line with this slope
Continue the above procedure until the limit of the y=f(x) curve has been reached. The curve formed by the many slopes of the projected lines is the integral curve of y=f(x). The same procedure can be used to get the second integral by integrating the ff(x)dx curve.
A baseline for 0 and 9 must be located so that the true deflections and rotations can be determined. The 6 baseline connects two points of known deflection. In the example in Figure 3.9, the ends of the beam have zero deflection t so the 6 baseline is located by connecting the two end points of the 6 curve with a straight line. To locate the baseline for 9, take the ordinate value of the 0 curve at x=L and divide this value by L. In Figure 3.9, let. 6 =1329 and L =20, then Q 1329/20 = 66.45. Draw a horizontal line at 9=66.45. This is the Q baseline. b ase The deflection and rotation at any point on the beam is the vertical distance from the respective curve to the baseline •
.3-44
/)
STRUCTURAL DESIGN MANUAL 3.8
CONVERSION FACTORS
Table 3.10 shows conversion factors for most technical units. Equations for converting from one unit of temperature to the other are shown below:
3.9
~R=1.8~OK-273.16)+491.69 R=l.8 C+49l .. 69 °R=oF+459.69
3.30 3.31 3.32
°K=5/9(oR-491 .. 69)+273.16 °K=5 /9,( °F_ 32 )+2 73.16 °K=oC+273.16
3.33 3.34 3.35
°C=5/9(oR-49l.69) °C=5/9(oF-32) °C=oK-273.16
3.36 3.31 3.38
~F=1.8&OK-273.16)+32 F=1.8 C+32 °F=oR-459.69
3.39 3.40 3.41
THE INTERNATIONAL SYSTEM OF UNITS (SI)
The purpose of this section is to acquaint the engineer with the inevitable, the Metric System. The International System of Units, or Systeme Internationale (S1), is sometimes referred to, in less precise terms, as the Meter-Kilogram-Second-Ampere (MKSA) system. The SI should be considered as the definitive metric system, since it is much broader in scope and purpose than any previous system. 3 .. 9.1
Basic SI Units
The following are the basic units for the SI: meter, m kilogram, kg second, s
)
ampere, A degree Kelvin, oK candela, cd
In addition, the amount of a substance is treated as a basic quantity" The basic unit is the mole, symbol: mol. The mole (mol), a unit of quantity in chemistry, is defined as the amount of a substance in grams (gram mole, gram molecular weight; pound mole, pound molecular weight) which corresponds to the sum of the atomic weights of all the atoms constituting the molecule. 3.9.2
Symbols and Notation
When using the 51, the following rules apply:
3-45
STRUCTURAL DESIGN MANUAL
TO CONVERT acres acres atmospheres atmospheres atmospheres Btu Btu Btu centimeters centimeters centimeters centimeter-grams centimeter-grams cen time ter- grams centimeters/sec centimeters/sec centimeters/sec centimeters/sec centimeters/sec centimeters/sec centimeters/sec/sec centimeters/sec/sec centimeters/sec/sec centimeters/sec/sec coulombs cQu1ombs/sq/in cubic centimeters cubic centimeters cubic feet cubic feet cubic feet cubic inches cubic inches cubic inches cubic meters cubic meters cubic meters cubic meters cubic meters degrees (angle) degrees/sec drams (u.s., fluid or apoth. ) drams drams drams
INTO sq. feet sq. meters kgs/sq. em pounds/sq. in. newton/sq. meter foot-1bs joules ki1owatt-hrs feet inches meters em-dynes meter-kgs pound-feet feet/min feet/sec kilometers/hr knots meters/min mi1es/hr feet/sec/sec kms/hr/sec meters/sec/sec miles/hr/sec faradays coulombs/sq. meter ell. inches liters ell. meters gallons (U.S. liq.) liters ell. meters liters quarts (U.S. liq.) ell. feet cu. inches cu. yards gallons (U.S. 1iq.) liters radians revolutions/min cubic em. grams grains ounces
TABLE 3.10 - CONVERSION FACTORS 3-46
MULTIPLY BY 43,560.0 4,047.0 1.0333 14.70 1.013 x 778.3 1,054.8 2.928 x 3.281 x 0.3937 0.01 980.7 10-5 7.233 x 1.1969 0.03281 0.036 0.1943 0.6 0.02237 0.03281 0.036 0.01 0.02237 1.036 x 1.,550. 0.06102 0.001 0.02832 7.48052 28.32
10
5
10- 4 10- 2
10- 5
10- 5
16,387~06
0.01639 0.01732 35.31 61,023.0 1.308 264.2 1,000.0 0.01745 0.1667 3.6967 1.7718 27.3437 0.0625
)
STRUCTURAL DESIGN MANUAL
INTO
TO CONVERT
)
feet feet of water feet of water feet of water feet of water feet of water feet/min feet/min feet/min feet/sec feet/sec feet/sec feet/sec foot-pounds foot-pounds foot-pounds foot-pounds foot-pounds foot-pounds foot-pounds foot-pounds/min foot-pounds/sec foot-pounds/sec foot-pounds/sec gallons gallons gallons gallons gallons (liq. Br. Imp.) gallons (U.S.) gallons of water gallons/min gallons/rnin grains (troy) grains (troy) grams grams grams grams grams grams grams/cu. em. grams/cu. em. grams/liter grams/liter grams/liter grams/sq. em.
meters atmospheres in. of mercury kgs/ sq/r:n.e terpounds/sq. ft pounds/sq. in. ems/sec kms/hr miles/hr ems/sec kms/hr knots mi1es/hr Btu gram-calories joules kg-calories kg-meters ki1owatt-hrs newton-meters horsepower Btu/hr horsepower kilowatts cu. ems. cu. feet cu. inches liters gallons (U.S. liq.) gallons (Imp.) pounds of water cu. ft/sec liters/sec grains (avdp) grams dynes grains joules/meter (newtons) ounces (avdp) ounces (troy) poundals pounds/cu. ft. pounds/cu. in. grains/gal pounds/l,OOO gal pounds/cu. ft. pounds/sq. ft.
MULTIPLY BY 0.3048 0.02950 0.8826 304.8 62.43 0.4335 0.5080 0.01829 0.01136 30.48 1. 097~ 0.5921 0.6818 1.286 x 10-3 0.3238 1.356 3.24 x 10- 4 0.1383 3.766 x 10-7 1.356 3.030 x 10-5 4.6263 1.818 x 10-3 1.356 x 10-3 3,785.0 0.1337 231.0 3.785 1.20095 0.83267 8.3453 2.228 x 10- 3 0.06308 1.0 0.06480 980.7 15.43 9.807 x 10- 3 0.03527 0.03215 0.07093 62.43 0.03613 58.417 8.345 0.062427 2.0481
TABLE 3.10 (CONT-D) - CONVERSION FACTORS 3-47
STRUCTURAL DESIGN MANUAL
-
r-----~------------------------------------------------------------------------~
INTO
TO CONVERT
horsepower horsepower horsepower horsepower (550 ft Ib/sec) horsepower horsepower horsepower-hrs horsepower-hrs inches inches of mercury inches of mercury inches of mercury inches of mercury inches of mercury inches of water (at 4°C) inches of water (at 4°C) inch-pounds joules joules/em joules/em kilograms kilograms kilograms/cu. meter kilograms/meter ki lograms/ sq •. ' cm •. kilograms/sq. em. kilogram-calories kilogram-calories kilogram-calories kilogram meters kilometers kilometers ki1ometers/hr kilometers/hr ki1ometers/hr kilometers/hr kilometers/hr kilowatts kilowatts kilowatts kilowatt-hrs kip kips/sq. in. knots knots knots knots knots
MULTIPLY BY
Btu/min 42 .. 44 foot-1bs/min 33,000. foot-1bs/sec 550.0 horsepower(metrie) (542.5 ft 1b/sec) 1.014 kilowatts 0.7457 watts 745.7 Btu 2,547. foot-1bs 1.98 x 10 4 eentime·ters 2.540 atmospheres 0.03342 feet of water 1.133 kgs/sq. meter 345.3 pounds/sq. ft. 70.73 pounds/sq.. in. 0.4912 inches of mercury 0.07355 pounds/sq. ft. 5.204 newton-meters 0.11298 kg-meters O. 1020 poundals 723.3 pounds 22.48 poundals 70.93 pounds 2.205 pounds/cu. ft. 0.06243 pounds/ft. 0.6720 atmospheres 0.9678 pounds/sq. in. 14.22 Btu 3.968 foot ... pounds 3,088. 426.9 kg-meters foot-pounds 7.233 feet 3,281. miles 0.6214 ems/sec 27.78 feet/min 54.68 feet/sec 0.9113 knots 0.5396 meters/min 16.67 BtO/min 56.92 foot-lbs/sec 737.6 horsepower 1.341 Btu 3,413. kilonewton 4.4482 megapascals 6.8948 ki1ometers/hr 1.8532 nautical mi1es/hr 1.0 1.151 statute mi1es/hr 1.689 feet/sec 0.5148 meters/sec
)
)
1-
--------------------~----------------------------------------------------------~
TABLE 3.10 (CONT'D) - CONVERSION FACTORS 3-48
~ STRUCTURAL DESIGN MANUAL ~u ....... 1C1iOfIt"'f ...
• )
• )
,~
TO CONVERT liters liters liters liters Megapasca1 Megapascal meters meters meters/min meters/min meters/min meters/min meters/sec meters/sec meters/sec meters/sec meter-kilograms miles (naut. ) miles (naut. ) miles (naut. ) miles ( statute) miles ( statute) miles ( statute) miles/hr miles/hr mi1es/hr millimeters mils Newton Newton Newton-meter Newton/sq. rmn ounces· ounces ounces poundals poundals pounds pounds pounds pounds pounds pounds pounds pounds of water pounds of water pound-feet pound-feet pounds/cu. ft.
INTO em. feet cu. inches quarts (U.S. liq.) pounds/sq. in. newton/sq .. rmn feet inches cms/sec feet/sec knots mi1es/hr feet/min .kilometers/hr miles/hr miles/min pound-feet feet kilometers miles (statute) feet kilometers mi 1 es (nau t .. ) feet/sec knots meters/sec inches inches pounds Dynes inch-pound Megapascal grains grams ounces (troy) grams pounds grams kilograms newtons ounces ounces (troy) poundals pounds (troy) cu. feet gallons em-grams meter-kgs kgs/cu. meter Cll. Cll.
MULTIPLY BY 1,000.0 0.03531 61.02 1.057 145.039 1.0 3.281 39.37 1.667 0.05468 0.03238 0.03728 196.8 3.6 2.237 0 .. 03728 7.233 6,080.27 1.853 1.1516 5,280. 1.609 0.8684 1.467 0.8684 0.4470 0.03937 0.001 0.22481 1 x 10 5 8.8507 1.0 437.5 28.3495 0.9115 14.10 0.03108 453.5924 0.4536 4.4482 16.0 14.5833 32.17 1.21528 0.01602 0.1198 13,825. 0.1383 16.02
TABLE 3.l0·(CONT t D) - CONVERSION FACTORS
3-49
STRUCTURAL DESIGN MANUAL TO CONVERT
INTO
pounds/cu. in~ pounds/ft pounds/in. pounds/sq. in. pounds/sq. in. pounds/sq. in. pounds/sq. in. pounds/sq. in. quadrants (angle) quadrants (angle) radians radians radians/sec. radians/sec. revolutions revolutions/min revolutions/min slug slug square centimeters square feet square inches square kilometers square meters square millimeters temperature (OC) + temperature (oC) + temperature (OF) + temperature (OF) tons (long) tons (long) tons (long) tons (metric) tons (metric) tons ( short) tons ( short) watts
kgs/cu. meter kgs/meter gms/cm atmospheres feet of water inches of mercury kgs/sq. meter megapascals degrees radians degrees quadrants revolutions/min revolutions/sec radians degrees/sec radians/sec kilogram pounds sq. inches sq. meters sq. ems. sq. miles sq. feet sq. inches (oC) absolute temperature 0 temperature ( F) absolute temperature (OF) temperature (OC) kilograms pounds tons (short) kilograms pounds pounds pounds (troy) Btu/hI'
-
273 17.78 460 32
TABLE 3.10 (CONT'D) - CONVERSION FACTORS
3-50
MULTIPLY BY 2.768 x 10 4 1.488 178.6 0.06804 2.307 2.036 703.1 6.8948 x 10-3 90.0 1.571 57.30 0.6366 9.549 0.1592 6.283 6.0 0.1047 14.5939 32.17 0.1550 0.09290 6.452 0.3861 10.76 1.550 x 10- 3 1.0 1.8 1.0 5/9 1,016. 2,240. 1.120 1,000. 2,205. 2,000. 2,430.56 3.4192
)
.') /
STRUC1~URAL
DESIGN MANUAL
(1)
Symbols for units of physical quantities shall be printed in Roman upright type.
(2)
Symbols for units shall not contain a period and shall remain singular; e.g., 7em, not 7ems.
(3)
Symbols for units shall be printed in lower case Roman upright type. However, the symbol for a unit derived from a proper name shall start with a capital Roman letter; e.g.: m (meter); A (ampere); Wb (weber); Hz (hertz).
(4)
The following prefixes shall be used to indicate decimal fractions or multiples of a unit. Equiv.
deci
(10- 1 )
d
centi
(10- 2 )
c
milli
(10- 3 )
m
micro
(10- 6 )
nano
(10- 9 )
picD femto
(10- 12 ) (10- 15 )
atto
(10- 18 )
deka hecto kilo mega giga tera
)
Symbol
Prefix
1
(10 ) 2 (10 ) 3 (10 ) 6 (10 ) 9 (10 ) 12 (10 )
Jl n
p f
a da h k
M G T
(5)
The use of double prefixes shall be avoided when single prefixes are available.
(6)
When a prefix symbol is placed before a unit symbol, the combination shall be considered as a new symbol. A numer~cal prefix shall never be used before a unit symbol which is 2quared, thus cm is never written, nor is it written 2 O.Ol(m ) but as (O.Olm) •
(7)
The following are SI units for various commonly used factors.
3-51
STRUCTURAL DESIGN MANUAL SI Unit
Symbol
acceleration
meter/second squared
m/s2
area
square meter
m
density
kilogram/ell meter
kg/m
energy
joule
J=N~M
energy/area time
watt/sq meter
W/m2
force
newton
N=
length
meter
m
mass
kilogram
kg
power
watt
pressure
newton/sq meter
W==J/s 2 N/m
speed
meter/second
m/s
time
second (mean solar)
s
viscosity
newton second/sq meter
Ns/m
volume
cu meter
m
2
3
kgrnls
2
2
3
Table 3.10 shows the alphabetical listing of conversions from the English system to SI.
3.10
WEIGHTS
3.11
SHEAR CENTERS
The shear center is defined as the point on the cross-section of a beam through which the transverse load on the beam must pass in order that no rotation of the beam will occur. A pure twist with no moment produces no deflection at the shear center, only rotation. In the case of a beam of variable cross-section, the shear center may be determined for each section, but these points will not connect to form a straight axis. For instance, a cantilever beam of non-uniformly varying cross-section may have a load so placed on the end that the end section will not rotate, but all other sections of the beam may rotate. The axis formed by connecting all of the shear centers is called the axis of rotation or the elastic axis. For any doubly symmetrical section or a section with point symmetry, as a zee, the shear center is at the center of gravity.
3-52
)
STRUCTURAL O'ESIGN MANUAL For any singly symmetrical section, the shear center is somewhere on the
a~ls
of
syrnmetry~
For any section made up of two intersecting plates, for example an angle or a tee, the shear center is at the point of intersection of the plates. Shear Centers of Open Sections
3.ll.1
)
The coordinates of the shear center position for the general case of an open section ,are defined by equation 3.42 and 3.43 as shown in Figure 3.10.
y
t
.1
~
x.~ FIGURE 3.10
SHEAR CENTER OF OPEN SECTIONS
L
Xo
,)
1 IX1y-( Ixy) 2
fly
10
Ws ytds - Ixy
v
.0
Ws
xtdsJ
3.42
0
L
1 [- Ix = Ix1y- (Ixy) 2
I
L
1o
Ws
L
xtds
+
Ixy
j0
Ws
ytds J
s
Ws
1o
rds;
The double area swept by the radius vector when the distance along the midline increases from s=O to s. Ws is taken as positive when ,the area is swept in a counter-clockwise direction. Coordinates of the shear center with respect to the axis through the section centroid.
3-53
STRUCTURAL DESIGN MANUAL Revision F
IX' Iytl xy ;
Inertias of the section.
L;
The limit of s or the developed length of the cross section.
r;
Radius vector measured from the section centroid to median line of the cross section.
s;
Circumferential distance measured along the median line of the section.
t;
Thickness.
If x and yare the principal axes of the section, Ixy=O and equations 3.42 and 1.41 become:
1 L
xo=l / Ix
r
Yo=-l/Iy
Wsytds
3.44
wsxtds
3.45
c;)
Examples of the previous procedure are shown in Figures 3.11 and 3.12. Table 3.11 and Figures 3.13 through 3.17 give shear center locations for some common sections.
3-54
STRUCTURAL DESIGN MANUAL 3.11.1.1 Example of Shear Center Location for Hat Section 2.0
a...---- 1.169 - - -.....----..
)
o.625Yl t t
I
I 0.1875
-0.6
till HAT SECTION
(c) PLOT OF •
a w. s
. 0.4
Q=f:B_ _...:.Ir=.2'-.............:lr.6:....-_-=2r=.0~...-,...lII:.i2. I
r-
I'
iii
f w.lds· -O~698 I
-0.1
: I I I
: •
-0.2
I• t
I I
(b1 OOUBLE
.
A
Ix I
sw£p r
AREAS
(en PLOT OF Iw.d
2
= 0.1835 In.. = 0.0345 In. 4
Double Swept Area
= 0.0894 In. 4 y
y-
= 0.447 ill.
)
w
8
t
0
0
2
-0.212
-0.212
3
-0.523
-0.7·10
4
+0.279
-0.461
5
+0.234
-0.227
..
~
FIGURE 3.11 - EXAMPLE OF SHEAR CENTER LOCATION FOR HAT SECTION The section has one axis of symmetry and consequently the shear center will be'on this axis. The plot in (c) above shows the variation of Ws and x along the developed length of the hat section. Due to symmetry only one half of the cross section is considered. The result will be mUltiplied by two. The expression wsx is shown in (d) and th~ area under this curve is ~wsXdS. Then Yo I y =
-f
L
.
wsxtds == 2(O.698){.040)
==
0.0558
o
.0558/.0894
.624
3-55
STRUCTURAL DESIGN MANUAL 3.11.~.~
. Example of Shear Center Location for Open Shell I
I.e.
~.~~-+------~-
te)
..
SECTION PROPER11U
•
:t nili'll)
'It
An
Ar II
(3,_(t)
1
U
1 I) I)
,•
..
a
12
I
12
6.5
~D(Jf,1
A ..CT)t Allry"
(l).(4)
t~)lI{lI)
.,
281
122
.Sf;
9.
122
26'
S
.1
t
lSI
48
O·
II
0
0'
.,
I
S
S
,
0
0
0
0 124
«II
..,.. ..
10.33
LOC~11OJf
(I)
",sA
• •
'.}a(S)
t4
U
DEAR CUITER
,
Qi)C5){'l)
+1.5
...130
.LST
+41'
-1.1I
-114
+1.8'
.17.
-$..5
-~$t
~.SS
.... 5-1
-5.5
-1410
2
I'"
0
,.
0
.1
...
II
.8
2''-50
0
0
If.c.o
ltO.O$
-7.»
-2S50
1M
0
0
3&.80
1116.85
-10.33
-M30
-2.5 .1.5
Z.... to
-n1{)
.... lI
T88
t.1~U
-IO•.U
1910
0
1:
•
.'1$1
".&1
12
-1H80
-800
d95 .teU -801
FIGURE 3.12 - SHEAR CENTER LOCATION FOR UNSYMMETRICAL OPEN SKIN-STIFFENED SHELL In this example, the method is applied to an unsymmetrical open section composed of stiffeners. The dimensions of the cross-section are indicated in (a) above. The numerals within the squares indicate the cross-sectional areas of the stiffeners. The skin has been assumed to be ineffective in bending for the subject problem. The sketch in (b) above shows the double areas swept by a radius vector as it moves counter-clockwise from element to element. Due to the introduction of the $tiffener areas, Equations 3.44 and 3.45 have to be modified. For this case, tds is replaced by A, the stiffener area. n 124
y
=
x
I I
Y x
66 12
10.33 5.5
= 243 12(10.33)2 = 680
606 - 12(5.5)2 1960 -
•
STRUCTURAL DESIGN MANUAL
•
Revis:lon B TABLE 3.11 - LOCATION OF SHEAR CENTER LOCATION OF SHEAR CENTER, Q, FOR SECTIONS HAVING ONE AXIS OF SYMMETRY
FORM OF SECTION 1. Triangle
= 0.47
e
)
for narrow triangle (~< 12°) approx.
a
2. Sector of thin circular tube
2! Sin0Cos ~
e
=
( 7r
e
=
<_8_ 3 + 4JL)R 1511' 1 +jL
-0)
r
(7r
-~)Cos0+Sinil1
J
3. Semicircular area
Ii}
•
4.
= moment
II
of inertia of leg 1 about Yl (central axis) 12 = moment of inertia of leg 2 about Y2 (central axis) -- ~:.e
Y
Leg 1
I
=~h(
H
r
t
Leg 2
=
t2h2
1
6. I Section
=
product of inertia of the half
xy section (above x) wi th respect to axes X and Y; and I = moment x of inertia of Whole section with respect to axis x.
x
t
tl hi
+
where H
e = h(~) I
If
=
l) 12 (for ex use Xl and X2 central axes) t2 are small'~x=e =0 (practically) and Q is at 0 . 1 II
If tl &
)
Q is to right of centroid
, e
12
+
e = b(I
1
1
) where
I
2
II and 1 , respectively, denote moments of 2 inertia about X-axis of flange 1 and flange 2 7. Tee Section
•
-It -.
t-t21
I
hi..
I -L
Th2
For a T - beam of ordinary proportions,
Q may be assumed to be at 0 3-57
STRUCTURAL DESIGN MANUAL TABLE 3.11 (CONT'D) - LOCATION OF SHEAR CENTER 8. I with unequal legs
r[[
-tblte'" b +1...1..
h
W
t
with lips.
nj.e
10. Sector of arc
e
=
~~ t - ----=:. -- -
9. Right angle
.t
.,.
j
2
2 - b1 (w/t}h + 6(h + hI)
3 [b
t
~_
~~~lb~ 45°
--gOO
~b(bl)21
(3b - 2b l )
e -
- J2
r
2b 3 - (b - h l )3J
e = 2 R:Sin~ - 0Cos~ ~ - Sin g Cos Q
when
Q
=
~/2
4R
e =~
(semicircle)
.,I,,'Y.
_".~
.L-l - -'. _\ _\,
,,~. ~
STRUCTURAL DESIGN MANUAL
•
"1
(.)1-'= . N
)
-:
FIGURE 3.13 - SHEAR CENTER OF A LIPPED CHANNEL SECTION
3-59
W I
'"o
~
1-1
C)
c:: ~
t'!1
~
I-'
~h
. 5 m~Hml -J..a..-ll-fi~
.,.
~tl U ffilHm1
CI'l
_~ ::r,~~ .,
.1
'
::I::
.~
i:'E:1
4
Z
t-3
t:T:1 :::0
:
r;
~ t-3
en
;j o z
d
'T-
t"'i-!iiJ
H
~liT'lftt
H
•
'
J-
t~
H
[.
I' rl
'I T'.... ~ ,
'.
'.: -
R.' .
.I-Ir.U
It
: I.
"
-lH.~~ ..rtft ,~'
.,
.: 1
. . . ".
H
1:11
~
;
II
t·
t'n; '1
.~
ii! 1 f
'L;'
1
,,~.
Viii-"Il 11~l-- I'tI ". f¢" E :,- Ir~t
.~!. ±
If.f.+,..
It.
,HIIl'·rl
,ft' .. P'
tJ1 ~1t
. - _,:', : , '
,tt ~ , t],.~ J0
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STRUCTURAL DESIGN MANUAL
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1
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o
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40
80
~20
160
a .- DEGREES
FIGURE 3.16 - SHEAR CENTER OF CIRCULAR ARC SECTION
3-62
200
STRUCTURAL DESIGN MANUAL-
)
--~-~~~~+-i~
9t ts
0.5
~) 3.0 5
6
7
FIGURE 3.17 - SHEAR CENTER OF D-SECTIONS
3-63
STRUCTURAL DESIGN MANUAL I
xy
= 788 - 12{5.5XIO.33)
Revision C
= 106
Using Equations 3.44 and 3.45 the following shear center location is determined:
xo
=1
1 I -(1 )2 x y xy
[ 1 y 1: (ws yAn ) - I xy k( ws xAn )
I
Xo =
(680)(24~)-(106)Z [(243)(-6460)-(l06)~08)1
Yo =
1 1 -(1 )2 x y xY
Yo
(680)(24~)-(l06)~
1
=
1
... 1
x
l:(wxA)+I
s
n
xy
=
-9.60
(WYA)l
s
n
[(-680)(-908) + (106)(-6460)
I
= ... 0.45
3.11.1.3 Shear Center of an Open Cell Box Beam The shear center of an open cell box beam such as the one shown in Figure 3.18 is found by determining the internal loads
~ ~
r-- 7.5 -----i
I ~25 .5 a,b g-' T 10 I 6.~67 1-- --. - -$C!g____ --L L c ~: ~ . f 3.333 ~ 5 ~ ! 4:' 5 ~l.OT 1.5 b
~5
FIGURE 3.18
Ix = 100 i4t4Iy = 178.13 ",,4 EA= 4.5".'1.
OPEN SINGLE CELL BOX BEAM
distribution for an arbitrary applied load first vertically and then horizontally and then equating the internal moments to the external. The shear center is the point at which the load is placed to create zero in-plane moment or zero rotation. It is assumed that the beam is composed of axial members capable of carrying tension and compression loads and thin skins capable of shear only. The beam can be any length wi th the members changing area and thickness. Figure 3.19 shows a tYrical cross section with a load of 100 lbs. applied. The axial stress per inch of span is calculated using:
3 .. 64
•
STRUCTURAL DESIGN MANUAL -Vv
f =
3.46
--..:.-L.
I
where
v = arbitrary
applied load
= distance from centroid to axial member I = moment of inertia resisting the bending created
y
by V
The axial load is then obtained by multiplying equation 3.46 by the axial area. Figure 3.19 shows the distribution of axial loads for the 100 lb. shear. The loads are calculated as follows:
3.333 1.667 1.667 3.333
FIGURE 3.19
Pa = Ph =
Ph
=
)
= Pe
Pd
Ix -V~Ab
Pg
Pc = Pf
-Vra.A~ =
Ix
AXIAL LOADS DISTRIBUTION
... 100(6.667)(.25) 100
= -100(6.667)(.5) = 100
= -Vl:'~AC = Ix
-1.667 lb/in -3.333 1b/in
-100(-3.333)(1.0) = 3.333 lb/in 100
-100(-3.33)(.5) = 1.667 1b/in = -V:tdAd = Ix
100
The shear flow distribution is determined from the axial loads by
= t1PIL
q
3.47
where ~p
L
=
the change in axial load the length over which 6P occurs
3-65
STRUCTURAL DESIGN MANUAL Beginning at point nau and summing forces to put each element in static equilibrium: qab = -PaIL = -1.667/1.0 qbc
= -Pb/L
= -1.667
+ qab = -3.333/1.0 - 1.667 3.333/1.0
qde
1b/in.
= -5.0
1b/in.
5.0 = -1.667 1b/in.
= Pd/L + qcd = 1.667/1.0 - 1.667 = 0
This procedure is completed around the cell until the internal loads as shown in Figure 3.20 are all calculated.
/~66~ ~
3.333
5.0
1.667
I
/&6~ 1.667
3.333
~
J 5.0 r
FIGURE 3.20
INTERNAL LOADS DISTRIBUTION
The sign of the shear flows is determined by their direction on the forward face of the cup. If q is clockwise, it is positive. The next step is to check the internal balance by ~Fx, ~Fy and ~Fz. If these sum to zero, proceed. If they do not sum to zero, there is an error and it must be found before proceeding. Sum moments about any point. !;M !;M
c c
0
= -100e-10(5)(1.667)+10(5)(1.667)+15(5)(10)
= -100e+750·= e
3-66
The lower LH corner is convenient.
7.5
0
o
STRUCTURAL DESIGN MANUAL The horizontal location of the shear center is then 7.5 inches to the right of point c. This is an axis of symmetry and the previous calculations could have been avoided by recognizing the axis of symmetry. They were shown to demonstrate the method. The vertical location of the shear center is calculated in the same manner. A horizontal load is applied and the shear flows are determined. Figure 3.21 shows the shear flow distribution for the horizontally applied 100 lbs. The load must he .,
)
.351
.351
--a
-.::....-
J
b
.
-~
Ii
g~U)
It)
:t1
It)
J~
c .2 ~ .......:---...---........-. 6.667
FIGURE 3.21 ....)
...... 100
7.368 6.667
SHEAR FLOW DISTRIBUTION FOR HORIZONTAL LOAD
applied so that no rotation occurs.
=0
~M C
= -IOOe+IO(5)(.351)+lO(5)(.351)+l5(lO)(2.456)
IOOe e
= 403.49 = 4.035
The shear center is located as shown in Figure 3.22
b
--a
I hI
g
II
) c
!!- I •e -
~f4.035
I
I
I
s.c.®
FIGURE 3.22
SHEAR CENTER LOCATION
3-67
STRUCTURAL DESIGN MANUAL 3.11.2
Shear Center of Closed Cells
Figure 3.23 shows a typical closed cell. It is the same as the Fig.3.18 example except the cell is closed. The first step is to assume that one web is cut. Web
Tb
t = .05 canst..
10
~~C ~d~__-4b~. ~f. __
FIGURE 3.23
__
CLOSED SINGLE CELL BOX BEAM
ah will be cut since much of the previous solution can be used. The horizontal position of the shear center lies on the axis of symmetry. If no axis of symmetry existed, the procedure would be the same as for the forthcoming analysis.
with web ah cut, the resisting shear flow distribution for a horizontal load is shown in Figure 3.21. The web is then assumed to be closed and a constant shear flow of qo is applied arbitrarily in the counter clockwise direction. The shear flow at any point in the cell is then defined as: q = qo
qo
=
+ q'
3.48
the balancing shear flow
q' = the shear flow with one web cut The shear frow qo is that which will make the angle of twist, 9, equal zero. twist is: 9 - ~q&L - L 2AtG
o
The
3.49
L = 1 inch and 2AG is constant for all webs and may be taken outside the summation and then cancelled. Equation 3.49 becomes:
~~ere
3.50 Substituting equation 3.48 into equation 3.50 and taking qo outside the summation sign because it is constant for all webs, the following equation is obtained: 3.51
STRUCTURAL DESIGN MANUAL The numerical solution can be put into table form as shown in Table 3.12.
Web
~s
~s/t
q'
ql ~s/t
(2)
(1)
a-b b-c c-d d-e e-f
5 10 5 5 5
f-g
10
g-h h-a
5 5
100 200 100 100 100 200 100 100
--
1000
Total
e)
(1)
Figure 3.21
(2)
Figure 3.24
.351
2.456 6.667 7.368 6.667 2.456 .351 0 --
2Aql
2A
35.1 491.2 666.7 736.8 666.7 491.2 35.1 0-
3122.8
50
a
0 0 0 150 50 50 300
17.55 0 0 0 0
368.4 17.55
a
403.5
TABLE 3.12 - NUMERICAL SOLUTION FOR SINGLE CELL
CLOSED BOX Substituting values from Table 3.15 into equation 3.51 1000q
qo
=
o
+ 3122.8 = a
-3.123 lbs/in
The moment about any point can be obtained from the relation
)
T
= l:2Aq
3.52
Substituting equation 3.48 into 3.52 yields 3.53 where A is the area enclosed by a web and the lines joining the end points of the web and the center of moments as shown in Figure 3.24.
3-69
STRUCTURAL DESIGN MANUAL
c ~ d q;""" e "liT" f FIGURE 3.24
ENCLOSED AREA DEFINITIONS
The values from Table 3.12 can be substituted into equation 3.53 with moments being taken about point c. lOOe+300Qo+403.5
0
lOOe+300(-3.123)+403.5
=0
lOOe = 533.4 e - 5.334 The shear center is 5.334 inches above point c on the vertical axis of symmetry. 3.12
STRAIN GAGES
A strain gage is a small device which is attached to a structure to measure the strain. They are usually attached to test articles but in some cases they are permanent fixtures on operational aircraft. A strain gage measures the change in length of the structure over the length of the gage. The gage itself changes length along with the structure and the electrical resistivity of the gage changes. This resistance change is measured and by proper calibration the resistance can be related to strain in the structure. The strain gages most commonly used are (1) the wire gage, (2) the foil gage and (3) the weldable gage. 3.12.1
The Wire Strain Gage
The wire gage as shown in Figure 3.25 is made of a grid of very fine wires. The wires are usually made of copper-nickel alloy and bonded to a lacquered paper base which is subjected to a slight initial tension. The paper base is bonded to the specimen. A felt cover is placed over the grid in the longer size gages for protection. The felt cover also helps to minimize temperature changes. This type of gage is called a ItDueo" gage and is the least expensive and most convenient gage to use.
3-70
)
STRUCTURAL DESIGN MANUAL
.)
.-"'~
,
-
( )
(
r---
)
1/ 16
to 1
~--
FIGURE 3.25
"
-1
GRID OF VERY FINE WIRE ABOUT .. 001 DIA. -HEAVIER WIRE
WIRE STRAIN GAGE
Another type of wire strain gage has the wire grid molded in a thermosetting phenol resin. It is called a "bakelite" gage and the molding process makes this type of gage much more time consuming. Bakelite gages are useful though, when temperatures are between 150°F and 450°F and when humidity presents stability problems to Duco gages. Otherwise the Duco gage is sufficient.
3.12.2
The Foil Strain Gage
The foil strain gage is shown in Figure 3.26. It is etched from a foil sheet. The width of the element is increased at the end of the loops to reduce the effect of the transversely oriented parts of the conductors. The gage is mounted in a thin cement which is applied to the specimen. Adhesives are available which permit the use of the gage up to 700°F. Gage lengths are available from 1/64 inch. These gages are easy to mount, inexpensive and very accurate.
THIN LACQUER
)
r.=======::::!.J.......------f--- FOIL LEAD
FIGURE 3.26 3.12.3
FOIL STRAIN GAGE
The Weldable Strain Gage
The weldable strain gage is a length of fine wire surrounded by high temperature insulation encased in a flanged metal tube. It can be spot welded to a test specimen in less time than any of the other gages can be installed. It can be located
3-71
STRUCTURAL DESIGN MANUAL on a curved surface. Very small gage lengths are available and they can be used in 0 static tests up to 850°F and dynamic tests up to 1600 F. 3.12.4
The Strain Gage Rosette
When the direction of the principal strain is known, it is usually sufficient to mount a single gage along the axis of the strain. If the direction of the principal strain is unknown or if it is not axial, it is necessary to make several measurements along different axes or to use a Urosettelt consisting of three strain gages 0 on the same paper or bakelite backing. A 45 rosette is shown in Figure 3.27. In this rosette, three gages are mounted with their axes intersecting at a common
FIGURE 3.27
45
0
STRAIN GAGE ROSETTE
0
centerpolnt and are 45 apart. Each gage is electrically insulated so that the effect is that of three separate gages. When the direction of the'principal stresses are known, a "Visette" which is shown o in Figure 3.28, is used. It consists of two gages mounted 90 apart on a single mount.
+-----FIGURE 3.28 '#
3-72
90
0
STRAIN GAGE ROSETTE (VISETTE)
STRUCTURAL DESIGN MANUAL When the direction of the principal strain is unknown, the delta rosette can be used. It is shown in Table 3.13. It has the maximum possible angle between gage axes. The angles are 0, 60 and 120 degrees. A T-delta rosette, shown in Table 3.13, is identical to a delta rosette except that a fourth gage is mounted at right angles on top of the other gages. This fourth gage is used as a check.
-j
3.12.5
Strain Gage Temperature Compensation
Two methods are available for eliminating the strain due to thermal expansion; the dummy gage and the self compensating gage. The dummy gage consists of a gage mounted on a separate bar and a gage mounted on the specimen. The separate bar is made of the same material as the specimen. The two gages are connected to the readout equipment in such a way that equal resistance changes in the two gages will cancel. If the gages are identical and the bar is subject to the same temperature conditions as the test specimen, the temperature effects will cancel and the readout will be in terms of actual mechanical strain in the test specimen. The self compensating gage is made in such a way that they are uneffected by thermal strain. This gage is expensive and each gage is limited to one temperature range and one material. Its advantage is simplicity. 3.12.6
Stress Determination From Strain Measurements
The purpose of strain measurements is to obtain stress levels in a structure. If the directions of the principal stresses are known, only two strain measurements are required. It can be shown by Hooke's law that if the directions are known, the stresses can be found using the following equations:
)
E 0"1 = 1- JJ2
(€l +JJ€2)
3.54
E =-2
(E 2 +JJ€l)
3.55
CT
2
l-1l
where p. = Poisson's ratio
0"1 = Principal stress in one direction 0"2 = Principal stress at right angles to €1 = £2 E
CI
1
Strain in the direction of 0"1 Strain in the direction of 0"2 Modulus qf elasticity
3-73
(...,)
I
..... .I:'-
Rosette Types
L
Required Solution
Rectangular
Two-Gage
.
Max. Normal Stress (7max Min. Normal Stress
E - ' ( £ 1 + 11€2)
Et
V<€1- "3)2 +! 2E2 -(E1 + E3>1
~(€2+1J.€1) 1-J,L
2
J
Y(€1-€3)2+[2E 2 -("-1 +
Max. Shearing Stress
_1_
+ 1+1-'
Vh- E1HrE~2+(~E~2 E
E3)r~j
E1 2 +E 3(I-#-'
(1'1-
1 1 + IJ
4
en
..... :::a
1<.[E1+q +.....!2 I-I-' 1+J.L
c n ~ c:
\/("-1- E4)2+1-(€2" (3)2]
_-1-
1<.[€1+€4 2· 1-1l
e1+E~+€~2 +f€Sr€3)~
1+1l
::c:I
"-4)"J+~ (E2-€3)2]
V(E 1 -
:D-
r-
I
E 2 (1 + W)
E 2 ( 1 + IJ.) (£1- € 2) T
max
Angle From Gage 1 Axis To Max .. Normal Stress
3(1-f.L)
-
T- Delta
·t +? _
.1L[E1 + E3 _ ~ 2 I-p. 1+[J.
O"min
1
Delta
E1 + E:4 H 3
4€1+ Ejl +--L 2 1-11 1+ IJ.
1-P. E
a
3~
(E1- "-3)2+ [2E 2 - (E1+ €3)t
2
2(1+[J.)
\"-1- €1+'t €312+(S€3t
Vh-
€1- €3
I
1£4)2+
..., C
en
~(€2- €a?
-z
en
31:
t
:DZ
..l..tan-1 2 (€2-€3) ] 2 3(€1-€4)
1 -It *(1£2 - (2 3) -tan 2 € _ €t+ £2+€3 • 1 3 _
..l.. t an- 1 [2 €2 -(€1+€3) ]
0
E
E 1+ ,.,.
c:
Axis .. Qo
:D-
r-
TABLE 3.13 - RELATIONS BETWEEN STRAIN ROSETTE READINGS AND PRINCIPAL STRESSES
'-....-
--.
--------'
-/
STRUCTURAL DESIGN MANUAL The maximum shearing stress will occur at 45 as follows:
0
to the principal stresses and is found 3.56
)
If the directions of the principal stresses are unknoWD t the problem is more complex. The axes of an element are shown in Figure 3.29. y
----=-===-:....----x FIGURE 3.29
STRESSED ELEMENT . a
The general equation for strain at an angle 9 using the results of a 45 _ Ex+€y
EQ -
2
+ (EX_Ey) 21'\ 2 cos ~
+'
'Yxy s1"n2f'lo
2
rosette is 3.57
~
where Yxy
= the shearing strain
In Figure 3.30 the strain measured along the (A), (B) and (C) axes can be used to
)
2
1
FIGURE 3.30
STRAIN AXES
3-75
STRUCTURAL DESIGN MANUAL 0
calculate the principal stresses which act along axes (1) and (2). When a 45 rosette is used, the strain along (B) is measured on an axis at an angle of 45 0 to 0 the (A) and (C) axes. The angles (AOB) and (BOC) are 45 • Axis (2) is at an angle 0 of 90 to axis (1). The strains along axes (A), (B) and (C) will have the following relationships with the (x) and (y) axes of Figure 3.28.
EA
Ex+E:;t: . Ex_EX = Ex 2 + 2
3.58
EB =
Ex+EX Yxy 2 + 2
3.59
€C =
Ex+€X 2
-
€x-€~
2
= Ey
)
3.60
or E"x
= €A
3.61
€y = EC 'Yxy 2 EB-(€A+€C) .
3.62 3.63
By using the equation for maximum shear and Hooke's law, the principal stresses can be obtained from the following: 0"1 =
~ [E\~IlEC
+
1~'"
J2(€A- EB)2
+
E
2(€B- C)2]
3.64
3.65
3.66
3.67
3.13
ACOUSTICS AND VIBRATIONS
Most periodic waves J regardless of the form, can be represented by two or more sine waves. Most waves can be reduced to simple harmonic or sine wave components which generally form harmonic series. They have frequencies which are integral mUltiples of the lowest frequency. The lowest frequency is called the fundamental and the higher ones are called harmonic.
3-76
)
STRUCTURAL DESIGN MANUAL Revision F 'I'll('
rn~tIUl:ncy
or a vibrating hody is the number uf cycles of motion in n unLt tinll'.
The period of a wave is the time elapsed while the motion repeats itself. the reciprocal of the frequency.
It is
The amplitude of a wave is the maximum distance the vibrating particles of the medium in the path of the wave are displaced from their position of equilibrium.
)
The wavelength of a wave is the shortest distance between two particles along the wave which differ in phase by one cycle. The number of independent coordinates necessary to describe the motion of a system is called degrees of freedom. Examples of systems with various degrees of freedom are shown in Figure 3.31. Example (d) in Figure 3.31 is a single degree of freedom with a mass "mil supported on frictionless and massless rollers attached to a spring and a dashpot. This is a representation of a fairly common situation occurring in aircraft and helicopters.
(a) SINGLE
DEGREE OF FREEDOM
(c) MULTIPLE
(b) TWO
DEGREES OF FREEDOM
FIGURE 3.31
DEGREES OF FREEDOM (d) ONE DEGREE OF FREEDOM LINEAR VIBRATION
EXAMPLES OF DEGREES OF FREEDOM
If a force IIF" which is a function of time Itt" acts on the mass, the differential equation dx
+ c dt
+
kx = F(t)
3.68
must be satisfied at all times,
3-77
STRUCTURAL DESIGN MANUAL Revision F \vhere m c
=
k x F( t)
If
~fter
the the the the the
mass of the system coefficient of viscous damping spr{ng constant displacement from rest external force as a function of time
an initial displacement of the system the external force ceases to act,
the equation becomes:
+
c
dx + kx = 0 t
2m"
If the quantity
/4m2 >k/m, the m2ss will not oscillate but will gradually r('turn to its rest position. If c /4m < kIm, there will result a decaying oscjll:~ tion of circular frequency, radians/sec
3 .. 70
for which the corresponding linear frequency will be in = 1/2
1r
y;./m_c 2 14m2
3.71
cycles/sec
where "nIl denotes na tural frequency wi th damping, and 2
1f
f
3.72
n
I
2 2 If c /4m kIm, this is the limiting case for which no oscillation occurs and the system is said to be critically damped. This particular value of c is designated c cr where C
2rnjk/m
cr
=
2~
2mw
n
2k/w
n
3 .. 7 3
)
If the driving force is sinusoidal F( t)
= Fosinwt
3.74
equation 3.68 becomes 2
dx d x + m-cTt+ kx 2 dt
3-78
Fosinwt
3 .. 75
•
STRUCTURAL DESIGN MANUAL Revi s ion F wll(:re Fo is tilt: maximum value of the sinusoidal force and Th(; general solution to equation 3.75 is
x
)
=
(-~~)
(AsinWnt+Bcoswnt) +
e
W
Is the forced frl'C)uclH:Y.
FQsin(wt-~)
2 2 2 (cw) +(k-mw)
3.76
where. is the phase angle and A and B are arbitrary constants depending on the initial conditions. As before: 3.70
and
3.77
The: first term on the{!it?~ ~and side of equation 3.76 vanishes in time due to the fae t tha t the term, e c. m, constantly diminishes and is called the transient term. The second term gives the amplitude of the forced vibration in terms of the system constants and driving force and is called the steady state term. The amplitude of the steady state vibration is x = Fol
y,
2 -2 2 (cw) +(k-mw )
3.78
This is also expressed in the convenient form (Fo/k)1
x
V
2 2 + [2(c/c fl-(wIWQ)]
cr
) (wIWn)] 2
).79
where Folk is the displacement that would be produced by a static force FoFollowing are equations which predict deflections and mode patterns for beams .. They are based on simple beam theory and are accurate for beams having a length to depth ratio of the order of 10 or more. Uniform Beams
J.l3.l
Uniform Bar With Free Ends The equation for finding the deflection for different mode patterns is as follows: y
=
1/2.04 (-sinanX+l. 02cos(Vnx- sinhO'rk+l .. 02 coshatnX)
3.80
3-79
STRUCTURAL DESIGN MANUAL .
"
Revision F
__
where an is the characteristic number for the nth mode and is the root of the equation cosancoshan 1. The characteristic numbers for the first three modes of this beam are: 4.73, 7.853 and 10.996. Frequencies of higher modes for this beam are given in Figure 3.32) in which w is the weight per unit length of the beam and g = 386 in/sec 2 •
AMPLITUDE PROFILE
FIGURE
,--..,;..-> "',,- __ ,> tC . . . . _ ..-"
2 NODES
-.. . '5"--' ,'-', .----
3 NODES
....
........
_
..... ..;
( ' ....... lfIIII"
""" _ _
NATURAL CIRCULAR FREQUENCY
MODE"
;x., ... , ,
22.4 L2
j
\
EwIg ,
4 NODES 5 NODES FIGURE 3.32
9.00 Wi.
UNIFORM BAR WITH FREE ENDS
Uniform Bar With Simple Supports at Ends The equation of deflection for the fundamental mode is y = sin(~x/L)
3.01
if the amplitude is taken as unity at the center. this beam are shown in Figure 3.33.
AMPLITUDE PROFILE
FIGURE
r -~4 1=-- ---~ "
.........
~x-----
X:=-----.--~ .................... ----~.,."
""" ...-:
2 NODES
NATURAL CIRCULAR FREQUENCY w.1.==
~ L
3 NODES
4wl
*--.. . ,. --, ",.-.. *
4 NODES
9wt
5 NODES
16w 1
.......... """---""
' - -.-" IIII: ........
~
,--------
-~",,---
-,,,-, '- ........
,.,--~
... "" '--''''''''' J''' "'_-'"
FIGURE 3.33 3-80
MODE
/----:*
)::----~
.
Frequencies of higher modes for
..
UNIFORM BAR WITH SIMPLE SUPPORTS
/
EI 9 \ W
!
STRUCTURAL DESIGN MANUAL
e)
Revision F Uni rorm Cantilever Beam
The frequencies of modes for this beam are shown in Figure 3.34. Uniform Beam With Clamped Ends The frequencies of various modes for this beam are shown in Figure 3.35. Hinged Fixed Uniform Beam The frequencies of various modes for this beam are shown in Figure 3.36. Uniform Cantilever Beam With Mass at the End See Figure 3.37. Uniform Beam Simply Supported With Central Mass See Figure 3.38.
3.13.2 Tt~
Rectangular Plates
general equation for the frequency of a plate with simply supported edges is
w
3.82
where g = 386 in/sec 2 , p is Poisson's ratio. W is the weight per unit area of the plate, and M and N are integers depending on the number of nodal lines. Figure 3.39 shows normal modes of rectangular panels with values for M and N. "L 13. 3
Columns
The equation for the natural frequency for an axially load member is given by W
l
= woJl-P/Pcr
~ radians/sec.
3.83
where P
Per Wo
= axial =
load 2 2 Euler buckling load = n EI/L natural frequency for zero load
For the pin-ended column, Wo is given by Wi for an uniform bar \l1i th simple support at ends. For the fixed-end column, Wo is given by WI for an uniform beam with champed ends.
3-81
STRUCTURAL DESIGN. MANUAL Revision F
FIGURE
AMPLITUDE PROFILE
~
---E -
-- --- -
• t!!::: -- ""..
............
I'--L
-;---.-
MODE
f- -. . """"... --.. . ,,-
3 NODES
"' ....... - ,' .......
,->'--
4 NODES
t->.,-. . . , ,_.... ,_,,1',,_ .... " "'_'"
5 NODES
,...-
..........
t-,_,,..... . -.- . ....
,.\ '-' ......
:;c __
AMPLITUDE PROFILE
FIGURE L
1 NODES 2 NODES
i
.....
I"
NATURAL CIRCULAR FREQUENCY
~ ....
"I
~
f==~-~~~:~
-----..-- .... .........
MODE ......
~
.-;:
2 NODES
NATURAL CIRCULAR FREQUENCY
~= 22.4 /
E1 9'
L2
\'1
3 NODES
4 NODES 5 NODES
AMPLITUDE PROFILE
FIGURE
r--_~ --=-=-=1
lr:=--:---~ ,Jf---)/-l, , -..... ---
Jf: ___ -~1a,
......
,
3-82
...-.../
........ ""'-
Jf-........-~'"' -......... . '~.,-ll '-;
MODE 2 NODES
NATURAL CIRCULAR FREQUENCY WJ=
15.4 L2
j
3 NODES
3 . 24 U1
4 NODES
6. 76 W,
E~g'
. -.- 7Jr':::-~ , I'
'
,
•
,
\\
"I+~.~ STRUCTURAL DESIGN MANUAL \\ :-\ .... ~'"'' ,
~l:..
,"
Revision F
f. .-: . . ---- -..
......0.
CI
nJ
m
IN'1
==
EI
• 5611
33
3
(m + 140 -8-)L
....... '0
)
EI 17 g 3 (m + )5VL
• ) ~=1,
N=l
~1
M=2, N=l
=2,
N= 2
M=3,
~=1,
N=2
N=2
3-83
STRUCTURAL DE,SIG·N ':MANUA~ 3.13.4
Stress and Strain in Vibra
Plates
The determination of stress and strain in this section is based on a rectangular plate, perfectly elastic, homogeneous, isotropic with uniform thickness small with respect to its other dimensions. It is assumed that the deflections are small compared to the thickness with no stretching of the mid plane. The strain in a thin layer indicated by the shaded area in Figure 3.40 located a distance Z from the mid plane is given by equations 3.84 through 3.86.
e xx = Z/R 1
a2 6 = -z ax 2
e
a2 {j = -z ay 2
yy
= Z/R
2
3.84
)
2
'Yxy
=
a y
-2z ax
ay
3.86
e and e are unit deflections in the :x: and y directions" 'Yxy is _shear deformation i~Xthe xyYplane and 0 is the deflection of the plate. Rl and R2 are curvatures of the plate in the xz and yz plane. Stress is obtained by E
fx
(e
1- ,.,.2 E
fy = - 2 1- ,.,.
T
tvhere
J.l
Ge
(e
xx + 11 e yy )
3.87
+ J.le xx )
3.88
yy
3.H9
xx:
= Poisson's
ratio
\,
\ )
z
I
dx
FIGURE 3.40 - ELEMENT FOR STRAIN DETERMINATION 3-84
STRUCTURAL DESIGN MANUAL 3.14
BELL PROCESS STANDARDS
The list of subjects covered by Bell Process Standards (BPS) is given in Table 3.1~. The BPS should be consulted because most likely procedures have already been established for a particular aspect of structures design. A
)
Acrylic Lacquer, Application of Acrylic Plastic: Working and Maintenance of Adhesion Promoter, Application of Adhesive Bonding (Nonstructural) of Silicones Adhesive Bonding (Nonstructural) with Film Type Cloth Supported Epoxy Adhesive Adhesive Bonding (Nonstructural) with Rubber Phenolic Adhesives Adhesive Bonding of Nameplates Adhesive Bonding of Polycarbonates Adhesive Bonding (Structural) with Eldstomer Modified Epoxy Adhesive Bonding (Structural) with Epoxy Resin-Based Adhesives Adhesive Bonding (Structural) with Film Type Modified Epoxy Adhesive Adhesive Bonding (Structural) Using Intermediate Temperature Curing Modified Epoxy Film Adhesive Bonding (Structural) Using Intermediate - High Performance Supported and Unsupported Adhesive Film Adhesive Bonding (Structural) with Rubber Phenolic Adhesive Adhesive Bonding (Structural) with Vinyl Phenolic Base Adhesives Adhesive Bonding Using Epoxy Based Adhesives Adhesive System (Structural) for Honeycomb Sandwich Construction Alcoholic Phosphoric Treatment Aluminum Foil Identification Plates, Application of Anaerobic Sealants Anaerobic Sealants, for Bearing Retention Anodizing, Chromic Acid Anodizing, Hard Anti-Fretting Treatments for Titanium Anti-Seize Compounds, Use of Anti-Static Coating, Epoxy, Application of Application of Adhesion Promoter Application of Flame-Resistant Silicone Application of Powdered Coatings Application of Suede Coatings Application of Urethane coating
FW 4386 FW 4302
FW 4398 FW 4392
Bonding, Composite Bonding, Nonstructural Bonding Structural, with High Temperature Resistant Epoxy Phenolics Brazing, Silver Brush Cadmium Plating
FW 4429 FW 4401 FW 4335
FW 4434
FW 4415 FW 4402 FW 4408
FW 4423
FW 4458 FW 4400
FW 4328 FW 4403 FW 4449 FW 4300 FW 4l6B
FW 4421 FW 4426 FW 4001 FW 4387 FW 4456 FW 4396
FW 4413 FW 4398 FW 4447 FW 4465 FW 4479
FW 4464
FW 4446
FW 4171 FW 4448 FW 4098 FW 4312
C
Cable, Control, Fabrication and Testing Cables and Terminals, Electrical, Preparation and Installation of Cadmium Coating (Vacuum Deposited) Cadmium, Fluoborate, Plating, High Strength Steels Cadmium Plating, Brush Cadmium Plating (Electrodeposited) Carburizing and Heat Treatment of Carburized Parts Casting Impregnation, Process for Castings, Aircraft Castings, In-Process Welding of Chemical Cleaning of Aircraft Materials Chemical Film Treatment Chemical Machining of Metals Cleaning and Preparation of Mate. rials for Resistance Welding Cleaning, Mechanical, of Metals Clutch Linings, Bonding of Coating, Urethane, Application of Coating, Walkway, Nonslip Coatings, Powdered, Application of Coatings, Suede, Application of Coatings, Tungsten Carbide, Deposition of Color Identification of Rivets, Bolts, Nuts and Washers Composite Bonding Compounds, Corrosion Preventive, for Aircraft Assemblies Compression Molding of Plastic Parts Copper Plating Corrosion and Abrasion Resistant Coatings (2216 Mix) Corrosion Preventive Compounds, for Aircraft Assemblies Countersunk or Dimpled Screws, Installation and Inspection of Covering for Model 47 Fuel Tanks Cyclewelding Metal to Wood
FW 4108 FW 4332 FW 4436 FW 4466
FW 4312 FW 4006
FW 4420 FW 4432
FW 4163 FW 4470
FW 4139 FW 4182 FW 4389
FW 4113 FW 4343
FW FW FW FW FW
4121 44 64 4306 4465
4479
FW 4463 FW 4064 FW 444 b
FW 4362
FW 4469 FW 4l1D
FW 4435 FW 4362
FW 4039 FW 4433 4028.
B
D
Barrel Finishing of Metals Black Oxide Treatment of Steels Blind lIigh strength Steel Fasteners, Pull Type, Installation of Bonding Clutch Linings
FW 4326 FW 4084 FW 4472 FW 4121
Decal, Application of Dimpling, Hot Coin, of Aluminum, Magnesium, Stainless Steel, and Titanium
FW 4158
FW 4135
TABLE 3.14 - BELL PROCESS STANDARDS 3-85
STRUCTURAL DESIGN MANUAL Dopes to Fabric Surfaces, Application of Dual Purpose Laminating Glass Cloth Dynatube Fluid System Fittings, Installation and Assembly of Dzus Fasteners, Installation of
4024' FW 4437
FW 4473 FW 4365
E
Edgings, Thermoplastic, Bonding to Acrylic Assemblies Elastomer Modified Epoxy, Bonding with Electrical Wiring; Installation of Electrical Connectors, Potting of Electrical Connections, Soldering of Embrittlement and Stress Relief for Plated and Pickled Parts Epoxy Based Adhesives, Adhesive Bonding with Epoxy Primer Surfacer, Application of Epoxy Resin Based Adhesives, Adhesive Bonding with External Roll Threading of Metals
FW 4351
FW 4415 FW 4332 FW 4311
FW 4439 FW 4044 FW
FW 4471 FW
4402
FW 4445
Impregnation of Castings Inert Gas Tungsten Arc Welding of Aluminum Alloy Inserts, Installation of Internal Threading of Swaged Aluminum Tubing
FW 4472 FW 4408
Lacquers, Acrylicj Application of Lacquers, Cellulose Nitrate, Application of Linseed oil Treatment of Closed Steel Tubular and Hollow Parts LQw Density Insert Material in Conjunction with Honeycomb Sandwich Construction, Application of Lubricants, Solid Film
FW 4466 FW 4089 FW 4399 FW 4017
FW 4397 FW 4339 FW 4425 FW 4305
FW 4432 FW 4404 FW 44 2"" FW
441~
FW 4386 FW 4441 FW 4317 F~v
4383
FW 4310
M
Magnesium Alloy; Chemical Treatments for Corrosion Resistance of Magnetic Particle Inspection Marking Aircraft Parts Materials for Adhesive Bonding, Surface Preparation of Mechanical Cleaning ·of Metals Metal-Cals, Application of Metal to Wood, Cyclewelding of
FW 4473 FW 4474 FW 4468
f'W 4140
L
FW 4366 FW 4364 FW 4380 FW 4365
F\'i 44 2\.
I
4403
F
Fabrication of Heat Resistant Structural Components of Glass Fabric Reinforced Plastic Materials Fabrication of Structural Components of Glass Fabric Reinforced polyester Materials Fairing Compound, Application of Fasteners, Dzus, Installation of Fasteners, Installation of Blind, High Strength Steel, Pull Type Film Type Modified Epoxy Adhesive, Structural Adhesive Bonding with Fittings, Dynatube Fluid System, Installation and Assembly of Fittings, Permaswage, Installation of Fittings, Rosan Fluid System, Ins tall_a tion 0 f Fluoborate Cadmium Plating, High Strength Steels FLuorescent Penetrant Inspection Forgings and Wrought Stock, Ultrasonic Inspection of Forgings for Aircraft Application
Heat Treatment of Carburized Parts Heat Treatment of Steel High Visibility Paint System, Application of Honeycomb Core Material; Handling, Cutting, Forming, and Adhesive Priming of Honeycomb Cores, Splicing of Hydraulic Fluids, Handling of
FW
400~
Fly 4 a 7 FW 4'05\.
FW 4352 FW 4343 FN 4168 4028
N
Nameplates - Adhesive Bonding of Nital Etch Process Nitriding of Steel Nondestructive Testing of Bonded Components Nonstructural Bonding, Using Rubber Base Cements
FW 4335 FW 4092 Fly 4 )04
FW 4424 FW 4171
a
G
Oil, Hot Linseed, Application of Glass Fabric, Pre impregnated , Fabrication of Glass Fabric Reinforced Polyester Materials, Fabrication of Glycol, Polyethylene, Use of
FW 4366 FW 4364 FW 4444
H
Hard Anodizing of Aluminum-Alloys Hardness Testing and Requirements of Hetals Heat Treating (Annealing) of Titanium
T~LE
3-86
FW 4387 FW 4467 4212
FW 431.
) ,-'
P
Packaging and/or Preservation of Aircraft Parts for Stock Paint, Corrosion Protective Coating Paint, Epoxy Enamel, Application of Paint Finishing of Polycarbonates Paint Stripping of Metal Parts and Assemblies Paint System, High Visibility, Application of Passivation
3.14 (CONT'D) - BELL PROCESS STANDARDS
FW 4102 F'w'J 4428 FW 4427 FW 4438
rw
4357
FW 4397 FW 4007
-
e)
STRUCTURAL DESIGN MANUAL Inspection Permaswage Fittings, Installation of Phosphatizing of Steels Plastic, Acrylic; Working and Maintenance of Plastic Parts, Compression Molding of Plating, Cadmium, Electrodeposited Plating, Cadmium, Vacuum Deposited Plating, Copper Plating, Fluoborate Cadmium, High Strength Steels Plating, Selective Brush Cadmium Plexiglas Panels, Fabrication of Polycarbonates, Adhesive Bonding of Polycarbonates, Palnt Finishing of Polyethylene Glycol, Use of Polyurethane,- Abrasion and Corrosion Resistant Coating Polyurethane Enamel Application of Polyurethane Striping Material Application of Potting of Electrical Connectors Powdered Coatings, Application of Preparation and Application of Fuel Resistant sealant Compounds Primer, Epoxy, Application of Primer, Epoxy polyamide, Application of Primer Surfacer, Epoxy, Application of Primer# Zinc Chromate, Application of Proof Testing of Lines and Tanks Protective Coating (Strippable) Protective Compound (Strippable) Hot Dip
Penetr~nt
FW 4089 FW 4.4 74 FW 4384 FW 4]02 FW 4469
F.W 4006 FW 4436 FW 4110
FW FW FW' FW
4466
4312 4351 4434 FW 4438 FW 4444
FW 4457 FW 4442 FW 4452
FW 4311 FW 4465 FW 4407 FW 4325 FW 4451
FW 4471 FW 4367 4020 FW 4381 FW 4055
R
)
Radiographic Inspection Rivets, Aluminum and Aluminum Alloy, Processing, Use, and Storage of Rivets, Blind (Hollow and SelfPlugging Type) Riveting Procedure for Rivets,.Chobert Blind, Riveting Procedure for Rivets, Hi-Shear, Riveting Procedure for Rivets, Universal, Round Head, and 100 Degrees Flush, Riveting Procedure for Roll Threading, External, of Metals Rosan Fluid System Fittings, Installation of Rotor Blades, Wood Control and Bonding Procedure for Rubber Phenolic Adhesives~ NonStructural Adhesive Bonding with Rubber Phenolic Adhesives, Structural' Adhesive Bonding with
FW 4309 FW 4316
FW 4315 4038
FW 40.39 FW 4407 FW FW FW FW FW FW
4409 4012 4098
4439
4310
4115
FW 4440 FW 4162
FW 4381 FW4055 FW 4479 FW 4352 FW 4471
FW 4419
T
Tanks, Fuel, Covering for Testing; Conductivity, to Determine Heat Treat Condition of Wrought Aluminum Alloys Threading, Roll, External, of Metals Tightening Procedures for Threaded Fasteners and Fittings Titanium, Fusion Welding of Titanium, Heat Treatment (Annealing) of Titanium Sheet, Forming of Tubing Assemblies: Metal, Fabrication and Installation of Tungsten Carbide Coatings, Deposition of
FW 4433 FW 4453 FW 4445
FW 4018 4207
4212 FW 446.2
FW 4149 FW 4463
U
FW 4036
Ultrasonic Inspection of Forgings and Wrought Stock Ultrasonic Inspection of Rotor Blades Urethane Coating, Application of
FW 4399
FW 4424 FW 4464
FW 4019 V
FW 4445
FW 4468 4083 FW 4401
Vacuum Cadmium Coating Vibratory and Barrel Finishing of Metals Vinyl Phenoiic Base Adhesive, Structural Adhesive Bonding with
FW 4043 FW 4358
FW 4436 FW 4326 FW 4328 .
W
FW 4400
S
Safety Wiring Methods Scotcheal Film, Application of
Screws, Dimpled or Countersunk, Installation anu Inspection of Sealing Compound, Fuel Resistant, Application of Shot P.eening of Steel, Aluminum, and Titanium Shrink Fit Units, Method of ASSY. Silver Brazing Soldering of Electrical Gonnections Solid Film Lubricants Spotwelding Aircraft Parts Spray-Up Fabrication of structural Components Staking of Bearings Strippable Protective Coating, Application of Strippable Protective Compound, Hot Dip, Application of Suede Coatings, Application of Surface Preparation of Materials for Adhesive Bonding . Surfacer, Epoxy Primer, Application of Swaged Aluminum Tubing, Internal Threading of
Walkway Coating, Nonslip Waterproofing Electrical Connectors Using Flexible Compound Welding l Electron Beam
FW 4306 FW 4311 FW 4455 '
TABLE 3.14 (CONTln) - BELL -PROCESS STANDARDS
3-87
STRUCTURAL DESIGN MANUAL Welding, Fusion, of Titanium Welding, Inert Gas Tungsten, of Aluminum Alloys Welding, Inert Gas Tungsten, of Steel Welding (In-l'rocess) of Castings Welding Operators, Oual.:i,.fication Procedure for Welding J. Resistance, Cleaning and Preparation of Materials for
4207
FW 4404 FW 4359 FW "4470 FW 4431
FW 4113
Welding, Resistance, of Aircraft Parts Welds, Electron Beam, Ultrasonic Inspection of Wood Control and Bonding Procedure for Rotor Blades
FW 4115 FW 4454 4083
z Zinc Chromate Primer, Application of
FW 4361 .
) TABLE 3.14 (CONT'D) - BELL PROCESS STANDARDS
)
3-88
STRUCTURAL DESIGN MANUAL SECTION 4 INTERACTION 4.0
GENERAL
When a structural element is subjected to combined loadings, such as tension, compression and shear, it is often necessary to detemine the resultant maximumi stresses and their respective principal axes. When a body is subjected to a combination of tensile and compressive stresses., it is usually designated as a biaxial or triaxial stress condition. When a combination of tensile, compressive and shear stresses are present, the body is usually referred to be in a combined stress state. The material at the inner surface of a thick pressure vessel is subjected to triaxial stress (radial compression, longitudinal tension and circumferential tension); a shaft bent and twisted is subjected to combined stresses (longitudinal tension or compression and torsional shear). 4.1
Material Failures
Fracture of a material is very complex. Generally, failures can be grouped into two categories, ductile or brittle, depending upon the state of stress and the environment. Metals with high strength exhibit low ductility prior to failure. Failure can occur after elongation of the metal over a relatively large uniform length or after a concentrated elongation in a short length. Shear deformation will also vary depending on the metal and the stress state. Because of these variations in magnitude and mode of deformation, the ductility of a metal can have a profound effect on the ability of a part to withstand applied loads.
)
Brittle failures are characteristic of the high-strength materials and are accompanied by little or no plasticity. The lack of plastic strain generally results in the brittle failure occurring without warning, and in service can lead to catastrophic results. The fracture surface is usually distinguished by a cleavage or rough crystalline texture which appears bright and granular. The fracture surface of steels will have a herringbone appearance with the chevrons pointing to the origin of the fracture. A micro analysis of the fracture surface shows a direct separation of the crystalline planes with the plane of separation normal to the applied load. Ductile failures will show substantial amounts of plastic strain. The loaddeflection curve will show the failure occurring well out of the linear region. The fracture surface is generally distinguished by a "hilly" appearance with some reduc tion in cross sec tion. A micro --analysis wiil show- that tll-e- frac ture is· a result of slippage between crystalline planes with the plane of failure oblique to the applied load. The ductile failure is, thus, an action of shear stresses. There are ductile materials whose fractures appear brittle and brittle materials demonstrate ductile failures. Fracture surfaces often have the appearance of both brittle and ductile. There are no precise equations to define the mechanism of fracture. High or low temperatures affect the mechanism of failure. The strain history and rate of load application will affect the fracture.
4-1
STRUCTURAL DESIGN MANUAL 4.2
Theories of Failure
When one principal stress. exists at a point, the stress is frequently referred to as a uniaxial or one-dimensional stress; when two principal stresses occur, the state of stress is frequently called biaxial stress, or two-dimensional stress, or plane stress, and when all three principal stresses exist, the state of stress is triaxial or three-dimensional stress. It is not always possible to produce in a test the exact biaxial or triaxial stress which exists during operation. Sometimes it is not even possible to test at all. In such eventsrational analysis procedures must be used. Such procedures are called theories of failure and require that the general mode of failure of the member under the assumed service conditions be determined or assumed (failure is usually yieldin~ or fracture) and that a quantity (stress, strain, energy) be chosen which is associated with the failure. This means there is a maximum or critical value of the quantity selected which limits the loads that can be applied to the member. Generally, an ultimate load test of the material with the resulting stress-strain cuive is a suitable test for determining the quantities associated with theories of failure. The six main theories of failure for a material that is considered to fail by yielding under static loading are: (1)
Maximum principal stress theory - Rankines theory - Inelastic action at any point in a material at which any state of stress exists begins only when the maximum principal stress at the point reaches a value equal to the tensile ~ (or compressive) elastic limit or yield strength of the material, regardless ~ of the normal or shearing stresses that occur on other planes through the point.
(2)
Maximum shearing stress theory - Coulomb's or Guest1s law - Inelastic action at any point in a body at which any state of stress exists begins only when the maximum shearing stress on some plane through the point reaches a value equal to the maximum shearing stress in a tension specimen when yielding starts.
(3)
Maximum strain theory - St. Venants theory - Inelastic action at a point in a body at which any state of stress exists begins only when the maximum s train at the point reaches a value equal to that which occurs when inelastic action begins in the material under a uniaxial state of stress as occurs in a specimen in a tension test.
(4)
(5)
4-2
Total energy theory - Beltrami and Haigh theory - Inelastic action at any point in a body due to any state of stress begins only when the energy per unit volume absorbed at the point is equal to the energy absorbed by the material when subjected to the elastic limit under a uniaxial state of stress as occurs in a simple tensile test. Energy of distortion theory - Inelastic action at any point in a body under any combination of stresses begins only when the strain energy of distortion per unit volume absorbed at the point is equal to the strain energy of distortion absorbed per unit volume at any point in a bar stressed to the elastic limit under a state of uniaxial stress as occurs in a simple tension or compression test.
'I
/
~,
~ ~
STRUCTURAL DESIGN MANUAL (6)
4.3
Octahedral shearing stress theory - Inelastic action at any point in a body under any combination of stresses begins only when the octahedral shearing stress becomes equal to 0.47 times the tensile elastic strength of the material as determined from the standard tension test. Determination of Principal Stresses
When an element is subjected to combined stresses such as tension, compression and shear, it is often necessary to determine resultant maximum stress values and their respective principal axes. These stresses ~~d their angles may be obtained by the use of Mohrls circle. This is a convenient graphical representation of the relation between principal stresses at a point and the shearing and normal stresses at the same point on planes inclined to the planes of principal stresses.
fx
FIGURE 4.1
STATE OF STRESS AT A POINT
The following are definitions and sign conventions for terms in Figure 4.1:
fy: applied normal stresses applied shear stress fnmax,fnmin: resul ting principal normal stresses fsmax= resu 1 ting principal shear stress Q: angle of principal axes Sign Convention: Tensile stress is positive (+) Compressive stress is negative (-) Shear stress is positive (+) if its action is a tendency to rotate the element c"iockwise Positive Q is countercloc~wise fx,
)
is:
The procedure for constructing Mohr's circle is as follows: (1)
Make a sketch of an element for which the normal and shearing stresses are known and indic,ate on it the proper senses of the stresses. Such a sketch is shown in Figure 4.2.
4-3
STRUCTURAL DESIGN MANUAL ,fy
-
~
-\
FIGURE 4.2
fs
.
-fX
-
r ight
h and f ace
1r
TYPICAL ELEMENT IN BIAXIAL STRESS
(2)
Set up a rectangular coordinate system of axes where the horizontal axis is the normal stress axis and the vertical axis is the shearing stress axis. Directions of positive axes are taken as upward and to the right.
(3)
Locate the center of the circle, which is on the horizontal axis at a distance (fx + fy)/2 from the origin. Tensile stresses are positive and compression stresses are negative. See Figure 4.3.
(4)
From the right-hand face of the element prepared in step (1) read the values for fx and fs and plot as point ItAu • The coordinate distances to this point . - . are measured from the origin. The sign of fx is positive if tensile, negative if compression; fs is positive clockwise.
(5)
Draw a circle with center as found in step (3) through point ttA n found in step (4). The two points of intersection of the circle with the normal stress axis (fn) give the magnitudes and sign of the two principal stresses. If the intercept is positive, the principal stress is tensile and vice versa.
(6)
The circle will also pass through point liB" which has coordinates of fy and fs taken from the upper surface of the element in step (1).
(7)
Construct a line through the center of the circle connecting points A and B. The angle formed by this line and the normal stress (fn ) axis (abscissa) is 29, twice the angle of the principal axes.
(8)
Construct the biaxial state of stress at the point in question as shown in Figure 4.4.
The following equations can now be written using Figures 4.3 and 4.4. 2
-~(fX ; fy)
2
+
f nmin =
4-4
fx
+ 2
ff
J(f
; fJ!)
X
+ fs
+
fs
2
2
4.1
4.2
STRUCTURAL DESIGN MANUAL tan2e =
2fs
. fx - f y ·
The solution results in two angles representing the principal axes of f max & fmin
J
4.3
4.4 The angles located on Mohr's circle are twice the angles on the biaxial stress element. The maximum principal stress (fnnuud occurs at point I in Figure 4.3. Point I is located at a 29 counterclockwise rotation (about point G) from applied stress point A. In the element, Figure 4.4, face I is located on angle 9 counterclockwise . (about point 0) rotation from face A. The minimum principal stress (fnnrtn) occurs 0 at point H in Figure 4.3. It is located 29 counterclockwise from point.S and 180 counterclockwise from point I. In the element, face H is 90° from face I or 9 counterclockwise from face B. The maximum posi tive and negative 'shear stresses (fsmax) occur at points J and K in Figure 4.3 and their magnitudes are equal to the radius of the circle. Point J is the maximum positive shear stress and is located (90° - 29) clockwise from point B or 90° clockwise from point H. Face J, in the element, Figure 4.4, is located 0 0 (45 - 9) clockwise from face B or 45 clockwise from face H. The shear stress on face J is positive ·which is shown producing a clockwise rotation of the element. Point K is the maximum negative shear stress and is located in the same manner as point J. The planes of maximum shearing stress are always at 45° to the principal planes, regardless of the applied stress conditions. Figure 4.5 shows some typical loading conditions with the resulting Mohrts circle and the state of stress on an element in the body. 4.4
)
Interaction of Stresses
The means of predicting structural failure under combined loading without determining principal stresses is known as the interaction method. The critical strength of structural members with a single type of load is generally defined. That is, the yield, ultimate or buckling of a member load in tension, compression, shear or bending can be determined. The critical strength of a member subjected to simultaneously applied combinations of loads is often difficult to determine. This is especially true if local or overall stability, plastic bending or torsion are involved. The interaction method was developed for predicting ultimate strength of members subjected to combined loads. It is the most satisfactory method of predicting structural failure without determining principal stresses. The basis for the interaction method is: (1)
The allowable strength for each simple loading condition (tension, shear, bending, buckling, etc.) is determined by test or theory.
(2)
Each load of the combined load conditions is represented by a ratio (R) of applied load or stress to allowable load or stress.
(3)
The interaction relationship is the effect of one condition on another (or others) and is 'determined by theory, test or both.
4-5
STRUCTURAL DESIGN MANUAL
STRESS CONDITION ON 'UPPER FACE OF ELEMENT
fy •
fX;fy -trX;fY fx
STRESS CONDITION ON RIGHT SIDE~AC~ .QF EL~MENT FIGURE 4.3
MOHR'S CIRCLE REPRESENTATION OF BIAXIAL STRESS AT POINT "Qn IN FIGURE 4.4
• \
4-6
STRUCTURAL DESIGN MANUAL
MINIMUM PRINCIPAL STRESS PLANE fy fs
-----"- -----=...,...-
i
MAXIMUM PRINCIPAL STRESS PLANE
fx
fx
MAXIMUM SHEAR STRESS_-ld-+_~
PLANES
fy
FIGURE 4.4
BIAXIAL STATE OF STRESS AT POINT "Q"
4-7
STRUCTURAL DESIGN MANUAL LOADING
ELEMENT STRESS
MOHR'S CIRCLE f
Y
"fy/2
fy/2 PURE TENSION
PURE COMPRESSION f
5
f
smax
X
x 0
UNEQUAL BIAXIAL TENSION +f f
s
f x
0
s
=0
!x=:j
f
+f
x
n
y
EQUAL BlAX IAL TENSION +fs
fsmax=£x
-fx EQUAL TENSION AND
=fx= -fy
f
max
f.
mln"
PURE SHEAR
---- FIGURE 4.5 - ELEMENT STRESSES
4-8
=
f
=-f
s s
STRUCTURAL DESIGN MANUAL Revision F The stress ratio, R, is expressed as! R
=
applied load or stress allowable load or stress
4.5
The margin of safety is then
4.6
MS = l/R - 1
)
Generally, for a combined system of loadings the interaction relationship is expressed as 4.7
where R , R2 and R3 are stress ratios and x, y and z are exponents defining inter1 action relationships. ' When only two loading conditions exist, such as bending and torsion, equation 4.7 can be plotted as a single interaction curve of Rb and R. When three or more loadins conditions exist, the equation of interaction become~ a surface and can be plott~d as a family of curves. When the exponents are equal, the interaction curve is' a straight line and indicates the maximum interaction. This might occur when bending is present with tension or compression. Making one exponent equal 2 gives a parabola. With both exponents equal 2 the interaction curve is a circle. Complete independence or zero interaction is obtained when the exponents are infinite.
,)
The amount of interaction between two loads is detennined by theory or test. The analyst must use good engineering judgment and common sense to determine the relationship of one load to the other. For instance, if torsion and bending are present and torsion is the predominant stress, then the interaction equation using the maximum shear' stress theory should be used. If bending had been the dominant stre~s, then the interaction equation based on maximum principal stress should be used. The end points of the interaction curves are always correct; at least they represent failure under simple loading. This reduces the probable error when one type of loadins dominates. The prime advantage of this method is that it yields good results ~h~n anyonE loading condition dominates and exact results when only one loading condition is present. 'rho effect of one loading R1 on another simultaneous loading R
is represented by Z an equaLion or interaction curve like that shown in Figure 4.6. This curVE represcnLs all the possible combinations of Rl and R2 that will cause failure. The curve is used as follows: (1)
Let the value of Rl and R2 locate point "a". indicated because It is inside the curve.
(2)
Failure can occur at three points a. b. c.
A positive margin of safety is
Point lid" by a proportionate increase in Rl and,R2 Point "h" by an increase in Rl with R2 constant Point "g" .by an increase in R2 wi th Rl constant
4-9
STRUCTURAL DESIGN MANUAL
z
-8 :z
1.0
tit 1.0
Ul
0
Z
H
~
U)
z
f-l
.. R e ~ «
~R2 ~ :r: CI) ,
W £:-l
~ R2
""
b
U)
'-" N
0
.
0
.
'.
z
z
)
0
H
I
0
f-l
~
H
~ p::
~
0 t-l
/ /
o
I
0
0
C
I
Cl
If
0
Rl RIa
1.0
Rl RIa
LOAD RATIO NO. 1 (BENDING, SHEAR) FIGURE 4.6
1.0
LOAD RATIO NO. 1 (BENDING, SHEAR)
TYPICAL TWO-LOADS-ACTING INTERACTION
FIGURE 4.7
TYPICAL THREE-LOADS-ACTING INTERACTION
•
y
fy
I--.f--~~
fx
I-~.J---~-X
z FIGURE 4.8
4-10
BIAXIAL STRESS CONDITION
•
~.\\
)( \'::tP~11 ~'\ ,; ............ ~~/
( 3)
STRUCTURAL DESIGN MANUAL Revision F
The margin of safety for the loading .represented by point "a three ways a. b. c.
u
can be found in
MS = od/oa - 1 MS = bh/ba - 1 MS cg/ca - 1
Values of od, bh and cg are referred to as allowables (load or stress) and oa, ba and ca are applied load or stress. Using this procedure and equation 4.7 procedures for two loads acting and three loads acting ~an be determined.
)
4.4.1
Procedure for Margin of Safety for Two Loads Acting
(1)
Using ~uckling, yield or ultimate criteria and equation 4.5, calculate the stress ratio for each load acting alone.
(2)
Using the calculated stress ratios locate point tla" on the proper interaction curve (using Figure 4.6 as an example).
(3)
Draw a straight line from the origin "0" through point u a" and intersect the interaction curve at point "d". Read the stress ratios R (ed) and R (fd). 1a 2a
(4)
_Compute the applied stress ratios R (ba) and R (ca). 1 2
(5)
Compute the margin of safety · ~S
4 .. 4.2
=
R1a/Rl - 1
=
R2a /R 2 ... 1
4.8
Procedure for Margin of Safety for Three Loads Acting
(1)
Using buckling, yield or ultimate criteria and equation 4.5, calculate the stress ratios for each load acting above.
(2)
Using the appropriate interaction family of curves locate point "a" corresponding to the calculated stress ratios Ri and R2 as shown in Figure 4.7.
(3)
Draw a straight line from the origin "0" through point I'a".
(4)
Extend this line to locate the allowable point "x" which must satisfy the following relationships: 4.9 or 4.10 Point Ilxtl is obtained by trial and error in the following manner: (a)
Select an arbitrary value of Ria"
(b)
Calculate R3 from equation 4.10 using the known value of Rl and R3 and a va I ue 0 f Rla' ' h b ltrary tear 4-11
~ STRUCTURAL-· DESI.GN. M:ANUAL '~~~.~ .
,
(c)
-
./,~-
Locate point "x" on the line "oa" using the calculated R3
and compare the corresponding Rla with the assumed R . 1a Cd)
(5)
"e
Repeat steps (a) through (c) until the assumed Ria and the "x" value of Ri converge. At convergence, Ri ,R and R3a wIll be at a common 2a II a a pOInt on 1.Ine It oa.
Compute the margin of safety MS
4.5
from step (b)
a
=
R1 a IR....L
-
4.11
1
Compact Structures
,
)
A compact structure is one in which failure does not occur by crippling or buckling. This section presents interaction criteria for compact structures with biaxial stress in a rectangular volume such as in plates, membranes and shells and with uniaxial stress in a plane such as in beams, round bars and ,bolts. 4.5.1
Biaxial Stress Interaction Relationships
Tests have been conducted to determine the failure theories of biaxially loaded isotropic ductile materials. The maximum shear stress theory and the octahedral shear stress theory adequately predict the yield and ultimate strengths. There are a few cases where convenient-margin of safety calculations a~e possible. These are shown in Table 4.3. A general interaction method is required. It is shown in Figure 4.8. The method is applicable to stress conditions which combine in a two-dimensional manner like that shown in Figure 4.8. This condition exists in a rectangular volume and not on a single plane. Tension is positive, compression is negative. The int(!raction equations and curves are applicable for ultimate and yield by use of the parameters given in Table 4.1.
~ ~
The interaction equations contain certain factors which relate one stress to the other. They are defined as follows: The constant relating interaction in terms of tension or shear strength allowabies: K == F
su
IF tu
4.12
Tests show this value to vary from 0.5 to 0.75.
) /
The transverse shear and torsional stress ratios combine as R
S
= R
S5
+
R
st
4.13
The directional tension and bending stress ratios combine as R
4.14
R Y
4.15
x
The directional compression and bending stress ratios combine as R
4-12
x
4.16
(
STRUCTURAL DESIGN MANUAL Revision A 4.17 Usin~
equations 4.1 and 4.2 and substituting the previous relationships the following
is de ri ved.
R
Rx
nmax:'::
\
Ry
+
2
Rnmin = RX )
4.5.1.1
+
~
~ tRx
4.18
~RX
4.19
"".:.:
Ry -
Maximum Shear Stress Theory Interaction Equations
The maximum shear stress theory states that yielding or fracture occurs when the maximum shear stress in a combined stress element equals the maximum shear stress in a pure tension test specimen subjected to the yielding or ultimate stress of the material. The substitution of fX = Ftu' fy = 0 and fs = 0 into equation 4.4 results in the maximum shear stress in a pure tension specimen at fracture equal to 1/2 Ftu. Dividing equation 4.4 by Ftu' substituting equations 4.12 and 4.13 and parameters conLained in Table 4.1 and setting f smax = 1/2 Ftu results in 4 .. 20 Usin>~
K = 1/2, which is the theoretical value of K determined for a specimen loaded in tension, the equation 4.20 yields 4.21
This equalion is plotted as the dashed lines in Figure 4.9. The maximum shear stress is obtained in terms of the three principal triaxial stresses (f , f2 and f3) as: 1 f
smax
+
(f 1 ~
f 2)
4.22
)
4.23
4.24 [Ising Figure 4.9 and testing equations 4.22, 4.23 and 4.24 in each quadrant for the hiaxial slress state (f = 0) results in the maximum shear stress theory interac3 lions. These arc shown in Table 4.2. 4.~.1.2
OClahedral Shear Stress Theory Interaction
E~uations
The ()ctahedral shear stress theory states that yielding or fracture occurs when the octahedral shear stress in a combined stress element equals the octahedral shear sLrf'SS in a pure tension test specimen subjected to the yielding or fracture stress 4-13
STRUCTURAL DESIGN MANUAL
TABLE 4.1 STRESS RATIO PARAMETERS
STRESS RATIO PARAMETER STRESS TYPE
YIELD CONDITION
Tension
u x"
Direction·
Tension
uy"
Direction
ULTIMATE CONDITION
Compression "x" Direction
R
Compression U ylt Di rec tion
R
I Bending
tlXU
Direction
Bending
.t y "
Direc tion
ex
C
(2 )
Transverse Shear
RS5 =f ss IF S
Torsional Shear
R
Subscripts:
t =- tension c = compression b = bending
=f
(2)
=f
=f
ex C
(1)
If? Cll
IF Cll (1)
Rss =f ss IF su R =f 55 = transverse shear st = torsional shear s - shear
Subscripts x and y on Rand f refer to x and y directions, respectively. Subscripts y and II on F refer to yield and ultimate strength conditIons, respectively. Notes:
(1) Assume Feu (2) Assume F K
4 ... 14
=
F tu
= F = KF sy sty ty 0.5-0.75 for most isotropic ductile materials
STRUCTURAL DESIGN MANUAL
TABLE 4.2.
SHEAR STRESS THEORY INTERACTION EQUATIONS
MAX~MUM
CONDITIONS
QUADRANT
£
(Figure
4 ~)
£ I
f
f
I
f
f
n n
n n
n
II f
n
III
n
>
IV
n
f
+,
=
>
£
= -,
f
max
= -,
f
max
>
-, max
>
= +, max
INTERACTION EQUATION
=+
min
+f
n
n
/2
=
R
n
max
1
max
n
=+ min
n
n
/2
= 1
R
n
min
max
max
n min
n
+f
min
=
+
-f
n
max
-
f
n
min
R
n
2
=-
Ifnminl f
EQUATION max
n .. min
f
min
Ifnminl f
n
max
III
.)
f
max
\fnmaxl f
+,
=
max
s
= -
min
I nmax I
max
- f nmax /2
R
-
R
f
n
n
/2
n
min
-
R
n
= -1
min
= -1
max
=
-1
min
f
f
n
= min
-
+f
n
-£ n
max 2
min
R
n
max
- Rn
= -1
min
4-15
LOADING DESCRIPTION
LOADING PICTURE fs
CASE
--
Uniaxial Tension
f
--~DI~ ....--
1
fs
+
Uniaxial Compression
-~Dt~
2-
+
---
R 2+ R2 s
x
:;;:
1
4.9 or 4.10
JR
x
2
~
(I)
REMARKS
til
0 ::1
-1
+ R 2
tD
s
~
~ ~
P.I
1 R 2
x
+
4.9 R 2 = 1
s
or 4.1"0
JR 2 + R 2 x
..c:-w
-1
s
(")~(")
OZ~ ZO t:='
I-I(")~ ....,:;otJ
t-rt-r1-3
Biaxial Tension
OIltJ
4.9
ZtoCCf.I
Cf.I~;1
\ ozc:: ~G')(")
-+
~f
...,
!
T
4
<: ...,.
...,-
1
Shear
-Df~
3
MARGIN-OF-SAFETY EQUATION
Shear
f
...;;...a,
INTERACTION CURVE EQUATION FIGURE
{J)o~
1-3:;0
I~
Biaxial Compression
~
tr:I
t'11t:.DCf.I
4 .. 9
I
ZC::.
; G')(")t;.D
""pH :t:Hs< t ZH !
C":l>
-t""1
All other states of biaxial stress
5
Refer to section 4.5.1 and use: (1) Table 4.1 ( 2) Equations 4.18 and 4.19 (3) Figure 4.9
4.9
(3) (4)
(5)
R
= R
+
r<~ Htr:I t:r:1~
1:"'"1).>-
t:='~
~~
oz
c:::n r~
t-lt-l t-It-l
NOTES: (1) Tension is positive, compression is negative. (2)
it-r
~~ ...:3H
R
s ss st Rx = Rtx + Rbx (tension) ; Rx = Rex + Rb x (compression) R = R + Rb Y (tension); R = Rey + Rb y (compression) Y ty Y :;;: R = f IF R f IF t fb/F by , R :: f IF f R c y, ~ t ty, c t st IF sty s Sy, st ss R :: f IF = f t 1Ft u, Rc ;: f c 1F t u, Rb = fb/Fbu, Rss = f IF st stu st :;;:
Sj' ,-'
tr:I>
Vt)
--I
:::a c: n
--I
= :::a ::Da
r-
=
.."
-
(I) .
CD
Z
:I: ::Da Z
= ::Da r-
(YIELD)
(ULTIMATE)
e
.''- /T\;""~·
fL.,' I' \ \\ "- \ \\ .B~II , ,., ........ '"
\
.'
'
s-rRUCTURAL DESIGN MANUAL
'
",-: .... iL" ." \
Revision A Ftu' fy =
By substituting fx
of the materiaL. stress equation
fs
0,
=
0
into the octahedral shear
thc' octahedr.al shear stress in a pure tension specimen at fracture equals
IT/3 Ftu'
By proceeding as described previously the octahedral shear stress theory interaction equation becomes RX
Uslng K
2
+
Ry2 - RxR y "+ 3K2Rs2 = 1.
4.25
1/ ['3' equation 4.25 becomes 2 Rx
2
+ Ry - RxRy + Rs
2
4.26
= 1.
Equation 4.26 can be rc:arranged to form (R
n
)2
+
max
Thts
equation is plotted
4.5.1.3
(R
n .
)2
min
+
(R
_ R
n
max
)2
n .
m1n
2
4.27
as the solid line in Figure 4.9.
Margin of Safety Determination
The procedure for determining the margin of safety where orient.ation of applied slress wilh respect to the grain direction is unknown is as follows: (1)
Assume the allowable stresses in each direction of applied stress are the same as in the weaker direction.
( 2)
Evaluate K
(3)
If 0.5 ~
I :=
F
su
IF tu •
~ .577, use K = 0.5 in calculating the principal stress ratios 4.18 and 4.19) and use the maximum shear stress interaction curve,
KI
(e~uations
Fiburc 4.9.
(4)
If Kt ~ 0.577 , use K = 0.577 in calculating the principal stress ratios (equations 4.18 and 4.19) and use the octahedral shear stress interaction curve, Figure 4.9.
(5)
[n evaluating Rx and Ry , the allowable stress is taken to be Fty (yield) or Ftu (ultimate) regardless of whether the applied stress is tenslon or com-
)
pression. (6)
Applied tension is positive, applied compression is negative.
Calculate the Inargin of safety using the two-parameter procedure outlined in sccLion 4.4.1.
procedure [or deLermining the margin of safety where orientation of applied stress with respect to the grain direction is known is as follows~
'fhe
(1)
Determine applied stresses on an element with sides parallel and perpendicular to the grain direction.
4-17
1/;1\,\,
STRUCTURAL DESIGN MANUAL
"""'-1~Bell \ \\ , ," ..•. " .. \.'\.
J
.
I
..... ~. 1/.--::""
!,.
·
2
'2
I
'
·
'2
.....= (Rn ma x) + (Rn min} +(Rnmax-Rnmin) = 2 :
e = p:: +
/
RnmaxRnmjn=-l ~
I
FRnm~ X
-1
A / //
11/
/1
I...
/
p
I:
(Rnrnax-l I
\ I
\iL~
I I
~
~"
~
'/ l/
V'
O.!'!
-~
V
.=
I
I
Rnm~x= 1~
J
,
-
"
4-
I
o4
~
fnmaJ
0.6
IV ~
-
0.8
!l
"
r-
l/
I)(, ~
!V
~
nmalx
Octahedral Shear Stress_
k'~J7'~~ . 1
It:-R.
+1
1/1 .IV
if; • / I
-1
......
.
tfnI nin
:r-2 ~ ~ ~
~- I - - - - I -
FIGURE 4.9
O.~
-o. e
Rnmi
/
/
~.2
I I I f-.f-~IL--O ..:
'"
,I'
~V
-02 /
/ . . 7:!Y /. I \
/ /
V2
/
~ r---
Rnminl - ~-
0.6
t /
I
4-18
"1
-0 :6 .-0.4
-
to-- - 1 - - "
O.!.
i
..
-0..8
k
V/ / V/ 0..8
Jvy /
--........
V--~
17
r-RnmaxRnmin= 1
T~~r~
va -
Maximum Shea.r Stress The.oTr 1<:=\ 2
-
INTERACTION CURVES FOR BIAXIALLY STRESSED STRUCTURES
.
STRUCTURAL DESIGN MANUAL (2)
where:
Evaluate'
T and L refer to the
transv~rse
and longitudinal grain direction.
J
(3)
If 0.5 ~ K ~ .577, use K = 0.5 and the appropriate transverse and longitudinal allowable stresses in calculating the principal stress ratios (equations 4.18 and 4.19) and use the maximum shear stress interaction curve, Figure 4.9.
(4)
If Kt ~ 0.577 • use K = 0.577 and the appropriate transver·se and longitudinal allowable stresses in calculating the principal stress ratios (equations 4.18 and 4.19) and use the octahedral shear stress interaction curve, Figure 4.9.
(5)
In evaluating Rx and Ry the allowable axial stress is taken to be Fty (yield) or Ftu (ultimate) regardless of whether the applied stress is tension or compression. Applied tension is positive, applied compression is negative.
(6)
Calculate the margin of safety using the two-parameter interaction procedure outlined in section 4.4.1.
)
4.5 .. 2
Uniaxial Stress Interaction Relationships
When compact structures such as beams are loaded by axial load, bending moment and shear, methods other than those presented in section 4.5.1 must be used. Such conditions where shear does not combine in a simple two-dimensional manner like that shown in Figure 4.8 and conditions where shear, tension and bending must be combined in the plastic region. Table 4.4 shows conservative interaction equations to be used when combining these stresses. It is sometimes convenient to combine the maximum bending stress and maximum shear stress even though these stresses do not occur at the same point~ It is recommended that several points in the section be checked for their actual conditions of stress to reduce the conservatism. 4.5.3
)
Thick Walled Tubular Structures
The interaction stress ratios for the design and analysis of thick walled tubular structures must be determined from critical tube strength and stability criteria. Bending stress ratios must include the effects of secondary bending, if any, and compressive stress ratios must be based on column stability criteria. For cases involving combined tension and bending the exponents of 1.5 is conservative. Also for combined shear and bending the exponent of 2 is generally conservative. Table 4.5 shows the applicable interaction equations and margin of safety equations along with the applicable interaction curve. 4.5.4
Unstiffened Panels
The interaction stress ratios for flat rectangular and curved unstiffened panels are based on elastic initial buckling. If a panel is subjected to direct axial stress, tension is considered as negative compression using the critical compression allowable. Table 4.6 shows the interaction relations •
.
4-19
STRUCTURAL DESIGN MANUAL 4.5.5 Unstiffened Cylindrical Shells The shell structures for which interaction is shown must have a radius to thickness ratio greater than 10. Otherwise the interaction relationships given for tubes should be used. The interaction stress ratios must be based on initial buckling criteria. If direct axial stresses are present, tension is treated as negative compression using the compression buckling allowable. Table 4.7 shows the interaction relations. 4.5.6 Stiffened Structures Table 4.8 shows combined loads interaction data for the design and analysis of stiffened panel and cylindrical shell structures. The interaction stress ratios must be based on stability criteria.
• I
4-20
I
~
TABLE 4.4
COMPACT STRUCTURES-UNIAXIAL INTERACTION CRITERIA (NO CRIPPLING OR BUCKLING)YIELD AND ULTIMATE CONDITIONS OF STRENGTH
Axial, bending (simple and complex), and shear (snnple and complex) on beam cross section (symmetrical and unsymmetrical) LOADING DESCRIPTION
LOADING
CASE
INTERACTION CURVE EQUATION FIGURE
MARGIN· OF-SAFETY EQUATION
REMARKS
CI)
Axial Tension or Compression
+y' 1
"'-
+
.....
Ra + Rb
;::
4.10
1
::a
-1
1
C.
Ra + Rb
n
..... c::
Bending
::a Axial Tension or Compression
2
+ Shear
R 2 a
+R
s
2
=1
4.10
....
:D-
1
JR2+R2 a s
-1
t:I ...,
-
(I)
C)
Bending
+
3
Shear
1
R 2 + R 2 b
s
;::
1
4.10
•
+ Bending
4
N
......
+
Shear /
Axis
~
:DZ
,.c:r-
Axial Tension or Compression +'"
jR2+R2 b s
Z
... 1
1
(Ra + Rb)2
+R2 s
=1
4.10
+
1 R 2
s
.~
~
+"
Axial, bending (simple and complex), and shear (shnp1e and complex) on beam cross section (symmetrical and unsymmetrical)
~
I
tTl
N N
.
+" +"
.-«()
LOADING
CASE
INTERACTION CURVE FIGURE EQUATION
Tension
2
MARGIN-OF-SAFETY EJ2UATION
REMARKS
ROUND BARS
1-10
~~
t;> 0
~:t-3
LOADING DESCRIPTION
fst
rf:ft
~
5
C::~
1
R
+
t
Torsion
+ Rst
2
:::
1
4.10
JR 2 + R 2
-1
st
t
~c:
t-30
1-11-"3
RECTANGULAR BARS
~~
"""t'r.l pjen
•
OC: 02:: 2::1-1
fst
~ft
~
6
Ss<
1
Tension
+
R
t
Torsion
2
+ Rst
2
:::
1
4.10
-1
VR t 2 + R st 2
.-iH
~~
BOLTS (FINGER-TIGHT NUTS)
'Z
fs
CIlH
~ J~IIIIII-- ft
!Z
0;-3
~~
7
en>
;i3g
R 2 + R 3 = 1 t s
Tension
+
4.24
Shear
tTlH
ZO
G')Z
NOTES~
t-3 ::J:O
(1)
Rb ;: Rbx ' + ~
(2)
R = s
~
,-. H
(');-3
~~ ()H t;'"'4> c: t:I ".....,
-
trlZ
( 3)
2 R 2 + R sx' sy'
8 = 1/2 tan
.. 1
00 0
(4)
(~~~yy)
-
M (cos a cos 8 + sin O! sin e ) M' (-cos (l sin e + sin a cos 9 ) M = y Vx t = V (cos f3 cos e + sin t3 sin 8 ) V (-cos {3 sin e + sin (3 cos () ) Vy '
M,,'
<
( 5)
::;;;:
~
1-4 "'t1 "d
(6)
R
t
t-I
=
f
t
,
R
c
f
= ......£,..,
H
F ty
F
G')
f
f
Z 0
(7)
:;d IJj
c.: (')
F H
z ~ .......
I
f
(8) (9)
Rbxt
M.
:::_K_,
M'
cy
x'y
M
M
Rby '
= ....L...., M. Y
y
R
sx' =
f F
sx'z::,
R sy' =
sy
f
F
sz::'z:: sy
Rbxt ;; _x_, Rby' --~ , Rsx' = fsx' R sy' ;;;:~ M M F ' F tu x'u y'u su su M M M and M are yield and ultimate allowable bending moments. x'y, x·u, y'y, y'u R;;;: t -
f
F
t, R c tu
= ......£...,
f
t
F
sx • y, f sx'u, f sy'
and f
t
sy u
(YIELD)
(ULTIMATE)
are yield and ultimate plastic bending shear stresses.
-
<--
e
LOADING CASE PICTURE ROUND TUBES 1
2
3
4
5
6
;b Gee .~ @-)b tI
tst
~h
(0
Ge Gjb ~
~)b
8
EQUATION
FIGURE
MARGIN-OF-SAFETY EQUATION
Compress.ion
+ Bending Tension
+ fl,pniiinc
Bending
+ Torsion,
1
Rc + Rb R 1.5 + R t b
+ Bending
Let Rt
+
1
4.10
J
Rb 2
. Bending
+
R1
(I)
+ Rst 2' Kbt - it/R t a
-1
=
4.10 IjR
1
bt
-4 ::.:I
.. 1
1
L+R2 s
vhcre Rt
•
= n .....
~/Rb
and Rb
at are ob-
c: :::a
a t:3 talned ft:om figure 4.k' (R -;t R • ib • R2t n • 1.5) by 4
tAe tvo.loads-acting procedure as outlined in
l:II
r-
eection 4.4.1
Let
,
(Rc
+
Rb)2 + R s
i
1
4.10
j
Shear
Q~)fb
1
Shear Compression + Bending
~
Rj) ;;: R2
~2 + R 2 = st
R 2 + R 2 bt s
REMARKS
-1
4.23
1
:::::
Rc + Rb
4.10
== 1
Tension
+
fs
fs 7
TNTF.RAr.TTON r.lJRVF.
LOADING DESCRIPTION
1 R 2 b
+R 2 s
=
_] Rc
+ Rb
Rs
= R2
;;:
C
R1
rI1
-z
(I)
CD
-1
4.10 }R2+R2 b s
1
3:
Shear
Gt
--BC @
ft
~~ 00~
Tension
+ Torsion
1
R 2 + R 2 t st
=
1
4.10 .
1
Compression
+ Bending
+ Torsion
R 2 + R 2 b st (l-R )2 c
=
4.11
l:II Z.
JR t 2 + Rst 2 -1 R c
+jR 2 + b
In using figure -1 4.11 follow twoR 2 loads-acting prost cedures as au tlined in section t...
Q 1
:;c CD
< ..... til ..... 0
::s to
= l:II r-
TABLE 4.5
LOADING PICTURE
CASE
9 .;,
THICK-WALLED TUBULAR STRUCTURES-INTERACTION CRITERIA-YIELD AND ULTIMATE CONDITIONS OF STRENGTH, INCLUDING THE EFFECTS OF COLUMN STABILITY (CONCLUDED)
~
ru)
+ Bending
+
Torsion
+
INTERACTION CURVE FIGURE EQUATION
MARGIN-OF-SAFETY EQUATION 1
Rc + R st
.5
+~2 + Rs~~
a
4.12
-1
1
8C·····00 \.J.u~..
t
Tension
+
+
R 2
+
+ Rp 2
Torsion Internal Pressure
t
R
st
2
1
4.13
= 1
~R t 2
+ R st
REMARKS
In using figure 4.12, follow two-
loads-acting proced':Jre as outlined in section 4.4.1
Rc+R st + Rb+Rs
Shear fst
10
LOADING DESCRIPTION Compression
-1 2 + R 2 p
STREAMLINE TUBES 11
CC
p~;b
Bending
+ Torsion
1
Rb + Rst = 1
4.10
IJRb + R
-1
SQUARE TUBES
ec
12
CD
~st
Cqmpression
+ Torsion
4.14
Let R c R s
= Rl :::
R2
NOTES: (1) R must be based on the tube column allowable. (2) R~ must be based on the material strength allowable. (3) Rb and R t must be based on tube strength al1owables. (4) Rb must fnclude the effects of secondary bending. (5) For shear-bending analysis use f = f and fb = fb even though the locations of the two maxima do not coincide. The al18wabl~mt~ansverse sheW~Xstress is equal to the lower of 1.20 times the allowable torsional shear stress and the material allowable shear stress. (6) R = pd/2t F • d = tube mean diameter, t wall thickness. p
-z
tu·
!I:
:lit Z
=
:lit
r-
'~
CASE
1
LOADING DESCRIPTION
LOADING
(
I
,
J f ex
or ftx
B8DB 8 f f f t fey
f f
f 1'ty
tII
2
3
or
~fcv
cx
1 I .1
III
BBDsO'a fs
j J f ftv or f + I I
t
fev .
1CJt r-n-t I
.r:-
I N
t.n
~
Longitudinal Compression or Tension
MARGIN-OF-SAFETY
REMARKS
4.15
+
Transverse Shear
CI)
..... ::a c:
+
c-)
Transverse Compression or Tension
.....
4.15
c: :::a ::a:-
j
--~.....-
4
INTERACTION CURVE EQUATION FIGURE
Longitudinal Compression·
§D§f f
-.-/'
f
s
r-
Longitudinal Compression or Tension
+
2
R
ex
+R
s
2 = 1
4.14
or
R
tx
+
R 2 S
=
1
(1)
Shear
Transverse Compression or Tension
Rex -Vexort + 2
Rt~Vt}
+
4R
R
s ~i 1
s
;;;;; R
2
-z
CI)
CD
:i: ::a:z c: ::a:r-
4.16
+
Shear
.
TABLE 4.6. UNSTIFFENED PANEL INTERACTION CRITERIA (INITIAL BUCKLING)
ex or Rtx = Rl
R
s
I Il
..., et
~~l 4R
-
LOADING DESCRIPTION
LOADING
CASE
f
bx
~~D~~ ------......:-
S
f
s
I i ~
6
f
ex
l
f
f cx
• * I Jfcy
BCJ8f~
8
fst~r'J
L £:[Jf
CY f
bx
t=r=n
+
Longitudinal Bending
4.17
REMARKS
In using Figure 4.17, follow twoloads-acting as outlined in Section 4.4.1.
+
f~-""'--
Longitudinal Compression
+
Transverse Compression
4.18
In using Figure 4.18, follow three~ loads-acting as outlined in Section 4.4.2.
+
Longitudinal Bendink! Longitudinal Compression
+ Transverse Compression
4.19
Transverse Compression
+
Longitudinal Bending
4.20
In using Figure 4.20, follow three~ loads-acting as outlined in Section 4.4.2.
+
Shear bx
~jr~l~~ s
Longitudinal Compression
MARGIN-OF-SAFETY
+
f
9
~IGURE
Shear
~lDi[ fa
bx
~.
fey
.
fIIRVI(
c:::l-t':::'!:l""
~~CJ~ t f f t
7
TN'T'F.RAr.TTON
EQUATION
Longitudinal Bending
+
Shear
4.20
In using Figure 4.20, use R /R c~ s = 0 curve an follow two-loadsacting as outlined in Section 4.4.1.
TABLE 4.6 (CQNT'D). UNSTIFFENED PANEL INTERACTION CRITERIA (INITIAL BUCKLING).
I .
-
~/.
ttl
A~[]~t f I I
12
,
f ex or
'tl(
.
t t
If
tftv
I
lfCY
J
or
~D~fcx t 1ft I II J
FIGURE,
REMARKS
MARGIN-OF~SAFETY
n using Figure use R = 0 ~nd follow E~ooads-acting of ~ection 4.4.1.
(I)
~bx
~
~.18,
!Bending 4.18
+
,trransverse Compression ~ongitudinal
Compression
+
R ex
+
R 1.75=1 bx
~.23
jL.ongitudinal ~ongitudinal
~ompression
or
.
= Rl
;) :\cx =
Bendin~
8 BDB B f ~fCY
INTERACTION CURVE
EQUATION
,~ongitudinal
rension
f f
13
ffCY
~Dg:fbX I· I , t'
" 10
11
LOADING DESCRIPTION
LOADING
CASE
~.21
+
R2
-I
=-a c:
..... C :::IC
::1:1 rC
rr v.
transverse r:omnression
-
a: :2
.... ongitudinal t;ompression
:I
+
~ransverse
Compression or trension
~.21
TABLE 4.6 (CONT'D). UNSTIFFENED PANEL INTERACTION CRITERIA (INITIAL BUCKLING)
:1:1 :I C 21
,.
LOADING
CASE
, I II '~v t
"tv
j
~----
JCJ~fs
~
14
----~-
f
LOADING DESCRIPTION
INTERACTION CURVE "EQUATION FIGURE
MARGIN-OF-SAFETY
Transverse Compression or Tension
REMARKS
R
=
R
= R
cy
+
s
R
1
2
4.14
Shear
(2)
II t
I I I J f ex or f tx
fs
15
BBiD:BB --~~,~ :l~ ~fbX .
16
Q
f
tfex
-----, I I J
!
s
I I I Jfcy -~
-
17
I
e--
~CJI ~ ~ ~fl
n,-,
Longitudinal Compression or Tension
+
~
+ R 2 = 1 R ex s or - 2 R + R tx s
;:;
1
ex
+
4.14 (1)
Shear
R
ex or 2
Rtx+JRtx
,
Longitudinal Bending
2 2
2
-1
+ 4R z s
+ 4R s "L
-1
R
= R
R
= R
ex 1 R = R s 2 or tx 1 Rs = R2
+
Transverse Compression
4.22
+
Shear Transverse Compression + Shear
R
+ R 2 = 1 cy s
TABLE 4. 6 ( r ....'TT t D). UN STIFFENED PANEL
4.23 ~ey + Rcy 2 + 4R s
2,-1
R = R s 1 R
ey
1.1'rERACT I ON CRITERIA (I NITI AT. BUCKLING)
e T
2
= R
2
-
CASE
LOADING DESCRIPTION
LOADING
INTERACTION CURVE EQUATION FIGURE
18
to I
19
bx!
,
B
f
e
.1
+
: Shear
.
'ex
Of'
R
ex
+
R '2 = 1
s
4.23
~
+1R 2 + 4R ex l-cx
2 S
REo = R1 R
cx
= R2
fhl
2
Longitudinal Compression
~[J~~ :::~:on
> r-
,., CI
-
CI)
PAHF.T.~
f S
r---_-~--_-
Compresslon
,~
:::a
2
JDt8LOngitud~nal
, _ ...... _
n ..... c:
4.14 ( 3)
ex
1
LUMVr..;U RECTAN~ttT,AR
21
+
\Compression
f ....... --. • ..a..
c:
Bending
, Transverse
itt t
CI)
..... ::a
ri::: ~ Longitudinal
•
,
REMARKS In using Figure 4.22, R = 0 and follow ~~o-loads acting as outlined in Section 4.4.1.
4.22
~
20
MARGIN-OF-SAFETY
r---~--p-----~-
~
ex
IR 2 + ~ ex
-+:
or
Rt x + Rs
4R 2
s
or 2
4.14
=1
(1)
2
~t + x
R
tx
2
+
4R 2'-
s
TABLE 4.6 (CONT'D). UNSTIFFENED PANEL INTERACTION CRITERIA (INITIAL BUCKLING)
G)
Z
s::
>
z c:
> r-
INTERACTION CURVE
LOADING LOADING
CASE 22 f
DESCRIPTION
~D~
ex
p
f
f
...:....
..... $
~
pl)[]\
23
ex
Longitudinal Compression
+
EQUATION R 2 cx
-
FIGURE
MARGIN-OF-SAFETY
4 .. 24 ·1 Rpl =1
Internal Pressure Shear
+ Internal Pressure
REMARKS R = Rl cx R
p
= R2
CI)
..... ~
c:
n R 2 s
-
I
Rp\
=
1
4 .. 24
R
~
R
= R2
s
p
~"""'II.'-'---
Rl
--I
c:
::a :r:-
.... ~
NOTES: (1) R, R , and R must be based on panel initial buckling allowables. c s b (2) R is negative and is based on compression allowable.
rn
(3)
R is negative and is based on external collapsing pressure.
(4)
S - simple supports, C
z
(5)
Dimensions; a
t
p
=
long
= clamped supports, side, b = short side.
e
= elastic supports.
-
CI) C i)
31:
:r:z
TABLE 4.6 (CONT'D). UNSTIFFENED PANEL INTERACTION CRITERIA (INITIAL BUCKLING)
c:
:r:-
r-
CASE
~fb 1
~
Longitudinal Compression
"-
Longitudinal Bending
+
./
~b f ~
1
R + R ;;: 1 c b
4.10
~fc
2 Longitudinal R + R t c s Compression
'-.-fst
3 .-/
~
4.23
3 Longitudinal Rst -IRtl Tension
st
=
1
4.24
+ Torsion
./
~tb
~C~ 5
-'
fSo ~
6
-
Longitudinal Bending
+
ill.
b
1.5
+
R
-2
st
=
1
+
R
c
\fR
2 c
2
+ 4R st
i-I
::a
c:
n
~
= R1
c:
IRtl < 0.8
R
st R =R 2 c
,..::ar-
Rst
= R1
CJ ,..,
IRtl
= R2
(I)
4.24
Rb = Rl R
=1
-I
= Rl = R2
-z ,..z = ,..rCD
Torsion Longitudinal ~_3+R 3 Bending b 5
Rt Rb
1
;:;;
Torsion
~/
\.....
- 1
+ Rb
4.25
+
4
Rc
REMARKS
(I)
+ = 1 Longitudinal Rb + O.9R t Bending
~
'-
MARGIN-OF-SAFETY
Longitudinal Tension
t
2
INTERACTION CURVE FIGURE EQUATION
LOADING DESCRIPTION
LOADING
st
4.10
Transverse Shear
TABLE 4.7. UNSTIFFENED CYLINDRICAL SHELL STRUCTURES (INITIAL BUCKLING)
;;: R
2
51:
LOADING CASE
LOADING
DESCRIPTION
INTERACTION CURVE WIGURE
E~UATION
Longitudinal Compression
I~
or 7
Tension Lon;itudina1 Bending
MARGIN-OF-SAFETY 1
RC + Rb = Rl Rst :; R2
or
or
11: +
2
+..., +J
REMARKS
4.14
Rb + Rst
2
=1
(1)
2
", + IlJ
+.J
'
or
-1
Tor~i()n
Longitudinal Compression
8
Longi tudinal Bending
1Rc + ~ Rb 3 + Rs 3
1
i ...
R
+
10
2
RbP+(R
+
+ Rst ) s
Transverse Shear
= 1
Longitudinal Compression
R + R
2
+ ~ I~
+
+
+
VR
c
" - 1
st 3
b
R
4.23
st $' Rs P = 1, R < R st s P ; ; ; 1.5
Transverse
-
If R 3
s .
i
+
R
+ ~ IR
3
R
c
Torsion
+
+
For
+
Longit:udinal Bending
+ 3 + R 3~1 cbs
MS ;;::
Longitudinal Bending 9
R1
If
Transverse Shear
Torsion
=
Rb
Rst ;; R2
+ +
+
Rt
"
MS
st
+
b
+
2
R 3 '< 1 s
C:'ho.!:IT'
TABLE 4.7 (CONTID). UNSTIFFENED CYLINDRICAL SHELL STRUCTURES (INITIAL BUCKLING)
--
A.
I
',,-.
CASE
LOADING DESCRIPTION
LOADING fst p
11
EQUATION
FIGURE
MARGIN-OF-SAFETY
REMARKS
Torsion
+
Rst + Rl
Internal Pressure
R
P
· uP
f~'
12
= R2
Transverse Shear
+ Internal Pressure
Longitudinal Bending
+
13
Torsion
Rb
2
+
(R
S
+
R ) 2 "" 1 st
4 • 10
+
.. / Rb 2
l
+
(R
1
s
+
R )2 st
Transverse Longitudinal Bending
+
14
4.10
Transverse Shear Longitudinal Bending
15
+
R
b
2
1
+ Rx
.. 1 I
Rb :;; Rl Rs + Rst
= R2
2 -1
1
"Ji""="Jiii=I=--IIIJIIII-l
4 .. 10
Torsion
NOTES: (1) (2)
(3)
(4)
R, R , R , and R must be based on cylindrical shell initial buckling allowables. b c st s R is negative and is based on compression buckling allowable. t For shear-bending analysis, use f = f and fb :;; fb even though the locations of the maxima s smax max do not coincide. The transverse shear buckling allowable stress is equal to 1.2 times the torsional buckling allowable. The interaction equations are applicable to both internally pressurized and unpressurized cylinders. , TABLE 4.7 (CONT'n). UNSTIFFENED CYLINDRICAL SHELL STRUCTURES (INITIAL BUCKLING)
-
LOADING CASE +'"
T.OAnTN~
nF.~r.RTPTION
INTERACTION
CURVE
JFT~nRF.
EOtTA1'TON
MARG~N-OF-SAFETY
'RF.MARJ(~
FT.A'l' RF.C'.1'ANc:nT.AR r.ORRTTI"!A.TED PANELS
•
rfIJ -----
f t f t'e
t...J
+'" ~
)II
rl• --tit·f'kfi;.!V
1
to
t-t t;Ij
(X)
CURVED
~
Ig: 1
~ ts.
2
~
Z
I:lj
t='
R
s
1.7 = 1
4.23
Shear
PANELS
~fc
til H I"1j I":£j I:lj
+
R + c
, I I J
~
•
Longitudinal Compression
Longitudinal Compression
+ Shear
+R 2 = s c
R
2
4.23 1
R
2 4R 2,-1 +JR s c + c
CIRCULAR CYLINDERS
CIl
'b
8
~ ()
Longitudinal Bending
~
' ..... r- .........
t--3
3
c::
+
'r--- ...... i/ ' .... ~~"" ,........... ~
eltil
-
Longitudinal Bending
...LL
.-3
" .... _""',,!
rnH
+
'r-. -"","
4
t.-«
,'!"0-.............
.-3 ~
'c
r-.r-. ,...,.... ~
5
~~ .... """i--" r-.,_""",~
,..... --~
ROO b
+R
00
s
=
fit
Longitudinal Compression + Torsion
R +R c st
Rb +: Rb
I
1
4.10
(Zero Interaction)
~
c.:,)
4.23
Shear
r- .... "
H
• 1
Torsion
fb
CIl
Rb + Rst
2
Rb-
I
2 2
-
+ 4R st
2
-1
I
or
-Rs1 - 1
1.5 .... 1 4.23
(2)
e
R, R , and R
:z
,..:z
!I:
,..c: r--
NOTES:
(1)
-
C i)
must be based on stiffened structures stability criteria.
c p st For shear-bending analysis of circular cylinders use f = f and fb = fb even though the s smax max locations of the maxima do not coincide. For circular cylinder general instability failure cridilif the allowable trans·~se shear stress assumed~be 0.8 times the allowal~ .. torsional shear str
STRUCTURAL DESIGN MANUAL
m
1.0
00
20 10
)
8
0.8
6 5
0.6 3.5 3
R2
-
2.5
0.4
1.6
0.2
m
m
Rl + R2
1.4 =-1
1.2 1
0
)
0
0.2
0 .. 4
Or6
0.8
1..0
Rl
FIGURE 4.10 GENERAL INTERACTION CURVES
4-35
STRUCTURAL DESIGN MANUAL
,.~ , c
-------;f." ': ":,
-
,',
.' I - -,,';
• FIGURE 4.11. INTERACTION CURVES FOR THICK-WALLED ROUND TUBES BENDING, AND TORSION (REF. TABLE 4.5, CASE 8).
4-36
COMPRESSION,
)
tt
~
v
---
~.-/
ttl[JJ} fs
Cf)
.....
1.0
:=tt
c:
2 Rb t A/
RC
n
0.8
-I
c:
,..-:::a
0.6
=
=
a:
", Cf)
0.'1
-z
a -,
1.0 ..r;:..
O.B
0.6
0.4 Rc
0.2
0.2
o
o
o
0.2
0.4
0.6
0.8
2 Rb + R,/
1.0 .
,.:I: Ie:: ,. Z
r-
•
l,..)
-...J
FIGURE 4.12. INTERACTION CURVES FOR ·THICK-WALLED ROUND TUBES: COMPRESSION, BENDING, TORSION AND SHEAR (R~F. TABLE 4.5, CASE 9).
STRUCTURAL ~DESIGN MANUAL
)
O.S ! - - _...........;L.
""':----
...
a:
O.4~--~-----+----~---~----+-----+-~--~--~~1-~~--~
R
For: Rp
< RSt' use ..,.,J! lOst
with R l
S -+----~----~--~~~--~--~~~~
Rst For: Ru < R • useR with R p p p O.2~--~-----+-----r----~----~----+---~~-+
0.2
0.4
0.6
__~__-4--~~
0.8
1.0
) FIGURE 4.13 INTERACTION CURVES FOR THICK-WALLED ROUND TUBESTENSION, TORSION AND INTERNAL PRESSURE (REF. TABLE 4.5, CASE 10)
4-38
STRUCTURAL DESIGN MANUAL
e
i
T
1 i-
'"
\
I
I-
r-
@...../
.f.. N
~ /
V
a:: - 1.4
~~
~
~
,
l-
1.2
CURVE ~NTERACTION
CD ®
I-
....
.
I
-1.4
R1 + R2
2
1
4.6 4.7 4.6
4.5
12
-1.0
~(3)
lr-
~~
. 1- ~0.4
_GJ -1.2
0.6
.
V / / ~@"~ ~K
\-
14 19
•
~
7
4.6 -I
l- I--
/
~
I.
3,15 ;23
@
I
,
-
REFERENCE TARLE NO. CASE NO.
EQUATION
V
/
0.8
-1.6
~
. ~~
NO.
I
1.6
f-
t-
.
•
....
Io-
r:::::: ~ K~ ~
.
1.8
(
I
0.2
I
I
·0.8
-0.6
.
I . -0.4
-0.2
,\
.
0
o
0.2
0.4
0.6
,
0.8
1.0
) FIGURE 4.14. GENERAL INTERACTION CURVES
4-39
"8
·STRUCTURAL DESIGN MANUAL .... -
--- .
.
alb
r~
L~ ~
~ ~
-....
.........
f-~
'f
tJ...J, i', J "
-4-1.4 •
co
""l (!~
1
I ~,~1.2 ~
r. . =-- - --.
i-'"
~~
/(
~~~~
> u cc
"""" ~
" l'\',I"" ~
,
I I 0.6 ! -'~4 -
f fey f f f , I
...
l":~'~,~ ~
--0.8
t-
cy f '
~
BEB8\~ =B~8fc. , I I I • r
-0.8
-0.6
-0.4
I Rtl(
.l
0
-0.2
-0.2
I J
-
l: ~,.4 1-a:
-
I0.6
.'"
0.2
I
0.4
fty
I
8~8t f I I I
I
m-vb ,,~
", ",
~ ,r ~
l
N
0.6
0.8
l\
L'Rcx
.1.2
i\.
.\ f
ex
-
i---
~
'----
FIGURE 4.15. INTERACTION CURVES FOR FLAT RECTANGULAR PANELS (REF. TABLE 4.6, CASES 1 AND 2)
alb !'-
1
!-
2
l~ ....j4
I~~
~~ ~ --
I
',.0
4-40
1
.~ ...... ""
-0.8 r---~-
2
I .. -1-1 T7-~ I
-
I
3
"- J'tr/-"
"
I
V~__
J~
,
- '1111 -1.0
aI b co, 4
3 4 co '~ I
---,
~
~
tit
"_/
1.4
'
1.2
--+---+--+--1----1---+---+---+---+------1
r-I
~'J: ,-~ ).-1
;;:-
1.0 .........
I alb
I
(I)
-I
:::u c:
O.S
'" 1----+---......- - fc:y
7.c:: '<:
0::
t---+--I---
Et .....
t t t
........
--f-----I
......
......1
t---+--'+---
~~t + * + +~l ......-
___
.....
_
1----+I
f.
0.4
of---
n
..... c:
-. ~ __
~ ~~
i + iTI
0.6 1
1!+!=II
'-I
:
i
__
ED' ~ I ,
b
"
..
......--
-
:=a
_
I""''''''''''~
rf l S
:Da ret
""'l1lI:'"
.."
t f J t t
-
(I) t :i)
z B:
0.2
-0.8
-0.6
-0.4 R
-0.2
tv
.
o
o
0
0.2
0.4
0.6
O.S
Rev
1.0
:Da Z
=
:Da
r-
~
•
~
.....
FIGURE 4.16. INTERACTION CURVES FOR FLAT RECTANGULAR PANELS (REF. TABLE 4.6, CASE 4)
f$ +::-. I .+::-.
f ex
fb~
~~D~
tv
1.0
(I)
......
0.8
'< 'Y"
I
'"
I'" 1-<:
.l',,"~
I~O.Z--l
0.6
~
c: n ...... c: ~
:.:.
)(
I
r-
0.4
..,,-
C
-
(I)
Q2
t.o
0.8
0.6
0.4
0.2
o
o
Rcx
IL
--~~+-~~-+---rl-\rilr\-1'tl~\r1I~\-rI~~~1
0
0.2
0,4
0.5
0.8
to
R,.
~
z
E
~l.,.~'
.:.:.. z c::.:. r-
FIGURE 4.17. INTERACTION CURVES FOR FLAT RECTANGULAR PANELS (REF. TABLE 4.6, CASE 5)
~
cu
a
.)
s-rRUCTURAL DESIGN MANUAL alb = 1.0
alb" 0.8 1.0
0.8
0.8
0.6
0.6
~
~
u:
tt:
')
0.'"
0.4
0.2.
0.2
0
0 0.2
0
•
1.0
------p---...--....--~-..,.----,--.,...__,r__,
0.6
0.4
0.8
0.4
0.6
Rcy
Rey
alb" t.2
alb c 1.6
1.0
1.0
0.8
0.8
0.8
0.6
0.6
I
')(
.D
II:
~
0.4
0.4
)
0.2
0
1.0
0.2.
0.2.
0
0 0
()'2.
0.6
0.4-
0
0.2
0.6
0.8
Re:y
Rcy
FIGURE 4.18.
0.4
INTERACTION CURVES FOR FLAT RECTANGULAR PANELS ,(REF. TABLE 4.6, CASES 6 AND 10)
. 4-43
1.0
.\
~~.~~ ~
STRUCTURAL DESIGN MANUAL
I,f/
alb -3.0
alb· 2.0 1.0
t::~==F:::::r:::L-r-T-r---'-:::::J
1.0
0.8
0.6
0.6
')
lit
11
.D
a::
a:: 0.4
0.4
0.2
0.2
a
o a
0.2
o
0.2
0.4
0.6
0.8
1.0
0.6
0.9
1.0
a
0.2
0.4
0.6
0.8
1.0
1.0
0.8
0.6
0.4
0.2
o 0.4 Rcy
FIGURE 4.18. (CONT'n) INTERACTION CURVES FOR FLAT RECTANGULAR PANELS - (REF. TABLE 4.l C~.sES 6 AND 10)
4 .. 44
.1 ,
STRUCTURAL DESIGN MANUAL alb
alb" 1.1ft
1
1 ,0
c: 1IE"""T"~T"""T'"..,.....T""""T..,.....'T"""'I'""""'r"""T"""'I"""'T""""T"""'1r-T'--y--T"""1
1. 0
1Ir'""T"--y--,-,.-rT""""!-.-"'T'I-.-.--T""'T'-r-.....-,r-r~
0.9
~~-+-t---+---t-+---+--
0. 0
~~:-+-+---+--i--+--+-
0.8
~""k-'~:---t---+---t-~-A--
O. 7
~+Jo,,~~k--+--+~I'h'I~=
0.6
~-f-l'i~~~~~~,f--+---If---+--t
> 0.6
~ri-~~~~~~-Y--+-t--t--i
CJI
a:
~
~ 0.5 ~~r+~~~~~+--+--If---+--t
0.3
)
0.5~~-~~~~~~~-+-+-~~
0.3
~+---+-t-"II:-~IIr"""+.a.r-~~---t~-I---I
0.2
0.2
0.1
0.1
~'f---+-,
0.1 0.20.3 0.4 0.5 0.6 0.7 O.B 0.9 1.0 R
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
ex
Rex
alb .... 2 1.0 1:"""'T..,.....T""T-.-.,.........,....,....~I"""'T""T"""T"""I'...,.....,.....,r--r"'T""l
1 .0
~__'l""""T"""'T""""T""'""!~"""T"""'Ir--"I"'........--T"""'I"'............T"""""I"""T'"~
0.9 ~~~-+--+--I--+---+~~-+--1
0.9
~~~~~"OI:f----tr---I-_+-
0.8
r--+-...:a..too:::---...,....,-P'~~~.-A=
0.8
~~~~R.oc-+---I--t--,~=
alb· 3
O. 7 I---+-..........-+-""rt---""'od~~~'""'" O. 6 > u
a::
>
0.5 ~........-+~t--~-
a:
0.4 .....-+----to.r--fo-
t--~-+---.::~-+-~-
U.5~-+-~~+-~~~~~~~-+~
u
C.3
.....-~-I--'f
0.3
0.2
~-+--
O.2~n-~~+-~~~w-~~~~~
0.1
0.1
o
~-+-~,--+___J_........-----~~Ir-+~~po\Ip___l
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
o
o
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Rex
R
ex
alb • 4 1.0
)
~
a::
_~"r"T..,.....r_r""""i'_""'!""""'T"""'T"""T"""1I"""T'"-r:::---r"-r-r_1
0.9
~~~~~J-l
O.B
I---p..,~---"'~......po~~~~-
O. 7
I--~_+~,j______P'I...-+~..;...;;....~--'r---I-_I
0.6
Ir---+-----...-+--'I...-+--'
f
cv
0.5 ..........Iri---+-~--:= 0.4 I---+-'Ir-+--
0.3
0.2
0.1 0.2 0.3 0.4 O.S 0.6 0.7 0.8 0.9 1.0
G' ex
FIGURE 4.19. INTERACTION CURVES FOR FLAT CASE 7)
~ECTANGULAR
PANELS (REF. TABLE 4.6, 4-45
~ I ~
0'
fey
~
CD
t Jt=tf
< ..... en ..... 0 ::s
...-. ................ s
~~lEJ:~ - - - ........ ---
tJj
f:- '\ ~~ ;;;J -
;:--1111
'--..." \\
'--~
"~ ~CI'~" ,~-
~-
f.
4 albJ == I '"
"en
bx
......
CD
:=tel
1.0
c: c-J
......
c:
O.S
::a
:rO.S
I
l(
I
I I
I
I
ret
..., -
Rex
R. 0
.0
«
(I)
I-
o.i
0.+
.I r
r-to
I
I
J
C i)
0.4
:z
0.6
31:
-
I-
:r:z c: :r-
0.2 t-
1.0
0.8
0.6
0.4-
0.2
o
o
Rey
o
O.S
0.2
0.4
0.6
0.8
1.0
RS
FIGURE 4.20. INTERACTION CURVES FOR FLAT RECTANGULAR PANELS (REF. TABLE 4.6, CASES 8 AND 9)
e
e
e
r-
STRUCTURAL DESIGN MANUAL
e)
fey
_1--+--+---+-'-'
0.4-
§~ ~ f~ ~2 4 I I 444
4--1--1-
alb
-1.0
)
-0.8
·0.6
-t--+-+---+--+-!--+--+--O.61--+-+--I--+--+---+--+--+-I--4--....J.+""'---I
FIGURE 4.21. INTERACTION CURVES FOR FLAT RECTANGULAR PANELS (REF. TABLE 4.6, CASES 12 AND 13)
4-47
STRUCTURAL DESIGN MANUAL
) t.O
La.
I
I
I
"--....
r--....
~
0.8
~
"'"
I
~
". " J.
~
I-
-r--...... r--- r-----.
----
I-
0.6
~
l-
0.94 i-
-
0.4
I-
l-
I
o
I
I
J
I I
L~
I
0.4
0.2
V
I
f'
oJ
-
K
-
'"1\\
-
/
f
/
I-
V -
I
I
V i
J
/R cy -
/
/
I
~
J
I
0.1
I
"-'['\5 r\ .\0.8 \ I W
I
l-
I
I
~
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FIGURE 4.22 INTERACTION CURVES FOR FLAT RECTANGULAR PANELS. (REF. TABLE 4.6, CASES 16 AND 18)
.. 4-48
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STRUCTURAL DESIGN MANUAL
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FIGURE 4.23. UNSYMMETRICAL GENERAL INTERACTION CURVES
4-49
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STRUCTURAL DESIGN MANUAL
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I
2.0
R, FIGURE 4.24. SPECIAL INTERACTION CURVES
A
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2.0 1,8
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0.4 0.2
-1.0
4-50
·0.8
·0.6
-0.4
-0.2
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FIGURE 4.25. LINEAR INTERACTION CURVES.
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STRUCTURAL DESIGN MANUAL SECTION 5 MATEPIALS 5. L GENERAL
)
This section describes the materials corrunonly used at Bell Helicopter. The forms, temper designations, properties, design limiations and testing methods are discussed. The information contained herein is in agreement with MIL-HDBK-S and MIL-HDBK-I7. For information on materials not shown in this section the appropriate material specification or the previously referenced handbooks should be consulted. 5.1.1 Material Properties The physical properites of some materials are shown in Table 5.1. This is a summary of commercially pure eJ ements- and is presented for comparison only. Primary strength properties of metallic materials are minimum values at room temperature, established on an A, B, C or S basis. Elongation and reduction in area properties presented in the referenced handbook property tables are minimum values at room temperature, established on an A or S basis. Elongation and reduction in area at other temperatures, as well as elastic properties (E, Ec, G and IJ), physical properties (w, C, K and a), creep properites and fracture toughness properties are average unless otherwise specified. A Basis - The mechanical-property value indicated is the value above which at least 99 percent of the population of values is expected to fall, with a confidence of 95 percent. B Basis - The mechanical-property value indicated is the value above which at least 90 percent of the population of values is expected to fall, with a confidence of 95 percent.
)
S Basis - The mechanical-property value indicated is usually the specified m1n~ mum value of the appropriate Goverrunent·specification, or SAE Aerospace Material Specification for this material. The statistical assurance associated with this value is not known. C Basis - The mechanical property value indicated is the value developed by tests at Bell Helicopter.
5.1.2 Selection of Design Allowables Specification MIL-S-8698 contains the general requirements which in combination with specific model detail specifications set the requirements for structural design, analysis and test of helicopters. It is required by these specifications that minimum guaranteed values (A values) be used with nominal dimensions. Nominal is the average between tolerances. The use of B values must be approved by the procuring agency. If conditions such as crash, rollover or impact produce loads in excess of maneuver design loads, consideration should be given._to requesting permission for use of B values.
5-1
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STRUCTURAL DESIGN MANUAL Revision C ~
MATERIAL
.
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ALUMINUM BERYLLIUM CADMIUM CALCIUM CHROMIUM COBALT COPPER GOLD GRAPHITE INCONEL IRON LEAD MAGNESIUM MANGANESE MERCURY
MOLYBDENUM NICKEL PLATINUM SILVER trIN IrITANIUM TUNGSTEN URANIUM VANADIUM ZINC
27 9 112 40 52 59 64 197 12 56 56 207 24 55 201 96 59 195 108 119 48 184, 238 51 65
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0
co
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C:::::!-...,,;
.098 .. 067 .. 313 .. 056 .258 .322 .323 .697 .081 .300 .284 .410 .063 .269 .489 .369 .320 .775 .379 .264 .163 .697 .676 .205 .258
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1215 2345 608 1564 2822 2696 1981 1945 6740 2600 2795 621 1204 2268 -38 4748 2646 3224 1761 449 3272 6098 3075 3110 787
3733 5036 1411 2719 4500 5252 4703 5371 8730
1Z..e'
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-
5438 3171 2025 3904 675 8677 5252 7932 3634 4118 9212 10701 6332 5432 1665
168 581 24 143 .J8
107 92 29
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119 11 118 118 5
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135 50 44 26
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80
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2912
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.055 .157 .12 .099 . 092 . 031 .17 .109 .108 .03 .249 .107 .033 .065 .112 .032 .056 .548 .142 .034 .028 .115 .093
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Ul
1215
-924
924 5169 3959 1800 200 1064 1652 2106
-
112 1938 784 932 5454 879 223 2666
-
373
1501
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13.1 6.3 27.8 12.9 4.2 6.2 8.5 7.4 1.2 6.2 6.1 15.2 13.2 11.7
2.66 18.5 7.6 4.6 13.1 9.7 1.7 2.4 78.0 98.0 9.8 20.7 4.5 185 95.8
10 42 10
2.8 7.1 4.5 9.} 11.5 3.7 2.3
4.8
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-
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-
..
5 23
6.9 9.8 1.6 11 .. 5 3 .. 0 5.5 60.0 26.0 6.2
-
36 30 16 12 •7 31 29.7 2.6 6. 3 23
-
30. 24 .. 2 11.2 4.0 15.5 )1.4
-
19 17.0
TABLE 5.1 - COMPARISON OF PHYSICAL PROPERTIES OF ELE.MENTS
5-2 /
/
STRUCTURAL DESIGN MANUAL· C values must also be approved by the procuring agency. These values should be developed in accordance with the procedure outlined in MIL-HDBK-5 for allowable development. Regardless of the values used the structure must sustain static ultimate load without failure. The use of any value in excess of A values does not justify static test failures.
)
5.1.3 Structural Design Criteria It is the responsibility of the Airframe Structures group to prepare a Structural Design Criteria report for each model helicopter. This document is based on the requirements of the detail specification for the helicopter. It contains the structural criteria to which the helicopter will be designed and tested. This report shOUld specify the materials, limitations and allowable basis. 5.2 MATERIAL FORMS The metallic materials most cmnmonly used at Bell Helicopter come in a variety of forms. They are available as pla'te, sheet, bar, extrusion, forging and casting. The selection of a form for a particular part should be based on material properties, machining costs, compatibility with part shape and manufacturing methods, total manufacturing cost and availability. All of the rolled, drawn, extruded and forged forms exhibit anisotropic properties. These properties often differ considerably along the principal axes particularly in forgings. The directional characteristics are produced by moving the material during forming. Castings, since they are formed in the molten state, do not exhibit these directional properties. The anisotropic properties are defined as longitudinal (L), long transverse (LT) and short transverse (ST). These terms define the direction the grain is formed within the material. Figure 5-1 shows typical examples of grain directions in various forms of materials. 5.2.1 Extruded, Ro.lled and Drawn Forms
)
These forms are produced by rolling and by forcing or drawing metal through dies to the proper shape. In these processes, the grains are elongated in the direction of extrusion or rolling and are parallel to the longitudinal direction of the finished product as it comes from the mill. These processes can be used to make many different products. Some common products and forming methods are defined as follows: Sheet - rolled flat products less than .250 inch thick (most materials) Clad -
a thin coating of metal bonded to the base alloy by cold rolling. The purpose is to improve the corrosion resistance of the basic metal. Thickness of the clad is generally 2~ - 5% of the total. thickness per side.
Plate - rolled flat products generally .250 inch or greater in thickness.
5-3
~ STRUCTURAL DESIGN MANUAL
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= LONGITUDINAL
B
= LONG
C
= SHORT
SHEET, PLATE, RECTANGULAR BAR, HAND FORGING
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TRANSVERSE TRANSVERSE
DIE FORGING
PARTING PLANE --7 / /
BAR (EXTRUDED, FORGED, DRAWN, ROLLED, WIRE)
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ROUND SQUARE HEX
TUBING
FIGURE 5.1 5-4
EXAMPL~S
OF GRAIN DIRECTIONS IN VARIOUS MATERIAL FORMS
STRUCTURAL DESIGN MANUAL Extrusion - formed by forcing metal at elevated temperature through a die. Stepped extrusion - a single product with one or more abrupt changesin cross section. Impact extrusion - a product formed by a punch striking unheated metal in a confining die and consequently extruded through an opening or around the punch. Hot impact extrusion - the same as the previous impact extrusion except the metal is preheated. Drawing - forming the cross section by pulling it through a die. Wire - a drawn form with a diameter or width across the flats of less than .375 inch. Rod - a drawn form with a diameter greater than .375 inch. Bar - a drawn form with a width across the flats greater than .375 inch. 5.2.2 Forged Forms
)
Forged forms are produced by impacting or pressing the material into a predetermined configuration. Pre-forge stock has generally been pressed, rolled, hammered or extruded to produce a well wrought material. Pressure is then applied to force the pre-forge stock into the desired shape. Pre-forms and multiple dies are often necessary to produce the desired configuration and tolerances in die forgings. The mechanical properties of forgin~are maximum parallel to the grain flow. This is the primary advantage of a forging. Parts should be designed to fully utilize these properties. However, the disadvantage of a forging is that the mechanical properties transverse to the grain flow and parallel to the compression forces exerted on the material are minimum. These are called the short transverse properties and the part should be designed to minimize the occurrence of areas where short transverse can occur. Terms normally used to describe forgings are as follows: Forging - plastically deforming metal into desired shap'es. not be used.
Dies mayor may
Hand forgings - a product formed by hot working the material, usually between simple flat dies, into the desired shape. This process requires the least expensive dies but most expensive machining. These forgings come in two types: Hand forged billet - a-forged block with basically unidirectional properties. Must be completely machined to shape. Shaped hand forgings - a product forged into a shape that generally outlines the basic contour of the desired final configuration.
5-5
STRUCTURAL DESIGN MANUAL
e -"-
Blocker die forgings - the final forging res~mbles the machined configuration and is generally machin~d on all surfaces. Approximately .25 to .50 inch excess material is allowed on all surfaces of the blocker forging depending on part size, configuration and material. Low die cost. High material cost. Conventional die forgings - made in caosed dies. The final forging more closely resembles the machined part than the blocker die forging. Usually some lias forged" surfaces are left on the finished part. Medium die cost, __ m~dium machine cost. Close-to-form die forging - the forging approximates the finished part as closely as possible. Machining is minimized and most surfaces are used ttas forged". Highest die cost. Lowest machine cost. Flash - metal which is extruded through the space be.tween dIe halves along the parting line. Mismatch - the die halves will not match perfectly. at the parting line.
The offset of the dies
As Forged - the term "as forged" is used to describe the surfaces that are not machined. Forged surfaces of steel and titanium are usually chern-milled to remove decarburization and alpha case but are still referred to as "as forged" surfaces. All forgings produced at Bell Helicopter must satisfy the requirements of Bell Process Specification BPS FW40l7. This specification must be shown on the drawing. 5.2.3 Cast Forms A casting is a product made by pouring molten material into a mold of predetermined shape. The material is allowed to solidify and then removed from the mold. There are generally four types of castings available: Sand casting - a mold of compacted sand is made for each individual part to be poured. The mold is broken away from the part after solidification. Permanent mold casting - a mold of high strength steel alloy is most commonly used. The mold is reusable. Castings produced in these molds usually yield high quality parts due to the chilling action of the mold and core. Investment casting - a mold of plaster is formed around an expendable or plastic pattern. The pattern is burned out of the mold leaving the desired cavi ty into which the mol ten met.al is poured. More intricate parts can be cast by this method. Die casting - molten metal is injected into metal dies under pressure. Die castings are not used in structural applications. 5-6
)
STRUCTURAL DESIGN MANUAL Casting strengths are not significantly directional but they do vary from part to part, lot to lot, and even within the individual part. This is usually caused by cooling rate differences due to thickness changes. It can also be caused by poor manufacturing techniques; such as improper location of chill blocks, risers and gates. Cooling rate can be controlled by use of chill blocks which tend to improve the structural quality of the casting. Bell Process Specification BPS FW4163 should be specified on the drawing of all castings except die castings. The drawing should also contain the casting classification. The Structures Engineer should make sure the stress analysis contains the casting factors specified in the Structural Design Criteria and check to see that the casting is properly classified according to one of the following: c'lass IA - A class IA casting is a casting, the single failure of which would result in the loss of the aircraft, one of its major components, or loss of control. Class IB - Class IB includes all critical castings which are not included Class IA and which would cause unintentional release of or inability to release anyarmament/store, failure of gun installation components, or failure of which may cause significant injury to occupants of the aircraft. Class IIA - Class IIA castings are those not included in either Class IA or Class IB, which have a margin of safety of 200 percent or less (MS ~ 2 .. 0). Class lIB - Class lIB castings are those castings having a margin of safety greater than 200 percent (MS > 2.0), or those for which no stress analysis is required .. The Structures Engineer is responsible for insuring that an X-ray diagram is shown on the drawing. All engineering drawings for Classes IA, IB, and IIA shall contain an X-ray diagram. This diagram shall contain the following information:
)
(a) Areas of casting that mayor may not be weld repaired per BPS 4470 when applicable. (b) Location of critically stressed areas. (The critically stressed areas shall be shown by encircling those areas with phantom lines.) (c) Designate X-ray views required. The X-ray laboratory shall specify the required views and shall initial the diagram if acceptable for X-ray views. The Structures Engineer shall make certain that the Structural Materials Group and the X-Ray Laboratory have approved the. casting drawing. This shOUld be accomplished before the Structures Engineer approves the casting drawing.
5-7
STRUCTURAL DESIGN MANUAL Casting materials commonly used at Bell Helicopter are steel and aluminum. Although magnesium is readily castable. its properties vary significantly. Magnesium should not be used without the approval of the procuring agency_ 5.3 ALUMINUM ALLOYS Aluminum is a lightweight structural material that can be strengthened through alloying and, dependent upon composition, further strengthened by heat treatment and/or cold working. Among its advantages for specific applications are: low density, high strength-to-weight ratio, good corrosion resistance, ease of fabrication and diversity of form. Wrought and cast alloys are identified by a four-digit number, the first digit of which generally identifies the major alloying element as shown in Table 5.2. For casting alloys, the fourth digit is separated from the first three digits by a decimal point and indicates the form, i.e., casting or ingot. Alloy Number lXXX 2XXX 3XXX 4XXX
5XXX 6XXX
7XXX BXXX 9XXX
Wrought Alloys Major Alloying Element 99°/.. Min. Aluminum Copper Manganese
lXX.X 2XX .• X 3XX.X
Silicon Magnesium Magnesium and Silicon Zinc Other Elements Unused series
4XX.X 5XX.X 6XX.X 7XX.X BXX.X
TABLE 5.2 5.3.1
Cast Alloys Alloy Number
9XX·.X
Major Alloying Element 997.. Min. Aluminum Copper Silicon with added copper and/or magnesium Silicon Magnesium Unused series Zinc Tin Other Elements
BASIC DESIGNATION FOR WROUGHT AND CAST ALUMINUM ALLOYS
Basic Aluminum Temper Designations
The temper designation appears as a hyphenated suffix to the basic alloy number. An example would be 7075-T73 where -T73 is the temper designation. Four basic temper designations are used for aluminum alloys. They are -F: as fabricated; -0: annealed; -H: strain hardened and -T: thermally treated. A fifth designation, -W, is used to describe an as-quenched condition between solution heat treatment and artificial or room temperature aging. Following is a list of tempers which define aluminum alloys.
5-B
)
STRUCTURAL DESIGN MANUAL -0:
annealed.
Applies to wrought products which are fully annealed ..
-F: as fabricated: No special control over thermal conditions or strainhardening is employed. For wrought products, there are no mechanical property limits.'
)
.)
-H: strain-hardened (wrought products only). Applies to products which have their strength increased by strain-hardening, with or without supplementary thermal treatments to produce some reduction in strength. The -H is always followed by two or more digits. The first digit following the H indicates the specific combination of basic operations as follows: -Hl: strain-hardened only. Applies to products which are strain-hardened to obtain the desired strength without supplementary thermal treatment. The number following this designation indicates the degree of strain-hardening. The numeral 8 has been assigned to indicate tempers having an ultimate tensile strength equivalent to that achieved by a cold reduction of approximately 75 percent following full anneal. Tempers between 0 and 8 are designated by numbers 1 through 7. Materials having an ultimate tensile strength about half way between that of the 0 temper (annealed) and the 8 temper are designated by the number 4; about midway between 4 and 8 by 6; and midway between o and 4 by 2. Any of the odd number designations can be obtained in the same manner, i.e., midway between the adjacent designations. The following generally defines the two-digit tempers: -HlO: -H12: -H14: -H16: -H18:
annealed strain-hardened strain-hardened strain-hardened strain-hardened
to to to to
1/4 hard 1/2 hard
3/4 hard full hard
-H2: strain-hardened and partially annealed. Applies to products which are strain-hardened more than the desired final amount and then reduced in strength to the desired level by partial annealing. The second digit indicates the same as in the -Hl tempers. Temper -H24 would be strain-hardened and partially annealed to 1/2 hard.
)
-H3: strain-hardened and stabilized. Applies to products which are strainhardened and then stabilized by a low-temperature thermal treatment to irtcrease ductility and prevent stress corrosion (applies only to alloys containing magnesium). The same second digit rules as described for -HI tempers apply here also. The third digit, when used, indicates a variation of the two-digit temper to which it was added. The minimum ultimate tensile strength of a three-digit H temper is at least as close to that of the corresponding two-digit H temper as it is to the adjacent two-digit H tempers.
5-9
STRUCTURAL DESIGN MANUAL -HIll: Applies to products which are strain-hardened less than the amount required for a controlled Hil temper. -HI12: Applies to products which acquire some temper from shaping processes not having special control over the amount of strain-hardening or thermal treatment, but for which there are mechanical property limits. The following H temper designations have been assigned for wrought products in alloys containing over a nominal 4 percent magnesium.
-H311: Applies to products which are strain-hardened less than the amount for a controlled H31 temper. -H321: Applies to products which are strain-hardened less than the amount for a controlled H32 temper. -H323: Applies to products which are specially fabricated to have acceptable resistance to stress corrosion cracking. Products which are thermally treated with or without supplementary strainhardening are designated with a -T temper. The T is followed by a digit or digits which designate the specific thermal treatment. Temper designations for aluminum alloys are as follows: -Tl: Cooled from an elevated temperature shaping process and naturally aged to a substantially stable condition.
-T2:
Annealed (cast products only).
-T3: Solution heat treated and then cold worked. Applies to products which are cold worked to improve strength or in which the effect of cold work in flattening or straightening is recognized in mechanical property limits. -T31: Solution heat treated and then cold worked by flattening or stretching. Applies to 2219 and 2024 sheet and plate per MIL-A-8920. Also applies to rivets driven cold immediately after solution heat treatment or cold storage. 2024 rivets are an example. -T351: Solution heat treatment and stress relieved by stretching. This is equivalent to -T4 condition. It applies to 2024 plate and rolled bar and 2219 plate per MIL-A-8920. -T3511: Solution heat treated and stress relieved by stretching with minor stretching allowed. This is equivalent to -T4 condition and applies to 2024 extrusions. -T36: Solution heat treated and then cold worked by a reduction of 6 percent. Applies to 2024 sheet and plate.
-T37: Solution heat treated and then cold worked by a reduction of 8 percent. Applies to 2219 sheet and plate.
. 5-10
STRUC1·URAL DESIGN MANUAL -T4: Solution heat treated and naturally aged to a substantially stable condition. Applies to products which are not cold worked after solution heat treatment, or in which the effect of cold work in flattening or straightening may not be recognized in mechanical property limi ts. -T42: Solution heat treated and naturally aged by the user to a substantially stable condition. Applies to 2014-0 and 2024-0 plate and extrusions which are heat treated by the user from the annealed condition. -T451: Solution heat treated and stress relieved by stretching. Equivalent to -T4 and applies to plate and rolled bar stock except 2024 and 2219. -T45l1: Solution heat treated and stress relieved by stretching with minor straightening allowed. Equivalent to -T4 and applies to all extrusions except 2024 and 2219. -T5: Cooled from an elevated temperature shaping process and then artificially aged. -T5l: Cooled from an elevated temperature shaping process, stress-relieved by stretching and then artificially aged. -T52:
Cooled from an elevated temperature shaping process, stress-relieved
by compressing and then artificially aged.
-T54: Cooled from an elevated temperature shaping process, stress-relived by stretching and compressing and then artificially aged. Applies to die forgings which are stress-relieved by restriking cold in the finish die. -T6: Solution heat treated and then artificially aged. Mechanical property limits not affected by cold working. Most alloys in the -Wand -T4 conditions artificially aged to -T6.
)
-T61: Solution heat treated and then artificially aged. Applies to forgings which receive a boiling water quench to avoid internal quenching stress. Applies to solution heat treated and artificially aged castings when more than one aging cycle is available for that alloy. -T61l: Solution heat treated and artificially aged. forgings which are quenched in 175 0 to 1850F water.
Applies only to 7079
-T62: Solution heat treated and then artificially aged by the user. Applies to any temper which has been heat treated and aged by user which attains mechanical properties different from those of the -T6 condition. -T65l: Solution heat treate~, stress-relieved by stretching and artificially aged. Equivalent to -T6 and applies to plate and rolled bar except 2219.
5-11
s-rRUCTURAL DESIGN MANUAL -T6510: Solution heat treated, stress-relieved by stretching and artificially aged with no hard straightening after aging. Applies to extruded rod, bar and shapes except 2024. -T65l1: Solution heat treated, stress-relieved by stretching and artificially aged with minor straightening. Equivalent to -T6 and applies to extruded rod, bar and shapes except 2024. -T652: Solution heat treated, stress-relieved by compressive defonnation and artificially aged. Equivalent to -T6 and applies to hard forged squares, rectangles and simply shaped die forgings except 2219. -T7: Solution heat treated and then stabilized. Applies to products which are stabilized to carry them beyond the point of maximum strength to provide control of growth and residual stress. -T73: Solution heat treated and then specially artificially aged. Applies to 7075 alloys which have been specially aged to make the material resistent to stress-corrosion. -T7351: Solution heat treated and specially artificially aged. Applies to 7075 alloy sheet and plate which have been specially aged to make the material resistant to stress-corrosion. -T735ll: Solution heat treatment and specially artificially aged. Applies to 7075 alloy extrusions which have been specially aged to make the material resistant to stress-corrosion. -T7352: Solution heat treated and specially artificially aged. Applies to 7075 alloy f~rgings which have both compression-stress relief and special aging to make the material resistant to stress-corrosion. -T8: Solution heat treated, cold worked and then artificially aged. Applies to products which are cold worked to improve strength, or in which the effect of cold work in flattening or straightening is recognized in the mechanical property limits. -T81: Solution heat treated, cold worked and then artificially aged. Applies to 2024-T3 artificially aged to T-8l. -T85l: Solution heat treated~ stress-relieved by stretching and artificially aged. Applicable to plate, rolled bar and rod. -T851l: Solution heat treated, stress-relieved by stretching and artificially aged. Applies to 2024 extrusions and 2219. -T86: Solution heat treated, cold worked by a thickness reduction of 6 percent and then artificially aged. Applies to 2024 sheet and plate. -T87: Solution heat treated, cold worked by a thickness reduction of 10 percent and then artifically aged. Applies to 2219 sheet and plate.
5-12
~"
"'-III1IIIII!!!!!I!!II!-Bell .WJ:tK:7
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STRUCTURAL DESIGN MANUAL
.~
-T9: Solution heat treated, artificially aged and then cold worked. to products which are cold worked to improve strength.
Applies
-TIO: Cooled from an elevated temperature shaping process, artificially aged and then cold worked. Applies to products which are artificially aged after cooling from an elevated temperature shaping process, such as casting or extrusion and then cold worked to further improve strength. 5.3.2 )
Aluminum Alloy Processing
The processes through which various aluminum alloys must be subjected to achieve a particular temper are shown in Figures 5.2 through 5.4. 5.3.3
Fracture Toughness of Aluminum Alloys
Typical values of plane-strain fracture toughness, KIC' for several aluminum alloys are shown in Table 5.3. These are average values for the alloys and tempers for which valid data are available and are thus representative of the various products. They do not have the statistical reliability of the room temperature mechanical properties shown in subsequent sections. 5.3.4
Resistance to Stress-Corrosion of Aluminum Alloys
The high strength heat treatable wrought aluminum alloys in certain tempers are susceptible to stress corrosion to some degree, dependent upon product, section size, direction and magnitude of stress. These alloys include 2014, 7075, 7079 and 7178 in the T6 tempers and 2014, 2024 and 2219 in the T3 and T4 tempers. Other alloy temper combination~ notably 2024 and 2219 in the T6 or T8 tempers and 7049, 7075 and 7175 in the T73 tempers, are decidedly more resistant and sustained tensile stresses of 50 to 75 percent of the minimum yield strength may be permitted without concern about stress-corrosion cracking. The T76 temper of 7075 and 7178 provides an intermediate degree of resistance to stress-corrosion cracking, i.e., superior to that of the T6 temper, but not as good as that of the T73 temper of 7075. A measure of the degree of susceptibility of various products of these alloys and tempers is given in Table 5.4.
)
Where short times at elevated temperatures of 150 0 to jOCpF may be encountered, the precipitation heat-treated tempers of 2024 and 2119 alloys are recommended over the naturally aged tempers. Alloys 5083, 5086 and 5456 should not be used under high constant applied stress for continuous service at temperatures exceeding 150 0 F, because of the hazard of developing susceptibility to stress corrosion cracking. In general, the H34 through H38 tempers of 5086 and the H32 through H38 tempers of 5083 and 5456 are not recommended, because these tempers can become susceptible to stress corrosion cracking.
5-13
STRUCTURAL DESIGN MANUAL
ALL PRODUCTS ALL TEMPERS
SHEET, PLATE, WIRE, ROD, BAR, EXTRUSIONS, TUBE
PRODUCER ANNEALED
PRODUCER SOLUTION HEAT TREAT
USER SOLUTION HEAT TREAT
FLAT SHEET AND PLATE STRESS RELIEVE STRETCH MINOR HAND STRAIGHTENING
PLATE, BAR AND ROD
ROLLED SHEET, ROD, BAR STRETCHED, STRAIGHTENED AND ROLLED
EXTRUSIONS, PLATE, SHEET ROLLED ROD AND BAR TUBE
6% COLD
WORKED NATURAL AGING
ARTIFICIAL AGING
Figure 5.2--Temper Processing Chart for 2024 Alloys 5-14
COLD WORKED
STRUCTURAL DESIGN MANUAL
ALL PRODUCTS ALL TEMPERS
SHEET. PLATE, ROD, BAR, EXTRUSIONS, TUBES, FORGINGS
PRODUCER ANNEALED
PRODUCER SOLUTION HEAT TREAT
USER SOLUTION HEAT TREAT
STRETCHED, STRAIGHTENED, OR ROLLED
COMPRESSION STRESS RELIEVE
NATURAL AGING MINOR. HAND STRAIGHTENING ALL PRODUCTS
PLATE, BAR, ROD, TUBE
EXTRUSIONS
EXTRUSIONS
ALL PRODUCTS
FORGINGS
)
ARTIFICIAL AGING
Figure 5.3--Temper Processing Chart· for 6061 Alloys
5-15
STRUCTURAL DESIGN MANUAL
ALL PRODUCTS ALL TEMPERS
PRODUCER SOLUTION HEAT TREAT
PRODUCER ANNEALED
USER SOLUTION HEAT TREAT
FORGINGS EXTRUSIONS ROLLED ROD AND BAR
STRESS RELIEVE STRETCH
SPECIAL ARTIFICIAL AGING ARTIFICIAL AGING
PLATE BAR AND ROD < 2 IN
ALL PRODUCTS EXCEPT FORGINGS AND ROD-BAR < 2 IN
FORGINGS
COMPRESSION STRESS RELIEVE STRETCHED, STRAIGHTENED, OR ROLLED
ARTIFICIAL AGING
')
Figure 5.4--Temper Processing Chart for 7075 Alloys 5-16
~
> '"" ttl
--
\,-,,'
Plane-Strain
trl
(,JJ
.-3
r:: ~ H
c:: ~
I:"L1
(')
Alloy
> ~
2014
<:
2024
,",,')I-
C/) C/)
0
1'71
» t""'4
en
0
~
:;tI
2219
~
7075
t-:l
Z
""
M
~ ~ '"" E:: trj ~ t""'4
0
r<
en
"t:I
,....... ~ z :;x1
trj -
'"%j
7079
1 - 2
3
2 - 6
4
3/4
Plate
T3S1
.. --
1-1/2.
Shapes
T3510,1
Plate Extruded Shapes Forgings ' Plate
T8S1 T8510,1 T852
~~
1 - 2
3/4- 4 2. - 6
4 5 5
1-3/8 3/4
TB5l T37
1 - 2 3/4- 1
2 '3
Plate Extruded Shapes Forgings
T651 T6510,1 T652
1/2- 2
6
1/2- 4 2 - 6
10 2
1-3/8 3/4 1/2 1/2 1/2
Plate
T7351
1
1-3/8
Extruded Shapes Forgings
T73510,1
1-3/8 l/Z- 4 1 - 5
2
5/8
31
5
3/4
27
Plate
T651 T652
1 - 3
3
2 - 6
2
3/4
22
Plate
Extruded Shapes
T7352
3/4
1
T651 T6510,1
1/2- 2 1/2- 1-1/2
3
1
1
1
1/2- 2 1/2- 2
3 5
T7671
T7650,1
-
22 22 23
31 2.6
1/2 1/2
Avg
Max
No. ' of' Thick. Lots Inch Min
23
24
4
29
34
4
1
3/4
23
25
28 26
32
2 4
30
4
1-3/8 3/4 3/4
33 27
36 28
---
2
1
19 19
No. of Avg
21 23
.-.
20
30
2
26
-~~
23
---
4 I
25 22 23
29 24
33
25
28 26
24
24 23
24 • 25
19
21
16
19
23 20
1
22 28
1 1
33 31.
3S
3
27
29
30
30 31
2
28
23 25
26
4
1/2
11
4
1/2
30 31
2 3
1/2 5/8
22
22
18
22
29 29
1/2 1/4
23
22
2 5
3/4
l/Z
22
--34
1
3/4
20
30
1/2 3/4
1 I
19
~--
26
1/4
30
':8
21
16 19
3
29
26
15 18
. --
1
20
4
23
1
18
10 1
20
---
17
27
18 19
2
---
17
2. 1
(I)
--18
1/2 1/2
1
.--
1/2
19
1/2 1/2 1/4 1/2
1
3 1
--- ---- - ---17 --16 16 --- 2.0 --- 20
1/2 1
3
1 1/2
1/2
~CI
---19
---17 ---------
11
20 20 21
16 18
IS 14
17 16
Max
-.-
19 26
-------17 ----18 22
--2.1 ---25 -18 --
-I
::a
c:: n
..... c: ::a ,.,.. r-
et
"'
---
-z
-----
:r: ,.,..
(I) c :i)
-~-
z <=
•..... ,
I...n
...., i-"
/,,'
,~-~ i-
Avg
1
20 18
32
1
\{
Thick. Inch Min
2.
20 16
26
26
23 30
------- ---
28
3
Max lLots
--- --- --1/2 1/2 1/2
, ''", -~' .- _/1/
(ST)(L)
25 26 24
---
.,/'in:
Tougnness, KIc ' ks1
(T)(L)
----- ----- ----- 46 ---
1-1/2
Plate
H
2225
2.
Extruded Shapes
Z
1-1/2 1-1/2-
1
2.
C'll
0'~
~ots
Thick. Inch Min
T651
en
•
No. of
T652
Forgings 7178
Temper
Product Thic.kness Range, Inch
Plate Forgings
0
~ ~
1-4
Product
E~truded
0
§ 8 t'!'j ~ t'!'j
Frac~ure
(L)(T)
•l.U
() ~
-.,.,
--
t""'4
"Zj
e,
.~
STRUCTURAL DESIGN MANUAL Revision E Estimate of Highest Sustained Tension Stress (ksi) at Which Test Specimens of Different Orientations to the Grain Structure Would Not Fail in the 3~% NaCl Alternate Immersion Test in 84 Days
Alloy and Type of Temper 2014-T6
Test Direction L
LT ST 2219-T8
L
LT
ST 2024-T3, T4
2024-T8
7075-T76
7079-T6
7178-T6
50 27
45 22 8
30 25 8
·......
35 35
35 35 35
38 38 38
50 37
50 18
·.
45 30 8
45
40 38 38
·15.
·..
·.
·.
·.
30
8
10
·.
L
50 50 30
47
60 50
43
43
·.
60 50 45
L LT ST
50 45
50 .. 15
60 50
60 32 8
35 25 8
L
49 49 25
·.
52 49 25
·. ·.. ...
·.. ...
50 48 43
54 48 46
53 48 46
50 48 43
... ·..... ...
60 50 ..
60 35 8
50
65
65 25 8
·.. ..
8
LT ST
50 48 43
L
L
55
LT 8T
40 8
L
55 38 8
LT ST 7178-T76
Hand Forgings
35 20
LT ST 7075-T73
Plate
Extruded Shapes Section Thickness, Inch 0.25-1 1-2
L LT ST LT 8T
7075-T6
I
Rolled Bar and Bar
L
LT 8T
52 52 25
... "
.
·
·.. ·..
·..... .. ..
·....
·.
.
45 ..
.
55 52 25
8
·.. .. · ..
·.
43 15
·..
30
8
·. .. "
·. ••
TABLE 5.4--COMPARISON OF THE RESISTANCE TO STRESS CORROSION OF VARIOUS ALUMINUM ALLOYS (REF. 1) 5-18
•
•
'
STRUCTURAL DESIGN MANUAL In the recommended tempers, Hll3 and H321 for sheet and plate, cold forming of 5083 and 5456 should be held to a minimum radius of 5T. Hot forming of the 0 temper alloys 5083 and 5456 is recommended and is preferred for the Hll3 and H321 tempers in order to avoid excessive cold work and high residual stress. If the HI13, H321, H323 and H343 tempers are heated for hot forming a slight decrease in mechanical properties, particularly yield strength may result. In order to avoid stress-corrosion crackinB, practice~ such as the use of press or shrink fits; taper pins; clevis joints in which tightening of the bolt imposes a bending load on lugs; and straightening or assembly operations; which result in sustained surface tensile stresses, should be avoided in these alloys: 2014-T451, T4, T6, T651; 2024-T3, T35l, T4; 7075-T6, T651, T652 and 7178-T6 and T651. Where straightening or forming of heat-treated materials is necessary, it should be performed when the material is in the freshly quenched condition, or at an elevated temperature 'to minimize the residual stresses induced. Where elevated temperature forming is performed on 2014-T4, T45l or 2024-T3, T351, a subsequent precipitation heat treatment to produce the T6 or T651, T8l or T851 temper is recommended. Specific guidance on safe stress levels for avoiding stress-corrosion cracking is shown in Table 5.4. These stresses represent the algebraic sum of all the continuous tension and compression surface stresses resulting from any source such as quenching, forming, assembly and design. These stresses should be kept below those given in Table 5.4. It is particularly important to consider clamp-up stresses and pressfit stresses. If stress levels cannot be kept within the Table 5.4 limits t the Airframe Structures Group Engineer should be consulted. 5.3.5 Mechanical Properties of Aluminum Alloys The mechanical properites of aluminum alloys are specified in MIL-HDBK-S. description of some common aluminum alloys follows:
)
A
2014 is an AI-Cu alloy available in a wide variety of product forms. It is useful for application over the range from cryogenic to elevated temperatures'. Resistance to stress corrosion is discussed in Section 5.4.4. 2024 is a heat-treatable AI-Cu alloy which is available in a wide variety of product forms and tempers. The T3 and T4 tempers have high toughness while the T6 and TB tempers have high strengths. The T6 and T8 tempers also offer good resistance to stress corrosion cracking, while the T3 and T4 tempers should be considered in light of the guidelines in Section 5.3.4. 2024 alloy is not weldable by commercial practices.
5-19
STRUCTURAL DESIGN MANUAL 5052 is a low-strength AI-Mg alloy. It is very ductile and very readily welded and is usually used in applications where these characteristics are more important than strength. It is highly resistant to corrosion. It is extremely tough at low as well as room temperatures. 6061 is a very readily weldable AI-Mg-Si alloy available in a wide range of product forms. It has high resistance to corrosion. 7075 is a high-strength AI-Zn-Mg-Cu alloy and is available in a wide variety of product forms. It is available in several types of tempers, the T6, T73 and T76 types. T6 has the highest strength and lowest toughness, and is susceptible to stress-corrosion cracking. Since toughness decreases with a decrease in temperature, the T76 temper is not generally recommended for cryogenic applications. The T73 temper has th~ lowest strength, but is relatively tough and very resistant to stress-corrosion cracking and exfoliation attack. The T76 temper is a compromise providing higher strength than the T73 temper and higher resistance to stress corrosion than the T6 temper.. 7075 is not corrunercially weldable. 201.0 is a high-strength, heat-treatable Al-Cu-Ag casting alloy. It is readily weldable. In the T6 (aged) temper, it possesses high strength and good ductility, but is not recommended for use in environments conducive to stress~corrosion cracking. In the T7 (over-aged) temper, it possesses a high strength and moderate ductility and optimum resistance to stress-corrosion cracking. 224.0 is a heat-treatable Al-Cu-Zr casting alloy. When solution heat treated and over-aged, it possesses excellent mechanical properties at elevated temperatures, good fatigue properties and toughness. 295.0 is a heat-treatable Al-Cu casting alloy with high strength at elevated temperatures. Casting characteristics are only fair and it is very readily welded. 354.0 is a heat treatable Al-Si-Mg alloy having among the highest strength of commercial casting alloys. It has good casting characteristics and is readily weldable. Its use is generally restricted to permanent mold castings. 355.0 is a heat-treatable Al-Si-Mg alloy that is readily cast, very readily weldable and has good pressure tightness. C355.0 is an Al-Si-Mg alloy similar to 355.0 but has impurities controlled to lower limits resulting in higher strengths. It is very readily weldable and has good casting characteristics.
5-20
STRUCTURAL DESIGN MANUAL
e)
356.0 is among the easiest of alloys to cast by a variety of techniques. It is heat treatable, has intermediate strengths, is very readily weldable and has high resistance to corrosion. A356.0 is an Al-Si-Mg alloy similar to 356.0 but with impurities controlled to lower limits resulting in higher strengths and ductility. It is very readily weldable, has good casting characteristics and high resistance to corrosion. A357.0 is an Al-Si-Mg alloy generally used ror permanent mold and premium quality castings in which special properties are developed by careful control of casting and chilling techniques. It has excellent casting characteristics, is heat treatable and provides the highest strengths available in commercial castings, together with good toughness. The alloy also has excellent corrosion resistance and is very readily welded. 359.0 is a relatively high-strength, permanent mold casting alloy. It is heat treatable~ very readily weldable, and has good corrosion resistance. 5.4 STEEL ALLOYS One of the major factors contributing to the general utility of steels is the wide range of mechanical properties they are capable of attaining. Softness and good ductility may be required during fabrication of a part and very high strength during its service life. Both sets of properties are obtainable in the same material. All steels can be softened to a greater or lesser degree by annealing, depending on the chemical composition of the specific steel. Annealing is achieved by heating the steel to an appropriate temperature, holding, then cooling it at the proper rate. Likewise, steels may be hardened or strengthened by means of cold working, heat treating or a combination of these. The basic classifications of steels are as follows:
)
A.
Plain carbon steels - contain carbon as the only major alloying element.
B.
Alloy steels - contain small percentages of alloying elements, thus modifying properties of the steel. Alloy steels include: The AISI alloy steels, the alloy tool steels, the high-strength steels and silicon steels.
c.
Corrosion resistant steels - contain significant additions of chromium (greater than 12 percent by weight) plus other additions. The corrosion resistant steels are further broken down into: Martensitic, Ferritic, Austenitic and Precipitation Hardening. The martensitic and precipitation hardening grades are hardenable by heat treatment, the austenitic grades are hardenable by cold work and the ferritic grades are essentially unhardenable. By definition, steels containing 14 percent chromium or more are classified as corrosion resistant. Alloys with lesser chromium'content require protective finishing.
5-21
STRUCTURAL DESIGN MANUAL D.
Maraging and 9 Ni-4Co - New alloys containing substantial additions of nickel, cobalt and molybdenum. The maraging alloys are machined and formed in the solution treated condition, then hardened by aging at approximately 900 o F. Fabrication and heat treatment procedures for 9 Ni-4Co are similar to those used for alloy steels. All usage of these steels should be used only with the guidance of the Structural Materials Technology Group.
Cold working is the method used to strengthen both the low-carbon unalloyed steels and the highly alloyed austenitic stainless steels. Only moderately high strength levels can be attained in the former but the latter can be cold rolled to quite high strength levels, or "tempers". These are commonly supplied to minimum strength levels. Heat treating is the principal method for strengthening the remainder of the steels (the low-carbon and the austenitic steels cannot be strengthened by heat treatment). The heat treatment of steel may be of three types: martensitic hardening, age hardening and austempering. Carbon and alloy steels are martensitichardened by heating them to a high temperature, or ttaustenitizing", then cooling at a recommended rate, often by quenching in oil or water. This is followed by "tempering" which consists of reheating to an intermediate temperature to relieve internal stresses and to improve toughness. A relatively new class of steel is strengthened by age hardening. This heat \ trea tmen t is designed to dissolve certain cons ti tuen ts in the steel, then preci- . _ pitate them in some preferred particle size and distribution. Special combination'll' of working and heat treating are being employed to further enhance the mechanical properties of certain steels. \
Another process used in the heat treatment of steels is austempering. In this process ferrous steels are austenitized, quenched rapidly to avoid transformation of the austenite to a temperature below the pearlite and above the martensite formation ranges, allowed to transform isothermally at that temperature to a completely bainitic structure and finally cooled at room temperature. The purpose of austempering is to obtain increased ductility or notch toughness at high hardness levels, or to decrease the likelihood of cracking and distortion that might occur in conventional quenching and tempering. Steel bars, billets, forgings and thick plates, especially when heat treated to high strength levels, exhibit variations in mechanical properties with location and direction. In particular, elongation, reduction of area, toughness and notched strength are likely to be lower in either of the transverse directions than in the longitudinal direction. In applications where transverse properties are critical,requirements should be discussed with the steel supplier and properties in critical locations should be substantiated by appropriate testing. 5.4.1
Basic Heat Treatments of Steel
The mechanical properties of alloy steels are largely dependent on the use of proper thermal treatment. Some of these treatments and a description follow.
5-22
"~" .\,,~\
;r 1I 1/ / / '-/' I I -i' , .
\
_1_
STRUCTURAL DESIGN MANUAL ...Bell ..... '# ~ 0-." __" ~
')
,
!
,.. ''''-_
A.
Annealing - A heating process which places the material in its softest condition to enhance formability and produce a desirable microstructure. Annealing consists of heating to temperatures of l500-1600oF and slow cooling.
B.
Normalizing - A homogenizing treatment used to improve machinability and response to hardening treatments. Normalizing consists of heating to l600-1700 0 F and cooling in air.
c.
Hardening (Quench)- A controlled cooling of parts heated above the transformation temperature to produce martensite. Parts are normally quenched in oil from a temperature of l550-l600 oF. The quenched material is extremely hard and brittle, so it must be tempered prior to use.
D.
Tempering - Reheating a quench hardened or normalized part to a temperature below the transformation range to restore ductility and toughness. Hardness is reduced during tempering, so the temperature selected is a compromise to yield the optimum combination of strength and ductility.
E.
Stress Relieving - A heating process which reduces the residual stresses occurring during machining, grinding, forming, etc. and reduces distortion during hardening.
F.
Heat Treat Range - Steels used at BHT are usually heat treated between 125 ksi and 200 ksi tensile strength. The strength range usually has a 20 ksi spread, e.g., 145-165 ksi, 180-200 ksi, etc. When alloys are to be used above 200 ksi tensile, the Structural Materials Technology Group should be consulted.
5.4.2
)
\
,\.,
Fracture Toughness of Steel Alloys
Steels when processed to obtain high strength or when tempered or aged within certain critical temperature ranges may become more sensitive to the presence of small flaws. The usefulness of high strength steels for certain applications is largely dependent on their toughness. It is generally noted that the fracture toughness of a given alloy product decreases relative to increases in the yield strength. Typical values of plane-strain fracture toughness, KICt for several high-strength alloy steels are presented in Table 5.5.
5-23
STRUCTllRAL DESIGN MANUAL FTY Alloy
Produc t
4340 5-Cr-Mo-V 5-Cr-Mo-V 17-4 PH
Plate Plate Bar Plate
17-4 PH
Bar
(a) (b)
(KSI) 260 260 275 190 (H900) 190 (H900)
KIC'
.-
KSI. 1/2 -~n.
t (in .. )
(L)(LT)(a)
(LT)(L)(b)
3/8 1/2 1 1/2
53 34 23 42
53 32 --
5/8
52
--
38
Longitudinal grain direction normal to the crack plane and long transverse grain direction parallel to the fracture direction. Long transverse grain direction normal to the crack plane and longitudinal grain direction parallel to the fracture direction.
TABLE 5.5 --TYPICAL VALUES OF ROOM TEMPERATURE PLANE STRAIN FRACTURE TOUGHNESS FOR AIR MELTED ALLOY STEELS. (REF. 1) S.4.3
Mechanical Properties of Steel Alloys
Table 5.6 shows the maximum diameters to which various alloy steels may be through hardened consistently by quenching. The values shown are based on through hardening to at least 90 percent martensite at center. Table 5.7
~
~
shows temperature exposure limits for various alloy steels.
Material specifications for alloy steels are shown in Table 5.B. and physical properties are specified in MIL-HDBK-S.
Mechanical
5.5 MAGNESIUM ALLOYS Magnesium is a lightweight structural metal which ~an be strengthened greatly by alloying, and in some cases, by heat treatment or cold work or both. Magnesium alloys are highly susceptible to corrosion and proper protection must be included in all designs. Proper drainage must be provided to prevent entrapment of fluids. Dissimilar metal joints must be properly and completely insulated. Magnesium alloys must not be used in elevated temperature applications since annealing can result after exposure to elevated temperatures. The use of magnesium must be approved by Structural Materials Technology Group. Mechanical and physical properties of magnesium alloys are specified in MIL-HDBK-S. Standard temper designations for magnesium alloys are shown in Table 5.9.
5-·24
)
-
e
~.
\~
"--"'"
e
-"",.,./'
DIAMETER OF ROUND OR EQUIVALENT ROUND, inch 0.5
0.8
1.0
1.7
2.5
3.5
5.0
280 ksi
-
-
.....
-
-
-
300M 98BV40
260 ksi
...
.....
AISI 4340
AISl 4340
AlSI 4340
......
220 ksi
.....
-
.....
AMS Grades
AMS Grades
D6AC
D6AC
AISI 4140
AISI 4340
AISI 4340
AISI 4340
AMS Grades
AMS Grades
D6AC
AISI 4340 AMS Grades
AISI 4340 AMS Grades
AlSI 4340 D6AC
F
tu
.'
200 ksi
-
-
AISI 8740
ofrl~
~1i~ a\<~ ;-
(I)
-I :::.:J
D6AC
c:
c-)
180 ksi AISI 4130 and lower and 8630
AISI 8735 and 8740
AlSI 4140
D6AC
..... ~
::u
Table 5.6 - Maximum Round Diameters for Alloy Steel Bars (Ref 1)
:a:r-
!
Ftu ksi Alloy AISI 4130 and 8630 AISl 4140 and 8740 AISI 4340 AISI 8735 D6AC AMS 6418 4330Si and 4330V 4335V 98BV40 300M
125 925 1025 1100 975 1150 875 925 975
......
....
150 775 875 950 825 1075 750 850 875 .. ....
...
180 575 725 800 675 1000 650 775 775 .. ..
.
.....
200
rPI
220
240
260
. ... ·... .. .. . ·.. ...... ..... ... 900... ........... 950
.. .. 625 700
..
550 700 700
450 500 500
.. .. .. " ,.
....
..
...
. .. ..
..
...
·.... ..... ·......
500
.....
·.. ......
..
·.. 350
,.
.
'
... ·... ..
Table 5.7 - Temperature Exposure Limits for Alloy Steels (Ref 1) VI
•
N VI
c::.
280
·.. ··....
(I)
-
c :i)
z
3:
·.. .......
:a:-
...
c:
·. ..... ..
400 475
z
:a:r-
STRUCTURAL DESIGN MANUAL TYPE OF PRODUCT Allov
Sheet t strip. and plate Bars and forgings
Tubing
.
4130 ••••.• MIL-S-18729 •••••••••••• MIL-S-6758 ••••••••••••• 4140.""""" ••••••• " .•• " ........... " .... MIL-S-5626 ............ "" 4340 . " ........ AMS 6359 ........ " ........ MIL-S-5000 ... "." ••• " ..... MIL-S-8844 8630 •••••• MIL-S-18728 •••••••••••• MIL-S~6050 •••••••••••••
MIL-T-6736 ••••••• AMS 6381, 6390 ••• AMS 6415 ....... " .... " MIL-S-8844 ••••••• MIL-T-6732 ••••••• MIL-T-6734 ••••••• 8735 •••••• MIL-S-18733 •••••••••••• MIL-S-6098 ••••••••.•••• MIL-T-6733 ••••••• 8740 ........ AMS 6358 ........... " ........ "" MIL-S-6049 ........... " •• ". AMS 6323 .......... . D6AC ......... MIL-S-8949 ................. MIL-S-8949 ........ "............ .. ...................... . 4330 S i. .. ............................ AMS 6407............ ... ~. ................ ••
AMS 641 8.. ......................... .AlviS 6418................. . 4330 V• • .. •• • ............. ., •• ~ • • • . . .. ... 4335V. • • ... ~s 6434.,.................
ms ~s
*' • • • • • • • • • • • • • • •
6427.................. • ................ . 6428 •. ,. ...... .,......... . ................. .
300M ••••••••...•.•••••••••••••••• MIL-S-8844 •••.•.••••••• MIL-S-8844 ••••••• 98BV40 ............................. .AMS 6423 ................ A'tJIS 6423 •••••••••
Table 5.8
- Material Specifications for Alloy Steels (Ref 1)
)
5-26
STRUCTURAL DESIGN MANUAL
e
J
Definition
Temper
')
F
As fabricated
0
Annealed, recrystallized (wrought products only)
H
Strain hardened (wrought products only)
H2, plus one
T
Treated to produce stable tempers other than F, 0, or H
T4
Solution heat· treated
TS
Cooled from an elevated temperature shaping process and then artificially aged
T6, T61
e)
Strain hardened and then partially annealed
or more digits
Solution heat-treated (T4) and then artificially aged
\
T7
T8, T8l
Solution heat-treated (T4) and then stabilized Solution heat-treated (T4) cold worked t and then artificially aged TABLE 5.9 --TEMPER DESIGNATIONS FOR MAGNESIUM ALLOYS
)
5-27
STRUCTURAL DESIGN MANUAL 5.6 TITANIUM ALLOYS Titanium is a relatively lightweight, corrosion-resistant structural material that can be strengthened greatly through alloying and in some cases by heat treatment. It has good strength-to-weight ratios, low density, low coefficient of thermal expansion, good corrosion resistance, good oxidation resistance at intermediate temperatures and good notch toughness as well as other metallurgical advantages. The material properties of titanium and its alloys are determined mainly by their alloy content and heat treatment, both of which are influential in detennining the allotropic form in which this material will be found. Under equilibrium conditions, pure titanium has an "alpha" structure up to l620oF, above which it transforms to a "betal! structure. The properties of these two are quite different. Through alloying and heat treatment one or the other, or a combination of these two structures, can be made to exist at service temperatures. Titanium is susceptible to creep deformation in its unalloyed form below 300°F o and above 700 F at stresses above 50 percent F ty . This stress level should be avoided. Alloyed titanium at stresses above 60 percent F ty will also be susceptible to creep deformation. Mechanical and physical properties of titanium are specified in MIL-HDBK-5. cription of some commonly used titanium alloys follow:
Des-
Commercially Pure Titanium is unalloyed and is available in a familiar product form and is noted for its excellent formability. Unalloyed titanium is readily welded or brazed. It is used mainly where strength is not a requirement since it cannot be heat treated to high strength levels. Property degradation can ~e experienced after severe forming if as-received material properties are not restored by re-annealing. Ti-8Al-lMo-lV is a near-alpha composition alloy with improved creep resistance 0 and thermal stability up to about 500 F. It is available as billet, bar, plate, sheet, strip and extrusions and is usually used in the single annealed or duplex annealed condition. Room temperature forming is difficult, and for severe operations hot forming is required. It can be fusion welded with inert gas protection and spotwelded without protection. Ti-6Al-4V is an alpha-beta alloy and is available in all mill product forms as well as in castings and powder metallurgy forms. It can be used in either the annealed or solution treated plus aged (STA) conditions and is weldable. For maximum toughness, Ti-6Al-4V should be used in the annealed or duplex annealed condition whereas for maximum strength, the STA condition should be used. The full strength of this alloy is not available in thicknesses greater than I inch. This alloy can be fusion welded and spotwelded, but stress relief annealing after welding is recommended.
5-28
.)
STRUCTURAL DESIGN MANUAL
5. 7
STRESS-STRAIN CURVES
Many useful properties. are obtained from a stress-strain diagram of a material. Figure 5.5 shows a typical stress-strain diagram for a metal with no definite yield stress. Such metals as aluminum, magnesium and some steels fall into this category .. 5.7.1
Typical Stress-Strain Diagram
The curve in Figure 5.5 is composed of two regions; the straight-line portion up to the proportional limit where the stress varies linearly with strain; and the remaining portion where the stress is not proportional to straIn. Most analysis methods assume stresses to be elastic below the ultimate tensile stress (Ftu); however, some analyses will employ a plasticity reduction factor to correct for the nonlinearity of the plastic range. The properties shown in Figure E
5.5 are described below:
Modulus of elasticity; average ratio of stress to strain for stresses below the proportional limit. In Figure 5.5, .E = tan ¢ • Secant modulus; slope of the stress-strain curve at any point; reduces to E in the proportional range. In Figure 5.5, Es = tanq,r Tangent modulus; slope of the stress-strain curve at any point; reduces to E in the proportional range. In Figure 5.5~ E t = df/dE = tan f/> 2.
Fty Fey
Tensile or compressive yield stress; since many' materials do no exhibit a definite yield point, the yield stress is determined by the 0.2% offset method. A straight line is constructed with a slope E passing through a point of zero stress and a strain of 0.002 in./in. The intersection of the stress-strain curve and the constructed straight line defines the magnitude of the yield stress. Proportional limit stress in tension or compression; the stress at which the stress ceases to vary linearly with strain. Ultimate tensile stress; the maximum stress reached in tensile tests of standard specimens. Ultimate compressive stress; taken as F tu unless governed by· instability.
5-29
STRUCTURAL DESIGN MANUAL
\
Elastic ___. . . - - - - - - Plastic - - - - -.....
"J
I
,
f
I
(psi)
I I I
I--- .002
I EU E
(inches/inch)
FIGURE 5.5 - A TYPICAL STRESS-STRAIN DIAGRAM
5-30
E
Fra.cture
•
STRUCTURAL DESIGN MANUAL The strain corresponding to F tu • Elastic strain. Plastic strain. Efracture
Fracture strain; a relative indication of ductility of the rna terial.
There are other properties and terminology used by the Structures Engineer which are not shown in Figure 5.5. These are defined below. Yield and ultimate bearing stress; determined in a manner similar to those for tension and compression. A load deformation curve is plotted where the deformation is the change in hole diameter. Bearing yield (Fbry) is defined by an offset of 2% of the hole diameter while bearing ultimate (Fbru) is the actual failing stress divided by 1.15. Ultimate shear stress.
Fsu
Proportional limit in ~hear; usually taken as 0.577 times the proportional limit in tension for ductile materials. Poisson1s ratio; the ratio of transverse strain to axial strain in tension or compression. For materials stressed in the elastic range, ~ may be taken as a constant but for inelastic strains, J.l. becomes a function of axial strain. Plastic Poisson's ratio; unless otherwise state d, IJ.p may be taken as 0.5.
)
G
Modulus of rigidity or shearing m~dulus of elasticity for pure shear in isotropic rna terials. G = E/2 (l + J.l. ).
Isotropic
Elastic properties are the same in all directions.
Anisotropic
Elastic properties are different in different directions.
Orthotropic
Distinct material properties in mutually perpendicular planes.
Stress-strain curves for various materials can be found in MIL-HDBK-5 5.7.2
(Ref. 1).
Ramberg-Osgood Method of Stress-Strain Diagrams
Many structural problems involve inelastic instability. The solutions require information from a compressive stress-strain curve. Often it is desirable to represent this curve analytically. A method has been developed by Walter Ramberg and William R~ Osgood and reported in NACA TN902 (Ref. 9).
5-31
STRUCTURAL DESIGN MANUAL The Ramberg-Osgood method uses three parameters to represent the stress-strain relations in the inelastic range. The resulting equations are:
;~.7 = F~.7 + t (F~.7r n
=
....................................
1 + In (1 7 / 7 ) / In ( F 0 • 7/ F 0 • 85 ) .............. ..
where: e E f FO.7
FO.8S
« .. "
..
(1)
.o........... (2)
= strain modulus of elasticity = stress
the stress at which a line of slope O.7E drawn from the origin intersects the stress-strain curve (see Figure 5.6) = the stress at which a line of slope O.8SE drawn
from the origin intersects the stress-strain curve (see Figure 5.6)
The curves expressed by Equations 1 and 2 are plotted in Figure 5.7 and 5.8. Consult the stress strain curves of Ref. 1 for FO.7andFO.85 for various materials.
5-32
STRUCTURAL DESIGN MANUAL
FO.7 FO.8S
STRESS-STRAIN CURVE
e-z...STRAIN Figure 5.6 - Ramberg-Osgood Parameters
)
5-33
STRUCTURAL DESIGN MANUAL 1.5r-~-r~--~~~~--r-~~~~--r-~-r~--r-~-~-T~--~~~V~--~~~ __ ~~
rnc;"V
=kV'
I I
/ o
/ 0.5
1.5
1.0
Eel
Figure 5.7
- Ramberg-Osgood Constant, n. as a function of f/F • O7
:Ft=i :i{Jt~::tI ~
o
~
I" I I
-! ""
I
,~
¢
! I!
.,
I
~.
I I
I
!
~K
!
I
I' -I
•
: I
II
'
iii I I!
I!
i.1
I
t,
I
;
I:; !
I
I
!
i I I I I i 1; l~n~/7 I I I Il_+-I-<--+-~-4-!-t
h
I
i,"i
lOb rro,'(/T.)'B5,Jill, ~
~...
~
!
i I
j
I:
"L
-
1
=
\ : Iii
f
1
i
i
I
,;
',...~
I
I
I
I
! ml i , I L
I
Ii'
II
r
\.1 .1 ..1,
I
11
I
+-i I
i
111
~-+--;. I
!_
I
I1II
lilTI
i:
I
: ~+ I
!
n'-" :
I i ~ I l~-D+ 11 itll III !! I .. 11.1.,1..1.1.,111
i! l I I ! II I
: i J 1.:1
I
I! I II
I
T rt\
,,,-.~,
:1 I
i
II I; 1
I!.,'dd: i
iii
IJJ...!,U"J",!""". Ll
1'0. i /FO.8S
5-34
1\\1
I!!l,
I 1,1
I
I
I,
l l F i_I__ L-L lit'S. I I I ' I I : II I il'(iJ1 I
If!
Lill
1
i:
-=
l.r :
IlL
1 : I
j{: III! I ~~
Ii.!:
I
I: I: ~I i
j
j'
j
3d
?
I
i
illI
I
I
Figure 5.8
2.5·
2.0
ro,'?
- Relation between nand FO 7/F • 0.85
I
3,0
STRUCTURAL DESIGN MANUAL
•
Revision A SECTION 6 FASTENERS AND JOINTS
6.1
GENERAL
This section presents BHT policy on design allowables for mechanical fasteners, metallurgical joints and mechanical joints. Mechanical fasteners include solid and blind rivets and nuts, bolts and pins. Spotwelds and fusion welds are shown in th£~ metallurgical joint section, while lugs, sockets, bearings and bonding are discussed in mechanical joints. 6.2
Mechanical Fasteners
The aclual statE:! of stress in a joint is complex. Such items as stress concentrations at the edgesof the holes, non-uniform distribution of shear stress across lhe section of the fastener, and bearing stress between fastener and plate (installation stresses) are generally ignored in the sizing of fasteners. Simplifying assumptions are made for riveted and short bolted (no bending present) joints and are summarized below:
•
(1)
The applied load is assumed to be transmitted entirely by the fasteners; friction between connected plates is ignored •
(2)
When the center of the cross-sectional area of each of the fasteners is on the line of action of the load, or when the centroid of the total fastener area is on this line, the fasteners of the joint are assumed to carry equal parts of the load if of the same size; otherwise loaded proportionally to their section areas.
(3)
The shear stress is assumed to be uniformly distributed across the fastener section.
(4)
The bearing stress between the plate and fastener is assumed to be uniformly distrihuted over an area equal to the fastener diameter
(hole diameter for rivets) times the plate thickness.
)
(5)
The stress in a tension fastener is assumed to be uniformly distributed over the net area.
(6)
The stress in a compression fastener is assumed to be unifonmly distri-
buted over the gross area. No matter how well structural components are designed to carry their intended loads, a poor use of fasteners joining the components can cause the entire
struclure to fail with catastrophic results. Joint failures can be grouped in three general categories or combinations of th(, three:
6-1
/,;,,>7'1<
~.-,
,,~\\\ \\. @!.I'-
~ "
J
... ', I ~ ...
STRUCTURAL DESIGN MANUAL
/
Revision A (1)
Fastener shear
(2)
Sheet bearing
(3)
Sheet tearout
The first category is primarily a failure of the fastener, whereas the last two are primarily failures of the material being fastened. The material being fastened can have an effect on the shear strength of a fastener and the fastt!nt.'r geometry can have an effect on the bearing strength of the sheet being fastened. 6.2.1
Joint Geometry
In addition to the more obvious considerations of fastener and sheet material, an equally important consideration, joint geometry, is necessary to provide a joint capable of developing the fastener and sheet strengths. The three geometry parameters to consider are:
D,
fastener hole diameter
t,
sheet thickness'
e,
~dge
distance (center of hole to edge of sheet)
Joint allowables are influenced by e/D and O/t. Small values of e/D will produce shear tearout failures and lower bearing allowables. In addition, where fatigue is a consideration small values of e/D will result in fatigue cracking. For design, e/D shall not be less than 2.0. In fatigue critical areas, e = 2D + .06 should be maintained. For repair and manufacturing discrepancies the e/D ratio shall not go lower than 1.5. The nit ratio influences the sheet bearing stress distribution and fastener shear allowable. A high O/t indicates a fastener is too large for the sheets being joined. Above O/t = 3 the bearing stress distribution changes significantly enough to require a reduction of the basic allowable because the sheets tend to cut into the rivet. A low D/t indicates a fastener that is too small for the sheets being joined. This situation would produce fastener shear failures rather than sheet bearing failures. This is a very undesirable situation. The joint can literally "zip" open if a failure of a single fastener should occur. If the joint is properly designed the sheet will fail in bearing before the fasteners shear. As a joint is loaded the end fasteners in a pattern will load first. As the first line loads the sheet deflects at each fastener in the line and a portion of the load transfers to the next line and so on until the whole pattern is carrying the load. If the joint is shear critical,the first line of fasteners will shear, then the second will overload and shear and so on. If the joint is bearing critical, a load path will remain after the sheet yields in bearing.
6-2
.)
STRUCTURAL DESIGN MANUAL The nit ratio shall not exceed 5.5 nor be less than 1.0. Fasteners shall not be designed closer than 4D nor farther apart than 8D. repair and manufacturing discrepancies spacing can be reduced to 3.5D.
For
Design for sheet bearing to be reached prior to fastener shear.
Do not mix non hole filling fasteners with hole filling fasteners.
If a mixture
cannot be avoided the non hole filling fast~net should be installed in an interference fit hole. 6.2.2
Mechanical Fastener Allowables
MIL-HDBK-5 allowables should be used for all mechanical fasteners. of fastener types follows.
A description
6.2.2.1 Protruding - Head Solid Rivets The load per rivet at which the shear or bearing type of failure occurs is separately calculated and the lower of the two governs the design. Table 6.1 shows the standard rivet hole drill size and nominal hole diameter. Determination of the design bearing stress for hole filling fasteners is based on the nominal hole diameter as specified in Table 6.1. The yield and ultimate bearing stresses for various ma'terials are given in MIL-HDBK-5, and are applicable to riveted joints where cylindrical holes are used and where O/t < 5.5. Where D/t > 5.5, tests to substantiate yield and ultimate bearing strengths must be performed in accordance with MIL-HDBK-5. These bearing stresses are applicable only for the design of rigid joints where there is no possibility of relative motion of the parts joined without deformation of the parts.
)
In computing the design shear strength of a protruding head rivet, the shear strength allowable should be multiplied by a correction factor for sheet thickness. This compensates for the reduction in rivet shear strength resulting from high bearing stresses on the rivet and D/t ratios in excess of 3.0 for single shear joints and 1.5 for double shear joints. A further shear reduction factor is required if the fasteners are exposed to elevated temperatures. Figure 6.1 shows the reduction factors applicable to protruding head rivets at elevated temperatures. 6.2.2.2 Flush-Head Solid Rivets Ultimate and yield allowables are specified in MIL-HDBK-5 for both machine coun0 tersunk and dimpled sheet using solid flush rivets with a head angle of 100 • These strength-values are applicable when the edge distance is equal to or greater than two times the nominal rivet diameter (e ~ 2D). Other strength values for different edge distances must be substantiated by test per MIL-HDBK-S. The yield allowable is the average load at which the following permanent set across the joint is developed:
6-3
STRUCTURAL DESIGN MANUAL DRILL SIZE 80 79 1/64 78 77 76 75 74 73 72 71 70 69 68 1/32 67 66 65 64 63 62 61 60 59 58 57 56 3/64 55 54 53 1/16 52
51 50 49 48 5/64 47 46 45 44
DEC EQUIV .. 0135 .0145 .0156 .016 .018 .020 .021 .0225 .024 .025 .026 .028 .0292 .031 .0312 .032 .033 .035 .036 .037 .038 .039 .040 .041 .042 .043 .0465 .0469 .052 .055 .0595 .0625 .0635 .067 .070 .073 .076 .0781 .0785 .081 .082 .086
DRILL SIZE 43 42 3/32 41 40 39 38 37 36 7/64 35 34 33 32 31 1/8 30 29 28 9/64 27 26 25 24 23 5/32 22 21 20 19 18 11/64 17 16 15 14 13 3/16 12 11 10 9
DEC DRILL EQUIV SIZE .089 8 7 .0935 .. 0938 13/64 6 .096 5 .098 .0995 4 .1015 3 .104 7/32 .1065 2 .1094 1 .110 A .111 15/64 .113 B .116 C D .120 .125 1/4(E) .1285 F G .136 17/64 .1405 .1406 H .144 I .147 J .1495 K .152 9/32 .154 L .1562 M 19/64 .157 .159 N 5/16 .161 .166 0 P .1695 .1719 21/64 .173 Q .177 R 11/32 .180 S .182 T .185 23/64 .1875 U .189 .191 3/8 V .1935 .196 W
DEC EQUIV .199 .201 .2031 .204 .2055 .209 .213 .2188 .221 .228 .234 .2344 .238 .242 .246 .250 .257 .261 .2656 .266 .272 .277 .281 .2812 .290 .295 .2969 .302 .3125 .316 .323 .3281 .332 .339 .3438 .348 .358 .. 3594 .368 .375 .377 .386
DRILL DEC SIZE EQUIV 25/64 .3906 X .397 .404 Y 13/32 .4062 Z .413 27/64 .4219 7/16 .4375 29/&4 .4531 15/32 .4688 31/64 .4844 1/2 .500 33/64 .5156 17/32 .5312 35/64 .5469 9/16 .5625 37/64 .5781 19/32 .5938 39/64 .6094 5/8 .625 41/64 .6406 21/32 .6562 43/64 .6719 11/16 .6875 45/64 .7031 23/32 .7188 47/64 .7344 3/4 .750 49/64 . 7656 25/32 .7812 51/64 .7969 13/16 .8125 53/64 .8281 27/32 .8438 55/64 .8594 7/8 .875 57/64 .8906 29/32 .9062 59/64 .9219 15/16 .9375 61/64 .9531 31/32 .9688 63/64 .9844
•
FOR HOLES DRILLED WITH A DRILLING MACHINE USING SUITABLE JIGS AND FDITURES, THE HOLE TOLERANCES DEPEND UPON THE DIAMETER OF THE HOLE AND INCREASE AS THE HOLE DIAMETER INCREASES. THE FOLLOWING ARE STANDARD TOLERANCES FOR GENERAL MACHINE WORK AND APPLY IN ALL CASES EXCEPr WHERE GREATR'R. OR LESSER ACCURACY IS REQUIRE' BY THE DESIGN:
HOLE DIA .0135 THRU .. 125
+.
.126 THRU .250 .251 THRU .500 .501 THRU .750 • 751 THRU 1.000 1.001 THRU 2.000
(a )
REFERENCE AND 10387 TABLE 6.1 - STANDARD DRILL SIZES AND DRILLED HOLE TOLERANCES 6-4
TOLERANCES
+.
+. +.
..' -:r'\,-- ~, /.
(,.
'
,.
, \
\\
\\
", aFBeli \\
, \\ ,,:''''. '*" •• -
STRUC1·URAL DESIGN MANUAL
~~.
1.0
'\
.9
"'\
.8
\.
~
~
~
'\
'"" '\
.7
'\
~
~
~
l-
.6 ~
........
o
Figure 6.1
100
200 300 Temperature of
\. 400
REDUCTION FACTOR FOR ALLOWABLES OF PROTRUDING HEAD, MS20470AD RIVETS AT ELEVATED TEMPERATURES FOR FIVE MINUTES
6-5
STRUCTURAL ,DESIGN ,MANUAL Revision F i t
(1)
0.005 inch, up to and including 3/16 diameter rivets.
(2)
2.5 percent of the rivet diameter for rivet sizes larger than 3/16 diameter.
6.2.2.3 Solid Rivets in Tension Solid rivets are not to be used as a primary tension load path. They are to be used in shear. A tension Load will loosen the rivet. Tests have shown this loosening will occur at a low load level. When loosening occurs the shear valu~ for the rivet no longer applies since this value was obtained for a tight joinl. The joInt with loose fasteners is highly susceptible to fatigue and can fail at a low number ~f cycles of the design shear. . Often it is impossible to keep tension loads out of rivets. If this condition is unavoidable it should be held to a minimum. When secondary axial loads are imposed on a protruding head aluminum rivet such that it is loaded in tension and shear, the margin of safety is based on the the following interaction:
where
R
s
R
t
applied shear/allowable shear applied tension/allowable tension
6.2.2.4 Threaded Fasteners Bolts, screws, nuts and nutplates commonly used at Bell Helicopter are shown jn Tables 6.2 and 6.3. In shear joints the load per fastener at which the shear or bearing type of failure occurS is separately calculated, and the lower of th~ two values designs the joint. Two types of thread forms are available on threaded fasteners. Cut threads, which conform to MIL-S-7742, are generally acceptable for' use in' shear applications or areas where high tension or repeated loads are not present. Rol led threads which conform to MIL-S-8879, controlled root radius threads, arc acceptable for use in any appiication. They should be used where tension loads design the joint or where fatigue is present. Tensile strengths are based on the basic minor diameter of the thread.
.I I
6-6
STRUCTURAL DESIG,N MANUAL
•
Revision C
0
MATERIAL
DESCRIPTION
FASTENER
g}=J3
RECOMMENDED USAGE
HEX HEAD BOLT ALUMINUM: SHEAR DRILLED HEAD/SHANK OPT. Ftu=62ksi t Fsu=35ksi ALUM STD. TOLERANCE STEEL, CRES: STEEL CRES Ftu=125ksi.Fsu=75ks!
300 0 p
450°F 800°F'
AN3-AN17
8
a
tPl
~LEVIS BOLT PRILLED SHANK OPT. ~TD.. TOLERANCE
SHEAR STEEL: Ftu=125ksi,Fsu=75ksi
4S00F
AN 23-AN27
0
(f_I3
~EX HEAl> BOLt ~LUMINUM: PRILLED HEAD/SHANK OPT. Ftu=62ksi,Fsu=35ksi CLOSE TOLERANCE ~TEEL, CRES
~tu=125kslJFsu=75ksi
SHEAR ALUM STEEL CRES
300°F 450°F 800°F
AN173-AN185
0
HEX HEAD BOLT
~
G
NAb 4M
·~o·: ~ ©
DRILLED SHANK OPT. CLOSE TOLERANCE SHORT THREAD HIGH STRENGTH
HEX HEAD BOLT
SHEAR STEEL: Ftu=160ksi,Fsu=95 ks Fsu=95ksi
450°F
DRILLED HEAD/SHANK [-'.~~ CLOSE TOLERANCE
STEEL: OPT. Ftu=160ksi, Fsu=95ksi
INTERNAL WRENCHING HIGH STRENGTH DRILLED HEAD OPT.
STEEL: Ftu=160ksi t Fsu=95ksi
TENSION 450°F
EXTERNAL WRENCHING HIGH STRENGTH
STEEL: Ftu=180ksi. Fsu=108ksi
TENSION 450°F
lOOoCSK HEAD SCREW PHILLIPS OR HEX RECESS PLOSE TOLERANCE ~IGH STRENGTH
STEEL: Ftu=160ksi, Fsu=95ksi
SHEAR
lOOoCSK HEAD SCREW PHILLIPS RECESS CLOSE TOLERANCE
STEEL: lFtu=160ksi, Fsu=95ksi
SHEAR
SHEAR AND TENSION 450°F
NAS130J-NAS1316
C@ If] :::.1'
[-.-.~
MS20004-MS20017
CO ID MS21250
.>
@@ D=G
450°F
NAS333-NAS3ljO
® p:].
450°F
NAS517
TABLE 6.2 GENERAL DESCRIPTION OF BOLTS, SCREWS AND NUTS
6-7
STRUCTURAL DESIGN MANUAL Revision C FASTENER
DESCRIPTION
1000 CSK HEAD SCREW PHILLIPS RECESS CLOSE TOLERANCE DRILLED SHANK OPT. SHORT HEAD
STEEL: Ftu=160ksi. Fus==95ksi
SHEAR
PAN HEAD SCREW PHILLIPS RECESS STANDARD TOLERANCE SHORT THREAD
STEEL: F tu= 160ksi, Fus=95ksi
SHEAR
PLAIN NUT CASTELLATED
ALUMINUM STEEL CRES
tENSION ALUM STEEL CRES
PLAIN NUT CASTELLATED
ALUMINUM STEEL CliES
~HEAR ALUM
BARREL NUT SELF LOCKINC FLOATING
STEEL
~HGH
SELF LOCKING NUT
STEEL CRES
D=-~~
®
NAS120J-NAS1210
(}==Lj
@
RECOMMENDED USAGE
t-1ATERIAL
~.-.
450°F
•
4500 p
~"
NAS623
© ©
~J
lOOoF 4500 p 450°,
ANliO
~
-
.
. -
STEEL CRES
3000 p 450 0 p 450°F
AN320
~
0 0 0
L]J
STRENGTH 450°F
NAS577
t).
!Low HEIGHT
tLIGHT WEIGHT
SUPpLEMENTS MS21042 & MS21043 HIGH STRENGTH 450°F
NAS1291
f)
~ELF LOCKING NUT tREDUCED HEX !REDUCED HEIGHT
STEEL
~ELF LOCKING NUT ttEDUCED m:x tt,EDUCED HElGHT
CRES
HIGH STRENGTH HIGH n:MPERATURE SOooF
SELF LOCKING NON-METALLIC INSERT RECULAR HEIGHT
ALUMINUM STEEL
TENSION
HIGH STRENGTH 0
450 p
MS21042
t1 --
MS21043
g
@
250°F
MS21044
l'A1H,f':
6-8
6.2
(CONT'l))
GE;,\TI':HAL DESCRIPTION OF BOLTS, :~CR1':~·lS AND NUL'S
•
e
e-
.\~'
'--'
MS21047
MS21Oq8
MS210f19
MS21050
MS2 )(151
e MS21052
@@@ ~ ~
MSZL053
MS210:i4
0 0
0 0
0
... -~
~O;
I
S l' tTl
.
0'
w
~-....,
lm
tTVET SPACING 1ATERIAL t-<.AX. TE~fP. ISIZE RANGE FIXED FLOATING
\:') t'Tj
t:::I
i!-.._..-:
t'Tj
C/.l
~ ~ H o
~YPE
ot'%j
~IZE :RANGE
IPrxED
z
IFLOATING
c:
~IVET
SPACING
~TERIAL tAX. TEMP.
E t'%:l
en
TYPE RIVET SPACING iMATERIAL : ~.
TEMP.
SIZE RANGE
FIXED FLOATING
I
100 0 CSK TWO LUG STANDARD CRES
CSK 1:'"1'10 LUG STANDARD STEEL 450 0 450°&800°
{~8
-5/16 X
MS21060
(F8 -5/16 X
MS21061
ONE LUG STANDARD STEEL 450 0 -7/16
"6
X
MS31062
{t6 -7/16 X
MSZ1069
I
\
I.
~ ~
,....." /,
1000 CSK ONE LUG STANDARD CRES 450°&800 0 IF8 ·5/16 its -5/16 100° CSK ONE LUG STANDARD STEEL 450 0 X
MS21070
X
MS21071
MS2105 7
~ ~ t§h CORNER CORNER STANDARD STANDARD STEEL CRES 450 0 &800 0 450 0 ff6 -7/16 X
MS21072
100° CSK CORNER STANDARD STEEL 4S0 0
ff6- 7/16 118 -5/16 X
MS21073
X
MS21074
£JI~
~.~ \~;::!II ,;;1; ~~CI- ,_~;1
~=~
(I)
--I
::a
c::
c;-)
c: @9 ®9 ::a @ @ @ @ ®®~ ~ •r-
TWO LUG STA.NDARD STEEL
TWO LUG STANDARD CRES 450"&800 0
ONE LUG ONE LUG STANDARD STANDARD STEEL CRES f+ 50 o&8"OOo 450 0 4fo4 -3/8 ft4 -5/16 #4 -5/16
TWO LUG REDUCED STEEL
450 0 in -3/8 X
X ..1S21076
®~
X MS21086
X
I
X
X
MS21087
~@
SIDE BY TWO LUG rrWO LUG SIDE. REDUCED REDUCED REDUCED STEEL STEEL CRES 450 0 450°&800° 4500 i,4 -5/16 i/4 - 5/16 118 ·3/8 X
ONE LUG STANDARD CRES 450 0 &800°
I
MS21056
-I
450 0 &800° 450 0 #8 -5/16 4t4 -3/8
MS21075
~
0"-
100° CSK CORNER STANDARD CRES
MS21059
0
X
.."
\D
X
~
~
Z
X
MS21058
ztrj
H
TWD LUG TWO LUG STANDARD STANDARD STEEL CRES 450 0 450°&800° fi4 -7/16 il4 -7/16
100 0
~ ""-"
MS21055
TWO LUG REDUCED CRES f+500&800 0 /;2 -3/8
ONE LUG REDUCED STEEL
ONE LUG REDUCED CRES
CORNER REDUCED STEEL ~500&8000 450° 450° 1f2 -3/8 ft2 -3/8 4il -3/8
X
X
X
X
90-00B
90-033
90-036
90-037
CORNER REDUCED CRES 4.50°&800 0 tt2 -3/8 X
X
NASl473
NAS1474
0 00
90-038
\0 6 §Q e fJ
SIDE BY DOME SIDE DOME REDUCED STANDARD REDUCED CRES STEEI,CRES STEEL,CRES 0 450°&800° 1450°&800 450°&800° IF8 -5/16 1J4 - 1/4 4f8 -3/8
DOME REDUCED STEEL 0 450 #6 -5/16
DOME STANDARD STEEL 450° D8 -5/16
X
X X
X
X
DOME REDUCED STEEL 0 450 (16 -5/16
x
DOME
DOME
STANDARD REDUCED STEEL STEEL 0 4500 450 16 -3/8 /16 -5/16
x:
x
CJ rY1
en
-
C i)
z
B:
•c:z :D-
r-
\
STRUCTURAL DESIGN MANUAL In addition the nut or nut plate allowable must be used to determine the tension capability of the joint. The nut or nut plate will generally be the limiting item in a nut/bolt joint. Determination of the design bearing stress for non-hole filling fasteners is based on nominal shank diameter of the fastener. The design bearing stresses for various materials are given in MIL-HDBK-5 and are applicable to joints with fasteners in cylindrical holes and where D/t < 5.5. Where D/t > 5.5, tests to substantiate yield and ultimate bearing strengths must be performed in accordance with MIL-HDBK-5. These bearing stresses are applicable only for the design of rigid joints where there is no possibliity of relative motion of the parts joined without deformation of the parts. 6.2.2.5 Blind Rivets MIL-HDBK-5 gives the ultimate and yield allowable single shear strengths for protruding and flush head blind rivets. These strengths are applicable only when the grip lengths and hole tolerances are as recommended by the respective manufacturers and may be substantially reduced if oversize holes or improper grip lengths are used. The strengths have been established by test using e/D equal to or greater than 2. Where e/D values less than 2 are used, tests to substantiate yield and ultimate strengths must be made. In view of the wide variance in dimpling methods and tolerances for aluminum and magnesium alloys, no standard or uniform load allowables are recommended. Allowables for ultimate and shear strengths of blind rivets in dOUble-dimpled or dimpled, machine countersunk application should be established on the basis of specific tests. In the absence of such data, allowables for blind rivets in machine-countersunk sheet may be used. Blind rivets are primarily shear type fasteners. They should not be used where appreciable tension loads on the rivets will exist. They should not be used to attach heavily loaded fittings, in engine air inlets or on the tail rotor side of vertical fins. 6.2.2.6 Swaged Collar Fasteners The Ultimate allowable shear and tensile strengths of "Hi-Shear" rivets, lockbolts and lockbolt stumps may be obtained from MIL-HDBK-5. For all lockbolts under combined loading of shear and tension installed in material having a thickness large enough to make the shear cutoff strength critical for shear loading, the following interaction equation is applicable: Steel lockbolts, R
t
+ Rs 10
=
1.0
where Rt and Rs are ratios of applied load to allowable load in tension and shear, respectively.
6-10
I
STRUCTURAL DESIGN MANUAL Revision B 6.2.2.7 Lockbolts Lockbolts and lockbolt stumps shall be installed in properly drilled holes. The ultimate allowable shear and tensile strengths for protruding and flush head lockbolts and lockbolt stumps are shown in MIL-HDBK-5. When both tension and shear are present the lockbolts should be designed to the same interaction criteria defined in Section 6.2.2.6. 6.2.2.8 Torque Values for Threaded Fasteners
)
Proper installation torque is of utmost importance if a threaded fastener joint is to function properly. The torque values to be applied to threaded fasteners and fittings are specified in Bell Standard 160-007. Torque values for commonly used nuts and bolts are shown in Tables 6.4 through 6.6. All drawings showing installations of threaded fasteners should specify "torque per Bell Standard 160-007. H Two types of torque values are given; one for shear nuts and one for tension nuts. Type III (shear nut) values are based on developing a nominal 24,000 psi in the shank of the bolt while Type IV values produce 40,000 psi in the bolt. Table 6.6 shows maximum tightening values. These produce 54,000 psi and 90,000 psi in shanks of shear and tension bolts respectively. When repeated tensile or bending stresses are present, preload should be applied to the fastener by torqueing. The amount of preload should equal the maximum tensile stress expected. This eliminates cycling of the stress in the bolt since the prestress will remain constant. Often the preload will need to be greater than that developed by the torques in Table 6.6. An equation has been developed which gives a reasonable estimate of the torque necessary to produce preload. That equat'ion is:
•
a
T ::= .2 DL
where T is torque, D is the mean diameter of the thread and L is the preload produced by the torque. This is an empirical equation for dry threads and is a func tion of many variables. For lubricated threads the coefficient may
)
reduce as much as 50 per cent. For large threaded connections, such as mast nuts, the following equation has been developed.
•
where Dr is the diameter of the contact ring, y is the friction c~efficient, d p is pitch diameter, B is the lead angle (lead/~dp), ~ - arc tan Y I and a = ~ thread profile angle • '
6-11
STRUCTURAL DESIGN MANUAL
TYPE III BOLT
NUT AN316 AN320 AN345 AN150401-AN150425 MS25082 MS35650 MS35691 MS51968 NASI022
AN3-AN20 AN42-AN49 AN173-AN186 AN525 MS20004-MS20024 MS20033-MS20046 MS70073-MS20081 MS24694 MS27039 NAS144-NAS158 NAS333-NAS340 NAS464 NAS517 NASS83-NAS590 NAS623 NAS1003-NASI020 NAS1202-NAS1210 NAS1218 NAS1297 NAS1303-NAS1320 NAS1351 (NON-LOCKING) NAS1352 (NON-LOCKING) ALL THREADED STUDS
TYPE III CONSISTS OF ANY COMBINATION OF NUT AND BOLT SHOWN REFERENCE BELL TABLE 6.4
6-12
STD
160-007
- TYPE III FASTENERS
STRUCTURAL DESIGN MANUAL
.)
Revision E
TYPE IV NUT
BOLT AN3-AN20 AN42-AN49 AN173-AN186 AN525 MS20033-MS20046 MS20073-MS20081 MS24694 MS27039 NAS333-NAS340 NASSl7 NAS623 NASI003-NASI020 NAS1202-NASl210 NAS1297 NAS13S2 (NON-LOCKING) ALL THREADED STUDS
AN256 AN310 AN315 MS9358 MS20365 MS20500 MS21042 MS21043 MS21044 MS21045 MS21047 MS21048 MS21049 MS210S1 MS21052 MS21053 MS21054 MS21055 MS21056 MS21058 MS2fo59 MS21060
MS21061 MS21062 MS21069 MS21070 MS21071 MS21072 MS21073 MS21074 MS21075 MS21076 MS21083 MS21086 MS21208 MS21209 MS21991 MS122076
NAS577 NAS1291 NAS1329 NAS1330 NAS1473 NAS1474 80-006 80-007 80-013 90-002 90-003 110-061 110-062
thru
MS12227S MS124651 thru MS124850 NAS509
TYPE IV CONSISTS OF ANY COMBINATION OF NUT AND BOLT SHOWN REFERENCE BELL STO 160-007
TABLE 6.5
- TYPE IV FASTENERS
6~13
STRUCTURAL DESIGN MANUAL
•
Torque, In-Lbs TYPE III NUT AND
BOLT THREAD SIZE
TYPE IV
SHEAR Recommended Max Allowable Installation Tightening Torque Range Torque (a)
(b)
TENSION Recommended ~ax Allowable Installation Tightening Torque Range Torque (d) (c)
\
/ 10-32
12-15
25
20-25
1/4-28
30-40
60
50-70
40 100
5/16-24
60-85
14'0
100 .. 140
225
3/8-24
95-110
240
160-190
390
7/16-20
270-300
500
440-500
840
1/2-20
288-408
660
480-700
1100
9/16-18
480-600
960
800-1000
1600
5/8-18
660-780
1400
1100-1300
2400
3/4-16
1300-1500
3000
2300-2500
5000
7/8-14
1500-1800
420'0
2500-3000
7000
1 - 12
2200-3300
6000
3700-5500
10000
1 1/8-12
3000-4200
9000
5000-7000
15000
1 1/4-12
5400-6600
15000
9000-11000
25000
(a)
TYPE III RECOMMENDED TORQUE RANGE IS BASED ON A NOMINAL STRESS OF 24 KSI IN THE BOLT.
(b)
TYPE III MAX ALLOWABLE TORQUE IS BASED ON A STRESS OF 54 KSI IN THE BOLT.
(c)
TYPE IV RECOMMENDED TORQUE RANGE IS BASED ON A NOMINAL STRESS OF 40 ~I IN THE BOLT.
(d)
TYPE
IV MAX ALLOWABLE TORQUE IS BASED ON A STRESS OF 90 KSI IN THE BOLT.
REFERENCE BELL STn 160-007 TABLE 6.6
- TORQUE VALUES FOR THREADED FASTENERS
AND FITTINGS
6-14
•
STRUCTURAL DESIGN MANUAL
•
Revision B 6.3 METALLURGICAL JOINTS
In the design of metallurgical joints, the strength of the joining material (weld metal) and the adjacent parent material must be considered. Wherever possible, joints should be designed so that the welds will be loaded in shear. The allowable strength for both the adjacent parent metal and the weld metal is given in MIL-HDBK-5. Two types of metallurgical joints are discussed; welding and brazing. Welding consists of joining two or more pieces of metal by applying heat,pressure or both, with or without filler material, to produce a localized union through fusion or recrystallization across the joint. Common welding processes include: fusion (inert gas, shielded arc with non-consumable tungsten electrode - TIG and inert gas shielded metal arc welding using consumable electrodes-MIG), resistance (spot and seam) and flash. Brazing consists of joining metals by the application of heat causing the flow of a thin layer, capillary thickness of nonferrous filler metal into the space between pieces. Bonding results from the intimate contact produced by the dissolution of a small amount of base metal in the molten filler metal without fusion of the base metal.
•
6.3.1 Fusion Welding - Arc and Gas
In the design of welded joints, the strength of both the weld metal and the adjacent parent metal must be considered. For materials heat treated after welding, the allowable strength in the parent metal near a welded joint may equal the allowable strength for the material in the heat treated condition; ho~vever, it should be noted that the weld metal allowables are based on 85 percent of these values. 6.3.2 Flash and Pressure Welding
The ultimate tensile allowable strength and bending allowable modulus of rupture for flash and pressure welds are specified in MIL-HDBK-5 . .)
6.3.3 Seat and Seam We lding
Design shear strength allowables for spot welds in various alloys are given in MIL-HDBK-5j the thickness ratio of the thickest sheet to the thinnest outer sheet in the combination should not exceed 4:1. Table 6.7 gives the minimum allowable edge distance for spot welds and seam welds. Combinations of aluminum alloys suitable for spot welding are given in Table 6.8. 6.3.4 Effect of Spot Welds on Parent Metal
In applications of spot welding where ribs, intercostals~ or doublers are attached to sheet, either at splices or at other points on the sheet panels, ~ the a] lowabLe ultimate strength of the spot welded sheet shall be detenmined ~b~ multiplying the ultimate tensile sheet strength by the approximate effi. c1ency factor shown in HLL-HDBK-5.
6-15
STRU'CTURAL DESIGN MANUAL
•• Nominal thickness of thinner sheet, in. 0.016 0.020 0.025 0.032 0.036 0.040 0.045 0.050 0.063 0.071 0.080 0.090 0.100 0.125 0.160
Edge distance, E, in.
............... ···............... ............... ··............... ...............
3/16 3/16 7/32 1/4 1/4 9/32 5/16 5/16 3/8 3/8 13/32 7/16 7/16 9/16 5/8
·........... ,. ... ··............... ................ ·................ ·................ ·............... '!!
·............... ·............... •
•••••••••• 0 ••••
• • • • • • • • • • o • • ., • •
{a}
Intermediate gages will conform to the requirement for the next thinner gage shown.
(b)
For edge distances less than those specified above, appropriate reductions in the spot-weld allowable loads shall be made.
, .'
?
, (
,
•
i
!l
J~i t
TABLE 6.7
6-16
- MINIMUM EDGE DISTANCES FOR SPOTWELDED JOINTS
e
\
e·
e
..........
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~
~ ~
Specification ••••• ".
co
Material
.'"
~Q-A- ~S-
b QQ-'b RQ-A- RQ-A- t QQ-A- QQ-A- QQ-A- QQ-A- RQ-A- QQ-A250/1 4029 4014 250/3 250/4 250/5 250/2 250/8 250/11 250/1: 250/l~ Bare Bare Clad Bare Clad Bare Clad 1100 2014 2014 2014 2024 2024 3003 5052 6061 7075 7075
;;II~;} C-~~ ".
: ~CI--~
"~-~ ~.
Material (I»
..
I'lJO 0(")
t-3
~
I'lJ
~t-3
t;;G;
~ (l)tJ:j
tz:!
~~ c: :r:~
[ljl--l ~z
t;:!c
H::3:
Z G)~ CI
QQ-A-250/l 1100 AMS-4029 Bare AMS-4014 Bare OO-A-250/3 Clad QQ-A-250/4 Bare QQ-A-250/5 Clad QQ-A-250/2 3003 QQ-A-250/8 5052 QQ .. A... 250/ 1] 6061 QQ-A- 250/ I ~ Bare QQ-A- 250/1' Clas
2014(b~
2014(bj 2014 2024(b~
2024
···...... ·... ·.... ··.... * * ·.. ..... ··*.... ··*..... • •• ·.. • •• · ·• ..•• •*..•• ··*..." 1:
';':
..'t
•
7075(b) 7075(c)
oJ.
Q
•
·... ·." ·.. ·.. ·" . ·.. ·.. * • •• ·.. * • •• ·.. ·.. • •• • •• .. .. · ... ·.. ·.. ·... ·.. ·.. ·.. ·.. • •• ·.. ..... · .. * " .. ·.. ·.. .".
"It't
i(
-1(
0
•
(II
o
0 •
"
•
;':
1/
...'r
•
II
•
000
..... ·.. ·*.. • •• .. .. ···....... .. **.. ·.. .... ·.... ··.... ··...... ·.. .. ... . ... ··"" .. ·• ••" . • •• ··......
··.... ·.. ·.. ·.. ··..... ·.. 000
• •• "
0 •
~'<
•
0 0
"
"It'f;
~
t""I
~
()
(a)
The various aluminum and aluminum-alloy materials referred to in this table may be spot welded in any combinations except the combinations indicated by the asterisk (*) in the table. The combinations indicated by the asterisk (*) may be spot welded only with the specific approval of the procuring or certifying agency.
(b)
This table applies to construction of land- and carrier-based aircraft only. The welding of bare, high-s th alloys in construction of seaplanes and amphibians is prohibited unless fically authorized by the procuring or certifying agency.
(c)
Clad heat-treated and 7075 material in thicknesses less than 0.020 inch shall not be welded without specific approval of the procuring or certifying agency.
o
3: t:tt
1--1
Z
~ o
H
Z
en
Q:>
'-" ~
o
~
.... n ....c:
(I)
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r-
= =
rn
(I)
:z 31: :I> :z
=
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r-
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STRUCTURAL DESIGN MANUAL
/
Revision B 6.3.5 Welding.of Castings In-process repair welding of rough castings is permissible when accomplished in accordance with drawing requirements or other authorized documentation. BHT designed castings are repair welded to BPS 4470 procedures. These basic procedures, as listed below, shall also be considered when supplier designed castings are to be repair welded. A. Dpfects shall be carefully and completely remo.ved by grinding t routing or filing and reworked area shall be thoroughly cleaned. B. Fluorescent penetrant or radiographic inspection shall be used to insure complete defect removal prior to. welding. C. Acceptable heat treat conditions of castings at time. of welding shall be in accordance with BPS 4470 or as otherwise authorized by the BHT Metallurgical Laboratory. D. Welding method shall be tungsten inert gas (TIG) and shall be specified per specifications MIL-W-8604 for aluminum alloys, MIL-W-86l1 for ferrous alloys and MIL-W-18326 for magnesium alloys when applied to supplier design parts. E. Applicable filler material shall be specified if parts are not welded per BPS 4470. F. Zoned sketches shall be required when specific areas or defect sizes to be welded are limited. Procedures shall confo.rm to BPS 4470 requirements.
•
G. Welding shall be performed by welders qualified in accordance with BPS 4470 test requirements or as otherwise authorized by BHT Metallurgical Laboratory. H. Welded castings shall be properly identified to indicate that the casting was repair welded. I. Casting shall be completely heat treated after all welding is completed.
J. All weld repaired castings, except Class lIB, shall be 100% radiographically inspected after all welding and heat treatment is completed. K. When supplier designed castings are to be weld repaired, the Structural Materials Technology Group should be consulted for specific requirements .
6-18
•
~~:-'T--_ I - '-
Ll.
~
.'
\ Bell \\- . "\~'" . 11.~-L~K:""'T'" ~
6.4
STRUCTURAL DESIGN MANUAL
MECHANICAL JOINTS
This section contains design information for lugs, sockets, pins, cables, pulleys, bearings, bonding, etc. 6.4.1
Joint Load Analysis
The analysis in this section is a thermo-elastic analysis of mechanical joints based on the procedures set forth in Reference 10, WADD-TR-60-S17. The analysis presented is directly applicable to problems where the stress' levels lie in the elastic range. Certain aspects of the analysis are shown to be of value in the solution of problems with stress levels in the inelastic and plastic ranges. 6.4.1.1
One-Dimensional Compatibility
The problem is simplified if the overall geometry of the joint does not allow the joint to bend out of plane. Such a case is shown in Figure 6.2. This results in joint displacements and attachment loads essentially dependent on axial flexibility of the joint components in the direction of the applied external loading.' . The load distribution is obtained by satisfying compatibility conditions for the joint displacements and the equilibrium equation. This analyis is applicable to a mechanical joint composed of two dissimilar elastic materials. It is assumed that the attachments initially fill the holes and that each attachment hole combination deforms elastically under load. The presence of "slop" and its influence upon the load distribution is then considered. Figure 6.3 shows a longitudinal section through a typical joint. Denoting the shear load at the jth attachment by Pj, equilibrium of forces requires that N
X
)
=
:E
j=l
(l)
p. ]
where a tensile applied load X and attachment loads acting to the right on the upper sheet are positive. As shown in Figure 6.4, compatibility of displacements requires that the axial contraction or expansion of the plate material at the common surface, measured from a datum (defined by the unloaded, unheated spacing between the centerlines of adjacent pins) must be identical for the upper and lower plates. For the jth general bay the compatibility equation is (2)
where T and B denote top and bottom sheet. For the one-dimensional case with no tlslop" there are basicially two types of deformation which contribute to the Ll.js of the joint.
6-19
"~I
~~n
STRUCTURAL DESIGN MANUAL LOAD
/ )
FIGURE 6.2
FIGURE 6.3 6-20
TYPICAL SPLICE FOR RIGID BEAM
ONE DIMENSIONAL JOINT EQUILIBRIUM
STRUCTURAL DESIGN MANUAL
·)
j
j+l
FIGURE 6.4
ONE DIMENSIONAL COMPATIBILITY
) P.
j-l
! . '
_1=1
P.
1
S
m
FIGURE 6.5
J
r-J-ji, 4IIIIIIf •
.",.
~·B .J P. 1 J+
4 I
JlJ
S'~ r.;1 ~ P.
DEFORMATION OF THE JOINT DUE TO LOCAL DISTORTIONS OF THE HOLES AND ATTACHMENTS 6-21
STRUCTURAL DESIGN MANUAL
•
The first type is uniaxial stretching or contraction of the sheets due to the combined effects of temperature and mechanical loading. From Figure 6.5 the uniaxial stretching for the jth bay is
I
djT =
(x - t p~(~) i=l
+
AE jT
(tl Pi) (lE)
I
d jB
~
E
0
+
jB
where a positive
fLj fLj
€
jT
jB
dX
(3a)
dx
(3b)
0
increases the spacing between adjacent attachments and h
-
E
L
f
EaT dy
h
E dy
o
If the thermal gradient is linear through the thickness, then € is approximately equal to the value aT at the plate midplane.
•
The second basic type of joint deformation occurs because the internal joint loads create local distortions of the holes and attachments as shown in Figure. 6.6 • The deformation is expressed in terms of an experimentally determined attachment hole flexibility factor f for the given attachment-sheet flexibility (see Section 6.4.3). Thus, for the top sheet A\l II
jT
=
Pj + 1
f
f
( j +1 ) T - P j j T
(4a)
f
(4b)
and for the bottom sheet A\\
L.ljB = -Pj+l
Substituting tljT = ~jT equation (2) Ylelds
where
and
6-22
+~jT
(j+l)B
and
Ll cPo
J
f. J
+
PjfjB
ajB
= ~jB +
6."jB from equations (3) and (4) into
~cP.J
-
P. +1 f . +1 J J
!
P. f. J J
+
Lj (fjT - EjB ) dx
+
X (ALE)
jT
(5)
If?F~
•
~ ~
"I
Bell '.''''''''''''''
STRUCTURAL DESIGN MANUAL
/,' '/ -. ~. J r~.>:7
..
\
L. J
P.f.
J JB
FIGURE 6.6
DEFORMATION OF THE JOINT DUE TO LOCAL DISTORTIONS OF THE HOLES AND ATTACHMENTS
)
• 6-23
,...~
'!1",
,~
/:/1', \\'-
t.
j
l·~
-
~~~~'" STRUCTURAL DESIGN MANUAL
Revision B ---
e/
The compatibility equation (5) which constitutes N-l equations together with the equilibrium equation (1) provides N linear algebraic simultaneous equations for unknown attachment loads, Pj. Since equation (5) is in the fo~ of a recurrence equation, it can be used to express all the attachment loads in terms of Pl, the load in the first attachment. Solutions can also be obtained by solving the simultaneous equations directly or by iteration and relaxation techniques. Often in design, the sheet thicknesses are constant; the attachments are all of the same type, size and spacing and the temperature through the splice thickness does not vary appreciably in the direction of mechanical loading. This case of constant bay properties alters equation (5) somewhat. The coefficients of the Pj's become constant and a general solution takes the form of (6 )
(j=1,2, .•• , N-l) where
ilcp
=
f
L
a
(E'
T
- 'l ) dx B
and f = fT
+ fB
Combining equations (1) and (6) yields the following: [AjN
+
BjN
(~)T (i)] X + BjN (¥)
(7)
where the subscript jN refers to the j th attachment in a joint of N attachments. Values of the coefficients AjN, BjN and Z are plotted in Figures 6.7 through 6.20, where
By interchanging the designation of the top and bottom sheets, the curves of Figures 6.7 through 6.20 can be used to obtain all of the loads in joints having' as many as ten attachments. When the total number of attachments exceeds ten, the curves give the loads in the first five attachments from either end of the splice. The first term on the right-hand side of equation (7) represents the contribution of mechanical loading and the second term the contribution of thermal loading. For constant bay properties the thermal load at the center of the joint must be zero because of symmetry. Thus, B23 B35 = B47 = •••• = o. In addition the thermal loads, in joints with constant bay properties, are symmetrical about the center of the splice, BIN = -BNN; B2N = -B(N-l)N; etc.
6-24
"I
STRUCTURAL DESIGN MANUAL
v ) L
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....-4
o o
.
FIGURE 6.7
JOINT COEFFICIENTS FOR CONSTANT BAY PROPERTIES (A
1N
, j=l) 6-25
STRUCTURAL DESIGN MANUAL o o
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FIGURE 6.8 6-26
r-I
JOI~T
COEFFICIENTS FOR CONSTANT BAY PROPERTIES (BIN' j=l)
•
STRUCTURAL DESIGN MANUAL ii
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6-27
STRUCTURAL DESIGN MANUAL o o
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.
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o o
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STRUCTURAL DESIGN MANUAL o
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JOINT COEFFICIENTS FOR CONSTANT BAY PROPERTIES (B 2Nt j=2) 6-29
STRUCTURAL DESIGN MANUAL o o
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JOINT COEFFICIENTS FOR CONSTANT BAY PROPERTIES (A
.
~
o
3N
, j=3)
-:-70:-- . . " /
~
f
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, FIGURE 6.13
JOINT 'COEFFICIENTS FOR CONSTANT BAY PROPERTIES (B
33
, B , j=3) 34 6-31
0"
I W
~
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ro
f~~": /'
<: /,,~ --
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rn
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1
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100
Z FIGURE 6.14
-
=(I) ..... -I
~'" ~ ~" "'"",,,
JOINT COEFFICIENTS FOR CONSTANT BAY PROPERTIES (B
-
3N
, j=3)
e
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FIGURE 6.15 JOINT COEFFICIENTS FOR CONSTANT BAY PROPERTIES (A
100
4N
, j=4)
-~-~4
.....
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: ;-
I
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e
Z
\
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1
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1\
1\
JOINT COEFFICIENTS FOR CONSTANT BAY PROPERTIES (B
e
"-
""
::.:. z c::
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31:
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100
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r-.- 12
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1
c: :J>
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100
10
JOINT COEFFICIENTS FOR CONSTANT BAY PROPERTIES (B
<: ,...en ,...-
o ::s
4N
, j=4)
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r-
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100
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--
FIGURE 6.18
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n
r-....
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r-....
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1\
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--
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= :1:1-
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10
JOINT COEFFICIENTS FOR CONSTANT BAY PROPERTIES (B
f',
100
Z FIGURE 6.19
:1:1Z
1'\
SN
' j=S)
cr-
•
~
(Xl
10
J
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100
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FIGURE 6.20
JOINT COEFFICIENTS FOR CONSTANT BAY PROPERTIES (B
-
SN
' j=5)
•
:::l
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\ ~-. ..
c:
r--.r-I
BSN
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1
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S.
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I
STRUCTURAL DESIGN MANUAL Revision E An example problem best illustrates the procedure. Figure 6.21 shows a bimetallic splice, titanium and aluminum sheets joined by six steel bolts. The titanium and aluminum have uniform temperature rises of 300 0 F and 70 0 F respectively. The results of the example show that the maximum load occurs in the first attachment and that the two end attachments carry more than half of the total applied mechanical load. When plastic deformations occur in the vicinity of the bolt holes, the bolts tend to carry equal loads. Example Problem: f
=
.900 x 10-
6
in/lb (Section 6.4.3)
(L/AE)T
6 1/(1.5)(,125)(15)(10 )
=
.356 ( 10" 6) in/lb
(L/AE)B
6 1/(1.5)(.250)(10)(10 )
=
6 .267 (10- ) in/lb
At.
( a
AT)T - (a AT)B
Substituting into equation (7)
(Ajn
+ Bjn
20,000 A. In
(.356/.900»)
+ 9135 B.
The coefficients A.
In through Figure 6.20.
z
= ((
N
=
~) + Ae T
(20~OOO) + BJN (1110/.900)
In
and B. are now determined from Figure 6.7 In
(..h-) 1(1.)= Ae B f
(.356 + .267) (1/.900)
.692
6
A16 A 26
.0140; Figure 6.7
B16
.8000; Figure 6.8
.0239; Figure 6.9
B26
.3200; Figure 6.11
A36
.0500; Figure 6.12, B36
.0880; Figure 6.14
A46
.1090; Figure 6.15, B46
-.0860; Figure 6.16
AS6
.2450; Figure 6.18, B56
-.3200; Figure 6.19
The curves give values of Ajn and Bjn up to j = 5, but the splice under consideration has 6 fasteners. In order to obtain the coefficients for the last attachment, the designation of the top and bottom plates must be interchanged as shown.
6-39
STRUCTURAL DESIGN MANUAL X-20000
TITANIUM. E=15(106~6 .125xl.SO. a=6.5(lO ) T=300oF
h
20000
ALUMINUM, E=lO(10
.250 OIA. STEEL BOLTS
6
• 25 Ox: L 5 0, a;;;< 12 (l 0
26
)
T=700,
12410 9010 7210 5820 3840
LOAD IN TOP PLATE •
•
I
20000
1390+ 1980 7590
~ -
-
3400
BOLT LOADS
4030
-,.------19~80 _
__.-______ 1800 1390
I
FIGURE 6.21 - EXAMPLE PROBLEM, COMPATIBILITY As shown above, the last attachment (j - 6) in the original designation becomes the first attachment (j 1) in the interchanged position.
= .900 (10. 6 ) in/lb
ft = f
= (L/AE)B
.267 (10- 6 ) in/lb
(L/ AE)tB = (L/ AE)T
.356 (lO-6) in/lb
(L/AE)~
Z'
)
Z = .692
from equation (7)
+ B!In (.267/.900»(20,000)
p~
(A~
P~
20,000 A! + 4693 In
JU
JU
In
from Figures 6.7
B~
Jll
and 6.8
.BOOO
6-40
= B~ (1110/.900) In
•
STRUCTURAL DESIGN MANUAL
•
Revision B Substituting coefficients A. and B. into the equations for p. In In J n
r 16
= 20 t OOO (.0140)
+ 9135 (.8000)
75901bs.
r 26 = 20,000 (.0239) + 9135 ( .. 3200) = 3400 lbs. P36 = 20,000 (.0500) + 9135 (.0880)
+
1800 lbs.
P46
= 20,000 (.1090)
9135 (-.0860)
1390 lbs.
PS6
= 20,000
(.2450) + 9135 (-.3200)
P oo
= P'16 =
20,000 (.0140) + 4693 (.8000)
= 1980
Ibs.
= 4030
lbs.
Equilibrium Check
n
~Pj
= 7590 + 3400 + 1800 + 1390 + 1980 + 4030 =20190 lhs.
j==l 6.4.1.2
Constant Bay Properties - Rigid Sheets
The special case in which the sheets have negligible axial defo~ation as compared to the deformations caused by local distortions of the holes and attachment will be considered now. This condition is possible when the sheets are thick and the attachments are small or when local yielding causes the effective attachment/hole flexibility to become large as compared to the axial flexibility of the sheets. In this case (L/AE)T--P' 0 and (L/AE)B~ 0 and the compatibility equation (5) in Section 6.4.1.1 reduces to
P.
J
(8)
PI -(j-l) A
PI is obtained by summing equation (1) over the total number of fasteners: N
L
j:=l
X
P. J
PI
NP 1 -
c/>/ f [
t
J=1
= X/N + (N-l)!2 (Ac/>/f)
(j - 1 >j
NP
l
- N/2(N-l) ~¢/f (9)
Substituting (9) into (8) P
j
= X/N
+
(N+l/2 -;- j)A4>/f
( 10)
6-41
STRUCTURAL DESIGN MANUAL The solution shows that for the case of constant bay properties and infinitely rigid sheets, the mechanical load distributes equally to the fasteners while attachment loads due to thermal effects vary symmetrically about the transverse centerline of the joint with magnitudes inversei y proportional to the attachment/ hole flexibility. For high loads which cause extensive plastic defo~ation in the vicinity of the attachment holes, the effective attachment/hole flexibility may become large as compared to the sheet flexibility, in which case the solution of equation (10) is approached. If these plastic effects become large enough, the increase in f tends to wipe out the effects of thermal loading with the result that P. -- X/N J
This indicates that near the failure of ductile materials the mechanical load tends to distribute equally to the attachments regardless of temperature distribution. 6.4.1.3
Constant Bay Properties - Rigid Attachments
The attachments may be considered rigid if the attachment/hole flexibility is negligible when compared to the axial flexibility of the sheet. This situation seldom occurs in practice since it represents the limiting case of f~O. 6.4.1.4
The Influence of IIS10pn
The presence of fls1op" due to manufacturing tolerance and differential thermal expansions between the plate holes and attachments affects load distribution through the basic joint compatibility equation. The slop at each attachment is indicated by the difference in diameters of the plate hole and attachment and is expressed by e=e
mf g
+e
t
emp
;
(e>O)
where e
initial room temperature manufacturing tolerance (clearance or interference. Interference is negative.)
mfg
)
and e
temp
=
thermal slop (clearance or interference due to differential thennal expansion between plate holes and attachments) (11)
6-42
STRUCTURAL DESIGN MANUAL The::' joint displacements (for compatibility purposes) have been measured from a datum dQfined by th<:. initial spacing between the centerlines of adjacent attachments. When slop is present and thermal and mechanical loads are applied to the joint, the attachments are displaced from the centers of the holes until they bear up against sheet material as shown in Figure 6.22. The algebraic sign of the slop displacements depends on the direction of the joint loads. For the top sheet, the displacement between adjacent attachments is given by
) and for the bottom sheet
positive 0 increases the spacing between adjacent attachments and
wh~re a
p
TPT
+1 for a positive attachment load and -1 for a negative attachment
load
rhe incompatibility due to slop is therefore ~S.
J
S J"1'
-
( 12)
S.J B
Equation (12) is significant in that it brings out the very nature of the slop problem. Consider, for example, that the slop is the same at all attachments. In this case equation (12) reduces to (13)
)
As the mechanical loads applied to the joint increase, all the attachment loads t('nd to act in the same direction (opposite to the externally applied load) or
P.
1'1
=
+1
The bracket quantity in equation (13) becomes zero and ~O.
J
=
a
(14)
'6-43
STRUCTURAL DESIGN MANUAL
..
r~- L j+ 0jT ------:a.....1
..1
~i I
•
:!JIIo
,
....
P. J
P'+l J ')
...
P.
Pj + 1
J
l~Lj L.......-_ _ L.+O. J JB
I"
---l~'",,,,,,~ e(j+1)B/
2
..I
FIGURE 6.22 - ATTACHMENT DISPLACEMENTS DUE TO SLOP
Since the
~OIS
deteLmine the influences of slop on the load distribution, equation
.(14) indicates that for high joint loadings the effects of uniform slop are
elimina.ted. To solve for the load distribution with slop, the basic one-dimensional compatibility expression equation (5) must be modified by the addition of ~o .. .The compatibili ty equations become: J
)
Equation (15) and equation (1) provide N equations for the N required attachment loads Pje The solution of these equations, however, involves more than simply solving a set of simul taneous algebraic equations. The values of aO i on the right side of equation (15) are given by (12) from which, in order tn determine thp aB.'s, thp si~~n of th(\ ALtachm{~nt loads (positiv(' or nugnt iv(') musl be uelC:!flnlncLi. J Hul Llll~ is nol known ill advance. 'l'hl!:> Irrescnls one of the major difficulties the slop problem. The method is as follows:
of
6-44'
•
STRUCTURAL DESIGN MANUAL (1)
Assume a set of directions for the attachment loads.
(2)
Detenmine the dOj*S from equation (12) and solve the simultaneous capability and equilibrium equations (1) and (15).
(3)
If the directions of the att~chrnent loads as obtained from the solution agree with the initially assumed directions, the solution is correct.
(A)
If the directions of the attachment loads as obtained from the solution do not agree with the initially assumed directions, the solution is incorrect. The procedure must be repeated with a new set of attachment load directions, preferably the oneS obtained from the solution.
The solution is the correct one when the assumed set of attachment load directions yields a solution with the same set of directions. An example best illustrates the procedure. Figure 6.2.3 shows scarfed steel and aluminum plates bolted together with three attachments. The steel and aluminum plates are subjected to unifonn temperature rises of 6400 F and BooF respectively and a mechanical load of 5000 pounds is applied. The manufacturing tolerance is emfg = 0.0003 inch for all bolts.
5/16 steel bolts STEEL a = 6(10 6 )in/in/oF
5000
= 30
10 PS1 a = 6.5(10- 6 )in/in/oF 6T = 640 0 F E
\_______~~~~.~21o~O~--~----~--~--------~=r~~--~ X ~
= 500
.200
)
1--1.25 I
---+---
I
1.25
I I
- _ .... _ _
2.00
~
ALUMINUM E = 107psi a = 12(10- 6 ) 6T
= BOoF
I
+--
L
FIGURE 6.23 .. SCARFED SPLICE 6-45
STRUCTURAL DESIGN MANUAL
Example Problem: 6 fl = 1.300 (10- ) in/lb 6 f2 = 1.200 (10- ) in/Ib f3
=
6 1.300 (10- ) in/lb
Using average thicknesses for each bay;
(l/AE)lT
=
1.25 (.175)(2)(30){10 6 )
=
.119(10- 6 ) tn/lh
(l/AF)2T
=
1.25 (.125)(2)(30){10 6 )
=
.167(10- 6 ) in/lb
(I.I flE) IB
=
1 1 25 (.225)(2)(10)(10 6 )
=
.278(10- 6 ) in/lh
0.1 AE) 2B
=
1.22 (.275)(2)(10)(10 6 )
=
6 .227(10- ) in/lh
Since the temperature rise in each bay is uniform, the incompatibilities due to unrestrained thermal expansion is .6 f> 1 = A q., 2 = [ ( a.6 T )T - (a .6 T)B] L'
=«6.5)(64~}-(12)(80»(1.25)(10-6)
= 4000(10- 6 )
In.
Assuming the temperature of each bolt to be the same as the surrounding sheet material, the slops due to temperature are given by
) (e
teMI1
)t
op
= ( (a( 6.T )
h
top s eet
= «.
.
12 ) ( 80) -
)
top of attach.
6 (10- ) In.
= 150 (IO-6) 6-46
(a 6.T )
nh ole
(6)(640)} (,3125)(10- 6 )
= «6.5)(640) -
= 100
-
(e) (
in.
80) ) ( .3125) (10- 6)
STRUCTURAL DESIGN MANUAL Since emf g
~ 300
6
(10. ) inch, the total slop is given by
=
(300 + 100) (10)-6
= 400
= (300
(1 0 - 6) in' :"
+ 150) (10- 6 )
= 450 (10- 6 ) in. The incompatibilities due to slop, equation (12) are
substituting into equation (15)
) and for equilibrium
1st. Assume all attachment loads are positive 1;697 PI - 1.200 P 2 .394 PI + 1.594 P
2
= 4595 a = 4835
- 1.300 P
PI + P2 + P = 3
5~OO
._ PI = 3880, P2 = 1850, P,.a = -530
6-47
STRUCTURAL DESIGN MANUAL The above solution would be correct if no slop were present , however- since J'oint . IS present, the solution contradicts the initial assumption that the bolt loads are all positive and it is therefore incorrect. ~
S 1op
2nd. Assume that PI & P2 are positive and P
a
is negative
1.697 PI - 1.200 P = 4595 2
+ 1.594 P2 - 1.300 P a = 3985 PI + P2 + Pa = 5000
.394 Pi
PI .= 3730, P2
)
= 1440, Pa = -170
This is a correct solution since the directions agree with those assumed. 6.4.1.5" Two-Dimensional Compatibility When the boundary conditions are such that the joint is allowed to bow out of its own plane, the solution is much more complicated. Additional factors such as rotational and out-of-plane displacements, beam column effects, moments at the attachments, etc., enter into the solution. An exact analytical solution will not be attempted here. Instead, an analysis method is presented which obtains the first approximation to the solution of the two-dimensional problem by modifying the equations of the one-dimensional solution. Bowing of the joint, Figure 6.24, occurs due to the combined effects of nonuniform temperatures and externally applied mechanical loadings. The solution presented givestthe shear loads in the attachments for a known set of applied mechanical and thermal loads where the following simplifying assumptions are made:
ELASTIC AXIS OF PLATE
·1 I
) ,/
I
j
j+l
FIGURE 6.24 - JOINT WITH BOWED CONFIGURATION 6-48
_I )
STRuc-rURAl DESIGN MANUAL (1)
The bay properties are constant (sheet thicknesses, attachment size and spacing, stiffnesses, etc., are the same for each bay). The thennruloading is assumed not to vary in the longitudinal direction but may vary through its thickness.
(2)
Vertical out-of-plane- deflections and clamping loads are assumed to have a negligible effect on the load distribution (negligible beam column effects).
(3)
Moments at the attachments have a negligible effect on, or ate included in the attachment/hole flexibility.
(4)
The contact faces of the top and bottom plates of the joint are initially plane; the external axial loading is applied parallel to this plane in the direction of the line of attachments.
(5)
As in the one-dimensional case, the joint materials are assumed to deform elastically under load.
Under the above assumptions t the requirements of compatibility at the attachments yield the following j
fA
2: i=l
P. 1
~- Pjf + Pj+lf
+ XfAT
(j
= 1,
2, •••• N-l)
(16)
where
fA = l!lq:, =
)
U~)T + U~)B 6rfJ.- L
fAT = (L )
+
L (YT + YB) EIT + EIB
-]
EI W + EI 101 ~ T _ B ,.B (y EI + E1 T
[-
T·
ML [ YT
AT T + X EIT
B
+ +
2 (17)
+y )
(18)
B
B1
Y ErE
(19)
and W is the curvature due to temperature. If the thermal gradient is linear through the thickness t then W app.roxima tely equals CiA.T/h where AT/h is the linear thermal gradient through the plate thickness (positive for higher temperatures) on the upper face of the plate. Equation (16) and equation (1) combine to fonn N equations and N unknowns.
6-49
STRUCTURAL DESIGN MANUAL A comparison of equation (16) with the one-dimensional compatibility equation, equation (6), shows that the two forms are identical. Thus, when the bay properties are constant, the procedure for the two-dimensional solution is exactly the same aS,for the one-dimensional case if the one-dimensional coefficients
are replaced by the expressions on the right side of equations (17), (18), and (19), respectively. Coefficients AjN and BjN can then be obtained, from Figures 6.7 through 6.20. 6.4.2
Joint Load Distribution - Semi-Graphical Method
This is a semi-graphical method for determining the load distribution in a joint., This method is based strictly on geometry. If a more precise load distribution is required, the method of strains in Section 6.4.1 should be used.
6.4.2.1
Fastener Pattern Center of Resistance
Locate the center of resistance of the fastener pattern, G, on the basis of bearing or shear area. If the fasteners are critical in sheet bearing, the bearing area should be used. If the fasteners are shear critical, use the shear area. Equal Areas Figure 6.25 shows a typical fastener pattern. Assume each fastener has equal areas. Connect any two of the fasteners and bisect this line. The point of bisection is the centroid of the first two fasteners. Join this centroid with the third fastener and locate a point one-third of the line distance from the previous centroid to obtain the centroid of the three fasteners. Join this centroid to a fourth fastener and locate a point one-fourth of the distance from the previous centroid. Continue adding a fastener at a time until all areas have been included.
FIGURE 6.25 - FASTENER PATTERN CENTER OF RESISTANCE 6 ... 50
)'
STRUCTURAL DESIGN MANUAL Unequal Areas Add a fastener at a time as described previously. At any stage where the centroid of n bolts has been found and is joined to the (n+l) fastener, the fractional part of the connecting line measured from the previous centroid is
+ "
\
6.4.2.2
IJ
••••
Load Detennination
Figure 6.26 shows a typical joint with an applied load P and three fasteners AI, A2 and A3. Draw the joint to scale and locate the center of resistance G. Extend the line of action of the applied load P, and from this line erect a perpendicular that passes through the centroid G and extends a distance GQ' away from P, so that GQ
where
e)
A = area of fastener in shear or bearing r = radial distance from G to fastener e = distance from G to line of action of P
FIGURE 6.26 - TYPICAL JOINT
6-51
~ STRUCTllRAL DESIGN MANUAL
,,~,J. Revision E
Next determine the radial distance Ll of the number one fastener from Q., The load P on that bolt is PeAl Ll
P
1
=--~A 2 - r
and is directed perpendicular to radial line L • 1 Repeat this procedure until the loads for all fasteners are detennined.
6.4.3
Attachment Flexibility
The flexibility of an attachment/sheet combination should be detennined experimentally. If load-deflection curves for a particular fastener/sheet combination aTe available, the flexibility is the slope of the curve at the estimated load level. If load-deflection test data is not available for the exaat fastener/sheet combination, two methods can be used to deteumine a spring rate. 6.4.3.1
Method I - Generalized Test Data
Some test data is available to develop generalized stiffness curves. Figure 6.27 shows a curve of tiD versus K for a single shear joint with a steel fastener. The procedure for determining joint stiffness is as follows: DIA
1/8
5/32
1/4
3/16
5/16 SRxlO-
ALUM
STEEL TITAN OTHER
3/8
1/2
9/16
5/B
1.21 1.51 1.81 2.4r: 1 0" 1 h':l 6. ? r 4.B" 5.17 6.0 3.62 4.53 5.44 7.25 9.06 10.9 12.6 14.5 15.5 18.1 1.93 2.42 2.90 3.87 4.83 (Eother/Esteel)xSRsteel
SHEET SPRING RATE JOINT SPRING RATE TABLE 6.9
= K x SR
=
l/(l/SRu
5.81
+ l/SRl)
- BASIC SPRING - --------- _-_.-RATES
--'-.''''-~-~---.--.--.- ..
.
....
1.
Calculate tiD for upper sheet
2.
Calculate tiD for lower sheet
3.
From Figure 6.27 determine K for upper sheet
4.
From Figure 6.27 determine K for lower sheet
6-52
7/16
6
6.'72
7.73
8.27
9.65
STRUCTURAL DESIGN MANUAL
1.6 ~
'"
1.4 ./
1.2
L.O K
-
.8
.6 .4
,,
,"
,
.,. "
0
, , ,
/ I I I I
"-
NOTE: CURVE FOR STEEL ATTACHMENTS IN STANDARD HOLES. CORRECT FOR OTHER MATERIALS AND HOLES AS FOLLOWS: ALUMINUM - .59K TITANIUM - • 72K CLOSE HOLES - 1.33K COUNTER SUNK - .67K
,,
.2
o
('
, ""
,,
,, "
, , ,
, ,,
,,, .,
/
,I
I
1
I
I
I
I
I
I
I I
I
I
I
I
I
I
I
t
I
I I I I I I I I I II I I I II I I I II I I I I
.1
.2
.3
,,4
.5
.6
.7
.8
.9
1.0
tID
FIGURE 6.27 - EFFECTIVE SPRING RATES FOR STEEL PINS IN SINGLE SHEAR
6-53
STRUCTURAL DESIGN MANUAL Revision C 5.
If the fastener is steel and in a hole drilled to nonnal tolerance (Table 6.1), proceed to step 6. Modify the K factors of step A by the following factors Aluminum Fastener, K X .59 Titanium Fastener, K X .72 Close Tolerance Hole, K X 1.33 Countersunk Hole, K X .67 Example: Titanium fastener in close tolerance hole. From step 4: K Correct K: .72 x 1.33 x K
6.
Detennine SR from Table 6.9. k
Calculate spring rate for each sheet by
K x SR
n
n
where: k
n
spring rate of sheet
constant from step 4 or 5 n = SR = value from Table 6.9
K
7.
Calculate joint spring rate k ..
I
J01nt
6.4.3.2
Method II - Bearing Criteria
If load deflection data is not available, the limit bearing load criteria of Reference 1 may be used to obtain an estimate of the attachment-hole flexibility. These criteria result in an overestimate of the attachment-hole flexibility and an underestimate of the maximum attachment load at load levels below yield. As an example of the way the criteria of MIL-HDBK-SB (Reference 1) may be used to detennine the attachment-hole flexibility factor, consider a joint in which the bolt diameter is 0.25 inch t the upper sheet is 0.125 inch titanium and the lower sheet is 0.25 inch aluminum. Assuming that the aluminum is 2024-T6 and the titanium is 6 Al-4V, the respective bearing yield stress allowables from Reference I are 78,000 psi and 198,000 psi. The yield loads are then calculated to be Pal
= 78,000(.250)(.250) = 4875
P. . = 198,000(.125)(.250) t1tan1um
6-54
lbs
6200 Ibs
)
STRUCTURAL DESIGN MANUAL
e" The average load is then (6200 + 4875)/2 = 5590 lbs.
P avg
The flexibility is calculated for a deformation of 2 percent of the hole diameter per Reference 1. f
6.4.4
avg
=
~/p avg ::; (.02)(.250)/5590~900(lO-9)in./lb
Lug Design
This section presents a basic method of analysis and procedure for the design of lug-pin combinations loaded axially, obliquely or transversely.
• )
An accurate analysis of a lug-pin combination under load is difficult because the actual distributions of stresses in the lug and pin involve a combination of shear, bending and tension of varying amounts, which are a function of the ratio of lug edge distance and thickness to pin diameter t shape of lug, number of lugs in a joint, material properties, stress concentrations, rigidity of adjacent s t ru c tu r e , etc • The various modes of failure for a lug are: 1.
Bearing of pin, lug or bushing
2.
Tension across minimum net section. The full P/Anet stress cannot be carried because of the stress concentration around the hole.
3.
Hoop tension failure of the lug across the section in line with the load.
4.
Shear tearout failure of the lug.
5.
Shear and bending of the pin.
Shear tearout and bearing are closely related and are covered by shear-bearing calculations based on empirical data. Also, the shear-bearing criteria precludes hoop tension failures. Yielding of the lug is also permanent set of 0.02 times checked as it is frequently from the ratio of the yield material.
a consideration. It is considered excessive at a the pin diameter. This condition must always be reached at a lower load than would be anticipated stress, F ' to the ultimate stress, F tu ' for the ty
6-55
~ STRUCTURAL DESIGN MANUAL "---!jal~'.' /
/
Revision E Since lugs are elements having severe stress concentrations, the ductility and/or impact strength of the material is of importance. For this reason, attention should be paid to the longitudinal, long transverse and short transverse material properties. Lugs are a small weight portion of a structure and are prone to fabrication errors and service damage. Since their weight is usually insignificant relative to their importance, the following criteria should be used. 1.
2.
Design lugs for a minimum margin of safety of 0.15 in both yield and ul timate. If no bushing is included in the original design, design the lug so that one can be inserted in the future; however, express margins of safety with no bushings.
6.4.4.1 F
tu
Nomencla ture Ultimate tensile strength; F tuw with grain, F tux cross grain. When the plane of the lug contains both long and short transverse grain directions~ F tux is the smaller of the two. Tensile yield strength; Ftyw with grain, Ftyx cross grain. When the plane of the lug contains both long and short transverse grain"directions, F tyx is the smaller of the two.
= Compressive yield strength
= Ultimate p
y
M max p'
u
load
Yield load Maximum bending moment on pin = Allowable ultimate load
p' bru
Allowable ultimate shear-bearing load
p' bry p. tu
Allowable yield bearing load on bushing
p' tru pi y A
Allowable ultimate tensile load = Allowable ultimate transverse load = Allowable yield load of lug = Area; Abr projected bearing area, At m1n1mum net section
for tension, Aav weighted average area for transverse load.
6-56
)
STRUCTURAL DESIGN MANUAL Efficiency factor; Kbr for shear-bearing, Kt for tension, Ktru for transverse load (ultimate), Ktry for transverse load (yield).
K
:;:: Yield factor
c
Load ratio; Ra for axial, Rtr for transverse
R
= Thickness of lug
t
L, T, ST
Grain direction; (L) longitudinal, (T) transverse and (ST) short transverse. Angle of d:>l ique load; ex and 0 < a < 90 oblique.
Y
r 6.4.4.2
•
o
for axial, a
=
90 for transverse
:;:: Pin bending moment reduction factor for peaking
[e - D/2] It Analysis of Lugs with Axial Loads (~= 0°)
The determination of the allowable ultimate and yield axial loads for lugs of the type shown in Figure 6.28 is as follows:
p
FIGURE 6.28 - AXIALLY LOADED LUGS
6-57
STRUCTURAL DE.SIGN MANUAL" Revision
A.
Compute:
B.
P' b ru 1. 2.
c.
E,.
= allowable
ultimate shear-bearing load
Enter Figure 6.30 with e/D and D/t and obtain P ~rAbrFtux
P'
allowable ultimate tension load
tu
Enter Figure 6.32 with vv/n and obtain K for proper material. t P~u KtAtF tu
pt Y
allowable yield load of lug
1.
Enter Figure 6.31 with e/D and obtain Kb
2.
p' y
pi
bry
1.
~r'
bru
1. 2. D.
e/D, D/t, W/D t ~r = Dt, At = (VV- D) t
=K
A
• ry
F
bry br ty
= allowable yield bearing load on bushing
Pbry
= 1.85
FcyAbrb
Where Abrb is the smaller of bearing area of bushing on Din, or bearing area of bushing on lug. Latter value may be smaller due to effect of external chamfer of bushing. F.
Margins of safety 1. 2.
6.,4.4.3
•
.1S for ultimate shear-bearing and ultimate tension
Minimum M.S. Minimum M.S.
o
for yield of lug and bushing.
Analysis of Lugs with Transverse Loads (a = 90
0
)
The determination of the allowable ultimate and yield transverse loads for the type shown in Figure 6.29 is as follows:
l~gs
of,
Dt A
6
av
Aav lAb I' (1)
Al, A2 and A4 are measured on the planes indicated in Figure 6.29(a), Al and A4 should be measured perpendicular to the local centerline •
• 6-58
STRUCTURAL DESIGN MANUAL
•
Revision E
-1
--.
~
Ai (a) j
Equivalent Lug
Equivalen Lug
A4
(c)
(d)
FIGURE 6.29 - TRANSVERSELY LOADED LUGS
B.
(2)
A3 is the least area on any radial section around the hole.
(3)
At, A2t A3 and A4 should adequately reflect the strength of the lug. For lugs of unusual shape, such as severe necking or other sudden changes in cross section, an equivalent lug should be used such as shown in Figure o.29(c) and (d).
P tru
= Allowable
1.
2.
c.
pi
y
1.
2.
ultimate load for lug failure
Enter Figure 6.33 with A /~ and obtain K • t ru p' = K A F av r tru tru br tux
= Allowable
yield load of lug
Enter Figure 6.33 with A /~ and obtain K p' K A F av -Dr try y try b~ tyx
6-59
~ STRUCTURAL DESIGN MANUAL
~~II ~.M D.
Check bushing yield per 6.4.4.2(E).
E.
Margins of Safety Minimum M.S. = .15 for ultimate transverse load Minimum M.S. for yield of the lug and bushing
1. 2. 6.4.4.4
°
Analysis of Lugs with Oblique Loads (0<0:<90°)
In analyzing lugs with oblique loading it is necessary to resolve the loading.into axial and transverse components (denoted by the subscripts tla and "trl l respectively), analyze the two cases separately and then combine the results using the interaction equation. The interaction equation: lt
R 1.6 + R 1.6 = 1 a tr where, for ultimate load, R
Axial component of applied ultimate load Smaller of P ' or p' (6.4.4.2 B or 'C) b ru tu
a
R
tr
= Transverse component of applied ultimate load Pt'
ru
(6.4.4.3.B)
and for yield load
Ra R
tr
=
Axial component of applied yield load Pt Y
(
6 • 4 .. 4. 2D)
= Transverse pi
y
component of applied yield load (6.4.4.3C)
The margin of safety should be 0.15 minimum and is calculated using the following equation:
MS
6;.4.4.5
= (R
1 - I 1.6 + R 1.6)0.625 a
tr
Analysis of Pins
The ultimate strength for a pin in a single lug/clevis joint as shown in Figure 6.34 will be analyzed first.
6-60
en
-I :=r:J
c:
c-)
-I
c: ::cJ )aI.
1.4
r-
1.2
c::I
1I
.."
I
-z
1.0
CI)
.8
CD
:r:
.6
)aI
z c:
.4
.2
~ (1)
0
~.~~l~~__~~~~~~~~~__~~~~·_·~'~e/D ~"~j:~~:~~~~"~"~'''~-L~~~I__~~__~~ .5 .6
'" I
...... '"
I
.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
3.0
3.2
3.4
FIGURE 6.30 - SHEAR - BEARING EFFICIENCY FACTORS OF AXIALLY LOADED LUGS
3.6
3.8
4.0
~
...,en
1--'"
0
~
t;Ij
)aI
r-
STRUCTURAL DESIGN MANUAL Revision B
Curve (A) is a cutoff to be used for all aluminum alloy hand forged billet when the long transverse grain direction has the general direction uC ti in the sketch. Curve (B) is a cutoff to be used for all aluminum alloy plate, bar, and handforged billet when the short transvers e grain di ree tion has the general di rec tion "C' in the sketch, and for die forgings when the lug contains the parting plane in a direction approximately normal to the direction tiC". NOTE: In addition to the limitations provided by curves (A) and (B), in no event shall a K greater than 2.00 be used for lugs made from .5" thiN or thicker aluminum alloy plate, bar, or handforged billet.
FIGURE 6.30 (CONT'D) - SHEAR - BEARING EFFICIENCY FACTORS OF AXIALLY LOADED LUGS
•
e· 6-62
STRUCTURAL DESIGN MANUAL Revision A
2.5
2.0
1.5 K
bry 1.0
.5
o
o
1.0
2.0 e/d
3.0
4.0
FIGURE 6.31 - BEARING YIELD EFFICIENCY FACTORS FOR AXIALLY LOADED LUGS
6-63
STRUCTURAL 'DESIGN- MANU'AL _Revision E
1.0
--
-~ ~ ..:- ;:-
-- -- -::::: "
\" """" ~ ~ ~ '-.. '---
.9
\,\ r\
~
~
.......
,
~~
'-
\ ~'" '...::(~ ~ ~
.8
\
•7
Z ~~
\
.6
.
~ ".......
~
...........
'I\.
" '\~ t\.
.5
~ ~
~\
\
,
-
\
'\
1\ ,
"'-,
.4
.3
I----
.2
-
.1
f--
.-
S~
®~ .......,;
............
(,
""P'" \.
~
~
\
'-...,
~
~
t i . ) C - ,\LOAD
l±\-~:;~~ ~ e :i ~ l i! I t t
~
!\.
\~
""
'\
" "" l'....·t ~
-
-, ,
\ ~ -...... «~ \ \' ""- ~.................. \
'\
Kt
----
\
--......
r.:"t.
"- ~
"" "
t:; ~
~rT ~
~"
-~
I
~
" '" "
r-....
"'",
-
.
o 1.0
2.0
FIGURE 6. ,32a -
3.0
4.0
5.0
TENSION EFFICIENCY FACTORS FOR
AXIALLY LOADED ALUMINUM AND STEEL LUGS
6-64
\ I
STRUCTURAL DESIGN MANUAL
•
Revision B
L, T and ST indica te grain in the "eft di ree tion. Material
•
•
Curv.e
4130, 4140, 4340 and 8630 steel 2014-T6 & 7075-T6 plate ~. 51t (L, T) 7075-T6 bar and extrusion (L) 2014-T6 handforged billet ~ 144 sq in (L) 2014-T6 & 7075-T6 die forgings (L) 2014-T6 & 7075-T6 pla te >. 5" ~ itt (L, T) 7075-T6 extrusion (T, ST) 7075-T6 handforged billet ~ 36 sq in (L) 2014-T6 handforged billet > 144 sq in (L) 2014-T6 hand forged billet ~ 36 sq in (T) 2014-T6 & 7075-T6 die forgings (ST) 17-4PH & 17-7 PH-THD 2024-T6 plate (L, T) 2024-T4 & 2024-T42 extrusion (L, T, ST) 2024-T4 plate (L, T) 2024-T3 plate (L, T) 2014-T6 & 7075-T6 plate > 1" (L, T) 2024-T4 bar (L, T) 7075-T6 handforged billet > 36 sq in (L) 7075-T6.handforged billet ~ 16 sq in, (T) 195-T6, 220-T4 & 356-T6 aluminum castings 7075-T6 hand forged billet > 16 sq in (T) 2014-T6 handforged billet > 36 sq in (T) Aluminum alloy plate, bar, handforged billet & die forging (ST). NOTE: ST direction exists only at parting plane 7075-T6 bar (T) 18-8 stainless steel, annealed 18-8 stainless steel, full hard. NOTE: for~, ~ & 3/4 hard interpolate between Curves 7 and 8 7075-T73 Die Forging (L) ~3tf 7075-T73 Die Forging (ST) ~3"
1 1 1 1 1 2 2 2 2 2 2 2 3 3 4 4 4 4 4 4 '5
5 5 6
6 7
8 9
10
FIGURE 6.32a (CONT'D) - TENSION EFFICIENCY FACTORS FOR AXIALLY LOADED ALUMINUM AND STEEL LUGS
6-65
STRUCTURAL DESIGN MANUAL Revision B
1.0
.9 .8
.7
•
.6 K
t
.5
.4 .3
.2
.1 0
1.0
2.0
3.0
4.0
5.0
WID FIGURE 6. 3Z,b -
6-65a
TENS ION EFFICIENCY FACTORS FOR AXIALLY LOADED TITANIUM LUGS
•
STRUCTURAL DESIGN MANUAL Revision B Material C.P. Ti Type I Comp. B t ~ 1.0 Ti-6Al-4V Mill Ann. Plate t S 4.0 Ti-6Al-4V Mill Ann. 8&F t S 3.0 Ti-6Al-4V Ann. Ext. Ti-6Al-4V STA B&F.5 < t ~ 2.0 Ti-6Al-4V STA B&F t S .5 Ti-6Al-4V STA B&F 2.0 < t < 3.0 Ti-6Al-4V STA Plate t $ .75 Ti-6Al-4V STA Plate .75 < t ~ 2.0 Ti-6Al-6V-2Sn Cond. A-I Die Forging t s 2.0 Ti-6Al-6V-2Sn Ann. Ext. t ~ 2.0 Ti-6Al-6V-2Sn Mill Ann. Plate t s 2.0 Ti-6Al-6V-2Sn STA Plate t ~ 1.5 Ti-6Al-6V-2Sn Cond. A-l Die Frg. 2.0 < t S 4.0 Ti-6Al-6V-2Sn STA Die Frg. t ~ 1.0 Ti-6Al-6V-2Sn STA Plate 1.5 < t S 2.5 Ti-6Al-6V-2Sn STA Die Frg. 1.0 < t S 3.0 Ti-6Al-6V-2Sn Cond. A-l Die Frg. 4.0 < t $ 8.0 Ti-6Al-6V-2Sn STA Die Frg. 3.0 < t S 4.0
Curve 1 2 2
2 3
4 5 6 8 4 4
-7 I
7 8 9 9 9
10
FIGURE 6.32b (CONT'n) - TENSION EFFICIENCY FACTORS FOR AXIALLY LOADED TITANIUM LUGS
6-65b
~ STRUCTURAL DESIGN MANUAL ~., Revision B
.
j.
1. 6 'f
,
:
I.
., "
1. 4
.
~
"
I
.
1 2
:.1'
~.
::i ,
'
1.0
"
".
:
K
*K
~
,j 10.,.
:;'1
tru
t ry
'.
, ... ,
.8
.
"
::l::
:
T-
,
"
"
.,
"
-.:d~::-r-: '.
'1
'f
f
"
.2~_~
.2
.4
.6
.8
1.0
1.2
1.4
A'av I A br FIGURE 6.33a - EFFICIENCY FACTORS FOR TRANSVERSELY LOADED ALUMINUM AND STEEL LUGS
.,
6-66
Il/!~\-~
"~.!
4it)
\
,I
-
STRUCTURAL DESIGN MANUAL
Revision B
.-7_ tru .4 • • • • •_ _ __ •
• 2 _ __
o
.. 2
.4
.6
.B
1.0
1.2
FIGURE 6.33b - EFFICIENCY FACTORS FOR TRANSVERSELY LOADED TITANIUM LUGS 6-67a
1.4
STRUCTURAL DESIGN· MANUAL Revision B L, T I and ST indicate'
gra:fn
in the
II
C It direction
Material 17-7PH-THD 2014-T6 pIa te ~ .5" 2014-T6 pIa te > • 5" ~ Itt 2014-T6 pIa te > Itt 2014-T6 die forgings 2014-T6 hand forged billet ~ 36 sq in 2014-T6 hand forged billet > 36 sq in 2q24 ... T3 plate ~ .5 11 2024-T3 pIa te > .5" 2024-T4 pI a te ~ .5" 2024-T4 plate > .5 11 2024-T4 bar 2024-T4 & 2024-T42 extrusion 2024-T6 plate 7075-T6 pIa te ~ .5 u 7075-T6 plate > .5 ~ 1" 7075-T6 pIa te > 1tt 7075-T6 extrusion. 7075-T6 die forgings 7075-T6 handforged billet ~ 16 sq in 7075-T6 handforged billet > 16 sq in 195-T6 & 356-T6 aluminum! casting' 220-T4 aluminum casting 4130, 4140, 4340 & 8630, Ftu = 125 ksi 4130, 4140, 4340 & 8630, Ftu = 150 ksi 4130, 4140, 4340 & 8630, Ftu = 180 ksi 7075-T73 Die:Forging (L) ~3tt 7075-T73 Die Forging (ST) ~3" Ktry, all materials All curves are for Ktru except the one noted as Ktry.
Curve
7 8
11
13 11 11
14 5 9 5 9 9
12 12 8 11 13 11
11
13 14
@
6
2 3 4
15 16 1
.
In no case should the ultimate transverse load be taken as less than that which could be carried by cantilever beam action of the portion of the lug under the load. The load that can be carried by cantilever beam action is indicated approximately by Curve A. S.hould Ktru be below Curve A, separate calculation as a cantilever beam is necessary_
FIGURE 6.33a (CONT'D) - EFFICIENCY FACTORS FOR TRANSVERSELY LOADED ALUMINUM AND STEEL LUGS
STRUCTURAL DESIGN MANUAL Revision B
L, T and ST indicate grain in the "C" direction Material
Curve
Ti-6Al-4V Ann. Cond. A Die Forging (T) t $ 5.0 Ti-6Al-4V Ann. Cond. A Hand Forging (T) A ~ 16 Ti-6Al-4V Ann. Cond. A Hand Forging (T) A > 16 Ti-6Al-4V STA Die Forging (L) t ~ 5.0 Ti-6Al-4V STA Die Forging (T) t ~ 1.0 Ti-6Al-4V STA Die Forging (T) 1.0 < t ~ 3.0 Ti-6AI-4V STA Hand Forging (L,T) t ~ 2.0 Ti-6Al-4V STA Hand Forging (T) 2.0 < t < 3.0 Ti-6Al-6V-2Sn Ann. Plate (T) t ~ 2.0 Ti-6Al-6V-2Sn Ann. Die Frg. (ST) t S 2.0 Ti-6Al-6V-2Sn Ann. Hand Frg. (T) t < 2.0 Ti-6Al-6V-2Sn Ann. Plate (T) 2.0 < t ~ 4.0 Ti-6Al-6V-2Sn Ann. Die Frg. (ST) 2.0 < t ~ 4.0 Ti-6Al-6V-2Sn Ann. Hand Frg. (T) 2.0 < t ~ 4.0 Ti-6Al-6V-2Sn STA Die Forg. (L) All Ti-6Al-6V-2Sn STA Die Forging (T) All Ti-6Al-6V-2Sn STA Hand Forging (L,T) t ~ 4.0 Ti-6Al-6V-2Sn STA Hand Forging (T) t > 4.0
1 1 2 2
2 3
1 2
4 4
4 5 5 5 6 7 6 7
In
no case should the ultimate transverse load be taken as less than that which could be carried by cantilever beam action of the portion of the lug under the load. The load that can be carried by cantilever beam action is indicated approximately by Curve A. Should Ktru be below Curve A, separate calculation as a cantilever beam is necessary_
) FIGURE 6.33b (CONT'D) - EFFICIENCY FACTORS FOR TRANSVERSELY LOADED TITANIUM LUGS
6-67b
STRUCTURAL DESIGN MANUAL
P/2 'r------!.... P / 2
1-----,.- P / 2 1----'-_---'---1
P/2
FIGURE 6.34 - SINGLE LUG/CLEVIS JOINT A.
Obtain moment arm "bit. For the inner lug of Figure 6.34 calculate r = {(e/D)-~}D/t2. Determine e and (Pi) min as noted below, and compute (Plu) min/AQrFtux. Enter Figure 6.36 and obtain the reduction factor lIyU whl.ch compensates for the "peaking" of the distributed pin bearing load near the shear plane. Calculate
(t l /2) + 9 + y (t /4) 2 where "g" is the gap between lugs as shown in Figure 6.34 and may be zero. Note that the peaking reduction factor applies only to the inner lugs. b =
1.
2.
Determination of e and (P' ) min u For lugs loaded axially. Take the smaller of P'bru and P' tu for the inner lug as (P'u)min and e as the edge distance at ~ = 90 degrees. For lugs loaded transversely 'fake (P
I
u
)
min = P
t
tru
and e as the edge distance at
Ci.
= 90
degrees. 3.
For lugs loaded obliquely Take (P I
u
)
min
P
1.6
1.6 0.625 + R tr ) and e as the edge distance at the value of to the direction of load on the lug. (Rft.
B.
a. corresponding
Calculate maximum pin bending moment, "M", from the equation M = P(b/2)
c.
6-68
Calculate bending stress assuming a Me/r distribution.
STRUCTURAL DESIGN MANUAL
•
Revision E
P/2
P/2
FIGURE 6. 35
.~
ACTIVE LUG THICKNESS
D.
Obtain the ultimate strength of the pin in bending by use of Section 9.4. If the analysis should show inadequate pin bending strength it may be possible to take advantage of any excess lug strength as follows.
E.
Consider a portion of the lugs to be inactive as indicated by the shaded area of Figure 6.35. The portion of the thickness to be considered active may have any desired value sufficient to carry the load and should be chosen by trial and error to give approximately equal margins of safety for the lugs and pin.
F.
Recalculate all lug margins of safety with allowable loads reduced in the ratio of active thickness to actual thickness.
G.
Recalculate pin bending moment, H = P(b/2) and margin of safety using value of rib" which is obtained as follows:
•
Take the sma.ller of Plb ru and pit u for the inner lug, based based upon the active ..t~ckn_:~ss I as (P I u) min and compute (l?~)min/Abr
F tux where Abr =2t D.-'>Enter Figure 6.36 and 4 obtain "y". Then
b = t3/ 2 + g + y(t 4 /2). This reduced value of lib" should not be used if the resulting eccentricity of load on the outer lugs introduces excessive bending stresses in the adjacent structure. In such cases pins must be strong enough to distribute the load uniformly across the entire lug. Lug-pin combinations having multiple shear connections such as those shown in Figure 6.37 are analyzed as follows. 6-69
STRUCTURAL DESIGN MANUAL
1.0;
)
.8
.7 .6 'Y
Lr. =
.5:
L .... -.. -
.
•
~.. - .:..:
.55:.
4'I
.3 .2 .
r
=
e-D/2 t
! ..
.. 1
0
a
.2
.4
.6.8
1.0
1.2
1.4
1.6
(P~)min/A br F tux FIGURE 6.36 - REDUCTION FACTOR FOR PEAKING OF BEARING LOADS ON PINS
6 .. 70
1.8
2.0
STRUCTURAL DESIGN MANUAL
p
p
"}---__ P2
tit (typ)
FIGURE 6.37 - MULTIPLE SHEAR JOINT A.
The load carried by each lug is detennined by distributing the total applied load "p" among the lugs as shown in Figure 6.37 and the value of ttC" is obtained from Table 6.10.
B.
The maximum shear load on the pin is given in Table 6.10.
c.
The maximum bending moment in is given in Table 6.10.
6.4.4.6
the
pin is given by: M = P l b/2 where Itb tl
Lugs with Eccentrically Located Hole
If the hole is located as in Figure 6.38 (el less than e2), the ultimate load and yield lug loads are determined by obtaining Pbrul P tu and P for the equivalent lug shown and multiplying by the factor (e + e + 2D)/(2e + 2D). l 2 2
y
)
6.4.4.7
Lubrication Holes In Lugs
When lubrication holes are present, the lug may be analyzed as follows. A.
Axially loaded lugs. Modify the calculation of P~ or pending upon the location of the hole. (Fig: 6.39)u
Ph
or both, deru
If Ptu requires modification, obtain the net tension area using a thickness given by t-minus lube hole diameter. If Pbru requires modification, obtain Abr using a thickness given by t minus the lube hole diameter. Obtain Kt from Figure 6.31 for WiD = 1.75 using the weakest grain direction occurringin the plane of the lug. Then Pbru
~ru Kt ~r F tux 6-71
~ "@'.'
STRUCTURAL DESIGN MANUAL
• Total number of lugs including both sides
C
Pin Shear
5
.35
.50Pl
.28 (t t ; tit)
7
.40
.53P
1
.33(t
9
.43
.54P
1
.37(tt ; til)
11
.44
.54P
1
• 39 ( t t + til)
00
.50
.50Pl
b
l
til)
;
2
• 50( t
t
;
I
tit)
TABLE 6.10 - PIN BENDING FACTORS
)
6-72
STRUCTURAL DESIGN MANUAL Revision E
D
FIGURE 6.38 - LUGS WITH ECCENTRICALLY LOCATED HOLES
/
)
FOR HOLE IN THIS REGION MODIFY P tu
FOR HOLE IN THIS REGION MODIFY Pbru
"FOR HOLE IN MODIFY P
THIS REGION
tu
FIGURE 6.39 - LUBRICATION HOLES IN LUGS
6-73
STRUCTURAL DESIGN MANUAL B.
Transversely loaded lugs. Obtain pi neglecting lube hole and mUltiply tru by 069 (1 _ lube hOl~ diameter).
c.
Obliquely loaded lugs. Obtain P~ut Pbru' and Ptru according to A and B above. Then proceed according to Section 6.4.4.4.
6.4.5
Stresses Due to Press Fit Bushin&s
Pressure between a lug and bushing assembly having negative clearance can be determined from consideration of the radial displacements. After assembly, the increase in inner radius of the ring (lug) plus the decrease in outer radius of the bushing equals the difference between the radii of the bushing and ring before assembly ..
8 =
u.
rlng
- u
b us h'lng
where
8
= difference
u
= radial
between outer radius of bushing and inner radius of the ring
displacement, positive away from the axis of ring or bushing.
Radial displacement at the inner surface of a ring subjected to internal pressure p is u
D p .
=E
rlng
Radial displacement at the outer surface of a bushing subjected to external pressure p is B u
=-
P bush
E
where: A inner radius of bushing B = outer radius of bushing C = outer radius of ring (lug)
D E ~
=
inner radius of ring (lug) modulus of elasticity = Poisson's ratio
Substitution of the previous two equations into the first yields:
6-14
STRUCTURAL DESIGN MANUAL p
=
c2 + n2
2 2 + fLring C - n
E .
1;'lng
+
E
bush
Maximum radial and tangential stresses for a ring (lug) subjected to internal pressure occur at the inner surface ,of the ring (lug) and are fr
= -Po
= P (~~~
ft
:~).
Positive sign indicates tension. f
f
s
=
t
-
f
The maximum shear stress at this point is
r
,2
The maximum 'radial stress for a bushing subjected to external pressure occurs at the outer surface of the bushing and is f
r
=-p
The maximum tangential st~ess for a bushing subjected to external pressure occurs at the inner surface of the bushing and is f
t
=
2 P B2 B2 _ A2
The allowable press fit stress should be based on stress corrosion, static fatigue, fatigue life, and the ultimate strength. Any questions concerning the limits of the press fit stress should be directed to the Airframe Structures Group Engineer.
)
6-75
STRUCTURAL DESIGN MANUAL
10 ____--~--~--~--
9
Steel 8
= Tangential
7 A L.l
stress at inner radius of aluminum ring = Bhush ... Brtn
B/A
5
r-1-~~~j-~~~~~=-~'--t==F=~=t==t=j1.3
L-Jl-1~L-~~~~~-l--L=J==t==~~~--~d1~25 ::t::===l====t 1. 2 3
2 ----I---..J.
1
o 1
2
3
cIs
4
FIGURE 6.40 - TANGENTIAL STRESSES FOR PRESSED STEEL BUSHINGS IN ALUMINUM RINGS
~-76
1. 05
STRUCTURAL DESIGN MANUAL
r. .1.
0000
I
\ \
\
.)0000
,
\
\
"-
' " ~
f\
B
f
~tALUM.
RING ,
~~ c ~:=NAS75 BUSHING
1\\
\\
""""'...
BE ..L ~sr rANDAl D
.......
~~".L
~
~~
~~
: T f = Tangential stresses @ t Inner radius,Al.Ring INT =(Bbush - Bring)(2)
~
~
~ r--
BS2~
-005
----r----- ---- r---
r-
NAS75 -3
-----r--\ ~ '" ~~~ -------- -~ ~0 ----- ----"'~~ ---------~ ~ --... 006
r--- I---
~ ~
'-
'"
'"
~
.........
""'----
~
-5
.008
-6
----------
~ ~ ~ r----~- F==:::::: ": ~ ----I---~ -----r---- r--I'----
r--
-007
~
~~
r-......
-4
r--::: t::::r---
I--.
r---
I---- -009 -010 -012 -011 ::--- _ () 1
-7
:~
'~
~14
-016 -018
-uzo -ULL
10 11 12 i- 14 16 i-
I
!'"
-
- ~S
o 1
2
FIGURE 6.41
C/B
TANGENTIAL STRESSES FOR PRESSED NAS75 BUSHINGS
6-77
STRUCTURAL DESIGN MANUAL
6
I'J\
o
r-I
r£~ ~\ + :-.f-~ R
.X
I
'~~' ---
5
4
3
V
/
L
I V
/
/~
V
~~
~
.5
.6
I---___
~
--
~
j-.--
BUSHING O.D • BUSHING I.D.
./
! -~
~
~
l----
./
-----------
~
~
-
~
~
...-
-
1.2C
-
1.3C J
2 .. 0 .1
f--::::::::
.8
1.0
1.2
1.4
OUTSIDE RADIUS OF RING (R) DIA. OF HOLE IN RING FIGURE 6.42
6-78
1.05 ~
1.le
V
V
/'"
r--
V
BUSHING
---~ -L~ ~
2
o
STEEL
;
I
1
t
,....f MAG. 1,1- RING
MAXIMUM INTERFERENCE FITS OF STEEL BUSHINGS IN MAGNESIUM ALLOY RINGS
1.6
1.8
~. ,
" L--.
\'
"~-:-a:aell..~ . -,'/3
STRUCTURAL DESIGN MANUAL
The presence of ha~d brittle coatings in holes that contain a press fit bushing or bearing can cause premature failure by cracking of the coating or by high press fit stresses caused by build-up of coating. Therefore, hardcoat or HAE coatings should not be used in holes that will subsequently contain a press fit bushing or bearing.
:J
Figures 6.40 and 6.41 pe~it determining the tangential stress, ft, for bushings pressed into aluminum rings. Figure 6.40 presents data for general steel bushings and Figure 6.41 presents data for NAS75 class bushings. Figure 6.42 gives limits for maximum interference fits for steel bushings in magnesium alloy rings. 6.4.6
Stresses Due to Clamping of Lugs
Joints which are clamped should be checked for the residual stresses developed in' the lugs. This ftclamp-up" stress can be determined in the following manner. Figure 6.43 shows a typical lug/clevis joint subject to clamping.
r-
L
t
.:•
i
! I
I I
I
I I
.
i I
I I
1
I I I I
I
l
I
l
I
FIGURE 6.43 - TYPICAL LUG/CLEVIS JOINT The stress produced in the lugs by clamping is f = 3Eot/L2~ where 0 = (x r - d)/2. This assumes that the total clearance x - d is equally divided between each lug. If some other distribution of clearance is required, the stress in each lug must be calculated ..
e
The allowable stress t Fall = Fty /2, should not be exceeded in order to minimize the possibili ty of s tx:ess corrosiog. failure~ r ' 6.4.7
Single Shear Joints
In single shear joints lug and pin bending are more critical than in double shear joints. The amount of bending can be significantly affected by bolt
6-79
•
STRUCTURAL DESIGN MANUA.L The presence of hard brittle coatings in holes that contain a press fit bushing or bearing can cause premature failure by cracking of the coating or by high press fit stresses caused by build-up of coating. Therefore, hardeoat or HAE coatings should not be used in holes that will subsequently contain a press fit bushing or bearing. Figures 6.40 and 6.41 penmit detenmining the tangential stress, ft, for bushings pressed into aluminum rings. Figure 6.40 presents data for general steel bushings and Figure 6.41 presents data for NAS75 class bushings. Figure 6.42 gives limits for maximum interference fits for steel bushings in ,magnesium alloy rings. 6.4.6
Stresses Due to Clamping of Lugs
Joints which are clamped should be checked for .'the residual s tresses developed in' the lugs. This "clamp-up" stress can be determined in the following manner. Figure 6.43 shows a typical lug/clevis join~/subject to clamping. /
t--
L
:
f
t
I
I I
•I
·
I
I I
I
)
I
I
I
I
•
: t
I
.: , I
I
FIGURE 6.43 - TYPICAL LUG/CLEVIS JOINT
2
The stress produced in the lugs by clamping is f = 3Eot/L ; where 5 = (x - d)/2. This assumes' that the to tal clearance x - d is equally divided between each lug. If some other distribution of clearance is required, the stress in each lug must be calcub;'ted. The allowable stress, Fall = Fty/2, should not be exceeded in order to minimize the~,possibility of stress corrosion failures. 6.4.7 •
Single Shear Joints
In single shear joints lug and pin bending are more critical than in double shear joints. The amount of bending can be significantly affected by bolt
6-79
STRUCTURAL DESIGN MANUAL Revision B clamping. In the case and the bending moment Therefore, even though is applicable to solid
considered in this section, no bolt clamping is assumed in the pin is reacted by socket action in the lugs. Figure 6.44 shows a bolt/bushing arrangement, the analysis pin/sockets.
p M
w
FIGURE 6.44 - SINGLE SHEAR LUG JOINT In Figure 6.44 a representative single shear joint is shown with centrally applied load (p) in each lug and bending moments M and Ml that keep the system balanced. Assuming no gap be tween the lugs, M + Ml = p( tl + t)/l. The individual values of M and Ml are determined from the loading of the lugs as modified by the deflection, if any, of the lugs. The following analysis procedure is applicable to either lug. The joint strength is determined by the lowest margin bf safety of the various failure modes. The bearing stress distribution between lug and bushing is assumed to be similar to the stress distribution that would be obtained in a rectangular cross section of width, D, and depth, t, subject to load, P, and moment, M. At ultimate load the maximum lug bearing stress, fbr, is approximated by
where kbr is the plastic bearing coefficient for both the lug material and is assumed to be the same as the plastic bending coefficient for a rectangular section which may be found in Section 9.6. The ultimate allowable is found by the methods defined in Section 6.4.4 for shear bearing.
6-80
e!
STRUCTURAL DESIGN MANUAL Revision A The ultimate tensile stress in the outer fibers in the lug net section is approximately
where kb is the plastic bending coefficient for the lug net section. The allowable ultimate is found by the methods defined in Section 6.4.4 for axial tension. The bearing stress distribution between bushing and pin is assumed to be similar to that between the lug and bushing. At ultimate bushing load the maximum bushing bearing stress is approximated by
where kbr, the plastic bearing coefficient, is assumed the same as the plastic bending coefficient for a rectangular section. The allowable ultimate value is Fey for the bushing material. The maximum value of pin shear can occur either within the lug or at the common
shear face of the two lugs, depending upon the value of M/Ft. At the lug ultimate load,the maximum pin shear stress (£s) is approximated by fs = 1.273 P/Dp2; (M/Pt ::;;2/3) fs
1.273 P
Dp2
j
2M/P t) 2 + 1
- 1
(M/Pt
)t
2/3)
«2M/Pt)+1- v(2M/Pt)2
The first equation above defines the case where the maximum pin shear is obtained at the common shear face of the lugs. The second equation defines the case where the maximum pin shear occurs away from the shear face. The allowable ultimate is Fsu of the pin material. The maximum pin bending moment can occur within the lug or at the common shear faces of the two lugs, depending on the value of M/Pt. At the lug ultimate load,the maximum pin bending stress (f bu ) is approximated by 10.19 M
kb Dp3
10.19 M
kb Dp3
(~~
- 1) ; (M/Pt
(vi 2M/Pt) 2 + 1 2M/Pt
~ 3/8)
(M/Pt > 3/8)
where kb is the plastic bending coefficient for the pin. 6-81
STRUCTURAL DESIGN MANUAL Revision E The equation for (M/Pt < 3/8) defines the case where the maximum pin bending moment is obtained at the common shear face of the lugs and the equation for (M/Pt > 3/8) defines the case where the maximum pin bending moment occurs away from the shear facet where the pin shear is zero. The allowable ultimate value is Fbu for the pin or if deflection or fatigue is critical Ftu should be used. 6.4.8 Socket Analysis The method presented here applies to sockets or sleeves made of aluminum or steel alloys. It is based on the assumption that the socket or sleeve walls (section cut by a plane parallel to the beam or pin centerline) are rectangular or nearly rectangular. The method for obtaining bearing pressures within the socket or-sleeve is also applicable to sockets or sleeves whose wall cross-sections vary appreciably from rectangular. An analysis suitable to the wall configuration must be used for the determination of the wall strengths. This method may also be used for the analysis of single shear lug joints by considering the lug as a socket and the bolt as the beam.
The maximum allowable wall strengths of sockets or sleeves having rectangular or nearly rectangular wall cross sections (section cut by a plane parallel to the beam or pin center-line) may be determined from the following equations. Note that \vhen e/D approaches 1.0 Pall may be larger than the allowable lug load as determined by Section 6.4.4. In all cases the lesser of the two allowables should be used for the margin of safety. (D/t ~10)
1~~~~~~1oKtFtu;
(D/t >10)
t t
J
e
-----1-- -rt T-
--+---
--------1- --- -- _. . _-
L-I
the above result in pounds per inch e = edge distance of socket, inches D = diameter of beam or bolt, inches K = tension efficiency factor, Figure 6.32 t Kb = bearing rupture factor, Figure 6.45 ' ' tensl'1 e strength , PSl F TU = u ltlmate tt~ wall thickness of socket, inch 6-82
•
STRUCTURAL DESIGN MANUAL
• 2.6
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2.4
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eOw-------~2~--------4------~--6~--~~~8~~~--~1~O------~~12
D/L
FIGURE 6.45-BEARING RUPTURE FACTOR .... K b ru
6-83
STRUCTURAL DESIGN MANUAL 90------------~----------~-------
e
80
.,
:
7 0 -::. _... :
60
I
.
.~.-
.. 1.. .....
~ h_~_
! . "
(PaIl)e
.
,
,I. .
#
~,.
~'
20 . '-.. -~. .
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.;
1 O·~~~~·l_·_·L._} ~.
;..
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--
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•
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.5
FIGURE 6.46
.. -
_.
••
~
"
__ f
.6
f
'
~_,.~j
~ -..,
__.i_,
,,,,. t ,~ :
;",- .. ~.
t- ,
t
~_.~I~'.~._I_.
_'
_l._,~
_____
1.0
•
ALLOWA BLE LATERAL LOAD
• 6-84
I
8
7
P .... unit lo.ad -kips/in
-
G ')
M ..... moment .... in-kips
:z 31:
at end of socket
:III
PI = KIS/L
Z
c:: :III r-
P2 == K 2 S/L a =KaL Smax== -KsS
'¥max= Km SL
o M/SL
.2
,4
.6
.8
0"-
•
00
0"
__------__----.----~~----~----~--~~~~~10
9 /-.--..r.---+ .....---+.-~-::..001
fI
L
P"-'unit load'"-lkips/in
Ml'oJmoment
in-kips at 'end of socket
P
1
P2 a
=K 1M/L2 2
K2M/L
= KaL
Smax=-KsM/L Mma x = Km Ivl
IIOJ
-
~ ~\~
~ell ~.. ~
STRUCTURAL DESIGN MANUAL Revision B
Figure 6.46 shows the correction factor for obliquely loaded sockets. Multiplication of the axial allowable loading of the socket or sleeve by this correction factor gives the allowable loading when the load is applied at an angle to the axis of symmetry. Figures 6.47 and 6.48 show the methods for determining the bearing pressure loadings in the beam - socket. 6.4.9 Tension Fittings The analysis methods presented here apply to the most common types of tension fittings. The fitting design should also comply with the following design practices:
•
A. Bolts highly loaded in tension should be assembled with a washer under both the head and the nut. B. Eccentricities in fitting attachments should be kept to a minimum. C. In order to keep deflections to a minimum and to increase fatigue life, it is desirable that the fitting end pad thickness be not less than the bolt diameter for aluminum alloy fittings, nor less than .70 of the bolt diameter for steel fittings. D. Fitting and casting factors as specified in Structural Design criteria shall always be used in the analysis. If in any application both a fitting and a casting factor are applicable, they shall not be multiplied, but only the larger shall be used. 6.4.9.1 Wall Analysis Tension and bending stresses in the fitting wall may be determined by conventional methods •
.)
6.4.9.2 End Pad Analysis Tension fitting end pads should be analyzed for both shear and bending. The shear surface area is the bolt head periphery through the end pad. Then the shear stress is p
21Tro t e
tit
where r is the bolt head outer radius and te is the end pad thickness. ~nd pad bending in common types of tension fittings may be analyzed as shown in Figures 6.49 through 6.55.
6-87
STRUCTURAL DESIGN MANUAL Revision B r.~ Bolt Hole
Radius (Nominal
Wall
Thickness tw
d
.. '- ,·X:o
P
Applied Load
Effective
Bearing Area
Bolt Head Radius
Ref. Figures 6.51 and 6.52 FIGURE 6.49 BATHTUB FITTING - END PAD BENDING ANALYSIS
Wall . Actual Fitting Bathtub Fitting
P Applied Load
a d
:::'I
A
=
+
B
TT
a -
(C
+ 2
f D)
b
=
'p
-:-1Kl K2 t e
Ref. Figures 6.51 and 6.52
FIGURE 6.50 ANGLE FITTING - END PAD BENDING ANALYSIS 6-88'
•
STRUCTURAL DESIGN MANUAL r./a J.
.4
3.-
Revision B .3.....,. .2 .1 t-
2
1
•
o 1.0
1.5
2. 0 ' 'a'-d
2•5
3.0
ro
FIGURE 6.51 BATHTUB
FITTING - END PAD K1 VALUES
1.0 .,:
.8
KZ
.6
.4
•
.3
o
FIGURE 6.5Z BATHTUB
2
4
6
8
10
FITTING - END PAD K2 VALUES
6-89
STRUCTURAL DESIGN MANUAL Revision B
=
f
b B- B
P(2d - tb)K3 t
e
2a
Ref. Figure 6.55
Section A-A
a
ct
~
Wall
FIGURE 6.53 CHANNEL FITTING - END PAD BENDING ANALYSIS
Ref. Figure 6.56 FIGURE '6.54 PI FITTING
6-90
Base
STRUCTURAL DESIGN MANUAL Revision B .10
1.0
.13 .. 9 .16
.8 .20
.7
.25
.30
.6
.5 .40
) .4
t r./a l.
.3
1.0
1.5
2.0
2.5
3.0
b/a FIGURE 6.55 CHANNEL FITTING
END PAD K3 VALUES
6-91
STRUCTURAL DESIGN MANUAL Revision
B
Figure 6.54
~et.
t~
,!~~CIRI _C"
First Calculation
A
o
t~
mil iRB:
2"
'(6I t ~
l!.~ C
Center BqltlLoad Assumed Distributed I I I I I
I
•
I I
• -
I
I
M
l~
YR'A Second Calculation Assuming 50% Fixity
Moment Diagram
1. Determine the fixed end moment for a beam continuous over three supports assuming all loads as concentrated. 2. Assume 50% of the above values as-end moments and determine bolt loads (concentrated). Analyze the end bolts for the combined loading (moment plus tension) and the center bolt for direct tension. 3. To determine the bending moment curve, assume the center bolt load computed in (2) is uniformly distributed over the bolt head flat. FIGURE 6.56 PI FITTING - END PAD BENDING ANALYSIS
6-92
STRUCTURAL DESIGN MANUAL
e
Revision C
i
ALLOWABLE *LOADS FOR SINGLE ANGLES - -I, I
24S-T
i
p
= ~30~(~)~·7(3.3
400
+ lo910Ll
t
p
, • 09 '. '102
200 t
1 I
L
I I
1.00; til
IX'!
..,:1-
e
8_0_
I
J--- ·
032
-l-- . .-.
X _DJL HH
= LENGTH
1,---------·
?25---.. . .
!
--~
(IN)
..
: --,!
: ~
-+ . . :.. 020.+
~
!:
40·
-
! I I
IL
= 1 JL = 2 L = 3 L = 4
-1
___ 20.:
TYPICAL
_____ i,
)
,
, ;
I
i
,
___ .In________
i
J _______.__.___ i ___ .,._ ."
Notes:
t
~J
' .1.___.. L".__L.J ' ."
,::.
..'"
J Ctf _.;.1 I~CHES
L_ ..
'.
'1,
"'.
b.
*These are limit loads. This design is not recommended where the angles are subjected to repeated loads which produce high stresses. These curves are for leg A restrained, for little restraint use half these values. Beaded corners increase the failing BM approx. 6.5 times. Deflection is decreased about 50%. Ref. LAC 8M 63 P 2. For thick angles, the bolt may be critical. . Ref-. Rpt. R-900, pg. 11 Figure 6-57 Vega A.C. 6-93
STRUCTURAL DESIGN MANUAL Revision C 'r,
DESIGN
j 1
I
!! 1
. t
1 ... -;. J
,
•
i .,.
i.l.t:
!
--.....-----~I
·f· ;
~c t~
I
'L
=
1.
.L
~-~-~trT·
=
t
~
i
.,
! : I length, (in) l \
!
I
l.St > R > tl
I
2000
,j i
1500
In
== ~/8
t
1000
,
. ~ ..
. •
.-- r
..0 ~
t
e
I
. I -'l --,. .:.,I .._~I__ !L... L ...
800
i
; - .!i -..
."" ~
Po.
r'
C"T"'-
1...
j.
:i .!. '... I I
'
. ! .-'~.-.
6.Q,0
I :
t·--
·1
I
~'-f," .... j.-i
400
'-f --- ~
=
t
I
.1I I
~L
200
= 1
.i
i
i·: j
". -t- -, -- "! .-,- .
.L = 2 L
=
3
L = 4 1
.
-
........,--
1 "
.-
typical
.
.
~
.:..._. j ......_.- .-l __ --L-.i.. . _J~..-,--->O~_30...-..:L.-J'-,
"s
.!t
.w
.!
.h
.~)
C -
1.
Note: These curves are for leg For little restraint use values .. Ref. Rpt. Vega . Figure 6 -58 6-94
:-1.
4.
E...
6.
inches A restrained. half these R-900, pg. 10 A.C •
STRUCTURAL DESIGN MANUAL Revision C ALLOWABLE COMPRESSION IN L[GS OF FORMED ALUMINUM CHANNELS
tIS
UJ
a:
t;;
UJ
20~~+-~~·~~~~~~~~~~~~
>
u:;=
LEG 1 IN. WIDE AS TESTED
CJ')~
wo::U')
t
0..0 ~w
0....J
Uz
R'= 1.50t
w..J al <{
:s:o j
<{
0.50 IN.
~4::::=:::::rt=i .....:: '--::'"""~"-v" -I N. 80 L TIN T E5T
Dcflectic' ,,; At limit load 12 w..u.L::.:J.;:..L!J.I.L.I.J..U 0.04 . 0.08
0.12
~
0.02 in.
At ultimate load ~ 0.05 in. NOTE: No allowance need be made for tensile
0.16
strength of material.
THICKNESS OF CHANNEL t (in.)
FIGURE 6-59
ALLOWABLE TENSION IN LEGS OF FORMED ALUMINUM CHANNELS 40
.:;.
.:.:
(/)
U'"J LU
0:::
l-
V)
30
til
...J (/)
Z
w
I-
oUJ ....J
20
on:
10
<.9 ut -1
co
~
o .~
...J
..J <{
0.04
0.24 TOTAL THICKNESS OF BASE t (in.) (includes filler)
NOTE: Deflections at limit load up to 0.025 in. per channel; up
to 0.07 in. at ultimate toad (applicable through range of chart).
For radius larger than l.5t, increased bolt spacing or, bases or fillers less than required for minimum deflections, conservative allowances must be made.
STRUCTURAL DESIGN MANUAL Revision C 6.5
CABLES AND PULLEYS
The strength of MIL-W-83420 flexible wire rope is shown in Table 6.11 for var.ious size cables. The load/deflection relationship of these cables is show~ in Figure 6.49.
NOM. DIA.
1/32 3/64 3/64 1/16 1/16 3/32 3/32 1/8 5/32 3/16 7/32 1/4 9/32 5/16 11/32 3/8
)
TYPE
OF CONST.
MIL-W-83420, TYPE I (NON-JACKETED)
COMPOSITION A COMPOSITION B APPROXIMATE WEIGHT CRES CARBON STEEL LBS/IOO FT MIN. BREAKING MIN. BREAKING STRENGTH (LBS) STRENGTH (LBS)
7x19
7x19 7x19 7x19
.16 .33 .42 .75 .75 1.60 1.74 2.90 4.50 6.50 8.60 11.00 13.90 17.30 20.70 24.30
110 230 270 480 480 920 920 1760 2400 3700 5000 6400 7800 9000
110 230 270 480 480 920 1000 2000 2800 4200 5600 7000 8000 9800 12500 14000
3x7 3x7 7x7 7x7 7x19 7x7 7x19 7x19 7x19 7xl9 7x19 7x19
--
12000
Cable Terminal Efficiency (% of Cable Strength) Five-Tuck and Nicopress Type Splice-Flexible Cable ••.. 75% Shackle and Ferrule Loop Terminal - Hard Wire .•..••.•• 85% Wrapped and Soldered Splice - 19 Strand Wire •.•..••..• 90% Swaged Ends ..........
TABLE 6.11
It
••••
III
...........
I
...........
.,
••••
'"
.1007D
STRENGTH OF STEEL CABLE
6-97
STRUCTURAL DESIGN MANUAL Revision C t= 1r tJ P r. G. all "'''T~
8000
~n~
,. n ~ f' - f" r
~,
(',I\RPnt-' ST~F'.
F r T I (V·, r/\f~1
r-s
(RFC. ,111.-llf)r.y 5)
3/8" 7xlr'1
7000
6000
5000
til
.c
-
C
/4H 7x19 4000
0
c.n C Q.;
tQ)
..r
3000
It! L
2000
1000
7x7
o
.002
.004
Elnny,ation 6-98
.OOG
.008
Inches/Inch of rahle
•
STRUCTURAL DESIGN MANUAL
•
SECTION 7 PLATES AND'MEMBRANES 7.1 Introduction to Plates This section covers the analysis of plates as commonly used in aircraft structures. In general, such plates are classified as thin; that is deflections are small in comparison with the plate thickness (y ~ t/2). These plates are capable of carrying compression, bending, and shear loads; however, critical values of these loads produce a wrinkling or buckling of the plate. Such buckling produces unwanted aerodynamic effects on the surface of the aircraft, and also may result in the redistribution of loads to other structural members, causing critical stresses to develop. Thus, it is essential that the initial buckling stress of the plate be known. In addition, if the buckling stress is above the proportional limit, the panel will experience ultimate failure very soon after buckling.
.)
The critical buckling of a plate depends upon the type of loading, plate dimensions, material, temperature, and conditions of edge support. This section considers the various loadings of both flat and curved plates, with and without stiffeners. Single loadings are considered first followed by a discussion of combined loadings. 7.2 Nomenclature for Analysis of Plates plate length stiffener area plate width effective panel width core thickness, signifies clamped edge compressive buckling coefficient for curved plates strain modulus of elasticity secant modulus tangent modulus
a A
b b
st ei
c C
e
)
E E
E
S
t
secant and tangent moduli for clad plates ratio of cladding thickness to total plate_thickness F stress FO. 7 ' FO. 85 secant yield stress at O.7E and D.8SE F critical normal stress cr E , E
fS
F
cr
F
t
critical normal stress, clad plates critical shear stress
crs Fpl
stress at proportional limit
Fcl
proportional limit of cladding
F
compressive' yield stress
F
cy f
FR
crippling stress free (refers to edge fixity)
7-1
STRUCTURAL DESIGN MANUAL g k k C k
number of cuts plus number of flanges buckling coefficient compressive buckling coefficient shear buckling coefficient
kc L' n p P r R ss
equivalent compressive buckling coefficient effective column length shape parameter~ number of half waves in buckLed plate rivet pitch total concentrated load radius of curvature stress ratio simply supported thickness skin thickness web thickness total cladding thickness unit load total load, potenti~l energy deflection 2 2 ~ length range parameter b (1 - v ) Irt ratio of cladding yield stress Eo core stress crippling coefficient ratio of rotational rigidity of plate edge stiffness plasticity reduction factor cladding reduction factor buckle half wavelength inelastic Poisson's ratio elastic Poisson1s ratio plastic Poisson1s ratio radius of gyration .
s
t
t t
S
W
t 1 w
w y
Zb ~
Pg
E
~
n A
v v ve
!
7.3
Axial Compression of Flat Plates
The compressive buckling stress of a rectangular flat plate is given by Equation (7-1).
Fer =
~~(
k
1T
2 E
2
l2(1-Ve)
)(
t b
r
(7-1)
The relation is applicable to various types of loadings in both the elastic and the inelastic ranges and for various conditions of edge fixity. The case of unstiffened plates is treated first and then stiffened plates are discussed. The edge constraints which are considered vary from simply supported to fixed. A simply supported edge is constrained to remain straight at all loads up to and including the buckling load, but is free to rotate about the center line of the edge. A fixed edge i6 constrained to remain straight and to resist all rotation.
7-2
STRUCTURAL DESIGN MANUAL These two conditions define the limits of torsional restraint and are represented by E = 0 simply supported edges and E =
Plates are frequently loaded so that the stresses are beyond the proportional
)
limit of the material. If such is the case, the critical buckling stress is reduced by the plasticity reduction factor ~ , which accounts for changes in k, E, and v. This allows the values of k, E, and v to always be the elastic values. The second reduction factor in Equation (7-1) is the cladding factor Tf. In order to obtain desirable corrosion resistance, the surface of some aluminum alloys a~e coated or clad with a material of lower strength, but of better corrosion resistance. The resultant panel may have lower mechanical properties than the basic core material and allowance-must be made. Values for the factor if are given in Table 7.1.
Loading
FcI
Short plate columns
1
+ (3{3f/4) 1 + 3f
or
1 1 1
+ 3f
1
1
+ 3f
1
+ 3f
+ 3{3f
+
Fcr>Fpl 1
1
Long plate columns Compression and .shear panels
F crs
3f
1 1
+ 3f
Table 7.1 - Simplified Cladding Reduction Factors
)
7.3.1
Buckling of Unstiffened Flat Plates in Axial Compression
The buckling coefficients and reduction factors of Equation (7-1) applicable to flat rectangular plates in compression are presented in this section. Figures 7-1, 7-2, and 7-3 show the buckling coefficient kc as a function of the ratio alb and the type of edge restraint; and, in the case of Figure 7-2, the buckle wave length and number of half waves. Figure 7-4 shows kc for infinitely long flanges and plates as a function of the edge restraint only. The edge restraint ratio € is the ratio of the rotational rigidity of plate edge support -to the rotational rigidity of the plate.
•
The condition of unequal rotational support can be treated by Equation (7-2) •
kc =
l:z (kc1 k c 2 )
(7-2)
7-3
STRUCTURAL DESIGN MANUAL Revision C The coefficients kCl and kC2 are obtained by using each value of Figures 7-5, 7-6, and 7-7 present kc for flanges. a long rectangular plate with one edge free. The plasticity reduction factor given by Equation (7-3).
~
E
independently.
A flange is considered to be
for a long plate with simply supported edges is
2
0.500
+
0.250
[
1
1 - V
For a long plate with clamped edges, the factor is given by Equation (7-4).
~
2 =
[(:s )( 1:::2 )] j
0.352
The value of the inelastic Poisson ratio
+
J)
0.324
[
(~s
1+
E
t
)
]
I~
(7-4)
is given by Equation (7-5).
(7-5)
•
The tangent and secant moduli can be determined from the Ramberg-Osgood relation as shown in Equations (7-6) and (7-7).
E
E
1
1
+
+
(--~ ) (
(7-6)
F
F
0.7
-F_F_ _ )
n-l
(7-7)
0.7
Figure 7-8 shows the characteristics of stress-strain curves used to determine the shape factor n. Table 11.1 lists values of E, FO.7, and n for various materials. Figures 7-9 and 7-10 present values of kc for plates restrained by stiffeners. This data is included here instead of in the section on stiffened plates because the stjffeners are not a part of the plate. To be noted is the effect of torsional rigidity of the stiffener on the buckling coefficient of the plate.
7-4
•
STRUCTURAL DESIGN MANUAL
•
Revision C
16r----------------------------------14
c
c
A ____
B
c
S C
3___ E 3 E ~3~c r E 3~D---....-E 3 §} s
12
S
a ---,
FR
s
E
10
FR
Type of
- - - - Loaded Edges Clamped
•
8 k
c
Support Along Unloaded Edges
- - - Loaded Edges Simply Supported
-----....-..--.........-...------.-.... A
6 B
4
)
~--~------~----------~c
2
--D ----~~------~-----~-:-~-~-~-=JE --_
o~ __~------~----~------~ o ____~---1 2 3 4
5
alb
•
FlrflPE 7.1
COtH)rES~f'lE
Rl1CKLPfG rOEFFIf,(fNTS F0P GlAT REfTI\"Jf.IJL/\R rLI\TfS
7-5
\.
\
STRUCTURAL DESIGN MANUAL
• 10.0.-------------------------------------------. Maximum kc at transition from 1 buckle to 2 buckles
)
\ A is ~ buckle wave length n is no. of half waves in buckled plate
k
c
• 5.
o 4.~--~--~--~~--~~~------~--~~--~
.4.5
.7.8.9
1.0 1.1
1.2 1.3
1.4 1.5
'Alb
FIC"HE 7.2
7-6
COnPRESSIVE R"C~(L'r'r rnEr-FICIE~TS OF PLATES I\S.A F"~lCTln~j OF Vb FO R 'I AR I nus At10llNTS OF E()GE ROT 1\ T I 0 ~f\ L RE STRI\ I NT
•
\
STRUCTllRAL DESIGN MANUAL
•
15~------------------------------------~
I I
,
,
\
', . ·\ x \
k
c
\
Loaded edges clamped
,, _'""" ....
" ........ _, ' .....
~
__
~
~~
____
ro
____
~
~
____
~--~~
______--.50
~
______
~20
____------~lO 5 2 1•5
o
O~--~--------------~--~--~--~--~--~ o . 4 . 8 1 . 2 1 . 6 2. 0 2. 4 2. 8 3. 2 3 . 6 4 .0 alb
FrGUrE 7.3
COtlPRESSIVE R"r.I
7-1
...... I
(X)
Flange c 1. 4
.01 i
•1
1 i
10
1.21-
1. a l-
100 • 14 .kc = 1. Z8C
~12
I~
.-1 10
I~
f
::a
-I .81-, kc
.6L
Flange
. 41-
\V
.,.."
...
Y
~'~I
~
,..",
kc at
~ t
F'
Ie:
-I 8
=6.96 £=
<:0
~
6
-I
4
::a
I~
kc Pla.te
C
.""enCD
IZ
2
o·
.1
,
I
I
1.0
10
100
Plate
FlbUP,E
e.
7.4
i:
•0 1000
IZ Ie:
€
I~
Co~,~ PRE SS I V E ~ PC KL 'H1 CO E F Fie rENT S , Fn R r NFl NI TE L Y l ON G F L l\ NGES l\ ~ D PL ATE S AS FUN CT ION S f) F F~ r, E RO TAT I 0 NA L RES T RA PIT S
e
e
. .7\, '..
STRUCTURAL DESIGN MANUAL Revision F
1.9
\
\
1.8
\ \
\
1. 7
\ \
\
1.6
\
Loaded Edges \
1. 5
Clamped
\
1.4
,, ,
, "" "
1.3
50
1.2
20 kc
ttY
1• 1
-
10
1.0
5
.9 2
1
-
•7
.5 .2
.6
)
. 1
.. 5
.4
o 0
3.0
3.5 4.0
4.5 5.0
alb rlf"fq;1
7.';
r:()Hf>,:r~r;r"f." ~flr.!~Llr'r. r.nr:r-r.lrfr.~T«) 0r rL,\~'rF~ .,\r; "r:ll~'rT'f)~1 nr. a / h r- n R ~ //H~ r () t' S f~ , In U ~!T 5 rq: r n r r ~ n T '\ T I ()' 1:\ l 11 F ~ T P~. , fiT <)
7-9
STRUCTURAL DESIGN, MANUAL,
1.9
1.8
1.7 1.6 '1. 5 1.4
1. 3 k
c
1.2
Maximum kc at transition from 1 buckle to 2 buckles
1.1 1.0
•9
.8 ·7
.6 ·5
.4
-----------------0 0
1
2
3
4
5
6
7
8
9
10
Alb rlrpr.[ 7.G
COnrrESS1\Jr ()~
7-10
AI . . r-nn
I1l'C~(Llt'r CrJEFFIr.IEnrS f)F FlArr:FS AS ," r.r'rtrT,n~! nfl~ 1\'1np·IT~ nr ~]trrE rflTAT t n'lt\L. f"r.r;Tn.1\ Inrs
\,,,0,
• 1. 20
2.5
.i 1. 16 At L!b=O
(f)
2.0
1. 14 'V
1. 12
e
1. 5
--t
kc at a /b ==
::a' c::
0.456 kc
kc
1. 08
1. 0
1. 04
.5
.s
1
.30
.• 426
.35
.395
c:-)
-I
c: ::a ::DI
r-
=
10
fI1
-
(f)
alb
L/b
(b) Hinged flange s
(a) Plate columns with hinged. loaded edges 2 . 2 Fer = kc TT E/(L/t)
CD
z
12
Fir" PE 7. 7
r. () ~
it:
PH~ ~ s , V E Po VC~~ L t r r !\ S ,"\ r- t!'! r. T I f) ~ 1 fl r- pn, 4
I
r: f) E r. 1= I r. I r: ~ T ~ I") S().! t ~ r :\ T , (1
f)
r.
::DI Z
r L 1\ T F. Cn L tI ~ ~ r'\ f\ ~ 'f) r: t.." ~ ~ F ~
c:
~
I'D
< .....
1-'fI.l
0
::1 '"rj
::DI
r-
....., I
20.0
\
15.0
\
10.0
9.0
FO.74-____S_1_0p_e__=~E~---O~.8-5-E~O_.~7~0-E~ FO.85~--------~~--~
"-
6.0
F pl
n
~
"'"
8.0 7.0 5.0
4.0
,,
,--""-
3.0 2.5 2.0
t F
""' "-
~
~ r--.....
--....
1. 5 1.0 1.0
e
...
1.1
l.2 1.3
1.4 1.51.61.71.&1.92.0
FO. 70/ F O. 85
(a) Significant stre s s quantities on a typical stress-strain curve
(b) Dependence of shape factor on ratio FO. 70/ F O. 85
n= 1 +loge(l7/7)/loge(F O. 70 /FO.8S}
r- I ("f I r~ [: 7. 3
C1' l\ P/\ r. T [ n 1ST I C~ '1 r. ST r. ~ ~ ~ - S T P!\ I tJ C I , r: \.) [s r. nr r; T P" CTIt PAt '\ L Ln Y ~ nr n 1(' T t ~':G n t "T f TIE ~ (I SEn I r TI I r THrEF r /\ fV\' ·ETr:: r t 'E T' !() n
1;'
-
STRUCTURAL DESIGN MANUAL
6p-------~--------_P--------._------~
a - - -....
f Fx bIZ
--1
5
Fr
:i Fx
at
0
0 •2
4
.05 k
c 3 F
..:t..= F x
(~Ar )
Ar
•5
•1 · 15
1 .. 0
.2 .25 .
2.0 5.0
•3
Q)
(1 + :~) 1
-2
3
4
a~
)
FIGURE 7.9
COHPRESSIVF.: RnCKLIN~ COEFFICIENT OF FLAT PLATES PESTRAINEO AGAINST LATERAL EXPANSION (Poisson's ratio = 0.3)
7-13
STRUCTURAL DESIGN MANUAL
2~--------------~--------~------~------~------~ o 50 100 150 200 250 300 bIt
FIGURE 7.10
CO~~PRESSIVE
BUCKLING COEFFICIENT FOR LONe RECTANGULAR STIFFENED PANELS AS A FUNCTION OF bIt ANn STIFFENER
TO RS , 0 f--.i\ l R r Glor TV
7-14
)
•
STRUCTURAL DESIGN MANUAL
7.3.2
Buckling of Stiffened Flat Plates in Axial Compression
The treatment of stiffened flat plates is the same as that of unstiffened plates
except that the buckling coefficient, k, is now also a function of the stiffener geometry. Equation (7 .. 1) is the basic analysis tool for the critical buckling stress. As the stiffener design is a part of the total design, Figures 7-11 and 7-12 present buckling coefficients for various types of stiffeners. The applicable critical buckling equation is indicated on each figure. A plasticity reduction factor ~ applicable to channel and Z- section stiffeners is given by Equation (7-8), which is taken from Reference 7. 2
11
=
.95
(_:s)(_I-V:)
(7-8)
1- V
.)
For other structural elements such as hat and rectangular sections, no specific plasticity correction factor has been established. However, Reference 7 recommends using the correction for a long clamped flange. Valuesfor the buckling coefficient, k c , for axially stiffened, infinitely wide plates are given in Figure 7-13. Figure 7-14 presents curves for finding a value for k for plates with transverse stiffeners. It is noted that in these plotst the torsional rigidity, GJ, of the stiffener itself is used, whereas in most data for longitudinal stiffeners, no torsional properties of the stiffeners are included.
)
In this brief section on buckling, an attempt has been made to present data that is most often used for routine analysis. Should the user require a more comprehensive treatment of buckling, References 2, 11, and 12 are excellent sources of additional data.
7-15
STRUCTURAL DESIGN MANUAL
7~----~----~~----~----~------~~---'
6
5
kw 3
Z
.1
tw/tf
Lu TLwe[j
0.5
bf
.6 .7
.~
.8
.9 1.0 1. Z 1.4
'1.6
2.0"'- '1.8 0 0
.2
.. 4
.6
.8
1.0
1. Z
bf
bw
(a)
FIGURE 7.11
7-16
Channel and Z Section Stiffeners
BUCKLING COEFFICIENTS FOR STIFFENERS
STRUCTURAL DESIGN MANUAL
1
Flange Buckles First
6
--------
Fer
=
--
.
kw lT2 E
tw
2
:--z
12 (l-'Je 2)
bw
5
4
-
k
w
3
l~f
r-
tw/tf
0.5
2 tw
.6
Web " -
1
t{
0)
V
bw
.1
J
.9
.8 !.JLl .. 2 - 1 .. 4 ............... 1.6
Flange 0
.2
0
.4
.6
.8
1.Q
1. 2
hf bw (b)
.-
F t Gun E 7.11
H Sec t ion
(COt1T to)
Stiffeners
RUCKL I NG COEFF rei E NTS FOR ST I FFE NE RS
7-17
STRUCTURAL DESIGN MANUAL
7
tb/th 2.0 1.8
-==:1.6
6
,1. 4 5 1.2
4
h Side BUCkle First
~
X
--
),0
.9
b Side Buckles First
.8
3
•7
.6 th
2
T b
tb
1
0
0
.5
I.-- h--..J .2
.4
---L kh rrZ E Fcr =
.6
12 (l-'Je 2 )
.8
2
C~ ) 1.0
1. 2
bin (c)
Rectangular Tube Sec t ion Stiffeners
FIr,URE 7.11 (CONT'O>
7-18
BUCK.!.. I NC CO EFF I C I EnTS FOR 5T I FF EN ERS
..
STRUCTURAL DESIGN MANUAL
•
Revision E
6~--~----~----~----~----~----~----~----, \
1
5
4
Flange Buckles , First .",;,
Top Web Buckles First
r---- -, Side Web Buckles First
3
Top Web Side Web
z
t 1
.8 1 .0
o0
)
FIGtH1E 7.12
+
2
BIiCKlI
.4
~G
.6
.8
1.0
1.2
1.4
1.6
STRESS FOP. HAT SECTION STIFFENERS (t=tf=t",=tt)
7-19
STRUCTURAL DESIGN MANUAL
• 7------------------------------------~----~ Buckling of Skin Restrained by Stiffener/<
,
6
Buckling of Stiffener Restrained by Skin
(:5)
kc 1'2 F. 2 12(1 • Ve
5
)
2
S
4 k
c 3,
Z.O 1.8
z
1.6 1.4 1.2
1.0
1
.. 9 .. 8 "7 .5. 6
.2 .4 .6 .8 1.0 1.2 o o..------~------------------------------------~
b /b w
s
a) Web Stiffeners. 0.5 < FIGURE 7.13
<
2.0
CO~,1PRES5tVE LOCAL RUCKLI ~!G COEFFICtENTS FOR INFINITElV "II £) E In EJ\L r ZEO 5T I FFE ~JE n FLAT PLATE S
•
STRUCTURAL DESIGN MANUAL
7
6
5
4
k
c
0.3
3
.4
z
·5
•3 .4
1
·5
) .2
.4
.6
.8
1.0 .
1• 2
bw/b f b) Z-Section Stiffeners.
FI~tH~E
7 .. 13
t
It s =
w
0.50 and O.S9
(Cn~JTln)
COMPRESSIVE LOCAL BlJCKLIN(1 COEFFICIENTS FOR PfFfr-r,rELY NinE IDEALIZEO STIFFENEO FLAT PlATE~
7-21
STRUCTURAL DESIGN MANUAL
7------~------~------~----~------_r------~
OCr :::
6
k '1r 2 E _~c:_~_
12(1 - Ve 2)
t
( bs )
2
s
5
.63 4
k
•5
c
3
tf 2
1 {j b
.3
.4
J bw
ts tw
t
•5
Lbs~ •2
.4
.6
.8
1.. 0
1.2
wIb s c) Z-Section Stiffeners. t wIt s = 0.63 and 1.0 b
FIGURE 7.13
7-22
(CONT 'n) COt·1PRESS I V E LOCAL RUe Kl 'NG COE FF r erE NT~ FOP INFINITELY \-'JIPE IDEAlIZE[) STIFFENED FLAT PlJ\TF.<)
•
STRUCTURAL DESIGN MANUAL
7 tw t8 bf
6
bw
0.3) .4, .5 1.0
5
.7
4
k
c
3
Buckling of Skin Restrained by Stiffener.
......
.3
f
.. 4
~
Buckling of Stiffener Re.trained by Skin
.5 .. 3
.4
2
.. 5 1
k OC=C _r
)
1T2E
(t)2 _ 5_
12(1 _ Ve Z)
bs
. °O~------~.~2------~.4~-----.~6-------.~8----~1~.~O----~1.2 b
Ib
w s
d) T-Section Stiffeners. tw t
F I r.tJn E 7.13
f
=
1.0; b f t
f
> 10; bw > 0.25 b
s
(CONT If) COMPRESS I V f LOCAL RUCKll ~-G COE FF reI E NTS FOP. ItfFfNITELY \'JlDE IDEALIZED STIFFENEf' FLAT PLATES
7-23
STRUCTURAL DESIGN ,MANUAL
lOO----------~~--------~----~----~
GJ
-EI O. 1
75
k
c 50
.01
o
25
~
II
Stiffener
7
]IJ
a
zoo
100
I l
JEJ 300
EI bD
. (a) alb
FI~URE 7.14
==
0.20
l()~-'G'TUnJNAl COf1PRESSIVE qIlCKl.ING COEFFICIENTS FOP.
SlnPLY SUPPORTED PLATES ~',IJTH TRANSVERSE STIFFENERS
7-24
STRUCTURAL DESIGN MANUAL
50~----------~-----------r-----------'
)
40 GJ
EI
) Stiffener
o
25
50
75
EI bD·
(b) alb = 0.35
FIGURE 7 .14
(CONT' n) LONG I TUn I NAL COI-1PRESS I V E BUCKL I NG COE FF I C IE NTS FOR S I '1PL Y SUPPORT EO PLA TE S HI TH TRA NSVE RS E ST IFF E f'.'E RS
7-25
STRUCTURAL DESIGN MANUAL ZO~----------~------------~----------~
GJ
EI _
k
c
Stiffener
10
5
15
EI bD (e)
F1GtHtE 7.14
7-26
a/b=O· .. 50
(CO NT I [l) LONG' TUD I ~·lA L cor,1PRESS I V E RUC KL J Nfl COE FF Ie, E NTS FOR S H1PL Y SliP PORT ED PLATE S \'/1 TH TRA I\'S IJE RS E ST I FF E ~IE P.S
_
STRUCTURAL DESIGN MANUAL Revision E 7.3.3
Crippling Failure of Flat Stiffened Plates in Compression
For stiffened plates having slenderness ratios Ljlp ~ 20, considered to be short plates, the failure mode is crippling rather than buckling when loaded in compression. The crippling strength of individual stiffening elements is considered in Section 10. The crippling strength of panels stiffened by angle-type elements is given by Equation (7-9). Ff
--;;:. f3g Fey
[
gtwts A
(
"
Fey
E
) r,85 ~
(7-9)
For more complex stiffeners such as Y sections. the relation of Equation ( 7 -10) is used to· find a weighted value of twa
= (7-10) where 8i and ti are the length and thickness of the cross-sectional elements of the stiffener. Figure 7-15 shows the method of determining the value of g used in Equation (7-9) based on the number of cuts and flanges of the stiffened panels. Figure 7-16 gives the coefficient ~g for various stiffening elements.
If the skin material is different from the stiffener material, a weighted value of Fcy given by Equation (7-11) should be used. Here :ris the effective thickness of the stiffened.pane1. (7-11)
The above relations assume the stiffener-skin unit to be formed monolithically; that is, the stiffener is an integral part of the skin. For riveted construction, the failure between the rivets must be considered. The interrivet buckling stress is determined as to plate buckling stress, and is given by Equation (7-12).
Fi
=
2 E_) (_E7r_ YJ_YJ
2 12(1- v )
( _ t _ s )2
(7-12)
P
Values of (, the edge fixity, are given in Table 7-2. After the interrivet buckling occurs, the resultant failure stress of the panel is given by Equation (7-13). )
(7-13)
)
7-27
STRUC,JURAL DESIGN MANUAL
i
F
ccF
J
C
Fyi
FFIFF
c
I
g= 1 9
c
5 cuts 14 flanges 19 = g
g= 19
g= 19
Avera.ge g
g= 1 9
g= 18
= 18.85
(a) Y-stiffened panel
F
i,gl 1 1-! 1- !g). c
'2 cuts 6 flanges 8 = g
5 cuts 12 flanges 17 = g
Average g = 7.83 (b) Z ... stiffened 'panel
Average g = 16.83
(c) Hat-stiffened panel
FIGURE 1.15
7-28
t1 ETHOD OF CUTT' NG ST 1 FF EN EO PAN ElS TO OET ER~H NE g
STRUCTURAL DESIGN MANUAL
e\
1.0 \
.9
1
.8 .7
.6
--
.5
.)
f3 g
- -L
.4
e .3
-0
0
• •Z .3
r-
,r
~Z-panels
~
.,."",.- ~
-
AI ~~ ~ .".
-
Hat 8. ,Y -panels I
I
I
I I I
V - groove plate, extruded angles,. tubes, multicorner formed sections Formed Z-.panels Formed Hat panels Extruded Y -panels
I
I
I
.4
.5
.6
I .7
.8 .9 1.0
1.5
2.0
3.0
twIts
) FIGURE 7.16
CRIPPLING COEFFICIENTS FOR ANGLE-TYPE ELEMENTS
7-29
STRUCTURAL DESIGN MANUAL
Fastener Type Flathead rivet
st
Ff st
3
Counters.unk rivet
1
EN D F I X r T yeO E FFie r EN T S FOR AN r, L E- TV PE E l E~,~ E ~'T S
>Fw
-
<
- unstable
Panel Strength
stable
J"w bsts + Fist Ast Fw
TABLE 7.3
7-30
4
Brazier-head rivet
Stringe r Stability
Ff
E:
3.5
Spotwelds
TAB L E 7. 2
(F ixity - Cae [fie ient)
hsts + Ast
RrVETEn PANEL STRENGT~ DETERt1lrJATtflN
)
\
STRUCTURAL DESIGN MANUAL Revision E Here the value b ei is the effective width of skin corresponding to the interrivet buckling stress Fi The failure stress of short riveted panels by wrinkling can be determined. The following quantities are used: Ffst crippling strength of stringer alone .(Compressive Crippling Section 10) Fw
wrinkling strength of the skin
Ff
crippling strength of a similar monolithic panel
F fr
strength of the riveted panel
The wrinkling strength of the skin can be determined from Equation (7-14) and Figure 7-17. Here f is the effective rivet offset distance given in Figure 7-18. This was obtained for aluminum rivets having a diameter greater than 90% of the skin thickness.
(7-14)
Now, based on the stringer stabil'ity, the strength of the panel can be calculated. Table 7-3 shows the various possibilities and solutions. It is noted that in no case should should be used.
Ff~
> Ff • -Thus, the lower of these two values
The use of the coefficient k w is based upon aluminum alloy da ta for other materials. The procedure is to use Equation (7-15) for the panel crippling strength.
~)
17.9
I
\
(
)l~ (7-15)
7-31
STRUCTURAL DESIGN MANUAL
\
lOr-~~-r~~~----------'---------------------
k\lJ
5 4
3
2
FIGtH~E
7.17
EXPEPH1EtrTALLV nETE"R~1INEr COFFFlrlENTS FOR FAILURE IN
7-32
un I NKL I NG
~·1nDE
.
)
e
STRUC1~URAL
DESIGN MANUAL
1
)
.P. d
) ... J' F'~lJnE
7.18
EXPERlr1ENTALlY OETERMINEO VALUES OF EFFECTIVE RIVET OFF S ET ( P .= R t ve t Sp ac 1n g )
7-33
STRUCTURAL DESIGN MANUAL 7.4
Bending
of
Flat Plates
The bending of a flat plate can be caused by either in-plane or normal loads. In the former, the plate is subject to various buckling modes depending primarily on boundary conditions and aspect ratio. Here the critical parameter is the magnitude of stress at which buckling occurs, since redistribution of load also starts at that time. Thus, as with axial loads, it is important to know when local buckling due to bending may be expected. An exact analysis for a flat plate loaded transversely involves a very complex mathematical treatment. A plate can be considered as a two-dimensional counterpart of a beam, except that plates bend in all planes normal to the plate, whereas a beam bends in one plane only. Also, plates exhibit varied behavior depending on thickness and have therefore been classified into four types: thick, mediumthick, thin, and membranes or diaphragms. Since most aircraft applications of transversly loaded plates involve either medium-thick plates or membranes, only the two types are included herein. Further information is available from many sources. 7.4.1
Unstiffened Flat Plates, In-Plane Bending
The general buckling relation for plates subjected to in-plane bending is given by Equation 7-16, which has the same form as Equation 7-1. The only difference is in the coefficient, kb2
11 T]
2 l2( 1- ve )
(-~ )
(7-16)
Values of bending coefficient, kb ,are given in Figure 7-19 for various edge restraints and the number of buckles versus Alb, the buckle wave length ratio, and in Figure 7-20 for various edge restraints versus the ratio alb.
• 7-34 I
/
STRUCTURAL DESIGN MANUAL 53~------------------------------------51 CD
49
Maximum kb at Transition from 1 Buckle to 2 Buckle,s
'Alb at e= 5
kb /
VB
,5
) 2
Z3~----~--~------~------~----------· 2.5 '3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.511.512.5 'Alb FIGURE 7.19
'1ENntrH,-RtJCKlt~lr,
OF
COEFFIC1E~rrS OF PLATES AS A FUNCTION A/h FOR VARIOUS AMOUNTS OF ROTJ\TtOt!AL RESTRAINT
7-35
STRUCTURAL DESIGN MANUAL
60
...............;.,..."",,---.......-----............... 100
50 20 10
5 3
1
o
23
alb
F l(~lJRE 7. 20
7-36
BENDING-RlICKlINr. COEFFICIENTS OF PLATES AS A FIHJCTION OF a/ b FOR VAR I OUS Ar1011tJTS OF EnGE ROT AT I () Nt\ l RE STRA I NT
STRUCTURAL DESIGN MANUAL 7.4.2
Unstiffened Flat Plates, Transverse Bending
The data presented in this section are predicated on the following a'ssumptions: 1)
The plates are flat, of uniform thickness, and of homogeneous isotropic material.
2)
The plate width, b, is
3)
All forces (loads and reactions) are normal to the plane of the plate.
4)
The plate is nowhere stressed beyond the elastic limit.
5)
Poisson's ratio = 0.3; however, no significant error is introduced if these coefficients and formulas are used for materials with other values.
~ 4t
and the deflection,y, is
~
.St.
For unstiffened flat plates with various types 'of loading, the maximum stress and maximum deflection can be represented by simple relations by the use of a series of constants which depend upon the plate geometry and loading. Tables 7-4 through 7-9 present loading coefficients for use with Equations (7-17) through (7 -.22) • Equations (7-17) (a) and (b) pertain to rectangular, elliptical and triangular plates. Loading coefficients are presented in Tables 7.4, 7.7 and 7.8 respectively. (a)
(b)
F
Et 3
=
Kl w L t 2
2
(7-17)
Equation (7-18) (a) and (b) pertain to corner and edge forces for simply supported rectangular plates. Loading coefficients are presented in Table 7.5. (b)
)
V
=
KwL
(7-18)
Equations (7-19) (a) and (b) pertain to partially loaded rectangular plates with supported edges. Loading coefficients are presented in Table 7-6. (a)
y =
KwL 3
(b)
Et 3 Equations (7-20), (a), (b), and (c) pertain to circular plates. are presented in Table 7.9. (a)
y:::
KWa
2
Et 3
(b)
F
:::
(7-19) Loading coefficients
(7-20)
Equation (7-21) (a), (b), and (c) pertain to circular plates with end moments. Loading coefficients are presented in Table 7.9.
7-37
STRUCTURAL DESIGN MANUAL .
•
(7-21)
Equations (7-22), (a) and (b) apply to trunnion-loaded plates only. coefficients are presented in Table 7.9.
Loading
)
7-38
I
--
--
HANNER OF LOADING All edgea supported surface. ~-b
¢4f111JV t---
a
load avel:' entire
All edges aupported, distrlbuted load varying linearly along length. L • b
____
~V Ii. t----
unllo~
All edges supported. distributed load varying linearly along bread th.
alb
l.0
1.2
1.4
1.6
1.8
2.0
2.5
l.O
3,5
4.0
Location of Maximum Stress and Deflection
Kl
0.287
0 • .376
0.453
O• .>L7
0.569
0.610
0.661
0.713
0.727
0.741
at the center
K
0.044
0.062
0.077
0.091
0.102
0.111
0.122
0.134
0.137
0,140
at the center
Kl
0.16
0.21
0.31
0.34
0.38
0.43
0.47
0.49
K
0.022 - I - - - -
PltO
0.28 0.048
0.053
0.060
0.070
0.078
0.086
0.091
1(1
0.16
0.21
0.25
0.28
K
0.022
0.031
0.0)8
0.045
0.051
0.056
0.063
0.067
0.069
0.070
1(1
0.)1
0.38
0.44
0.47
0.49
0.50
0.50
0,50
O.~O
0.50
center of long edges
K
0.014
0.019
0.023
0.025
0,027
0.028
0.028
0.028
0.028
0.028
at the center
K.
0.754
0.894
0.962
0.991
L.OOO
1.004
l.005
1.006
1.007
1.Ooa
at the' centeT
K
0.061
0,071
0.075
0.078
0.079
0.079
0.079
0.079
0,019
0.079
at the center
Kl
0.418
0.463
0.466
0.491
0.497
.498
98
0.499
0,500
Center of long edges
28
0,028
0.028
at the
0.733
0.742
0.750
center of short edges
-
-
0.25
38
0.38
L • b
b----to!
All edges fixed. unIform load over entire surface. L • b
AIL edges fixed. uniform load over small concentric 11 ... "1'0 2 v. ~lr~ular area of '['adiull r . o Wb 2 Max .!I .. Kl -;z- • Hax y • K i"tJ Lon& edges fixed, short ~dle8 supported, uniform load over entire surface. L .. b
"
,
Short edges fixed, long edge. supported. unLform load over entire .urf~ce. L • b
One long edge fixed, other free, short edges supported, unUoTlll load over entire surface. L - b ~..
-_.
One long edge fixed, other thTee ed&ea supported. L •. b unifo~ load over entire surface,
One short edle fixed. other three edges supported. uniform load over entire surface. L - b
one ahort edge free, other three edles aupported, unlform load over entire surface. L. b
0.021
0.ol4
0.026
0.027
0.028
0.02.8
Kl
0.418
0.521
0.599
0.654
0.691
0.715
K
0.02l
0.03'
o.o~o
0.066
0.080ll 0.092
Rl
O.H4
0.973
1.232
1.482
1.693
1.914
2.285
2.~68
2.780
3.000
K
0.123
0.230
D.330
0.435
0.535
0.636
0.860
1.027
1.196
1.365
0.50
0.58
0.63
0.68
0.71
0.74
0.74
0.74
0.75
0.75
center of fixed edge
0.038
0.042
0.047
0.050
0.054
0.056
0.OS7
0.058
0.058
at the center
0.58
0.63
.11
0.74
0.74
0.75
0,75
0.75
center of fixed edge
0.100
0.122
O.lll
0.137
0.139
at the center
0.79
0.80
0.80
0.80
at center of free edge
0.16
0.17
0.17
0.17
at the center
K1
ltE K
0.030
0.050
0.068
Kl
0.67
0.74
0.76
• VJ
I,/:)
~ 0.77
0.78
,-
0.724
,"
K
~
~enteT
K
0.14
O. L5
0.16
0.l6
0.16
t
TABLE 7.4 LOADING COEFFICIENTS FOR RECTANGULAR FLAT PLATES UNDER VARIOUS LOADINGS
at the center center of fixed edle
-
CD
:z
I ......
: I
:.p.
o
...... alb
l.0
1.2
1.4
1.6
1.8
2.0
2.5
3.0
3.5
4.0
Location of Haximum Stress and Deflection
Kl
0.20
0.24
0.17
0.Z9
O.ll
0.32
0.3)
0.36
0.37
0.:J7
at center of free edge
K
0.040
0.04$
0.048
0.051
0.053
O.OSS
0.064
0.067
0.069
0.70
at the center
One long edge free. other three edge. supported, uniform load over entit& surfaee. L - •
Kl
0.67
0.57
0.48
·0.42
0.38
0.36
~.~6
0.36
0.36
0.36
at center of free edge
K
O.llt
0.12
0 .• 11
0.10
0.09
0.08
O.~8
0.08
0.08
0.08
at the center
One long edge free, other three edges &upported. distributed load varying linearly along
Kl
O.ZO
0.18
0.11
0.105
O.lJ
O.ll
~
..
..
..
at center of free edge
bnadth.
K
0.025
-
-
..
.
at the center
MANNEl OF LOADING
(;.o-ru" ~ • Free !dge
---c
(jl11111 Uv b
Iorf---.::.
----IJoI
{F211lV !.---
b
----e
~ 4 I Z / FA t-
III
-to!- a
-III
On~
sho£t edge free, other three edges aupported. Distributed load varying linearly along 1ength. L • b
L - •
All edge. supported. d18trlbute4 load In fo~ of • trl.ngular prism. L '" b
t-; --if I-x
a
--.i
K
0.0)6
0.03l
0.028
0.030
0.204
0.262
0.111
0.352
0.38'
0.411
0.450
0.476
0.029
0.040
0.050
0.058
0.065
0.071
0.082
0.085
0.476
0.500
at the center
0.085
0.091
at the center
CI)
length.
::a n
..... c:
1
)
:
b/a
All edges fixed, dIstributed Load varying linearly along
.....
c:
-.
HANNER Of' LOADING
~
. Kl
0.040
I I
0.6
0.8
LO
1.2
1.4
1.6
L8
2.0
,.::ar...,
COIIlIIIenta
Kl
0.ll08
0.1434
0.1686
0.1800
0.1842
0.1872
0.1902
Kl
0.0636
0.0688
0.0762
0.0715
O.Of112.
0.0509
0.0415
0.1908 {Hax Fb 18 at x • ± O.Sb, f v • 0.5.5. 0.0356 F~ at x • 0; y = O.~a
Kl
0.0832
0.1718
0.2365
0.2561
0.3004
0.3092
0.3100
0.3000
Hax F• ia at x - O~
Kl
0.0206
0.0497
0.0898
0.1249
0.1482
0.1615
0.1680
0.1709
F a at x
Kl
0.0410
0.0&33
0.0869
0.1038
0.1128
0.1255
0.1157
0.1148
'a at x • 0; y = 0.6a
K
0.0016
0.0047
0.0074
0.0097
0.0111
0.0126
0.0[33
0.0\36
Hax Deflection
r•
CI
-
CI) f :i)
a
L • a
= 0; y •
Z
0
,.=-= ,.=
.
Z
TABLE 7.4 (CONT'D) LOADING. COEFFICIENTS FOR RECTANGULAR FLAT PLATES UNDER VARIOUS LOADINGS
r-
-
e
It
STRUCTURAL DESIGN MANUAL (a) Uniform Loading K
b/a
V x(max.)
Kl Vy(max. )
R (a) Uniform Loading
)
1
0.420
0.420
0.065
1• 1
0.440
0.440
0.070
1.2
0.455
0.453
0.074
1.3
0.468
0.464
0.079
1.4
0.478
0.471
0.083
1.5
0.486
0.480
0.085
1.6
0.491
0.485
0.086
1.7
0.496
0.488
0.088
1.8
0.499
0.491
0.090
1.9
0.502
0.494
0.091
2.0
0.503
0.496
0.092
3.0
0.505
0.498
0.093
4.0
0.502
0.500
0.094
5.0
0.501
0.500
0.095
0.500
0.500
0.095
0)
Vx
:2. 2
(')
L
max
R/
,
M'R
'Vy(max)
y
Remarks L:::aforVxandV y
.. TA'lLE 7.5
LOAn I NG CO EFF ref ENTS FOR COR NE R AND EDGE FORCE S FOR FLAT sn'1PLY SUPPORTED RECTANGULAR PLATES UNOER VARIOUS lOADINGS'
7-41
STRUCTURAL DESIGN MANUAL Distributed Load Varyin~ linearly Alon~ Len~th
(h)
1<,
K
Rema.rka
'0/ ..
"xl
V x2
"'y
RI
1\2
1.0
0.126
0.294
0.2 I 0
Q.026
0.039
1.1
0.136
0.30.
0.199
0.026
0.018
1.2
0.144
IS9
0.026
D.On
Uee L .. b for Vy
0.178
0.02.1.
0.1)]6 0.035
Bec:a.Ll.ae the lOad i. not .ymmetricai, the re.ac:tiona Rl are diUerent (rom lhe I'ea.ctiona RZ, &1.0 Vx l i. different than V x2- The ...me appliea to ea. •• V.3.
v••
(a
L" .. lor V xl'
Vx 2
t.l
0.150
1.4
O. 1,5
0.lL3
0.169
0.025
1.5
0.159
O.3Z1
0.160
0.024
o.on
).6
0.162
O. :nO
0.151
0.021
O.OH
1.1
0.164
0.H2
0.144
0.022
0.0)0
\.8
0.166
0.333
0.136
O.Oll
0.029
1.9
0.161
0.334
O.
no
O.Oll
0.028
2.0
0.168
0.335
O.lU
O.OZO
O. D26
3.0
0.169
O. 336
0.083
D.!)!4
0.018
4.0
0.168
o. H4
0.063
0.010
0.014
5·0
0.107
0.334
O.OSO
0.008
O. all
0.167
0.333
.
D.3111
.-
0.Z50
~-
..
o. on
0.250
0.002
0.017
O. Oil
0.112
0.251
0.004
0.020
3.0
0.023
0.143
0.252
0.006
0.025
2.0
0.050
0.197
0.251
0.013
0.033
1.9
0.055 0.205
0.251
0.014
0.034
1.8
0.060
0.213
0.249
0.016 O.03S
1.7
0.066
D.2Z1
0.248
0.017
1.6
0.073
0.230
0,24S
1.S 1.4-
.-
--
5.0
O.OOB
4.0
10.
0.037
0.080 0.240 0.243
0.020
0.037
0.088
O.ZSO
0.2.39
0.021
0.038
L3
0.097
0.260
0.234
0.OZ3
0.039
1.'2
0.106
0.271
0.221
0.024
0.039
1.1
O. lib
0.281 0.220 O.Ol!>
0.03<)
1.0
0.126
0.29-4
0.2.10
0.039
0.02b
(CONT'n)
In thil ca.", only. tbe fDl'mul& {V} for tne corner force R c .. n be u.ed when lub.Ututlt'll .. (or '0 V.. (ml.x. J .. nd Vy(max,) Are .t th.e middle oC .lde. blind .. r.upectlve1y III .hown La the ligure for tbi. t.ble
Remark.
nistributerl Load Varying Linearly Along Rrp~rlt~ (a)h)
Use'L
a fot' Vxl'
V:xZ
Uae L
b lor Vy
/y /V X1
bIZ
/
V"2 " -
x
0.036
018
TABLE 7.5
I.
(c) RZ
Vy
Note.:
l
RI
V xl
,
Kl
VxZ
.
y
--
--
1<
alb
V1I.l
.. x
~--~~~~~
VV
~JZ
R/
alZ
y
aJ'l.~"R2
y
LOADING COEFFICIENTS FOR CORNER ANn EDGE FORCES
FOR FLAT S 1t·1PLY SUPPORTEO RECTANGOLAR PlJ\T FS fiNO Fr1
VARIOUS LOAI1INGS
- - - - " . - - - - - - - . - - - - - - _. . _
7-42
..• • • _ _ _ _ w .
_ _ • • • _ _ _ • _ _ • _ _ _ . _ . _ _ .• _ _ _ _ _ .•... _ . _ . _
STRUCTURAL DESIGN MANUAL Cd)
loarl Distribut@d As Trian~ular frisM Across Rreadth (a(b)
K
)
b/ ..
Vx(m.ax. )
1.0
0.147
1.\
0.161
1.2.
O. 173
1.3
D.Is",
0.202
0.1>30
\.4
0.191
0.189
0.035
\.S
0.202
0.17&
0.034
1.6
. 0.208
o. loa
o.on
1.7
0.214
0.158
0.031
t.8
o.no
0 ISO
0.030
'.9
O.ll"
0.142
o
Z.O
o.ns
0.115
0.028
3.0
0.245
0.090
0.019
Remarks Uu' 1.:: • lor V.
U.e L = b lor V y
02.9
y
0.2S0
.)
(e) K
.)
KI
Vylmu.}
R
o.~a
Loan
nistrihutp.rl As Triangular
Pri SM Along len~t""
(a)b)
Remark. U.e L; .. lor V:II
).0
o.on
I)
0.010
Z.O
G.on
0.3650
0. DB
I. 9 .
O.OU
O.lSI
O.Ol4
1.8
0.098
D.HO
a.OU
"7
0.07"
O. .J42
O.l>l8
j.~
O.OtU
L'S
0.09D
O.lolZ
0.011
1..
0.0''9
0.111
o.on
1.3
0.109
0.2'8
O.OlS
U . . L", b for Vy
A
X
)
R
, y
1.2
0.12.0
O.2U
0.036
E.I
(1.13)
a.us
0.031
L::..'I
0.147
O.lSO
0.038
TI\RlE 7.5
" " V,(maxt
R
-(Cr)~lT'·n) LOAn'~lG ·t-6·fF·FTc~IENTS-·FOR fORflER-ANf)' EnGE FORCES FnR FLAT S 1t1PL Y SUPPORT ED RECTAt·J(1l1lAR PLATES UNOER VARIOUS LOADINGS
. 7-43
....... I .(:'~
K t factor (or maximu.m stress at center (F :: F b )
a./b
"lIb
0.4
t
0.6
alb =: t.4
0.8
alb
b1/b
0
0
--
o.a
I. 82
0.63 1 78 1. 43 1. 23 0.95 0.74 0.64
0.4
0.2
1.0
0
0.2.
--
1 82 1. 38 1. 12 0.93 0.76
0.4
1.2
0.8
0
I 4
0.8
I.Z
1.6
1.20 0.97 0.78 0.64
1. 71
I. 31
t.03
1.39 1. 07 0.84 0.72 0.62 0.52 1. 39 1.13, 1. 00 0.80 0.62 0.55
l. 32
1. 08
0.88 0.74 0.60 O. 50
0.6
1.12 0.90 0.72 0.60 0.52 0.43
1. 04
0.90 O. 76
0.8
0.92 0.76 0.62 0.51
1.0
0.76 0.63 O. 52 0.42 0.35 0.36 0.75 0.62 0.51 '0 .. 47 0.38 0.33
Note:
1.28
1.08 0.90
0.76
L 10 0.91
L SS
0.82
1. I Z 0.84
0.68 0.53 0.47
0.42. 0.36 0.90 0.16 0.68 0.57 0.45 0.40
0.B7 O. 76 0.63 0.71
0.61
b>a.
L
a
L
b
Note:
0.64
o
b/s.
2-
1. 5
1 4
1 '.3
I.Z
K
O. lOB
0.099
0.096
0.092
0.OB7
0.081
1.3
1.4
1.5
2.0
1.1
a>b K
1. 1
0.088
1. Z 0.101
0.114
0: [26
0.137
0.178
l.0
0.074
..
0.227
I
~CI--
1-
CI)
-i
:=a c:
0.54 0.44
54 0.44 0.38
r
All edges supported Uniform load over central rectangular area shown shaded
n
~
=
:=a
~~b~l
2:a
rc:J
rn
a
l
-
(I) G)
All .edge s supported. Uniform load along ~he axis of symmetry pa rallel to the dimension a (b. very ~mal1)
Z
s: 2:a
Use unit applied load w in this case (lb. lin. )
Z
c: TARLE 7.6
e
2:a
LOAnING cnEFFICIEtTS FOR PARTIALLY LOAnFn
nECTA Nr,ULI\R
r-
FLAT PLAT ES
e
,__r
~-
0.613 0.57
Total load W =
"
b
0.84
0.53 0.45 0.38 0.30
K hlctor (or maximum deflection
~.~:~
2.0
I 64
2.0
0.75
0.4
= 2.
e
.
STRUCTURAL DESIGN MANUAL Edge Fixed Edge Supported Uniform Load Over Unifo rm Load Ove r Entire Su rfac e Entire Surface Manner of Loading
K
Kl
K
Kl
1.0
0.70
1. 24
O. 171
0.75
1. 1
0.84
1. 42
0.20
0.90
1.2
0.95
1. 57
0.25
1. 04
1.3
1. 06
1. 69
0.28
1. 14
1.4
1. 17
1. 82
0.30
1. 25
1.5
1. 26
}. 92
0.30
}. 34
1.6
1. 34
2.04
0.33
l. 41
1.7
1. 41
2.09
0.35
1. 49
1.8
l. 47
2.16
0.36
1. 54
1.9
1. 53
2.22
0.370
1. 59
2.0
1. 58
2.26
0.379
1. 63
2.5
1. 75
2.45
-0.40
1. 75
3.0
1. 88
2.60
0.4'2
1. 84
3.5
1. 96
2.70
0.43
1. 89
4.0
2.02
2.78
0.43
1.9
alb
)
b
b
F max.- at center Y max. at center
F max. at end of shorter principal axis. Y max. at
L
I
I
Locations of and deflection ~ t !" e
g"
center
:TA~lE
7.7
---
- ---------
~'-~-----------'
------- -
~
.. -- ----.
--
--.--~.---
-------,._-
LOADING COEFFICIENTS FOR FLAT ELLIPTICAL PLATES UNnER UN I FORt1 LOAD
7-45
'....., I
..po .0\
Plate nelcriptlon and Type of Edge Support
a
~
-qe Circular Se.ctor
All edges supported. L • a
I
Straight edges supported. Circular edge Uxed. L '"' a
0
Locations of Stress and Deflection
9
4.5°
60°
90°
l80
IC L
0.102
0.147
0.240
0.522
Max radial stress. on line to midpoint of circular edge
Kl
0.114
0.155
0.216
O. )L2
Hax tangefitlal stress, at midpoint of circular edge
K
0.0054
0.0105
0.0250
0.0870
Hax deflection is at midpoint of circular edge
Kl
0.1500
0.2040
0.2928
0.4536
Hax radial stress is at curved boundary
K
0,0035
0.00&5
0.0144
0.0380
en
......
:=ct
c:
n
...... c: ::a
,. Plate Description and Type of Edge Support
~1,i
t
Y
y
... 0
~ -to
8
j
Coefficients
Equilateral Triangle
Kl
0.1488
Max Fx at y
All edges supported.
~1
0.1554
Max Fy at y • 0, x • 0,129.
K
0.0112
Max deflection is at 0
Kl
0.131
Max Fx t
Kl
O.llZS
Max F • no data available on location
K
0.0095
Hax deflection, no data avallable on location
L• a
Right Angle Isosceles Triangle All edges supported. L .. a
r-
Locations of Stress and Deflection
DO
~
0, x - -0.062.
data available on location
~
rn en
-
l
C i)
Z
y
,.iI: ,.c: Z
TABLE 7.8 LOADING COEFFICIENTS FOR CIRCULAR SECTOR AND TRIANGULAR FLAT PLATES UNDER UNIFORM LOAD
r-
e
e
e
~
..
e
.~;
@.
~a
~ ._ -
-
W
Out.,. edge lupporled Uniform load along inner edge
:0
and .upported Uniform load oY~r entire actual surface
- __ ._
b Z)
,. "J
(il' •
lr'-
"""'tl.
W
8-
~
l
~
= \101" (a l . b Z ,
U,U
,'l+'
,.......8
a~
~ W
O\lte r edge fixed and liupported Uniform loa.d along inne r edge
~
Inner edge Cixed and su.pported Uniform lQad over entire actual s'~rface
W
=wn (a l
~
,
I
~
Outer edge supported. inner edge fixed Uniform load ove r entire actual surface
W::: wn(a Z _ b Z,
~
TABLE 7.9
or
I
I':
··_..
1.
~c;
I.S
Z
"
3
S
F "ClOrs' 1< ftlr
df."flt!'("!
;.lM
O.l'J7
o
o. S.U O.5SQ
O.blZ
0 .. 613
3.704"
0.7Z5
K 11f t
0.186
0.679
0.0"0
P·HI
0.401
0.379
KZ
0.919
O.HI
:l.61,!
0.447
0·3>.5 O.ZqO
·. ·.
a.HI
~.;19
::I.
on
0.114
O.714
0.704
1.1:)
1. Z6
J. ·HI
1. 81:1
Z.17
2.34
1<1 (F t at inn(-l' ,,',:;d
-
I 646
1. .f7::l
1.237
1.006 1.2'8
0.895
O.8ll
K.2. at ullter
..
-
0.163
o. Z37 O.Z90 0.295
-
I 7!;8
1.6S0
I.HS
Q. 17~
0.21:11
0.383. O. -137
~).
··. -.
O.S84 0.884
0.682
0.867
0.794
!'l.606
1.01l
0.910
·--
0.0018 0.0015 0.024
0.046
O.rt5S
O.OS7~
O.OZS 0.069
;).1-45
O.IC)2.
O. UZ
·
..
.. ·. ·. ·. -.
O.
O!H~
,'I
inner t"dgel
OIl ouh,,.
~.
!ft'
K2: at .nner edlklt" K
O.93Z
K? al inn("l' ,·dgC'
444
0.434
K (at oute r
1.4 Sij
1.6SH
K 11F t at inner t'dx,e)
a.S6' n.492 0.8&2 0.791 n.lz6
0.448
KZ at out!:'l' .. dge
O.6~2
K~
l·d~l')
0.069
;).096
0.096
0.08,9
0.204
O. ZlS
O.NO
0.242
. KI (F r at ollter edge)
0.005
!>.O2:4 0.01:17
O.OBI
O. )71 0.670
0.215 I. (l18
0.237
0.269
O.
K K I {F t at inner e'dge} K I (F r at UIII('... 't"dgd
'J. 1 q:,
0.320
0.41)5
:1.539
n. C"IS
0.1[5
O. Z69
O.44!:i
D.SIO 0.522
KZ at inIn:r edl?E.>
0.002
0.010
D.040
O. lOS
rl. I S2
0.18i
". I .• t
~
O.l'"
0.442
0.770
1.014
Lll
K I IF r al inot" r t>dgl·l
O.rlll
1'.040
0.097
fl.171
O.lZl
().ZH
"2 al Ollie-r
0.0051
O.02S
O.OI:Hl
0.209
0.293
O· 350
K ta! uute-r I'dt,td
~.2Z7
O.-IZ8
0.153
). lOS
I. 51-4
!.745
K I .I~I inne!' t"dl!"\
--
['1.046
0.10:'
0.239
0.403
0.499
O:~44
t
_.
0.003
0.018
O.OS)
0.104
0.141
0.163
..
0.108
O.19Z
0.l14
0.431
0.491
O. !)Z6
0.09!
0.ll3
0.197 0.Z60 0.2.97 0.30&
·.
-·. ·-
--
O. II
::a
Kl at ioner edg~
O. OJ!)
::I. !l-l""
r--'
..... c:
11Ft at innl:"l' eod¥f')
0.1-48
HZ
c:-)
K ~~
O. Oil
1.300
=
at innt·1' t"dge
0.093
o. OZS
en ..... ::a
f'd~1"
1.082
1.196
K I for stre •• "K2: ror ~ lope
1<
Z&9
--
• b Z)
'Itl.
H.U
W
Inner edge fixed and supported Uniform load along outer edge
.c:-....,
___
W =U,tTT{a Z _ b Z )
Out. r edge fixed
""-J I
\II"
HH.J
W
Inner edee lupported UnHof'm load ove r ennre actual .uda<:e
alb or afro
witb ~
Outer edge lupported Uniform load over ,plare actual lurCaee
e
'-,
itt
:DrC ITI (I)
outf'r edgl")
oult'r edg ...
K K 1 {F'r at innf'r ('d£f:'1 K Z at ,outer
-
("d~ ..
f'd~t'
LOADING COEFFICIENTS FOR CIRCULAR FLAT PLATES UNDER VARIOUS LOAOINGS
CD
:z
-
31:
:DZ
c: :III r-
.....
@ '"
I
,.
,t-. CO
~_,
b~/a
ro
I
alb or a/ro Circular l~te with~------r------.------.---~~------,------,,------r---------------------------------------------. Concentric;: Hole or Factor~: KI for: l5tre&& 1.2.5 I 1. S 2. 3 4 s Circ;ular Flange K (or deflection KZ for slope
ci rcular S~lid Plate
Loading'
Outer edge fil(f"d Uniform moment along inner edge
lnne r edge fixed Ulluorrn moment illong outer edge
o. ZO
~ (.
M
~
M
~
~'B
0.B5
0.94
0.80
0.66
K mal(. at inner edge
0.37 In.86
2.44
4.10
4.84
5. II
KI mal(, at inner edge
l.20 13.16
3.88
1.31
3.IZ
Z.72
K Z at inner edge
O.l3 10.66
11.~9
12.55
13.10
13.41
6. B7 17. SO
1B, 14
IL 71
B.94
9.04
2.3013.84
Is.67
16.94
17.8Z
18.!7
10.
10.37 Inner edge lupported Uniform moment ... Iong outer edge
,.....M
\....
M
B
"'"'\
~/
~
M
~
Outer edge fixed ~nd lupported. inne r edge fixed Uniform lo.a.d "'long inne r ed~e
(M".",.
42.70
W ::. WTT (a Z _ b Z )
~ w
~ W ':
Both edge s fixed Balance loading (pi.ton')
wn
(a 2 _ b Z )
~ M
No .upport Ubiform edge moment
c:.-
M
~
I
I 4.65 I Z.StJ I 1.()9 I 1.21 I I 7.50 I 6.80 I 6.50 I 118.75 I 7.81 I 2.93 I 1.56 I 0.94 I I 2Z. j 5 Ill. 27 I 5. 52 1 4.08 I 3. 17 I 6.92
27.36 115.60 110.0 44. 90
Outer edge fixed and Il.Ipported. inne r edge {lll:ed Uniform load over entire actual .urface
16.31
S1.00 12.8.00 116.40 111.60 110.2.3 I 9.611 53.30 127.80 115.60 18.78 I 6.N I 4.86 I 8.87
Outer edge supported Uniform moment along inner edge
r.so
9.23 21.616.013.5
t-----+33 . 3
~D ._-'i."-.:
yr
i--.J \'0..~1I
.f..!.J-
~~,,--,
~-
K molX, at outer edge
K 1 max. at inner edge
en
K Z a.t outer edge
--I
K max. a.t outer edge
:::a
Kl max.(Ft at inner edge)
c:
KLatouter edge KZ at innH edge
c-)
KI max at inner edge
--I
KI max. F t at inner KZ at ou.ter edge
c: ::a :.:-
K 2 at inne r edge
10·000410.00310.01'0
O.OZ3 10.031 fO.031
10,036 ,IO.!J65 10.104
(r.
151 10. 176 I~, 192
KI (F .. t in"er edge)
O. D6Z 10. IDS' 10,153
0.195 10. ZJZ 1'1. Z21
K I (F at oute r edge)
~.OOI31().'064ID.OZ4
0.062 10.09Z 10.114
10.115 10 220 10.405
0.70310.933 11.130
KI (F r .t inner edge)
CJ ...,
10.098 10,168 -IO.l57
O. 347 10, 390 10. 4 I 5
K I (F r lit inner edge)
(I)
K
10.08010.156 10.30Z 10.55110.75610.927
r-
K
10.000.1 0 .00310.014 1-0.03919·06110.077
-
CD
K
Z
K 1 (F r at inne r edge)
4
K mal(. at cenler
6
KI (FI
8
At edge
31:
:.:-
Fl at any point)
z
c:
:.:-
TABLE 7.9
e-
(CONT'O)
r-
LOAniNG COEFFICIENTS FOR CIRCULAn FLAT PLATES UNDER VARIOUS LnAD I ~,r,s
a
'----'
:
. , ,
a
I
a
•
"'-"
It ,,,--;~~I:::~:\
l;
,~-
..... - ......
!(-Xi!
~i." --~;1 : ~CI- .-/A/
fA>(a l~ te '" ith
\2J
role or
W " '1I1f
concentric circular
area of
Ir
rl.dha 1'0
ro l
'.''''"l -r.
1
\.25
I
1.5
I
Z
]
..
I
K.for denection
0.350
0.413
0.469
O.49Z 0.50)
0.568
0.700
0.900
1. 161
l.lS3
0.636
Factorf;:
S
O.lIZ 0.295 0.)Q8
LrolroJ
,~ =:::::::/
alb or a/ro
_,_------ __ nge
Edge. Iu.pported Uniform load ovet'
K 1 Cot' stre •• K'Z (or .lope
(I)
K
1.500
-I
1 (F I:' at c::enter,
::a c:: n
KZ
-
-I
c: ~
rola ManDer of lo.din,
O. I
0.15 O.ZO 0.25 0.30 O.}S
0.40 0.45
0.50
0.55
0.60
0.65
0.70
0.75
:.:-
KZ slope K I .tr ....
0.80
I
!:dgu lu?ported Ceatral cou.ple, tru.a.nion loading
~,
Edgu fixed Central couple. trunnion loadina
~ M~t
21:'0
0.71 0.97 L ZZ I. 60 l.O
2.5,0 3.53 5.60
B. S4
BZ.C
I Z. 00 16.30 24.10 41.4
I&Z,O
1<2 max. at c::eonter I
4.36 3.80 3.27
2.80 2.17 2.. 10 I. 84
0.67
}'23 1. 68 l.31
!..06
5.40 4.52 3.40 2.£0,5 2.. IS
3. 10 4,00
1.58
1." J
5.45
8.20
12.40
75
I.'H ,-----
L l4
J,
L 16 18.0
0.B9
l. 07 lB.t,
0.68
o
90
44.0
0.57
0.70
0,78
77.9
15&.0
0.47
0.35 ,
0.S7 f14,Q O.Z~
K,
m'llI;. F a.t center
Kl max. at center
K I max. 'F at center
-----
r-
CJ
rn
-
(I)
en
z
:I: :.:TARLE 7.9
(C'lrIT'!")
L n A,", I ~ I G CO E FFIr. I Er·1 T S FOP. C I Pr. " LA R F L J\ T PL l\ T ~ S urlOER "AR t OUS L('I.I\f\ I fJ~S
z c: :.:r-
...... I
+"" -.D
\1
~
'r
STRUCTURAL DESIGN MANUAL
•
Revision A 7.5
Shear Buckling of Flat Plates
The critical shear-buckling stress of flat plates may be found from Ks rr
2
E
11 11 - - - - - - -
(~)
2
Figure 7-21 presents the shear coefficient kg as a function of the size ratio alb for clamped and hinged edges. For infinitely long plates, Figure 7-22 presents ks as a function of A/b. Figure 7-23 (a) presents k s.oo for long pIa tes as a function of edge restraint, and Figure 7-23 (b) gives ks/kg 00 as a function of b/a t thus allowing the determination of ks.
)
The nondimensional chart in Figure 7-24 allows the calculation of inelastic shear buckling stresses if the secant yield stress, FO• 7 ' and n the shape parameter is known (Table 7-10). The plasticity-reduction factor 11 for shear panels can be obtained from Equation 7-23. 11 =
1- ve
(
Es E
2
I- v 2
)
(7 .. 2.3)
Cladding reduction factors, 11 , are given in Table 7-1. 7.6
Axial
Com~ression
•
of Curved Plates
The radius of curvature of curved plates deter~ines the method to be used to analyze their buckling stress. For large curvature (b /rt < 1), they may be analyzed as flat plates by using the relations in Section 7.3. For elastic stresses in the transition length and width ranges, Figure 7.25 may be used to find the buckling coefficient for use in Equation (7-24). 2
=
2 12 ( 1- ve )
(+)
(7-24)
)
2 For sharply curved plates, (b /rt > 100), Equations (7-25) and (7-26) can be used.
Fer
11
E E
=
?leE
Et Es
(
t
r
)
(1-
ve )
(7 - 25)
2
(7-26)
2 (1- v )
Figure 7.26 gives values of C in terms of r/t. Figure 7.27 gives 11 dimensional form. Here the quantity Eer = Ct/r.
in a non-
•
STRUCTURAL DESIGN MANUAL
e) '-'--"~'
3
5
10
.--
---
-- ---"--"--.---"
~'~.T"-
Material
n
)
--
One-fourth hard to full hard 18-8 stainless steel. with grain One-fourth hard 18-8 stainless steel, e 1'os-s grain
One - half hard and three - fourths hard 18-8 stainless steel, cross grain Full hard 18-8 stainless steel, cross grain 2024-T and 7015-T aluminum-alloy sheet and extrusion 20Z4R-T aluminum-alloy sheet
20 to 25
2024-T80, 2024-T81, and 2024-T86 aluminum-alloy sheet 2024 - T aluminum -alloy extrusion SAE 4130 steel heat-treated up to 100, 000 psi ultimate stress
35 to 50
2014-T aluminum-alloy extrusions SAE 4130 steel heat-treated above 125.000 psi ultimate stress
QO
fABLE 7 .. 10
SAE 1025 (mild) steel
VALUES OF SHAPE PARA~1ETER n FOR SEVERAL ENGINEERINn
t-,1AT ER' AlS
)
7-51
STRUCTURAL DESIGN MANUAL
15
13
11
Clamped Edges
Symmetric Mode
9 Symmetric, Mode 7
Hinged Edges
z
3
4
5
alb
FIGURE 7.21
7-52
SHEAR-RUCKLING-STRESS COEFFICIENTS OF PLATES AS A FUUCTION OF alb FOR CLA~1PED ANn HINGED EOGES
STRUCTURAL DESIGN MANUAL
lZ.5
~----------------------------~
12.0 11. 5
)
11. 0
Maximum ks at Trans ition from 1 Buckle to 2 Buckles 1-2
F::U· ------
.) kd VB Alb at e= 5
5.0
.4
.6
.8
1.0
1.2 1.4
1.6
1.8 2.0
2.2
2 4 .
Alb FrGur.E 7.22
SHEAR-BLJCKLI~G-STRESS COEFFICI'ENTS FOR PLATES OBTAINEfl FR011 ArJALYSIS OF INFINITELY LONG PLATES AS A FUNCTION OF Alb FOR VARIOUS Ar10UNTS OF EDGE ROTATIONAL RESTRAINT
7-53
_
~ STRUCTURAL DESIGN MANUAL
~,'I~.~ /
lO~------------------------~
o
1
(a)
Ksoo
as a functIon of
E
I.8~------------------------~
Long Edges Simply Supported
1·~O--~·.·2--~-.~4--~~----~--~1.O
b/a
(b)
FIGURE 7.23
7-54
~s/KSoo as a funtion of b/a
CtJRV ES FnR EST U1AT ION OF SHEAR-RUCK L t Nfl CO EF Fie I ENTS flF ~ PLATES NITH VARIOIJS M10lHITS OF Enr.E ROTATImlAL rESTRAlh •
STRUC1·URAL DESIGN MANUAL
.7p-----------------------------------------------~
n
.6
.5
F ere F 0.7
. I
.2.
.3
1.0
.4
ksn2:
(~)Z
lZ(l-ve )F O• 7
F rGURE 7.24
CHART OF NOND 'MENS IONAl SHEAR BUCKll NG
PANELS WITH EDGE ROTATIONAL RESTRAINT .... _
~·
STRESS FOR .. -··· ..
·'r'---~·
...· -
7-55
STRUCTURAL DESIGN MANUAL.
1000
r/ t 300
/
/
100
/
v/ V V .~ v/
k
~
/ ~/ ./
V L
500
V ~ 000
VV V
V
V// /
c
1/1/ v/ V
20
V
10
~
V VV V
/.V / ~ ~V
8 6 4
3 2
2
3
4
6
z _ b -
100
20
8 10
b
2
rt
1000
(1 _ v 2).5 e
FIGURE 7.25 - BUCKLING COEFFICIENTS FOR CURVED PLATES
•
STRUCTURAL DESIGN MANUAL
-
.6 ·5
.4
r-- r--Io...
·3
-........ ~
'"
•Z
c
·1
;:::t:"
~
'"~
~
lIlIlI-
'"
~
~
~ ~ .01
~ll"UIt
I
l
10
I
3
I
.1111
I'll
5
7
2
10 2
') r
FIGUI1E 7.26
/t
f10DIFIED CLASSICAL RUCKLING COEFFIC'E~'T AS A FUNCTfON OF r/t FOR AXIALLY COMPRESSEn CYLINDRICAL PLATES
1.4--------------------------------------------, n 1.2 n
Fer ')
F!).7.6
.4
o
.2
.4
.6
.8
1.0
2.0
E
.7
FrGURE
7.27
~!f)~nn1PJSIONAl
11 = ( Es / E)
RDCKLING CHART FOR AXIALLY CURV ED PLATE S
CO~·~PRESSEf:'
( E t / E) ( 1- vPo 2 ) / ( 1- v 2 ) ) • 5
7-57
STRUCTURAL DESIGN MANUAL 7.7
Shear Loading of Curved Plates
Large radius curved plates (b 2 /rt < 1) loaded in shear may be analyzed as flat plates by the methods of Section 7.5. For transition length plates (1
=
ers
12(1-
(-~-/
v:)
(7-27) 1
For (b 2 /rt > 30), Equation (7-28) may be used. 0.37 (Zb
)~
,/
(F
crs
)
flat plate
(7 ... 28)
Curved plates under shear loading with stiffeners can be analyzed by using Figure 7-29 for the value of the buckling coefficient ks. Both axial stiffeners and circumferential stiffeners are treated, 7.8
Plates Under Combined Loadings
In general, the loadings on aircraft elements are a combination of two or more simple loadings. Design of such elements must consider the interaction of such loadings and a possible reduction of the allowable values of the simple stresses when combined loads are present. The method using stress ratios, R, has been used extensively in aircraft structural design. The ratio R is the' ratio of the 'stress in the panel at buckling under combined loading to the buckling stress under the simple loading. In gen'eral, failure occurs when Equation (7-29) is satisfied. The exponents x and y must be determined experimentally and depend upon the structural element and the loading condition. (7-29)
7.8.1
Flat Plates Under Combined Loadings
Table 7-11 gives the combined loading condition for flat plates. Figures 7-30 and 7-31 give interaction curves for several loading and support conditions. It is noted that the curves present conditions of triple combinations. 7.8.2
Curved Plates Under Combined Loadings
For curved plates under combined axial loading and shear wi th 10 < Zb< 100 and < 3, the interaction relation of Equation 7-30 may be used.
1 < alb
2 Rs
+
1
(7-30)
This may be used for either compression or tension with tension being denoted by a negative sign.
.a ~
STRUCTURAL DESIGN MANUAL
•
TABLE 7.11 COMBINED LOADING CONDITIONS FOR WHICH INTERACTION CURVES EXIST Theory
Interaction Equation
Loading Combination
Figure
For plates that buckle in square waves, R + R = 1 x y
7.31
Longitudinal compression and shear
For long plates, R + R 2 = 1 c s
7.30
Longitudinal compression and bending
None
7.31
Bending and shear
RL+R2= 1
7.30
Bend shear~ and transJ verse compression
None
7.30
Longitudinal compression and bending and transverse compression
None
7.31
Biaxial compression
Elastic
Inelastic Longitudinal compression and shear
b
s
R2+R 2 c
s
=:
1
)
7-59
STRUCTURAL DESIGN MANUAL !
l03~------------------------------------~
10
5
2 1 1
2
5
10
(a) Long si:mply supported plates
FIGURE 7.28
)
SHEAR BUCKLING COEFFICIENTS FOR VARI()US CURVEt) PLATES
• 7-60
STRUCTURAL DESIGN MANUAL
lo3------------------~----------------~
el 1 ~1--~~~~~lO----------~10~~------~103
) (b) Long clamped plates
F I GlJRE 7.28
(CONT t
[)
SHEAR BUCKL I NG CO EFF I C I ENTS FOR VAR I OOS CURV EO PLATES
7-61
STRUCTURAL DESIGN MANUAL
l03--------------------------------------~
10 2 a b
ks
1.0
1.5 2.0 Cylinder 10
5
2
1 1
10
(c) Wide, simply supported plates
F I GORE 7.28
(CO ~,1' 10)
SffEAR BUCKlltJG CO EFF rei ENTS FOR V J\R r Ott~ ClJRV Et') PlA TE S
• 7-62
STRUCTURAL DESIGN MANUAL
.)
3 10 _ - - - - - - - - - - - - - - - - - ,
Cylinder
ks
.)
/
10
~a~. . t It~j~ f . ~
b
'" --- ~
~-.l
'"'--
10
10
2
10 3'
Zb
-)
(d) Wide clamped plates
FIr.UPE 7.28
(COtJT'O)
SHEA R RUe KLING CO EF F rei E NT S FOR VARIOUS ClJRV En PLATES
7-63
STRUCTURAL DESIGN MANUAL
•
,... a Zb
b l-
200
--
100 ~
60 40
:=...
1
I, 000
1.5 2
-,
--'
1 1.5
I-
100
2
~ ~
20
10
30 20 k
s
10
~
=~
l-
-
1
~
1.5
2
30
I-I--
E
30
1. 5
1
2
20 10
6
E1 bD
(a) Center axial stiffener; axial length greater than circumferential width
FIr, 11 RE 7. 2 9
7-64
SHE AR- BtJ CKLI NG C0 EF Fie I EN T S F OR 5 n 1PLY SUP PORT F. n CURV Ell PLATE S \'II TH C ENTE R 5T' FFENEJ1
STRUCTURAL DESIGN MANUAL
-a 200
b
Zb
1
1. 000
Cylinder
100 50
1. 5 1
100
2
\
!
20
Cylinder
10
-----
30 l20 ~ ks
e)
10
2 _ylindert
~ to~
30
I'
I-
30 20
1
.10
10
Cylinder
6 30 20
11
10
r- b 1
CY,Under
6
)
I
1.5
1
o .1
1
~---
lB(! ~ \1
10
1
EI bD (b) Center axial stiffener; circumferential wiatu greater than
axial length.
F r GUR E 7.29
.
(CONTI D) SHEAR-BUCKll Nl COEFF ref ENTS FOR S 1r1PLY SlIPPORT EO ClJRVEO PLATES Nt TH CE NT ER STI FF EN fR
7-65
STRUCTURAL DESIGN MANUAL ~
a
300 '200
b
~ ~
~~-'
Zb
1
],000
2 -1.5
~ ~~
-100 f:= 60
~
i!~
30 ~ ~
20
~
-.0
-~
-
~
2
1.5
1 100
a;I
10 l=I-
40
.__-t---- 1
20
---r-l.5 2
30
10
ks
10
11 ~
la~ ---
_ _ _ _-+-r:o
fiITJII ~ -- ----\1
10
.I
E1 bD
(c) Center circumferential stiffener; axial length greater than circl1l'r1ferential width.
F'I GURE 1. 29
7-66
( eON T 1 0 ) SHEA R- Bue K L I rr; Co E F Fie lEN T S FOR S Ir 1PLY SUPPORTFn CURVEt" PLATES NITH CEtlTfR STIFFEtlEP.
STRUCTURAL DESIGN MANUAL ~
a
b
300 200
I-
I-
!-
100 :::60 :::
--
-
20
.-
10
-
40 ")
::;
Zb
~ 1.L-
L..!..:..5 2
Cylinder
1 1.5 2 Cylinder
20
•
100
,...
40
ks
1,000
Cylinder
30
10
10
30 20 10
Cylinder
6 30 20 10
6 4
) EI bD
(d) Center circumferential stiffener; circumferential width greater than' axial length
FtC:tJHE 7.29
(CONTln) SHEAR-BUCKLING COEFFICIENTS FOR SH1PlY SUPPORTED CURVED PLATES WITH CENTER STIFFENER
7-67
STRUCTURAL DESIGN MANUAL
rn--====---, 1.
0
.i .4
Rt:'
.6 1.0
.2
o~----~~~~~
o
.2
,4
.6
.2
.8 .8
1. 0
1.....-_ _"'----01:_ _- - '
o
1.0
R~
(a.) Uppe r and lower edge s simply s uppo r~ed
.2
.4
.6
.8
1.0)'z'
}, 0 r----~----...,
.8
.6
.2
.8
.2
1. O......IIIIIICiiii:.::::....-_ _ _ _...J 1.0
"Ii
•b
.4
.2
o
o
Rc
(b) Upper edges simply supported. lower edges clamp.ed
Flr,unr 7.30
JUTERA.CTt0N CUnVf.S FOR LONG FLAT PLl\TE~ "N(,)ER \',-\P.IOPS NAT I () NS OF CO~'1PRESS I () r, R FtHl' ~G, ANn SH FJ\P
CO~1'31
7-68
STRUCTURAL DESIGN MANUAL
e\ J.
0r-=-------___.
.1:1
.6
.6
.z
.z
.2. D~~~~~~~~~
.2
.0
4
.B
o
1.0
.2.
.4
Ry
.6
.8
1.0
.l
4
fly
(b) alb = 1. 0
(a)a/b=O.8
l..!)
'.1;;
.8
I. ()
It)'
(c) a /b
= 1. 20
.---====--------,
.IS
.6
(d) alb = 1. 60
(e) alb
= 2.0
(£) alb::: 3.0
F'y
)
Ry
(g) alb =
~ I CoUP. E 7. 31
nlTE RACT InN C rlRV ES FOR FLAT n ECTANGUllR PLAT fS l.t tln ER CO~AR I ~Irn
RIAXI
AL-COt1PRESSIO~J
AN£' LONGITUf') PJAl-R Er·Jf'f Nr, LOAt1INGS
7-69
STRUCTURAL DESIGN MANUAL 7.9
•
Triangular Flat Plates
Figure 7-32 presents buckling coefficients for isosceles triangular plates loaded under shear and compression. Equation 7-31 is the interaction equation for shear and normal stress on this type of plate.
2Fs
(--Fcrs+
+
+
u)
2 F
+
2 (l-u)
=1
(7-31)
Fcrs -
The + and - subscripts refer to either,tension or compression along the altitude upon the hypotenuse of the triangle caused by pure shear loading. Table 7-12 contains values of kc' ks+ and k s - for various edge supports. 7.10
Buckling of Oblique Plates
In many instances, the use of rectangular panels is not possible. Figures 7-33 and 7-34 give buckling coefficients for panels which are oriented oblique to the loading. Figure 7-33 covers flat plates divided into oblique parallelogram panels by nondeflecting supports. Figure 7-34 covers single oblique panels.
Edge Supports ( a)
k
c
ks+
k
s-
AIl edges simply supported
10.0
62.0
23.2
Sides simply supported, hypotenuse clamped
15.6
70.8
34.0
Sides clamped, hypotenuse simply supported
18.8
80.0
44.0
aHypotenuse = b in Figure 7.32
TABLE 7.12 BUCKLING COEFFICIENTS FOR RIGHT-ANGLE ISOSCELES TRIANGULAR PLATES LOADED INDEPENDENTLY IN UNIFORM COMPRESSION, POSITIVE SHEAR, AND NEGATIVE SHEAR
7-70
--
-
',-",
301.---------------------------~
6
"""".,r --y----.--., . . . . .. "','
25
)'1- - - - - - - - -
""'/ F
...........
"',,..
r
... ,
a
3:")
F
•••.
~ \' /,
F
0:.••\
/.'
s
~b~
2a
~
",
,
kg
IS
3)
IO
2·,)
,II;
--
::a n ..... c: :::a :.:c:
Sim pI y Suppa rted
r-
5
10
~
rn
-
(I) O'
0
I
.J
1.0
I
/
1 . :;
2.0
°0
.. 5
r.lrl'~r::
C(YV)r05sion
7.32
2.0
CD
z:
aiL
1 . :;
1.0
r"r~LPlr.
(~)
rnFr.Tlrr F"TS r:nn,
~ ~~~n
~
r
l~nSr~LFS TR1.l\t!r.I'L-\n
PLATE'S ~
ro <:
1-"'
C/l 1-"-
"-J I
"-J I--'
I~
~rr' '!-~ ;-
~
Claf"!":ped
K~
..
(I)
~ b ---J
-l~
",
2._~' ,~
:: Fb
I..,{
tF t t t t t
~'
"/
/../' \ ..~
s
,----
'
.>\\ L I\, f"f"t"T"'t",
'.,.#,~,
\ ,.."", ... ""'''''\ ... iIIo'''-..\" .. ', ....."....
20
..
---/....\
F , ../
"'·.'.. .· ,,", L it
e
"-'/
o ;j
'Tj
:.::z c: :.:r-
... /
STRUCTURAL DESIGN MAN,UAL 25r---~-'----------------------------------~ ZQ
Fx 15
kc x 10
)
5
o~------~------~--------~------~------~
o
3
2
4
5
alb (a) Loading in x-direction
2Sr-----------------------------------------~
F
Y
~deg
2
3
4
5
alb (b) Loading in y-direction
r: f
rt
7-72
I
f~
r.
7. 3 3
rr
r
n c r ric
r; (l H f'l r; S I ',1 1: - fH J f~ l. I r! n rI PIT;' FOR r- L':\ T SI ~ EE T f) ~ ! 'I t') if f'; r: r L Fe: rIr~ r; I r r r 0 RT S f\ I V I r F. r I: IT () P1\ r 1\ L L [ L n nn1\" ~ C;I!Arrn D/\td[LS (/\11 nilnpl sidf's rlrf" f"n~lill.)
T'
STRUCTURAL DESIGN MANUAL
• ZO~----~------------_
ei
r ;>== =!6Y e -
20.-----------------__~
c
-. C
::
c
'I -
15
10
-....ooiI8.98
2
alb (a) Compressive loading
FIGURE 7.34
3
.1 .
.2
3 .
(b) Shear loading
BUCKLING COEFFICIENT OF ClAMPEn ORllQUE FLAT PLATES
)
7-73
STRUCTURAL DESIGN MANUAL 7.11 Introduction to Membranes
A membrane may be defined as a plate that is so thin that it may be considered to have no bending rigidity. The only stresses present are in the plane of the surface and are uniform throughout the thickness of the membrane. This section consists of methods of analysis of circular membranes, long rectangular membranes (alb> 5), and short rectangular membranes (alb < 5). 7.12 Nomenclature for Membranes
length dimension of membrane width dimension of membrane diameter modulus of elasticity calculated stress = calculated maximum stress
a b D E f f
max
n
l
,coefficients given in Figure 7.40
n7
pressure outside radius of circular membrane = cylindrical coordinate = thickness of membrane = rectangular coordinates deflection center deflection of circular membrane Poisson's ratio
p R r t
x, y {j
Oc p.
7.13 Circular Membranes Figure 7.35 shows two views of a circular membrane with the edge clamped under a uniform pressure, p. The maximum deflection of this membrane is at the center and is given by
DC = 0.662
R
\j3/ P R Et
The deflection of the membrane at a distance, r, from the center is
·7-74
(7-32)
"8~
STRUCTURAL DESliSN MANUAL
FIGURE 7.35. CIRCULAR MEMBRANE WITH CLAMPED EDGE The stress at the center of this membrane is
n
Ep2R2
f = 0.423
t2
(7-34)
while that at the edge is
)
Vilf Ep2R2
f = 0.328
t2
(7-35)
7-75
STRUCTURAL DESIGN MANUAL
7.14 Long Rectangular Membranes Figure 7.36 shows a long rectangular membrane (alb> 5) clamped along the two long sides.
a thic knes s ::.: t
FIGURE 7.36. LONG RECTANGULAR MEMBRANE CLAMPED ON TWO LONG SIDES The deflection and stress at the center of a long membrane clamped on all four sides are approximately the same as those in a long membrane clamped along the two long sides. The maximum stress and center deflection of the membrane in Figure 7.36 under uniform pressure p are given by Equations (7-36) and (7-37).
f
max
(7-36)
(7-37) These equations are presented graphically in Figures 7.37 and 7.38 for A
~
= 0.3.
long rectangular plate may be considered to be a membrane if P/E(b/t)4 is than 100.
7-76
4l. 70
,,1 IO~Ot I
I
60 i
e
~
\,~,/
= .25 I
I'
.' I
L I"
.....-'
>~.5
I
7'
b..
.75
I........."' 1.0
50 l
f
I§"
I'"
40 I
max (KSI)
A;,,'
30 I
1/
I
I
:>-1':
~
en -I ::a c: n ~
,.<
Ate
A
c:
::;iII'I"=
::a
..,"'F
•r-
~
,..,
et 20 I
}
J' bI'
"
V"
J....""-=
en
-
1 5 ."0
a) 10.0
10
o0
"2' ..
4
6
8
fa p J10E/10
.....
I ..... .....
i2
14
16
18
•:z = •
20
4
FIGURE 7.37 MAXIMUM STRESS IN LONG RECTANGULAR MEMBRANES (alb> 5) HELD ALONG LONG SIDES (
z 31:
r-
~
= 0.3)
....... I
...... (X)
::::aJ.
.071
I~
.061
I
= .25
1GOOt b
I.
en ----I
:::;;P'l:;li#"'"
· 051
r=
::::l;;...--~
"
:::a c:
75
n
----t
.04
0
b
c:
:::a :w:. r-
I -.
Y / ./I
f
• 0 2 L-«.u::
L
w:::
~
I
CJ ,..,
5
-z
en
• IJ
~
t :i)
. 0111" 7'::J»P
---i
i:
:w:.
2
4
6
8
10
12
14
16
18
z
c
20
:w:.
l07p E FIGURE 7.38
e
r-
CENTER DEFLECTION OF LONG RECTANGULAR MEMBRANES (alb> 5) HELD ALONG LONG SIDES (
-,
~
= 0.3)
-
STRUCTURAL DESIGN MANUAL
•• 7.15 Short Rectangular Membranes Figure 7.39 shows a short rectangular membrane (alb < 5) clamped on four sides under a uniform pressure p.
y
thickness
=t
FIGURE 7.39. SHORT RECTANGULAR MEMBRANE CLAMPED ON FOUR SIDES The deflection at the center of such a.membrane is {j
= n1 a
Vtf na
(7-38)
.,t;.;;;.
Et
where 01 is given in Figure 7.40.
)
The stresses at various locations on short rectangular membranes are given by the following equations for which the values of the coefficients TI2 through n7 are given in Figure 7.40. Center of plate (x f
f
x
Y
D2
D3
b/2, Y = a/2)
Vp2E(~)
2
Vp2E(~ )2
(7-39)
(7-40)
7-79
STRUCTURAL DESIGN MANUAL
Center of short side (x = b/2, Y f
f
x
n
4
f
0)
Vp2E({ )2
(7-41)
(7-42)
Y
Center of long side (x = 0, y f
•
x
Y
n6
= n7
V
p2E({ )
V
=
a/2)
2
p ZE({ ) Z
(7-43)
~
(7-44)
It should be noted that the maximum membrane stress occurs at the center of the long side of the plate.
7-80
STRUCTURAL DESIGN MANUAL !
.40~--------~--------~---------+--------~
.32~--------~~------+---------~--------~
.24Hr~~----~~------~----~~-+--------~
.u ~
(l)
'H (j
-H 4:-1
4:-1
o
~.16~--~-+--~--------~---------+~~----~
.08~~------+---~--~~-=~~--~--------~
4
5
alb
4IIJ
FIGURE 7.40
COEFFICIENTS FOR EQUATIONS (7-38) THROUGH (7-44)
7-81/7-82
-, .~ ~'1"
• ii
-.~~
STRUCTURAL DESIGN MANUAL SECTION 8 TORSION 8.0
GENERAL
This section presents the analysis methods and allowables for members torsionally loaded. Members subjected to torsion are categorized according to their cross sections for analysis purposes, i.e. (1) solid sections, (2) thin walled closed sections and (3) thin walled open sections. 8.1 Torsion of Solid Sections
The torsional stress (fs) and resulting angle of twist (0) for an applied twisting moment can be determined when the material and section properties of the bar are known. The torsional shear stress (fs) distribution on any cross section of a circular bar will vary linearly along any radial line emanating from the geometric centroid and will have the same distribution on all radial lines. The longitudinal shear stress (fx) which is equal to the torsional shear stress (is) produces no warping of the cross section when the stress distribution is the same on adjacent radial lines. For non-circular sections the torsional shear stress distribution is nonlinear (except along lines of symmetry where the cross section contour is normal to the radial line) and will be different on adjacent radial lines. When the torsional and longitudinal shear stresses are different on adjacent radial lines, warping of the cross section will occur. When the warping deformation induced by longitudinal shear stresses is restrained, normal stress (u) are induced to maintain equilibrium. These normal stresses are neglected in the torsional analysis of solid sections since they are small, attenuate rapidly and have little effect on the angle of twist. Restraints to the warping deformation occur at fixed ends and at points where there is an abrupt change in the applied twisting moment.
)
The torsional analysis of solid cross sections is subject to the following limitations: 1) The material is homogeneous and isotropic. 2) The shear stress does not exceed the shearing proportional limit and is proportional to the shear strain (elastic analysis). 3) The stresses calculated at points of constraint and at abrupt changes of applied twisting moment are not exact. 4) The applied twisting moment cannot be an impact load. 5) If the bar has an abrupt change in cross section, stress concentrations must be used.
The basic equation for determining the torsional shear stress at some arbitrary point on an arbitrary cross section is f
s
8.1
8-1
STRUCTURAL DESIGN MANUAL where T(x) is the applied torque at some distance x along the beam and Q is the torsional section modulus at the same place. The basic equation for determining the angle of twist between two points x distance- apart is dx
8.2
where K(x) is the torsional constant. The area-moment technique can be used to determine the angle of twist between any two sections by plotting T(x)/GK(x) for the be~. Table 8.1 shows equations for calculating stress and angle of twist for some commonly used cross sections. The equations are for points of maximum torsional shear stress. Some cross sections have torsional stress equations shown for more than one section. The angle of twist equations are for a bar of length L and constant cross section. When a circular beam of nonuniform cross section is twisted, the radii of a cross section becomes curved. Since the radii of a cross section were assumed to remain straight in the derivation of the equations for stress in uniform ' circular beams, these equations no longer hold if a beam is nonuniform. However,. the stress at any section of a nonuniform circular beam is given with sufficient accuracy by the equations for uniform bars if the diameter changes gradually. If the change in section is abrupt, as at a shoulder with a small filLet, a stress concentration must be applied. In nonuniform circular beams-having gradual diameter changes, the angle of twist can be determined using equation 8.2. This equation is used to determine the equations for 0 in Table 8.2 for various beams of uniform taper.
8.2 Torsion of Thin-Walled Closed Sections A closed section is any section where the center line of the wall forms a
cl6sed curve. The torsional shear stress distribution varies along any radial line emanating from the geometric centroid of the thin-walled closed section. Since the thickness of the thin walled section is small compared to the radius, the stress varies very little through the thickness of the cross section and is assumed to be constant through the thickness at that point. The angle of twist of a thin-walled closed beam of length, L, due to an applied torque, T, is given by
J dtU
8-2
8.3
\
STRUCTURAL DESIGN MANUAL Revision E Mod~lus
T = Appii ed Torque(in-Ib)
G=
L
f/J= Angle of
Length of Beam(in)
K= Torsional Constant(i~) Q= Section Modulus (in3 ) SE-eTlON
of Rigidi ty(psi) Twist (rad)
fs= Shear Stress(psi)
K
MAX: STRESS
Q
SOLID CIRCLE
4
'IT
2r (ro-r i
4 )
o
HOLLOW CIRCLE
O.1406a4
at
~idpoint
of
each side SOLID SQUARE A ..--------.
®
)
.-------t
T
Ba
B
'-----IAto------.-I.-l
I..
b
{j b a3 {j=
f.333-.21 (1- o.o83~\1 L (b/a) (b,Al) ~J
@A: fs=
~
@B: fs= Ta
Qb
-----1
SOLID RECTANGLE @A: fs:::
~
@B: {s= Ta Qb
SOLID ELLIPSE TABLE 8.1 - EQUATIONS FOR STRESS AND DEFORMATION IN SOLID SECTIONS LOADED IN TORSION 8-3
STRUCTURAL DESIGN MANUAL
@M
Q
SECTION
~
a3
. a4 (3' 80
C B
MAX STRESS
atA,B & C
20
~a~ SOLID EQUILATERAL TRIANGLE
0)
0.1045 d
4
0 .. 1704 d 3
at midpoint of each side
SOLID HEXAGON
®
0.1021 d4
O.. 1751d
3
at midpoint of ~ach
side
SOLID OCTAGON
fg\ E
I
F
\2)m:· I ':
BID I ;c : ;
I
I
H
I
G
I
~
Form equivalent rectangle through points Band D. Then use equations for rectangle to determine stress and twist. To locate Band D, construct perpendiculars from centroid (c) to each side (B and D).
SOLID ISOSCELES TRAPEZOID O.0261a4
at center of long side
SOLID RIGHT ISOSCELES TRIANGLE
rra3b3(1- q4) a 2+b 2
at A
q = ao = l\, a: b
HOLLOW ELLIPSE TABLE 8.1 (CONT'D) - EQUATIONS FOR STRESS AND DEFORMATION IN SOLID SECTIONS LOADED IN TORSION 8-4
STRUCTURAL DESIGN MANUAL K
SECTION
Q
For cases 12 through 18, f occurs max. at or very near one 0 f t h e pOlnts where the largest inscribed circJe touches the boundary unless there is a sharp re-entrant at some other point on the boundary causing hi local stresses. Of the points where the largest inscribed circle touches the boundary, f occurs at the one where the bo~~aary curvature is algebraically least. Convexity represents pos~tive, concavity negative, curvature of the boundary. At a point where the curvature is positive (boundary of section straigh or convex) the maximum stress is given approximately by:
4Ix (
1 + 16Ix) AL2
©~:;'Area ::::...-;::--~ ..
t
"-
dL = element length along median
@
Revision C IMAX STRESS
Urea
SOLID FAIRLY COMPACT J=Polar Moment SECTION WITHOUT of Inerti a REENTRANT ANGLES
D)1J
2r
D = dia. of largest inscribed circle r = radius of curvature of boundary at the point (convex) A = area of section
fs max C
~=Cd3[i _ O.105d(1_
[3
r. D. t,. t~same as case ® t~
Q!=
C
1,,0.. 15 + 0.1
d
4
_~
192~
=
=
G0C/L or fs max D '2 4 [1 +
l+~ 16A
I
= TC/~
0 .11Btn( 1 _ D)
2
-0. 238D/2r
I tanh
2! ]
t).
TABLE 8.] (CONT'D) - EQUATIONS FOR STRESS AND DEFORMATION IN SOLID SECTIONS LOADED IN TORSION
8-5
STRUCTURAL DESIGN MANUAL Revision C SECTION
Q
K K l +K2+ an4 Kl& K2 per 0'=
=
D
-
®
~(O.07 + 0.076-li)
MAX STRESS
The angle 0 is the angle through which a tangent to the boundary rotates in turning or traveling around the reentrant portion, measured in radians.
2(3r + b + d) +b) (2r + d)
[~(2r
Sum of K's of consti tuent 'L'Mctions c·om'put.ed per@
e/d := A
d/D=n
f
r
smax
A; • ¥~,
r"f't
Q:=C r3
@ a
60° 90° 1200 450 .0825 .148 .0349 .. 0181 C 0 0 a 180 270 3000 360 0 C .296 .528 .686 .878
TABLE 8.1 (CONT'D) - EQUATIONS FOR STRESS AND DEFORMATION IN SOLID SECTIONS LOADED TN TORSION
'
. 8-6
•
STRUCTURAL DESIGN MANUAL ANGLE OF TWIST
TYPE OF BEAM
~ e\\., I I~
L
-
8TL 7TG(P2-D].)D33
2 arctan (nD32'-2arctan(DDl'
1
r
~
+Ln-r~l~
INSIDE TAPERED, OUTSIDE CONSTANT
SOLID BEAM, OUTSIDE TAPERED
1......... . . . . . - -
)
L
INSIDE UNIFORM OUTSIDE TAPERED
~L THIN TAPERED TUBE WITH CONSTANT WALL THICKNESS
TABLE 8.2 - EQUATIONS FOR ANGLE OF TWIST FOR NONUNIFORM CIRCULAR BEAMS IN TORSION
8-7
STRUCTURAL DESIGN MANUAL where A is the area enclosed by the median line of the thickness, t, and u 'is the length along the median. The shear flow is constant around the tube and is q
T
8.4
2A
The shear stress is assumed to be constant through the thickness and is q/t = T/2At
8.5
If the cross section is of nonuniform thickness, the shear stress will be maximum where the thickness is minimum.
)
Table 8.3 shows the angle of twist and shear stress for thin-walled closed sections subject to an applied twist, T. 8.3 Torsion of Thin-Walled Open Sections An open section is one in which the centerline of the wall does not form a closed curve. Channels, angles, I-beams, Tees and wide flanges are among structural shapes characterized by a combination of thin-walled rectangular shapes. Additionally many thin-walled open sections are curved. This section presents the means to calculate the stress and twisting angle for these sections. For a bar of rectangular cross section of width b and thickness t the equations for maximum shearing stress and the angle of twist are f
s
8.6
max
8.7 where aand f3 are defined in Table 8.1, Case 4. When the ratio bl t becomes very large, a and ~ become 0.333. Equations 8.6 and 8.7 become f
s
=
3T/bt
2
8.8
max 8.9
These equations, 8.8 and 8.9, are applicable for narrow rectangles. They also apply to an approximate analysis of shapes made up of thin rectangular members such as those shown in Figure 8.1
8-8
)
fflp·\~
a L- -~-31--\\
STRUCTURAL DESIGN MANUAL
III
~ ~
_,t"ftELfCQP"t'1:1I
,
SECTION
K
47r 2 t
Q
rCa -.5t)2(b-.5t)2]
MAX
STRESS
21ft(a-.5t)(b-.5t)
constant if t is small
2tA
const ant if t is
U U:::: length of median U== 7T(a+b-t)[l +a27(a-b)2] b)2
ta+
2 4A t -U-
small
U==length of median A:::: mean of areas enclosed by two boundaries
e)
®~
4A2
~
2tA
rd~' U & A same as
2tt,(a- t)1b- tl)2 2 I- tat + b t, - t - t,2 A
®
@A: 2t(a-t}(b-t,)
There wi 11 be
@B! 2t l (a-t)(b-t.)
higher stresse s at inner corners unless fillets oJ fairly lavge
)
radius are used
Equations for twist and stress are shown in Table 8.1
TABLE 8.3 - EQUATIONS FOR STRESS AND DEFORMATION IN HOLLOW CLOSED SECTIONS LOADED IN TORSION
8-9
~ STRUCTURAL DESIGN MANUAL '\ti.~.L~"~ . /...,?
~----------------~-
b
FIGURE 8.1 - BEAMS WITH THIN RECTANGULAR MEMBERS WITH CONTINUOUS CENTERLINES For the members shown in Figure 8.1, b can be taken as the continuous centerline of the member and equations 8.8 and 8.9 used to determine stress and angle of twist. Shapes with a member of thin rectangular members such as T and H sections shown in Figure 8.2
t---
b2
J
r-
I
I
Il
I
t
Tt2
-1-.L I
b3
t 2-
bl
~l
~ ~t]
t3
b2
J~ I
r
r--
bl
--1T tl
FIGURE 8.2 - BEAMS WITH THIN RECTANGULAR MEMBERS OF COMPOSITE SHAPES
8-10
.-
f--
•
STRUCTURAL DESIGN MANUAL Revision A can be analyzed using equations 8.10 and 8.11 (
s
3Tt / J: bt max
3
8.10
n
n
3TL/G
~ bt 3
8.11
For the T s('ction in Figure 8.2 the angle of twist is 8.1.2
and the shear stress is f f
s s
3T t I I
2
AD J t 13
3Tt Z/(b] tl.
3
+
3
b 2 t2 )
3
+ b 2 t2 )
8.13 8.14
The same procedure is applicable for any type of shape; however, the accuracy is considerably improved when sharp corners are avoided by the use of liberal
•
T a eli
i .
8.4 Multicel.l Closed Beams in Torsion tube with an externally applied torsion. The torsion is reacted in the tubE! by internaL shear floylS acting around each cell.
FigUT(' 8.3 shows a multiceLl
)
FIGURE 8.3 - MULTICELL TUBE IN TORSION
8-11
STRUCTURAL DESIGN MANUAL The tube has n cells with a pure torsion, T, applied. The torsion apptit'd externally must be reacted internally. This is expressed by 8. L5
where Al through A are the areas enclosed by the median lines of cells t through n. ql Ehrough qn ~re the reaction shear flows acting on cells 1 through n. For elastic continuity, the twist of each cell must be equal, or
o2
= .•. = 0n
The angular twist of
8.16 8
cell is
o = q/2AGfdS/t
8. j 7
= q/A §ds/t
8.18
or 2G0
Thus for each cell of multicell structure an eXPlession for q/A !dS/t can be written and equated to 2G0. The line integral Ids/t is represented by a .. Then 8KL is the value of the integral along the wall between cel] s K and L wher('! the area outside the tube is designated as cellO. The shear flows acting in a clockwise direction are assumed to be positive. Using this rotation, equation 8.13 Can be applied to each cell resulting in the following: cell (1) : 1/A1[q1a 10 + (ql - q2)al21 = 2G0
8.19
cell (2): 1/A21~2 - ql)a 12
8.20
+ Q2 a 20 + (q2 - q3)a 23 ] = 2G0
cell (3): .t/A [(Q3 - q2)a 3 23
+ q3 a 30+ (q3 -
q4)a 34 J
2G0
8.21
8.22 8.23 The shear flows, ql through qn' may be found by solving equations 8.l5 and 8.J9 through 8.23 simultaneously. From these shear flows, the shear stresses may be found using f = q/t. s As an example, consider the multicell beam shown in Figure 8.4.
8-12
STRUCTURAL DESIGN MANUAL
A
t=.03
t=.03
t=J)4 10
FIGURE 8.4 - EXAMPLE OF MULTICELL BEAM IN TORSION Cell Areas:
Al = 39.3 Line integrals for each cell: a a
a
lO 12 20
= 1/2(rr)(10)/.025 = 628 = 10/.05
=
200
a 23 = 10/.03 - 333 a
30
= 2(10)/.03
+ 10/.04
=
917
= 2(10)/.03 = 667
Equate external torque to internal reactions:
)
T = Zq1A 1 + 2q2AZ + ZQ3A3' Equation 8.15
= Z(39.3)Ql + Z(lOO)Q2 + 2(100)q3 100000 = 78.6Ql + 200Q2 + 200Q3 100000
Write the expression for angular twist of each cell: Cell (1): 1/39.3 ( 628q l + 200(ql - Qz)]
= 2G0 = Zl. 07q l
- 5. 09Q2
Cell (Z): 1/100 [200(Q2 - Ql) + 667q2 + 333(q2 - q3] = 2G0 Cell (3): 1/100 [333(q3 - Q2) + 917Q~
=
= - 2q l +
12q2 - 3. 33Q3
2G0 = -3. 33Q Z + 12. 5q 3
8-13
STRUC1-lIRAL DESIGN MANUAL Solving the equations simultaneously: 234 #/in, q3 = 209 #/in, 0
=
.0002288 Tad
8.5 Plastic Torsion The previous methods of analysis are based on stress levels in the elastic range. These stress levels are based on limit loads. For ultimate loads, it is often desirable to allow the section to operate in the plastic region. All types of cross sections not subject to local crippling can be analyzed for allowable torsion by use of the plastic torsion theory at ultimate load. The method of analysis is called the It sand heap" analogy. If the maximum amount of dry sand is heaped on a level platfonm having the same shape as the cross section of the beam in torsion, the slope of the heap represents the shear stress. The shear stress for this condition has the same magnitude over the entire cross section. The torsional moment) T, is related to the volume of the heap, V, by T
2VF su
8.24
where F is the ultimate allowable shear stress. The difficulty of the sand heap an~Yogy is determining the volume of the heap. This is simplified somewhat by constructing contour lines. A contour line defines the contour of the heap at some constant elevation. It is a plane passed through the parallel to the torsional section. Also, it is assumed that the-maximum possible slope of the heap is achieved, i.e. slope is equal unity. It is easy to construct a contour map of the sand heap surface.
Contour lines intersect normals through the section boundary at right angles and at a distance from the boundary equal to the elevation of the contour line.
I'.
/ '\
/
"-
/
"-
/
"v/
vk6, 1/
II
/ V
....
"
RECTANGLE
'\
"
I
1
\ \
8-14
-r " --
I ,/
NOTCHED CIRCLE
) ./
STRUCTURAL DESIGN MANUAL Revision E It is possible to determine the volume of the sand heap for any cross section by integration. Figure 8.5 'shows equations for sand heap volumes with various bases. For a surface with a hole, subtract the volume of sand that could be heaped on the hole alone.
I1- -------"I-1 I L----
a
V= Ar 3'
A= Area of triangl e FIGURE 8.5 - SAND HEAP VOLUMES
)
8.6 Allowable Stresses For limit load conditions, the applied stresses should be kept below the ultimate shear stress, Fsu' These are defined for various mate'rials in MIL-HDBK-5. The torsional failure of tubes may be due to plastic failure of the material, instability of the walls, or an intermediate condition. Pure shear failure will not usually occur within the range of wall thicknesses commonly used for aircraft tub- . ing. Torsional allowable stresses are shown in Figure 8.6 through 8.22. These curves take into account the parameter LID and are in good agreement with experimental results. Interaction data of Section 4 should be used when other stresses are combined with torsion.
8-15
s-rRUCTURAL DESIG
MANUAL t
T
T
11
f1u: 55 ~
40
~
~~ .....
,
30
o
.... "
....tn
20
,
I' ,,-
~
"-
-
I-
t-
LID:: 10 JiI ~ ~
LID =5
l/O:O
iIIII,
:"/0:2
o
" 1
I - k/D:20
-"
....
...... ~
h..
~
L/D:I
"
~
.
.
"or
I ' t--.. "'-
r-
LID: 1/2
'"
~ ~
....
-~ r-
I
~'r-
LID: ~~
.... r--.."'"
~
.....
~
-
I "1"'-0- .....
--
....
1"-00 ~ 1-.. ....
r--
)
,
._-
'1- I--
10
0
-~-
-
"-
,0
I,
ksi - 1-
.
o
10
20
40 Olt
30
50
70
60
80
FIGURE 8.6 - TORSIONAL MODULUS OF RUPTURE - PLAIN CARBON STEELS
F
i'
0
= 55 ksi
tu
'
o
,
~
r-f-
~
"-
60
,,~
~
~
'u..'t 5Of- I "- L--
~
--.
" """
LID ......
"'" ...... ~
~
;: ~ r-... 1-0...
....... ~ ~ ......
r--..
40
30
o
10
2.0
--
r--.. ~ .... """" ~ 1-00.. .... ~
"""
r--. t--.. ... ~
'-
""
30
IE
1""-0 ~
r-- ..... ~ ,..... ""..... ~ ~ ""~ ,..... 100...
...
r--
i"""'" r-
.... t-
40
......
--
roo- r--
.....
-
r- '-
r- r- 1"-- .... r-
r--.
-
50
tu
= 90·ksi
1/2
r-
roo- I- ~ ~r-
r-- .....
r- ....... ~
-~ I"- r.;..,. i"""'oo ~
60
"-
-
I
2 -I"-
""'" r- ........
70
FIGURE 8.7 - TORSIONAL MODULUS OF RUI'TJ·<.T.. - ALLOY' STEELS HEL\.1 '!"'REATED
TO F
w
o
l"- I""-
r- l"- I'"-
r-r-- r--- ~
r-
O/t
8-16
kSI
... ~
\
oo
.
'
: Ftu =90
\
5 10
-r- ""- ~ 20
30
STRUCTURAL DESIGN MANUAL 80
..
,
\.
70
o '0
, o
-
,'-
t
--
-
-
, Ft =95 ksi
"_ -
_ u
\.
' ...
""",,:00-. ~
60
'-
~
"~
~"
'" ... ,"" f' .....
~
,"
.....
~~ ill..
~
50
~
"""'-...
"""'--.
....... ......
....
~ ~ .......
'"
.....
---- - -.... , .... '- -- - -- -.... - .... -- -- -- -- - - .... -- - .... - -- m ~
.......
....
..... """"'- -...
-I-- ......
...... ....... ....
....
.........
~
....
.......
IiiiiiiiO ...... ..... ...... '- I"'Iiiiii ......
...... -.... -. ......
o
10
20
40
30
-,
-
~-
.....
.......
30
114
.....
.....
4/" .......
o
...... ~ .....
..... ...... ..... -.
.... .....
L/O
.....
.... .... ....
...... ......
60
50
-'-'III!
112
2
- -.... JO
.... .....
5
....... ......
..... ...... ......
,70
20
80
O/t
FIGURE 8.8 - TORSIONAL MODULUS OF RUPTURE - ALLOY STEELS HEAT TREATED TO F
= 95 ksi
tu
.
,
',90
~
)
"-
. LL'" "
.
J,
I
I
=125 ksi LID
" .... -l'---.. ~
0 0 0-.
-~
Ftu
70
, - -- -- r-r--
........ "~ ....... ~~ ~
~
-... !-..o ........ .....
......
~
~
-~
......... ~ """""" r-..... ....... ~
-
.... ....... .......
-
~ ....
...... ~ I"-'ii ~ .......
50
....... "....... ......
....- .....
-- ---- - ..... -- -- ...... ...... ........
20
30
40
- 1/2 •
~
i
10
o
50
- --
60
...... ......
~
~
...... .......
70
2
-
S r-- 10
-
20
80
CIt FIGURE 8.9 - TORSIONAL MODULUS OF RUPTURE - ALLOY STEELS HEAT TREATED TO F
tu
:<::
125 ksi
8-17
STRUCTURAL DESIGN MANUAL 21 0 Ftu =150 ksi
190
I
170
I
15 0
130 ~!\..
II 0
\ j
I\~ ....
L/O
~ ~ ...
~~
""
90
"""'I~ ....
-- .....
~ ..... t--- ~ ......,...... ~ ..... ...... I'" ...... ...... ~ Iioo.. ~ ~
~ ~ ~ iii...
~
~
~
100..
70
--
~
r--Iooo.
~
-~
r---. .........
r-
r--. "-- -""""" ~ ~ loo-. -.....
r--
50
-~ ~
~
.... .........
r--.. ~ r"
30
o
10
20
30
40
50
- ,
1/2
-- r--. r-- """'......... r-.. -- r--- ... r-- r--. ......
.... ~ ""'"l1lI ~ ~
0
60
...... I'-- ""j""'"oo
~~
~ .....
r-- 2
I'-- '""- "--
--
I'- ~
~
.........
.......
70
FIGURE 8.10 - TORSIONAL MODULUS OF RUPTURE - ALLOY STEELS HEAT TREATED TO F 150 ksi
8-18
--
r----. ""'"
D/t
tu
5 10
20
80
•
STRUCTURAL DESIGN MANUAL ...
21 0
-
Ftu =180 ksi - ~-
~.
- ......-
1910
. -.
~
1---
•.. -_. r17'0
'&_
-
-
15,0 ~
~.
--
_.. ' -
,
[\
o
130
~
"
oo
-
....
• t.r" ""
~ ...... ~ ""I~
LID
t
,
r\ ~ ~ .... ~ Il1o..
r-- ~ ,..."
- r-- ...... --
r-- t-- ~ I ' l ' ~ ~ ...... I ' ... i' ~ , r-....... ~ ..... ~ ~
9~O
" ...... --. , ....... , '" .......
r0- t--
...... .....
~-
-- --...
r-- ....... ..... ...... '1iIo.. ~ ...... ..... ....... r--- ..... ...... ~ ......
'" '-- ...... ...... ;"'II
...
....
~
-~ ~
......
-
~
~
~
,
~ ......
.....
~ ...
.... ..... ......
- ..... ---.. ......
~
"-
..... ~
)
o
10
20
30
40
50
60
10
.....
r-- ~ ..... 30
2
..... ~ ....
~
50
112
r-
~
....... .......
'- ~
~
7'0
- "4o
70
20
80
Olt FIGURE 8.11 - TORSIONAL MODULUS OF RUPTURE - ALLOY STEEL HEAT TREATED TO F
tu
=
180 ksi
8-19
STRUCTURAL DESIGN MANUAL 2 10
-
Ftu =200 ksi
190
~
I .~
- i-
_.
!
.-
----
I
170
)
, 15;n ' \
--
13.0
...
,,
\.
..... "'II~
'" "~ ~~ i'
b.U)
""- , .....
-1:t::
~
~
~~ ~
to
LID
..... -.
~
8o ::::
\
"""'III
~
:-- "- i'-- "
..... ....... .....
~
"""~ i ' ..... I""- "-
9~O
,
~~ ~
-
~
~
...........
--..
~
70
~
........ ........
"""'" ~
f' ~
50
- ....... -......-
~
.....
l'\
- ....... ~
...... ~ ......
" ....... ........ """" " '- "'- " -,...... , ~
...
310 0
~
"-
,
I
~
~~
........
- ......
20
30
......
-~
~
.......
'"",
-- ---.. .......
r-- .......
...... ............ .......
,
I'~
~,
""- ......
1'" ~
.... ~ .....
50
2
~
.......
'
.....
r--.. . . .
, ..... r--- ............
40
,/2
~
60
70
0/1
FIGURE 8.12 - TORSIONAL MODULUS OF RUPTURE -"ALLOY STEELS HEAT TREATED TO F 200 ksi tu
8-20
-
"'II
R 10
-
.... ......
0 1/4
Io
20
80
--
STRUCTURAL DESIGN MANUAL 2(0
±i
190
\
170
)
t
t
, ~
,
\ ISO
\ ~
~~
, '" ""
"-
~ ~~
130 0 0 0
•
F"tu =220 ksi
"" LL1n
...... ~
........
........
......
""'
"- ~ i ' r-... '"'- ~ "-
~
atO
~
~~
r--
~
''"" , , '" ~
f'- "-
~
~ ......
.....
"""
90
~
L
~
-
~
t-... l"""'- I"~
"'- '"""""
--
~ ..... ~
r-...... ~
"""III.. .....
o
1'-0 ......
,....., """" -... r---....'""""" '""""" -.. r-.... ~
70
it
LID
,
.......
~ ~
..... ~
"" ,
...... r-- ~ "'"-
"""" --........
..
-.. ........
r-......
'-
'-
..... --
~~ ..
~--
.....
r--
""~
r"""-
~
"' "-
......
...... ..............
~
50
)
~
....... ....... -.
\..
- 1,4
...... ~ ~ .....
'IlL
....
~ ~
.....
,
Iro..
i
30
o
10
20
30
40
50
60
r-.. ~ 2
"- -...
' ...
1/2
r-.. "'-
~
f'~
~
"'-
-~
'-.
~ 5
-
10
~ 20
70
80
Olt FIGURE 8.13 - TORSIONAL MODULUS OF RUPTURE - ALLOY STEELS HEAT TREATED TO F
tu
= 220 ksi
8-21
STRUCTURAL DESIGN MANUAL
•
2 10
~
90
1\ \ 70
1
,
\
'""",,-~ i-.. ... ...... r--..
I" ~
50
,
~ ~""~ ~~ ~
.... ~
'""~"~,-"""'" 'f'
~
~~
o o
,o....
.~
~
""" ~
IlO
-
.....
LID
II ~
~
30
'1u =240 ksi
--... ~ 100.....
~ """"-
r--..
r--. ~ ....
,
~
"-
....... r-.....
".~
f'
~
r-- .......
-:---. ~ ~
........ ,....."
r--... 1'0......
.........
"' ~~
~
~
-
~
-r--., ~
!"""I'III ~
.... ~
"""'"
........
r--... ...... I'
~~
........
.....
-I....
"""'"
-r-.. r-
..... 1'-- .....
, :""III, -
"I'r\. " "" , '",
90
~
-".....
~
""
~
~ 5
~
l
"'-
~~
~
30
40
50
60
70
Ott FIGURE 8.14 - TORSIONAL MODULUS OF RUPTURE - ALLOY STEELS HEAT TREATED TO F 240 ksi tu
8-22
10
....... ~
r--.. r--... 20
....... 2
""- ~ ,,~
"' 10
112
~
".~
50
o
r--
i'... ~
30
V4
~
~
70
"'-
~ ~ ....
~ ....
""- ~
""',-
-......
~~~
.... ~ ~ ~
o
-
r-- "-
20
BO
~ STRUCTURAL DESIGN MANUAL "~U
.:
210
, ~
190
,
\
Ftu :.260 ksi
,
I
~
170 )
\
f'
....... ~
~
150
~
\.'~\
~
r\ ~ !\. 1\" I\-
"-
or\.
•
~
"-
~,
~
I'
~
~
i'"' . . . ~~
~
I'
~
0 0
..
LID
" "' "fII.
130
"-
,
--
~f'..
"
Q
r--
~
,
~'" "~
, I0
~'"
t'-. "--
""""
r--- ~
f""", ~ ~
90
I'
I' ..... ~
~ ....
r--.
'"'" ~
... ~ ......
~
""
~
~ io...
~~
~
-~
~~
""~"
70
~
~
,
'-... '--
~
"-...
~
"-
0
~
~
10
~~ ~~
)
~" ......
10
20
30
40
50
60
2
j"..~
.... ~
50
~ ~ ........
~
'" "-....
-
~~
t'-.... 1000... ~
""'"
112
~
~
r"."",
~
... -..
....... t--..
~ 1000...
I' ~
~
- -.. ~
~
r-.. ~~
", " ', '- '" "'"' ~
o
.....
!'
~--
70
20
80
Olt
FIGURE 8.15 - TORSIONAL MODULUS OF RUPTURE - ALLOY STEELS HEAT TREATED TO F = 260 ksi tu
8-23
STRUCTURAL DESIGN MANUAL
60
I
1
I
J
I
I
I
I
I
I
I
,
I
2014-T6 rolled rod Ftu =65 ksi 50
,
-..
f\. ~
40
.~
I ' """"" ~
r--.~
o
;:- ~ i""'" 1--0""", r- ~ ~~ !'-oo. l""- t-... -.. r-- ~ """' ... I""~
"
en
.!II!
-.
LID
..... t....
~
30
f"""ooo
"""" ~
""""
~
I"""-
""""
1/4
1/2
""'"'-
-1'-0 I--,",""
-I""'"
""'-
r-
"-
r- I""""'"' ~ I!!. ~
r- Io-. .... ~
~
~
i' ~
-..
r-
-.." ......
2C
~
""'"
.....
!'o .....
-,.-. r--
""""'.... r""II"",,-
r-- ~ i"""
\
r-
........ : -
-. ~ ......
I 2
5
""'-
!o
~~
20
to
tm o o
,
I
10
20
30
40
50
60
70
80
D/t
FIGURE 8.16 - TORSIONAL MODULUS OF RUPTURE - 2014-T6 ALUMINUM ALLOY ROLLED ROD
8-24
STRUCTURAL DESIGN MANUAL 60
I
, " '-
50
I
I
I
I
~-"""' t'~ :--."""
-!---
.... ~~
.... !'""""ooo.
"
"C;;
....:
~
-
:--.~ ............
30 1J...v,
- - ------ .... - -
-"""'t- "--
......... ....
......
•
~r-
!
I"-r-.
.....
1"-_
.....! .
r- r-_ -~
....
......
-~
'--t 1-00.
J'he curve representing material failure (L/o=O) is computed for fO ~~ Ftu =39,OOO, and does not allow for p.the possibility of reduced strength : . along the porting plane.
I
LID o
.........
"
. 1111 . . I I 11 .I ..I I •
___
I
J I
10
i
i
-
40
30
20
l
"-
1/2
I 2
r-_ ~ ....
....
5
-""'"
-
-r--.~
~
10
..... 20
70
60
50
...
.....
D±f
I
1/4
r-
--- --
-
o
I
-
r-- """
:r- Note:
o
r
I
i"'-..
40
20
I
2014- T6 forging Ftu=65 ksi
80
D/t
FIGURE 8.17 - TORSIONAL MODULUS OF RUPTURE - 2014-T6 ALUMINUM ALLOY FORGING 60
2024-T4 tubing Ftu =64 ksi
50
1'\
~\.m
40
)
:3 0
.......... ......
........
..... ..... ~ r--..
...... to--
'"', ............ "I'
-
....
-~
20 ~-
-p.--
m
OR __ I
- -.........-
............... j
...........
r-.'~
r-- ....
1
~r-.J.
!'ooo. ' '-1""'-
_
10
20
1/4
-~
I"-r-._
r-_
r-
...
~
-r- r- ...
-~ ""'"
.....
....
....
r-""",
.... r--_
...........
r"'oo!'ooo.
--~
""",,,-
-~ ....
r--. .... """
-r- r-_ -~
....
1/2 '
I
2 5 10 20:
I I I I I t I I I I I I i
-,-""
o
""",I
o
r-r--. 1"-"-
:t$ -,- .... 1----- -
~:-J.
"""
LID
----
-r-_
-;..,
-~
....
0.
m
t I
I ~ I 1
30
40
50
60
70
80
D/t
FIGURE 8.18 - TORSIONAL MODULUS OF RUPTURE - 2024-T3 ALUMINUM ALLOY TUBING
8-25
STRUC1~URAL
DESIGN MANUAL ,
l=ti 50
.....
l'iii:r-
1
40
--
" "" ..... """ """
tu
F ;62 ksi
1..... 1'0
I r"
I"
30
!'
I E
r---"""
...........
f""o ....
!""-
.~
2024-T3 tubing
~
"
11 I f
I T
I"-~'-
1""'1"- ....
I""'i"oo 1""'0""" ..... r"'oo r"'1'-0~ -~ ~I- .... r"'oo .... ........ :--. r-_ r"'oo ~""" --1"'-",,", I""' .... ~ -~ ,...""" --~~
....
-1'-0 .... r-- ....
~
20
-"-"'"
-to- ~
LID
o
1m=-
1/4
........
1/2
I
..........
2
r-
I"""!- 'f"oo. ....
.....
....
5
r-
r--!o-
10
1""'01-"""
20
r-~iooo.
i
10
.0
r- ~
1--
- I - .....
-t-io....
It "
-;;;;;;;
i"""~ ...
-I"-..... ~
r-- ......
.... r-.
ttt ---
o
80
70
60
50
40
30
20
10
Olt
FIGURE 8.19 - TORSIONAL MODULUS OF RUPTURE ,i
i
I
i
I
!
•
I
2024-T4 ALUMINUM ALLOY TUBING rjTj~rrlT
i
,
606! - T6 tu bing Ftu =42 ksi
50
40 0;;
10..
..:ac
..
Im
30
lLV,
....
....-. ~~
~t-..
...;;;~ ~
....
i""i""
20
--
r-;",.._
.....
'- - f l- e--
LID
:.
r- r--
o
-
!
1/4
t/2
1
....
I""'!-
........
2
5-
~t- 10-.
..... r-. ~ ....
to 20
10
o o
10
'20
30
40
50
60
:m
70
..
80
D/t
FIGURE 8.20 - TORSlONAL MODULUS OF RUPTURE - 6061-T6 ALUMINUM ALLOY TUBING '8-26
)
STRUCTURAL DESIGN MANUAL 1'\ 11 1
I I I "
i\ ~
......
50
LID
Iiii::!_
'" "" ""-
~,
I
..... ""'- r" ""'- .... i'o ""'"' ' '
"
I
Io..l"""oi'o
r-- ,.....""",
-~ to-.
:--j....
r"1'-..
[""Iii
r--.
"
~
30
1/4
-~"-
~;....
r"""j....1.....
I
u;
~.... 1-- ...
:"'01-- ....
.....
40
o
1""'0 ......
!'oo.-.. r-...."" """'10.. r--""",
U;
1111 , , I
7075- T6 rolled rod Ftu= 77 kSI
~r...
....
-I- ~
-"""~ .... r---..
1-0 ......
:""0001'0..
~
~
.....
r-..
........... 1
)
1/2
r- 1--,-
.... 1
LL
....--r-_ ....
-
P""'
--
I'-~
..... ~
....
i'oloo...
....
:""0001-0...
to-.
~
.... ioo..
~
""'IIi"",,-.. ~
.
I
10
,
,
I
o
o
i0
20
.... 10
.... 1-- ~
20
t
70
60
50
40
30
5
i'
20 1
2
80
Olt FIGURE 8.21 - TORSIONAL MODULUS OF RUPTURE - 7075-T6 ALUMINUM ALLOY ROLLED ROD ..
--
I
!
,-'
, , \1
,
~
! I
,
I
: :
,
t
,
i
i
!
)
l'
§
...
' ..
30
i
:
j
!
I
i
t
I I
: :
.,.....
... r---.i.-.
i'oo.i.
,.... r-..
['i....
1""'0 ....
~
I
I
I
I
1-0-...' :
, T
,~;
:
1 ,I :
~
~i
1 i
-r:....
t
~
!
"
,
~
,
!~:
. I
T
~I-
!
J
~ ..... ""', r--,..
•
,
I
(l/J--v) IS '::OMPU1EO FOR 10 : i Fl -= 45,000, AND DOES NOT A.LLOW fOR ~ T~'E W5SI!llIlY o~ RE:>UCED STRENGTH ~ ALONG THE PARTI~G PLANE.
o
i
o
1
I
10
T -I I I i
I f
j
2
-
~!....
.......
.,..
....
........
1"--100... ~
10
""1--
T
!
20
1 I
i
,
I
30-
20
.....
.........
"'"'
~
i""of..,
~ F~llljn
J I
...... i'oo"
i
r-l- NO TE, jHE CURVE liE PR ESE N TlNG MATERIAL
!
:-0 ....
.... ,.....0-, I
I
i
'-I-- -I--
I
.
......
I
-~I...
i ,.... 1"--
T
I
I
!'-H-L.
~
i ~
J
20
I
I
r-o-;......;..
i
;;
1
I
I
i
~;
I
I
!
1'1"""--__
l""'o ..... ....
1""'0 .... !'~ .....
!
i
I
~x,~
, r
t
I
I
•
i
,"'-~ I
f-I-I-
I
' t
~"-i ..... 10....
i I
,....i--~
Ftu =75 ksi
I I
;
T
I 7075-T6 fORGING
I
~;
50
I
i-t
i
i 40
T
so
60
70
80
DIt
FIGURE 8.22 - TORSIONAL MODULUS
Or
RUPTURE - 7075-T6 ALUMINUM ALLOY FORGING 8-27/8-28
STRUCTURAL DESIGN MANUAL SECTION 9 BENDING 9.1 GENERAL
This section presents methods of analysis of beams in bending. Formulas for shear, moment, and deflection are shown in simple beam analysis. Beam column data for use in the three-moment procedure are included ,in this section, as are strain energy methods, plastic bending analysis, curved beam correction factors, and a bending analysis of bolt-spacer combinations. Allowable bending moments for 'tubes and channels subject to local crippling are also presented in this section. 9.2
S~ple
Beams
Shear, Moment, and Deflection
9.2.1
The general equations relating load, shear, bending moment, and deflection are given in Table 9.1. These equations are given in terms of deflection and bending moments. y
Title
M
Deflection
t:.
::::
y
t:.
Slope
Q
=
dy/dx
9
M
2 = EI d Y/dx 2
M
Shear
V
=
3 3 EI d y/dx
V
Load
W
= EI
ding Moment
)
4 4 d y/dx
=
=
JJ El dx dx f ~dx (M
dM/dx
W = dV/dx
2 2 = d M/dx
TABLE 9.1 SHEAR, MOMENT, DEFLECT ION EQUAT IONS
Sign Convention a) b) c) d) e)
x is positive to the right. y' is positive upw:ard
M is positive when the compressed
-x
+x
w '--......-_--="'--{f X
....
fibers are at the top. W is positive in the direction of negative y. V is positive when the part of the beam to the left of the section tends to move upward under the action of the resultant of the vertical forces.
9-1
-
~ STRUCTURAL DESIGN MANUAL \s:s\M.L~'" -\-\~
/--,/-
_ -4 -
..
-
---
,
r/
\
_ \
//
"/#
The limiting assumptions are: a) b) c)
The material follows Hooke's Law. Plane cross sections remain plane. Shear deflections are negligible. d) The deflections are small. The deflection of short, deep beams due to vertical shear may need to be considered. The differential equations of the deflection curve including the effects of shearing deformation is: dxdx
dx
E1
9.1
(K) is the ratio of the maximum shearing stress on the cross section to the average shearing stress. The value of (K) is given by the equation:
a
K
A
I
b
f
b'ydy
o
(I) is the moment of inertia of the cross-section with respect to the centroidal axis and (a), (b), (b ' ), and (y) are the dimensions shown in Fig. 9.1 (A) is the area of the cross-section
FIGURE 9.1
DEFINITION OF VARIABLES FOR DETERMINING K
Table 9.2 presents beam formulas for several different types of load cases. 9-2
•
Notation: W = load (lb); w = unit load (lb. per linear in.); M is positive when clockwise; V is positive when acting upward; y is positive when upward. Constraining moments, applied couples, loads; and reactions are positive when acting as shown. All forces are in pounds, all moments in inch-pounds; all deflections and dimensions in inches. Q is in radians and tan Q = Q •
Cantilever Beams Type of loading and Case number
React ions, Vertical Shear, Bending Moments, Deflection y t and Slope
RB
= +W;
Mx
= -Wx;
y
=-
i
V = -W
CI)
-I :0
c:
n --I c: :::a >rc;,
Max M =-WL at B
1 WL~
3
3 2 :1 (x _3L x+2L ); Max y
3'
E1 at A;
Q
1 WL
= "2
E1
rn
2 at A
-z
CI)
C ')
RC : +W; (A to B) V = 0; (B to
2.
(A to B)
x
M = 0; (B to C) M
= -W(x-b); Max M =
322 E1 (-a +3a L-3a x):
:::;;
(B to c)y
E1 = - 6lW'
= -
V = -W
1 W
-6
(A to B)y
Max y
C)
~
(x-b) 3 -3a 2 (x-b)
1 W 2 3 - -- (3a L-a)· 6 E1 '
TABLE 9.2
1 Wa
0 = -2 -E1
BEAM FORMULAS
+ 2a3] ; 2 (A to B)
-Wa at C
31: >z
= >r-
Cantilever Beams
Type of loading and -case number 3.
Y
w
W= wL
Illlll111l11~
A~~-X IL "
i RB 4.
y
r.L
k..J
~
W=w(b--a)1. a 4 ~
IIIIIIIDw
~
C D~ x
Reactions, Vertical Shear, Bending
RB = +W; V = M = -
I
x
1I x2 ; Max M = - i WL at B 1
=-
Q
=
Rn
= +W;
4
3
3
1 WL 24 EiL (x -4L x+3L ); Max y = - '8 EI
Y
W
4
1 WL2
+ '6" EI at A (A to B)V = 0: (B to C)V
(A to B)M = 0; (B to C)M
=-
1
-2
= ~~a
W b~a
(x-L+b); (C to D)V ; -W 2
(x-L+b) ;
(C to n)M
= - ~ W (2x-2L+a+b);
(A to B)y
= -
1 W [2 i4 EI 4(a +ab+b 2 ) (L-x)-a 3
( B to C) Y
= -
i4 EI 6 (a+b) (L-x)
(c to D)y
=-
A. ~I
B
Max y
1 W [
Max M =
2
-
-4(L-x)
i
W (a+b) at D
2 2 3J -ab -a h-b ,
3
+
(L_X_a)4] b-a
[3(a+b)(L-x)2 -2(L-X)3]
= - ~f ~I ~(a2+ab+b2)L_a3
() =
+i :1 e+
ab+b2 )
TABLE 9.2 ( CONT t D)
I
Deflection Y, and Slope
Moments~
2 3 _ab _a2 b_b ]
(A to B)
BEAM FORMULAS
at A;
;
-
~
..
Cantilever Beams Type of loading and case number
Reactions. Vertical Shear, Bending Moments. Deflection Y. and Slope
5.
RB
=
+W; V = - ;
x
2
L
3 13 ~L 2 x • '
= = _
Max M
l-~2 60 EIL
1 WL = + 12 EI
= -
13
WL at B
(x 5.SL4X+4L S)6, M 1 WL ax y = - T5 EI
3
A.,
at
2
at A
Rn = +W; (A to B)V = 0; (B to C)V = - W(X-L+~)2; (C to D)V = -W
6.
(A to B) M = 0; (B to C) M = y
J~b~ r- a ~~
W= ~ (b-a)
I~
A
r
B
w
~
~
- ~--x
- C D
_tR L----, D
_! 3
(b:"a) W(x-L+b)3 (b-a)2
1
1
'3
-
~20a+lOb)(L-x}2 ~
(B to C) y = - 610 EWr
(C to D) y
= -
1 W
6
EI
0r: 2a+b )
(L-x)
2
-lO(L_x)3 +5 (L-x-a)4 _ (L-x-a)SJ;
3l -(L-x) J;
b-a
1 W 2 +lOba+15a 2 ) L-4a 3 -2ab 2-3a 2b-b 3J at A Max Y = - 60 EI [(5b
8
=
1 +T2
W
EI
2
2
(3a +2ab+b ) (A to B)
TABLE 9. 2 ( CONT • D)
I
-
.
W(3x-3L+b+2a); Max M = - 3' W(b+2a) at D 1 W I; 2 2 3 2 2 3J (A to B) y = - 60 EI !iSb +10ba+l5a ) (L-x) -4a -2ab -3a b-b
(C to D) M =
BEAM FORMULAS
(b-a)2
Cantilever Beams Type of loading and case number
Reactions, Vertica.l Shear, Bending Moments, Deflection Y , and Slope
en 1 W - 60 BIL2 g
8.
4 EI
4
4
5
(-x -15L x+5LX +IIL ); Max Y
=-
11 WL 60
EI
(C to n) V W;;;~
w(b .. a)
t.--b~~If I_---:-___ w~a~~ --===-_..,iI-!X :
r...._B-
n ~ L ----iRn
(b-a)
= -w
(A to B) M :;: 0; (B to C) M :;: _
1
W[3(X-L+b)2 _ 3 b-a
(C to D) M
= - 31 W(-3L+3x+2b+a); Max
(A to B) y
= - 1~ 60 EI
(B to C) y
=-
(C to D) y
=-
Y
=-
::a c:
at A
!O
~I [(~~~:;;
i ~I r
1 W 60 EI
M- -
31
~
2
3
-2a 2b-Jab 2 -4b 3J
(A to B)
TABLE 9.2 (CONT I D)
BEAM FORMULAS
=cJ ):II
rc:::::.
W (2b+a) at D
-10(L-x)3+(lOa+20b) (L_x)2]
2
c:
(b-a)2
Ea+2b)(L-X)2 -(L_x)3] 2
-I
(~-L+b)3J
5
~5a +10ab+15b )L-a
n
J;
~5a2+10ab+15b2)(L-x)-a3_2a2b-3ab2-4b3J I~ .
C
Max
-I
L2 at A
Rn :;: +W; (A to B) V = 0; (B to C) V :;: - W [1 _ (L- a - x y
A
+! ~
:;:
5
3
",
;
en
-z
CD
!I: at A
):II
z
c: ):II r-
I
e '--
Cantilever Beams Type of loading and case number
Reactions, Vertical Shear, Bending Moments, Deflection Y, and Slope
9~_1 Y .. ~
r~_ L r
10.
RB
=
jB M :K
0; V =
-I
:::a c:
0
(A to B)
M ; Max M = M
o
0
2 LIMo 2 2 __ 1 MoL By=- - - (L .. 2Lx+x ); Max y - - at A; 2 E1 2 E1
s_
MoL Q - -
EI
(A to B) M = 0; (B to C) M =- Mo; Max M = Mo (B to C) (A to B) y
~ ~a
(B to C) y
~ {~
~~
at A
n ..... c :::.::11
RC =- 0; V ;;; 0
Max y ;
(I)
(.. -
ta-
C
x)
.....
J
[
(L - { a
)at
A;
:Dr-
Q
=-
Maa EI
(A
-
(I) to B)
CD
z
:c:
TABLE 9.2 (CONTID)
BEAM FORMULAS
:DZ
c: :Dr-
Simply Supported Beams Type of loading and case number 11.
Reactions, Vertical Shear, Bending Moments, Deflection Y, and Slope
RA =
21
1
= 2 W; (A
W; RC
1
to B) V = 2 W; (B to C) V = -
21
W
1 1 1 (A to B) M = - Wx· (B to C) M = 2 W (L-x); Max M = 4' WL at B (A
to B) y 1
Q
12.
RA
(A
=
.=
-
16 Wb
+L
:=
WL
2 ' 1 w 2 3 48 EI (3L x-4x ); Max y 2
EI
;
at A;
L
(A to B)
Y :=
X'
'
-Wbx 6EIL
48
Er
en -t :::a c:
at B·,
2
+ IT EI
n
at C Wb
c: :::a
Wa
; (A to B) V = + L ; (B to C) V = - L
+L
Wb
:=
WL
=-
wr. 3
-I
Wa
RC =
to B) M := + -
Q
1
1
(B to C) M = +
LWa
Wab
(L-x); Max M = + -
L
at
B
= ,..,
[ZL(L-X) -b 2 -(L-x) 2 ] ;
en
2 (B to C) Y = _ Wa(L-x) [2 Lb_b _(L_X)2]. 6 ElL '
Max y
Q
=-
Wab
=-
6'1 W EI (
27EIL (a+2b) bL -
3 b L)
~3a(a+2b) at A;
Q
at x
:=
~~3
(a +2b) When a > b;
3 1 W ( b + - -- 2bL + - - 3b 2) 6 EI
:J:a r-
L
at C
-:z
C)
51: :J:a
:z
TABLE 9. 2 ( CONT I D)
'------
BEAM FORMULAS
.,
c: :J:a r-
S imply Supported Beams Type of loading and
case number
13.
Y
W=wL
Reactions J Vertical Shear, Bending Moments, Deflection YJ and Slope
w RB ::;; 2; w V ::;; + 2w ( 1 RA = 2; Hax. M
= + -WI, 8
L = -' 2'
at x
_£:). L' Wx
RA ::;; ~d; RD ::;; ~
(a
+
(x _
(I)
2
c:
......
2!:2 )
L
3
::a
3
Y ::;; - 24EIL (L -2Lx +x );
5WL 3 L Max. y ::;; - 384EI at x ::;; 2; 14.
~1 = + ~2
Q
I) ; (A
WL
2
WL 2 Q = + 24EI
= - 24EI at A;
to B) V = R : (B to C) V
A
= RA
n
_ w(~-a)
(C
to D) M
= RAx
(A to B) Y ::;;
W= wc d= L-1:2b-~a
1
48fi
wG - ~ - ~);
Max. M =
{8R (X 3-L 2X) + Wx A
~d
(a +
[S~3_ 2~c2
~~) at
x
+f3+
r-
=a
CJ rt1
48Ei
[8~3
_
2~c2
+
~3
1 :-48EI
{8R (x 3-L 2 x ) + wx A
[8~3
_
2~c2
+
~3 ]
-
1
3
'b) 3 + W(2bc 2-c 3:\r 8W ( x - '2 - 2" a
+
~d
-
(I)
2c2J~
{SR (X -L 2x) + Wx A
(B to C) Y ::;; (C to D) Y
-
2c
:=0 ):II
2
(C to D) V ::;; RA-W; (A to B) M ::;; RAx; (B to C) M = R x _ W(x-a) A
..... c:
at B
+2C 2]
CD _
:z
2W~X-a)4}
if:
:.:-
z
" (1)
1-'-
en
1-'"
0
::s TABLE 9. 2 (CONT' D)
BEAM FORMULAS
~
c: :.:r-
~imnlv
Type of loading and case number
~l1nn()rted
RpamJO:
Reactions, Vertical Shear, Bending Moments, Deflection Y, and Slope
14. (Cont d)
1
I
Q = 48EI
Q =
en
1 48EI
15.
-I
E!.
3'
V
2
(1:.3 _x 2 '
=w
). M
3 ex _ x ). 3 2'
= li
L
r:-
L
4
224
L ~3 Y = -Wx(3x -10L x +7L ) 0 128 WL a t x::: -3Max. M =. ; 2 180EIL 3 0.01304 WL Max. y = EI at x = 0.519 L Q
=-
2 7WL 18DEI
--.;..;.~
at A ;
Q ==
2 8WL 18cEI
at B
:::c n -I c:: :::c :Dr-
c:
-=
rn
-
(I) C i)
TABLE 9.2 ( CONT I D)
BEAM FORMULAS
z
31:
:DZ
= :D-
r-
•
'\~,
Simply Supported Beams Type of loading and
case number
16.
Reactions, Vertical Shear,
w(
w R = -W* (A to B) V = R = _. A 2' C 2' 2 2 (B to C) V = [1 ]
~
(B to C) M =
(A to B) y;;
4(~;X)
~ ~(L-X) Wx 6EIL2
(L
-
RA
=:
w "2;
1 - -4x2) • 2 f L 3 W( 4x ) ; (A to B) M ;;"6 3x - L2 Max. M =
~
4 x 5
2
5WL 2
Moments, Deflection Y, and Slope
4(~;X)3l
2x2 _
Q = - 96EI at A ;
17.
Bendin~
Q =
_ SL4). Max y 16 ' •
at B
= _ WL60E3I
at B
5WL 2
-
+ 96EI at C
W (A to B) V ;; '2 W (L-2X)2 RB ;; 2"; -L-
(B to C) V = ..
!! 2
(B to C) M =
~
(2X-L\2 ; (A to B) M = L
1
r(L-x) -
l
2
2(~_x)2
t-+
Q = -
+
4(L_~)3 3L
W (3 x4 (A to B) y = l2EI x -
WL 2 32EI
l'!
at A;
TABLE 9. 2 ( CONT t D)
X)
(x _ 2x2
J;
+ 4X
L
Max, M =
3
3L2
~
)
at B
2 3 5 3L 2x 3WL SL2 .. -8- ; Max. y = - 320EI at B
WL2
Q = 32EI
at C
BEAM FORMULAS
I
Simply Supported Beams Type of loading and
case number 18.
Reactions, Vertical Shear, Bending Moments, Deflection Y, and Slope
M RA = - ~ L '. RB
y
Y=
:~I
Q =
R = A
~3 -2L~ ; Max.
(?x 2 M L o
- 3E1
r-;o
y = _ 0.0642
RC =
Q
M0 ,
r-;
o 6E1
a
(A to B) y
RAr=e;:~::=-rf;X A
I_
L
=
(B to c) y =
1e
:~I
just right of B
3~2 20
[(6a -
-
:~I [3a 2 +
3x2 _
~3
x -
E1
at A
M~~2 at x = 0.422L
just left of B
]
3~2)xJ
g=6~I L_3~
M . (
(a _ .!!) 2
Mo
~3
-(2L +
g=-:~I(2L-6a+3~2)atA;
Q =
0
(A to C) V = + R ; (A to B) M = + RAx A
Max. (+M)= + RAa + Mo Mo
= M
at B
(B to C) M = + RAX + Mo; Max. (-M) = + RAa
~ -1
Max · M
M L
at A;
M
19.
M
=~ x· L ·~ V = RA·· M = M0 + RA'
a
L
_
3
TABLE 9.2 (CONT'D)
at B
BEAM FORMULAS
2)'
atC
-
,
e
\
''-../
'"--,,,,'
Simply Supported Beams Type of loading and case number
Reactions, Vertical Shear, Bending Moments, Deflection Y, and Slope
20.
RA
~
-
Wb a-;
RB =
(A to B) M
WL a-;
(A to B) V = + RA; (B to C) V = +W;
+RAx; (B to C) M= + RAa + W(x-a); Max. M = + RAa at B Wbx
2
2
(A to B) y
; - 6aEI (x -a );
(B to C) y
= - 6~I [(L_x)3 - b(L-x)(2L-b) + 2b 2LJ
Max. y Q
2 Wb L
= - 3EI
=-
at C;
9
Wah
= 6EI
at A;
9
=-
Wab 3EI at B
Wb (2L+b) at C 6EI .
TABLE 9. 2 ( CONT I D)
BEAM FORMULAS
-
Statically Indeterminate Cases Type of loading and case number
Reactions, Vertical Shear, Bending Moments, Deflection Y t and Slope R
21.
A
=!! (3a
2
2
(A to B) V
3 L_a ).
L3
=
(B to C) M =
'
R
:=
C
W -R • M
A'
c
3 (a +2aL 2_ 3a 2L )
=!!
L2
2
RA; (B to C) V RA - W; (A to B) M = RAx RAx - W(x-L+a); Max. (+M) = RA(L-a) at B max. possible value = 0.174 WL when a = 0.634 L
y
W
... t..
Max. (-M) = -M at C c max. possible value = -0.1927 WL when a = 0.4227 L
(A to B) Y = (B to C) Y =
6~I
'
3 2 2 [RA(X -3L X) + 3wa x]
6~I {RA(x3-3L 2x ) + W ~a2x_
(x-b) 3J} J
If a<0.586 L, max. y is between A and B at x 2
2
2
2
= L ~J'1
• - ~ 3L-a
_ L(L +b )
If a > O. 586 L, max. y is at x -
3L .. b
If a = 0.586 L, max. y is at Band deflection Q =
~ 4EI
3 (a _ a 2) L
at A
TABLE 9 .. 2 (CONT' D)
I
:=
3 WL -0.0098 EI '
BEAM FORMULAS
m~.
possible
-
I
Statically Indeterminate Cases Type of loading and case number
Reactions, Vertical Shear, Bending Moments, Deflection Y, and Slope
W==wL
:;~U1U~B1 ~Tx I'
-f
L
~
(+M)
Max.
3L
9~
128 at x ~ ]i; Max. (-M) ; -
~ s-
at B
W 3 4 3 ~3 y; 48EIL (3Lx -2x -L x); Max. y == -0.0054 EI at x ; 0.4215 L
RB Q = -
2 WL 48EI
at A
(I)
..... ::.::»
c::
n
.....
23.
= ~
". r-
CJ
rn
-.:z (I)
a)
3:
".
:z c:: ". r-
(C to D)
TABLE 9.2 (CONT'n)
BEAM FORMULAS
Statically Indeterminate Cases Type of loading and case number 24. Y
==~L
~ L --4 RB
RA
Y
=.!'!:. 5t
=
R == 4W. MB 5 ' --B
=
2WL. V ==
60~IL ~:x3 - L3x - { 9 == - 60EI
25.
(1 _
y~
and
Slo~e
(~
3 x2) . M = W _ x ) 15' 5 2' 5 2 L 3L w x Max. (+M) = 0.06 WL at x : 0.4474 L; Max. (-M) = -~
~
A_ RA
Reactions, Vertical Shear, Bending Moments, Deflection
) ; Max. Y
L 2)
(B to C) M = +RAx
Max. (-M)
=
-0.00477
(L 2_a 2 ).
3Mo ; RC = 2L
L2
(A to B) V == +R ; (B to C) V A
==
=
+
= +R
M • c
==
2 L3 a ) 2
; (A to B) M = +RAx A
Mo; Max. (+M) = Mo ( 1 _
3a;~~-a2»)
just to the right of B -Mc at C when a < 0.275 L RAa to the left of B when a > 0.275 L
~ ( L:~{
2 3 (3L x_x ) - (L-a)x)
(Bto C) y =
~ rL~~;2
2 3 (3L x_x ) _ Lx
:~
(1 _
Mo 2
(A to B) y =
Q =
~3 at x = L f i
at A
3Mo / 2_ a RA = - 2L ~ L2
Max. (-M)
w
( a _
*_~~2)
TABLE 9.2 (CONTID),
-
+
at A
BEAM FORMULAS
2 (x2;a
»)
-
•
•
Statically Indeterminate Cases Type of loading and case number Reactions, Vertical Shear. Bending Moments, Deflection y, and Slope 3 2 2 llW 9W 7WL (11 2x x ) (llX x x ) RA ;: 20; RB :: 20; ~ = 60; V = W 20 - r- + M:: W W - r-+ 3L 2
17 ;
Max. (+M) :: 0.0846 WL at x ;: 0.329 L; Max. (-M) :: - ~~ at B y =
12~EIL
(ULX
Max. y :: -0.00609
27.
R
:: _ 3Mo. R
A
2L'
Max. (+M) y_
Mo 4EI Q = -
3
- 3L 3x _ lOx4
~3
~
:: 3Mo.
B
+
2~5 )
at x :: 0.402 Lj 0 ::
~
2L' -13
::
Mo.
V == _ 3Mo. M ::
2'
:: M at A; Max. (-M) o
2L '
MoL
No 2
(2 _
3X)
L
2
= -
L
27EI
at x :: 3
-
at A
TABLE 9. 2 (CONT' D)
at A
= - Me at B
x ; M ax. y (2x2_~L3 - L)
ill
~2
40EI
BEAM FORMULAS
~~ <
~ '~
~
CD
1-'til 1-'''
0
1.jw ,,~ 2
28·ly
(. r----..MA · A
RA
, ( A to B) M = W(4~-L); (B to C) M = W(3~-4X); Max. (+M)
i-")...!
B L
c,~
WL = - g-
MC Max. (.M)
at A and C; (A to B) y
l~~I
RC Max. y = -
and ~loDe
en -I
2
RA
Wb = -3L
W
=-
=~
2
c:
n
at B
......
3
c:::
48EI (3Lx -4x )
::a :z:-
at B
r-
2
(3a+b); RC
= -Wa3 L
(3b+a);
(A to B) V = R -, (B to C) V A Wab
= RA
c::J
2
M
Wab = -_. 2
a
L
M c
'
__ Wba
rn
2
en
-:z
L2
2 L2
'
C i)
+R x
- W· (A to B) M = - Wab
A
2
(B to C) M = - ---- + R x - W(x-a); L2
Wab
Max. (+M)
=-
Max. ( ... M)
= -MA
Max. (-M)
==
2
~
31:
A
WL
:z::z c: :z:-
L
+ RAa at B, max. value = g- when a = 2
when a
< b,
L max. possible value ;:; -0.1481 WL when a =-
r-
3
2L -M when a > b, max. possible value = ... 0.1481 WL when a;:;c 3
TABLE 9. 2 ( CONT ' D)
,
'-i_
::c:I
1-- --.f
29.
ttt
Vertical Shear. Bendine: Moments. Deflection Y
Reactions~
-r--
\'t -~. J1 iCl-~
::s
Statically Indeterminate Cases Type of loading and case number
'- -
e
BEAM FORMULAS
e--
I
and ( Con t t d)
29 •
2 2 Wb x
(A to B) y
(B
to C) y
Max. y ;;;; -
Max. y ;;;; -
=
(3ax + bx - 3aL) 3 6EIL Wa 2 (L_x)2 [(3b+a) (L ... x) - 3bV 6EIL3 ::J
3 2Wa b 2 3EI(3a+b}2 2Wa2b
at x
= -2aL
at x
=L
3 2
3EI(3b+a)
>b
if a
3a+b
2bL
- 3b+a
if a
30.
IY M
2
W=wL
Cjl!fE LlITf!lf)-~ f.-- -----4
A
RA
A
B
L
RB
(x - Lx 2 ~
1). 6'
M = .!'!.
-
Max. (-M) ~ -
12
Max. y
=-
WL
WL
Max. (+M) = WL24
-
= .!;2
Wx 2
2
2
at A and B; Y ~ 24EIL (2Lx-L - x )
3
384EI
at x
at x ;;;;
L
2 ~
CD
< ..,. CIl .....
TABLE 9.2 (CONT'D) BEAM FORMULAS
o:;
t:d
..0 I N
o
Statically Indeterminate Cases Types of loading and case number
Reactions, Vertical Shear, Bending Moments, Deflection Y. and SloDe
31.
R = A
M
A
~ 4L2
= _
~ =
(12d 2 _ 8d
2
3
+ 2hc _ c
L
.JL 24L
(24d
3
L
W (24d
3
24I \:-L-
L
.. 6bc
2
L
2
(B to C) V = RA -
w(x~a);
L
+
6bc 3c -r+L
-
3
3c
3
+
; R = W ... RA D
+ 4c 2 _ 24d 2)
L
3
_ c2)
,
2 2 2c - 48d
+
) 24dL
; (A to B) V = RA
(C to n)v = RA - W; (A to
B)
M ~ -MA+RAx
2
(B to C) M = -M + R x - W (x-a) • (C to D) M - -M + R x - W(x-L+d) A
2c'
A
Max. (+M) is between Band C at x = a
= -MA
Max. (-M)
W=wc
y =
( B to C)
y
to
< (L-b); Max.
I
(-M)
132
(R
D) y = 6ir [R n (L-X)3 - 3Mu(L-X)2J
TABLE 9.2 (CaNTin)
A
when a
>
RAe +-w-;
6EI (RAx - 3MAx ) = __1__ 3 _ 3M x 2 _ w(x-a)4) 6EI AX A 4c
(A to B)
(C
when a
a
BEAM FORMULAS
= .. ~
(L-b)
-
e'-
.
\,-",
Statically Indeterminate Cases Type of loading and case number 32.
r
3W 7W WL RA = 10; RB = 10; MA = IS;
W=~L
~
(1 ~ ~
MA
RA
A
M
(
A
RA
Iy
M= w
r
W = 60EI
(3
~ C
M C
2
17 ) WL
2
+ RA
); V =
= -0.002617 WL EI at x = 0.525 L
;
to B) M = - MA + RAX
(A
= - MA + RAx + Mo
(+M) = M
(4a _ 9a 0 L L2
Max. (-M) :::; M
o
to B) y
3
6Mo 2 Me 2 2 = L3 (aL-a); MA = - L2 (4La-3a -L )
2
(aL-a); RC
(2La- 3a L2,
L ------iRe (B to c) M
(A
(3 x V = W 10 -
Slo~e
WL at x = 0.548 L; Max. (-M) = - TO at B
x ) - 2Lx2 - L2 ; Max. y
= -No
Max.
TO;
=
5
3x
RA = - L3
~ ABC
= 0.043
Max. (+M)
6Mo
~a~
WL
Deflection Y, and
3
B MB
t r -__ L ~~B
L_
~
Moments~
i~ - ~5 - ~L2)
(
y 33.
Vertical Shear, Bending
Reactions~
(4a _
2 2
9a
3
TABLE 9.2 (CONT t D)
::cJ
c:
n ...... c
,. ::cJ
r-
rt1
)
just to the right of B
3 6a
_ 1) just to the left of B
L L2 L3 123
= - 6iI (3MAx -
.....
CJ
+ 6a L3 +
en
RAx )
BEAM FORMULAS
-z
(J')
CD
,.3:z ,.c r-
,
...0 I
N N
(I)
-I :x:J
Statically Indeterminate Type of loading and Reactions, Vertical Shear, Bending case number 33.
(Cont'd)
(B to C) y
1 = 6EI
c:: n
Cas~s
Moments~
Deflection Y, and Slope
~(Mo - M ) (3x 2 ... 6Lx + 3L 2 ) - RA (3L2 x - x 3.. 2L 3~ ) A
Max (-y)
2Mc at x = L - Rc
Max (+y)
2MA at x = RA
if
if a
< 2L 3
a> -L3
-I
c:
::.::J
::III
r-
c::J ...,
-z
CI)
CD
TABLE 9.2 ( CONT • d)
BEAM FORMULAS
i: ::III Z
c: ::III
r-
STRUCTURAL DESIGN MANUAL 9.2.2 Stress Analysis of Symmetrical Sections
The bending strl'SS in a beam with at least one axis of symmetry in the cross .section is f :: Mc/I
"
where M is the bcnciing moment at the cross secti.on in question, c is the distance from the neutral axis of the cross section to the fiber in question and I is the moment of inertia of the cross section. This equation is valid when the tol lowing assumptions are satisfied: a) plane sections remain plane b) the material follows Hookefs law This t.'quation is called the flexure formula.
arc rarely loaded in pure bending.
B~ams
Generally the bending moment is a
resul t of a shear transfer. This means that the bending moment varies along tht: beam. This variation in bending moment produces a shear normal Lo a beam crOHS sc-'ction, i.E'., normal to the section resisting the bending moment.
•
The flexural [
v ]t
s
shear stress normal to the cross section is c
JY = 0
ydA
I n i t sin t e g rat e d f 0 rHl i tis f
s
=
VQ/lt
whcr(' V is the shE'ar paraU el to the cross section, Q is the area moment of the cross seclion, I is the moment of inertia and t is the section thickness.
)
9-23
~ STRUCTURAL DESIGN MANUAL L~ Revision' A 9.2.3 Stress Analysis of Unsymmetrical Sections The assumption for the flexural formula, f = Me/I, is that at least one axis of symmetry passes through the section centroid. This condition is not always possible. The section shown below is a general section
z
---~
Right Hand Moment Rule
----y
~
• with no axis of symmetry. The equation for the stress in a homogeneous with axial ~nd bending loads is
fx =
Px (\\:z + MzI~)y +{Mz\z + M/z)z
A -
I I - I y z yz
I I
y z
- I
yz
2
beam
9.2
• 9-24
STRUCTURAL DESIGN MANUAL 9.3 Strain Energy Methods 9.3. L Castigliano's Theorem Castiglianots theorem is used for determining deflections and rotations in structures. It is useful also in the analysis of statically indeterminate structures.
)
Castigliano's theorem states that if external forces act on a member or structure and cause small elastic deflections, the deflection of the point of application of the force in the direction of the force is equal to the partial derivative of the total internal strain energy U with respect to the force. That is
oQ =
auf aQ
9~3
where 00 is the deflection in the direction of Q of the point of application of Q. The deflection of any point due to any system of loads may be found by introducing a fictitious load at the point in question and writing the expression for the derivative of the strain energy with respect to the fictitious load at this point. The method of Castigliano requires that the internal strain energy U be expressed in terms of the loads in each member. Therefore, the strain energy expression for each type of loading is U =
JpL 2 dx/2AE;
9.4
Axial
o
u
fL
2 M dx/2EI; Bending
9.5
o L
u
.f V2dx/2AG;
9.6
Shear
o
where A and I are the area and inertia of the element and E and G ·are the modulus of elasticity and rigidity. P, V and M are the axial force, shear force and bending moment in terms of the applied load to an -element. 9.3.2 Structural Deformations.Using Strain Energy The deflection at a point can be obtained by the application of Castigliano's theorem. Consider the example shown below, a
( 9-25
STRUCTURAL DESIGN MANUAL Revision C simple beam of span L and constant cross section subjected to a concentrated load P at the center of the span. The moment at any point of the beam is
Px12, for x = 0 to x L/2 Px/2 - P{x-L/2), for x = L/2 to x = L From Equation 9.S L/2 (Px/2)2dx + 1/2EI M2dx = 1/2 EI 1/2EI f u M M
0
p2L3/96EI aUI oP =
U 6
L
2
f [P/2(L-x)] dx
L/2 )
2 3
81 ap(p L l96EI) =
3 2pr:/96EI = PL /48EI
The example is about as simple as possible, but the theory is applicable to problems of any complexity. The rotation of a point can be determined by employing Castigliano's theorem which becomes
aut
.Q. =
Be
9.7
where C is an applied or fictitious couple. Consider the following example.
It is
p
L
--1
reqoired to find the rotation of the free end of a cantilever beam. Since no couple acts at the end of the beam, one must be applied. It ~s denoted as C and since it is fictitious will be set equal to zero after the differentiation wi.th respect to C. Then the example is solved as follows.
au aC
,Q
=
a
Be
J'
2 M dx
2EI
aM JL EI ae M
=
dx
o M =
e+
3MI
ae = 1
~
rL (e
=
.
o
Now set C
Px
+ Px)dx(l)dx E1
o
since it is fictitious, then
2 PL -9-
=:
2EI
This procedure is applicable to slructures with any type of loading. O_?I..
STRUCTURAL DESIGN MANUAL Deflection By The Dummy Load Method
9.3.3
The unit load or dummy load method may be used to determine deflection in elastic or inelastic members. Deflection of inelastic members by this method is given in section 9.5.2. The theorem as applied to elastic beams is written in integral form as
o= o
f
L
~
dx • • • • . . • .. .. • . .. .. .. • • .. • . .. • • • . .. • • 9 .. 8
L
o= f ~~t
dx
• • • .. • • .. .. .. • • • .. • • • • .. • • • .. .. • • .. 9 .. 9
o
.)
Where (8) is the deflection at the unit load and (0) is the rotation at the unit moment. The moment (M) is the bending moment at any section caused by the actual loads. (m) is the bending moment at any section of the beam caused by a dummy load of unity acting at the point whose deflection is to be found and in the direction of the desired deflection. The bending moment (m r ) is the bending moment at any section of the beam caused by a dummy couple of unity applied at the section where the change in slope is desired. It is noted that although (mt) may be thought of as a bending moment, it is evident from the expression mt
=~~athat
it is actually dimensionless.
EXAMPLE: Find the elastic vertical deflection of point A (Fig. 9.2a) of the simply supported beam subjected to two concentrated loads.
r*-1 r -, PRIMARY LOADING
P
)
R=P
DUMMY LOADING
P
L
~L~R=P (a)
[M=PX M
V
r
r ~=PL/4
'{M=PX2
XI
x2~
r*1
4
1 LB
iLL 4
(b)
t
1
..I -4
~
m--.e=x2/4
~Xl
X2~ I
(c)
(d)
FIGURE 9.2
EXAMPLE OF DUMMY LOAD METHOD 9-21
STRUCTURAL DESIGN MANUAL
\
Solution: The actual loading is shown in Fig. 9.2a and the dummy loading is shown in Fig. 9.2h. The moment for the actual loading is shown in Fig. 9.2c and the corresponding moment diagram for the dummy loading is shown in Fig. 9.2d. The deflec tioD by use of equa tion (9.8) noting that xl starts at the left and x 2 starts at the right is L L
o = j~ EI
J EI eXl) 4
4
J
4
L
PX 1
dx = 0
dX l +
PX 2
0
(:2)
dX
2
3L
4"
+
1
PL
4EI
L 4
The problem of Figure 9.2 can also be solved using pictorial integration. To find any deformation in a structure due to any external loading, proceed as follows: (1)
Draw the moment diagram due to the actual loading. moments as M •
Denote these
o
(2)
Remove the actual loading and apply a fictitious unit load where the deformation is desired. This unit load must be of such a type and applied in such a manner that (load) x (deformation) :;: (work); i.e., if the deformation to be found is a rotation, the unit load must be a moment. Draw the moment diagram due to this unit load and denote these moments by M • a
(3)
Compute the deformation from
9.10 where 6 is given in TABLE 9.4 for various combinations of moment diagramgaM and Maa (TfBLE 9.4 is not applicable for curved members.) o Table 9.3 shows the solution of the problem in Figure 9.2 by pictorial inteThe moment diagrams of the actual and unit loads are shown in Table 9.3a. o is obtained from Table 9.3b, c t and d as,
~ration.
oa
7u
~ + 7v 22 + ~v33 = PL 3 j256 + PL 3 /64 oa :;: VII
+
PL 3 /768
The deflection at point A is:
o08 = 1 JEI 6"oa :;:
3 PL / 48EI
which agrees with the previous solution.
9-28 j
STRUCTURAL DESIGN MANUAL
• a. M
r---
o
_/;
~
Mo2
~ol~
_I
----,..,
M
a
~ ~ ~-rLL/l6
PL
f0 4 :Mo~
M =x /4
3L/16
2
a
I
;r-Ma~
: M2 a1 I a
:~
xl
MOMENT DIAGRAMS FROM FIGURES
9.2(a) and 9.2(b)
x2
UNIT
b.
c.
171 ~
TIMES
TGJ
MQ
~
Mo2
I-- L/2 --I
811 = k(Mo)(Ma)(L) k = .333: (TABLE 9.4) 011 = (.333)(PL/4)(3L/16)(L/4)
l~ j.-L/~
T~ TIMES
r
a
~
Ma2
....L
I-- L/2 ~ ~ab
822 = =
k k
=
3
PL /256
k(Mo)(Maa)(L) .50(1.0
= .667
+
M b/ M
);(TABLE
9.')
a aa 3 822 = (2/3)(PL/4)( 3L/16)(L/2)=PL /f4
833 = k(Mo)(Ma)(L) k = .333: (TABLE 9.4)
833 TABLE 9.3 - SOLUTION
OF FIGURE 9.2
= (.333)(PL/4)(L/16)(L/4) PL 3/768
=
BY PICTORIAL INTEGRATION
9-29
"8 .
STRUCTURAL DESIGN MANUAL
A.
CORRECTION COEFFICIENTS FOR EQUATION Elo
CASE
= k(Mmax )2L
DEFINITION OF SITUATION
..L I.
k
I
MI·
1.000
~I-I-L---...t ...L
II.
III.
Mmax~
TI-a
L ---r
---C-
M
Tf.e-L IV.
~IMmin
M. A=~ M
'~
.333
r.--L~
T
I---- L«::::::::::J ----1
.. T::::=:,.
J.. L-M max
L
Mmax
.333
..L
TI-- ~~ --...I
VI.
.333(1.0 + A + A2)
max
M~~
M max
)
...1
-*-~
V.
.333
.333 Mmax
~IMmax
VII.
~L--i
VIII.
Mmax~ Tt---L --t
2ND DEGREE
8 .533 (= - ) 15
-*-
2ND DEGREE
.200 i
.:L
IX.
Mmax~ Tt-- L--"';
1
.. 143 (= -) 3RD DEGREE
-*-
x.
Mmax~ T 111-- L -----I .tl
XI.
"
4TH
t
t--- L ---'1 TABLE 9.4
9-30
1 .111 (::; 9)
L~LM B min
Mmax
7
DEGREE
M •
A=~ M max
CORRECTION COEFFICIENTS
.333(1.0 + A + A2)
STRUCTURAL DESIGN MANUAL
e) B.
~J:o
CORRECTION COEFFICIENTS FOR EQUATION EI () = k(M ) (M ) L o
CASE
a
l---L~
DEFINITION OF SITUATION
lIMa
I
I.
k
1.000
I-t:---- L -----l
I
I I
II.
Mr~1 I
I
I
, r
III.
I I
:J
MIl
.f
,I
I
I
I I
I
I
-L~ a
• LC I
I
rE
a
.000
I
~I
I
I
l
•I
I
ML~ I
I I
I
... 333
2ND DEGREE
I
.250
•
I
I
I
I
, ML~ .
2ND DEGREE
I I
ML~
.667
I
•
I
x.
M
I
I
IX.
.000
•
I
I
VIII.
.500
" 2I t Mtl I
M~L
.50 (1.0 + A)
a
I
~
I
)
-M
TIMc::::::::::J T I
VII.
Ml
='
I I Ma
I
VI.
A
Mal~: I
v.
--.e- Ml
•
I
IV.
..500
3RD.DEGREE
I I
.200
I I
4TH DEGREE
•
TABLE 9.4 (Cant'd)
j
CORRECTION COEFFICIENTS 9-31
STRUCTURAL DESIGN MAN"UAL ~--------~------------~----------------------~---------------------.
(B Cont'd) XI.
+A)
• 5 ( B( 1-A)
XII.
A
=~
K
= arc length
M
a
.5
(!\
(K-l.)
+
-
1.)
L
C. CORRECTION COEFFICIENTS FOR EQUATION EIo = k (M )(M ) L o
CASE
a
DEFINITION OF SITUATION
k
I.
.333 I
•
I
I
~~~a
II.
I
.167 (=
•
I
I
I
I
•
III.
1 6')
.500
M1
A =-
IV.
M a
.167 (2.00 + A)
v.
.250
A='£L NOTE: b is dimension farthest from
VI.
.167 (1.00 + A)
Mr.
• • L'r:::::.......-. • rT II ac::::::::::::j T M I ___
VII.
a
1
.167 (="6)
I
I
•
TABlE
9-32
Ma
9 4 (Cout'd)
CORRECTIoN OQEFFICIENTS
I
-
•
STRUCTURAL DESIGN MANUAL CASE
DEFINITION OF SITUATION A=~
VIII.
L
.167 (1.00 - A)
Ml
A =M a
IX.
x.
k
I
I
I
I
I~]!a 2ND DEGREE
..333
2ND DEGREE
.250
XII.
.200
3RD DEGREE
1
DEGREE
.167 (=
6)
XIV ..
2ND DEGREE
.083 (=
12)
xv.
3RD DEGREE
XIII.
4TH
1
) 1 .033 (= 30)
XVI. 4TH
DEGREE
XVII.
A=~
XVIII.
A=.!
L
L
TABLE 9.4 (CONT'D)
2
.50 (1.0 - A )
CORRECTION COEFFICIENTS 9-33
/?C\\
STRUCTURAL DESIGN MANUAL ~.~" ~ ~
1/ , / ; 7
,"
w
Revision B
1--_ _C. ..... ~A~S;___ E_j___---J)E£--'lNITION
(C cant' d)
,..
XIX.
L
OF SITUATION
~=ta
:
t.--
A==!. L
*
!
B
MIT:~:TMa I
I
D.
a
A=L
a-tl·
xx. XXI.
k
--,1.
Ml
A =-
M a
I
.167 (2.0 - A - A2)
.167(A(2A2-3)+1) ( __1__ )2
A-I
CORRECTION COEFFICIENTS FOR EQUATION EId == k(M )(M ) L M
A
==
0
a
Ml o
CASE
I.
DEFINITION OF SITUATION
I
J.--
I~a
k
.50 (1.0 + A)
L----..r
II.
.167 (2.00 + A)
III.
2ND ~EE
.083 (3.00 + A)
• 333 (1. 0 + AB + O. 5
IV.
(A+B») i
V.
I
Mar~ , I
VI.
I
.05 (4.0 + A)
Mar~
4TH DEGREE
.033 (5.0 + A)
I I
I
I
I
I
I
TABLE 9.4 (Cont'd) 9-34
3RD DEGREE
I I
I
,
I
CORRECTION COEFFICIENTS
STRUCTURAL DESIGN MANUAL
•
CASE (D cont'd) VII.
DEFINITION OF SITUATION
MaLZ I+-
L
.333 (1.0 + A)
M~:~ •I
x.
J--a~
MaL~: I
I
• •
I
-LI
r----
T: a •
lIMa ~--r I
Ta :1
: I--.r-Ma
"":::::z:
I I
I
I~
XIV.
I
) xv.
i
L
.50« I-A) B2 + 2 AB)
~IMa
B=~
.167 «I-A) B2 + 3AB)
M
L
~a~
,
r:~ 21 I--a--i I
I
I
I
rMa
B
a
.167(t)(A(3Ef-6B+2) 2 + 1 - 3B )
L
f
:~.ra I
I I
I
•
I
I
.167 (1 + 2A)
I
M2Lr-----lIMa ~I
.167 (1 - A)(l - B)
B= ~
'~
I
XVI.
L
I
,I •
2-
I
I
I
B =
a
1
I
XIII.
,
t-t- a--t
I
XII.
.167 (1.00 - A)
I
~
I
• 167, (1+2A1 (1 - A»
I
l-+la~
XI.
.250 (1.0 + A)
I l
I
a
IX.
2ND DEGREE
---tI
t-.!!-i
I
VIII.
~
'k
I
M2 B :M
.333 (.50(AB+1)+A+B)
a
I
TABLE 9.4 (Cont'd)
CORRECTION COEFFIG:ENTS 9-35
STRUCTURAL DESIGN MANUAL 9.3.4
Analysis of Redundant Structures
The theories of strain energy and Castigliano can be used to determine the reactions on a statically indeterminate structure. A statically-indeterminate structure is one where there are more supports or more members present in the structure than are necessary to maintain equilibrium of the structure. The structure will still resist the loads if one or more supports or members are omitted. The total strain energy in a structure must be determined, i.e., the summation of the strain energy contributed by axial, shear, bending and sometimes torsion. Equations 9.4 through 9.6 are used to determine the energy in each element of the structure; Sometimes certain energy terms can be ignored because their contribution is small compared to other forms of loading. Such situation occurs in a ' shallow beam which is subjected to shear and bending. The shear energy can be ignored because bending energy is so much greater.
.~ !
To determine the redundant forces and moments in a structure for which it may be assumed that only bending deformations affect the magnitudes of the redundants, proceed as follows: (1)
Cut the structure at convenient points to make it statically determinate. Denote the redundant forces or moments as x , x ' x ,etc. The cuts b a c will generally be at reaction points but do not have to be there. For instance, because of symmetry, the structure could be cut at the line of symmetry.
(2)
Draw the moment diagram M for the actual applied loads acting on the statically determinate st~ucture.
(3)
Draw the moment diagrams for unit redundant forces or moments (1 lb or 1 in-Ib) acting on the statically determinate structure with the applied loads removed. Designate these moments as M , M , M , etc. b a c
( 4)
The expression for energy is U
= ~M2
dx/2 EI, (equation 9.5)
Write the moment equations for the structure
(5)
+
One Redundant:
M
Two Redundants:
M = Mo + xa Ma
+
xb Mb
Three Redundants:
M = Mo + xa Ma
+
xb Mb
M
o
x
a
M
a
+
Xc Mc
Evaluate the deformations for each redundant using Catigliano·s Theorem, i. e. ,
-\
9-36
STRUCTURAL DESIGN MANUAL
•
The resulting equations will take the form M 2 dx EI -+ xb
xaL a
MM a c xa L.- EI
,
There will be one equation for each redundant. The coefficients of x , a x ' and x can be determined using pictorial integration as explained b c in 9.3.3.
•
(6 )
Solve for the redundants, x , x ' x , etc. a b c
(7)
Calculate the actual moments using the moment equations developed in step 4 if they are needed. Examples are shown in Figures 9.3 and 9.4 with problem solutions.
9.3.4.1 Example Problem - Beam With Single Redundant
10,000
1
1- 5
A
1D~
Primary
Redundant
10,000
1
t
I
.5
2500
X
t=
a
1
+5
•
M
o~
M~ a
-37 500
FIGURE 9.3 - BEAM WITH SINGLE REDUNDANT
9-37
STRUCT-URAL DESIGN MANUAL Revision B
M=M
M2
+xa
= Mo 2 +
u =
x
o
M
a
2 x
a
JM o 2 dx/2
M M + x 2 M2 a 0 a a E1
+
J2
x
a
M M dx/2E1 a 0
' " k M 2 dX/E1 + '" k M M dx/EI aL a Lao
+
.
[x 2 M 2 dx/2 E1 a a
=0
where k is the constant of integration from Table 9.4. x
a
(.333)(5)2 (20)
+
+ (.333)(5)(-25000)(10) x
a
=
~611)(5)(-37500)(5)
(.333)(2.5)(-37500)(5) + = 0
(156094 + 577500 + 416250)/166.5
6906
~
The maximum moment is then
M = Mo + x a Ma = -37500 + (6906)(2.5)
= -20235
The beam balance and bending diagram is
10,000
f
l
f
6906
4047
I
953
9530
M ~------~--------~~
-20,235
9-38
)
STRUCTURAL DESIGN MANUAL 9.3.4.2 Example Problem - Beam With Two Redundants
Primary 10,000
f
2nd Redundant
1st Redundant
~
t
2500
500
~
.5
x
f a
1
cp= J
... 5
~
~
I
.05
.05
+5 M
1
M a
1.0
75
~
-37,500
FIGURE 9.4 - BEAM WITH TWO REDUNDANTS
166.531 xa
+ 24.994 xb - 989,219 = 0
24.994 xa + 6.333 xb - 218,672
=
0
1849 10,000 27272" 27,272
9
l
t
1849
7939
27,&
t
212
~-2120 -12,423 9-39
STRUCTURAL DESIGN MANUAL 9,3.5
Analysis of Redundant Built-Un Sheet Metal Structut"es
Section 9.3.4 deals with structures in which bending is the primary energy contributor. In this section, commonly-used ai.rframe structures are discussed. Such structures are composed of thin webs and axiallnembers. No bending members ate present since all of the loads can be transferred by shear, tension and compression. new energy term is necessary since shear flow, q, is now present. can be written in terms of shear flow.
A
Equation 9.6
)
It then becomes L
U
=fq2 a
dx/2Gt
9.11
o
where a is the panel dimension normal to the x direction and t is the web thickness. In most airframe structures the- shear flow, q, can be assumed to be constant over a given length. Since axial members are attached to the webs and must react the shear flow with a concentrated load the axial load will vary in an axial member. Such a situation is shown below.
L
•
....
,..
----
I ......
The axial load in the upper member varies from zero to Pf over the length L. load at any point x in a member is P = Pf(x/L). x Equation 9.4 is then written as U
= o
f
L 2 2 Pf x dx/2
L
2
AE
The
9.12
Pictorial integration of the above equation is possible. Figure 9.5 shows the integration constant for various axial loading conditions.
A redundant analysis of a sheet metal beam is shown in Figure 9.6 and the following example problem.
9-40
--
e
-
e
"-'-
-:: : i .~.
t ::::: 1 "':tj H (;")
C
~
tr.1 \.0 VI
I'~"-'<
Z
t-3
t:r3
c;":)
~
H
o Z
C"')
o Z
en
~
en
1-.... ; .•
~.: ::-=""p,: LO.~ ~_, 01"
r"
~-;.:.--:
t·: ~ .:
f.:;'.> I .. -.
i.. ·
~.
P,IA ~~AMI :. . ;_ rr-i: 1-:" -- ... -:- :--_.: . .
1
,.;
~
-~.: 'PRIM AR'y i',
Pll
I
.
•
~~.-:~~
~
-.
: .•. ~ r
_.;'-1''7'1'
':!:
~ ..
.'
~
:'.-
c:
..';----';'-I-
n
".~ ~. ~:t ~ ':- :- ':r~'" ~ r:: ;: - ['_
:,., "~.:...;.~::.:~~-Rp '-'-'<-~.n'~
/'
7
\._.; :-
~
• - :
t-' .. ~-r·::.:~
::tI
t
-"- r "
. "
.
:
•· ..
,..r-
·l...(~LJ-l=tt1.:;,J :~ kT'')l Pll ~ :-.; ·+··\·...:.--1- : .. k .
CI
1:
! • i
•
Y"
!-j
:. i ..
.; ..
~A)~. t-.~..
kPO l P11{A E
~
-.~'-
en
-
.
~F:'~:--
~
H
r.:
CD Z
t"""
o
~
~
H Z
:~ :.~::~~..:.1 -1.0 I
.p1-'
•z
.. -. 'fI T·---- ......
G'::l
\.0
~
:::Itt
,;-_._L! _.
-~l~ll?:t~rT~} ~-~ -~-+.~.-:. ;.1.... : : ~ 0"
-I
:t±rrt~-:-
•
-~~
~'IJ
-I
:~'~~+~--~--~-~·f'-·¥< ~·~~·~t··- -'. ~I-
~~;t~:j~~l
;L,;:-~~l.:~_. :;.T:~i-i ~. r ·~--:"tt"';"'.1.. ; :.~ ~ -11, U
~ '~~-- .
, \
en
:::-~-.; t·~ -::: :'1-'-: : -:'1 ...~.-: 11'~'
P12~+-p' ~Tj
·I.
---~~:,
;-
':~~~.. : st~-~: .~~' ~ l~~~~j+~<·~·Bf:~)~r~ .. ;.! . ,. ~:: ~
~
\~ ·~CI-~·/
.-
..
~ 1 .• - r . ; •. ; : ; : . I . , i. '. :
-"'-
"~
.t=p-t Rtrp,°2 .-~
:,~.~,
i
• --+
r'
~-.
.<:~'...-- ~_-P-~2 ~~+~~.--~ ~
'":I:j
o
."
'----,
'r~rr~\':l~)i:t~s-'i'}~~~~ .~ t·
H
-+--_. , r._,_,.1.,-.-',....,1.\.,...
1;' .. :
'
--- 1
__ ~.J. .___ .
-.8
~6
,..=
!' Rp
-.4
-:-2
°
.2
~
.4
Rp'
.6
.8-
1.0
r-
\
STRUCTllRAL DESIGN MANUAL
I
•
!
FIGURE 9.5 (Conttd) - INTEGRATION CONSTANTS FOR AXIAL LOADING 9-42
•
STRUCTURAL DESIGN MANUAL 0000
T 10
t=.05
t=.05
t=.05
Af = .1 in tw
E G
1t=s
---I
PRIMARY
~ ~
~
~ 4
~ ~
r~
~~ ~ i
~i t,
~~ ~ ~ t~ ~ i
~ ~ ~i
t~ ~
r~ ~ ~
7500 REDUNDANT
~
-- ----
1 ~~ ~~
+
~ ~
-- ----- ------
,, +
+
~ ~ ~ ~ t~ t~
t~ ~ i ---- ----, ---- -
q=.05
-;;iJI1"" __
~i ~
-.25
---:III" --;:;000
~
-==-
q=.05
., Compression in Flanges
~ ~ ~ V
~
.5 x
a
P a
q = qo + x a qa
V~ ----::;111"---. . . . ---=. ....:::......::..... --=-. , ....c
a
+, Tension in Flange
t~ V~ ~~
~ ~ ~1
~----
+ x
25 0
~....c._~"""""'--
~~ H ~ ~ ~i t~ ~~
,~ ~
q=.05
-+-~
0
..........
+.5 '!lilt
p = p
~ ~
q=250
~ ~ ~{
~~ ~ i ~
) psi
_ _ -:l1000.
+2500
+.25 ill' ........
t~ ~
~ ~ ~ ~
~~
....lIlIo.
-~-
~-- .-..
~ ( ~t
gSi
~ ~ ~
~~ ~~
q=250
+3750
= 4(10
---.......---
~~ ~i
,
t~
-""'"'--'::::""
~
___ ....- _ _ --~.......-
--------- --....... ---- ------- . --- -- --- --- -----~
..-..
•
-J5.Q.Q ~ ~ ~~
~ ~
)
10000 3750
--+1
q=750
~ 1
.05 in
= 10 7
10----1
---------- --- ------ -- ~ ~1 ~ ~ ~~ ~ ~ ~~ t + It.
.J
,-
10
2
=1
---lIIoo"
HE]r ----
~
H~~ -
FIGURE 9.6 - EXAMPLE OF REDUNDANT SHEET METAL BEAM 9-43
STRUCTURAL DESIGN MANUAL
Example Problem:
= UAX1AL
+
U =l:k (p ,T 0
+
U TOTAL
U = T
Ik
=
USHEAR
o x
p) a
a
2
J
L 2
L/2AE
P
dx/2AE
+
JL Zadx/2Gt q
0
+ I(q 0 +
x
a
q) a
+ Iq o 2
aUT/ax a = Ik P 0 P a L/AE + X a
Ik
P 2 L/AE
a
+ Iq o q a ab/Gt + x a~ ~q a 2 ab/Gt aUT/axa = 8xa = 0
- Ik
x
.=:
a k
9-44
P
P
L / AE - Iq
q
a bIG t
o a 0 a ----=----------------
Ik
P 2 L/AE
a
= constant
ab/2Gt
+ x a Ik P 0 P a L/AE + x a 2 I k P a 2 L/2AE ab/2Gt + x Iq q ab/Gt + x 2 I.q 2 ab/2Gt a 0 a a a
P 2 L/2AE
o
2
+ 2q
a
2 ah/Gt
of integration from Figure 9.5.
)
STRUCTURAL DESIGN MANUAL FLANGE
P
P
0
AE
L
a
x(10+371)0
+3750 .1.2500 +2500 0
be cd hg gf fe
-
0 -3750
+ .25
-1750
+
-2500 -2500
+ .5 + .5
0 'II)
{)
bg
+
5
0
0
0 ·10000
0 0 _1
f)
ef
0
+
) _/
5
0
-2500
WEB
x
qo
qa
a
AE Ix (10-
.333
-1560
.0156
.333
.026
5
1.0
.61
-5719
.0156
.7
.055
10
1.0
.333
-4163
.. 25
.333
.833
5
1.0
.333
-1560
.0156
.333
.026
5
1.0
.61
-5719
.25
.7
.875
10
1.0
.333
-4163
.. 25
.333
.833
10
1.0
.333
-12488
.25
.333
.. 833
10
1.0
0
0
0
.333
0
10
1.0
0
0
1.0
0
0
10
1 .. 0
.. 333
-4163
.25
.333
.833
L=
... 38135
L=
4.314
ab
qoqa ab
Gt
x(10-6)
Gt
6
2 qa
x (10 )
q Z ab a Gt 6 x(10 )
-750
.05
50
.. 2
-9375
.. 0025
.625
2
250
.05
50
.2
3125
.0025
.625
3
250
-.05
100
.. 2
-6250
.0025
1.250
-12500
2:=
2.5
=
6)
1.0
1
6
a
k P L. L
k
5
L= x
P 2 a
()
0
0
de
P L a
f)
-71)(){)
ah
0
AE (10- 6 )
6
-
0 - _25 ... .. 25 - 5 .5 0
0
ab
k P
k
6
... [-38135(10- )] ... [-12500(10- )] 6 6 4.314(10- ) + 2.5 (10- )
50635
7431
6.814
9-45
STRUCTURAL DESIGN MANUAL 9.3.6
Analysis of structures with Elastic Supports
The previous example, Figure 9.6~ could also be on spring supports instead of the rigid ones ~ssumed. The additional energy of the springs must be added to the total strain energy of the system. The energy in a spring is
u
.
spr1ng
:: ~F2/2K L.
where K is the spring rate in lbs/in. 9.3.7
Figure 9.7 shows an example.
Analysis of Structures with Free Motion
In the previous sections the partial derivative of the energy with respect to the' redundant (au/ax ) was set equal to zero. This the theory of least work which states the red~ndant does no work. This is a condition of minimum strain energy and no relative motion exists between the beam and the support. If, however, a definite free motion exists at some support the partial derivative is set equal to the free motion. The cantilever beam shown below illustrates
a beam with free motion. The beam is statically determinant until the deflection at the end of the beam is equal the free motion. Then the beam is statically indeterminant. The total energy in the beam prior to contact of the support is U = t
L: p2 L/2AE + L: q2
ab/2Gt
Selec ting the end support as the redundant,' the total axial load in any member is p = p
o
+
x
a
p
a
where P is the primary load in the Dlemher, x is the unknown redundant and P is the ~ember load due to a unit redundant lo:d. The shear flow in· the web i~
where the terms are similar to the previous equation. beam is
9-46
The total energy in the
.)
STRUCTURAL DESIGN MANUAL 10000
K, SPRING RATE Ibs/in
UT
= UAXIAL + USHEAR + USPRING +, TENSION IN SPRINGS
PRIMARY
-, COMPRESSION IN SPRINGS
(F ) o
7,500
2,500
F = Fo +
REDUNDANT (F )
x
a
Fa
a
) .5
-:~
a
=
+ x
k (F 0
-~
USPRING
x
x
a
a
=
1
F)2 a
2K
-I,k PoPa L / AE - ~ q 0 q a a b /G t - I F 0 F a /K
Ik
P 2 LIAE
a
+Iq
a
2 ab/Gt
+2F
a
2/K
FIGURE 9.7 - BEAM ON ELASTIC SUPPORTS 9-47
STRUCTURAL DESIGN MANUAL ut = 'L" k
(p
0
+
x
a
P) a
2
L /2AE
+ , (q + xa L" 0
q) 2 a b / 2G t a
Expanding and takingout/8xa gives an equation of deflection which is set equal
au/ax a
'" k P Pa L/AE + x a LkPa2L/AE+ L" 0
:i::
' " q 2 ab/Gt = 8 L" a a
Solving for x
a
gives
oa - I. k
x
a
Po P a L/AE - '"' q ql ab/Gt ~ 0
= ------~----------~----------------I. k
P 2 L / AE'
a
+ .I
q 2 a b /G t
a
This equation is applicable only if x is applied in the direction to close the gap_ If x = 0, the condition exists where the applied load deflects a ,the beam to the support but no further. For this case
which is the same as the virtual load equation.
9-48
8a -
STRUCTURAL DESIGN MANUAL
•
9.4
Continuous Beams by Three-Moment Equation
The process of three moments can be applied to a beam with redundant supports and any type of loading. The procedure is to equate the slopes of the spans at a support. An equation for any type of loading can be derived and by superposition the slope of the span can be determined. The slope o~ either span at a support is the sum of the slope produced on a simply-supported span by the given loading (found from Table 9.2). Consider the beam shown below.
~Ml
JT F-
.)
At x
M20
1
L
-I
= 0,
o = llEl
(-M
l
L/3 - M2
L/6)
9.13
At x = L,
o = llEl
(M l L/6
+ M2 L/3)
9.14
At x, 9.15
If the end supports settle or deflect unequal amounts of Yl and Y2' the increment of slope produced is the same at both ends and is
)
where y is positive when upward.
The
follow~~~
example illustrates the procedure:
Ll--------------~~M~~
~
The slopes at support B for each of the adjacent spans are
9-49
~ STRUCTURAL DESIGN MANUAL ~! Span 1:
°B
GMl A
M2~
~.8B
~
=
l/ElI l (M l L /6 + M L /3) 2 l l
= 5 WlL
2 l /96 Ell!
-
(Table 9 .. 2)
A
A
B
8B
= (Y - YA)!L! B
A
A
B
A
Span 2:
M
G A
B
2
M30 A
.B
l/E 2 I 2 (-M 2L2 /3 - M3L2/6) (Table 9 .. 2)
C
r
BB
2
h
BB
=
-w 2 (bL2- b 3 IL 2 )/6E 2 I 2
A
C
A
)\
B
C
°B
= (Yc -
YB)!L 2
The slopes for span 1 are set equal to the slopes for span 2.
) If Ml and M3
are determinate (ends simply supported or overhanging) the equation can be solved at once for M2 and the reactions then found by statics. If the ends are fixed, the slopes at those points can be set equal to zero; this provides two additional equations, and the three unknowns M , M2 and M3 can be found. If the two spans I are parts of a continuous beam, a similar equation can be written for each successive pair of contiguous spans and these equations solved simultaneously for the unknown moments.
9-50
• \
STRUCTURAL DESIGN MANUAL r
9.5 Lateral Buckling of Beams Beams in bending under certain conditions of loading and restraint can fail by lateral buckling in a manner similar to that of columns loaded in axial compression. However, it is conservative to obtain the buckling load by considering the compression side of the beam as a column since this approach neglects the torsional rigidity of the beam. In general, the critical bending moment for the lateral instability of the deep beam, such as that shown in Figure 9.8 may be expressed as M
GJ KJEI Y__ = ____
L
cr
where J is the torsion constant of the beam and K is a constant dependent on the type of loading and end restraint. Thus. the critical compressive stress is given by F
cr
=
Mcr c I
x
where c is the distance from the centroidal axis to the extreme compression fibers. If this compressive stress falls in the plastic range, an equivalent slenderness ratio may be calculated as
L' P
IE = 1rV-Fcr
1
h
FIGURE 9.B - DEEP RECTANGULAR BEAM 9-51
~ .STRUCTURAL .D·ESIGN MANUAL ~p. Revision E
The actual critical stress may then be found by entering the column curves of Section 11 at this value of (Lt/p). This value of stress is not the true compressive stress in the beam, but is sufficiently accurate to permit its use as a design guide. 9.5.1 Lateral Buckling of Deep Rectangular Beams The critical moment for deep rectangular beams loaded in the elastic range loaded along the centroidal axis is given by M
cr
= 0.0985
3 K E (b h)
m
L
where K is presented in Table 9.5 and b, h, and L are as shown in Figure 9.8. The cri~ical stress for such a beam is 2 b F KfE(Lh) cr where K is presented in Table 9.5. f If the beam is not loaded along the centroidal axis, as shown in Figure 9.9, a corrected value K t is used in place of K • This factor is expressed as f
f
K
f
t
=
K
f
(1 - n
(~») L
where n is a constant defined below: (1) For simply supported beams with a concentrated load at midspant n = 2.84. (2) For cantilever beams with a concentrated end load, n (3) For simply supported beams under a uniform
load~
(4) For cantilever beams under a uniform load, n
=
n
=
0.816.
= 2.52.
0.725.
Note: s is negative if the point of application of the load is below the centroidal axis. centroidal axis
FIGURE 9.9 - DEEP RECTANGULAR BEAM LOADED AT A POINT REMOVED FROM THE CENTROIDAL AXIS 9-52
\
)
."
~ "\DJ
STRUCTURAL DESIGN MANUAL
Type of Loading and Constraint K{
Km
1. 86
3.14
3.71
6.28
3.71
6.28
Top '(iew
Side View
~
~
-
-e
-
9
e
-
~
§::
e
-
~-
tl12~
5.45
9.22
~~
2.09
3.54
-1-=====~
3.61
6.10
4.87
8.24 -
2.50
4.235
&
~..
I
~
III j IIIIi 11 i IIi I~ ,
..
i
11 i 11111 I,I ; ;I i i1 I
,
II
L
II
I
II
t
IJ
i _.
t.
f
===i-
==§
t-L/2--1
~
~
k
-
=--§ __0
~
r'
=1
~=~
3.82
6.47
I
~ ~
6.57
11. 12 13. I
I
~
-j
.~=~
7.74
J
~
3.13
5.29
3.48
5.88
2.37
4.01
2.37
4.01
-::::::J
3.80
6.43
::s
3.80
6.43
i
-1-- !
.~
~U2*
I~II! 1I11I1H II ~~ •
L
~
-1--- --:J--
~ ~ == ;::::::::"i L t ~ L ~- ;:;;J
__ilH 1I!:1I 1I11l!1
-
~~
J--
~1II1I il WI! 1111
-
~
~
TABLE 9.5 - LATERAL BUCKLING CONSTANTS FOR DEEP RECTANGULAR BEAMS 9-53
STRUCTURAL DESIGN MANUAL
m = KIf. 591 3.0
2.0
I{~~-~-~-I .50 . elL
TABLE 9.5 (CONT'D) - LATERAL BUCKLING CONSTANTS FOR DEEP RECTANGULAR BEAMS 9.5.2 Lateral Buckling of Deep 1 Beams Figure 9.10 shows a deep I beam.
FIGURE 9.10 - DEEP I BEAM The critical stress of such a beam in the elastic range is given by
9-54
i
STRUCTURAL DESIGN MANUAL
•
where Kr may be obtained from Table 9.6 and a is given by
a =
.jEI Yh2/4
GJ
where J is the torsion constant of the I beam.
This constant may be approximated
by
J = 1/3 (2b t 3 f
+ h
t
3)
w
This method can be applied only if the load is applied at the centroidal axis.
)
9-55
"8
STRUCTURAL DESIGN MANUAL
•
Type of Loading and Constraint Top View
Side View
======iD
ml (E) 4
Ct=1
m2 (E,) 4
Ii. i L
,t
I
m3 (E)
16
m 4 (E) 32
i -----+--j
---+-1-
I.
L
ms
16 (E)
.1
m7 CE)
32
rnB (E)
32
*
Use Figure 9.11 to obtain m
,
TABLE 9.6 - LATERAL BUCKLING CONSTANTS FOR DEEP I BEAMS
9-56
!~q\~\
•
"~;,u 600 500 400
"'"
,
~~
~
""~"\
,,~
~
300
~
200
80 60
•
50
~
~ ~"
~
,m7
"\
150
100
STRUCTURAL DESIGN MANUAL
""-~ ~ ~~"
~
"-
""'-"
""" """"
~
,mL,.
"- ''-....
~
30
15
"~
~
'"
~~
~
.......
"-~
~
i'....... r--...
~
~ "-
"""-
8
'......... ...............
"'- "-r-...ml')
~
"" ""illl
6
--
:--. t'--..
"
"...........
~ .......
"""- ....... -..
'" K ~ "'"'- ITttt
4
.........
l.O
~
--
8 5
r--
4
---
3
i'.....
...............
~
r--~
I'--- r--
r--
3
.8
..... ........
6
......
"" ~
.6
------
""'--
~
.............. ~
5
.......
"""'-
'~
10
•
'-'" ~'-
~
"
,
~
.......
........
"-
"'"
'..m3
40
20
'"
~ ~ ~ '-
1.5
2
3
4
5
6
8
r----- r----.
-
10
.
'
2
1
15
20
30
40
FIGURE 9.11 - VALUES OF mFOR TABLE 9.6
9-57
STRUCTURAL DESIGN MANUAL 9.6
PLASTIC ANALYSIS OF BEAMS
9.6.1
Revision C
Bending About an Axis of Symmetry
•
The basic assumption of the classic theory of pure bending in the elastic range is that a plane cross-section normal to the longitudinal axis of the beam remains plane under bending deflections. This assumption is also valid for the plastic range. The strain is directly proportional to the distance from the neutral axis. The stress distribution across the plane of the cross section in a direction perpendicular to the neutral axis has the same shape as the material stressstrain curve. Such a distribution is shown in Figure 9.12. If a stress-strain
• FIGURE 9.12 PLASTIC BENDING STRESS DISTRIBUTION
curve is available from a tension test and if it is assumed that the stressstrain curve in compression is the same as tension, it is possible to determine the moment carried by a given section at a specified extreme fiber stress. In Figure 9.12 the actual stress-strain distribution across a symmetrical section in bending is shown where fm is the extreme fiber stress. The stress fro is equal to or greater than Fty and less than or equal to FtuSuperimposed onto the actual distribution is a trapezoidal distribution which passes through fm and has an intercept stress of fa. The intercept stress fa is defined as the stress required for the trapezoidal stress distribution to produce the same moment about the neutral axis as the actual stress distribution. This theory was introduced by Frank P. Cozzone and the following discussions are based on this paper in the May, 1943, Journal of the Aeronautical Sciences. Thi.s stress f is a fictitious stress which is assumed to exist at the neutral axis or at 2e~0 strain. The value of f is determined by making the requirement that the internal moment of the true st~ess system must equal the moment which results from the assumed trapezoidal stress-strain system. Figure 9.13 shows the trapezoidal stress distribution consisting of separate distributions, one rectangular and one triangular. The total moment Mb can be defined as
9-58 .
•
STRUCTURAL DESIGN MANUAL N.A.
To •
+
FIGURE 9.13- ASSUMED STRESS DISTRIBUTION where
~
= moment produced by the rectangular distribution
~
= moment produced by the triangular distribution
The triangular distribution is equivalent to an elastic distribution and can be defined as MT
=
(f
f
o
m - f 0 ) Ilc
For a symmetrical section, the moment produced by the rectangular distribution will equal f times twice the area above the neutral axis times the distance from the neutraloaxis to the centroid of this area t or
= where
A
A/2
2 f
(A/2)
o
I
Yo
Y
total cross sectional area the area above or below the N.A.
= distance
y = e/2
from N.A. to the centroid of the area above or
below N/A But Ay/2
= Qm
the static moment of the area above or below the N.A. about the N.A.
Then substituting Q
m
M_ = 2 f Qm -~ 0 and
=2
f
o
Qm + (fm - f 0 ) lIe
9-59
STRUCTURAL DESIGN MANUAL ell and k = 2 Qm 011 then
Mb Fb = f
m
+
f
0
9.16
(k - 1)
Fb is a fictitious M ell stress or the modulus of rupture for a particular cross sectIon at a given maximum stress level. Equation 9.16 is applicable only to sections symmetric about the neutral axis. The values of k vary between land 2.0. If the calculated value of k is greater than'2, use 2. Figure 9.14 shows the value of k for several typical shapes. k can also
FLANGES ONLY
THIN TUBE
HOUR GLASS
RECTANGLE
HEXAGON
SOLID ROUN
DIAMOND
Z -8k=l.O
k=1.27
k=1 .. 33
k=1.5
k=1.6
k=I.7
k=2.0
FIGURE 9.14 - SHAPE FACTORS FOR TYPICAL SECTIONS be calculated Jrom k
=
2 Qc I
=
2 c
lC
y d A
9.17
I
The modulus of rupture Fb may be yield modulus or ultimate modulus. For.yield modulus of rupture, the value of f in equation 9.16 is F of the material. If ty m ultimate modulus of rupture is desired, substitute F of the material for f in tU m equation 9.16. The modulus of rupture may be limited to some stress between yield and ultimate stress of the material because of local crippling or by excessive distortion. Regardless of what value is used for f in equation 9.16, the m
corresponding value of f
o
must be known before a value of Fb can be determined.
Figures 9.15 through 9.18 give curves for Fb and f versus k and strain for a various materials.
9-60
STRUCTURAL DESIGN MANUAL 2.0
160 k
1.7
140
120
100 Fb (ksi)
cc
60
40
20
o. 008 E
a.
)
F IGfJRE 9.15 -
0.009
. O. 010
(lnche s / inch)
Ti-8Mn Ti taniurn Alloy
MINIMUM PLASTIC BENDING CURVES FOR TITANIUM
9-61
STRUCTURAL DESIGN MANUAL 280
240
zoo
)
160 ,Fb
. (kat) 120
80
40
o 0.005
0.01
0.015
0.02 I:
b.
O. 025
(inche s linch)
o.
030
O. 035
O. O·t (n
Ti-8Mn Titanium Alloy
\
F I Ci lJ RE 9. 15 9-62
MINIMTJr-f PLASTIC BENDING CURVES FOR TITANI UM
STRUCTURAL DESIGN MANUAL
.) 200
180
/ 1&0
140
120
100
0.012 <:
'c.
finches/inch)
Ti-6Al-4V Titanium Al 1
oy-,
)
F'rtI'RE 9.15 _
MINIMUM PLASTIC BENDING CURVES
FOR'i'i'iANIUH---'9-63
STRUCTURAL DESIGN MANUAL
'!
. ct".
~ I GllRE 9.15 9-64
(inches/inch)
(u
Ti-6Al-4V Titanium Alloy
MINIMUM PLASTIC BENDING CURVES FOR TITANIUM
.:
STRUCTURAL DESIGN MANUAL k :::: 2.0
k ::: 1. 7
k ::: 1. 5
k
= 1. 25
k ::: 1. n
e/
E.
e.
(inches linch)
En
Ti-4Mn-4Al Titanium Alloy
F 1(;IIRE 9.15 - . MINIMUM PLASTIC BENDING CURVES FOR TITANIUM 9-65
STRUCTURAL· DESIGN MANUAL k
= 2.
0
k=1.7
k=1.5 k
= 1. 25
k=l.O
o. 002
o. 004
0.006
0.008
I
./
0.010
( (inches/inch)
a.
2014-T6 Aluminum Alloy Ex·trusions _. Thickness ~ 0.499 In,.
\
)
F I r,URE 9. 16 9-66
MINIMUM PLASTIC BENDING CURVES FOR ALUMINUM
(-" I.: i! :.i,l-'! ·~35c:.~+c:. ~LllM:INU~ IN~ES~~tN~ 2A~"I:NG~MAif.R'~l ~~OPERT;C~ -
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,
;!.
j
Ii"'!' I':
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•
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80 k
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60
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(inches/inch)
2014-T6 Aluminum Alloy Die Forgings - Thickness ~ 4 In.
MINIMUM PLASTIC BENDING CURVES FOR ALUMINUM
9-61
STRUCTURAL DESIGN MANUAL 12.0
100
k
= 2.0
k
= 1.7
k
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80
k ::; 1. 25
Fb (kai)
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40
20
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F rCURE 9.16 9-68
O. 04
o. 05
o. 06
(t.nche a I inch)
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2014-T6 Aluminum Alloy Die Forgings - Thickness ~ 4 In.
MINIMU1'1 PLASTIC BE~roING CURVES FOR ALUMINUM
)
I
STRUC1-URAL DESIGN MANUAL k
80
= 2.0
k=1.7 70
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d..
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2024-T3 Aluminum Ailo'y Sheet and Plate
- Heat F t GUn.E g .. 16 -
Treated - Thickness ~ 0.250 In. MINIMUM PLASTIC BENDING CURVES FOR ALUMINUM
9-69
STRUCTURAL DESIGN MANUAL
120
100
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k
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e.
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(inche s /L."lch)
0.12 EU
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2024-T3 Aluminum Alloy Sheet and Plate - Heat Threated - Thickness ~ 0.250 In.
FIGURE 9.16 9-70
0.08
MINIMUM PLASTIC BENDING CURVES FOR
ALUMI~lm
STRUCTURAL DESIGN MANUAL 70
= 1.7
60
= 1.5
= SO
= 1. 25 '~::k::
40
1.0
30
20 l
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f.
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0.008
0.010
(inches/inch)
2024-T3 Aluminum Alloy Clad Sheet and Plate - Heat Treated - Thickness 0.010 to 0.062 In •
FrGURE 9.16 -
MINIMUM PLASTIC BENDING CURVES FOR ALUMINUM 9-71
STRUCTURAL DESIGN MANUAL lZO
z.o 100 1.1 1.5
80
)
1 .. 25
60
1.0
.40
20
o
g.
2024-T3 Aluminum Alloy Clad Sheet and Plate - Heat Treated - Thickness 0.010 to 0.!.06~ I~.• _
FIGURE 9.16 .. 9-72
MINIMUM BLASTIC BENDING CURVES FOR ALUMINUM
)
STRUCTURAL DESIGN MANUAL 80
70
60 I------+-~-t·---'II
k "" 1.25
50
40
30
20
10
0 0.002
0.004
0.006 €
h.
0.008
0.010
(inches/inch}
2024-T6 Aluminum Alloy Clad Sheet - Heat Treated and Aged - Thickness < 0.064 In.
FtGURE 9.16
MINIMUM PLASTIC BENDING CURVES FOR ALUMINUM
9-73
STRUCTURAL DESIGN MANUAL 120 k
110
= Z. 0
.
100
k=L7
90 k
= 1. 5
80 k = 1. 25
70 Fb (kal) 60
k
= 1 .. 0
50
40 30
20 10
o E
(inches/inch)
) i.
.
2024-T6 Aluminum Alloy Clad Sheet - Heat Treated and Aged - Thickness < O.064·In •
F t GURE 9.16 -
9-74
MINIMUM PLASTIC BENDING CURVES FOR ALUMINUM
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AZ61A Mag'B:esiu~ "Alloy Extrusions tudinal) - Thickness ~ 0.249 In.
(i"ongi':"
)
F I GttRF. 9.17 -
MINIMUM PLASTIC BENDING CURVES FOR MAGNESIUM
9-85 .
STRUCTURAL DESIGN MANUAL
k = 2.0
40
30
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FIGURE .9.17 9-86
AZ6lA Magnesium Alloy Forgings (Longitudinal)
MINIMUM PLASTIC BENDING CURVES FOR MAGNESIUM
STRUCTURAL DESIGN MANUAL
2.00
60
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50
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AZ61A Magnesium Alloy Forgings (Longitudinal)
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FIGURE' 9.17
MIN~
PLASTIC BENDING CURVES FOR MAGNESIUM 9-87
STRUCTURAL DESIGN MANUAL 40
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::
;:
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0.01
)
If,-
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STRUCTURAL DESIGN MANUAL
.)
Fb (kai)
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e.
urn
m(3 iA ---~·---o-- Ma'-gn-e-s1
- Thickness
F I (llIRE 9. 17 -
~
All
oy--Sh"e e t-
0.250
MINIMUM PLASTIC BENDING CURVES FOR MAGNESIUM
9-89
STRUCTURAL DESIGN MANUAL
)
0.004
0.002
0.006
0.008
( (inches/in<:h)
f.
F I CaIRE' 9. 17 9-90
ZK60A Magnesium Alloy Forgings (Longitudinal)
,~INIMUM PLASTIC BENDING CURVES FOR MAGNESIUM
0.010
STRUCTU.RAl DESIGN MANUAL I
80
60
Fb (kai)
zo
o
)
g.-ZK60A:-Magensium Alloy Forgings (Longitudinal)
FIGPRE 9.17
~INIMUM PLASTIC BENDING CURVES FO~ MAGNESIUM 9-91
STRUCTURAL DESIGN MANUAL k :: 2. 0 70
k = 1. 70
60
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::
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50
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30
2.0
10
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,/
o 0.002.
0.004 ( (inches/inch)
a.
Carbon Steel AISI 1023-1025
F r G\tRE 9.1S -
MINIMUM PLASTIC BENDING CURVES FOR LOW @.ARBON
AND ALLOY STEELS 9-92
STRUCTURAL DESIGN MANUAL k = 2.0
100. k=1.7
k = 1.5
80
k
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60 h. - 1. 0 ,'"
40
20
o
o. 04
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'b.
0.12 (inches linch)
o. 16
0.20
Carbon Steel AISI 1023-1025 ~
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MINIMUM PLASTIC BENDING CURVES FOR LOW CARBON
AND ALLOY STEELS 9-93
STRUCTURAL DESIGN MANUAL 1-1-0
k
120
k=L7
¢:
Z. 0
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60
40
20
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c.
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AISI Alloy Steel, Nonnalized - Thickness > 0.188 In.
F , GU~ E 9. 18 9-94
o. 006
MINIMUM PLASTIC BENDING CURVES FOR LOW CARBON AND ALLOY STEELS
STRUCTURAL DESIGN MANUAL
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50
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--.
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F' (HIR'E
9. 15 -
MINIMUM PLASTIC BENDING CURVES FOR LOW CARBON AND ALLOY STEELS
9-95
STRUCTURAL DESIGN MANUAL k = Z.O
k
=L
7
k = 1. 5
k = 1.25
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e.
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0.008
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AISI Alloy Steel, Normalized - Thickness ~ 0.188 In.
MINIMUM PLASTIC BENDING CURVES FOR LOW CARBON AND ALLOY STEELS
.)
STRUCTURAL DESIGN MANUAL 280
240
200
k::: 2.0
160
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40
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f.
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0.14
(inchesJ inch)
0.16 €u
AISI Alloy Steel, Normalized - Thickness ~
F t GORE 9 .16 -
o. 1
O.1§8
:rn •
MINLMUM PLASTIC BENDING CURVES FOR LOW CARBON AND ALLOY STEELS 9-97
STRUCTURAL DESIGN MANUAL
k
= 2.0
k
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160 k = 1.5
k
= 1.25
120
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(kaf)
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0.004
0.006
0.008
0.,:010
. , (incheG/inch) g • . "~~SI Alloy Steel, meat Treated
FIGURE 9.18 9-98
MINIMUM PLASTIC BENDING CURVES FOR LOW CARBON AND ALLOY STEELS
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STRUCTURAL DES.IGN MANUAL 320
280
240
) 200 Fb (ks!)
160
120
80
40
c E
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EU
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• ,
FIGURE 9.18 -
MINIMUM PLASTIC BENDING CURVES FOR LOW CARBON AND ALLOY STEELS
9-100
STRUCTURAL DESIGN MANUAL k
c;::
2.
280
240
k = 1.. 50
2.00
160 Fb (kat)
120
80
ISO) 000 psi 163~~OO psi
4D
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0.004 £ -
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FIGURe 9. 18 -
(inches/inch)
AIS~ ~ilo~ Steel, Heat Treated
MINIMUM PLAST!C BENDING CURVES FOR LOW CARBON AND ALLOY STEELS 9-101
. em ~~
STRUCTURAL DESIGN MANUAL
320 k = 1.1
280 k = 1. 5
240 k
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k •. AISI Alloy Steer, Heat Treated
FIGURE 9.18 -
9-102
MINIMUM PLASTIC BENDING CURVES FOR LOW CARBON AND ALLOY STEELS
STRUCTURAL DESIGN MANU'AL320
k
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280 k = 1. 7
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80
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AISI Alloy Steel, Heat~Treated
MINIMUM PLASTIC BENDI.NG CURVES FOR LOW CARBON
AND ALLOY STEELS 9-103
STRUCTURAL DESIGN MANUAL
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Alloy Steel, Heat Treated
MINIMUM PLASTIC BENI)ING CURVES FOR LOW CARBON AND ALLOY STEELS
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."
.;
.:, ,. I '
--I~,
~
~
.Jl
~ ....
I' r
Q
. W If'., 4CI
~. :f
~
, . in I' '.
~ '\
I:
.
i '1"
'-:Tj!
11
. i-"
, I
;:.,'jeq . : i "
~~ 1~ ~~. ,-I .
• I '
t
.
"1:, " :' 10:. I
,',l .. --II ."j,,
i·
,_:j,. ,I"
,
"~T" ~ .. !_ . I
,,
.I.
!
.~ I. N :.0 •
t, ,
,'I
II
'
... ~ .. , ': I' .1. ...
I
I -
I ;
I
-
i .! ,t-, f. . . II·i ··· .. ~ I I.:J. .f~rJ I
'1
'0'
1
~:
';
;
I,
,
~
" " 1
I
.... 1
~=j
::!-·1
..
,30
;
too)-
I
II
• N'~
I
.'
._j .... ,
G't
,.tl~ ...... '1 I •~, .,..r-~~+-+-~~~+-~~~~~~~~~~~~~~--~--~~~~~--~~~--~~-+--- •.c,.·t~ I
. I'
.,' ..... (. .: .. ;. ;.
I
I'
\"oll-' f 1\
~i
I I
I
STRUCTURAL DESIGN MANUAL
e·./
9.6.2
Bending in A Plane of Symmetry
Equation 9.16 is applicable to sections with bending about an axis of symmetry. For bending in a plane of symmetry, the neutral axis is located as a line perpendicular to this plane which divides the total area into two equal areas. The static moments, Q and Q of the two equal areas of the cross section with respect to 2 the neutra axis are determined.
1
The value of k for unsymmetric sections is 9.18
where c is the largest value of c. ·1 and axis. 9.6.3
C
are computed with respect to the centroidal
Complex Bending
First consider the complex bending problem encountered when a section has two axes of symmetry, but the bending moment vector is not parallel to either. Denoting the axes of symmetry by x and y, the allowable moments M and Mare x y determined by the methods described in Section 9.6.1. Using equation 9.16 to determine Fb about both the x and y axes of the section M x
9.19
M
9.20
Y
The external bending moment is resolved into componentsm andm about the two x y axes. The condition of failure is defined by the interaction equation R n + R n x
=1
y
9.21
where R
x
::::
moment ratio for x axis =m 1M x x
R = moment ratio for y axis y
)
= my 1My
n
2 .. 0
-
for circular sections
n
1.7
-
for rectangular sections
n
= 1.5 - for all other sections
This interaction equation is plotted for all three values in Figure 4.10 in Section 4. The margin of safety is found by
.'
1.
Plot the point R., R
2.
Draw a line through the point R , R
3.
Determine the abscissa R
x
y
on Figure 4.10 x
xa
y
to the proper interaction curve
of the intersection with the interaction curve
9-105
STRUCTURAL DESIGN MANUAL 4.
Compute the margin of safety
MS = Rxa /Rx - I
9.22
•
For complex bending of sections with a single axis of symmetry the procedure is identical to that described once the reference axes have been located. If the x-axis is the axis of symmetry, then the y axis is located as described in Section 9.6.2. M is then determined from equation 9.16 and becomes the same as equation 9.1q. My is determined from equation 9.16 using the k value determined from equation 9.18. The appl ied momen t, m, is resolved in to m and m components. The M.S. is determined as previously described, usin~ equatron 9.21 (n = 1.5) and equation 9.22. For sections with no axis of symmetry, a different process is necessary_ Any convenient set of orthogonal reference axes x and y is chosen. Various positions of the neutral axis are determined to satisfy the requirement that this axis must divide the cross section into two equal parts. The correct position for the neutral axis is determined by the requirement that the ratio of the allowable moment about the x-axis to that about the y-axis must be the same as the corresponding ratio for the components of the external "moment with respect to these axes. This requirement is defined in equation 9.23. 9.23
•
N.A.
dy
M
MS
x
mx
-
1
M MS =-Y.. ... 1 my
FIGURE 9.19
9-106
NEUTRAL AXIS LOCATION - COMPLEX BENDING
•
STRUCTURAL DESIGN MA'NUAL 9.6.4
Evaluation of Intercept Stress, f
o
The preceding method of plastic bending analysis has made use of the intercept stress, fa. It has been shown that this stress is determined from the material stress-strain curve. Often stress-strain curves are not available for the newer materials, so two methods will be shown for the determination of f o ' one when the material stress-strain curve is known and one for when it is not known, both by the method developed by G. L. Hunt and C. A. Traylor. To begin, assume a stress-strain curve is not known. Assume that the material u.1 timate and yi~ld strengths ~F~u. and F ty }' ul timate strain (e u ) and elastic modulus of elasticity (E) are all that is known of the material. Now assume that the stress-strain curve of the material can be represented by a trapezoid using the four known quantities. Figure 9.20(a) shows this type of stress-strain curve. The point F. is the intersection of the two lines formed by the four known 1 quantities. Using the terms defined in Figure 9.29(a)
I(e - .002) = E Y Y e = Fty/E + .002 y
F
t
e.
1
m
u
F./E 1 = (F
tu
9.24 - F.)/(e ~
U
- e.) 1
Fi = - Inu (e u - FilE) + F tu Fi
- IDu e u
+ mu
F tiE
+ F tu
u F 1./E
F. - m 1
)
but then
-
e )
y
9.25
eo' 9-107
STRUCTURAL DESIGN MANUAL
STRESS
E = SLOPE --+--
) STRAIN (e)
eU' n
e
u
(a) STRESS-STRAIN CURVE
f
f
m
•
o
- - -----
• e
e /2
m
m
-...-_ _ 2e /3 m
(b) ASSUMED STRESS DISTRIBUTION
FIGURE 9.20 - STRESS DISTRIBUTIONS
9-108
STRUCTURAL DESIGN MANUAL Now, having equations 9.24 and 9.25 (e. and F.) the area under the assumed stressstrain curve can be evaluated. Proceea by di~iding the assumed curve into any number of segments (a minimum of 10 is recommended). The area under the curve is n
A =~
f
L...J o
n
l\e
9.26
n
where the distance to any element, n, is e • n
The intercept stress-strain This is shown the following
A
=
f
o
stress, f , may be determine~ by making the internal moment of the distribut~on equal the moment of the assumed stress distribution. in Figure 9.8 and in Figure 9.20(b). Referring to Figure 9.20(b) relationships can be established: e
m
+
(f
f ) e
-
mom
9.27
/2
Now equating the moment of the equivalent stress distribution to the moment of the stress-strain distribution, gives n
~fn ~en (en) = fo em ( e m/2) + (fm o
= f om e 2/2 + f e 2/ 6 o m
2 f
+ 2
(e 2/ 6 )(2 f
m
Solving this equation for f 2 f
m
+ f0 f
)
0
( 6/ e m2) = (6/e 2)
m
f
m
f o )(em/2)(2 em/3)
e 2/ 6 - 2 f
mm
e 2/ 6 om
m
+
f ) 0
yields:
tt 0
f n .1 e n ( e n )]
[i)n
Aen (en)]
-2 f m
9.28
Equation 9.28 can be used to determine the intercept stress, fo' from any stress-strain" curve regardless of its shape. Therefore, fo can be determined when no stress-strain curve is known from test results, provided an assumed stress-strain curve can be established as was done in Figure 9.20 (a). Figure 9.21 shows an example of the detenmination of f . It should be noted that in the preceding procedure the value of em for th~ first segment should be the strain corresponding to the material proportional limit. This will result in fo = 0 for the first point if f is the average stress. n
.' 9-109
STRUCTURAL DESIGN MANUAL 7075-T73 Aluminum Alloy Hand Forging, MIL-A-22771
< 3.00
t
mu
=
~
~
F ty
= 56
E
=
e
= 7%
Ftu e
u
u
66 KSI
10
- Ft
-
e
y
y
KSI 4
KSI
)
= 66000
- 56000 .070 - .0076
= 160256
e , Ftu -m u y :;: 66000 160256 (.070) 1 - m IE 7 u 1 - 160256/10
-
F.1 e.
F tu
in.
psi
= 55700
psi
~
= F./E = 557 7°0 = .0056 10
1
7o·
•
S 6a
~
55.7 .
6 a-
5 O· 0 0 0
O·
o
1
2 3 4
5 I
-r
0, ) .01 .0056..) .0076
7
6 .02
,•
9
8
.
T
10
I
.04 .03 STRAIN) IN.
.
11 12 , .05 I
.
.
13 14 ,I
.06
15
)
.
.07 eu
FIGURE 9.21 - SAMPLE CALCULATION OF INTERCEPT STRESS, f • o
9-110
STRuc-rURAL DESIGN MANUAL
n
CD
®
@
G)
en
~en
fn
f n e n.6.en
®
®
(j)
®
®
6~®
em
®/eem2
fm
fo
x(103)
x(103)
x(103)
x(103)
0
1 .0037*
.0056
27.85
.. 58
3.5
.0056
111.4
55.70
2 .0066
.0020
55.86
• 74
7.9
.0076 .
137.1
56.02
25.06
3 .0088
.0024
56.21
1.19
15.0
.010
150.4
56.41
37.58
4 .0125
.005
56.81
3.55
36.3
.015
161.5
57.21
47.11
5 .0175
.005
57.61
5.04
66.6
.020
166.5
58.00
50.47
6 .0225
.005
58.41
6.57
106.0
.025
169.6
58.81
52.01
7 .0275
.005
59.21
8.14
154.9
.030
172.1
59.61
52.85
8 .0325
.005
60.01
9.75
213.4
.035
174.2
60.41
53.36
9 .0375
.005
60.81
11.40
281.8
.040
176.1
61.21
53.70
10 .0425
.005
61.61
13.09
360.3
.045
177.9
62.01
53.92
11
~0475
.005
62.41
14.82
449.3
.050
179.7
62.82
54.07
12 .. 0525
.005
63.22
16.60
548.8
.055
181.4
63.62
54.20
13 .0575
.005
64.02
18.41
659.3
.060
183.1
64.42
54.29
14
0625
.005
64.82
20.26
780.8
.065
184.8
65.22
54.37
15
0675
.005
65.62
22.15
913.7
.070
186.5
66.02
54.43
*
e
1
= .0056
(2/3)
=
.0037333 .....
FIGURE 9.21 (Cont'd) - SAMPLE CALCULATION OF INTERCEPT STRESS, f .. o 9-111
STRUCTURAL DESIGN MANUAL
(0
0
..
•
C,!)
Z H
C,!)
)
p::: 0
~
A Z
an 0.
< ::r::.....1 r--. >""r--.
-..-I
ON
Cf.l
....:IN
:;1
I
<: I
• ....:1
.r-! II.)
0..
II.)~
0..
~O
0
0
X
0 0
r--.
~ ~
~
I
II
II
~
Ll"'\
r--. 0 r--.
0 .....I
~HO C")
0..
.r-!
:j ~
Ii-.
Ll"'\
O~ .....I r--.
II
III
::s
~
S
Z
~
1-1,
«p::
Ii-. (W)
0.
~~--~~---r-----T----~----~r-----r---~ o
FIGURE 9.21 (Cont1d) - SAMPLE CALCULATIONS OF INTERCEPT STRESS, f • o
9-112
Z
1-4
II
~~
~
"'d1 0.
0
q..j
°
~
rn
\
STRUCTURAL DESIGN MANUAL Revision E 9.6.5
Plastic
Bending Modulus, Fb
Figures 9.15 through 9.18 show curves for various materials. The curves are plotted as k = 2 QcJI versus Fb and strain. The strain versus Fb curves show f and Fb versus strain. The f curve is at k=l. The rest of the curves e:ftplo y equation 9.16 to obtain F~ at various s'trains. 9.6.6
Application of Plastic 'Bending
Consider a rectangular beam section which is .25-inch thick and l.5-inch deep. It is made of 7075-T6 extrusion and it is desired to find the yield and ultimate bending moment for the section.
= fm +
Fb
f
0
k = 2Q c/ I =
(k-1), equation 9.16
2~. 25) ( • 75 ) ( • 37 5 ~( • 75) J~ • 25 ) ( 1 " 5) 3 /12) = 1. 5
The value of k can also be found in Figure 9. 14. Ftu
= 75000
psi, Fty = 65000 psi
Find the yield bending strength: The value of f in equation 9.16, the maximum stress permitted on the most remote fiber, is 65~OO psi, the yield stress of the material. To find f , go to Figure 9.16 (r) to find the point on the stressstrain curve (k=l) tRat corresponds to a stress of 65000 psi. This point is projected downward to the f curve where a stress of 26000 psi is read. Then o Fb
= 65000 +
26000 (1.5 - 1)
= 78.000
psi
This same stress can be obtained by projecting up from the stress-strain curve in Figure 9.16 (r) to the curve labeled k=1.5 and reading Fbmrectly. The yield moment is then found to be
My = Fb I/c = 78000 (.0703)/.75 = 7312.5 ')
/
The ultimate moment is found the same way_ Fb = 75000 + 70500 (1.5 - 1)
Mu
= 110,250
(.0703)/.75
=
= 110,250
psi
10334
The previous example is for a section which is stable in compression and symmetrical about two axes. Consider now a section which is symmetrical about one axis and probably partially unstable. The Tee shown in Figure 9.22(a) is a
9-113
STRUCTURAL. DESIGN MANUAL euJJ24
e,-.055
FIGURE 9.22 - UNSYMMETRICAL EXAMPLE 7075-T6 extrusion.
Again, the properties of Figure 9.l6(r) are used.
First consider the maximum strain, e , in Figure 9.l6(r), e = .055in/in. It is apparent that the lower leg of the T~e will strain higher tHan the cap when the Tee is bent about the x axis, so set the lower extreme fiber strain equal 0.055. By ratioing the lower strain by the distances from the N.A. the strain in the upper extreme fiber is e = (.609/1.391)(.055) = .024 in/in. u
Equation 9.16 was derived for symmetrical sections about a neutral axis. The Tee can be made into two sections which are symmetrical about their neutral axis. These are shown in Figure 9.22(c) and (d). Figure 9.22(d) shows how the lower portion is made symmetrical about the neutral axis by adding the shaded portion above. The internal bending resistance is found for the entire section in 9.22(d). One-half of this amount will be the true moment developed by the lower portion. I
= '(.1)(2.782)3/ 12
=
.179
I/c = .179/1.391 = .129 k = 2 Qc/I = 2(1.391)(.1)(.6955)/.129 From Figure 9.16(r) at e = .055, Fb M = Fb (1/c)(1/2) = 7095 in-lb
=
= 1.5 110,000 psi
The 1/2 is because oply one-half the beam section is used in Figure 9.22(a).
9-114
•
STRUCTURAL DESIGN MANUAL Figure 9.ll(c) shows the upper half of the beam made symmetrical about the neutral axis by adding the shaded section. .
I
=
(1/12) (1.5) (1.218)3 - (1/12)(1.4)(1.018)3 = .226 - .123
llc = .103/.609 k
=
=
.103
.169
2 Q c/I = 2 «.509)(.1)(.2545) + (1.5)(al)(.559)1/.169 = 1.146
From Figure 9.16 (r) at e M = Fb (I/cXl!2)
=
.024, Fb
= 82,000
psi
= 6929
The total ultimate resisting moment is the summation of the two moments:
M TOT
= 7095 + 6929 = 14024
It might also be desirable to limit some portion of the section because of crippling or stability. That would be done after calculating Fb for the section. If Fb was higher than the critical crippling or stability stress, then Fb would be set equal the lower stress. This generally occurs when the yield modulus is being calculated.
9-115
STRUCTURAL DESIGN MANUAL 9.7
CURVED BEAMS CORRECTION FACTORS FOR USE IN STRAIGHT-BEAM FORMULA
When a curved beam ~s bent in the plane of initial curvature t plane sections remain planet but the strains of the fibers are not proportional to the distance from the neutral axis because the fibers are not at equal length. If (K) denotes a correction factor, the stress at the extreme fiber of a curved beam is given by f
K Mc I
in which
)
where
M A R c
is the bending moment is the cross-sectional area is the radius of curvature to the centroida1 axis is the distance from the centroidal axis to the extreme outer fiber I is the moment of inertia
Z
= - .!.A
J....:L R+y
dA
Values for K for different sections are given in Table 9.7
Section 1.
RIc
ct\R~
1.2 1.4 1.6 1.8 2.0 3.0 4.0 6.0 8.0 10.0
~I
Qdl
/
I
\.':::: l\ C
K is the same for circular and elliptical sections; independent of dimensions.
Factor K Insid ~ Outside Fiber Fiber 3.41 2.40 1.96 1.75 1.62 1.33 1.23 1,,14 1.10 1.08
0.54 0.60 0.65 0.68 0.71 0.79 0.84 0.89 0.91 0.93
Section
RIc 1.2 1.4 1.6 1.8 2.0 3.0 4.0 6.0
2.
8.0
0.0
Factor K Insid ~ OutsidE Fiber Fiber 2.89 2.13 1.79 1.63 1.52 1.30 1.20 1.12 1.09 1.07
K is independent of section dimensions.
.
Table 9.7 - K VALUES FOR DIFFERENT SECTIONS AND 9-116
RADTT OF rnlHTA,'T'TTRF.
0.57 0.63 0.67 0,,70 0.73
O.Bl 0.85 0.90 0.92 0.94
STRUCTURAL DESIGN MANUAL Section
Ric
Insid Fiber
1.2 1.4 1.6 1.8 2.0 3.0 4.0 6.0 8.0 10.0
3.01 2.18 1.87 1.69 1.58 1.33 1.23 1.13 1.10 1.08
1.2 1.4 1.6
1.8 2.0 3.0
4.0 6.0 8.0
10.0
1.2 1.4 1.6 1.8 2.0 3.0 4.0 6.0 8.0 10.0
)
R----;
3.09 2.25 1.91 1.73 1.61
1.37 1.26 1.17 1.13 1.11
3.14 2.29 1.93
1.74 1.61 1.34 1.24 1.15
1.12 1.10
1.2 1.4 1.6 1.8 2.0 3.0 4.0 6.0
3.26 2.39 1.99
8.0
1.12
10.0
1.09
1.78 1.66 1 .. 37 1.27 1.16
Section
0.54 0.60 0.65 0.68 0.71 0.80
RIc
7.
1.2 1.4 1.6 1.8 2.0 2.5 3.0
0.84
0.88
0.91 0.93
1.2 1.4
0.56 0.62 0.66 0.70 0.73 0.81 0.86 0.91
1.6 1.8 2.0 3.0 4.0 6.0
0.94 0.95
10.0
8.0
0.52 0.54 0.62 0.65 0.68
1.4 .1.6 I
6t
t
0.82 0.87 0.91 0.93
0.60 0.70 0.75 0.82 0.86 0.88
i
.
O. 76
0.44 0.50 0.54 0.57
1.2
3t +2t1
I
2. 0 3.0 4.0 6.0 8 .. 0 0.0
~ .....1
I
10.
1.8
I
1.2 1.4 1.6
1.8 2.0 3.0 4.0 6.0 8.0 0.0
3.65 2.50 2.08 1.85 1.69
1.49 1.38 1.27 1.19 1.14 1.12
0.53 0.59 0.63 0.66 0.69 0.74 O. 78 0.83 0.90 0.93 0 .. 96
3.63 2.54 2.14 1.89 1.73 1.41 1.29 1.18 1.13 1.10
0.58 0.65 0.67 0.70 0.72 0.79 0.83 0.88 0.91 0.92
3.55 2.48 2.07 1.83 1.69 1.38 1.26 1.15 1.10 1.08
0.67 0.72 0.76 0.78 0.80 0.86 0.89 0.92 0.94 0.95
2.52 1.90 1.63 1.50 1.41 1.23 1.16 1.10 1 .. 07 1.05
0.67 0.71 0.75 0.77 0.79 0.86 0.89 0.92 0.94 0.95
Table 9.7 (Cont'd) K VALUES FOR DIFFERENT SECTIONS AND RADII OF CURVATURE 9-117
STRUCTURAL DESIGN MANUAL Factor K
I!.
I I
d
-~c~ '--R
1.2 1.4 1.6 1.8 2.0 3.0 4.0 6.0 8.0 ~O.O
Table 9.7 (CONT'D)
l.a
Section
RIc Inside Outside
Section
v~
Factor K
Fiber 3.28 2.31 1.89 1.70 1.57 1.31 1.21 1.13 1.10 1.07
Fiber 0.58 0.64 0.68 0.71 0.73 0.81 0.85 0.90 0.92 0.93
I 4t ......
L
Ric InsidE Outside
r , 4t1 tE2
t~
---
t~
1--0--1
I-
1.2 1.4 1.6 1.8 I 2.0 3.0 4.0 6.0 t 8.0 10.0
Rl
Fibe 2.63 1.97 1.66 1.51 1.43 1.23 1.15 1.09 1.07 1.06
Fiber 0.68 0.73 0.76 0.78 0.80 0.86 0.89 0.92 0.94 0.95
}
K VALUES FOR DIFFERENT SECTIONS AND RADII OF CURVATURE
BOLT-SPACER COMBINATIONS SUBJECTED TO BENDING
When bolts and spacers are subjected to bendin& the allowable may be calculated in the usual manner. However, consideration should be given to the preload induced by the nut. Figure.9.23 shows the neutral axis location of a bolt-spacer combination subjected to bending. Figure 9.24 shows the effect of preload on a boltspacer combination.
9.9
\
STANDARD BENDING SHAPES
Standard bendi~g shapes, tubes a~d channels, which are subject to local crippling or crushing are presented in Figures 9.25 through 9.28. These figures present the _a~low~~1.e.~~ending moment for various materia~s ~~~_~r~ss sections.
tt'
STRUCTURAL DESIG-N MANUAL
•
"! a: ..... ...
.)
.n
~
o
q
~ I
•
QN\1l!
>t
Figure 9.23 - Neutral Axis Location and Moment of Inertia of A Bolt-Spacer Combination Subjected to Bending •
9-119
..... i. ' ~ _.:..1.. ___ ~ L
'.--:-::::-:-:I-:::::-:T:::=I::~_~r::·_-::::~~l-::':1==-:.r~.:
\0
•
'I'-.)
~ .....
o
','I
. f: ..
:...
: ,-
,1.
Allowable Moment Curves for Clamped-Up Bolt Bushing Combinations of 125 KS! Steel
, P-'
OQ
c:::
rc
HR= t·
(I)
'"
ALLOWABLE MOMENT OF COMBINATION ALLOWABLE MOMENT OF BOLT ALONE
N
.p..
NOTE:
Brinnelling effect of bushing on lug exceed a compressive stress equal to
>I-' I-'
o
~C"
2.0
......
,~~-J -:! •. --l~ljTJ : -r-~n~~·~t~::-~0 L-~-t--~!:;:I _ :rtt-L·:--
(I)
:s: ~ ~
"~-, ... ~.--!
C'1'
(")
c::
I"i
< ('1) en
R
o t1
I»
,t··
(I)
i . 1 5, .
,
c::::
"'0
I
i.'
o ..... 0..
.....
.. -
..
~.t-~.-~-.".".--t----""""'t",-+,-b.....;..,
-I
:0
i
c: n ..... c: ::a :.:.
~
CURVES ARE BASED ON THICK TENSION NUT
I-tl
.g
en
d
.
r-
-:f-4'~':"7~~'
C
.!
rn
M= P·e
en
-
"
r f'-"~
Ij;I
4
~
'---t,~-·-~--
'"'"'''''_.'/
o
C i)
Z
'...·~ .. -·f"--.. '-..-~ -r', ,
I-' C"T I
Ij;I
·i~···~~:
C CIl
=r..... ::s
1
~ 0'" .....
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STRUCTURAL DESIGN MANUA.L VOLUME I
• PROPRIETARY RIGHTS NOTICE
').
•
THUE DATA ARE PROPRIETARY TO BELL HILICOPTER COMPANY. DISCLOSURE, RIPRODUCTlON, OR USE OF THESE DATA FOR ANY PURPOSE IS FORBIDDEN WITHOUT WllnEN AUTHORIZATION FROM BELL HELICOPTER COMPANY.
Bell Helicopter i i 43 i it·] :I
STRUCTURAL DESIGN MANUAL This Structural Design Manual is the property of Bell Helicopter Textron and no pages are to be added or withdrawn except as directed by revision notices.
Information contained herein which is specifically identified may
not be reproduced or further disseminated without the approval of the Chief of Structural Technology, Bell Helicopter Textron.
\.
)
• ii
." -71, '.. /' : I ". \ '\
"~ '\ ,~~:~~ STRue TURAL DES IGN MAN UAL ...~:. ')1- ,,'.
OBLIQUE FILLET
A = (T A N 8 - 8 ) R 2
Y
=
[I -
2
e
SIN TANS] R 3 (TAN 8 - 8) 2
X =
p
•
8-
2
SI N 8 TAN 8 3(TANe -9) 2
] R
2
[SEC9 - SIN 8TAN 9 ] R 3(TAN8-8) 2
4
4
2
I
= [SIN 8 TAN8 + SINe case (1+ 2 SIN 9)- 8 _ SIN BTAN 8 ] R4 I-I 3 4 9(TANB -9}
I
=[TA N 9( TA N 29 + S IN 2-2 6
1
6
case
= [SEC! e-cos's _ 651 N 9
4-4
HOLLOW
TABLE 3. I
2
e )_ 38-
= [ SIN '8 _ 8 - SIN 8 cos 9 ]
a-! I
•
=
[TAN
4
4
SIN 8 cos8(3 - 2 SINi!8) _51 N STAN 8 ] R4 12 9(TAN 8.. 8)
R4
4 8+SIN9 4
cose _ SIN 2 8TANt ]R4 9(TAN9-8)
CIRCLE
(CONT'D) PROPERTIES OF COMMON SECTIONS 3-17
~1\'~~
,,~~
STRUCTURAL DESIGN MANUAL
Revision A ECCENTRIC HOLLOW CIRCLE 2
2.
HOLLOW SEMI- CIRCLE A = 1.5708 (R2-r 2
)
)( = R 2
,2
R +r )
Y
=
I
:; 0.3927 (R4- r·) -1.5708 (R 2 - r2)'9
I-I
0.4244 (Rf
12 -2 = 0: 392 7 ( R 4
-
r
4
4 )
4
Ia.a= O.3927(R - r } 1.- 4
= O.
2
3927 (R - r.
2
.
2
) (5 R +
r
2
)
,
\
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TABLE 3.1 (CONT'D) PROPERTIES OF COMMON SECTIONS 3-18
e--
~ s-rRUCTURAL DESIGN MANUAL
~.II ~'~R (3)
The margin of safety for the loading represented by point uall can be found in three ways a.
MS
b.
MS MS
c.
= od/oa = bh/ba = cg/ca
- 1 - 1 - 1
Values of od, bh and cg are referred to as allowables (load or stress) and oa, ba and ca are applied load or stress. Using this procedure and equation 4.-6- procedures for two loads acting and three loads acting ~an be determined.
4.4.1
•.
-,
Procedure for Margin of Safety for Two Loads Acting
(1)
Using buckling, yield or ultimate criteria and equation 4.6, calculate the stress ratio for each load acting alone.
(2)
Using the calculated stress ratios locate point curve (using Figure 4.6 as an example).
(3)
Draw a straight line from the origin "ott through point "al l and intersect the interaction curve at point lid". Read the stress ratios R (ed) and R (fd). 1a 2a
(4)
Compute the applied stress ratios R (ba) and R (ca) •
(5)
Compute the margin of safety
1
"a"
on the proper interaction
2
4.8 4.4.2
Procedure for Margin of Safety for Three Loads Acting
(1)
Using buckling, yield or ultimate criteria and equation 4.6, calculate the stress ratios for each load acting above.
(2)
Using the appropriate interaction family of curves locate point "a" corresponding to the calculated stress ratios Rl and R2 as shown in Figure 4.7.
(3)
Draw a straight line from the origin " 0 " through po'int "a".
(4)
Extend this line to locate the allowable point "x" which must satisfy the following relationships:
4.9 or
4.10 Point "x" is obtained by trial and error in the following manner: (a)
Select an arbitrary value of R • 1a
(b)
Calculate R3 from equation 4.10 using the known value of Rl and R3 and ' a va 1ue 0 f Rla. h b ltrary tear 4-11
~U STRUCTURAL DESIGN MANUAL
"%D (c)
Locate point "x" on the line "oa" using the calculated R3a from step
(b).l
and compare the corresponding R1a with the assumed R • 1a (d)
(5)
Repeat steps (a) through (c) until the assumed R1a and the "XU value of Rl converge. At convergence, R1 ,R and R3 w111 be at a common 2a pornt on line "oa".. a a
Compute the margin of safety
4.11 4.5
)
Compact Structures
A compact structure is one in which failure does not occur' by crippling or buckling. This section presents interaction criteria for compact structures with biaxial stress in a rectangular volume such as in plates, membranes and shells and with uniaxial stress in a plane such as in beams, round bars and bolts.
4.5.1
Biaxial Stress Interaction Relationships
Tests have been conducted to determine the failure theories of biaxially loaded isotropic ductile materials. The maximum shear stress theory and the octahedral shear stress theory adequately predict the yield and ultimate strengths. There are a few cases where convenient margin of safety calculations are possible. These are shown in Table 4.3. A general interaction method is required. It is shown in Figure 4.8. ~ The method is applicable to stress conditions which combine in a two-dimensional ~ manner like that shown in Figure 4.8. This condition exists in a rectangular volume, and not on a single plane. Tension is positive, compression is negative. The interaction equations and curves are applicable for ultimate and yield by use of the parameters given in Table 4.1. The interaction equations contain certain factors which relate one stress to the other. They are defined as follows: The constant relating interaction in terms of tension or shear strength allowables: K = F
su
IF tu
4.12
)
Tests show this value to vary from 0.5 to 0.75. The transverse shear and torsional stress ratios combine as R
s'
= R
ss
+
R st
4.13
The directional tension and bending stress ratios combine as R x
Ry
= R tx + ~x = Rty + ~y
4.14 4.15
The directional compression and bending stress ratios combine as R =R x cx
4-12
+R. -ox
4.16
..., ~ ~
e ~i----~--------------
LOADING PICTURE
t"Zl
f" VI
~~
ROUND TUBES
Gi
1
~ ~
E ~
H~
o Z > ~ C/)
~
~ ~~
~
~~ c.n
2
__
;b
{t
----------~--------------------~----------------~--------------~
LOADING DESCRIPTION
+
Bending
. Tenslon +
3
~ ~
.
~
t
oc.~
D
~
Bending +. TarSlon
~9\ ,~-
~
=
1
Rt1.5 + Rb ; 1
Let R t : Rl
4.23
en
Rb - R2
Rb
Z
+ Rst
2
;1
4.10JRb
Z
-I
:::a
-1
1
+R s /
c:
(III)
t-3
::t:C/)
v~ ~
4
c::::o
8C
t::'t:j
«i~b ;;
Tension
+ Bending
+
R..
--bt
2+ R 2=1 s
4.10
V
Rbt - Rt/R t • Rb/Rb • al where Rt and Rb a~e ob.
1
R 2 bt
+ Rs2
a
-1
Z
-t
c: :::a :.:-
a
tained from figure 4.~ (R a~. t Rl " Rb • R21 n • l.5) by tfie two-loods-acting procedure as outlined in section 4.4.1
Shear
HCIl
r-
C')I
1-31--1 :XZ t"Zl1-3
t:j::O
"rJ> ~CJ
5
t-x:I::.o
6
CJI-3 OP1
2~
.-<
7
1-3t't:1
~b H
~S; -< c::
~
t-3t:;1
... I N
V)
t'"'" t-3 t-I
~ t--:3
trl
~(
8
+
R )2
+
R 2
cbs
4.10
(Rc + Rb)2 + Rs2 1
b
fs
~::o
CIlH
+
(R
-l
+
c:;,
Rb ... R1
rn
R: = R Z
-
(I)
Shear
QC(Q1j
C::H
1
+
f
H
Let ~ R
Compression Bending
s
tz1t-3 CJH 1-30 CIlZ
on
~ C
b
t%j
Bending
+
R 2 + R 2 = 1 b s
4.10
1~~2 +
C i)
1
Rs2
:z
-
~
Shear
~
(6)\:fit fb
Efj\~
Tension
+ Torsion
R 2 t
'+
R 2;;; 1 st
J
1
R 2
4.10 I
t
Bending
+ Torsion
Rb
2
~ Rst 2
(l-R ) 2 c
=
4.11
R c
+
J
Rb
:.:-
z
R 2 -1 st
1
Compression
+
+
2
+
-----"
~~~CI~ -~i, '~ ~=~/
b
c
-~
.~.
+R- 1
R
4.10
Bpnn i no-
f4~
r-- ~
REMARKS
1
Rc
~fb
_________ __ _ ____ _ EQUATION
-ON r.lrRVR
co:preSSiOn
~
e
e
\"'--'-~
c:
In using figure 2 -1 4.11 follow twoR loads-acting prost cedures as outlined in section -'.J.Ql
:x:t (1)
< ell ..... ~.
o ~
t;xj
:.:-
r-
~
TABLE 4.5
I IV ~
THICK-WALLED TUBULAR STRUCTURES-INTERACTION CRITERIA-YIELD AND ULTIMATE CONDITIONS OF STRENGTH, INCLUDING THE EFFECTS OF COLUMN STABILITY (CONCLUDED)
"'~"
~D ~
LOADING DESCRIPTION Compression
LOADING PICTURE
CASE
~
9
~) fst
8"······00 \.L.~~. ~
10
t
+
Bending
+
Torsion + Shear Tension + Torsion
+
Internal Pressure
INTERACTION CURVE FIGURE EQUATION R
c
+R
+R
2 b
st 2 s
=1
2
1
t
t
4.13
jR2+R t
= 1
\L! _:
I
REMARKS
'~'iCJ .
"-.;-
i
In us figure 4.12, follow twoloads-acting procedure as outlined in section 4.4.1
4.12
+R
R 2 +
+ Rp 2
MARGIN-OF-SAFETY EQUATION
en ..... :=a
c:
n
-t c::
,..
-1 st
::ICJ
2+R2
p
r-
STREAMLINE TUBES
CCC p:@)h
11
Bending
+
Torsion
~
+ Rst
= 1
4.10
J
1
'J..b :;:
Rst
CI -1
.."
en
-
SQUARE TUBES
6(
12
t :i)
~st
z
Let
Compression + Torsion
R
;; R
,..z
c 1 ;; R s 2
4.14
!Ie
R
NOTES: (1) (2) (3)
(4)
(5) (6)
e
~,
must be based on the tube column allowable. R~ must be based on the material strength allowable. Rb and R t must be based on tube strength allowables. Rb must fnclude the effects of secondary bending. For shear-bending analysis use f = f and fb = fb even though the locations of the two maxima do not coincide. The alI5wabl~mt~ansverse she~Xstress is equal to the lower of 1.20 times the allowable torsional shear stress and the material allowable shear stress. R ;; pd/2t F ,d = tube mean diameter, t = wall thickness. p tu
R
'"'-'
e
"-
::t1 (t)
<:
r--
to ,..,..
o ~
()
e,
,......c:
e
e
e
',,--,",
~~ .-
.
W'I
(L) (T)
VJ
~
.
f;
.-<
Alloy
~
(")
c:: :> ::0
2014
<:
2024
t""4
t'J:I
1-3
0
'-:>
§ 8 ttj en
:J:
Z
t'::j
en en
0
Io%j
2219
:~.
0
Temper
Plate Forgings
T651 T652
No. of /-,ots
of
Inch
Min Avg
Max
No. of
Thick.
Inch
Lots
Min
Avg
Max
Plate E?'truded Shapes
T351 T3510,1
Plate Extruded Shapes Forgings . Plate
22 25
23
24
4
3/4
29
34
4
1-1/2 1-1/2- 2
--
1-1/2
---
2
1~1/2
-_.
---
46
-----
T851 T8510,1 T852
1 - 2 3/4- 4 2 - 6
4
1-3/8 3/4
22
2
3/4
32 30
4
5
22 23
23 28 26
25
5
T851 T87
1 - 2 3/4- 1
2 3
1-3/8 3/4
31 26
33 27
~
> ~
7075
Plate Extruded Shapes Forgings
T651 T6510,l T652
1/2- 2 1/2- 4
1/2
25 26
26 28
2 - 6
6 10 2
t-3
~ '1j ~
H
24
26
Plate Extruded Shapes Forgings
T7351 T73510,1 T7352
1-3/8 1/2- 4 1 - 5
1 2 5
1-3/8
---31
30
Z
p.j
33
---34
~ , t-3 ~ > t""4 fij 1:"""'1
trl
C/)
1:"""'1
0 t-<:
'
27
31
35
'"0
> -,Z ::0
tJ:I'
I-%j
en
rzj .
I
....-3 -
~
H
Z
7079
Plate Forgings
T651 T652
1 - 3
3
1
2 - 6
2
7178
Plate Extruded Shapes
T651 T6510,1
1/2- 2 l/Z- 1-1/2
3 1
Plate Extruded Shapes
T7671 T7650,1
1/2- 2 1/2- 2
5
1 - 2 2 - 6
3
3
1
1/2 1/2 5/8 3/4
3/4 1 1/2 1/2
3/4
-- -
---
4
36 2 28 --27 ' 4 32 10 28 1
1-3/8 3/4 3/4
1 1/2 1/2 1/2
2 5 3
1
2 3
1
19 19
23
.I.
23
30
2
--- --- .--
2
--- ---
---
---
- --
20
20
16 17
17
20 18
18
20
29
30
30
1
1/4
3
22
23
2
_A.
22 23
26
.. -
"'4 1
1/2 1/4 1/2
29
33
2
1/2
24
28
1
3/4
25 22 23
25
26
3
1/2
3/4
24 21
1/2 1/4
22
23
26 11
4
1/2
19
4
1/2
30 31
2 3
1/2 5/8
24
24
1
23
·25
3
21 19
23
1
16
20
1
22 18
22 22
22 28
1 1
..... :::a
Max
---
c:
19
26
c-)
--- ----- 17 ----16- ---16 ---17 --- 20 _.. ---
3/4
31
29
23
1/2
30
29
20
1
1
29
26
18 19
1
30
26
--18
1/2 1/2
20 19
1/2
(I) Avg
--- -- . ---
27
25
1
i-
Thick. Inch Min
,
21
28
---
1
1
Il-0ts
1
1/2 1
1/2 1/2
---
20
15
16
18
19
--19 ---19 ---17 -------
---
16 18
15 14
--._-
20 20 21
l7 16
r-
18 22
--21 . -25 ---18
17
--I
c: ::0 :.:-
---
-_. ---
c:J
rn
-
(I)
I I
G ')
z:
3:
:.:z c: :.:r-
VI I
1--'.
""-l
'\,
'---<-
""!It ;. \; ~~" -/{
(ST)(L)
No. Thick.
4
0
I"lj
:::
Product
Product 1:hickness Range, Inch
(T)(L)
H
~
J[n.
Plane-Strain Fracture Toughness, Krc' ksi
~
::lCI
~-- ~
,)/
C=' t:"""
"'Z1
"
,..-
1'-:3
:>
I
s-rRuc-rURAL DESIGN MANUAL Estimate of Highest Sustained Tension Stress (ksi) at Which Test Specimens of Different Orientations to the Grain Structure Would Not Fail in the 3~% NaCl Alternate Immersion Test in 81 Days Alloy and Type of Temper 20l4-T6
2219-TB
2024-T3, T4
2024-T8
Test Direction L LT ST L LT ST
7075-T73
7178-T6
717B-T76
8
·.
50 27
15
·.
40
·.. ..
35 35
45
30
38 38
45
"
45 22
Hand Forgings
B
30 25 8
35 35 35
38 38 38
."
.... 50 37 .
50
....
30
8
10
L
50 50 30
47
50 45
50
8
15
·.
L LT ST
59 49 25
·.. . .. . "
52 49 25
·.....
L
50 48 43
50 48 43
54 48 46
53 48 46
50
...
60 50
60 35
50
8
8
L
LT ST 7079-T6
and Bar
35 20
LT ST 7075-T76
Plate
Extruded Shapes Section Thickness, Inch 1-2 0.25-1
L LT ST LT ST
7075-T6
1 Rolled Bar
·. ·
"
43
."
L LT ST
55 40
L LT ST
55 8
·.. ..
L LT ST
52 52 25
.. .. .. ..
8
38
.... .. ..
· ..
.
·
18
...
·8
" "
60 50
...
60 50 45
43 43 15
60 50
60 32
35 25 8
·
·..
6 45
..
.
55 52 25
8
·.
65 25 8
·.. ....
·.
·· ...
·.
48
43
... ... "
..
·" .... ·..
TABLE 5.4--COMPARISON OF THE RESISTANCE TO STRESS CORROSION OF VARIOUS ALUMINUM ALLOYS (REF. 1) 5-18
)
30 /
STRUCTURAL DESIGN MANUAL
TYPE IV NUT
BOLT AN3-AN20 AN42-AN49 AN173-AN186 AN525 MS20033-MS20046 MS20073-MS20081 MS24694 MS27039 NAS333-NAS340 NASSl7 NAS623 NAS1003-NASI020 NAS1202-NAS1210 NAS1297 NAS13S2 (NON-LOCKING) ALL THREADED STUDS
)
-,-
AN256 AN310 AN315 MS9358 MS20365 MS20500 MS21042 MS21043 MS2l044 MS21045 MS21047 MS21048 MS21049 MS21051 MS21052 MS21053 MS21054 MS21055 MS210S6 MS21058 MS21059 MS21060
MS21061 MS21062 MS21069 MS21070 MS21071 MS21072 MS2l073 MS21074 MS2l075 MS21076 MS21083 MS21086 MS21208 MS21209 MS21991 MS122076 thru MS122275 MS124651 thru MS124850 NAS509
NASS77 NAS1291 NAS1329 NAS1330 NAS1473 NAS1474 80-004 80-005 80-006 80-007 80-013 90-002 90-003 110-061 110-062
TYPE IV CONSISTS OF ANY COMBINATION OF NUT AND BOLT SHOWN REFERENCE BELL STn 160-007 TABLE 6.5
- TYPE IV FASTENERS
6-13
STRUCTURAL DESIGN MANUAL
Torque, In-Lbs TYPE III NUT AND BOLT THREAD SIZE
TYPE IV
SHEAR Recommended Installation Torque Range (a)
TENSION
Max Allowable Tightening Torque (b)
Recommended Installation Torque Range (c)
Allowable Tightening Torque (d)
)
10-32
12-15
25
20-25
1/4-28
30-40
60
50-70
40 100
5/16-24
60-85
140
100-140
225
3/8-24
95-110
240
160-190
390
7/16-20
270-300
500
440-500
840
1/2-20
288-408
660
480-700
1100
9/16-18
480-600
960
800-1000
1600
5/8-18
660-780
1400
1100-1300
2400
3/4-16
1300-1500
3000
2300-2500
5000
7/8-14
1500-1800
420'0
2500-3000
7000
1 - 12
2200-3300
6000
3700-5500
10000
1 1/8-12
3000-4200
9000
5000-7000
15000
1 1/4-12
5400-6600
15000
9000-11000
25000
(a)
TYPE III RECOMMENDED TORQUE RANGE IS BASED ON A NOMINAL STRESS OF 24 KSI IN THE BOLT.
(b)
TYPE III MAX ALLOWABLE TORQUE IS BASED ON A STRESS OF 54 KSI IN THE BOLT.
(c)
TYPE IV RECOMMENDED TORQUE RANGE IS BASED ON A NOMINAL STRESS OF 40 KSI IN THE BOLT.
(d)
TYPE IV MAX ALLOWABLE TORQUE IS BASED ON A STRESS OF 90 KSI IN THE BOLT.
REFERENCE BELL STD 160-007 TABLE 6.6
6-14
~ax
- TORQUE VALUES FOR THREADED FASTENERS AND FITTINGS
)
STRUCTURAL DESIGN MANUAL An example problem best illustrates the procedure. Figure 6.21 shows a bimetallic splice, titanium and aluminum sheets joined by six steel bolts. The titanium and aluminum have uniform temperature rises of 30QoF and 70 0 F respectively. The results of the example show that the maximum load occurs in the first attachment and that the two end-attachments carry more than half of the total applied mechanical load. When plastic deformations occur in the vicinity of the bolt holes, the bolts tend to carry equal loads. Example Problem: f
=
.900 x 10-
(L/A~)T
6
in/1b (Section 6.4.3)
6
6
= 1/(1.5)(.125)(15)(10 )_ = .356 (10- ) in/lb
6 (L/AE)B = 1/(1.5)(.250)(10)(10 ) ~ ~
( at d T) T -
6 .267 (10- ) in/lb
(a.6 T) B
6 L = (605)(300) - 12(70») (1)(10- )
=
6 1110(10- ) in.
Substituting into equation (7)
Pjn = (Ajn + Bjn (.356/.900») (20,000) + BJN (1110/.900) 20,000 A.
In
+ 9135 B.
In
The coefficients A. and B. are now determined from Figure 6.7 In In through Figure 6.20.
z = [( ~) . + (~) J(.1)= Ae T Ae B f N =
(.356
+ .267)- (1/.900)
.692
6
A 16 A 26
.0140; Figure 6.7
B16
.8000; Figure 6.8
.0239; Figure 6.9
B26
.3200; Figure 6.11
A36 = .OSOO; Figure 6.12, B36 A46 = .1090; Figure 6.15, B46 AS6
.24S0; Figure 6.18, BS6
= = =
.0880; Figure 6.14 -.0860; Figure 6.-16 -.3200; Figure 6.19
The curves give values of Ajn and Bjn up to j = 5, but the splice under consideration has 6 fasteners. In order to obtain the coefficients for the last attachment, the designation of the top and bottom plates must be interchanged as shown.
6-39
~ STRUCTURAL DESIGN MANUAL
~
~--------------~. X-20000
TITANIUM, E=15(106~6
.250 DIA • STEEL BOLTS
• 125xl.SO, 4=6.5(10 )
T=300oF
20000
ALUMINUM. E=lO(106 .. 25Ox1.50, a=12(10 T=70oF
26 )
12410
9010 . 7210 5820 3840
LOAD IN TOP PLATE
+ •
20000
+
-+--
«
7590
I
3400
1~00
1390+ 1980
7590
~
BOLT LOADS
_
3400
4030 980
1800 1390 _ 1---,..-----r---1 . 1
I
FIGURE 6.21 - EXAMPLE PROBLEM, COMPATIBILITY As shown above, the last attachment (j - 6) in the original designation becomes the first attachment (j = 1) in the interchanged position.
f' = f = .900 (10. 6 ) in/lb 6
(L/ AE)'T =' (L/ AE)B = .267 (10- ) in/lb
.356 (10- 6 ) in/lb
(L/ AE)'B = (LI AE)T
Z·
=
Z
= .692
-
A~
= -1110(10 -6 )
from equation (7) pi
Jfi
p!
In
= (A! In
+
B! (.267/.900»(20,000) = B! (1110/.900) In In
= 20,000 A~ In
from Figures 6.7
6-40
+
4693 B~
In
and 6.8
)
STRUCTURAL DESIGN MANUAL Unequal Areas At any stage where the centhe fractional part of the connecting line measured from the previous centroid is
Add
~
fastQner at a time as described previously.
troid of n bolts has been found and is joined to the (n+l) fastener,
+ ..... 6.4.2.2
Load Determination
Figure 6.26 shows a typical joint with an applied load P and three fasteners AI, A2 and A3. Draw the joint to scale and locate the center of resistance G. Extend the line of action of the applied load P, and from this line erect a perpendicular that passes through the centroid G and extends a distance GQ away [rom P, so tha t GQ
where
•
.'
area of fastener in shear or bearing r ~ radial distance from G to fastener e = distance from G to line of action of P
A
FIGURE 6.26 - TYPICAL JOINT
6-51
STRUCTURAL DESIGN MANUAL' Next determine the radial distance Ll of the number one fastener from Q. ThQ load P on that bolt is
and is directed perpendicular to radial line L l Repea t this procedure UTIti 1 the. loads for all fas teners are de termined. 6.4.3
)
Attachment Flexibility
The flexibility of an attachment/sheet combination should be determined experimentally. If load-deflection curves for a particular fastener/sheet combination aTe available, the flexibility is the slope of the curve at the estimatf~d load level. If load-deflection test data is not available for the exaGt fastener/~heet combination, two methods can be used to determine a spring rate. 6.4.3.1
Method I - Generalized Test Data
Some test data is available to develop generalized stiffness curves •. Figure 6.27 shows a curve of tID versus K for a single shear joint with a steel fastener. The procedure for determining joint stiffness is as follows: DIA
1/8
5/32
ALUM
.163
STEEL
3.62
TITAN OTHER
3/16
1/4
5/16 SRxlO- 6
.203 .244
.325
.406
.487
.563
.650
.732
.813
4.53 5.44
7.25
9.06
10.9
12.6
14.5
15.5
18.1
1.93 2.42 2.90 3.87 4.83 (Eother/Esteel)xSRsteel
5.81
6.72
7.73
8.27
9.65
I
SHEET SPRING RATE JOINT SPRING RATE TABLE 6.9
= K,x SR = l/(l/SRu +
3/8
1/SR1)
- BASIC SPRING RATES
1.
Calculate tID for upper sheet
2.
Calculate tiD for lower sheet
3.
From Figure 6.27 determine K for upper sheet
4.
From Figure 6.27 detennine K for lower sheet
6-.52
7/16
1/2
9/16
5/8
)
•
STRUCTURAL DESIGN MANUAL The average load is then P
avg
= (6200 +
4875)/2
= 5590
lbs.
The flexibility is calculated for a defonnation of 2 percent of the hole diameter per Reference 1. f
6.4.4
avg
=
8/P avg
(. 02) ( .. 250)/5590 ~900( 10 - 9) in./lb
Lug Design
This section presents a basic method of analysis and procedure for the design of lug-pin combinations loaded axially, obliquely or transversely.
An accurate analysis of a lug-pin combination under load is difficult because the actual distributions of stresses in the lug and pin involve a combination of shear, bending and tension of varying amounts, which are a function of the ratio of lug edge di"stance and thickness to pin diameter, shape of lug, number of lugs in a joint~ material properties, ·stress concentrations, rigidity of adjacent structure, etc. The various modes of failure for a lug are:
'I.
Bearing of pin. lug or bushing
2.
Tension across minimum net section. The full plAnet stress cannot be carried because of the stress concentration around the hole.
3.
Hoop 'tension failure of the lug across the section in line with the load.
4.
Shear tearout failure of the lug.
5.
Shear and bending of the pin.
Shear tearout,and bearing are closely related and are covered by shear-bearing calculations based on empirical data. Also, the shear-bearing criteria precludes hoop tension failures. Yielding of the lug is also permanent set of 0.02 times checked as it is frequently from the ratio of the yield material.
a consideration. It is considered excessive at a the pin diameter. This condition must always be reached at a lower load than would be anticipated stress, F ty ' to the ultimate stress, F tu ' for the
6-55
STRUCTURAL DESIGN MANUAL
•
Since lugs are elements having severe stress concentrations, the ductility and/or impact strength of the material is of importance. For this reason, attention should be paid to the longitudinal, long transverse and short transverse material properties. Lugs are a small weight portion of a structure and are prone to fabrication errors and service damage. Since their weight is usually insignif'icant relative to their importance, the following criteria should be used. 1., 2.
Design lugs for a minimum margin of safety of 0.15 in both yield and ul timate.
)
If no bushing is included in the original design, design, the lug so that one can be inserted in the future; however, express margins of safety with no bushings.
6.4.4.1
Nomencla ture
= Ultimate
tensile strength; F tuw with grain t F tux cross grain. When the plane of the lug contains both long and short transverse grain directions, F tux is the smaller of the two.
= Tensile
yield strength; Ftyw with grain, Ftyx cross grain. When the plane of the lug contains both long and short transverse grain directions, Ftyx is the smaller of the two.
= Comparison p
y
M
max p' u
p' bru pI
bry
yield strength
= Ultimate load = Yield load = Maximum
bending moment on pin
Allowable ultimate load
= Allowable
ultimate shear-bearing load
) Allowable yield bearing load "on bushing
= Allowable
ultimate tensile load
pi
= Allowable
ultimate transverse load
pi
= Allowable
yield load of lug
pt
tu
tru
y A
6-56
= Area;
I
Abr projected bearing area, At minimum net section for tension, Aav weighted average area for transverse load.
STRUCTURAL DESIGN MANUAL
•
""
---". ./ _ _ _ Equivalent Lug
(c)
(d)
FIGURE 6.29 - TRANSVERSELY LOADED LUGS
B.
(2)
A3 is the least area on any radial section around the hole.
(3)
AI, A2, A3 and A4 should adequately reflect the strength of the lug. For lugs of unusual shape, such as severe necking or other sudden changes in cross section, an equivalent lug should be used such as shown in Figure 6.29(c} and (d).
P tru = Allowable ultimate load for lug failure 1. 2.
•
\
c.
Enter iigure 6.33 with Aav/~r and obtain K ru • t p' = K ~ F tru tru r tux
p. = Allowable yield load of lug y 1. 2.
Enter Figure 6.33-with Aav/~r and obtain K • try pt = K ~ F y try' r tyx
6-59
STRUCTURAL DESIGN MANUAL D.
Check bushing yield per 6.4.4.2(E).
E.
Margins of Safety
1. 2.
M.S. = .15 for ultimate transverse load Minimum M.S. 0 for yield of the lug and bushing
Mini~um
Analysis of Lugs with Oblique Loads (o
6.4.4.4
In analyzing lugs with oblique loading it is necessary to resolve the loading into axial and transverse components (denoted by the subscripts "a" and !'tr" respectively), analyze the two cases separately and then combine the results using the interaction equation. The interaction equation:
R 1.6 a
+R
tr
)' .
=1
1.6
where, for ultimate load,
R = Axial component of applied ultimate load a R
tr
Smaller of P '
b ru
dr p' (6.4.4.2 B or C) tu
Transverse component of applied ultimate load Pt· ru (6.4.4.3.B)
and for yield load R
a
R
tr
Axial component of applied yield load p' (6.4.4 .. 2D) y
= Transverse
component of applied yield load
pI (6.4 .. 4.3C)
y
The margin of safety should be 0.15 minimum and is calculated using the following equation:
MS
1
=
(R 1.6 + R 1.6)0.625a
6,.4.4 .. 5
1
tr
Analysis of Pins
The ultimate strength for a pin in a single lug/clevis joint as shown in Figure 6.34 will be analyzed first.
6-60
)
STRUCTURAL DESIGN MANUAL
•
Revision A
2.5
2.0
1.5 K
bry 1.0
•
.5
o
o
1.0
2.0 e/d
3.0
4.0
FIGURE 6.31 - BEARING YIELD EFFICIENCY FACTORS FOR AXIALLY LOADED LUGS
• 6-63
STRUCTU'RAL DESIGN MANUAL Revision B
1.0
t::-- """'~~~ .......
-, ::::: ~ " ' " " ~ \ , " (~ ~ ¥J '" --.....
~ ~ ~ ............. ~ \ \~ ~ ~~
\" '"
.9
.. 8
.....
\
•7
"'" "'-
\
,
~
\
K
'\
t
.. 5
~ ~
~
"-'\
I\..
'\ '\..
I---
"
f ..i..' c - ~ l.OAD
.2 -
10-
+
:;
Lt-;~~ ~e:i f ! i! 1ft 1
.....,;;;;
..............
2.0
............
~
~
~
I\.
~
""
~
(I) ;;
"\ \..
~
.......
-
~
3.0
~
''" " " ~I'
~t 6)
•
\
-.......
"'='
~ "-ts ru l\ " ~
~
~
'~
"" .~
I
o 1.0
®~
\
"" "- '-
W D
10-
.............
1\
~
10-
.1
"" \
'\
.4 .3
..... ~
(C ~ -\ ~ '" \ \"' "- ~ \
'\
.6
.............
~
4.0
'
!
" 5.0
FIGURE 6.32a - TENSION EFFICIENCY FACTORS FOR AXIALLY LOADED ALUMINUM AND STEEL LUGS
6-64
)
STRUCTURAL DESIGN MANUAL Revision B L, T and ST indicate grain in the "c" direction Material
•
Curve
Ti-6Al-4V Ann. Cond. A Die Forging (T) t ~ 5.0 Ti-6Al-4V Ann. Cond. A Hand Forging (T) A s: 16 Ti-6Al-4V Ann. Cond. A Hand Forging (T) A > 16 Ti-6A~'-4V STA Die Forging (L) t ~ 5.0 Ti-6Al-4V STA Die Forging (T) t < 1.0 Ti-6Al-4V STA Die Forging (T) 1.0 < t < 3.0 Ti-6Al-4V STA Hand Forging (L,T) t ~ 2.0 Ti-6AI-4V STA Hand Forging (T) 2.0 < t :s; "3.0 Ti-6AI-6V-2Sn Ann. Plate (T) t '$; 2.0 Ti-6Al-6V-2Sn Ann. Die Frg. (ST) t < 2.0 Ti-6Al-6V-2Sn Ann. Hand Frg. (T) t ~ 2.0 Ti-6Al-6V-2Sn Ann. Plate (T) 2.0 < t ~ 4.0 Ti-6AI-6V-2Sn Ann. Die Frg. (ST) 2.0 < t :S; 4.0 Ti-6Al-6V-2Sn Ann. Hand Frg. (T) 2.0 < t ~ 4.0 Ti-6Al-6V-2Sn STA Die Forg. (L) All Ti-6AI-6V-2Sn STA Die Forging (T) All Ti-6AI-6V-2Sn STA Hand Forging (L,T) t ~ 4.0 Ti-6AI-6V-2Sn STA Hand Forging (T) t > 4.0
1 1 2 2 2
3 1 2
4 4
4 5 5 5 6 7 6 7
In
no case should the ultimate transverse load be taken as less than that which could be carried by cantilever beam action of the portion of the lug under the load. The load that can be carried by cantilever beam action is indicated approximately by Curve A". Should Ktru be below Curve A, separate calculation as a cantilever beam is necessary.
FIGURE 6.33b (CONT'D) - EFFICIENCY FACTORS FOR TRANSVERSELY LOADED TITANIUM LUGS
• 6-67b
STRUCTURAL DESIGN MANUAL
~1
P/2
it T
P/2
P/2
d
Erg
P/2
)
FIGURE 6.34 - SINGLE LUG/CLEVIS JOINT A.
Obtain moment arm "b". For the inner lug of Figure 6.34 'calculate r = [(e/D) - ~] D/t2. Take the smaller of P~ru and P tu for the inner lug as (p min and compute (p~) min/Abr Ftux. Enter Figure 6.36 and obtain the reduction factor tty" which compensates for the Itpeakingtt of the distributed pin bearing load near the shear plane. Calculate
u)
where Itgtt is the gap between lugs as shown in Figure 6.34 and may be zero. Note that the peaking reduction factor applies only to the inner lugs.
B.
Calculate maximum pin bending moment, "M", from the equation
M = p(b/2) C.
Calculate bending stress assuming a M II distribution.
D.
Obtain the ultimate strength of the pin in bending by use of Section 9.4. If the analysis should show inadequate pin bending strength it may be possible to take advantage of any excess lug strength as follows.
E.
Consider a portion of the lugs to be inactive as indicated by the shaded area of Figure 6.35. The portion of the thickness to be considered active may have any desired value sufficient to carry the load and should be chosen by trial and error to give approximately equal margins of safety for the lugs and pin.
6-68
)
c
•
STRUCTURAL DESIGN MANUAL
P/2
t4
t:::=:::!::=~L~. . . P /2 P/2
FIGURE 6.35 - ACTIVE LUG THICKNESS
e;
F.
Recalculate all lug margins of safety with allowable loads reduced in the ratio of active thickness to actual thickness.
G.
Recalculate pin bending moment, M = P(b/2) and margin of safety using value of flb lt which is obtained as follows:
r
=
[(e/D) - ~] D/2t 4 -
Take the smaller of P'bru and Ptu for the inner lug, ~ased upon the active thickness, as (p~) min and compute (p~) min/Abru F tux where Abr = 2t4D. Enter Figure 6.36 and obtain "Y". Then
This reduced value of "b tt should not be used if the resulting eccentricity of load on the outer lugs introduce excessive bending stresses in the adjacent structure. In such cases pins must he strong enough to distribute the load uniformly across the entire lug. Lug-pin combinations having multiple shear connections such as those shown in Figure 6.37 are analyzed as follows.
6-69
STRUCTURAL DESIGN MANUAL
:".-"- T-: to , :,
1_' -
(.;
~-:--."' ;
,
~-
1'0
l.
1
• .1
". .. -":
,
... ,...
•
i
0
, ! '
L. •• - ......- ...~.
j
•~.- -.; ••
~
~
, .
! ..
. !
~
,'., ------;'-'•• - -+
)
'. .•. J.
f
"'f'
i: "
!:
-- ---rn -.------*e - -- e J2.
~
I '
:', ": "'- '
. L;~.L~-·t~5·5t· .
0'
,', rl...- '. ••
~
..
~ ,
.
~
..l . . . . . .
i ..
I
."
i'
t-
• .~;
. -
:
~ ~ t:" ~ ~--.
~
"
,
'
.... -,;I
... ;
•
.
, , ' '1"
!
!'
~ , I,
--;
•
r
I,
t"
~
.-
=
e-D/2 t
" -~ ..
'i'
)
'0
o
o
.2
.4
.6.8
1.0
1.2
1.4
1.6
(P~)min/AbrFtux FIGURE 6.36 - REDUCTION FACTOR FOR PEAKING OF BEARING LOADS ON PINS
6-70
!
~~
!
,. ~
_
l
f':·,::
•1
...r. '....
~
~
", .. __ :.. .; ...:-L.. '. ,: :: " . ':: l'
'! ......
,. ~"
. ~J ...~
I
t '"
(..
.. • • •
. 'f
: -... +-~: -.. -;.. ~ ~.. :. ~
'
) •
' )
i
I
-L~ t ~~.T ,:. J~_~.~.~,..L . . · . .:., _:, .........________-' . ,. ,." . ~o,J .
;.
'j
1.8
2.0
STRUCTURAL DESIGN MANUAL
1-18D
FIGURE 6.38 - LUGS WITH ECCENTRICALLY LOCATED HOLES . ~;
,
·':'r~
/
FOR HOLE IN THIS REGION MODIFY P tu
FOR HOLE IN THIS REGION MODIFY Pbru
FOR HOLE IN THIS REGION MODIFY P tu
FIGURE 6.39 - LUBRICATION HOLES IN LUGS
6-73
STRUCTUR.AL DESIGN MANUAL B.
Transversely· loaded lugs. Obtain by 0.9 (1 _ lube hOl~ diwneter).
c.
Obliquely loaded lugs. Obtain P~u' Pbru' and Ptru according to A and B above. Then proceed according to Section 6.4.4.4.
6.4.5
P~ru
neglecting lube hole and mUltiply
Stresses Due to Press Fit Bushings
Pressure between a lug and bushing assembly having negative clearance can be detennined from consideration of the radial displacements. After assembly, the increase in inner radius of the ring (lug) plus the decrease in outer radius of the bushing equals the difference becween the radii of the bushing and ring before assembly.
8=
u. r1ng - u b us h'.lng
where
8 = difference between outer radius of bushing and inner radius of the ring u
= radial displacement, positive away from the axis of ring or busHting ..
Radial displacement at the inner surface of a ring subjected to'interna1 pressure p is D
u =
P
E .
r1ng
Radial displacement at the outer surface of a bushing subjected to external pressure p is B
u == -
E
P
bush
where: A = inner radius of bushing B outer radius of bushing C = outer radius of ring (lug)
D E #
inner radius of ring (lug)
= modulus of elasticity = Poisson's ratio
Substitution of the previous two equations into the first yields:
6-74
')
•
STRUCTURAL DESIGN MANUAL Revision A The ultimate tensile stress in the outer fibers in the lug net section is approximately
where kb is the plastic bending coefficient for the lug net section. The allowable ultimate is found by the methods defined in Section 6.4.4 for axial tension. The bearing stress distribution between bushing and pin is assumed to be similar to that between the lug and bushing. At ultimate bushing load the maximum bushing bearing s tress is approximated by
where kbr, the plastic bearing coefficient, is assumed the same as the plastic bending coefficient for a rectangular section. The allowable ultimate value is Fcy for the bushing material. The maximum value of pin shear can occur either within the lug or at the common shear face of the two lugs, depending upon the value of M/Pt. At the lug ultimate load,the maximum pin shear stress (fs) is approximated by 2
fs = 1.273 P/Dp ; (M/Pt fs
=
1 .. 273 P
~2/3)
2M pt)2 «2M/Pt)+ 1-
+1 - 1 J( 2M/Pt) 2
(M/Pt> 2/3)
The first equation above defines the case where the maximum pin shear is obtained at the common shear face of the lugs. The second equation defineS the case where the maximum pin shear occurs away from the shear face. The allowable ultimate is Fsu of the pin material. The maximum pin bending moment can occur within the lug or at the common shear faces of the two lugs J depending on the value of M/Pt. At the lug ultimate load,the maximum pin bending stress (f bu ) is approximated by 10.19 M
kb Dp3
•
(~~
- 1) ; (M/Pt:s 3/8)
10.19 M kb Dp3
where
~
2M/Pt
(M/Pt > 3/8)
is the plastic bending coefficient for the pin.
6-81
fttt\-\, "-~,,,
STRUCTURAL DESIGN MANUAL
Revision B The equation for (M/Pt ~3/8) defines the case where the maximum pin bending moment is obtained at the corrmon shear face of the lugs and the equation for (M/Pt > 3/8) defines the case where the maximum pin bending moment occurs away from the shear face, where the pin shear is zero. The allowable ultimate value is Fb for the pin or if deflection or fatigue is critical Ftu should be used. U 6.4.8 Socket Analysis The method presented here applies to sockets or sleeves made of aluminum or steel alloys. It is based on the assumption that the socket or sleeve walls (section cut by a plane parallel to the beam or pin centerline) are rectangular or nearly rectangular.
)
The method for obtaining bearing pressures within the socket or-sleeve is also applicable to sockets or sleeves whose wall cross-sections vary appreciably from rectangular. An analysis suitable to the wall configuration must be used for the determination of the wall strengths. This method may also be used for the analysis of single shear lug joints by considering the lug as a socket and the bolt as the beam. The maximum wall strengths of sockets or sleeves having rectangular or nearly rectangular wall cross sections (section cut by a plane parallel to the beam or pin center-line) may be determined from the following equations.
(nIt >10)
t t
I
e
------I------ : ' - t --fo--- -.T --
--------1------ _.f. -.
L--I
the above result in pounds per e D K
in~h
= edge distance of = diameter of beam
socket, inches or bolt, inches = tension efficiency factor, Figure 6.32 K~ = bearing rupture factor, Figure 6.45 F ru= ultimate tensile strength, psi tt~ wall thickness of socket, inch J
6-82
)
•
STRUCTURAL DESIGN MANUAL
6~--~~--~----~----~----~~--,-----~--~
) Flange Buckles , First
Top Web Buckles First
....r---Side Web Buckles First
Top Web Side Web t
.8 1 .0
o oL-----~----~----~----~~--~~--~~----~~--~6 .2 .4 1.
\
-j
F' GURE 7.12
sue KL I
f\G STR ESS FOR HAT SECT ION ST I FFENE RS (t=tf=tkJ=t t)
7-19
STRUCTURAL DESIGN MANUAL
7-------------------------------------------. Buckling or Skin Restrained by Stiffener/<'
Buckling of Stiffener Restrained by Skin
6
t
.
kc '1T 2 E
12(1 - ve 2)
5
(~)
)
2
bs
4
k
c
3.
2.0 1.8
1. '6
2
1.4' 1 .2.
1.0
1
•9 .8 "7 .5' 6 •2
.. 6
.·4
b /b w
7-20
1.0
t
It
..JtiL ... S.
<
2.0
CO~~PRESSIVE LOCAL RIICKlt~r, COEFFICIENTS FOR
WinE InEALIZEn
1.2
s
a) Web Stiffeners. 0.5 < FrGURE 7.13
.8
)
STI~FENED
FLAT PLATES
HIFINITElV
STRUCTURAL DESIGN MANUAL 7.3.3
Crippling Failure of Flat Stiffened Plates in Compression
LJlp
For stiffened plates having slenderness ratios ~20, considered to be short plates, the failure mode is crippling rather than buckling when loaded in compression. The crippling strength of individual stiffening elements is considered in Section 11. The crippling strength of panels stiffened by angle-type ,elements is given by Equation (7-9).
(
E_)~
__1J__ Fey
]
0.85
(7-9)
For more complex stiffeners such as Y sections, the relation of Equation (7-10) is used to find a weighted value of twa =
(7-10)
where ai and ti are the length and thickness of the cross-sectional elements of the stiffener. Figure 7-15 shows the method of determining the value of g used in Equation (7-9) based on the number of cuts and flanges of the stiffened panels. Figure 7-16 gives the coefficient ~g for various stiffening elements •
•
! If the skin material is different from the stiffener material, a weighted value of Fey given by Equation (7-11) should be used. Here 1:is the effective thickness of the stiffened panel. (7-11)
CEI ts ) The above relations assume the stiffener-skin unit to be formed mono·1ithically; that is, the stiffener is an integral part of the skin. For riveted construction, the failure between the rivets must be considered. The interrivet buckling stress is determined as to plate buckling stress, and is given by Equation (7~12).
(7-12)
Values of c, the edge fixity, are given in Table 7-2. After the interrivet buckling occurs, the resultant failure stress of the panel is given by Equation (7-13).
•
(7-13)
7-27
STRUCTURAl'DESIGN MANUAL
I
g=19
g= 18
C 5 cuts 14 flanges 19 = _8
Average g = 18.85
(a) Y - stiffened panel
2 cuts 6 flanges 8 =g
Average g
= 7.83
(b) Z' .. stiIfened 'panel
) 5 cuts 12 flanges 17 g
Average g
= 16.83
=
(e) Hat-stiffened panel
F I GUHE 7 .15
7-28
f1 ETHon OF CUTT I NG ST I FF EN Fn PAN ElS TO n FT EFH~ I NE g
•
STRUCTURAL DESIGN MANUAL Here the value b ei is the effective width of skin corresponding to the interrivet buckling stress Fi The failure stress of short riveted panels by wrinkling can be determined. The following quantities are used: Ffst
crippling strength of stringer alone (see Section 11, Column Analysis)
Fw
wrinkling strength of the skin
Ff
crippling strength of a similar monolithic panel
Ffr
strength of the riveted panel
The wrinkling strength of the skin can be determined from Equation (7-14) and Figure 7-17. Here f is the effective rivet offset distance given in Figure 7-18. This was obtained for aluminum rivets having a diameter greater than 90% of the skin thickness.
(7-14)
Now, based on the stringer stabirity, the strength of the panel can be calculated. Table 7-3 shows the various possibilities and solutions. It is noted that in no case should should be used.
Ffr > Ffe
Thus, the lower of these two values
The use of the coefficient kw is based upon aluminum alloy data for other materials. The procedure is to use Equation (7-15) for the panel crippling strength.
)
17.9
(
( (7-15)
.' 7-31
~ STRUCTURAL DESIGN MANUAL
"~~.,
10~-'~~~~----~----------------~--'
3
2
o ,
FIGlH1E 7.17
.2
.4
EXPEPH'ENTAllY f')ETfR~l1NEf) COEFFIr,IENT~ FOR FAILURE IN
7-32
WRINKLING MOOE
)
STRUCTURAL DESIGN MANUAL Mod~lus
T = Appiied Torque(in-Ib)
G=
of Rigidity(psi)
L = Length of Beam(in)
(/J= Angle
K= Torsional ConstantCin4)
fs == Shear Stress (psi)
of
Twist (rad)
Q= Section Modulus(in3 ) SE-CTJON
CD /
K
G
I
Q
1rr4
11'
r3
~
"2
MAX: STRE.SS
at rmax
SOLID CIRCLE
r@0
@
r-1.
-
4
4
3 3 .J!:(ro-ri)
2!:(ro-ri)
2
2
a~
ro
HOLLOW CIRCLE
Q)
OI
0.208 a 3
O.1406a4
at ll?-idpoil).t of
each side
l--a-.a1
SOLID SQUARE
®-
A
T
B~
)
B
A
.I
r---
b
----1
a
.-l
O!ba2 .
/1ba3
~ = [333 -(~(l- 0iM:>\)]
1
a=
[3+~~
@A: fs=
L
Q
@B-· f s - -Ta Qb
SOLID RECTANGLE
~ B
.~
A
)1'Ii
7r b3
a3
16{b2+ a 2)
'If' b
a2 .
16
"b-4 .~
@A: fs=
~
@B"" {s --Ta Qb
SOLID ELLIPSE TABLE 8.1 - EQUATIONS' FOR STRESS AND DEFORMATION IN SOLID SECTIONS LOADED IN TORSION 8-3
STRUCTURAL DESIGN MANUAL SECTION
®
MAX STRESS
Q
K
a3
a4fi 80
atA,B&C
20
a~ SOLID EQUI.LATERAL TRIANG
o
.
0.1045 d
4
0 .. 1704d
3
at mjdpoint of."
)"
each side
SOLID HEXAGON
®
0.1021 d 4
0.1751d
3
at midpoint of ~ach
side
SOLID OCTAGON Form equivalent rectangle through points Band D. Then use equations for rectangle to determine stress and twist. To locate Band D, construct perpendiculars from centroid (c) to each side (B and D).
SOLID O.0261a4
at center of
long side
SOLID RIGHT ISOSCELES TRIANGLE
7r a
aha(! - q4) a 2+b 2
TABLE 8.1 (CONTtD)
8-4
at A
· EQUATIONS FOR STRESS AND DEFORMATION IN SOLID
SECTIONS LOADED IN TORSION
)
STRUCTURAL DESIGN MANUAL It is possible to determine the volume of the sand heap for any cross section by integration. Figure 8.5 shows equations fo~ sand heap volumes with various bases.
\
)
I
~II
2
- b (3a- b) V12
L---
f..
a
wi
v=. &
3
A== Area of triangl e FIGURE 8.5 - SAND HEAP VOLUMES
8.6 Allowable Stresses For limit load conditions, the applied stresses should be kept below the ultimate shear stress, Fsu. These are defined for various materials in MIL-HDBK-5. The torsional failure of tubes may be due to plastic failure of the material, instability of the walls, or an intermediate condition. Pure shear failure will not usually occur within the range of wall thicknesses commonly used for aircraft tubing. Torsional allowable stresses are shown in Figure 8.6 through 8.22. These curves take into account the parameter LID and are in good agreement with experimental results. Interaction data of Section 4 should be used when other stresses are combined wi th torsi on.
8-15
STRUCTURAL DESIGN MANUAL ,
,
40
.... ~
·0
o o
.... "....
r- LID: 10
-~,
~
~
20
"-
"-
~JIrr... ~
~
jiI ~
=::::
~ ......
~
! - f-l/D::20
Ii.
-r-
L/O=O
~
I I 30 r- I-L/D =5
I
F1u -
,
~/O::2
•
=55 kosi - I -
t ~
I
t
'
......
"- i""oo ......
~
~
L/O:hj
r
--- - -- -
i'"
~
.
t
~
r-..
'""~
I ' ......
-
""
LID: 1/2
"'
" .... ---- -.........
LID:
r--- ~ "-
1M -
,-
I
.--
I
)
I
I
.--
10 1-
0
o
20
10
40
30
50
70
60
80
D/t
FIGURE 8.6 - TORSIONAL MODULUS OF RUPTURE - PLAIN CARBON STEELS F = 55 ksi tu
0 L,\
"
i\.
l
\
o
8 ;;;
6 r.
-
,~t;;
" ""~ ~
"" -~
'"II.' "
"
.....
r- ,.....
~r-..
""",
-... ~
~ ~ i" ~ r" """ r-....
"
t-....
r-... ..... "'""" """'"
4 t"I
--
-r-- ,..... ...
r--- ....
1""-0
.... ~
,--u
lI"-
!oo...
--
l"""- t--
~
,......
r--. ~ ......
-
j
~'-
-
I""" t-
"- r-
"-
...... ......
......
....... ......
r-I-"-r....
1-
-- -
.....
20
30
40
.....
50
FIGURE 8.7 - TORSIONAL MODULUS OF TO F 90 ·,ksi tu
RUI'TU~H-,
1/4
,
r- r- .....
l
10
'-
- --- -------- ---
!" ~ .... ~
,.....
o ~
60
....
"""'"" ~ .....
70
Olt
8-16
0
1-1-
I- ~
o
0"
LID r-
r---. ..... ~ ~
30
- '
I\.
C\ ["
50
',-
F't =90 ksi
- ALLOY STEELS HEAl '!"REATED
1/2
-
1
2
r-
-~
- r~
5 10
r- ~ 20
GO
STRUCTURAL DESIGN MANUAL !
,
9.5 Lateral Buckling of Beams Beams in bending under certain conditions of loading and restraint can fail by lateral buckling in a manner similar to that of columns loaded in axial compression. However, it is conservative to obtain the buckling load by considering the compression side of the beam as a column since this approach neglects the torsional rigidity of the beam. \
)
In general, the critical bending moment for the lateral instability of the deep beam, such as that shown in Figure 9.8 may be expressed as M
= KjEly
cr
GJ
L
where J is the torsion constant of the beam and K is a constant dependent on the type of loading and end restraint. Thus, the critical compressive stress is given by
.)
M c F
cr
cr
I
x
where c is the distance from the centroidal axis to the extreme compression fibers. If this compressive stress falls in the plastic range, an equivalent slenderness ratio may be calculated as Lt
P
1 h
FIGURE 9.8 - DEEP RECTANGULAR BEAM 9-51
\
STRUCTURAL DESIGN MANUAL The actual critical stress may then be found hi entering the column curves of Section 11 at this value of (L·/p). This value of stress is not the true compressive stress in the beam, but is sufficiently accurate to permit its use as a design guide .. 9.5.1 Lateral Buckling of Deep Rectangular Beams The critical moment for deep rectangular beams loaded in the elastic range loaded along the centroidal axis is given by
)
3
M = 0.0985 K E (b h) cr m L where K is presented in Table 9.5 and b, h, and L are as shown in Figure 9.8. The cri~ical stress for such a beam is 2 b Fcr = KfE(Lh) where K is presented in Table 9.5. f
If the beam is not loaded along the centroidal axis, as shown in Figure 9.9, a corrected value K t is used in place of K • This factor is expressed as f f
K
f
t
= Kf
s
(1 - n) (-) L
where n is a constant defined below: (1) For simply supported beams with a concentrated load at midspan, n
(2) For cantilever beams with a concentrated end load, n (3) For simply supported beams under a uniform load, n (4) For cantilever beams under a uniform load, n
0.816. 2.52.
= 0.725.
Note: s is negative if the point of application of the load is below the centroidal axis. centroidal axis
FIGURE 9.9 - DEEP RECTANGULAR BEAM LOADED AT A POINT REMOVED FROM THE CENTROIDAL AXIS 9-52
=
2.84.
STRUCTURAL DESIGN MANUAL 9.6.5
Plastic
Bending Modulus, Fb
Figures 9.15 through 9.18 show curves for various materials. The curves are plotted as k = 2 Qc/I versus Fb and strain. The strain versus Fb curves show f and Fb versus strain. The f curve is at k=l. The rest of the curves e~ploy equation 9.16 to obtain F~ at various strains. 9.6.6 i
I
Application of Plastic Bending
Consider a rectangular beam section which is .25-inch thick and 1.S-inch deep. It is made of 7075-T6 extrusion and it is desired to find the yield and ultimate bending moment for the section. Fb = f k
m
+ f a (k-l), equation 9.16
2Qc/I =
=
2~.25)(.75)(.375~(.75)/~.25)(1.5)3/12)=
1.5
The value of k can also be found in Figure 9.10. F tu
-e)
= 75000 psi, Fty = 65000 psi
Find the yield bending strength: The value of f in equation 9.16, the maximum stress permitted on the most remote fiber, is 65ijOO psi, the yield stress of the material. To find f , go to Figure 9.16 (r) to find the point on the stressstrain curve (k=l) tRat corresponds to a stress of 65000 psi. This point is projected downward to the f curve where a ·stress of 26000 psi is read. Then o Fb
= 65000 + 26000 (1.5 - 1) = 78,000 psi
This same stress can be obtained by ptojecting up from the stress-strain curve in Figure 9.16 (r) to the curve labeled k=1.5 and reading Fbdrectly. The yield moment is then found to be M y
= Fb
I/c = 78000 (.0703)/.75
= 7312.5
The ultimate moment is found the same way_ Fb M u
= 75000 +
70500 (1.5 - 1)
= 110,250
psi
110,250 (.0703)/.75 = 10334
The previous example is for a section which is stable in compression and symmetrical about two axes. Consider now a ,section which is symmetrical about one axis and probably partially unstable. The Tee shown in Figure 9.22(a) is a
9-113
\
STRUCTURAL DESIGN MANUAL eu·024
) e,-.055
FIGURE 9.22 - UNSYMMETRICAL EXAMPLE
7075-T6 extrusion.
Again, the properties of Figure 9.16(r) are used.
First consider the maximum strain, e , in Figure 9.16(r), e = .055in/in. It is apparent that the lower leg of the T~e will strain higher tHan the cap when the Tee is bent about the x axis~ so set the lower extreme fiber strain equal 0.055. By ratioing the lower strain by the distances from the N.A. the strain 'in the upper extreme fiber is e = (.609/1.391)(.055) = .024 in/in. u
Equation 9.16 was derived for symmetrical sections about a neutral axis. The Tee can be made into two sections which are symmetrical about their neutral axis. These are shown in Figure 9.22(c) and (d).
)
Figure 9.22(d) shows how the lower portion is made symmetrical about the neutral axis by adding the shaded portion above. The internal bending resistance is found for the entire section in 9.22(d). One-half of this amount will be the true moment developed by the lower portion. .
I
= (.1)(2.782)
3
/12
= .179
I/o = .179/1.391 = .129 k = 2 Qc/r = 2(1.391)(.1)(.6955)/.129
= 1.5
From Figure 9.16(r) at e = .055, Fb = 110,000 psi = Fb (1/0)(1/2) = 7095 in-1b
M
The 1/2 is because, only one-half the beam section is used in Figure 9.22(a).
,
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Revision F
STRUCTURAL DESIGN MANUAL VOLUME II
RESTRICTED DISCLOSURE NOTICE THE DRAWINGS, SPECifiCATIONS. DESCRIPTIONS, AND OTHER TECHNICAL DATA ATTACHED HERETO ARE PROPRIETARY AND CONFIDENTIAL TO BEll HELICOPTER TEXTRON INC. AND CONSTITUTE TRADE SECRETS fOR PURPOSES OF THE TRADE SECRET AND FREEDOM OF INFORMATION ACTS. NO DISCLOSURE TO OTHERS, EITHER IN THE UN.TED STATES OR ABROAD. OR REPRODUCTION OF ANY PART OF THE INFORMATION SUPPliED IS TO BE MADE, AND NO MANUFACTURE, SALE, OR USE OF ANV INVENTION OR DISCOVERV DISCLOSED HEREIN SHALL BE MADE, EXCEPT BV WRITTEN AUTHORIZATlON OF BELL HEliCOPTER TEXTRON INC. THIS NOTICE WILL NOT OPERA TE TO NULlIFV OR LIMIT RIGHTS GRANTED BY CONTRACT. THE DATA SUBJECT TO THIS RESTRICTION IS CONTAlNED IN ALL SHEETS AND IS DISCLOSED TO PERSONNEL OF BELL HELICOPTER TEXTRON INC. FOR THE PURPOSES(S) OF INTERNAL USE AND DISTRIBUTION ONLY.
Bell Helicopter Li il i hI·] : I
STRUCTURAL DESIGN MANUAL
This Structural Design Manual is the property of Bell Helicopter Text-Ton:
•
and no pages are to be added or withdrawn except as direct.ed by revision notices. not
b~
Information contained herein which is
specif~cal1y
identified may
reproduced or further disseminated without t.he approval of
of Structural Technology, Bell Helicopter Textron.
th~
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)
•
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STRUCTURAL DESIG·N. MANUAL Revision F
TABLE OF CONTENTS VOLUME I
REFERENCES INTROOUCTION
xiii
xv
SECTION 1 - PROCEDURES General Design Coverage Critical Parts Flight Safety Parts Check List for Drawing Review Envelope, Source Control, and Specification Control Drawings Stress Analysis Structures Report Introductory Data Body of the Report General Information Structural Information Memos Purpose Preparation Published SIM's
1. 1
1.2 1.2.1 1.2.2 1 . 2.3 1. 2.4 1.3
•
1.4 1.4.1 1. 4.2 1.4.3 1. 5
1.5.1 1.5.2 1.5.3
1-1 1-1
1-1 1-2 1-2 . 1-5 1-6 1-8
1-8 1-9
1-10 1-11 1-11 1... 11 1-11
SECTION 2 - COMPUTER PROGRAMS General
2. 1 2.2 2.3 2.3.1 2 .. 3 .. 2
)
Facilities Finite Element and Supplementary Programs Finite Element Programs Fin;t~ Element Preprocessors and Postprocessors Approved Structural Analysis Methodology (ASAM) The LOOAM System The CASA System CASA System Features CASA System Architecture CAS A Programs Description The CPS2TSO System Other Programs Miscellaneous Programs Load Programs Unrelated to Finite Element Analysis Dynamic Structures Analysis Fatigue Evaluation C~mputer
2.4 2.4 .. 1 2.4 .. 2 2 . 4.2.1 2.4.2.2 2.4.2 . 3 2.4.3 2.4.4 2.5 2.5.1 2.5.2 2.5.3
2-1 2-1 2-1 2-1 2-2 5
2-5 2-5 2-6
2-7 2-9 2-12
2-16 2-17 2-17
2-17 2-17
SECTION 3 - GENERAL 3.1 3.1.1
Properties of Areas Areas ~nd Centroids
3-1 3-1 iii
/_
•
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STRUCTURAL DESIGN MANUAL,
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TABLE OF CONTENTS (Continued)
SECTION 3 - GENERAL (Continued) 3.1.2 3.1.3 3.1.4 3.1.5 3.1.6 3.1.7 3.2 3.3 3.4 3.5 3.6 3.' 7
3.8 3.9 3.9 .. 1
3.9 . 2 3.10 3.11 3.11.1 3.11.2 3.12 3.12.1 3.12.2 3 . 12.3 3.12.4 3.12.5 3.12.6 3.13 3.13.1 3.13.2 3.13.3 3 .. 13.4 3.14
Moments of Inertia Polar Moment of Inertia Product of Inertia Moments of Inertia About Inclined Axes Principal Axes Radius of Gyration Mohr's Circle for Moments of Inertia Mass Moments of Inertia Section Properties of Shapes Bend Radii Hardness Conversions Graphical Integration by Scomeano Method Conversion Factors The International System of Units Basic SI Units Symbols and Notation Weights Shear Centers Shear Centers of Open Sections Shear Center of Closed Cells Strain Gages The Wire Strain Gage The Foil Strain Gage The Weldable Strain Gage The Strain Gage Rosette Strain Gage Temperature Compensation Stress Determination from Strain Measurements Acoustics and Vibrations Uniform Beams Rectangular Plates Columns Stress and Strain in Vibrating Plates Bell Process Standards
SECTION '4.1 4.2 4.3 4. 4 4.4.1 4.4 . 2 4.5 4.5.1 iv
3-2 3-3 3-5 3-6
3-7 3-7
)
3-8 3-9
3 10 3-36 3-36 3-36 3-45 3-45 3-45 3-45 3-52
3-52 3-53 3-68
3-70 3-70 3-71 3-71 3-72 3-73 3 73 3-76 3-79
•
3-81
3-81 3-84 3-85
)
4 - INTERACTION
Material Failures Theories of Failures Determination of Principal Stresses Interaction of Stresses Procedure for M.S. For Two Loads Acting Procedure for M.S. for Three Loads Acting Compact Structures Biaxial Stress Interaction Relationships
4-1 4-2 4-3 4-5 4-11 4-11 4-12 4-12
•
STRUCTURAL DESIGN MANUAL Revision F TABLE OF CONTENTS (Continued)
SECTION 4 - INTERACTION (Continued) 4.5.1.1 4.5.1.2
)
4.5.1~3
4.5.2 4.5.3 4.5.4 4.5.5 4.5.6
Max Shear Stress Theory Interaction Equations Octahedral Stress Theory Interaction Equations M.S. Determination Uniaxial Stress Interaction Relationships Thick Walled Tubular Structures Unstiffened Panels Unstiffened Cylindrical Shells Stiffened Structures
4-13 4-13 4-17 4-19 4-19 4-19 4-20 4-20
SECTION 5 - MATERIALS 5. 1 5.1.1 5.1.2 5.1.3 5.2. 5.2.1 5.2.2 5.2.3 5.3 5.3 .. 1 5.3.2 5 . 3.3 5.3.4 '5.3.5 5 .. 4
5.4 . 1 5.4 . 2 5.4.3
)
5.5
5.6 5.7 5.7.1 5.7.2
General Material Properties Selection of Design Al10wables Structural Design Criteria Material Forms Extruded, Rolled and Drawn Forms Forged Forms Cast Forms Aluminum Alloys Basic Aluminum Temper DeSignations Aluminum Alloy Processing Fracture Toughness of Aluminum Alloys Resistance to Stress-Corrosion of Aluminum Alloys Mechanical Properties of Aluminum Alloys Steel Alloys Basic Heat Treatments of Steel Fracture Toughness of Steel Alloys Mechanical Properties of Steel Alloys Magnesium Alloys Titanium Alloys Stress-Strain Curves Typical Stress-Strain Diagram Ramberg-Osgood Method of Stress-Strain Diagrams
5-1.
5-1 5-1 5-3 5-3 5-3 5-5 5-6 5-8 5-8 5-9
5-13 5-13 5-19 5-21 22 5-23 5-24 5-24 5-28 5-29 5-29 5-31
SECTION 6 - FASTENERS AND JOINTS 6.1 6.2 6.2.1 6.2.2 6.2.2.1 6.2 . 2.2 6 . 2 . 2.3
General Mechanical Fasteners Joint Geometry Mechanical Fastener Allowables Protruding Head Solid Rivets Flush Head Solid Rivets Soljd Rivets in Tension
6-1 6-1 6-2 6-3 6-3 6-3 6-6
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STRUCTURALDE,SIG,N MANUAL
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Revision F TABLE OF CONTENTS (Continued)
SECTION 6 - FASTENERS AND JOINTS (Continued) 6.2.2.4 6.2.2.5 6.2.2.6 6.2.2.7 6 .. 2.2.8 6.3 6.3.1 6.3 .. 2 6.3.3 6.3.4 6 . 3.5 6.4 6.4.1 6.4 .. 1.1 6.4.1.2 6.4.1.3 6.4.1.4 6.4.1.5 6.4.2 6.4.2.1 6.4.2.2 6.4.3 6.4.3.1 6.4.3.2 6 .. 4.4 6 . 4.4.1 6.4.4.2 6.4.4 . 3 6.4.4.4 6.4.4.5 6 .. 4.4.6 6.4.4.7 6.4.5 6 . 4.6 6.4 .. 7 6 .. 4.8 6.4.9 6.4.10 6.5
Threaded Fasteners Blind Rivets Swaged Collar Fasteners Lockbolts Torque Values for Threaded Fasteners Metallurgical Joints Fusion Welding - Arc and Gas Flash and Pressure Welding Spot and Seam Welding Effect of Spot Welds on Parent Metal Welding of Castings Mechanical Joints Joint Load Analysis One-Dimensional Compatibility Constant Bay Properties - Rigid She~ts Constant Bay Properties - Rigid Attachments The Influence of Slop Two-Dimensional Compatibility Joint Load Distribution - Semi Graphical Method Fastener Pattern Center of Resistance Load Determination Attachment Flexibility Method I - Generalized Test Data Method II - Bearing Criteria Ltjg DeSign Nomenclature Analysis of Lugs with Axial Loads Analysis of Lugs with Transverse Loads Analysis of Lugs with Oblique Loads Analysis of Pins Lugs with Eccentrically Located Hole Lubrication Holes in Lugs Stresses Due to Press Fit Bushings Stresses Due to Clamping of Lugs Single Shear Lug Analysis Socket Analysis Tension Fittings Tension Clips Cables and Pulleys
6-6 6-10
6-10 6-11' 6-11 6-15 6-15 6-15 6-15 6-15
6-18 6-19 6-19 6-19 6-41 6~42
6-42 6-48 6... 50 6-50 6-51 6-52 6-52 6-54 6-55 6-56 6-57 6-58 6-60 6-60 6-71 6-71 6-74 6-79 6-79 6-82 6-87 6-93
6-96
SECTION 7 - PLATES AND MEMBRANES 7.1 7.2 vi
Introduction to Plates Nomenclature for Analysis of Plates
7-1 7-1
)
STRUCTURAL DESIGN MANUAL Revision F TABLE OF CONTENTS (Continued)
SECTION 7.3 7.3 . 1 7.3.2 7.3.3 7.4 7.4.1 7.4.2 7.5 7.6 7.7 7.8 7.8.1 7.8.2 7.9 7.10 7.11 7.12 7 .. 13 7.14 7.15
7 - PLATES AND MEMBRANES (Continued)
Axial Compression of Flat Plates Buckling of Unstiffened Flat Plates in Axial Compression Buckling of Stiffened Flat Plates in Axial Compression Crippling Failure of Flat Stiffened Plates in Compression Bending of Flat Plates Unstiffened Flat Plates, In-Plane Bending Unstiffened Flat Plates~ Transverse Bending Shear Buckling of Flat Plates Axial Compression of Curved Plates Shear Loading of Curved Plates Plates Under Combined Loadings Flat Plates Under Combined loadings Curved Plates Under Combined loadings Triangular Flat Plates Buckling of Oblique Plates Introduction to Membranes Nomenclature for Membranes Circular Membranes Long Rectangular Membranes Short Rectangular Membranes
7-2 7-3 7-15 7-27 7 34 7 34 7-37 7-50 7-50
7-58 7 58 7-58 7-58 7-70 7-70 7-14 7-74 7-74 7-76 7 79
SECTION 8 - TORSION Torsion of Solid Sections Torsion of Thin-Walled Closed Sections Torsion of Thin-Walled Open Sections Multicell Closed Beams in Torsion Plastic Torsion Allowable Stresses
8 .. 1 8.2 8.3 8.4 8.5 8.6
8-1 8-2
a-a
8-11 8-14 8-15
SECTION 9 - BENDING 9.1
9.2 9.2.1 9.2.2 9.2.3' 9.3 9.3.1 9.3.29.3.3 9.3.4 9 .. 3 .. 5
9.3.6 9.3.7
General Simple Beams Shear, Moment and Deflection Stress Analysis of Symmetrical Sections Stress Analysis of Unsymmetrical Sections Strain Energy Methods Castigliano's Theorem Structural Deformation Using Strain Energy Deflection by the Dummy Load Method Analysis of Redundant Structures Analysis of Redundant Built-Up Sheet Metal Structures Analysis of Structures with Elastic Supports Analys)s of Structures with Free Motion
9-1 9-1 9-1 9-23 9-24 9-25 9-25 9-25 9-27 9-36 9-40 9-46 9-46
vii
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Revision f TABLE OF CONTENTS (Continued)
SECTION 9 - BENDING (Continued) 9.4 9.5 9.5.1 9.5.2 9.6 9.6.1 9.6.2 9.6.3 9.6.4 9.6.5 9.6.6 9.7
9.8 9.9
Continuous Beams by Three-Moment Equation Lateral -Buckling of Beams Lateral Buckling of Deep Rectangular Beams Lateral Buckling of Deep I Beams Plastic Analysis of Beams Bending About an Axis of Symmetry Bending in a Plane of Symmetry Complex Bending Evaluation of Intercept Stress, fo Plastic Bending Modulus, Fb Application of Plastic Bending Curved Beam Correction Factors for Use in Straight Beam Formula Bolt-Spacer Combinations Subjected to Bending Standard Bending Shapes
9-49 9-51 . 9-52 9-54 9-"':58
9-58 9-105 9-105
9-107 9-113 9-113 9-116 9-118 9-118
)
viii
STRUCTURAL DESIGN MANUAL Revision F TABLE OF CONTENTS VOLUME II
SECTION 10 - BUCKLING
)
•
10.1 10.1 .. 1 10.1.2 10.1.3 10.2 10.2.1 10 . 2 . 2 10.3 10 . 4 10 .. 5
10 . 5.1 10.5.2 10.5.3 10.5.4 10.5.5 10 . 5.6 10.5.7 10.5.8 10.6
10.7 10.8 10.9
Shear Resistant Beams Introduction Unstiffened Shear Resistant Beams Stiffened Shear Resistant Beams Shear Web Reinforcement for Round Holes Doubler Reinforcement 45~ Flange Reinforcement Shear Beams with Beads Shear Buckling Incomplete Diagonal Tension Effective Area of Uprights Moment of Inertia of Uprights Effective Column Length Discussion of the End Panel of a Beam Analysis of a Flat Tension Field Beam Uprights Analysis of a Flat Tension Field Beam Uprights Analysis of a Flat Tension Field Beam Uprights and Access Holes Analysis of a Tension Field Beam with Inter-Rivet Buckling Compressive Crippling Effective Skin Width Joggled Angles
10-1 10-1 10-1 10-5 10-13 10-13 10-17 10-17 10-20 10-20 10-21 10-21 10-22 10-22
with Single 10-22 with Double 10-37
with Single Curved Panels
10-42 10-49 10-60 10-69 10-73 10-75
SECTION 11 - COLUMNS AND BEAM COLUMNS 11.1 11.1.1 11.1.2 11 .. 1.3
Simple Columns Long Elastic Columns Short Columns Columns with Varying Cross Section 11 .. 1 .. 4 Column Data for Both Long and Short Columns 11.2 Beam Columns 11.2.1 Beam Columns with Axial Compression Loads 11.2.2 Beam Columns with Axial Tension Loads 11.2.3 Multi Span Columns and Beam Columns 11 . 2.4 Control Rod Design 11 . 2.5 Beam Columns by the Three-Moment Procedure 11 . 3 Torsional Instability of Columns 11.3 . 1 Centrally Loaded Columns 11.3.1 . 1 Two Axes of Symmetry
11-1
11-3 11-3 11-18
11-33 11-72 11-72 11-79 11-79 11-100 11-100 11-101 11-101 11-101
ix
STRUCTURAL, DESIGN MANUAL TABLE OF CONTENTS (Continued)
SECTION 11 - COLUMNS AND BEAM COLUMNS (Continued) 11.3.1.2 11 . 3.1.3 11.3.2
General Cross Section Cross Sections with One Axis of Symmetry Eccentrically Loaded Columns
11-102 11-102 11-109
)
SECTION 12 - FRAMES AND RINGS 12.1
12.2 12 .. 2.1 12.2.2 12.2.3 12.3 12.4 12.4.1 12 . 4.2 12.4.3 12.4.4
General Analysis of Frames by Moment Distribution Sign Convention Moment Distribution Procedure Sample Problem Formulas for Simple Frames Analysis of Rings Analysis of Rigid Rings with In-Plane Loading Analysis of Rigid Rings with Out-of-Plane Loading Analysis of Frame Reinforced Cylindrical Shells Frame Analysis by the Dummy Load Method
12-1 12-1 12-1 12-2 12-5 12-7 12-22 12-22 12-53 12-53 12-91
SECTION 13 - SANDWICH ANALYSIS Materials 13.1 13.1.1 Facing Materials 13.1.2 Core Materials Adhesives 13.1.3 13.2 Methods of Analysis 13.2.1 Wrinkling of Facings Under Edgewise Load 13.2.1.1 Continuous Core Honeycomb Core 13.2.1.2 Dimpling of Facings Under Edgewise Load 13.2.2 13.2 . 3 Flat Rectangular Panels with Edgewise Compression 13.2.4 Flat Rectangular Panels Under Edgewise Shear 13.2.5 Flat Panels Under Uniformly Distributed Normal Load 13.2 .. 6 Sandwich Cylinders Under Torsion 13.2.7 Sandwich Cylinders Under Axial Compression 13.2.8 Cylinders Under Uniform External Pressure. 13.2.9 Beams 13.3 Attachment Details 13.3.1 Edge Design 13 . 3.2 Doublers and Inserts 13.3.3 Attachment Fittings
13-1 13-3 13-4 13-7 13-15 13-16 13-17 13-17 13-17 13-21 13-43 13-48 13-62 13-65 13-77 13-80
13-101 13-101 13-103 13-103
SECTION 14 - SPRINGS 14.1 x
Abbreviations and Symbols
14-1
)
STRUCT'URAL DESIGN MANUAL Revision F TABLE OF CONTENTS (Continued)
SECTION 14 - SPRINGS (Continued) 14.2 14.2 .. 1 14 .. 2.. 2 14 .. 2.3 14.2.4 14.2.5 14.2.6 14.2.7 14.2.8 14.2.9 14.2.10 14.3 14.3.1 14.3.2 14.3.3 14.3.4 14.4 14.4.1 14.4.2 14.4.3 14.4 . 4 14 . 4.5 14.4 . 6 14.5 14.5.1 14.6 14.7
)
14.7.1 14.7.2 14.7.3 14.7 .. 4 14.8 14.8.1 14.8.2 14.8.3 14.8.4 14.8.5 14.8.6
Compression Springs Design Formulas Buckling Helix Direction Natural Frequency, Vibration and Surge Impact Spring Nests Spring Index (Old) Stress Correction Factors for Curvature Keystone Effect Design Guidelines for Compression Springs Extension springs Design Formulas End Design Initial Tension (Preload) Design Guidelines for Extension Springs Torsion Springs Design Formulas End Design Change in Diameter and Length Helix of Torsion Springs Torsional Moment Estimation Design Guidelines for Torsion Springs Coned Disc (Belleville) Springs Design Formulas Flat Springs Material Properties Fatigue Strength Other Materials Elevated Temperature Operation Exact Fatigue Calculation Spring Manufacture Stress Relieving Cold Set to Solid Grinding Shot Peening Protective Coatings Hydrogen Embrittlement
14-2 14-4 14-4 14-7 14-7 14-8 14-8 14-8 14-8 14-9 14-9 14-10 14-10 14-10 14-12 14-13 14-15 14-15 14-15 14-17 14-18 14-18 14-18 14-20 14-20 14-22 14-22 14-22 14-26 14-25 14-27 14-28 14-28 14-30 14-30 14-30 14-31 14-31
SECTION 15 - THERMAL STRESS ANALYSIS 15.1 15.1.1 15.2
Strength of Materials Solution General Stresses and Strains Uniform ~eating
15-1 15-2 15-3
xi
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SECTION 15 - THERMAL STRESS ANALYSIS (Continued) 15 . 2.1 15.2.2 15.2.3 15.2.4 15.2.5 15.2.6 15.3 15.3 . 1 15.3.2 15 .. 4
15.4.1 15 . 4.2 15 . 5 15.6 15.6.1 15.6.2 15.6.3 15.7 15.7.1 15.7.2 15.8
Bar Restrained Against Lengthwise Expansion Restrained Bar with a Cap at One End Partial Restraint Two Bars at Different Temperatures Three Bars at Different Temperatures General Equations for Bars at Different Temperatures Non-Uniform Temperatures Uniform Thickness Varying Thickness Linear Temperature Variations Restrained Rectangular Beam, Uniform Face Temperatures Pin-Ended Beams Combined Mechanical and Thermal Stresses Flat Plates Plate of General Shape Square Plates Flat Plates with Uniform Heating Temperature Effects on Joints Preload Effects Due to Temperature Thermally Induced Loads in Material Thermal Buckling
15-3 1
3
1 4 15-4 15-5 15-5 15-5 15-6 15 6 15-6
15-7 15-7 15-8 15-8 15-9 15-9 15-9 15-12 15-12 15-13 15-15
,)
xii
STRUCTURAL DESIGN MANUAL SECTION 10 BUCKLING
to.
I
JO.'1.J )
SIIEAR RESISTANT BEAMS
Introduction
For shear resistant beams in bending, the simplifying assumption that all the mass is concentrated at the centroids of the flanges may be made if the web is sufficiently thin. The simple beam formulas may then be reduced to
fs =
•
10. I
for shear.
10.2
The bending is, therefore, resisted by the f]anges t and the shear is resisted by the webs. Figure 10. 1 may be used to determine If the pane1 in question 1s shear resistant or a state of Incomplete diagonal tension is developed. An estimate of efficient stiffener size and minimum stiffener moment of inertia are presented in Figur('s 10.2 and 10.3. to.I.2
)
v q ht' =T
for bending, and
Unst1[[ened Shear Resistant Beams
Failure checks must be made for both the web and flanges of the beam. The flange is usually considered to have failed if its bending stress exceeds the yield stress of the material, unless some permanent set is allowed. The allowable average stress at the ultimate load Fs is either 85% of the ultimate strength in shear or 12570 of the yield strength in shear, if the web is not subject to collapse. For thin webs (h!t >60), initial buckling does not cause collapse. The collapsing stress for two aluminum alloys is given in Figure 10.4. The required web thickness is t ::::
V
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FIGURE 10.1 -
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7075-T6 SHEET
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..
.
FIGURE 10.3 - MINIMUM STIFFENER MOMENT OF INERTIA FOR SHEAR RESISTANT WEBS 10-4
\,
.
1\
1-
.
.
o N
o
........
o
•
/
(/;T\'\'~
._11 STRUCTURAL DESIGN MANUAL
\-\
•
•• ,oe""""
:-..: I
\ 44
READ RIGHT SCALR
READ
42
22
LEFT SCALE
40
20
38
18
36 .
F
)
seD II
ksi
[~
7075
16
34
14
32
12
f~O
)0 t=.015
28
6
24
4
22
2
ks1
0
80
40
()
120
160 200 h height -,..., t web thickness
FIGURE lO.4 - COLLAPSING SHEAR STRESS, F 10.1.3
scol]
8
26
20
•
F
240
scoll t
280 hi;
FOR SOLID WEBS
Stif[ened Shear Resistant Beams
The vertica1 stiffeners of a shear resistant beam increase the web buckling stress. They resist no con~resslve load, but divide the web into smaller unsupported rectangles. An analysis of the flange,web and rivets is required. The yielding or ultimate strength of the flanges must be checked by using equa tion 10. 1.
)
In addition to the strength of the web panel, stability must aJso be checked. The strength of the web may be checked by using equation 10.2. Stability of the web may be checked by using the below equation in conjunction wi th Figures 10'.5 through 10. 11. F
-scr - - KsE 11 where
•
F
scr
1}
and
K
(~ )
2
critical buckling stress of the web plasticity coefficient
s
critical shear stress coefficient K = f(d/h t cngc restraint) s
10-5
~.--T., ;.~
i/' "\\\
\~F.'.~"" STRUCTURAL DESIGN MANUAL '-:: ,-.1/..;/ ---~
30.0
20.0 15.0
10.0
~
__~__~__+-__+-~~~~4-~--T+~~~--~~r-~
8.0
~--~--r---+---~+-~+-rr+-~--r+-+~~--r+--r---;
3. 0
f.--:---~-+---~~+----f-+---1--
2. 0
1---:---;-
)
• DESIGN CUTOFF
) .21--~1--+-~~~~4-~~--~--~--~--~----~----~
FOR EFFICIENT DESIGN Iu/ hl ' SHOULD BE BELOW CUTOFF
.15~~~~~~~~~--+r-4-r~~~~---,,---,---~--~
.10L-~LL~~--J-~~~~~~~--~--~~~----~--~~
o
1
2
3
4
<)
6
7
8
10
FIGURE lO.5 - CRITICAL SHEAR STRESS COEFFICIENT, Ks 10-6
j
I
12
•
STRUCTURAL DESIGN MANUAL Revision B Fscr rJ
(b)
03X30 28
26
24 20
)
18 16 14 12 10 8
30x106 28 26
26XI04 24
24
22 20 18 16 14 12 10
22 20 18
16 14 12 10
8
/'
6
6,
4 2
4 .......... .. 01 "::02,.03
o
•
E
(a)
./
'-
.002..004 JJ
(a) i (b) d 6 .008 ..01 ....... ' -
Ks
1 2 345 I
I
I
8 6
./
10/
Ilf~J11
/ /' ./
Fscr -= 11
(t)2 KsEd
)
FIGURE 10.6 - NOMOGRAPH FOR CRITICAL BUCKLING STRESS
• 10-7
,/
,./
~'
STRUCTURAL DESIGN MANUAL 90
I ITT T 180-200 KSI
85
75
/
65
if
60
/
/
55
/ /
/
50
V
b
45
71
40
~I
35
;/
) ~/
30
/
/
/
\
~~ V
~
--
.-- ---
-- -
125 KSI
~
-
,./
1,100 KS1
lS-- ~ r--
/ /
150 KSI
~
/ V
1/1
scr
25
\
I
/
~V
l,...-' ~
/
1/
/
70
KSI
1\
/
80
F
-
/'
/
v .,-
,/'
i.-""
V
"..
......... /
~ L.---NORMAL I ZED '- ~
,/
// t:/
1/
2°20
30
)
40
60
80
100
200
300
400
600
F
~ - KSI 11
FIGURE 10.7 - Fscr, VS. Fscr /ij ,
10-8
FOR ALLOY STEEL
•
/;11' ';,' \\
\{~:~~,.. STRUCTURAL DESIGN MANUAL 75
/
70
/ /
FULL HARD
65
)
\/;
3/4 HARD 55
VI
IV
45
1/ I
40
/~ / /
seT
KSI 35
/; 30
,/
~
/
V VV
l5
LO ,,/' ~
----6
5 5
8
~~
~
~/ V I---
L---- f---
./
./
/'
V
V
/
,/
/""
/
/
HARD
V /'
V
,/'/ V
I
V '/
f.- I--
I
~/
V j~ V V
20
•
/
1/
25
.)
/
V
r./l:i
V
50
•
/"'
/ V/ \/
60
F
:/
V
V'
../
/
V~ HARD
V
- --
i--
....- -
--c
~
-
I--- f----
-
ANNEALED
:,...-'
10
IS
30
20
F
40
60
80
100
200
400
scr _ KSI 'f1
FIGURE 10.8 - F
scr
VS. F
scr
/'fl
FOR STAINLESS STEEL
10-9
STRUCTURAL DESIGN MANUAL 40 35
30
25
j
--
20 F
scr
)
15 10
o
/
/
/
~
/
~
/
~
-
CLAD SHEET
2024-T3 2024-T4
CLAD SHEET
ALUMINUM J
'1
o
)0
20
30
40
50
60
70
80
90
JOO
Fscr/lJ 16 14
12
F
1/
10
scr
KSI
8 6
I
0
/
/
/ o
FSI-H24 SHEET
MAGNESIUM
)
/
\
5
10
15
FIGURE 10.9 - F
10-10
L---....---Ie ~
•
J
4
2
•
~
/
KSI
5
-~17075-T6
~
scr
20
VS. F
scr
25
]0
35
40
45
50
IlJ FOR ALUMINUM AND MAGNESIUM
•
STRUCTURAL DESIGN. MANUAL
20 .....--_--.---
•
Ii'
scr
KSI
5
) 1-+---i----4--+---I---4--+---+
o
•
20
40
60
FIGURE 10.lO - Fscr VS. Fscr I~
I MINIMUM
80
GUARANTEED
100
120
140
160
FOR 6061-T6 SHEET AND PLATE
10-11 '
If/F' \'\\
~~\~~" STRUCTURAL DESIGN MANUAL
• 12
~
10
/
8 i
6
F
scr
-- ---
~
/
4
/
KSl
2
o / o
/
/
V
~
---
...--
)
/
I
/
/
~1
-
5'.
25
20
15
10
F
FIGURE 10.11 - F
scr
VS. F
scr
/.,.,
•
MINIMUM GUARANTEED ]
scr
30
35
40
/.,.,
.
FOR 356-T6 SAND CASTINGS
)
10-12
•
s-rRUCTURAL DESIGN MANUAL K
~
I
(I
Is related to hand
3U ht
in Figure 10.5.
The moment of inertia of the upright. Iu, should be calculated about the base of the stiffener (stiffener-web connection). Knowing Ks , Fscr may be found from Figure 10.6. The critical buckling stress of the web Figures 10.7 through 10.11.
)
Fsc~
is then obtained from
SHEAR WEB REINFORCEMENT FOR ROUND HOLES
10.2 10.2.1
Doubler Reinforcement
The thickness of the reinforcing doubler may be obtained through the use of the equations below. The figure below defines the variables used in the equations.
q
q
) FIGURE 10.12 - SHEAR WEB REINFORCEMENT Rl
R+~ 2
D
q
Web shear flow
F
fb
Bending stress
f
W
= Doubler
width
=
2R
tu
= Ultimate
t
= Tensile
stress
td
= Doubler
thickness
tensile strength
10-13
STRUCTURAL DESIGN MANUAL The stresses at section A-A are: f
b
=
Moment Section Modulus
2q ( • 25R) (R ) l
t(t)
The stress interaction, assumed at failure, is:
Therefore,
q
)
FIGURE 10.13 - RIVET PATTERN LOAD
10 .. 14
STRUCTURAL DESIGN MANUAL Revision C For flanged doublers the total thickness, t web Figure 10.14.
ttot
~t======::J
L d
t
+ t d , may be obtained from
T
-talDJ.-
.36
/
.. 32
/
.28 ~'T:J
.24
~~
~ .20
V
Q).
!
.16
note 2/
II
"~ (
~ .. 12
o
~ ~
.08 .04
V
1
/
V
~3~\4'/5
~2
V
/
V
~v
/ /
6 DJin.
V
7
/
/ ./
~~ V
B
/
V
9
10
FIGURE 10.14 - TOTAL THICKNESS FOR FLANGED DOUBLERS
)
Note 1:
Not~
2:
The rivet pattern is to he uniform~ and develop a running load strength per inch (between the tangent lines) of
In this region, increase ttot to correspond to h
f
=
0 for the same D and omit the flange.
10-15
NOTE: The limits of the curve for Tall: 50
< hI t < 300
.15
cIt < 300
For cIt < 60, use correction factor, K, at right T
cor
:; K T
K
all
.2
(I)
.....
SECTION A-A
::::a
= .....
n
c:
::a
:D-
r-
..., CJ
en
-
C i)
z
:I:
:DZ
c::
:Dr.2
.3.4
.5.6
nIb
e
e
FIGURE 10.15 - ALLOWABLE SHEAR STRESS FOR 2024 WEBS WITH CIRCULAR HOLES HAVTNG 45° FLANGES
,
.~
.8
STRUCTURAL DESIGN MANUAL 10.2.2
4,)° Flan2c Reinforcement H
AllowabLes for panels loaded by pure shear (no addition bending forces) are given in Figure 10.15. Limited available data indicates that beaded lightening panels are more efficient than flanged panels. (Reference NACA RS No. 4B23, "Tests of Beams with Large Circular Lightening Holes".) 10.3
)
SHEAR WEBS WITH BEADS
Beaded panels are one type of non-buckling shear webs. Stiffeners must be added at load points to prevent premature collapse. Since the collapsing stress is only slightly' higher than the buckling stress, the buckling stress is considered the ultimate allowable. The critical shear stress T for a beaded web can be cr expressed as:
where K
5
K
l
Simply supported flat sheet, shear'buckling~onst~nt based on alb from Figure 10.18.
= Beaded-web
shear buckling coefficient'obtained from Figure 10.17.
Figure 10.17 is based on test results obtained from 2024-T4 clad panels ·with a bead spacing of 2 to. 5 inches, panel heights of.7 to,,12... .il1ches, and gages of 0.032 to 0.064 inches. It is suggested that. above the--· proporti.onal limit Tcr.:be reduced. by_ the factor Gt/G.
)
r"
a
,
I
I
-..J
~i
d ~
-r h
~
~ T-
~
If --r
FIGURE 10.16 - GEOMETRY OF BEADED WEBS
10-17
STRUCTURAL DESIGN MANUAL
• I
7
)
6 5
K 4 L
3
2
o .. 02
.04
.06- .08 -.1
.2
FIGURE LO.17 - BEADED WEB SHEAR BUCKLING CONSTANT
• 10-18
.3
STRUCTURAL DESIGN MANUAL Revision E
c\ c s
=
®
(3)
G)
c
c
s
s
9.2
9.2
c
c.
c\
\s
c
l
s
s
clamped
= simple
I
5.8
s
= long side = short side
a b
supported
Is
5.5
15 ~ef
)
(D \ \
14
,
13
TN
37 Bl
I
1\
12
1\\ \ \' ~, \ , \ ~ 'Or"
11
•
. Nk\CA
~3J
10
Ks
' " 1\ (2)
m\
9
'~
'\,
8
-
~ :::Ii ~
,
~
~
,,~
I\.
'\
7
::""'0...
.......... ~
"
~
6
" ~ f"'... ........
~~
~
""""
r-- r-- r--
r-- ~
.)
-
I'
5
o
1
2
3
4
5
6
PANEL ASPECT RATIO, alb FIGURE 10.18 - SHEAR BUCKL1NG COEFFICIENT, K
s
10-19
,.
STRUCTURAL DESIGN MANUAL Revision A 10.4
SHEAR BUCKLING
The critical shear stress at which a plate first buckles is given by the equation: K T
2
s
1r
l1E
cr
(b )2 t
where Ks (Fig. 10.18) is the non-dimensional shear buckling coefficient and is a function of the plate geometry and edge restraints. The values of Ks and ~ are always the elastic values since the plasticity correction factor, 11, contains all changes in those terms resulting from inelastic behavior. The term b is the smaller dimension of the panel.
A great deal of work has been done relative to the value of the plasticity correction factor. The expression for 11 must involve a measure of the stiffness of the material in the elastic and inelastic ranges. A simple means of obtaining a value of
~
is to take the ratio of the shear secant modulus to the shear modulus. G
~
s
= (f =
shear secant modulus shear modulus
10.4.1 CRITICAL BUCKLING STRESS WITH AXIAL LOADS When axial loads are present the actual shear buckling stress 10.4 will be different. The presence of compressive stresses stresses causes the panel to buckle at a lower value of shear were present. Tension causes the panel to buckle at a higher
defined in paragraph together with shear than if no compression shear stress.
When shear and compression are present the panel buckles according to the interaction f
c
IF c
+ cr
(f
s
IF
s
)2
1.0
cr
where Fccr and FScr are the critical panel buckling stresses for pure compression and pure shear. From chapter 7, section 7.3 the buckling stress for a panel under compression is F
c
cr
For any particular panel
IF s a A, (a constant) cr cr From conventional means the applied compressive stress, fc' and the applied shear stress, fs' can be calculated. These stresses will have a constant relationship with each other until the panel buckles, after which the compressive stress no longer increases. Thus F
c
10-20
)
STRUCTURAL DESIGN MANUAL Revision A f
c
If s
= B
Now the interaction equation can be rewritten as
2
f
s
=
Fs
cr
) where fs is the actual shear stress at which the panel buckles due to the presence of compression stresses. The expression in the brackets can be called Rc and the equation rewritten as f
wh~re
R F
s
c s
cr
Rc is always ],ess than 1.0 when compression stresses are present.
When shear and tension are present the panel buckles according to the interaction f
s
IF s
cr
- f t /2F ._ c
=
1.0
cr
where Fc and FScr are as before. The shear stress, fs' at which the panel buckles ~fth tension, f t , present is f
s
F
s
(1.0 cr
+
f /2F t
C
) cr
and can be rewritten, substituting R f
)
s
= R F t s
t
for the term in parenthesis, as
cr
Si nc (' Rt is always grea te r than 1.0 when tension is present, the ac tual shear buckling stress will always be greater than Fs ,the buckling stress for shear cr only.
10.5 INCOMPLETE DIAGONAL TENSION The incompLete diagonal tension theory is a usable engineering theory which is a combination of shear-resistant beam theory, the pure diagonal tension theory, and empirical results of tests. BHT computer program SSCS01 performs a computer analysis of incomplete diagonal tension. (See Section 2.6.) A physical description of what occurs during incomplete diagonal tension is given below. To a beam with a plane web, stiffened by uprights and free from large imperfections, apply a gradually increasing load. For low loads the beam behaves in accordance with the shear-resistant beam theQry. The web remains plane and no stresses are developed in the uprights. At a certain critical load the web will begin to buckle. Incomplete
10-20a
STRUCTURAL DESIGN MANUAL Revision A
diagonal tension has begun at this point. As the load is increased, the buckles become more distinct and the pure diagonal tension theory is approached. The state of pure diagonal tension is a theoretical limiting case, which can never be reached because some failure will occur preceding the limit. As the process of buckle fonmation progresses, axial stresses in the uprights deve1op. The portion of the total shear, V, carried by diagonal tension, VDT , is found by using the diagonal tension factor, k.
VDT
Vs
= kV = (l-k)(V)
)
STRUCTURAL DESIGN MANUAL Revision E The shear stresses are calculated from the shear flow equation:
f
v (h )(t)
s
q/t
c
(k)(f ); fs s s
=
(l-k)(f ) s
As the load increases beyond the initial buckling load, a higher percentage of the total shear is carried by tension field. This causes the ratio f Ifs to become cr s an important parameter. Methods of analysis for .three specific types of tension field beams are given: Flat tension field beams with single uprights. Flat tension field beams with single uprights and access holes. Curved tension field beams.
1. 2. 3.
The curves given for use in these analyses yield results with a reasonable assurance of conservative strength predictions, provided that normal design practices and proportions are used. 10.5.1
Effective Area of the Uprights
In order to make the design curves apply to both single and double uprights, it is necessary to define an effective upright area A ue For double uprights, which are symmetric with 'respect to the web: A
ue
= Au =
total cross-sectional area of the uprights.
For single uprights: A
)
A ue
= 1
u
+ (~t
where p= radius of gyration of the ,stiffener and e = distance from the centroid of the stiffener to the center of the web.
If the upright has a very deep web, Aue should be ta~n to be .the sum of the crosssectional area of the attached leg and an area 12 tu ,where tu is the upright thickness. 10.5.2
Moment of Inertia of the Uprights
The uprights must have a sufficient moment of inertia to prevent buckling of the web system as a whole before formation of the tension field, in addition to preventing column failure due to the loads imposed upon the upright by the tension field. Forced crippling failure, caused by the waves of the buckled web and possibly most critical, must also be prevented by the upright. The required moment of inertia of the upright may be determined by iterating through the appropriate Table 10.1, 10.2, 10.3, or 10.4. 10-21
STRUCTURAL DESIGN MANUAL Revision B 10.5.3
Effective Column Length
The effective column (upright) length is calculated by the equations:
10.5.4
h
If d < 1.5 h u' c
L
If d > 1.5 h , c u
L e
e
VI + k
2
(3 - 2d) h
=h
Discussion of the End Panel of a Beam
The following analyses are concerned with the "interior" bays of a beam. The uprights in these areas are subjected, primarily, only to axial compressive loads. The end panel, however, is a special case. Since the diagonal tension effect results in an inward pull on the end upright, bending, in addition to the usual compressive axial load, is also produced. There are three general ways of dealing with the edge member subjected to bending. 1.
Sufficiently strengthen the edge member so it can carryall of the loads (which is inefficient, weight-wise, for long unsupported leng ths) •
2.
Increase the thickness of the end panel either to make it nonbuckling or to reduce k, which would reduce the running load producing bending in the edge member. (This is usually inefficient for large panels.)
3.
Additional uprights may be provided to support the edge member and thus reduce its bending moment.
10.5.5
Analysis of a Flat Tension Field Beam with Single Uprights
Table 10.1 is a step-by-step procedure which yields the stresses in the flanges, webs, rivets, and uprights of a flat tension field beam with single uprights (Figure 10.19). Table 10.1 is based on a single web with parallel flanges and parallel uprights. Most beams consist of more than one;web. At various locations in the following table adjacent panels must be considered. Such a situation occurs for rivet load, stringer axial stress, upright stress and moment in stringer.
10-22
)
STRUCTURAL DESIGN MANUAL Revision A J
he
+ + 1-
+
i
r
f- ....
+
if+e
++++
+- +t- +- + + + + UPRIGHT -z-.... + +
l~ ++
+
++
+ +
L.±..
++ + +++ do
~
Rl.
~-E
WEB:z
Ii( + + +
!~f FLANGES
)
(
++++ .... + -..:+~I
IlL.
-<=1-
he
he
\ 1 J~-
-
---
I
dc'~
FIGURE 10.19 - FLAT TENSION FIELD BEAM WITH SINGLE UPRIGHTS Description
Variable and Equation
CD Elas tic
modulus
Ec
@ Upright
spacing, (NA to NA)
d
(]) Clear web between uprights (ri ve t to rive t)
dc
@Distance from median plane of web to centroid of upright
e
Numerical Value
(}) Clear web between flanges (rivet to rivet)
@ Di stance
be tween flange
he
centroids
(j) Length
hu
® Web
t
of upright between up· right to flange rivets thickness
(2) Upright
(9 Flange l!J)
thickness
tu
thickness
Upright area
~ Flange area
~
Radius of gyration of upright
1
Moment of inertia of upright
Iu
Moment of inertia of flange
IF
Applied load - upright
Pu
~ ~
iP Applied
load - flange
~ Applied web shear flow
~
Web shear stress
Pf q T=
q/t
=
@ /@
TABLE 10.1 - ANALYSIS OF FLAT TENSION FIELD BEAM WITH SINGLE UPRIGHT 10-23
STRUCTURAL DESIG·N MANUAL Revision E
Q9 Effective
@ Parameter @ Parameter
= @ 11+(0 21 © Aue/dct = @ /@® het =@® de/hu = G) I (J) tf/t = @ I@ tuft =@/@ he/d e = CD I G) de/he:::: 1 / @
@
tIde
@
Parameter
@
Parameter
area of upright
© Parameter @ Parameter
© Parameter Parameter
® Parameter Q9 Upright restraint
Aue
t/h c coefficient
2)
=@/Q) @I@
Rh, Figure lO.20(b)
QY
Flange restraint coefficient
Rd, Figure lO.20(b)
Q}
Theoretical buckling coefficient
kss' Figure 10.20(a)
@
Elastic buckling stress: de < he Tere @Q)@2l® + de> he Tcre
~(®
_®) ©31 @CD®21® + ~(® _@) ®3/
~ Initial buckling stress
Ter , Figure 10.21 (See Note 2)
@ Stress ratio
TITer =
Q9 Diagonal
k, Figure 10.22 @ 300tdc/12hc :::: 0
tension factor
@ /@
@ Parameter
~+l (l-k) =@+ (1- @)
~ Ratio of upright stresses
luu, Figure 10.23 max . UuIT, Figure 10.24
det
~ Ratio of upright to shear
2
2
Uu
stresses
~ Diagonal te~sion angle
€y web Stress
?_~ Tan~
Figure 10.25(a)
-@@@/@ uU avg = @@ I @
in median plane upright/ uu=
~ Upright average stress
@@
~ Upright maximum stress
O"u
~ Effective column length: If @< 1.5
Le=G) I 1 + ®2(3_2
If
@
>1.5
allowable
~ Proportional limit
@ Strain,
if
1
@)1 ~
®>@
Le/2p::::
@ /2 @
uco :::: 11"2(0/ @2 or Section 11 - F ) Fpl, Section 5 ( F PI tp Uu/E = (@ / (J)
TABLE 10.1 (cont'd) - ANALYSIS OF FLAT TENSION FIELD BEAM WITH SINGLE UPRIGHT 10-24
)
Le=hu=CD
~ Slenderness ratio
€9 Column
max
::::
•
.,' '.-.
'I\~ ~.
~..... ,:
<
-
"
•
\
\,
\'~
\ :- \ : Bell ,~,,,,,
~\
STRUCTURAL DESIGN MANUAL
~,~."
o
Revision E
From stress s train curve
® M;lrj::in .
of
Safety: column yield ©>(~)'
8
MS - Co'l umn
©
Parame te r
Fe, use ~ to determine all. MS=0/@)-l MS = ~ I ~ - 1 MS ::; @ / @-l k2/3(tu/t)1/3 ~ 2/3
@ Upright
(1ot Figure 10.26
~ MS - Forced crippling
MS ==
@
Wd
allowable (forced crippling)
Parameter
6 Parameter
@
Parameter
~ Maximum web stress ~ Web allowable
@ MS
- Web
~j) Pa ramt' ter
@
S(;C()n(j.~iry bending in flange
~ Oi~lancc from NA to extreme fib<::r of flange
@
== •
c
1/3
Q-1
7G)(®/2©®) 1/4
C , Figure 10.27 l C , Figure 10.28 2 T:nax =
+@@) a = 45
® /®
C , Figure 10.28 3 MSB == (1/12)(@®OO C
@)
f
stance - NA to near fiber of flange
Dj
Flange appl ied stress Diagonal tension stress-flange (camp)
q
@ /
Q}
(1
a;r
=@ /
©
-(@ @/@"[ 2
=
1
Secondary bending stress-flange uSB (comp)
= - ~~ I
Secondary bending stress-flange uSB (tension)
= ~~ /
Flange stress-inside fiber
(1tot
Flange stress-extreme fiber
CT
Allowable crippling stressflange
Fcc
tot =
Allowable tension stress-flange F MS -
Flange (tension)
MS - FLange (compression) Rivet
factor load-web to flange
=
tu
© /©+
.5 (1-
@)~
Q}
@)
<0 + ® + @
<@ + 0 + <@
or F
ty
= @ / @ -I, NS = @ / <§ -1 R = 1 + 0.414 ~ " = qR = @® MS
TABLE 10.1 (cont'd) - ANALYSIS OF FLAT TENSION FIELD BEAM WITH SINGLE UPRIGHT 10-25
STRUCTURAL DESIGN MANU,AL Revision F
Q) Allowable
rivet shear load
Pat
~ MS - Flange rivets
MS
=
Qj
Rivet load-upright to flange
P
=@@
@9
Allowable rivet load-upright
P
u
Qj) / @
-1
~ MS - Upright rivets
au MS =
~ Interrivet buckling allowable
Fir it Section 10.6
@) MS
MS =
- Interrivet buckling
@ /
®-1
@ / @-1
~ Ultimate tensile stress of web
©
Ftu t Section 5 R i v e't ten s i 1 e s t :r eng t h up- t1R = .22@@ right/web per inch*
~ Rivet allowable tensile load per inch @ MS - Rive t tension
F
RT
)
, Section 6
MS =
@ /
®
-1
NOTES:
If any of the margins of sa ety are less than zero, the design is inadequate. The« eficient area must be corrected and this table repeated. (2) If the web is subjected to ension or compression as well as shear, the initial buckling stress of the web must be modified according to the method described in Section 10.4.1. * See NACA TN 2661, "A SUf1r.J.ary of Diagonal Tension",1952, page 49 (1)
for explanation.
• )
TABLE 10. I (conL'd) - ANALYSIS OF FLAT TENSION FIELD BEAM WlTH SINGLE UPRIGHT 10-26
e--
·~
THIS CURVE IS SHOWN FOR d < h . c
c
IF d > h THE ABSCISSA SHOULD BE c c READ AS de
Rh = UPRIGHT COEFFICIENT Rd - FLANGE COEFFICIENT
~
c
~L
9
d
c
J
1.8
bK:- ~~ ,.
1.6
8
Y\;/?--
1.4
~
7
cf1
I
h
I
~d--J c
kss
1.2 c
_1
/
1.0 Rh , Rd
6
I I II I
.8 II
.4 /
/ 0 1
2
3 h d
4
I~~
/
~
/
-
/
/'
~
o
1
(B) EMPIRICAL RESTRAi;vT COEFFIClt<:NTS
l-'
o
•
N
3
c
PLATES WITH SIMPLY SUPPORTED EDGES.
......
2
c
(A) THEORETICAL COEFFICIENTS FOR
j
i
5
---~
,/
I
I
.2
/'"
J
I
.6 51----+--1--+----1----1-
/
/
L--- ~
FIGURE 10.20 - GRAPHS FOR CALCULATING BUCKLING STRESS OF WEBS
4
o
::::0 I'D
tv
-'.
I-'
I
<
ex>
40
ALCLAD
.VV
30
T
CI'
,KSI 20
10
o
V
o
V
~
!/'/~
~
~ l-----
/
v---
7075~
----~
------
2024-T3
-
-
-
V
v
10
20
30
40
50
FIGURE 10.21 - GRAPHS FOR CALCULATING BUCKLING STRESS OF WEBS
60
k
1
2
3
4
6
8 '10
20
30
40
60
il If h > d , REPLACE td /Rh c
c c c If d/h (or hI d) > 2, use 2
BY th c /Rd c
FIGURE 10.22 - DIAGONAL TENSION FACTOR, k
80 100
200
400
600
1000
to-'
o
I Ul
o
1.8~----~------~-----.-------.------.------.------'-------r-----~----~
1.6~~~-+------4---~~r------r------+-----~------~------+-----~----~
1.4~----~--~~~--~~r-----~~~--+---~~~-----r------~----~----~
~
........
~
1.2~--~~---+-----r----~~~+-----r---~~--~~--~--~
b'='
.2
.6
.4
d/h
.8
1.0
u
FOR CURVED WEBS, READ ABSCISSA AS d/h FOR ABSCISSA AS hid FOR STRI~GERS
RI~GS
AND READ
FIGCRE 10.23 - RATIO OF MAXIMUM STRESS TO AVERAGE STRESS IN WEB STIFFENER
,
•
•
1.0
!
J 7 - .05 .1
.9
!o%j
......
~
c::::
::tJ
t:rJ
. o N
.6
~ z
::.>
z
A
ue
/
/'
V V II V
V
/
V
.IV
/
/ /
, .I
1/
V I J V I / 7 /., ; 45 . 5 ' / / / / .I I II J . V V I I I / II .55 6/ / I V / I J ;1 /" / 1/ I , / I / / / / / I / / " I I I I / V V /7 8 v / V / / II I I / / / I 1/ / V 9 V v· / ' / / I/ II 1/ It 1/ v / ,I / 1/1 V / / V -I / VI / I / / / 1/ / V /1.0.1 / V JI VI / / I / / / / / / / / 1·,2 / ' / v I V V V " V 1 4 '" / V / V/ / / II ) I I I I / I I / / ~/ I ;' I v' / / V / / 1/ V/V v / '1.6/ V"""'" V/ VI I -I " II 1 L I II / / / 7 V /' ;,V 1.8 "" V V V/ ~/ V / V V/ / "/' / / /V / /' 2.0 / / I II " ..,;V .V/ V ~~ V, LT7 V7 /" I I II / / II / /, Ii / 1/ V ~ ! I / I/' I VI V/ V V / / '/: V/ V ~/'V/V . / 32.50 r---:; 1/7 / V/ V/ // v .(' '" ... V/ ~/v V 1./ • VI VI I I ,'J "" I 4.0 ~ / / / / II JI VI.; V/ / V/ V/ V 1/"",V, ~ v V 5.0 ; II / /) / / II V/ V/ V/ / / V V/ V/ V / I-- / V .l"V-:VI , I
~
,I
(I)
-t ::a
/
~r
CIt c
) If
/
.I •
,
c:
n
i.-
'"
-t
c:
/
::a
L
i
.4
01
.3
I
f!
CIl
j
/
to-<: H
~
• 35
~I
J
.5
Z
en
.3
/
V
1/
I
J
0
~
.25
l!
I
0
Ul H
.2
J
v
/
I
H
t-i trJ
.15
/
I
I
t::1
t'"'
T
./,
I
.7
+'-
II
II
./
.8
.t-
(iu
IJ
/
I
J
,
'"
:Dr-
L'
CI ...,
/
CI)
-
/'
.2
V
/
j
/
.1
J
I
J
/'
C i)
.,-
z
"....
/
if I
o
o
; ' . /
•1
.2
"3
.5
.4
.6
.7
.8
.1"
.....
.9
:I:
I---
:DZ
1.0
k
c: ::::c I'D
<
-'til -'a
0 ::3
I'T1
::a=-
r-
"
~ .B~II. . ,STRUCTURAL DESIGN MANUAL ~:r~~
~ 1.0
~~
'\
.9
~ ~, ~
.8 tan
~
>: ::::
"
~
'""'
r--....
~
~ r-....~ ............ ............
'"
""
"
;--.. .............
"' N
~
~
............
'<...
.. 7
.6
,
r......... -.........
r---..... ..........
k
i""----r-.,
r-....... ............. """'-- I--
r-.......
o
~
~
"""'-- """'--
-r--
"""'-- i--
--
1
----.. o
.5
2.0
1.5
.5
I,
)
INCOMPLETE DIAGONAL TENSION
(a)
48
2Af
V
44 /'
40
/
./
/" /'
/
/""
j'/ v L ~ / '
36
-- -- -- --- -
:-1 f-2 r--
:......-
8 r-C()
,-- r---r'"" ~
',--
:---
~ ~
/, r~V
~PDT
//J. ~ r/'lL
32
) '/
'LV
28
~
24
20
o
.4
.2
.6
.8
1.0
1.2
1.4
Aue
(b)
d t c PURE DIAGONAL TENSION
FIGURE 10.25 - ANGLE OF DIAGONAL TENSION 10-32
;;?7\~\'~
STRUC1·URAL DESIGN MANUAL
" \ :..'" B~II ' \ 'J '~'~'~'''.II '/ '. , "'--
....../
.... :..
60
1.0
10
70
I)
.9
50
60
8
.8
7 40
50
.7
6
40
.6
5
)
30
4 30
(]' , ksi
(]'
0
DOUBLE UPRIGHTS
.4
0
J
UPRIGHTS
0
20
..!:.u
2/3 ( . 1-) l/1 a --32 • r.k J tu l
t
0
'1.
15
.3
ksi
SINGLE
20
u =26k 2/3 (lu/ t )1/3
k
J
15
1.5
•
10
.2
9 B
7
5 •1
4 (b)
1.0
10
.9
9
.8
8
•7
7
.6
6
.5
5
,/
.4
7071 ALUMINUM ALLOYS
£,'lGlJl{g L0.26 (con ltd) - NOMOGRAPH I,'OR ALLOWABLE UPRIGllT STRESS (tj'ORCgU Cit I PPL [NG)
10-34
-'\
•
STRUCTURAL DESIGN MAN.UAL Revision E
.12
\
.10
)
\
\
.08 C 1
\
\
.06
\
\
.04
\
I'\.
/
"-
.02
"'" ,,~ ...........
0
.5
.6
.7 tan
"--
.9
.8
1.0
a
FIGURE 10.27 - ANGLE FACTOR C 1
•
1.2
-
1.0 C 2
C 3
I--
.8
--
C 3
r--... 7 /
.6
V
.2 ..
)
~
-7c 2
.4
0
o
1
-
~~
/
V
/
-
/
3
4
FIGURE 10.28 - STRESS CONCENTRATION FACTORS, C2 AND C 3
10-35
STRUCTURAL DESIGN MANUAL
•
30
'"~ ......
O!PDT' deg
~~
25
........... .....
~!oo..
- , , ---- - -'I-... ~t-..:
........ ~ ~
-..;
~ f...,.--
--
IOiIIi::::..
20
..........
* Tall' ksi
.......... ...........
...........
- - -- ----......
............. r-.
15
45
-
I--.
-...
I--. ""'-
40 35
-
---
-..... .........
---..;
30
25 20
10
o
.6
.4
.2
1.0
.8
k
2024 ALUMINUM ALLOY.
F
tu
=
•
62000 psi
DASHED LINE IS ALLOWABLE YIELD STRESS 35
I\..
~
~
O!pDT' deg
~~
~ ~'",,~ ~~
30
"-.. ~ ~ ~
"'"
25
* ksi Tall'
............ ............
-----r=-
~
-...
t--
......... .......
"'" "'
........
----
45
- --
-I-
,' -..... '-. f':
~
.....................
r--
r-I--
-""';:::::r-...
F::::.
20
-- ---
..... ~
----......
......
40 35 ")
30
/
25 20
15
o
.2
.4
k
7075 ALUMINUM- ALLOY.
.6 F
tu
.8
72000 psi
FIGURE 10.29 - BASIC ALLOWABLE VALUES OF TMAX '10-36
1.0
•
STRUCTURAL DESIGN MANUAL Revision A 10.5.6
Analysis of Flat Tension Field Beams with Double Uprights
Table IO.2 is a step-by-step procedure which yields the stresses in the flanges, webs, rivets and uprights of a flat tension field beam with double uprights as shown in Figure 10.30.
)
T h
l
UPRIGHT
+ +
e
T h
c
h
c
WEB
T T
+ +++ ..... + 1-++ -r-
~~
d
c
~
~
~,
FIGURE 10.30 - FLAT TENSION FIELD BEAM WITH DOUBLE UPRIGHTS Table 10.2 is based on a single web with parallel flanges and parallel uprights. Most beams consist of more than one web. At various locations in the following table adjacent panels nlust be considered. Such a situation occurs for rivet load, stringer axial stress, upright stress and moment in stringer.
)
10-37
STRUCTURAL DESIGN MANUAL
T h
•
UPRIGHT
+ +
e
WEB
-t
+
-+ -t-+-t--t-
~~
~
~
e~;==~~========~~========~~~
d
c
~
DESCRIPTION
CD CD
VARIABLE AND EQUATION
ELASTIC MODULUS
E
UPRIGHT SPACING, (N .. A. TO N.A.
d
G) CLEAR WEB BETWEEN UPRIGHTS
)
r
NUMERICAL VALliE
c
d
c
(RIVET TO RIVET)
®
DISTANCE FROM MEDIAN PLANE OF WEB TO CENTROID OF UPRIGHT
e
®
CLEAR WEB BETWEEN FLANGES (RIVET TO RIVET)
h
®
DISTANCE BETWEEN FLANGE CENTROIDS
h
CD
LENGTH OF UPRIGHT BETWEEN UPRIGHT TO FLANGE RIVETS
h
®
WEB THICKNESS
t
0)
UPRIGHT THICKNESS
t
FLANGE THICKNESS
t
UPRIGHT AREA
A
•
c
e u
u
f
FLANGE AREA
u Af
RADIUS OF GYRATION OF UPRIGHT
P
MOMENT OF INERTIA OF UPRIGHT
I
MOMENT OF INERTIA OF FLANGE
u IF
APPLIED LOAD - UPRIGHT
P
APPLIED LOAD - FLANGE
P
APPLIED WEB SHEAR FLOW
q
WEB SHEAR STRESS
T
)
u
f = q/t
= @ /@
TABLE 10.2 - ANALYSIS OF.FLAT TENSION FIELD BEAM WITH DOUBLE UPRIGHTS 10-38
STRUCTURAL DESIGN MANUAL EFFECTIVE AREA OF UPRIGHT
A
PARAMETER
A
PARAMETER
=(§)® de/hu = 0) 10
u
Id e t ue
=
~ 1f1vB' ~
=@/@ tu lt =@I@ h Id =®I@ e e dc/h e = 1 I @
PARAMETER
tf/t
PARAMETER PARAMETER PARAMETER PARAMETER
tIde
=0/0)
PARAMETER
t/he
=®/Q>
UPRIGHT RESTRAINT COEFFICIENT
Rh , FIGURE 10.20 (b) Rd t FIGURE 10.20 (b)
FLANGE RESTRAINT COEFFICIENT THEORETICAL BUCKLING COEFFICIENT ELASTIC BUCKLING STRESS: < h
d C d
C
k
, FIGURE 10.20 (a)
ss
TeTe
=@COO2 [® + ~ (0) _ @>@3
TeTe
@CD@2~+ ~ <® - ®>@l
C
> hc
Q3
INITIAL BUCKLING STRESS
T ,FIGURE 10.21 cr
®
STRESS RATIO
TIT
@
PARAMETER
k, FIGURE 10.22 @ 300td 112h c
~ ( 1- k> = Al\ +
Au e +
~
d t c
~ RATIO OF UPRIGHT STRESSES
RATIO OF UPRIGHT TO SHEAR STRESSES DIAGONAL TENSION ANGLE
(]'u MAXI (T
u
(T
u
,
=0
@)
2
IT , FIGURE 10.24
TAN a, FIGURE 10.25 (a)
®@® I @
0"
UPRIGHT AVERAGE STRESS
OUAVG
UPRIGHT MAXIMUM STRESS
~MAX=~
@>1.5
(1-
c
FIGURE 10. 23
STRESS IN MEDIAN PLANE UPRIGHT I WEB
~ EFFECTIVE COLUMN LENGTH: IF e<1.5
(See Note 2)
@I @
=
cr
~ DIAGONAL TENSION FACTOR
)
~
het
PARAMETER
Q)
= A
ue
= -
u
=
®®
I @
Le
=
[1 + ®2 (3 - 2
Le
=
=CD
hu
.
(3) J ~
1
TABLE 10.2.(CONT D) - ANALYSIS OF FLAT TENSION FIELD BEAM WITH DOUBLE UPRIGHTS 10-39
~
"~~.I!'! STRUCTURAL DESIGN MANUAL / //
Revision E
@
SLENDERNESS RATIO
L
e
= <9
/p
2
/ ©2 CD I ®
®
COLUMN ALLOWABLE
(Teo
(@
PROPORTIONAL LIMIT
F pI' SECTION 5
@
STRAIN, IF
@
FROM STRESS STRAIN CURVE
® ® ®
MARGIN OF SAFETY: CO~UMN YIELD
~
@ >@
(Tu
~~~
MS - COLUMN PARAMETER UPRIGHT ALLOWABLE (FORCED CRIPPLING)
~ PLASTICITY CORRECTION: IF
®>@ ~
MS - FORCED CRIPPLING
® © ®
PARAMETER
o
/E ,
or
SECTION 11
(F pI
@> I CD USE @ TO DETERMINE
Ft p )
=
ALLOWABLE
~~:~~~:~ MS=@/@-l k2 / 3( tu/t) 1/3 = ®2/3 @1/3 (To' FIGURE 10.26
PARAMETER
C , FIGURE 10.27 1
PARAMETER
C , FIGURE 10.28 2
MAXIMUM WEB STRESS
T'MAX =
WEB ALLOWABLE
T all' FIGURE 10.29 @ a pDT = 45 0
MS - WEB
MS
@
(1
+@2 @)(l +®®)
*
PARAMETER
= @ / ®-1 C , FIGURE 10.28
SECONDARY BENDING IN FLANGE
MSB
DISTANCE FROM N.A. TO EXTREME FIBER OF FLANGE
C
DISTANCE - N.A. TO NEAR FIBER OF FLANGE
D' f
FLANGE APPLIED STRESS
@ /© (TDT=-<@@I®jf2 @/@ +
DIAGONAL TENSION STRESS FLANGE (COMP)
9
= 7T
3
=
(1/12)(
®o@(}i@)
f
(Ta
=
=-@@ I @
SECONDARY BENDING STRESS FLANGE (COMP)
(T
SECONDARY BENDING STRESS FLANGE (TENSION)
erSB = @@
SB
.5(l-@~
/
e
TABLE 10.2 (CONTtD) - ANALYSIS OF FLAT TENSION FIELD BEAM WITH DOUBLE UPRIGHTS '10-40
)
STRUCTURAL DESIGN MANUAL Rpvi~ion
@
FLANGE STRESS - INSIDE FIBER
CT
@
FLANGE STRESS - EXTREME FIBER
(T
ALLOWABLE CRIPPLING STRESS -
F
@
= <ۤ) + @ + @
tot
FLANGE
tot =
® + @ + @>
cc
@
ALLOWABLE TENSION STRESS FLANGE
Ftu or F
@
MS - FLANGE (TENSION)
MS=@/@-l
MS - FLANGE (COMP)
MS=@/®-l
RIVET FACTOR
R
)~
Q) RIVET LOAD - WEB
@
TO FLANGE
F
ty
@ @@
1 + 0.414
=
=
Rft
qR =
ALLOl.JABLE RIVET SHEAR LOAD
P
MS - FLANGE RIVETS
MS=@/@-l
RIVET LOAD - UPRIGHT TO FLANGE
Pu
ALLOWABLE RIVET LOAD
P
MS - UPRIGHT RIVETS
MS=@J/@-l
STATIC MOMENT OF CROSS SECT.
Q
af
=@@
au
OF ONE UPRIGHT ABOUT MEDIAN PLANE OF \.JEB
€9
WIDTH OF OUTSTANDING LEG OF
b
IJPRIGHT
)
rJPRIGHT COLUMN YIELD STRESS
F
RIVET LOAD - UPRIGHT TO WEB
RR =
RIVET ALLOWABLE LOAD
P
MS ":' RIVET, UPRIGHT TO WEB
MS
=
ULTIMATE TENSILE STRESS OF WEB
F
' SECTION 5
RIVET TENS ILE STRENGTH
RIGHT/WEB PER INCH*
coy
UP~
, SECTION 11
2@@Q)/@§>
ar
tu
O"R
=
@I @ -
1
.15®®
~ RIVET ALLOWABLE TENSILE LOAD
F , SECTION 6 RT
@
MS=@/@-l
PER INCH MS - RIVET TENSION
*See NACA TN 266 ~ itA Summary of Diagonal Ten ion") 1952, page 49 for exp anation.
NOTES: ( 1)
If any of the margins of sa~ety are less than zero, the design is inadequate. The deficie~t area must be corrected and this'
(2)
If the web is subjected to
table repeated. _
~ension or compression as well as shear, the initial buckling stress of the web must be modified according to the method des~ribed in Section 10.4.1. - - - - - 1 -_ _ _- - - - - - ' _ - - - - '
i - - - '_
_
_
TABLE 10.2 (CONTID) - ANALYSIS OF FLAT TENSION FIELD BEAMS WITH DOUBLE UPRIGHTS 10-41
STRUCTURAL DESI,GN MANUAL Revision A 10.5.7
Analysis of a Flat Tension Field Beam with Single Uprights and Access Holes
The following step-by-step procedure (in Table 10.3) is an analysis of a flat tension field beam with single uprights and access holes (Figure 10.31).
FLANGE
T h
e
1 e
* T
1-
+
+++++ t ++ + + +
+
+ +
+ +
+ + ..,..
0
t
+
+ + ++ + +
~~
+
R-L d
c
T
eJ + + +
+
WEB
+
I h
)
c
1
~
~
FIGURE 10.31 - FLAT TENSION FIELD BEAM WITH SINGLE UPRIGHTS AND ACCESS HOLES Table 10.3 is based on a single web with parallel flanges and parallel uprights. Most beams consist of more than one web. At various locations in the followinb table adjacent panels must be considered. Such a situation occurs for rivet load, stringer axial stress, upright stress and moment in stringer.
)
10 ... 42
STRUCTURAL DESIGN MANUAL
)
DeSCRIPTION
®
)
VARIABLE AND EQUATION
[NUMERICAL VALUE
-E Elastic Modulus c Upright Spacing (N.A. to N.A.) d Clear Web Between Uprights (Rivet to d c Rivet) Distance from Median Plane of Web to Centroid of Upright e Clear Web Between Flanges hc Distance Between Flange Centroids he Length of Upright Between Upright to Flange h Rivets u Access Hole Diameter Web Thickness Upright Thickness Flange Thickness Upright Area Flange Area Radius of Gyration of p Upright Moment of Inertia of Upright
TABLE 10.3 - ANALYSIS OF FLAT TENSION FIELD BEAMS WITH ACCESS HOLE AND SINGLE UPRIGHTS 10-43
If~ ,,~~,,~ STRUCTURAL DESIGN MANUAL R~e~v~i~s_io_n __A__~________________~__________________________~______-,~
(9
Moment of Inertia of Flange
If
q)
Flange Applied Load
P
~
Upright Applied Load Applied Web Shear Flow Web Shear Stress
~
~
f P u q ,. =
@/®
@ ~ Effective Area of
Aue
@
Parameter
Aue/de t -
@ @
Parameter
het
@
Parameter Parameter Parameter Parameter Parameter Parameter Parameter Parameter
de/hu == (J) /(J) tf/t = ~ tult = (9 he/de = @ / Q) de/he = 1 / @ tide = Q) t/ h e = @ D/he = @
@
Parameter
D/d
Q)
Parameter
Upright
~ ~
@ @
®
®
= {)I 11+~/{921
@/GX2)
=C§X(2)
IGD IGD
GDI
GD / ®/
c =@ 1 + D/d c
/
CD
=1
+ @
o
Parame,ter
Aw
@
Parameter
@ / @
~
Upright Restraint Coefficient
~, Figure 10.20 (b)
~
Flange Restraint Coefficient
R ,
~
Theoretical Buckling Coefficient
k ss ' Figure 10.20 (a)
Elastic Buckling Stress: d
Tore =
d
Tore
Q}
c
>h
c
= t(d c
d
- D)
=®
cO) -@)
Figure 10.20 (b)
@
Initial Buckling Stress
T
@
Stress Ratio
TITer
@CD32 [®
+
~(o-@)~l
= ~C!)@2[@
+
~(@-@) ®j
cr ' Figure 10.21
(See Note 2)
=@ / @
L-----------'--~--~-----------~--~e , TABLE 10.3 (CONT'D) - ANALYSIS OF FLAT TENSION FIELD BEAM WITH ACCESS HOLE AND SINGLE UPRIGHTS lO-~4
STRUCTURAL DESIGN MANUAL Revision E
Diagonal Tension Coefficient
k, Figure 10.22 @
Parameter
@
Ratio of Upright Stresses Ratio of Upright to Shear stress
u.
Diagonal Tension Angle
tan a
Stress in Median PlaneUpright to Web
(Tu = -
Upright Average Stress
Upright Maximum Stress
o-urnax = ({])
Effective Column Length If @< 1.5 If ~ > 1.5
Le
Slenderness Ratio
Le/2p = @/2
Column Allowable
O"co
Proportional Limit
F pl ' Section 5
strain, If
O>@
From Stress-Strain Curve Margin of Safety: Column Yield
)
@ @ @) @ @ @
o
Q
300 td c/12 he = 0
+ ~(1
0)
-
umaxlu, u Figure 10.23 Figure 10.24
UU/T,
Le
Figure 10.25 (a)
I
0
= ({])@ I
©
02(] -
2
@>
P
3
2~Jo...2
=1(
0
@
=(2)1[1 + = hu =(2) \..:;;I'\?!J
o-u /E = ({]) Fc l
@ @I
or Section 11 (F
pI
-
F
tp
)
ICY
use ~ to determine
allowable MS
= @ / ({]) -
1
Parameter
MS=@)/({])-l MS=@/@-l k2/3(tu /t )1/3 = ~/3 ~/3
Parameter
C , Figure 10.32
Parameter
C S ' Figure 10.32
Parameter
@@@
MS - Column
4
Parameter
@+@
Parameter
@/Gm
Upright Allowable (Without Access Hole)
TABLE 10.3
(CO~TtD)
0"0'
Figure 10.26
- ANALYSIS OF FLAT TENSION FIELD BEAM WITH ACCESS HOLE AND SINGLE UPRIGHTS 10-45
STRUCTURAL DESIGN MANUAL 0
1
0"0'
MS Forced Crippling
Ms=O/@-l
Parameter
wdc =
Parameter
C , Figure 10.27 I
Parameter
C , Figure 10.28 2
Maximum Web Stress
T~ax
=
.7~/2~®J~
=
@ (1
web Allowable (Without Access Hole) Web Allowable (with Access Hole)
~
o
T* all, "..,
s
@~) +,@ @)]
[
+
Figure 10.29 @ ex PDT=45'0
=@@/@ =©
/ @-l
MS - Web
MS
Parameter
C , Figure 10.28 3
Secondary Bending in Flange
MSB = (1/12)
Distance From N.A. to Extreme Fiber of Flange
C
Distance - N.A. to Near Fiber of Flange
D
Flange Applied Stress
(Ta
Diagonal Tension Stress Flange (Camp)
(TOT =
Secondary Bending stress - Flange (Camp)
lTSB = -
Secondary Bending Stress - Flange (Tension)
USB
Flange Stress - Inside Fibers
0"
Flange Stress - Extreme Fibers
<0
@
Upright Allowable (With Access Hole)
<0 <9 @Gt
@)
f f
=@/@
-(O~ / ~ir2 @/ @ >]
+ .5 (1 - ~
@@/ @
=@@/@ ')
tot
=@+@+@
(Ttot
=@+@+@
Allowable Crippling Stress - Flange
Fcc
Allowable Tensile stress - Flange
F tu or F ty
TABLE 10.3 (CONTIO') - ANALYSIS OF FLAT TENSION FIELD BEAM WITH ACCESS HOLE AND SINGLE UPRIGHTS 10-46
/
.....
... ~ . . !'.. "....
/-,' r • \
\\
~~~f~~~ STRUCTURAL DESIGN MANUAL
411t)r-_______~_--_'____________~______~------------------~-------------R~e-V-l-·-S-i-o_n__F~ ~ MS - Flange (Tension) MS = ~ / €) - 1 o MS - Flange (Comp) MS == 0 1 <0 - 1 ~
Rivet Factor
R
@
Rivet Load - Web to Flange
R" == qR ==
~
Allowable Rivet Shear Load
Paf
~ ~
MS -
Flange Rivets
=1
MS
€} @@
+ 0.414
==@/O =@@
- 1
Rivet Load - Upright to Flange
p
Allowable Rivet Upright Load
p
MS - Upright Rivets
MS ==
Inter Rivet Buckling Allowable
FIR' section 10.6
MS - Inter Rivet Buckling
MS ==
Ultimate Tensile Stress of Web
F tu ' Section 5
u au
@/@
-
@I@-
1
1
Rivet ~ensile StrengthUpright to Web Per Inch
* Rivet Allowable Tensile Load per inch
(i00) HS - Rivet Tension
F
RT
, Section 6
MS=(§)/@-l
IJOTES: (1)
If any of the margins of safety are less than zero, the design is inadequate. The deficient area must be corrected and this table repeated. (2) If the web is subjected to tension or compression as well as shear, the initial buckling stress of the web must be modified according to the method described in Section 10.4.1.
*
See NAG A TfJ 2661, itA Summaril of Diagonal Tension ll , 49 for explanation.
TABLE 10.3
1952, page
(CONTlb) - ANALYSIS OF FLAT TENSION FIELD BEAM WITH ACCESS HOLE AND SINGLE UPRIGHTS 10-47
·8 .0
C 4 .4
.2
0
0
1.0
2.0
3.0
4.0
.!Y.L! d/h
1.5 1.4
1.3 C_ )
1.2
1. 1
1..0 0
D/h
FIGURE 10.32 - ACCESS HOLE REDUCTION FACTORS 10-48
•
STRUCTURAL DESIGN MANUAL Revision A IO.S.B
Analysis of a Tension Field Beam with Curved Panels
TabLe 10.4 presents an analysis procedure for a tension field beam with curved panels. The basic geometry of a sheet/stringer beam with curved panels is given in Figure 10.33. The applied loads are the shear flows (positive acting clockwise) on the edges of the bays and the axial stresses at each end of the stringers (tension is positive). Section properties and allowable stresses are required for each stringer and web.
---- --- ----------- ----/'
- - I/ I
G)
+
---I I I
I
I
r---\
0
'~
-
d
)
<=)
STRINGER NUMBER
o-
WEB OR RING NUMBER
FIGURE 10.33 - GEOMETRY OF BAY WITH CURVED PANELS Table 10.4 is based on a single web with parallel,flanges and parallel uprights. Most beams consist of more than one web. At various locations in the following tabt(~ adjacent panels must be considered. Such a situation occurs for rivet loud, strjnger axial stress, upright stress and moment in stringer.
10-49
STRUCTURAL DESIGN MANUAL DESCRITPION
CD Element
Number
EL
0Skin Thickness
t
G) Stringer
tst
Thickness
(i)Frame Spacing (Panel Length)
d
Height
h
® Panel
®Aspect Ratio
hid
(j)Aspect Ratio
d/h
® Stringer @ Average
Q9 Area Q)
Area Stringer Area
of Frame or Ring
Radius of Curvature
=@/(i) =(i)/®
AAVG = A
<@h
Z =@(1 -
02)~!@®
kg, Figure 10.34 or 10.35 E
of Web (See Note 5)
Tcr =
tT
2 3 @@/12 @2 ©2
@®
@ @
Parameter
300@(!)/
Parameter
RS
=@®I@
3
Parameter
RR
=G)@I @
@
paramet.er
(1
+ @)/(l +
3>
~ Ultimate Applied Shear Flow
qULT
~ Web Shear Stress
"ULT =
~ Shear Stress Ratio
TULT/TCR =
~ Diagonal Tension Factor
k, Figure 10.22
@
Parameter
~ Pure Diagonal Tension Angle
@
10 @/
3
<@)/ Q) (@ 1 @> ~ I a
pDT ' Figure 10.36
TABLE 10.4 - ANALYSIS OF CURVED TENSION FIELD BEAM 10-50
I
R
Q)
~ Elastic Modulus ~ Elastic Buckling Stress
+~+1) 12, Note
RG
/:L
~ Shear Stress Coefficient
)
AST
~ Poissonls Ratio Parameter
NUMERICAL VALUE
VARIABLE AND EQUATION
(I +
@)
z
STRUCTURAL DESIGN MANUAL ~ Web Allowable
@ @
r: l1 , Figure 10.29
Parameter
l/RR = 1 /
3
Parameter
l/RS = 1 /
@
Q9 Correction
Factor
II , Figure 10.38
QY ultimate Allowable Shear Stress
@ 'Margin of @ Parameter
QY
=@
Tall
Safe,ty - Web
(.65
+Q9>
MS=QY/@-l a/a pDT ' Figure 10.37
Tension Field Angle
a=
@{~~
~ Rivet Shear Flow
qRIVET' See Note 2
Q9
R"
Required Rivet Shear Strength
~ Allowable Rivet Shear/ ' MS -
Rivets
MS
~ Stringer Stress at the
@Cl/cos
0 - Ill·
= @~~ -
1
_ -~ @
Q) ,. -<@1®®)+·5(1.-@>
O"ST -
, Ring
@ €y
+
gall' Section 6
Inch
@
= @[l
Stress Ratio
O"STMAX/O"ST' Figure 10.23
Maximum stringer Stress
O"STMAX =
~ Average Maximum Stringer
@~
CTSTMAXAVG
Stress
= <@n
+ @n+l)/2, Note 1
~ Thickness Ratio ~ Avg. Diagonal Tension Factor
)
~ Forced Crippling Allow-
u, Figure 10.26
o
able
@
Moment in Stringer
MST
= @@~ 2tan
€) Allowable
Moment ·(Outside MACe of Stringer)
~ Allowable Moment (Inside
I2l
24@
MAC I
of Stringer)
t4P
"!31
@
Local Crippling Stress
cr Note 4 Bending Stress @ Center 01 fb Bay (From External Bendinc) C 0",
~
TABLE 10.4
(CONTln) - ANALYSIS OF CURVED TENSION FIELD BEAM 10-51
STRUCTURAL DESIGN MANUAL ~ Bending Stress at Ring
f
(From External Bending)
~ Stringer Tension Allow-
F
able
br tu
~ Local Crippling Allowable Fcc' Section 10.7 ~ Effective Length of
J,'
Stringer
~ Radius of Gyration of
p
Stringer
~ Slenderness Ratio ~ Johnson-Euler Buckling
)
~/® C, Section 11
Coefficient
~ Johnson-Euler Column
Fe' Section 11
Allowable
~
MS -
~
MS -
Stringer at Center of Bay Stringer at Ring
Figure 10.39 Figure 10.40
NOTES: (1) Average over stringer lelgth, d
(2) qRIVET along stringer is the difference in shear flow between. adjacent panels cr the shear flow in the outside skin of the lap' spl;ce if one exists, whichever is greater.
(3) Average of adjacent pane s (4) Portion of element adjacEnt to skin (5)
If the web is subjected ~ ..o tension or compression as well as shear, the in .. tial buckling stress of the web must be modified according to the method described in Section 10.4.1.
TABLE 10.4 (CONTID) - ANALYSIS OF CURVED TENSION FIELD BEAM
10-52
STRUCTURAL DESIGN MANUAL 1.0
3 r---
I--
I-I--
r-
f-f-I.....-
)
k
s
•
I--
d
r-
1
r-
1
-
1
10
Z
I
10 2
2 h Rt
2 1/2
= - (1- J.L )
) PLATES LONG AXIALLY (d2h),
T
cr
I I I
f-f-
1-1-
1-1-
10 3
=<0
, elastic
k
rr2Eh 2 s
2 2 12R Z
FIGURE 10.34 - CRITICAL SHEAR STRESS COEFFICIENTS FOR SIMPLY SUPPORTED CURVED PLATES
10-53
STRUCTURAL DESIGN MANUAL
-----
I--
>"--
I-~
I-I--
f-- I--
~ 0.-
s
I--
f--
I--
f-- I -
I--
~
"i'" I" 1\ f\ r\..
2 . 0 , '\,
inder\
I--
I-I--
~
~
~
.... v /
1
I
r'\
f'. l'
1'1\ ~I- ~f-"
~V ...
ti
.J""
il-
f-"
J. ~ 10""
-
'1-
'-
~
.1-
LJ}-=
I-I-I--
II1i
L..-
1
I
I 1.1 I II
10
d2
2 1/2
Z = Rt (I-Ii)
::
©
PLATES LONG CIRCUMFERENTIALLY (h2d), T , elastic cr
FIGURE 10.35 - CRITICAL SHEAR STRESS COEFFICIENTS FOR SIMPLY SUPPOR'L'EO CURVEt) PLATES
10-54
ffi
I-
~
1
)
I
.... 1.-' Vt::: ~VL.-v V""V ~ vV'
I'
./
r-T'
~V/
I
f\
~
//
I
.....
I
I--
v ... ./ v
I I I
'-
cy1 _
~~ ~""
/v
~
~~ 10
hId
1010- ~ 1.0 10- ~ I--1-1.5
I---
k
I-- 10I-- I--
•
STRUCTURAL DESIGN MANUAL 25 .. ,
•
1\
\
\
\
20
l\
,- ~-
'\
h E
)
"I'
10
'\
""
~
l+R R -- --
=R
"
2.0
1'\
1.5
I
~
f
r
f'..
(l+R )1/2
...... ~
I' ~
I
,
.5.-
R
~
I'-..
1.0 I
....
5
......
:-...
"'Iii;
~
;.....,
,
.....
.....
25
"
~
1\
~
\
"
I
I
l\ 2.0 \
1\
@=_'Rt_T_)_
I
\
1/2 LO
,
1.0 I\. ~' ~
.5
R
/
a
"-
, ......... '"
.......
i""
f'
""'-
o
"
-
~-: --
...... "-
, ......
.......
35
'" ...........'" ....."
' " ...... 1'00..
......
" ~
30
=
~I" 1\.."'-
.....
25
-
h"-
i'\
"
5
-
-
~
R
..
I'
20
~ -% ~
_\1 f\.
1\
(l+R ) I /2
)
...
1. ~
\
1\
50
~~
,
\
1\1
diE
.
,
[\
\
15
45
40
t I
.\
\
~
S 1+1)
1\
1\
III
~
"'- '" " 1\ '" f',.
,
35
...... I'
i..... "-
"I
l+R
,,
\
20
I
1\'
.....
r-...
1"10...
30
,
\
.....
l'o
o 25
,
!'o.
b-
.......
"" 20
--
a
l..
..
" " I
,
"'-
~\1~
f- .....
''\
I'
1/2
@=_idT_}_
,
"~.
- f-f.......
S -
";,
\
1\
15
l+R
40
~
......
i'-....
i' 45
~
"-
"
50
ct pDT
•
FIGURE 10.36 - ANGLE OF PURE DIAGONAL TENSION
10-55
STRUCTURAL DESIGN MANUAL 1.0 .. 9 .8
-Ci-
Ci
.7
pDT .6
)
.5
o
.1
.2
.3
.5
.4
.6
.7
.8
.0
1.0
k
FIGURE 10.37 - CORRECTION FACTOR F'OR ANGLE Or' DIAGONAL TENSION (1 IRS l/RR::;: 1.0)
00
.4
-:::
I~ ---:::
~ /' ~V
•3
~ //
v
,....... -"'"
i--'
'/' V . / V /' V '/ /. '/ V h v v ~ ~ 0 '/ ..- / 'V h ~ ~ V / ~/ ' , / j I/. 1'/ V / ~ ~ ~ 1/ V
6 .2
/J VI /J / V
/, 'lJ Z 7/ V
rl. '1/ /1 '/
•L
V"
/J ~ '/ /1 'JJ II //
v. J I
~I
a I o
.5
~
~ ~ ~ ~
i--- ~ ~
!--- I-~
::::: t-- r- 1.0 1.5 ~
...... i-- ' - -
!--- I--
-'
ttt
M
1.0
-
I
1.5
2.0
FIGURE 10.38 - CORRECTION FOR ALLOWABLE ULTIMATE SHEAR STRESS IN CURVED WEBS
10-56
-
1 1 • O.3tanh(RR)+O.ltanh(Rs) I
e
0
~
I
.8
.4
!--- 1--'
V
.6 .2
I
I
~
"" -~~ i-- r-
!--- I-i-- " . . '
!/ff
I
IJ /J /
- ----- -- --- ----- --- --- v -- --
:..v ,.... ,-
V
I--I-;:: ~ !---
l/RS
)
STRUCTURAL DESIGN MANUAL ~
NO
______~ Revision A
YES
YES
NO
") YES
NO
YES
) INNER
OUTER
DETERMINE M.S. FROM FIGURE 10.41
FIGURE
lO~39
- STRINGER MARGIN OF SAFETY AT CENTER OF BAY
10-57
STRUCTURAL DESIGN MANUAL Revision B YES
~ = 0
.) YES
-f br Rl = -Fcc.
-U
R
- CT
ST = S F cc
R
OUTER
S
=F-ST cc
INNER
DETERMINE M.S. FROM FIGURE 10.41
FIGURE lOe40 - STRINGER MARGIN OF SAFETY AT RING 10-58
•
STRUCTURAL DESIGN MANUAL
. R .
R
2
;R
1
2
COMP COMP
. 1
R
1.0 '
TEN
R~ COMP
)
t ..
,
I
",.,
.. '
CO~PRESSION!
. FAILURE
i i' ,
'\'
.
,
·R I
.
1
1
j
.~
(-R )1.5
1
,
l
_
,.'
i
,
•
,
,
..
OUTER FLANGE:
.. t
lo.. __
i
Rl TEN~ .. R2 TEN~
INNER FLANGE:
)
~
. TENSION FAlLURE
."
I R1
. ...... "1-' I
R
Z
COMP-
TEN
- .!
-.
=R +R 2 S M R =R -R 2 S M
R
FIGURE 10.41 - STRINGER MARGIN OF SAFETY INTERACTION DIAGRAM
• 10-59
STRUCTURAL DESIGN MANUAL 10.6
INTER-RIVET BUCKLING
The {!([eclive sheet an.l8 and stiffener are considered to act monolithically in most dL·signs. If, however, the rivet spacing is too large, the sheet wiLl buckle between the rivets before the crippling stress of the stiffener is reached. Thus, the sheet is Less effective in aiding the stiffener in carrying compressive loads. It is, thcrcfoye, necessary to calculate the inter-rivet buckling stress to ensure that inter-rivet buckling does not occur.
In caJculating the inter-rivet buckling stress, it is assumed that the sheet between adjacent rivets acts as a column with fixed ends. The general column ('quatian is
)
where C is the end fixity coefficient and is equal to' 4 for fixed end supports. Si.ne£:' the effective. column length Lt = L/~C'=
Fe = ~2 E /(L'/p)2 t
The rndius of gyration p for a unit wi.dth of
sht~L't 'is 0.20 L. Lelting lhe' dvel spacing p rcplflce the coLumn length L. the above equation becomes:
F. if"
=
2 (£. /0 29t) JC ·
•
The fixity coefficient C 4 is used for flat head rivets or bolts. For spotwelds it should be decreased to C = 3.5. For Brazier head rivets or screws usc C = 3 and for counter-sunk or dimpled installations C = 1. The inter-rivet buckling allowable stresses based on the column allowables of Section 11 and using the previous equalion aTe shown in Figure 10.42.
)
10-60
•
STRUCTURAL DESIGN MANUAL FIGURE 10.42 - INTER-RIVET BUCKLING
ALLOWABLES FOR ALUMINUM BARE COUNTER-SUNK OR DIMPLED (C=l)
~2Et/(p/O.29t)2
Fir =
t
•
I
,
;
i
.
!:
.
!
2024-T86
< .25 < .25 < .50 < . 063
A
5
2024-T86
> ~ 063 < .5
A
6
7075-T6
.016
.039
B
7
7075-T6
.04
- .249
B
1
2024-T42
2
2024-T3
3
2024-T36
4
P
I
r
Basis
:=
A B B
RIVET PITCH
t = SHEET THICKNESS I
;
i I
i I
\
1 ;
1-
30,
;
I
i ,
I
I
,
~
.. ..,
I
;
..•.
1-~
i
~
I
1
)
20~
..
10~
•
I
i . I '
•
} : 1
0:
o
j
10
10-61
STRUCTURAL DESIGN MANUAL FIGURE 10.42 (cont-d) - INTER-RIVET BUCKLING ALLOWABLES FOR ALUMINUM BRAZIER HEAD RIVETS IN BARE AL. (C=3)
~2Et/(P/.502t)2
F.
IT
,
;
i
•
·. :1
2024-T42 2024-T3 2024-T36 2024-T86 2024-T86 7075-T6 7075-T6
1
2 3 4 5 6 7 l
P
,J
~ I
.~~
4
r
Basis
< .25 < .25
A B
< .50
B
< .063 > .063 < .. 5
A A B B
.03£) .24£)
.016 .04
":)
RIVET PITCH = SHEET THICKNESS
I
i
-pol
t
t
,'.
i
i
!
!
i
°1'" 1
I
I
20fl~I , .:
~
... t,
I
i "
'j, ___ i,_
'
t
,
.
~
...
j
"~'
!
i ;
0'·;
o
, 10-62
•!
20
!
! !
! I
!
I
I
I
I ..
.L
L .!
,
I
I
!
•
Ii
~ 1
t
;
~
j
!t
j ,
J,,:_~. i :L} 30
I
f
i
j
;
.
~.> 1.J ;,;i
-.!,.~ t'
I :
)
I
I
i
I 1--
•
I
)
j
I I i
:-1-
I
1
j ,
I
:, !
j I
I
, I
i
I:
I
I
I
1 ,! •
• I :
I
!
,
1
,,',I
'I
t
10\; j
,.,'
.....
i
Ii :
~
I'!!'.l_ .i ! 1 'I
4
I
!
t ;'
I
'. ~ ~ !
II',
.1 PI t
;,
I
I
I
..
i
j
I
~
J.
40
50
-
!
, iI
i
I 60
. I
•
STRUCTURAL DESIGN MANUAL VIGURE
lO~42
80~ ~---
(cont'd) - INTER-RIVET BUCKLING ALLOWABLES FOR ALUMINUM
SPOTWELDS IN BARE ALUMINUM (0=3.5)
= W2Et /(p/.543t)2
P.
i
lr
70;
)
2024-T42 2024-T3 2024-T36 2024-T86 2024-T86 7075-T6 7075 .. T6 p
50~
t
t
Basis
< .25
A B B A A
< .25 < .50 < .063 > .063 < .5 .016 .039 .04 .249
B B
= RIVET PITCH = SHEET THICKNESS
...... til ~
~ J.t ......
~
40j
II
30!
- II I
· ! ! .
)
· ! / )
I
20t I
I •
I
I
, ! I ,
~
·
I
I
10.
o
o
10
20
10-63
STRUCTURAL DESIGN MANUAL FIGURE 10.42 (cent'd) - INTER-RIVET BUCKLING ALLOWABLES FOR ALUMINUM FLAT HEAD RIVETS AND BOLTS IN BARE AL. (C=4) , ..
l I
F.
1r Basis --A
t
2024-T42 2024-T3 3 2024-T36 4 2024 .. T86 5 2024-T86 6 7075 ... T6 7 7075-T6 1
2
, I
;
i ;.
--t-'
~
!
! I
,I
< .25 < .25 <.50 < .063 > .063 < . 5 .03<) .016 .249 .04
')
B B A A B B
j I
'
P t
RIVET PITCH
= SHEET
THICKNESS
• I ,
,
• • .,4
I
:I ; " .... -
-
I
•
i
••
j
I
~
)
;.
I
10
10-64
50
60
l
•
•
~_II STRUCTURAL DESIGN MANUAL ·..'. ·r· ..· . \§ ~
\_ . / ....... -~ ..
80~
I
FIGURE 10.42 (cont'd) - INTER-RIVET BUCKLING ALLOWABLES FOR ALUMINUM
j
, I
,I
,!
COUNTER-SUNK OR DIMPLED IN CLAD AL. (C=1 )
F. lr
= rr2Et/(p/.29t)2 t
..
I
I
2 60:
3 4
5 6 7 8
50
9
10
2024-T42 2024-T3 2024-T36 2024-T6 2024-T81 2024-T86 7075 ... T6 7075-T6 7075 ... T6 7075-T6
< .063 < .063 < .063 < .063 < .063 < .063 .016 - .039 .062 .04 .063 ... 187 .188 .249
Basis B
B B
A A B
B B B B
• 30:-
)
•
0-
o
10
20
i
I
PI t
30
40
10-65
STRUCTURAL DESIGN MANUAL , .,
! ;
1 • -.
:,
FIGURE 10.42 (cont'd) - INTER-RIVET BUCKLING ALLOWABLES FOR ALUMINUM
80! ... -, ;..- ~.. ~ ; i ;' , ' : '
BRAZIER HEAD RIVETS IN CLAD ALUMINUM
I ,
I
(C=3)
F. i
~2Et/(p/.502t)2
=
1r
,
Basis
t
1
2024-T42 2024-T3 2024-T36 2024-T6 2024-T61 2024-T86 7075-T6 707.5-T6 7075-T6 7075-T6
2 3 4
,i
5 6 7 8 9 10
< .063
< .063 < .063 < .063 < .063 < .063
.016 .039 .04 - • 06~
.063
.lB7
.188
.249
p = RIVET PITCH
= SHEET, THICKNESS
t .
,
! !
•
I
it I
•
I
...... -'r 1
I
i _: i
30~" L... ' ; I ~
l:-
!
, (
,
!
i
, I ;
I
20:'! : ~
~
,
,I
! .
I
I;
, i
;
~
.
,
..
I'
•
~
:
I -".
01.
!
!
,
I
,
, J
~ ~..
)
~
! . I
,
;.
.
: t ,
,, !,
I
!
.• I . •
r
~
: : i ;
I
I __ I ':
j
I
!
I~
...
...!
t
i.-': ; 'i'
•• ,I.
-
.j
--
.,
J
;~
,
,
.
.
~
;
. i
:
I
0' ,
10-66
,.! ~..
I .
•
o
~ • ;
j;
,
!
\
i
: r
,.i
i
I
~
I
: :
,
! ,,
!
:
:
.... ! :; •: j!
1
'j
•
i
!
,
'!
. i
, !
. .
I
, !
I
-, ... :, .. !_-
10
I I'
20
, I
.. __- ' .i, .'
30
~/
p/ t
40
50
60
1111\'\\,
"~~'" .'
-
"'''
/ i/
STRUCTURAL DESIGN MANUAL
1.,.;-:'"
FIGURE 10.42 (cont'd) - INTER-RIVET BUCKLING ALl.OWABtES I"OM
ALUMINUM
80
SP01VELDS IN CLAD ALUMINUM ( C=3.5
F.lr = ~2Et/(p/.543t)2 Basis
t
< .063 < .063 < . 063
1 2024-T42
70
)
2 3 4 5
. ,
2024-T3 2024-T36 2024-T6 2024-T61 2024-T86 7075-T6 7075-T6 7075-T6 7075-T6
6
7 8 9 10
60
< . 063 < . 063 < . 06'3
A
B
.. 016 - .. 039
B
.. 04 - .• 062 .063 - .187 .188 ... .. 249
B
B B
P = RIVET PITCH t = SHEET THICKNESS
!
! · I
50
B
B B A
!:
•
·
.....
. ,,
,
'1}
~
~40
· i .
.....~
•
I
Jl.:..
, ,
l
·
i
30
.! r I
)
.
· I 20.
.
I
" j .
I,
10'
I
i
:j I
, · i
0 0
,
!
I
· i · ,
I
: . : .
"
~
10
20
30
pit
,
· 1
, ;
,
,
!
. !
'
~.-
40
50
60
10-67
STRUCTURAL DESIGN MANUAL FIGURE 10.42 (cont'd) - INTER-RIVET BUCKLING ALLOWABLES FOR ALUMINUM
•
FLAT HEAD RIVE-TS AND BOLTS IN CLAD AL. (C=4)
Fir
2
rr Et /(p/·58t)
=
2 t
1
2024-T42
2
2024-T3 3 2024-T36 4 2024-T6 5 2024-T61 6 2024-T86 7 7075-T6 8 7075-T6 9 7075-T6 10 7075-T6iI
B
< .063
A
< .063 < .063
A B B B .B B
B B
)
= RIVET
t =
_i
< .063 < .063 < .063
.016 - .039 .04 .062 .063 .187 . 188 .249 p
i .
Basis
PITCH SHEET THICKNESS
: i -
-
•
;
, I
:
~
•
t
I:I
I
,
I
,
,
,~
,
I
,
,i , .
l :'
,
,
)
,
I .. •
t ~ ... _
10) , , : '
~
I
'
~
I
I
j
I'
••
,
'
!
,
~
j
,
10
20
1
,
I
J ..
10-68
"
I
- I
~ •
"
,
, :
I'
i,
. 30
pit
40
50
60
•
STRUCTURAL DESIGN MANUAL 10.7 COMPRESSIVE CRIPPLING Introduction
)
Compressive crippling or local buckling is defined as an inelastic distortion of the cross-section of a structural element in its own plane (rather than along the longitudinal axis, as in column buckling). The crippling stress, which is the maximum average stress developed by a structural shape, is a function of the cross-sectional area rather than the length. The crippling stress for a given cross-section is calculated by assuming that a uniform stress is acting over the entire section, Pce = Fcc' A. In reality, however, the stress is not uniform over the entire cross-section. Parts of the section will buckle at a stress below the gross area crippling stress, while the more stable areas, such as intersections and corners, reach a higher stress than the buckled elements. Method of Analysis The allowable crippling stress may be obtained from the procedure outlined below. 1. Divide the section into individual segments as shown in Figures 10.43 and 10.44. Define for each segment a width b and a thickness t. Each segment will have either zero or one edge free.
2. The allowable crippling stress, Fcc, for each segment is obtained from the compressive crippling curves of Figures lQ.43 or 10.44. 3. The allowable crippling stress for the entire section is found by taking a weighted average of the allowable stresses for each segment: bltlFccl + b2t2Fcc2 + ... ~bntnFccn F cc = b i tl + b t2 + ... .I; bnt n 2
=----.. . . .
The same procedure is used to analyze formed and extruded sections. Care must be taken in segmenting an unbalanced extruded section. When the thicknesses of the segments differ by a factor of 3.0 or more, the excess thickness should be discounted in calculating both the crippling stress of the segment and the effective load carrying area of the section. Also note that the bend radii of formed sections are ignored. For formed sections with lips, Figure 10.45 may be used to determine whether the lip provides sufficient stability to the adjacent segment.
10-69
~
~
?~r---------------------------------------------------------------------------------------~~,~
~'~
FIGURE 10.43 _ COMPRESSIVE CRIPPLING FORMED SECTIONS, GENERAL SOLUTION
o
1 EDGE FREE
1 EDGE FREE
t
-~
t
:~ ~r_
:E b N tN F cCN F = --.......;~~--.;;;.;. cc l: bN tN
t
2
o EDGES
.'. :-r
i" ~~. h" ~--~= .
FREE
.10r-~:,--rl!-·~~-'t----~---r--~~~~,·--~I.~!~·-~I.·-·:~~·_~I-_~·.~~
.OB't--.....-j-+---,--I·-~-·.;cutoff at IF .. , . . cy: .
F
j
.
i
; .
j . 1::"
.
I
.I
cc
\/Fcy
E
No edge free C
.67882
o
A o
=-
.81940
One edge free C
A 7
10
20
30
40 b t
50 60
80
100
150
1.1
o
o
.35728 = - .82571'
-
I
~
-.
"'"
"
FIGURE 10.44 - COMPRESSIVE CRIPPLING OF EXTRUSIONS, GENERAL SOLUTION 1 EDGE FREE t2
1 EDGE FREE
(I)
o
-I ::ICJ
c:
EDGES FREE
.10
~~~~~~----~--~!~~~!~~:~--~-~!.-:~I-'~'.:~'~:~!~·~:_--~
.OB
I---r--t-:-:-~...,""""""l~-- tuto~f at ~ ~
I ;I: !.1
cy'
.~.'
:.
c:-)
-I
,.= r-
/
I
.... _..
= Co (b/t)Ao
Fcc
::ICJ
./Fey 'E F
cc
No
JF CY E
..
~"'I~' ~ '~'-- ~. 1 . .:.~, -1--- -.... 1 .Dl~~~L~,~~·i~I~"I·~,~·_···~·I~· __~l__~i_'·~~_·__~~~_.~._,,~,,_. ~~~~~:_.-~. 5
o
·1
7
10.
I
I-'
20.
.....
3D
40
_.
=
.62153
(I)
A
= -. 73562
C i)
One edge free
,.iI:
~----------------------------------
0
C
=
A
=
0
z
.34881
z
.
50 60
80.
100
b t ____________________________
0
"......<: r('[)
150.
10-"
~
__________________________
,.
c:.
-.74858
rn
-
I-'
......
I,
rn
C
0
.. I
~
edge free
0
~
::3
tr:l
I-' .
o I
....... f',.)
FIGURE 10.45 - DESIGN RANGE FOR LIPS ON FORMED SECTIONS
.......
L
bL
-
1
c- --" "'I I I
-
~ II
-
I
-
b
L
F
Ll
'''-
""'" h/
.3 b
..I
ABOVE DESIGN RANGE: b BELOW DESIGN RANGE: F WITHIN DESIGN RANGE:
V .2
/
v"-
~
O~
~~
~
0p
0
10
20
30
-
~~
~ t--
~MINIMUM LIP SIZE
.1
0
V
BUCKLES BEFORE FLANGE TOO SMALL TO PROVIDE S~MPLE SUPPORT TO FLANGE PROVIDES SIMPLE SUPPORT TO ADJACENT ELEMENT IS TREATED AS A FLANGE IN CRIPPLING ANALYSIS
~
"-
'/MAXlMUM LIP SIZE
LIP LIP LIP AND
40
~
,
r---- r---.
50 b
- tF
k
t--- f----
60
70
80
90
STRUCTURAL DESIGN MANUAL Revision A ]().8 EFFECTIVE SKIN WIDTH The effective skin width is used to calculate the amount of skin that contributes to the stiffness of the attached flange. Figure 10.46 shows several types of skin-flange attachments and the corresponding effective skin widths. The skin width equations are based on the buckling compressive stress equation for sheet panels:
k
)
1f2E
c
12(1 _ 1l 2 )
(l)2 , where
b is the stiffener spacing.
b
If thp stiffener provides a boundary restraint equal to a simple support, then k
c
=
4~0
..
Assuming IJ.
.3,
When Fer is equal or less than the yield stress of the material, the ultimate strength of a sirnpJ y supported sheet is independent of the width of "the sheet. The term b may then be replaced with an effective width term, w.
Solving for w -
w = 1.9 t(F ,
E
)05
10.4
c
The constant 1.9 in the preceeding equation is valid for heavy stiffeners.
For
reJatively light stiffeners a constant of 1.7 is suggested. The radius of gyration of the stiffener should include the effective skin area.
)
For skin-stiffener attachments that develop a fixed or clamped condition -
w
=
2.52 t(E/F
c
).5
10.5
Note A - (Fig. 10.46-b) Staggered Rivet Rows In calculating the crippling stress of the itiffener, use a stiffener flange thickness of three-fourths the sum of the flange thickness plus the sheet thickness.
•
Note B - (Fig .. lO.46-c) t ~ t < 2t Find the crippligg s{ress for the tee section, assuming the vertical ITIE:'mber of the tee has both ends simply supported. For t (equ. 10 .. 4)
(
USE'
10-73
STRUCTURAL DESIGN MANUAL One rivet line
Two rivet
a. Constant Rivet Rows
lines
w
h.
Staggered Rivet Rows
NOTE A
t s< t
< 2t
f
c. Integral Sheet &: Stiffeners
s
I Sec Lion
Tee Section
~ t
s
~
w
.~
NOTE B
I
2
d. Free Edge Condition
2
I
Free
/ r
NOTE C
Edge
I
it
I I
+ I
~Wl ~~I
T
I
I
.
I
I
) w
2
FIGURE 10.46 - EFFECTIVE SKIN WIDTHS 10-74
STRUCTURAL DESIGN MANUAL Revision B
tf ~ 2ts - Find the crippling stress for the I section. The properties should include the I section plus effective skin. Note C -
(Fig. 10.46-d) For a sheet with one edge free, the buckling coefficient is 0.43. The effective width wI on the freeedge side of the attachment is: wI =
o. 62
t (E/F c
r5
) 10.9 JOGGLED ANGLES
Figure 10.47 shows crippling efficiency factors of each joggled leg of aluminum angles with joggle depth (D) relative to thickness (t) of 0 to 3, and joggle length-to-depth ratios (LID) of 4, 6, and B. 1.00
•
.80 K
8
_4_ 6
.60 • • •
.40
o
2
1
3
Depth to thickness ratio, D/t
)
FIGURE 10.47 - CRIPPLING EFFICIENCY FACTORS OF JOGGLED LEG OF ALUMINUM ANGLES
The allowable crippling load is P cc = Kbt Fcc for each leg, and Pcc = KlbltlFcCl + K2b2t2Fcc2 for the joggled angle. The allowable crippling stress equation on page 10-69 then KlbltlFcCl + K2b2t2FcC2
becomes Fcc = •
bit l + b 2 t 2
for joggled aluminum angles.
10-75/10-76
STRUCTURAL DESIGN MANUAL
SECTION 11
COLUMNS AND BEAM COLUMNS 11.0
GENERAL
The stresses that a structural element can sustain in compression are functions of several parameters. These parameters are:
)
(1) (2)
(3)
(4) (5) (6) (7) (8) (9)
•
The length of the element along its loading axis, The moment of inertia of the element normal to ks loading axis, The cross-sectional variation of the element with length, The eccentricity of the applied load, The continuity of the integral parts of the element, The cross-sectional characteristics of the elements, The homogeneity of the element material, The straightness of the element, and The end fixity of the element.
The effects of these parameters can be categorized by first establishing certain necessary assumptions. For the following analysis, it is assumed that the material is homogeneous and isotropic. It is further assumed that the element is initially straight and, if it is composed of several attached parts, that the parts act as integral components of the total structural configuration. The remajnder of the previously mentioned parameters dictate more general classifications of compression elements. If a compression element is of uniform cross section and'satisfies the previously mentioned assu:nptions, it is referred to as a simple column and is lreat(:'d In Section 11.1. On the other hand columns with non-rigid end supports, intermediate string supports or eccentrically loaded are called complex columns and are treated later in this section. 11.1
Simple Columns
In general, failures in simple columns may be classed under two headings: (1) (2)
Primary failure (general instability) Secondary failure (local instability)
Primary or general instability failure is any type of column failure, whether elastic or inelastic, in which the cross sections are translated and/or rotated but not distorted in their own planes. Secondary or local instability failure of a column is defined as any type of failure in which cross sections are distorted in their own planes but not translated or rotated. However, the distinction between primary and secondary failure is largely theoretical because most column failures are a combination of the two types.
11-1
STRUCTURAL DESIGN MANUAL
Figure 11.1. illustrates the curves for several types of column failure.
c,) ~
Fcc
r--_",,--
•
Tt)rs ional Ins tahU i ty
s
Johnson Parabola
o
,
a
Slenderness Ratio, LIp
Figure 11.1 - Types of Column Fai luC(~ /
L represents the effective length of the column and is dependent upon the manTlt!r in which the column is constrained, and p is the minimum radius of gyration of the cross sectional area of the column.
,
For a value of L /p in the range nail to lib It , the column buckles in the classIcal Euler manner. If the slenderness ratio, C'IP, is in the range of nO" to "a", a column may fail in one of the three following ways: (1)
(2)
(3)
11 .. 2
•
Inelastic Bending Failure. This is a primary failure described by the Tangent Modulus equation, curve mn. This type of failure depends only on the mechanical properties of the material. Combined Inelastic Bending and Local Instability, The elements of a column section may buckle, but the column can continue to carry load until complete failure occurs. This failure is predicted by a modified Johnson Parabola, "pq", a curve defined by the crippling strength of the section. At low values of L' / P the tendency to cripple predominates; while at L" / p approaching the point tl q", the failure is primari.ly inelastic bending. Geometry of the section, as well as material properties, influences this combined type of failu·re. Torsional Instability. This failure is characterized by twisting of the column and depends on both material and section propt.!rties. The curve tl rs " is superimposed on Figure 11.1 fOT iLlustration. Torsional instabilIty Is presented in Section 11.4.
•
STRUCTURAL DESIGN MANUAL t 1. I. I
\:ong Ela.stic Columns ;
A column with a slenderness ratio (L /p ) greater than the critical slenderness ratio (point Ita" in Figure 11.1) is called long column. This type of column fails through lack of stiffness instead of a lack of strength.
The critical slenderness ratio is given by: (
L 1/
P ) crit
(11.1)
The critical stress in the column that produces Euler buckling is 2 7r E F . cnl
(11.2)
Values for F . for various materials are shown in Section 11.1.4. crlt J
The value for L depends on the columnts end constraints.
The effectIve length
is Lt = L/
JC
(11.3)
The values of C for different end constraints are given in Figure 11.2.
11.J.2
Short Columns .I
A col umn wi th a slenderness ra tio (L / p) less than the cri tical slenderness
ratio is called a short column. This distinction is made on the basis that column behavior departs from that described by the classical Euler equation, Eq. 11.2.
)
Elements of a short column may buckle, but the column can continue to carry load until comptete failure occurs. This failure is predicted by a modified Johnson Parabola, tlpqlt, in Figure 11.1, a curve defined by the crippling strength of the section. The Johnson Parabola defines a column allowable stress as:
Fcrit Fcc [1 _Fcc «L 4 /p)2)] 1
=
~2E
(11.4)
Values of F . for various materials are shown in Section 11.1.4. crlt
11-3
STRUCTURAL DESIGN MANU-AL
END END COLUMN LOADING AND CONSTRAINT COLUMN LOADING AND CONSTRAINr~ ~~___E~ND~C_O_N_ST_RA __I_NT_S______~~~,~~::~~:~~-,.r~T·~~Nr~____E_ND__C-O-N-ST-RA--I-N-T-S-------+~n~n~IF.~FF~'T~r.,l~'F.~T +~ UNIFORM COLUMN C = .794 4L. UNIFORM COLUMN C = 1 ~----r DISTRIBUTED AxIAL I AXIALLY LOADED ~ I LOAD, 1 =1.12, PINNED ENDS L = 1 ~ ·ONE END FIXED, .[C ~ 111 ~ ----L . ONE END FREE
1~
I
tp
~?
II
P UNIFORM COLUMN' AX IALL Y LOADED FIXED ENDS
C
=4
1
=.5
JC
lp
UNIFORM - AXIALLY ONE END ONE END
COLUMN LOADED PINNED FIXED
UNIFORM COLUMN AXIALLY LOADED
C -2.05 1
= 0.7
"J=t C =0.25
ONE END FREE ONE END FIXED UNIFORM COLUMN AXIALLY LOADED PINNED. ENDS ONE INTERMEDIATE SPRING SUPPORT
UNIFORM COLUMN AX IALL Y LOADED,
ELASTICALLY RESTRAINED ENDS STEPPED COLUMN AX IALL Y LOADED
PINNED ENDS
c
UNIFORM COLUMN DISTRIBUTED AXIAL LOAD, FIXED ENDS
C = 7.5
UNIFORM COLUMN DISTRIBUTED AXIAL ONE END FIXED, ONE ,END PINNED
c :;:
= 1.B7 1, =.732
JC
1
=.365
J-c 6.08 (Approx. )
1
=.406
JC
SEE FIGURE 11.3
UNIFORM COLUMN AX IALL Y LOADED PINNED ENDS TWO INTERMEDIATE SPRING SUPPORTS
SEE FIGURE 11.4
AXIALLY LOADED
SEE FIGURE
PINNED ENDS
1.1.10
SEE FIGURE 11.10
TAPERED COLUMN AXIALLY LOADED PINNED ENDS
SEE FIGURE
(See Figure,11.6 for more loading cases.) FIGURE 11.2 - BUCKLING CONSTRAINT COEFFICIENTS 11-4
UNIFORM COLUMN, DISTRIBUTED AXIAL LOAD, PINNED ENDS
STEPPED COLUMN
SEE FIGURE 11.5
11.10
STRUCTURAL DESIGN MANUAL
FIGURE 11.3 - END CONSTRAINT COEFFICIENTS - UNIFORM SECTION COLUMNS WITH PINNED ENDS AND INTERMEDIATE SPRING SUPPORT.
')
KL3 EI
4.0 p
3.8 3.6
140
K 3.4
120
u 3.2
...
p
f-I Z
~ 3.0
K is the
tJ
spr ing cons tan t with units of
~
Ib
~
2.8 2.6
80
~
E-'
~
support is effectively rigid if
2.4
0
u ~
2.2
60
~
3
)
~ f-I
A center
KL
100
~
u
in
EI
H
= 16
rr
2
~
~ 2.0 E-e
u
DO NOT use this chart to solve for Required I.
~
~ 1.8
40
~
1.6
1.4
20
1.2 1.0 0
.1
.2
.3
-
.4
.5
a L
11-5
,,~~'~~~ .
.. '\
/
STRUCTURAL DESIGN MANUAL
//
~ ~/
DESIGN CHART
Torsional Spring ~constant (in-lbs/radian)
Do
not use to solve for P cr -<9
Ii.
J.t2
#J.2> J.tl
1.. 6
1.2
~1
PL
;~
H-+-1
.8
~t~V ,4
r--r-
~~/ ~~/
)
1/ ""
Y "-
~O
~,,~-,~ 0 .8 o .4
End Fixity Coefficient, C
V
/
A
...............
1.2
1.6
2.0
J.t2
PL ANAL YS IS CHART
Do not use to solve for I
t,,~
co
100 ~ 50
~~~~~
30 ".
1'1/ 15 ~"
"
20
10 8 I-'lL6 ./ 5
V
';'
ET 4
IX'~
I"
v'
"
V ./ V
"'"
L.oo"'"
1 ./
l..oo'~
.0 00
V
V
V
1/
~
I~·
V
)
" ~End "[>~
V
V
/
f03020 15 1 0 8 6 5 4 3 2 100 1-l2L EI
Figure 11.4 11-6
~'\I~ . V
...,V
3
2
'\
~
Fixity Coefficient, C
I~
1 .5
Fixity Coefficients-Single Span Columns With Elastic Restraints
;!?7\'\\\
•
STRUCTURAL DESIGN MANUAL
Bell ..... ,..... "
~ \\ \-
~,
,i",,''i
FIGURE 11.5 - END CONSTRAINT COEFFICIENTS - COLUMNS WITH PINNED ENOS AND TWO SYMMEtrnICALLY PLACEJJ SPRING SUPPORTS
p
K is the spring constant with units of Ib/in. DO
NOT use this curve to solve for required I
p
lO----~--~--~--~--~--~~--~--------------~----
~
8~--~~Hr--~--~~~~~~---r_--~---~--_+--~--~
~
H
U
~
7
J..-LI..u.u.Lf---+
13-.
'~
o u
)
6~~~~~+-~~~~~~~~~~~+---~----+--~--~
E-< Z
H
~ u 5 H---I-I--f1L-1;L--;.I'-:::",c-.~--f-lI.:--~--tor~~~~ I
E-< (/.)
:z
8
4~~~~4---~'-~~~~ ~~~~~~~~~~~4---~
Q
Z
~
(:il
:> H
E-'
U
(:il
13-. CL.t
~
.1
.2
.3
.7
.8
.9
1.0
11-7
STRUCTURAL DESIGN MANUAL FIGURE 11.6 - END CONSTRAINT COEFFICIENTS - UNIFORM SECTION COLUMNS SUBJECTED TO CONCENTRATED AXIAL- LOAD AND DISTRIBUTED SHEAR LOAD NOTE: Per found by using these coefficients is P, NOT P1 t shown in .1oading case diagrams.
LOADING CASE
8
7
/
~
tp
tp 3
~P1
·r-iI
~
D'
r
4
lqL
qL
6
_0" N II
u ~
H
5
H
~ ~ ~
/~
0
E-!
zH
/
4
/v
E-I
Ul
P
~Pl
Z 0
t
u
-I"
3
5
./
~
2
~q
V
11-8
/
V
V
~
-
1
~
KIPS INCH
o shear
o
~
.4
~
------- V.----- --- -~
= distributed in
/
V
/'
q
7
I
./
~
p
~/
/
V
U
u
V
/'
E-!
Z
/
qav L p
~
5
I.---- ~
...,..
L..--- V
V
.6
,8
~
1.0
)
•
STRUCTURAL DESIGN MANUAL Revision A In the short column range failure is often due to plastic crushing of the column. In other words, the column is too short to bow or buckle under end load but crLlshes under t.he high stresses. This' column range of stresses is usually referred to as the block compression strength. The influence of end supports on plastic buckling is the same as it is for elastic buckling. The allowable compressive stress is given by: F
. == cnt
)/E t
')
(11.5)
(L'/p)4.
This equation Is very simular to the Euler allowable except the Tangent Modulus is used in place of the Modulus of Elasticity. The values for ~ are calculated using Equation 11.3 and values for C are determined from Figure 11.2. The ratio of Tangent Modulus to Modulus of Elasticity (Et/E) is given by the Ramberg-Osgood relationship L
t
1
+17
n
(F 0.7 )n-1
(11.6)
The value'!": for E t n, and li'O 7 are material dependent. FO 7 Is a secant yield stH:'SS WhLCh is determined oy the intersection of the stress strain curve and a secant modulus curve for EO 7 ;::; .7E. This is shown below.. Another secant stress (F • ) Is needed to determine the constant n. O 8S
FO.7
) nal1lbcrg-O~good j';"rllmC'tf'I'.!I
(; ....STRAIN
The value for n is given by : n
I + rJog (> (t 7 / 7 ) I l o g e ( FO. 7 I F 0 • 8 5 )
1
(ll.7)
'STRUCTURAL DESIGN MANUAL Revision A Equation 11.6 is plotted in Figure 11.7. For a given material, n, Fa 7 and E must be known. Values of n, Fa 7 and E may be obtained from Table 11.1 for several materials. Then assuming values of F, E can be calculated. t
To use this approach an initial F . is calculated and E is determined froul t Figure 11.7.. This E is used incEij~ation 11.5 to determine a new F . which is used to o~tain a new i. This proceedure is repeated until the ~rit~€lf stress does not change after ~ach iteration. The Euler column E
F
)
equation can be rewritten as
(L'/p)2
t
=
crit
7r
(11.8)
2
The problem therefore resolves itself to obtaining an expression £o'r Et/F . from the nondimensional equation 11.6. Both sides of Equation 11.6 arc mucflt tlplied by (Fa 7/ F .). This yields • crlt .. 2 = B
(11.9)
Rearranging and substituting equation 11.8 into 11.9 yields: (L ' /p)2
(lL.lO)
FO.7
7[2
=
E
1
F. . )n crlt + _3n (F crlt FO.7
or
7
FO.7
~
'L'/P
B = -;- ~ r;---E-
(11.11)
'Figure 11.8 shows plots of this equation, F "t/FO 7 versus B for various values cr1 • o f n .. F
.
. cr1t Thus for g1ven values of E, n,and F . 7 ;-F---O Fcrit can be ca(l;Ul~te)d directly by 0.7
=F
F crit
11-10
0.7
crlt FO• 7 .
can be detennined from Fig. 11.8.
(11.12)
l?1\\~~
"'~'t~~,,~ STRUCTURAL DESIGN MANUAL Revision A
)
°o~~-+--~~~~~~~~~-1~~~~~~~ .1 .2 .3 .4 .5 .6 .7 ..8 .. 9 1.0 1 .. 1 1.2 1.3 1.4 L5 F
FO.7
FIGURE 11.7 - DIMENSIONLESS TANGENT MODULUS STRESS CURVES
11-11
STRUCTURAL DESIGN MANUAL Temp. Material
Exp ..
Hr. Stainless Steel AlSl 301 1/4 Hard Sheet Transverse Compo Longitudinal Compo AlSl 301 1/2 Hard Sheet Transverse Compo
1/2 1/2 1/2 1/2 1/2 1/2 Longitudinal Compo 1/2 1/2 1/2 1/2 AlSl 301 3/4 Hard Sheet 1/2 Transverse Compo 1/2 1/2 1/2 Longitudinal Compo 1/2 1/2 1/2 1/2 AlSl 301 Full Hard Shee 1/2 Transverse Compo 1/2 1/2 1/2 Longitudinal Compo 1/2 1/2 1/2 1/2 17-4 PH Bar & Forgings 1/2 1/2 1/2 1/2 17-7 PH(THI050) Sheet, Strip & Plate t 1/2 1/2 t = .010 to .125 in. 1/2 1/2 17-7 PH(RH950) Sheet, Strip & Plate, 1/2 t = .010 to .125 in. 19-9DL{AMS5526) & 19-9D~ 1/2 (AMS5538), Sheet, Strip & Plate 19-9DL(AMS5527) & 19-9D) 1/2 (AMS5539) Sheet, Strip & Plate
Temp. of
RT RT RT 400 600 1000 RT 400 600 1000 RT 400 600 1000 RT 400 600 1000 RT 400 600 1000 RT 400 600 1000 RT 400 700 1000
e, 70
25 25 15
F
tu'
12
8
6
125 125 150 118
80 43 118 108 .. 5 107.5 86 58 53.3 52.8 45.2 160 148 138 112 76 71 70.3 59.3 179 168 159 130
27.0 26.0 27 .. 0 23.2 20.9 16.2 26.0 22 .. 4 20.1 15.6 27.0 24.1 22 .. 4 18.9 26.0 23.3 21 .. 6 18.2 27.0 25.1 23.8 21.6 26 .. 0 24.2 22.9 20.8 27.5 25.3 23.1 21.2
86 150 118 86 175 148 138 112 175 148 138 112 185 168 131 185 l68 159 131 180 162 146 88
85
80.8 79.9 66.3 165 135 105.5 62.6
RT 400 700 1000
180 169 144 88
~62
RT
210
73 28.2 116.5 108.5 LOS.5 94.5 48 45.5 44 40 l63.5 153 152 l27 70 65 65.5 55 183 174 172 141.5 77.5 74 74 58 166
137 106 60
FO.85 ks·j
63 23 105 97 96.5 83.5 37 36 31
30.5 I.S 1 .5 ]42.5 ,140 1 21 61.5 56 56.5 46 L72 164 J 62 135.5 63 59.5 58 42.5 160 129
97 52
n
6.9 5.2 9.2 8.6 8.2 8.0' 4.4 4.7 3.5 4.3 13.2 1 3.2 J j .2 19.2 7.6 6.8 6.8 5.9 l6 1.6
21.5 5.2 5 4.6 3.9 24 16 l J
7.1
29.0 27.8 24.9 20 .. 3
166 146 117
145 126 104
56
47
~05
29.0
208
196
16.4
~44 ~18
61.5
)
,16
7.4 6.8 8.4 6
RT
30
95
45
29.0
' 36.5
32
7.6
RT
12
125
90
29.0
85
74
7.2
TABLE 11.1 VALUES OF RAMBERG-OSGOOD PARAMETERS 11-12
FO.7
lO6 psi ksi
159 8
E c
ksi
110
12
cy'
ksi
ItO 15
F
)
,.' -0" '.,
/111\\\\
•
~ \", \\', "
'::"
I
Bell ..... ....... '
STRUCTURAL DESIGN MANUAL
I ; ji ".,,'
Temp. Exp.
Ma t(' ri ill
Hr. Pili 5- 7Mu ('1'111050)
=
.020 to .L87 in. PHI5-7Mo(RH950) Sheet & Strip, t = .020 to .IB7 in.
E c
ksi
ksi
tu'
FO.7
FO. 85
10 ps
ksi
ksi
6
n
RT
5
190
170
28.0
171
l64
22.5
1/2
RT
4
225
200
28.0
218
IH9
7.3
'RT
22
55
36
29.0
32.7
31.5
RT
23
90 81 68 46
70 61.5 46.2 30.8
29.0
6L5 55 40 28
53 48 32.5 22
125
29.0 27.3 23.2 20.6
III
88 64
113 98.3 68.9 49.7
AISI 4130, 4140, 4340 Heat Treated
150 135 105 76
145 126 88.5 63 .. 8
29.0 27.3 23.2 20.6
145 126 88 62
15
180 162 126 92
79 ,56 09.3 77
29.0 27.3 23.2 20.6
179 176 IS) l56 109.4 105 68 75
50 46 22 9.8
AISI 4130, 4140, 4340 Heat Tre,ated
13.5
200 180 140 104
98 21 87.1
29.0 27.3 23.2 20.6
198 196 172.5 ~ 69 12 t .5 ~ 17 87 83
90 46 25
) Heat Resistant Alloys ~-286(AMS5725A) Sheet, Platc) & Strip
115
95 88.4 81.7
~9.0 ~4.4 ~9.8
52
50.3
SOD 800 1000 RT
t/2 1/2
500 850 1000
1/2
AISI 4130~ 4l40, 4340 Heat Treated
•
FcyJ
F
1/2
Low Carbon & Alloy Steels AISI 1023 & 1025 Tube, Sheet & Bar, Cold Ftnished AISI 4130 Normalized, t >. 18t)i n. 1/2 1/2 l/2 l/2 AISI 4130, 4140, 4340 Heat Treated 1/2
•
e, io
Slwc L
& Strip, t
Temp. of
l/2
RT
1/2 1/2 1/2
500 850 1000
1/2 1/2 1/2 1/2
RT
1/2 1/2 1/2 1/2
RT
1/2 1/2 1/2 1/2
RT
k-Monel Sheet, Age Hardened ~/2 Monel Sheet, Cold Rolled & Annealed 1/2
23
113
18.5
500 850 1000 500 850 1000
15
600 lOOO 1400
140 129
70
27.3
23.8 20.6
96 66.5 45.5
102 88 61.5
41 140 122 83.5 57
24 6.8
7.3 5.2 4.7 10.9
10.9 12 9.2 25 29 18.5
10.9
19
87 81
14
~4.2
93 87 81 50
75 47
12.5 15.3
L3.5
RT
15
125
90
~6.0
88
82
13.5
RT
35
70
28
~6.0
20
17
6.4
TABLF: II. L (CONT'D) - VALUES OF RAMBERG-OSGOOD PARAM'ETERS
11-13
STRUCTURAL DESIGN MANUAL Matt)ria 1
Temp. Exp. Hr.
Temp. of
1/2 1/2 1/2 1./2
RT 400 800 1'200
20
1000 1/2 1/2 1/2 1/2
RT RT
10 10
Inconel-X
Titanium All0;is Ti-8Mn Annealed Sheet, Plate &. Strip Ti-6AL-4V Annealed Bar & Sheet,
t~.
187 in.
1/2 .
Aluminum Allols 20L4-T6 Extrusions t~0.499 in.
2 2
7
1/2 2
RT
7
2 L/2
2 2 ·2
1/2 1/2 2024-T3 Sheet & Plate, 2 Heat Trt'ated, t~.250 .11:12 2
~ 2024-T4 Sheet & Plate, ~ Hca t T rea tE.:'d , t~O. 50 in .2' "" ~ ~
2024-T3 Clad Sheet & g Plate, Heat Tr{'ated, t = .020 to .062 in. 2 2 ')
~024-T6
Clad Sheet &
Plate,
2
Ift'a t T n'u l (·d ~
in.
11-14
F tu t
F
ksi
ksi
155 152 141 104
120 130 105 99 87 70 60 51 28 10 51 31
300 450 600 300 450
62 53 29 10
53 32
E
FO.7
FO. 8S
10 ps
ksi
ks i
105 95.6 90.2 83
31.0 28.9 26.4 23.2
1.04 100 94 89 88.6 84 82 78.6
2]
1.10
15.5 16.0
} 19.5 102 127 124.5 97 93 85.5 82 80.5 77 61 59.5
/3.7 43 22 22 21.5 36
cy'
L26
96 84.5 79.4 60.6 53 42.5 21
8.0 43.5 26 52 4J
22 7.5 43 25.5 40 37 26 7.5 38 34 24 7
c
6
14.1
13.0 11.8
7.7 )0.7 to.2
9.2 7.4 10.2 9.2 10.7 10.2 9.2 7.4 10.2 9.2 10.7 10.3 8.4
53 41.5 20.5 5.5 44.0 26 52.3 40.5 21.5 4.5 42.5 25.0
50.3 40
n 23.5 L7 18.5
50
13.S 24 25 5.4 25 29 20
3H.5 20
19 12.6
3.0
].2
19.5
4.5 42.5 25.2
40 23.5
10.7 10.3 8.4 6.4
35.7 24.8 6.2 36.7 32.5 23 60
15.8 15.6 36 If .5 33.5 - 15 22.8 10.9 5.5 8.2 34.5 J5.6 30 • .5 14.6 21 JO.2 5.7 '8.5
~0.7
35.7
33
1.2
34 24.5 6.5
0.3 8.4 6.4
33
30.3 20 5.5
JJ
22.7 5.8
J8.5
49
0.7
4()
45
II
45 22
10.3
44.3 31.5
700
6
H.4 6.4
40.7 2H 6.0
II
SOD
RT
12
6S
300 500 700 RT
12
65
12
60
300 500 700
RT 300 500 700
RT
H
62
37
6.4
39
)
•
7.9
l~O.O6·~
~
~ ~
TABLE 11.1
400 600 800 1000
RT 300 450 600 300 450
2 2014-'1'6 Forgings l~4 in.
e,
io
300
(CONT'n) - VALUES OF RAMBERG .. OSGOOD PARAMETERS
7.0
H.J 6.6
•
•
STRuc-rURAl DESIGN MANUAL Temp. ,Exp.
Material
Temp. of
e,
;0
Hr.
2024-'1'6 C J ad
SIH~C l
F
tu'
F
cy'
E FO.7 c lO6 psi ksi
FO• 85
ksi
ksi
60
47
J O. 7
47
43
10.6
43.2
42.3 29.5
6
10.3 8.4 6.4
5.0
38.7 26 4.0
]0.8 7.8 4.9
55 50.5
10.7 LO.3
56 5J .2
51.6 46.5
10
35 29.5 20.5 7.5
10.1 .,9 .. 5~ 8.5 7.0
35 29 19.3 6.6
34 28 17.7 6.2
67 54 25.5
10.5 9.4 8.1 5.3
63
10.5 9.4
70 55.8 25.4 7.2 34.5 72 58.5
54.5
7.8
2] .3
18.5
5.3 7.8 ]0.5 9.4 7.8 5.3 7.8 10.5 9.4
6.5 29 58.5 47.8 17.3 5.0 24 63.8 52.2 20.3 6.0 26.5
4.3
5.0 25.3
64.5 54 19.7 7.7 27.2 59.5 46.5 20 5.5
61.6 51.7 17.5 5.5 25.3 57.5 45 18.5 3.5
ksi
n
&
2
P I a tc',
RT
8
Beat TreatC'd, t<0.063
in.
2 2 2
2024-THl Clad Sheet, Heat Treated, t
2 2
300 500 700 RT
21
5
62
300
11.2
2 2 6061-T6 Sheet; Heat Treated & Aged, t < 0.25 in.
7075-T6 Bare
She(~ t
•
RT
2 2 2
RT 300 425 600 425
7
RT
7
in.
2
7075-'1'6 [':x lrus ions, t~O.25
in.
1/2 2 2 2 2 1/2
7075-'1'6 Die FOTgings, t~2 in.
2 2
'I. ~
1/2
)
7075-T611and Forgings, Area~16 sq. in.
~
Z ~
~
~/2
7075-T6 Clad Sheet Plate, t~O.
50 in.
7079-T6 Hand Forgings, t~6.0 in.
TABLI': 11.1
42
450 600
31
26 10.9 15.2
76
8
75
300 450
600 450 RT
7
7L
300 450 600 450 RT
4
72
300 450 600 450
30 70 54 22.5 8 25 58 47.6 18.5 7.0 23 63 51 .6' 20.2 7.6 24
8. I
7.B 5.3 7.8
9.2
52.5
15.6
23.5 5.2
12.1 3.7
32.5
16 16.6 J 3.4
68
26
55.1 45 J6
3.7 22 61.5 50 19
7.2 3.2 8.8 15.2 15.6 12 3.9 10.9 25 21.5 J 3. 7 5.8 19.5
& ~
RT
~
300
~
450 600 450
~
•
10
300
&
Plate, t~O.50
1/2 1/2 1/2 1/2
/2 /2 /2 /2 1/2
RT
300 450
600
8
4
70
67
64
50 20.5 7.7 23 59 47 21 7.0
~0.5
9.4 7.8
5.3 7.8 ~O.5
9.4 7.8 5.3
19.5 20 4.6 3.6 12.4 26 29 12
3.0
(CONTID) - VALUES OF RAMBERG-OSGOOD PARAMETERS
11-15
STRUCTURAL DESIGN MANUAL'
Temp. Material
Exp. Hr.
Temp. of
e,
i..
F tu t
F
ksi
ksi
38 30 20 15 10 34 22
14 12
E
cyt
c
FO.7
106 psi ksi
FO. BS n
ksi
~agnesium Allo~s
AZ61A Extrusions, t~O.249
in.
11K31A-0 Sheet ] /2 t = 0.016 to 0.250 in. l/2 1/2 l/2 HK31A-H24 Sheet~ 1/2 t<.25 1/2 1/2 t/2
RT RT 300
8
12
500 600 RT
3uO 500 600
4
11.1
9.3 4.9 19
17.7
17
14.8
1L
7.8
6.3 6.5 6.16 4.94 3.77 6.5 6.2 4.9 3.g
12.9 10
12.3
B.4
8.9
6.9
7.5
5.6
3.3 1.5.6
I .6 14.6 12.6
1.3.1
10.5
6.7
5.2
L7.3
19 6
4.S 4.2 2.2
)
6.2 5.1
4.9 4.5
• )
TABLE 1.1.1 (CONT'D) - VALUES OF RAMBERG-OSGOOD PARAMETERS
11-16
•
• 1.3
2.8
2 3 5 8
1 .. 2
2.6
:
'_J
~~-¥--*-~----------10
I
~~~~--~+---------15
~~~--~~-;----------20
1.1
2.4
~~~~~~~~--------25 ~~r~~~r-T-r--------35 ~~~~~~~~--------45
1.0
n
2.2
.9
F FO.7
.08
.8
. . . . . . < -.
1.6
_._ ..... - - - - •• _.
. L : !' _
FO.7
I ~
.06
1.8
,
.07 ,7 .
F
,,'.,
I'-~
~
I
•
4
.. - .. -~ ~'J :~:·-T·~ '": :" .. ,
.6
'I
,l.
t
,;
,',
_
;-' '
1.4
~,
:~ ::.;: ,: ~ •. ::: j':', ,:j~.: ,-'; ,, '1-"
:"~: ;~: ~:-:~r~;:~: ;' - .-
.05 .5
- , - ,.-:"-. ' .. -j:
1
,
~
1.2
.-~ .. -'; ~'~'
,04 .4
1 .. 0
.1
0
.2
.3
.4
.5
3.0 1.8.
3.2
.6
.7
3.4
3.6
B
.03 .3 .02 .2
.., .-:.--;.--_. SCALE ,.
1
~
,
.. 01 .1
•
II
••
1...-
;-
I
,I.
1--.2 I I !-' I-'
I I-'
......
.4
4. FIGFRE 11. 8 -
.6 6. COL0~
.8
1.0
8. BUCKLISG
1.2
1.4
10.
B
1.6
1.8
r
12.
=r:/p IT ~
O.7 '
E
2.0
2.2 14.
2.4
2.6 16.
2.8
2 O.
-
STRUCTURAL DESIGN MANUAL Revision C 11.1.3
Columns With Varying Cross Section
•
The conventional Euler critical column stress equation: 11"2 E
Fcrit
=
(L/pt
is only valid for a straight column under compression with constant bending rigidity (EI) and a constant area along its length. When the bending rigidity varies along the length of the column, determination of the Euler load becomes more difficult. In this section column buckling coefficient charts and the appropriate formulas for the Euler loads are given for numerous columns of varying cross section. CPS Program SCSOOI is a computer analysis of stepped columns.
)
The critical buckling load for variable section columns in the elastic range is
given by general equations of the form p
cr
= m E1/L 2 (11.13)
where m is the column buckling coefficient and is a function of the column geometry, bending rigidity and end restraint. Values of the column buckling coefficient, ~ for various stepped columns shown in Figure 11.10 are given in Figures 11.11 through 11.29. For tapered columns with the moment of inertia varying at the ends according to
I
x
=
12 (X/b)
n
•
(11.14)
where b, x~ Ix and 12 are defined in Figure 11.9, the values of the coefficient, m, to be used in Equation 11.13 are obtained from Figures 11.11 through 11.29 for the cases given in Figure 11.10.
r-
~~--------------L --------------~~
p
r_--- r
b
~-~-------.
...-.-_-_----"""!i-. ...- p
------
Figure fl.9 - Column with Varying Cross-Sections
1 1-18
•
e
e
\..t
e A1 : \I,TI
I-zj
H c;J
I ' f '
c:
I
\,
I
\
H-+-+-+: iT! '-+- ;
~ t'rj
..... "'""
\1 I
f--"
o
I
r H
l
a
I
'; I
1
'j
i
t-:3
o
n
FIGURE 11.11
!
r
"'\'
'
fs-~-"
I
- - ..............
FIGURE
FIGURE
FIGURE
FIGURE
FIGURE
FIGURE
FIGURE
FIGURE
FIGURE
11.12
11.13
11.13
11.14
11.14
11.15
11.15
11.16
. 11.16
~ ttl
~CD
en
en ~
::a
c-)
o
--I
~
8 t"'4
c:
3:
::.:.
c::
:=cJ
z
en o
r-
"'2j
<:
~ > t:J;;I H
t"""4 ~
..., ~
FIGURE 11.17
FIGURE 11.17
FIGURE 11.18
FIGURE 11.19
FIGURE 11.20
FIGURE 11.20
FIGURE 11.21
FIGURE 11.21
FIGURE 11.22
FIGURE 11.22
-
(I)
cu :z
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en C/)
I
s:::
en
tTl
n
::.:-
t-:3
H
o
z:
Z
c:
::.:FIGURE 11.23 I-'
.I
I-'
>.0
FIGURE 11.23
FIGURE 11.24
FIGURE 11.25
FIGURE 11.26
FIGURE 11.26
FIGURE 11.27
FIGURE 11.27
FIGURE 11.28
FIGURE 11.29
~
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c::
tzj
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(/)
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r-
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o
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c:
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:::a
):81
r1.0
CJI
a
Q
T
L
FIGURE 11.11 - BUCKLI~G OF VARIABLE SECTION COLl~S: STEPPED COLUMN WITH BOTH ENDS PINNED AND NO TRfu~SVER'SE AXIS OF SYMMETRY.
rrI
en
-:z
FIGURE 11.12 - BUCKLING OF VARIABLE SECTION COLUMNS ': STEPPED COLUMN WITH ONE END PINNED ~~D THE OTHER FIXED AND NO TRANSVERSE AXIS OF SYMMETRY.
en 31: ):81
:z
c: ):81
r-
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;=~.
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r
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0.1 0.2 0,3 0.4 0.5 0.6 0.7 0.8 09 10 o
T
eI ,."
FIGURE 11.13 - BUCKLING OF VARIABLE SECTION COLUMNS: STEPPED COLUMN WITH SMALLER HOHENT OF INERTIA AT ENDS, BOTH ENDS PINNED, AND A TRANSVERSE AXIS OF SYMMETRY.
FIGURE 11.14 - BUCKLING OF VARIABLE SECTION COLUMNS: STEPPED COLUMN WITH SMALLER MOMENT OF INERTIA AT ENDS, BOTH ENDS FIXED AND A TRANSVERSE AXIS OF SYMMETRY.
en
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z:
c: :Dr..... ...... I
N I--
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(I)
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z ~
8
o.~
c:I FIGURE 11.15 - BUCKLING OF VARIABLE SECTION COLUMNS: STEPPED COLUMN WITH LARGER MOMENT OF IKERTIA AT ENDS, BOTH ENDS PINNED AND A TRANSVERSE AXIS OF SYMMETRY.
FIGURE 11.16 - BUCKLING OF VARIABLE SECTION COLUMNS: MOMENT OF INERTIA FOR ENDS VARYING AS THE FIRST POWER (n=1) WITH BOTH ENDS PINNED AND A TRANSVERSE AXIS OF SYMMETRY.
rr1 Cf)
-
C)
:z !I: z c:
,.. ,..r-
e
"""--
e
tit
-
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e
e ---/
f:- ---" ~~ '-.'.
r '
~
= ! 28·-·-··-··----.t:
\ ;i-- -'
....
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I.t.J
t..
ov
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....
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~
::; ::r; ;,;
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:=a c:
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.:1,
e
!g
v
(I)
-I
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z
ffi
c-) "'i"";" ,~
.....
c::
:=a
:w:-
rFIGURE 11.17 - BUCKLING OF VARIABLE SECTION COLUMNS: MOMENT OF INERTIA FOR ENDS VARYING AS THE FIRST POWER (n = 1) WITH BOTH ENDS,
FIXED
AND A
TRANSVERSE AXIS OF SYMMETRY.
FIGURE 11.18 - BUCKLING OF VARIABLE SECTION COLUMNS: MOMENT OF INERTIA FOR ONE END V~YING AS THE FIRST POWER (n = 1) WITH BOTH ENDS PINNED AND NO TRANSVERSE AXIS OF SYMMETRY.
I:' .."
-
CI) C i.)
:z 31:
:w:z c: :w:r-
r~
I
N \..N
l
" ~=::;:::::' '
E
It'
-
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t--'
l-
I
N
.J:'-
~=,,_ . --'
/~~
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E I
\"-"ill
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Z
~ ::mii~~~ '-
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c"
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en
..... :=ct
i~
z
~
iJ
c:
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8
c:-)
..... c:
,.:=ct r-
FIGURE 11.19 - BUCKLING OF VARIABLE SECTION COLUMNS: MOMENT OF INERTIA FOR ONE END VARYING AS THE FIRST POWER (n = 1) WITH ONE END PINNED AND THE OTHER FIXED AND NO TRANSVERSE AXIS OF SYMMETRY.
FIGURE 11.20 - BUCKLING OF VARIABLE SECTION COLUMNS: MOMENT OF INERTIA FOR ENDS VARYING AS THE SECOND POWER (n = 2) WITH BOTH ENDS PINNED AND A TRANSVERSE AXIS OF SYMMETRY.
CI ...,
en
-
en
:z
,.51:
:z
,.rc:
e
e
~
i=~
I
e
\
-.~.
. I
e
',-"
e
-
-----"
~~ \,', ill --.-;- , ~
r""-
t?-fC:--.. \'
:-
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en -I
::a c:
n
i
3o
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o
c:: ::a :J:a r-
t FIGURE 11.21 - BUCKLING OF VARIABLE SECTION COLUMNS: MOMENT OF INERTIA FOR ENDS VARYING AS THE SECOND POWER (n = 2) WITH BOTH ENDS FIXED AND A TRANSVERSE AXIS OF SYMMETRY.
..., = en -
FIGURE 11.22 - BUCKLING OF VARIABLE SECTION COLUMNS: MOMENT OF INERTIA FOR ENDS VARYING AS THE THIRD POWER (n = 3) WITH BOTH ENDS PINNED AND A TRANSVERSE AXIS OF SYMMETRY.
C i)
Z
31:
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< 1-'to
...,.......
o ::s
I N
()
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c::
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r-
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t--' t--'
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t-..>
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w
1:)
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(I)
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i
c:
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-I C ~
4
>
rCJ n-I FIGURE 11.23 - BUCKLING OF VARIABLE SECTION COLUMNS: MOMENT OF INERTIA FOR ENDS VARYING AS THE THIRD POWER (n = 3) WITH BOTH ENDS FIXED AND A TRANSVERSE AXIS OF SYMMETRY.
-
(I)
FIGURE 11.24 - BUCKLING OF VARIABLE SECTION COLUMNS: MOMENT OF INERTIA FOR ONE END VARYING AS THE THIRD POWER (n - 3) WITH BOTH ENDS PINNED AND NO TRANSVERSE AXIS OF SYMMETRY.
c :i)
Z ~
> Z
c:
>
r-
e
~
e
.
-~=::::7'
E
e
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I
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11.1..::... 1 .,. '\ '. \ ! 't./{ :\;' ;A ..:.:. \: .:lL.;. ..LL!:JJ)j
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ei ",
FIGURE 11.25 - BUCKLING OF VARIABLE SECTION COLUMNS: MOMENT OF INERTIA FOR ONE END VARYING AS THE THIRD POWER (n = 3) WITH ONE END PINNED AND THE OTHER FIXED AND NO TRANSVERSE AXIS OF SYMMETRY.
FIGURE 11.26 - BUCKLING OF VARIABLE SECTIO~ COLUMNS: MOMENT OF INERTIA FOR ENDS VARYING AS THE FOURTH POWER (n ; 4) WITH BOTH ENDS PINNED AND A TRANSVERSE AXIS OF SYMMETRY.
-z
(I) C i)
3C
> z > r-
=
I-' I--'
I N '-J
';0 !'b
t-' t-'
<: ro-
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N
~.
co
~ /'
o
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~~~;[=~~-!J~:r~~7
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r
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w
U r:;;: u..
en ..... ::0
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U
t:l
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:J
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u ::>
c::
a;
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c;-)
::> ...J
o
--I
II
c:
:=c
» r-
09 1.0
C rTI FIGURE 11.27 - BUCKLING OF VARIABLE SECTION COLUMNS: MOMENT OF INERTIA FOR ENDS VARYING AS THE FOURTH POWER (n ; 4) WITH BOTH ENDS FIXED AND A TRANSVERSE AXIS OF SYMMETRY.
-
(I)
FIGURE 11.28 - BUCKLING OF VARIABLE SECTION COLUMNS: MOMENT OF INERTIA FOR ONE 'END VARYING AS THE FOURTH POWER (n = 4) WITH BOTH ENDS PINNED AND A TRANSVERSE AXIS OF SYMMETRY.
en
:z ~
»
z: c::
»
r-
•
•
"~'
•
,----/'
e
e
"-"
--
e e
1'1 ,.. I
'@\/-~ _
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-<~ ~
... , . . . . \
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;=~
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~ 12
~
en
u
~ 10
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i
f
o
::a
c:
;:) .j
u
:I
g I>
~ ~
.
c::
2
::a
o
r-
>
FIGURE 11~25 - BUCKLING OF VARIABLE SECTION COLUMNS: MOMENT OF INERTIA FOR ONE END VARYING AS THE THIRD POwER (n = 3) WITH ONE END PINNED AND THE OTHER FIXED AND NO TRANSVERSE AXIS OF SYMMETRY.
FIGURE 11.26 - BUCKLING OF VARIABLE SECTION COLUMNS: MOMENT OF INERTIA FOR ENDS VARYING AS THE FOURTH POWER (n = 4) WITH BOTH ENDS PINNED AND A TRANSVERSE AXIS OF SYMMETRY.
c:I rrI
en
-
C)
z: ~
> :z c: > rI-'
...... I
N
......
IO~,·.oN ..k(· R~;;:;S:,:;::::<;!, ;:::::::::;:::::>
I-' I-'
,
*5 ?t;2:2"~
N
CXl ~'-~:-~~ i',"j~~l ,~l: i',~
I~~~'~
Ht-i-'+i
~--
,,~:,
Z
~.
.
~~; ~~--~ ~~-=~:t~t~;~f~~
\ ~ il:ll
'~iCl-
E
.
;=~/
I
t-
Z UJ
D·
c;.:;
en
"~ u
'i
-I
..:>
z
;;C
==
:::i ::.c: u :::> a:;
:;)
z
~
:a c:
:Ii
5<..'
c:-)
::I ...J
0
-I
L'
c::
:a :D-
0,2
°
3 04 05 06 o
r-
°
1 0.8
T
FIGURE 11.27 - BUCKLING OF VARIABLE SECTION COL~S: MOMENT OF INERTIA FOR ENDS VARYING AS THE FOURTH POWER (n = 4) WITH BOTH ENDS FIXED _~~D A T~~SVERSE AXIS OF SYMMETRY.
c:J rrI
en
-
FIGURE 11.28 - BUCKLING OF VARIABLE SECTION COL~S: MOMENT OF INERTIA FOR ONE END VARYING AS THE FOURTH POWER (n ~ 4) WITH BOTH ENDS FIXED k~D A TRANSVERSE AXIS OF SYMMETRY.
C i)
:z :I: :D-
:z <= :D-
r-
e
e
. . .'
~"'
e
STRUCTURAL DESIGN MANUAL
•
a
7
FIGURE 11.29 - BUCKLING OF VARIABLE SECTION COLUMNS: MOMENT OF INERTIA FOR ONE END VARYING AS THE FOURTH POWER (n = 4) WITH ONE END PINNED AND THE OTHER FIXED AND NO TRANSVERSE AXIS OF SYMMETRY.
)
• 11-29
STRUCTURAL DESIGN MANUAL Revision C The significance of the value of the exponent n is as follows:
•
n= 1
For this condition, the column is of constant thickness with a linear taper in width. The thickness must always be smaller than the width at any section, so that actually this represents a thin, wide rectangular cross-section.
n= 2 For this case we obtain a braced polygonal or pyramidical column. Thus the moment of inertia for such a configuration is approximately proportioned to the square of the distance to the centroids of the concentrated outer areas. Often the moment of inertia of the individual sections about their own axes can be neglected. Also, when n = 2, the edge of the column is a parabola, the axis of which is perpendicular to the centerline of the column. The thickness is constant and smaller than the width at any section.
')
n= 3 For this exponent, the resulting column is a column of constant width and linearly varying thickness. In this case, although the thickness tapers, its dimension. is always smaller than the width. Thus a column of constant width is the result.
n= 4 A solid or hollowed cone, square or pyramid results for this exponent.
A column tapering linearly in width and depth, although not of the same taper ratio, falls in this class if extensions of the taper lines meet at a common apex.
Equation 11.13 is valid only in the elastic range and must be modified for stresses in the plastic range. The following procedure is used where Equation 11.13 is in the plastic range. A) B)
C)
D)
11-30
Determine the buckling coefficient, ~ from Figures 11.11 through 11.29. ' Calculate the equivalent slenderness ratio, (I/PJC)e' for each section corresponding to the smallest crosssectional area. If the column is composed of one material, then the equivalent slenderness ratio corresponding to the smallest column cross sectional area is all that is necessary. The equivalent slenderness ratio is obtained from Table 11.2 for the particular column. From the appropriate column curves of Section 11.1.4 and the equivalent slenderness ratios determine the critical compressive stress, (tr;r )1 and (qc,. )2Calculate the critical buckIing loads for each section
•
STRUCTURAL DESIGN MANUAL Revision C p
[ JCL] P
e1
m E212
cr =
L2
ITt
= P. ~E212
[ P
"
~ j 2 = P 7fL..rm e
·2
ElIl
mEl II P
[
P$
je 1
•
k
L2
lTL
=
~l
[ P
rm p
[p$
cr =
cr
=
,
P
1
e
, 7T
2
=
P2
L
~'El II
E212
mE I 2 2
4L2
21TL
=
Jc ]
[ P
m~212
$J
27fL e
2
=
Jffi
p 2
Elll
p
cr
=
mEl II 4L2
-
[ p}c )
Je 1 =
27fL
PI
Jm
L
r
P.[c
J e2
ffi 2
= P2
1T
L
E212
TABLE 11.2 - EQUIVALENT SLENDERNESS RATIOS
• 11-31
STRUCTURAL DESIGN MANUAL Revision C E)
The smallest critical load is taken as the critical load of the column.
It should he noted that in the elastic range use of the equivalent slenderness ratios and the column curves will give the same buckling load as that calculated by Equation 11.13. To illustrate the procedure consider the following example: Example Problem Determine the critical buckling load for the following stepped column where the larger section is made of 2024 aluminum and the smaller section alloy steel heat treated to 125000 psi.
A)
Section Properties Section 1 - Steel
Section 2 - Aluminum
= 0.2665 Al = 0.4462
= 1.549 in4 2 A2 = 1.964 in P2 = 0.888 in 6 E2 = 10.5 x 10 psi F = 42,000 psi cy
II
in
PI
= 0. 774 in
El
=
F
B)
in4
cy
29 x· 10
6
2
psi
100,000 psi
12
••
Column Buckling Coefficient - m
6 6 El Il/E2 12 = (29)(10 )(0.2665)/(10.5)(10 )(1.549) = 0.475
aIL = 26/40 :; 0.65 Using Table 11.2 and Figure 11.10, rn = 7.75, then 6 Per rr(E I2/L2) = 7.75 (10.5)(10 )(1.549)/(40)2 = 78,800 Ibs. 2 c)
Stresses in Each Section at Buckling )1 cr (O-cr)2
(u-
=
=
)l/ Al = 78800/0.4462 = 176,600 psi cr (Pcr)2/ A2 = 78800/1.964 = 40,100 psi (p
These stresses are in the plastic region so the Euler load must be corrected to include the effects of plasticity.
11-32
•
STRUCTURAL DESIGN MANUAL
•
Revision C D)
Equivalent Slenderness Ratio For Section 1 of the column, from Table 11.2 (L
/PJC>et rrL/Pi/rrF.212/Elli=40rr/6 .. 774y'7.75/0.476'=
40.2
For Section 2 of the column (L /P..jC)e = rtL/P2.[m= 40rr/O.888~ =50.8 2
E)
For Section 1, from column curves in Section 11.1.4, using the 125000 psi steel curve and (L~p~ ) = 40.2 the column stress is (O-cr ) 1 == 107,060. For Section 2, from the column curves in Section 11.1.4, using the 2024 curve and (L /p.JC ) = 50.8, the column stress is (O'"'cr)2;: 31,000 psi.
F)
Critical Column Load p
•
p
11.1.4
crl
=
(107,000) (0.4462)
=
47,7.50 1bs.
= (31,000) (1.964) = 60,900 Ibs. cr2 • The critical compression load of the column is p =47,7501bs. cr
Column Data for Both Long and Short Columns (FIGURES 11.30-11.65)
Critical buckling stresses for different materials and geometry are given. Given the slenderness ratio (L~ / p), and the material, the critical stress is determined. Both short and long columns are accounted for.
)
• 11-33
STRUCTURAL DESIGN MANUAL INDEX OF COLUMN ALLOWABLE CURVES FIGURE Johnson-Euler Curves Aluminum Alloys Magnesium Alloys Steel Alloys Titanium Alloys
11.30 11.31 11.32 11.33 ')
Euler Curves Aluminum 2014 2024 2024 2024 2024 2024 7075 7075 7075 7075 7178
Alloys Extrusion Bare Sheet and Plate Bare Plate Extrusion Clad Sheet and Plate Clad Sheet Bare Sheet and Plate Extrusions Die Forging Clad Sheet Bare Sheet and Plate, Clad Sheet and Plate and Extrusions 356 Castings Magnesium Alloys AZ63A-T6 Casting XK60A-T5 Extrusion AZ3lB-H24 Sheet HM21A-T8 Sheet 2 HM31A-F Extrusion, Area <1.0 in 2 HM31A-F Extrusion, Area::: 1.- 3.99 in Steel Alloys FTU = 180-260 ksi FTU = 90-150 ksi Stainless Steel 18-8 Cold Rolled - With Grain 18-8 Cold Rolled - Cross Grain AM350 Sheet PH13-8 Mo Plate and Bar PH14-8 Mo Sheet PH15-7 Mo Sheet and Plate 17-7PH Sheet and Plate 17-4PH Bar
11.34 11 . 35 11.36 11.37 11.38 11.39 11.36 11 .. 34 11.40 11.39 11.41 11.42 11.43 11.43 11.44 11.45 11.46 11.47
•
11.48 11.49 11.50 11.51 11.52 11.53 11.53 11.54 11.55 11.52
• 11-34
STRUCTURAL DESIGN MANUAL
fIGURE T i to Il i lIJII AU oy s Commercially Pure Sheet H Mn Annealed Sheet 4 Al-3 Mo-IV STA Sheet and PLate 5 Al-2.SSn Annealed Sheet, Plate, Bar and ForgIng () Af-4V AmwaJed Exlrusjon & AL-4V Annealed Sheet 6 Al-4V STA Sheet 6 Al-4V STA Extrusion 1) Al-lMo-IV Single Annealed Sheel and Plal(~ 11V-IlCr-3 Ai STA Sheet and Plate 13V-I1Cr-3 At Annealed Sheet and Plate
11..56 11.50
II •.57 J.I .58 1.'.59 11.60 11.61 11.62 j j,
.63
11.64
11.65
• )
• l1-35
.. " '" I \... •.
!',,/ I ~ ",
'\
\
'10
~
\ \'\
STRUCTURAL DESIGN MANUAL
.B~II ~
,4' I I' • U'"
I
":1~
j,
,',
80 Johnson-Eu)("f FormulAs For Aluminum Columns Johnson Fornlllla
70
. 2
F col = Fcc -
F~cc~)(p5E ) 4 n
c..~.
J:..
60 Where: C = restraint coefIicienl E = 10.3 x 10 6 psi 50 H U)
Euler Forn1uJa
~ (f.) tI)
r.:z.1
~
40
F
E-<
col
= nZE
/C )2 (LL
(f.)
•
~ ::..>
....l
0
u
30
~
,..J ::Q
<
~
0
,..J
~
20
,.....f
0 0 r:..:...
10 ~------~--------r-------~------~~----~~
20
40
60
80
__
~
100'
FIGURE .11.30 - JOHNSON-EULER COLUMN CURVES FOR ALUMINUM ALLOYS
11-36
•
STRUCTURAL DESIGN MANUAL
.'
L )2 ( p ./C
(FcC)l 2. "In E
F CCio 1':::: F 'cc -
60 ......:=-----+------+-
Where: C = coefficient of restraint E = modulus of elasticity;:; 6.5 x 10 6 H U'J
::.t;:
'"
-40
if}
U'J
w
F col
(:G
f:-I
(
U'J
...J
L
2
P/C )
~
,!-.
:..:.>
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10
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• Ll-37
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STRUCTURAL DESIGN MANUAL
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l~O
L F2ee F col ;' Fcc-
2
(p-7E)
4rr2 E
160 too--=-_ _~--l~~-+Whcrc C = restraint coclficicnt
E :::; 29, ODD, 000
Euler Formula 140
Applies to Corrosion and Non-Corrosion Resisting Steels
• )
40
lO
40
60
HO
100
L' /p :: L/p./C FH-:URE 11.32 -
11-38
JOHNSON- EULER COLUMN CIJRVES FOH. STEEL ALLOYS
•
STRUCTURAL DESIGN MANUAL
(Fcc)2 F col:: Fcc -
70
4n
2
E
( p~)
2
Where:
60
\
C :: rest raint coefficient E :: modulus of elasticity:: 15.5 x 10 6
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en
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=
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FIGURE 11.3'; - JOHNSON-EULER COLUMN CURVES FOR COMMERCIALLY PURE TITANIlIM
• 11-39
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Curve Designation 1 2
2014-TQ 2014-T6 2014-T6 2014-T62
3 4
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1
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B B
A B B B
B
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20 'j J
10
120 80 100 60 oo~~--------------------------------~----------~ 40 20 I
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11.34 - COLUMN ALLOWABLE CURVES
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FIGURE 11.]5 - COLUNN ALl.OWABLE CURVES
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90
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A B
-
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o o
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~~
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40
60
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I
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1
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100
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120
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40
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-
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FIGURE 11.45 - COLUMN ALLOWABLE CURVES LJ
-51
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AREA <1.000 in. 2
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40
60
80
100
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FIGURE 11.47 - COLUMN ALLOWABLE CURVES I I-53
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360
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------ ----
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120
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FIGURE 11.49 - COLUMN ALLOWABLE CURVES I j -55
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FIGURE 11.53 - COLUMN ALLOWABLE CURVES 11-59
STRUCTURAL DESIGN MANUAL
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STRUCTURAL DESIGN MANUAL I I I I I I I SAl-2.5Sn TITIANIUM ALLOY ANNEALED SHEET. PLATE. BAR AND FORGING
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STRUCTURAL DESIGN MANUAL
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STRUCTURAL DESIGN MANUAL
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L /P FIGURE 11.63 - COLUMN ALLOWABLE CURVES 11 .. 69
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STRUCTURAL DESIGN MANUAL I
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FIGURE 11.65 - COLUMN ALLOWABLE CURVES 11 ... 71
s-rRUCTURAL DESIGN MANUAL 11.2 Beam Columns
A beam column is a member subjected to transverse loads or end moments plus axial loads. The beam can be straight or have an initial curvature. Its cross sectional dimensions are small with respect to-its length. The axial loads (either tension or compression) produce secondary bending filoments because of the lateral deflection caused by the transverse loads or bending monlents. In the case of the axial compression load, the primary transverse bending moments will be increased, while the axial tension load will decrease them. 11.2.1
Beam Columns With Axial Compression Loads
Beam columns under axial compressive loads are far more critical than those with tension loads. Axial compressive loads increase the bending moment and increase the possibility of instability or buckling failure. The critical column load is defined in Sections 11.1 and is independent of the magnitude or distribution of the transverse loading. The cri tical load is the load under which the rnembe'r would be unstable if there were no side load. A beam column must satisfy both the criteria of a column ~nd a beam. Beam columns may be designed to function in the elastic and plastic range of the material but normal stiuc~ural design procedures must be f~llowed, i.e., no yield at limit load and no failure at ultimate. The following equatIons are used to calculate the combined effects on a single span beam column: Bending Moment:
M
= C1
Shear:
v
=
sin x/j
+
C cos x/j 2
C1!j cos x/j ..
+ f(w)
c2Asin x/j +
f'(w)
•
(11.,15)
(1l .. 16)
where ff(w) is 1st de'rivative of f(w) Def lee t jon:
o
= (M" - M) IP
(1l.17)
Slopc::
8
= (Vo - V)/p
(11.18)
In the above equations j = "EI/P' and Mo and Vo are the primary bending moment and shear, i.e., the bending'moment and shear that would be produced by the transverse loads and end moments acting without the axial loads. The constants C and C. and the expression few) depend on the type of transverse load, that ls, dL-Il"lhllf(.~d, point, IIIOIUt'nL, cLc. The lIlotllE~nl M L~ pot':dli.vl' when CUlllprC:-iHloll it> produc('d In the upper fibers and W or w Is pusItIve when upward. The load P and the distance X are as shown in Table 11.3.
11-72
•
•
STRUCTURAL DESIGN MANUAL Revision A The results of this method of beam column analysis are inaccurate when MIMo becomes less than 1.1. It is recommended that at least four significant figures be used in all computations. Table 11.3 shows the constants for use with Equations 11.15 and 11.16. Not all combinations of loads and moments are possible to present. The method of superposition can be applied provided each transverse loading is used with the total axial load for the systems which are being combined. The principle of superposition does not apply to a beam column when used in the conventional manner. The sum of the bending moments due to the transverse loads and the axial loads acting separately are not the same as the moments when they act simultaneously. To find the moments for several combined loadings, add the values of C , C2 ' and I f(w) for the several loadings using Table 11.3 for each individual case. Then substitute these values into Equations 11.15 and 11.16. In a beam column, the bending moments do not vary directly as the load increases. Thus, it should be noted that direct ratioing of moments to loads should be avoided and will result in inadequate structure. It is recommended that four significant figures be used in computations, making use of the so-called precise equations, since the results in many cases involve small differences in large numbers. In Equations 11.15 and 11.16 the terms x/j are in radians. The values of sin x/j and cos x/j can be found in several textbooks. They are not presented here since they are easily obtained on hand calculators. The equations presented previously assume that E is constant, that is, the stresses are in the elastic range. This is used for loads up to limit; however,
for ultimate loads the stresses may be in the plastic range. The method and equations used for the elastic analysis are also used for the plastic analysis with a plastic E used where previously E was elastic. A good approximation of the plastic E is as follows:
)
1) 2)
Compute Fe = PIA Enter,the ~asic column curve for the material at F and find L" I pJC corresponding to F. C = 1 for this step.
3)
Using these values of Fe andeL' / p
4)
Then
j
compu te
E/ =
:~( ~) 2
=(E ~l )%
Then proceed as explained previously •
• 11-73
r-----------------r-------~--------~----------------r-~---r------------------~ c1 C2 f (w) MAXIMUM MOMENT &. LOADING en
Unequal End Moments M2 - Ml Cos(L/j)
o
Tan(x!j)
n o
Equal End Moments
Z
=
~:;~;~~;1 ~I~~
Ml
Mmax
~
en
'0.}l
'~
Sin(L/j)
r c::
......
= Cos(x!j)
where
~
!~
Ml
Mmax
o
Ml Tan(L/2j)
= ----Cos(L/2j}
At Midspan
~
H
1-3
::r:
S< H ~
r
Uniform
()
~cn M
cn H o Z
-i:t3
w, 4F/in
I-&---
tf L
ftf"r-
.2
wj2 [Cos(L/j) -1 J Sin(L/j)
WJ
D2 - D1CoS(L/j)
wJ
M
max
- wj2[1-Sec(L/2j)]
At Midspan
.. , .2
Sin(L/j)
See Note 4
Concentrated Load
max
Dl
= Cos(x/j} +
Tan(L/j)
2
See Note 4
o
o
Mmax
c L+e L= ie 2 2
where
x > a:
Wj&in(a/j)
wj
where
x < a:
-WjSin(b/j) Sin(L/j")
M
-WjSin(a/j)
o
Tan(x/j)
C 1
=C
2
Cos(x/j)
-
,
\
"'-'"
;!
LOADING ~ [wo Symmetrical Concen· M trated Loads
C 1 x < a: _W.Cos(b/2j) JCos(L/2j) a < x < L .. a: -WjSin(a/j)Tan(L/2j)
o
o
-WjSin(a/j)
o
L-a
Concentrated Moment
x
MAXIMUM MOMENT
few)
W·Sin(L/j)Cos(b/2j) Cos(L/2i)
_w·Sin(a/j) JCos(L/2j) at midspan
o
- J
< a:
-mCos(b/j) Sin(L/j)
o
o
See Note (6)
o
mCos(a/j) ~
~
::r
~ixed
End Beam Uniform Load
At x
.2 wJ
-wLj 2Tan( L/ 2j)'
F1xed End Beam - Concentrated Load at Center
x
x
< L/2: -Wi
>
~ Fixed Simple Beam Unifonn Load
2
L/2:
x< L/2: ~Tan(L/2i)
w, #/in
• L/2;
Tan(L/j) - L/j
-wJ.2(L/2~ Sin 2j -1 )
j
L'
-w
ft u :::/~~S(L/P -w/ 1-fitHH pUxJ r t Sin(L/j)
~(l 2
-
- COS(L/2 j »)
Sin(L/2j)
o
Tan(L/2j) - (L/2i) Tan(L/j) - (L/j)
o
L' ( Tan(L/2;) - (L/2 j w J . Tan(L/j) _ (L/j)
J
x(Tan (L/
j»
:=1
») ~
~. Ui
b' ~
p
L
o
~(COS(L/i) - Cos(L/2i' 2 Sin(L/2 f}
( 2Cos(L/2j )-1)
wj2[ 1 ... L/2j ] Tan(L/2j) At x = L/2
- COS(L/2 ») Sin(L/2j)
~(l
2
= 0:
(Tan(L/j)(WLj»- wj 2
.2
',-
WJ
at x = 0
LOADING
-
.
w
f(w)
.2
Uniformly Increasing Load
~~
,2
o
-W]
wJ
Sin(L/j)
x
L
Occurs at
.2
W]
.2 -wJ
Tan(L/j)
Wj2{1_x/L)
Symmetrical Triangle
x
~x~
L---
L
x > L/2:
~
-4wj3Sin( L/2j)
2wj3Cos(L/j) L Cos{L/2 i) x~
,L
a:
Partial Uniformly Dis-2wj2Sin(z/i}SinCf/j) tributed Load Sin(L/j)'
r;.~~z~~
p
a
~
I
•
P
v
b
~L Symmetrical Partially Uniform Distributed In ~ L/2-+-L/2-'1 Loa~ ~ a + z..j....z -+- a1 " " " , w, 4f/in
r-~
L
-i
(j/L)Sin(L!y
<
x
< b:
P
b
<
'X
Tan(L/j) <: L:
- 2w j
-2wj3Tan(L/2j) + wj2 L
2wj2(1-X/L) at midspan
0
-
o
2 -wj Cos (a/ j)
Wj2
2S'~ n ( Z /')8' J 1 n ( e /') J
o
2wj 2Sin( z/ 1> Sin' e/ f) -wj ~Sin(b/j)
f t f t , tt W II/i n
C
.2 2Wl x
o
LCos(L/2j)
I
I'
=
.3
I~~ P
J
< L/2:
- 2wJ
-,
I
Cos(L~x)
Solve for x/j and x Substitute into Eg. 11.15.
p
--
Occurs at Cos(x/j)=(j/L)Sin(L!j) Solve for x/j and x Substitute into Eq. 11. 15.
Uniformly Decreasing ,- Load
L°i!L/2.....j
MAXIMUM MOMENT
See Note (6)
2wj2Sin(z/i)Sin(e/f) Tan(L/j) x
< a:
-wj2Sin(z/j)Sec(L/2j) a
0 -Wj2Cos(a/j)
o .2 wJ
2
-2wj Sin(z/j)Sin(L/2j
wj2Sin(z/j)Sec(L/2j)Cos{L/j~
o
w· 2 {1_Cos(a/ j ) } J . Cos(L/2j)
at midspan
~
LOADING
>to
~
Fixed Simple Beam
x
Concentrated Load
W
t
L
to
.
-Wj(jTanCL/j)SeC(L/2 j 2 jTan(L/j) - L x "> L / 2 :
WLf,Tan(L/j)(Sec(L/2j)- l)J'
)-1) 2 L •
2
o
jTan(L/j) - L .!iifLTan(L/j)Sec(L/2j} -1 2 [ j Tan ( L / j) - L
~ ~(L+2jSin(L/2j) -2LCOS(L/2i))
--1
r.:antilever - Concentrated Load
. ~
MAXIMUM NOME NT
f(w)
-2Sin(L/2j)]
WL 2
rj Tan ( LI
l
at x
j ) Sec ( L / 2j ) - ~ jTan(L/j)-L J
=0
o
jTan(L/j)-L
WjTan(L/j)
\.]
I ~-1~~x~ ~
at x = L
-L
Cf)
~ ~
~antilever - Uniform
Load w, 4F!in
Wj{j(l - SecL/j) + LTan L/ j} at x = L
aPTan(L/2j) 2(Tan(L/2j) - L/2j)
-
STRUCTURAL DESIGN MANUAL NOTES: (1) W or w is positive when upward. (2) M is positive when producing compression in upper fibers.
= ~~I
(3) j
with a dimension of length.
(4) D1 = M1 _wj2; D2
=
Ma _wj 2
(5) All angles for trigonometric functions are in radians. (b) When the formula
for the maximum moment is not provided in the table, methods of differential calculus may be employed, if applicable, to find the location of maximum moment; or moments at several points in a span Inay be computed and a smooth curve then drawn through the plotted results. The same principle applies in the case of a complicated combination of loadings.
(7) All points where concentrated loads or moments are acting should also
be checked [or maximum possible bending moments. (8) Before the total stress can reach the yield point, a compressjon beam column may fail due to buckling. This instability failure is independent of lateral loads and the maximum P that the structure can sustain fIlay he
computed pertaining to the boundary condition without regard to lateral loads. A check using ultimate loads should always be made to insure that P is not beyond the critical value.
•
TABLE 11.3 (Continued) BEAM COLUMNS WITH AXIAL COMPRESSION
)
11-78
•
STRUCTURAL DESIGN MANUAL
•
11.2.2
Beam Columns With Axial Tension Loads
Axial tension loads usually decrease the bending moment in the beam column. Bearn columns may be designed neglecting the axial loads; however, this will give conservative results. More precise results can be obtained with the following procedure: sinh x/J
+
+
Bending Moment:
M
C
Shear:
v
el/j cosh x/j + Cz/j sinh x/J + few)
\
1
C
2
cosh X/j
(11.19)
few) .I
( 11. 20)
" where f(w) is 1st derivative of few) Defleclion: Slope:
()
(M-Mo)/P
(11.21)
(V-Vo)/P
(11.22)
In the above equa tions j = J EI/P and Mo and Vo are primary bending moment and shear, i.e., the bending moment and shear that would be produced by the transverse loads and end moments acting without axial loads. The constants C and l C and the expression f(w) depend on the type of transverse load, that is, 2 dlslrihuted, point, moment, etc. The moment M is positive when cOlllpn:~ssjon is procluced in the upper fibers and W or w is positive when upward. The load P and lhQ dlstance x are as shown in Table 11.4. Trw r('t)ults of lh1s method of beam column analysis are inaccurate when M/Mo
becomes greater than 0.9. To maintain accuracy in the analysis [our significant rigur~s should be used in the calculations. Var.ious loading conditions are shown in Table 11.4. The method of analysis for other types of loading is the same as for compression loaded members described in Section 11.2.1.
)
11.2.3
Multi-Span Columns and Beam Columns
Multi-span beams are those with three or more supports and in general it Is not possible to develop simple equations as previously described. The determination of moments is more involved for multi-span columns and the method of "moment distribution" is used to determin~ the beam moments at each support. These moments are then used as previously described in Sections ll.2.1 and 11.2.2 in the single span equations to determine bending moments belween supports.
•
The moment distribution method is sometimes called the "Hardy Cross H method after the man who originated it. The method is simple and useful for the solutIon of continuous structures, i.e., multi-span heams. This method starts by assuming
11-79
C 1
....,~
... 6; J
~
LOADING
C 2
f(w)
MAXIMIM MOMENT
r
t"t::
nequal End Homents
M2 - MlCosh(L/j) ~
t;I:I ~
~
~l xJ
p
~ i
.....
Sinh(L/j)
MI
o
P
~~ \,~'~II=:;1 -. ::::~
Ml Cosh(x/j) where Tanh(x/j) = M2 - M1Cosh(L/j)
~.:-
'\~;CD~"-
;----~'
~-~
-MlSinh(L/j)
(")
en
o
.....
r
c
~
~qual
End Moments
til
~ I---'
rl
:r.:
~ 1-1 > r
~
~l
-M 1Tanh(L/2j)
M~
Z
o
Ml Cosh(L/2j)
c:
c-)
..... c:
::a
pnifonn Load Cosh(L/ j)] Sinh(L/j)
wj 2[ 1 -
Z
o
Ml
;a
P
t::
til H
~,,~.
w
4F/in
-J f f ~ I , f , ! !'Lex L---I
p
:III r-
wj2(Sech(L/2j) -1]
CI rY1
+-P tH ft fJ1 14+
ID2 - D1Cosh(L/j)
2
L
_
~oncentrated
~a4
P~
b
.. 51nh(L!J)
.2
-wJ
See Note 4
- Wj Sinh ( b / j) Sinh(L/j)
~
-r
x > a: Wj Sinh(a/
j) Tanh(L/j)
o -WjSinh(a/j)
e
-:z
en
Dl ,2 --.;;..-- -w J Cosh(x/j) where
CD
TaBh£n/~6sh(L/j)
o
o
iI:
2 1 - D1Sinh(L/j) See Note 4
:III
2
c:
x < a:
p
L
Dl
1'1
I,
Load
w
e
.2
-wJ
p
Uniform Load with End Momen~s. w, #/1n P
.2
wJ
- C1
Z
:III
Cosh(x/j)
r-
where C 1 Tanh(x/j) = 2
c:
e
;
tit r
C 1
LOADING
t-3
;J> o::l
~
.l>
o
z
C
.2
I-" I-"
2
.2
-WJ x
L/j)
l
MAXIMUM MOMENT
few)
o
WJ
.-
e
.~
Uniform Increasing Load
trJ
(')
--
'''"-'''''
L
p....-;F....,.a::;;IL--:~~L~--l~J----'II-"p
Occurs at: Cosh(x/j)=(j/L)Sinh( Solve for x/j and x, Substitute into Equation 11.19
-,
\\
--. "' \ -~ ,~ "~. ,~CD-~ '/' // ~ __.-;;.,r I
,-~
~
-t::'
t:::1:1 I;'l:j
~ o
o r
c::
~
Cf:l ~
H ~
. ::r::
~
H
;J>
r
...., I:Tl
Z
Cf:l H
~
Uniform Decreas
Load .2
t
L/2..I
,
P~P Partial Uniformly Distri bu teg ~d f--::-1
+dtd~C1
F 31tI !jw, t-----
Tanh(L/j)
PErU
.I I. '-' C· I ( ! n, 1
I
Sinh(L/2j)
• ") , I t, I
(J
iT!!l(b/j)
F 9HHt
...... ...... I
00
.....
~
p
~x~
~
L .
W,
11/1
t--
-.J
~
P
CD
z
See Note 6
wj 2CoSh(~/ j)
31:
< L:
2\\, j 2 SiT, h ( (1/ 2 j ) S lr; h ( ('I
[t
< X < L- a:
c:::::J
-
o
a
I
rrr1
2
n
'l'anh(L/j )
a~ I ~ .... a.,
o
'I, ',J',' ,! ~'. ( rl J. ) ., r l , -
Symmetrical Partial Uni f 0 rrn Dis t rib ute d L0 ad :x < a: .. \,1 j 2S 'j n 11 ( d /2 i )
L/a~ 'I'·
2 L ~T L 2 -2wj (l-x/L) L anh( 2j) - wj
~jnll(I,/.i)
= ......
n
(I)
x < a: ')
-I :=0
:r.-
- 2W] x
2 .3
-2wj3Cosh(L/j) LCosh(L/2j)
- ,. \-J
Solve Jfor x/j and x, Substitute into Equation 11.19
.2
o
x> L/ 2:
b
P
- Wj 2(1- x/L)
c: ::a
x < L/2: 2wj3 LCosh( L/ 2j)
ii/in
..........-f
Cosh(L:x)~(j/L)Sinh(L/j
.2 wJ
-WJ
~P Symmetrical Triangular Load
(I)
Occurs at:
-.2wj 2Sinh ( d/ j)S inh( e/f)1
o
CQ,<,h(L!lj)
-wj2Cosh(a/j)
wj2,sinh(J/2j)COSh(L/j) Cosh(L/2j)
-2wj2Sinh(d/2j)Sinh(L/2j):
z:
0 .2 -wJ
-w/Cost.(a/j)Tanh(I./2j) L-a
:r.-
0
o
. 2[Co s h ( a/ j
)
WJ [Cosh(L!2j)
_
0 J
c: :r.r-
ell
"""'--3 ......,~ I t::tI
r-'
.!:'
......... (J
o
z
t-3
o
LOADING
Two Symmetrical Concentrated Loads W W
-+a,
C1
C?
x
o
o
-WjSinh(a/j)
o
a
~ L-r-P ~at b
MAXIMUM MOMENT
f(w)
-WjSinh(L/t)COSh(b/2 j ) Cosh L/2j)
I[ ~.~~ tf YJtl.-~ "
.Sinh(a/j) -WJCosh(L/2j)
\
x
t;t1
t%1
>::=:: n
8
C::
2en ~
r
pl>~ L~P
~ ~
t""'
t-3
~
See Note 6
x>a: -MCosh(a/j) Tanh(L/j)
c-)
o
MCosh(a/j)
-I
c: At x
fI:~i tllt~ II/in~
J
x
P
L
-wLj 2
wLj 2Tanh(L/2j)
.2
-wJ
=
.2 rL/2j wJ
::::cJ
> r-
0
[r....:.a-n~h7":(L:---:/;-: :"2 j )
-~
~
P
ITI
~
Z
Z
o
Unifonn Load
~
en H o
......
Fixed End Beam -
H
~
~bi
a
o
Fixed End Beam-Concentrated Load at Center
p~
W
I
t
i+p
en x< L/2:
~rCoSh(L/2j) 2 Sinh(L/2j)
-Wj
2
~ICOSh(L/2j)-COSh(L/j) 2 Sinh(L/2j)
x> L/2:
!U. [2 Cos h ( L/ 2 j) 2
-1]
-
CD
o o
:z B:
!U.i1 - COSh(L/2j)j 2 Sinh(L/2j)
- 1]
> Z
Cantilever - Concentrated End Load
pJ~L ~ W
e
~; - .
en :::a
"-'"
Concentrated Moment
'--+.
'h...~". "-iCD !
at midspan
o
I~r
, m,,,
f:
>
=I-
r-
WjTanh(L/j)
P
--
e
e ~
>tl:'
..
.. -, .:::-(')
o
z
C l
LOADING Cantilever - Uniform Load
C 2
few)
MAXIMUM
MOME~T
wj[LTanh(L/j) .. j(l-Sech(L/j)]
~
p~I~Bn§ p w #/in
e
-
.........
'-,~ ~'- ~CD-
c
t:l:! ~
~
(")
o
t-
c::
~ V'l
.:c H ;-3
:J:
~ 1-1 > t"" ~
~
r.t'J H
~
...... I-' I
co w
~~ -t. ./
/
"!-~
~
'-"
..-
~
/ .:::::-~~'.
/~!
en
Fixed End Beam - Lateral Displacement
P-fx·~~~.~ L~
aPTanh(L/2j) 2 [ L/2 j - Tanh ( L 12 j )]
......
:::a c::
c-,
..... c:
t--
:::a
NoteS:
(1) W or w is positive upward. (Z) M is positive when producing compression in
upper fibers.
(3) j = IEf7V with a dimension of 'in~ (4) D1 = Ml + wj2; D2 = M + wj,2 Z
(5) All angles for hyperbolic functions are in radians. (6) wnen formula for ma~moment is not given in the Table, methods of differential calculus may be employed, if applicable,to find the location of the maximum moment; or moments at several points in the span may be computed and a curve drawn thru the results. The same applies to a complicated combination of loadings.
(7) Values given in Table 11.4 were obtained from Table 11.3 by the following substitution:
SinL/j = iSinhL/j: CosL/j - CoshL/j; Sinx/j = iSinhx/j; Cosx/j = Coshx/j and j = ij where i - J:T (8) All points where concentrated loads or moments are acting should also be checked for possible bending moments. (9) Axial tension helps to stabilize the structure. Usually instability need not be considered unless the beam is very thin for which bending buckling should be checked.
>r-
Ct ~
-z
(I) G)
i:
>z c:: >r-
~
STRUCTURAL DESIGN MANUAL
an arbitrary restrained state for the beam and then gradually releases these restraints according to definite laws of continuity and statics until every part of the structure rests in its true state of equilibrium. Certain terms are used in the Hardy Cross method and they are defined following: Fixed End Moments - The moment which would exist at the ends of a mcmber if these ends were fixed against rotation. The effect of a compressive axial load is to increase the fixed end moments (F.E.M.) while tensile axial loads will decrease the F.E.M. Values of F.E.M. for various loadings are shown in Figures 11 .. 66 th'rough 11 .. 72 .. Stiffness Factor - The stiffness factor (S.F.) is taken as the resistance to rotation at a joint between the subject joint and the adjacent joint. This Is the ratio of the change in slope at joint A to the applied moment at joint A for the beam between joint A and joint B if these are adjacent joints •. The S.F. of a beam at a joint caq be thought of simply as its effectIve' Lorsional spring constant at the joint. If any value of MA were applied and the resulting value of ~ computed (by using the equations of Tables 11.3 and 11.4) the value of S.F. could be calculated using:
This is' the method fOT spans without constant EI. ov(~ r a span: S.F. = S.C. (4EI/L)
For beams with constant EI
•
(1.J .23)
whe're S .C .. is the "Stiffness Coefficient" fOT the span and is obtained from TabJes 11.5 and 11.6. There are two cases in Tables 11.5 and 11.6, one for tlfull fixity" at the far end and the other for "pinned tt at the far end. Jolnt SUffness - When one or more beams meet at a joint or support, the total stiffness factor is the sum of the S.F.'s of each member at the joint.
Joint Stiffness = IS.F.
(1,1.24)
The total joint stiffness must be positive or the structure is unstable. Instability can exist even though the joint stiffness is positive. This occurs when members are not fully fixed at the far end but are somewhere betwecn pinned and fixed. Sometimes good engineering judgment is necessary to obtain a solution more exact than is possible if pinned and fixed are assumed. DistributIon Factor - The distribution factor (D.F.) is a measure of tlH' urnounL each of tlw 1Il(~lHh(.'rs 1II('l'lin).', al th(~ joint. I I is expressed as:
of lhe III()tIIelll at a joint that is resi~l(>d by
D.F.
11-84
= S.F./2,S.F.
(11.25)
•
STRUCTURAL DESIGN MANUAL
•
Factor - The carryover factor (C.O.F.) is the ratio of the moment generated at the "far end" of a span when a moment (M ) is applied at the A end. It is expressed as
Over
(11.26)
Valu(·s of CDI<' are given in Tables 11.5 8nd 11.6. The sign of the carryover factor 1.s given in the tables based on COF = -MB/M where positive M(+M) A produces compression in the upper fibers of the beam. The values of COF in Tables 11.5 and 11.6 are for, spans with constant E.I. If the ~pan does not have a constant E.I. the COF's in Tables 11.5 and 11.6 cannot be used. A unit moment is applied oc end A and resultant moment at end B is calculated; then COF = MB/MA"
• )
• 11-85
STRUC1·URAL DESIGN MANUAL
r
I
I
V
I
w, '#-/iAt.
17
~tIlm!t
/
J
16
Mg-MB-wIJ
/
c
15
14
13
c -~
12
~
~
V ~
11
/
V
/
V
" "'\
/
V
/
/
V
V
• ...,
COM PRES SIaN
\
]0
J
V
TEN ~ION
I\.
'.1
•
1\\
9
~
\, ~
8
o
1
2
3 L/j
4
\
5
6
FIGURE 11.66 - FIXED END MOMENTS FOR UNIFORM LOAD WITH TENSION OR COMPRESSION
• 11-86
,'" ,\ (;/.:')1\' ' \ \',
"\~~~~... STRUCTURAL DESIGN MANUAL .
•
I 30
............
I
" ""
·2
-
2 MB==_wlJ
-
'A
24
\
,
I I.
1\
\
I
•
-I"---
"-
18
16
)
14
\ \
~
eg
.-
CB
\c'
?O I
I
MA=WL
CA
~
26
I
~=
~
"-r\
28
I
"""
EEE '"
1\
\\
\
\ ~
1
2
3
L/1
4
5
6
FIGURE 11.67 - FIXED END MOMENTS FOR UNIFORMLY VARYING LOAD WITH AXIAL COMPRESSION
•
11-87
STRUCTURAL. DESIGN MANUAL I 54
I
I
I
•
I
r~
MA
B
2
M _wL ACA
50
2 wL M :-
B
CB
/
46
JV ,/
42
L
VI' 30
~ .--""""
./
~
/
•
...
./
T /
,,/
22
1
V
/'
26
..-- ~
V
~
V
V
V~
L
./
~B
~
~
2
3 L/j
4
5
6
FIGURE 11.68 - FIXED END'MOMENTS FOR UNIFORMLY VARYING LOAD WITH AXIAL TENSION
.
11-B8
•
STRUCTURAL DESIGN MANUAL
I---
J
~ I I
(£::~1 L
I I .. ~MB
1 31-- I- M
~
/
M --M _WL AB-C
I - - - i-
12
/ I
10
9
c l..,.........-
8
r---...
7
V
'"
V
I'.....
/
,
V
/
/
V
V I TEN~ION .
I I
r ~
'"
6
~OMP ~ESS [ON
~ 'I\. \
5
1\
~
,
4
/
V
11
)
/
V
o
1\ 1
2
3
L/J
4
5
6
FIGURE 11.69 - FIXED END MOMENTS FOR CONCENTRATED LOAD AT CENTER WITH TENSION OR COMPRESSION
11-89
....... 1-1
I
U)
2.0
o
i""~ \..~"
a
1.8 MA
MA- WL
L, ~ ~ (~tCA
~V
B
~L'
MB=-WL(~) 2bLCB /
1.6
/'
1.2
--.........
o
1~ ~
1----"---+--L
.. 1
.2
.3
1
--'"--
.4
.5
o./L or
.. 6
b/L
.7
\
(I)
--2 ~~ LL ~r ./V -~
n
5.0
;=~
..... =a
c:
c:-)
.....
~,~ 'I'. ~ _-.....~ -r--. ~ t'"': _::;:::;:;
1.0
.4.0
~~CI ~/
~~~
"'." 2.0
-"
____
,~·~ar
~1
~.v
/: ~f::::::: I---"" :=='-1--1--- ~ h?-.=-.__,.-
I
~
3.5
-,..-
~~ I~~~~,~ -----r-- - r - 0.8 1"::1'--... ~ ~~ r--- TENSION . ~ ~~ r---__ I............
0.6
~~
\
_
/ V V, ---I----~~~
/ / ' V...-<---
1.0
VI--
V/V _I--~~ / / v V~ ~~~
CA ~
J
r--..~
4.0
/ V ~PRESSfON
V
1.4
l.----::~4.5
I I"" -~~-J~~~~T-+--rl--r-~~1-l
---L:J
-
-
I
-~
> r-
...., ~
en
-:z
/'
V ~~ ~I
l.Y-~--J.8
c:: =a
.9
CD
. I
I
:r:
1.0
> c:: > r-
:z
FIGURE 11.70 - FIXED END MOMENTS FOR CONCENTRATED LOAD WITH TENSION OR COMPRESSION
e
""--- .'
e
'~.'
e
~'
I
1.6 f -
'-
I
I
w.#/t1r.
QfUU '. a
b
MA
I I
I I
1.5
I
I I
I I
I
~B l
I
~- MA= WL2(D2~ 12 - 6- 8(aL)+3 (8. tJ JCA 2
1.4
,V
V /
1.3
Y I
·
1.1
/ //
1/ /
V ./
.. .9
~
-
~ ~
~
....-
~-
~ r--
---
.............
~
~
I L. .7 .. 2
............... ~
~
t---
r---~
~
¥L
00 ~
~-
3.0
~
:E_ 0
2.5
0
-
--........
- 2.U H - ~Sz
--------
"'" "-r---
.3
00-
1.0 1.0
r-- r--
r--
3,,5
210
~
""'"
.1
L-----"
~
"'-
~ r-....~
l
..,..."
.--L..--'"
L---
z-
0
H
V ~ ",.,--
~H
./
Y
V
4.0
'"
/ ./
r---
~
o
V
/
/
~ ~~~
·8
)
V
/
----~ -~ -- ---- ----- - --~~ -V / ~ / ~ ~
1.0
/'
/
-
//
4.5
.4
r--..
.. 5
~.O
4.0
0jooo-I rn
Z_
5.0
~ ~
fLO
.6
FIGURE 11.7.1 - FIXED END MOMENTS FOR Ma WITH UNIFORM LOAD OVER PART OF SPAN
• 11-91
i!/!\'\'~ ~~~'JJl STRUCTURAL DESIGN MANUAL w,-::I/vtl
2 ..4
- c~ntB ~ - MA
2,2
.
I
I
b
I
I
B I
~ ~=-~;htHI) -3(j;n CB 2.0 V~
. 1.8
I
-~
--
r--- 4.5
/ I
5-
1.1>
V
L
c~
1.4
---1.2 . 1.. 0
.8
,6
4~r-
~
0
.....
v-
--
-----
-......
.'l 1:.
f---
3.0
I--
~
p..
0 0
nL'>.
"".V
~-
1.0
------ ---
1.0
---- --~--- -- ----------
2.0
i--
.1
~
2~5
-....
o
til 00 1
~
f.--- ~
--
z-
.. 2
3.0
'\: ...::I
4,,0
Z 0
0.0
UJ
l-oI
!--
Z
nJL I-~ ~. ~
.7-
~L
.4
.5
.. 6
FIGURE 11.72 - FIXED END MOMENTS FOR Mb WITH UNIFORM LOAD OVER PART OF SPAN
11-92
~ STRUCTURAL DESIGN MANUAL
"~,,,
1.2 I-----I----l---------i--
1.. 1 r----+---+--+----+-++---+----!---t---t-+---+---I
?
~
~
1.0 .-....!.!~---I-----+------+---I--
.9
.8 !----+----l----I-----+-+---+--+-~IIor-+--+----I----I
) o
1
2
3 L/j
4
5
6
FIGURE 11.73 - COLUMN DISTRIBUTION FACTOR, 28 -
Q'
11-93
••.. STRUCTURAL DESIGN MANUAL
1f/71";\\
"~V!//' . . L
C.D.F.
J 0.0 .1 .2 .3 .4
0,500000 1 .500243 .501001: .. 502260, .504034;
.6 .7
.509173 :512572 .516588 .521146
.8
.9
Far End Pinned 0.750000 .749505 .747996 .. 746408 .741963 .737410' .731812 .725149 .717398 .708528
S .. c. Far End Fixed 1.000000 .999664 .998663 .996996 .994656
s.c.
L /. J
3.05 3.10 3.15 3 .. 20
C.O.F. .945618 .974360 1.000000 1.00539 1.03897
f"'ar End Pinned ,0679879 .0318257 0 -.006639 :':.047650
.
S. Far End Fixed .642565 .628694 .616850 .614429 .599738
.987943 .983561 .978486 .972709
)
5 6
1.05 1.10 1.15 1.20
.529249 .532298 .525517 ~528929
1.35 1.40 1.45 1 5 1.55 1.60 1 .. 65 1.70 1 75 1.,80 1.85 1.90 1.95 2 2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2 .. 45
2.HO
.. 682268 .. 693454 .705272 .717765
.B29074
2.~5
.~49313
2.90
' .870941 .894096 .918930 '
2. <) 5
3.00
,693049 .687289 .681220 .674834
.962707 .959011 .955131 .951066
.620634 .611423 .601812 .591788 5 .570444 .559093 .547266 .534944
.917261 .911641 .905815 .8999781 .887077 .880400 .878502 .866379
.. 508733 .494798 .480276 .465139 .449357
.851444 .843624 .835563 .827255. .818697
.432895 .415718 .397785 .379058
.809090 .800809 .791468 .781854
.220HJl .193479 .164654 .134228 . 10.2060
.70fd51 .694300 .681906 .669160 .656050
365.751 188.680 -75.1658 -34.0630 -22.0401 4.60 4.65 4.70 4.75
5.10 5.15 5.20 5.25 5.30
-328.980 170.465 68.2040 31.1862 20.3312
-11 .. 7074 -7.97597 -6.05282 -4.88096
11 .. 0020 7.61877 5.86397 4.78531
-2.1117'J -1.96093 -1.88251 -1.72210 -1.67640
3.51784 3.10973 2.78587 2.52111 2. 2/)/) '~4 2 .10 C)7 'i 1.94475 1.79898 1.66844 1.55011
.128735 .097422 .064797 .030766 6 .0002459 -.004788 -.012074 -.026871 -.041940 -.080859 -.121674 -.164549 -.209664 - 257227 -.307471 -.360666 -.417117 -.477180 - .141267 - .. 60HH55
-.683511 -.762898 -.848813 -.94221'0
TABLE 11.5 - CARRY OVER FACTORS AND STIFFNESS COEFFICIENTS FOR BEAM COLUMNS WITH AXIAL COMPRESSION LOAD 11-94
STRUCTURAL DESIGN MANUAL
•
~.c.
L
5.35 5.40 5.45 5.50
C.O.F. -I .54291 -1.46967 -1.40514 -1.34009
Far End Pinne 1. 44166 f .34126 1.24748 1.15911
s.c.
s.c.
Far End - L. 04434 - I .15634 -l. 20024 -1.41816
Far End
s.c.
Far '
.
J~nd
2.1 2.2 2.3 2.4 2.5
.410548 .4036]0 .396616 .389588 .382544
.947214 .964.380 .982024 1. 00012 1.01863
1.13923 1• 1.5205 1.16534 1. J 7906 1.19325
2.6 2.7 2.8 2.9
.375502 .368480 .361492 .354553
l. 03754 1.05683 1.07646 J .09642
I .20785 1.22287
.5. I
5.2 5.3 5.4 5.5
.228713 .224418 .220239 .216176 .212224
1.58503 1.60940 1.62304 1.65674 1.68049
1.67836 1.69476 1.71629 1.73793 1.75974
5.6 5.7 5.8 5.9 6.0 6.1 6.2 6.3 6.4
.206381 .204645 .201012 .197481 .194048 .190711 .187466 .184312 .181245
1.70429 1.72814 1.75204 1.77399 1.79997 1.82400 I .84806 1.87215 1.89628
L.78166 1..80360 1 .82502 1.84006 1.87040 1 .89204 1.91837 1 .93799 1 .96069
j .23827 1.25406
-8.23357 i1 .4768 L8.590B 47.0647
TABLE 11.5 (CONTln) - CARRY OVER FACTORS AND STIFFNESS COEFFICIENTS FOR BEAM COLUMNS WITH AXIAL COMPRESSION LOAD
S.c.
L . 0.0 .I
C.O.F. 0.500000 .499757
.2 .3
.7 .8
.9
•
2.0
Pinned 0.750000 .750512 .75l990
.6
1. <)
f~nd
.754488 .757964 2 12 .767818 .774165 .781431 .789595
.4
)
Far
S.C.
Far End Fixed
1.00000 1.00036 l.00L33 1 .00300 1.00532 1 1.011.94 1.01623 1. 02116 1.02672
.443594 .437286 .430802 .424167 .417408
TABLE U.6 - CARRY OVER FACTORS AND STIFFNESS COEFFICIENTS FOR BEAM COLUMNS WITH AXIAL TENSION LOAD 11-95
i/(7\\~
"\~~ :j1/ .. .....
?'~
,~'/
STRUCTURAL DESIGN MANUAL
-:""
s.c.
L!' 7. I 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9
C.O.F. .162013 .159556 .157164 .154836 .152570 .150363 .148213 .146119 .144079
Far End Pinned 2.06598 2.09032 2.11468 2.18906 2 16 2.18788 2.21231 2.23676 2.26123
s.c.
s.c.
Far End Fixed 2.18167 2.14493 2.16824 2.19160
C.O.F. .068965 .066666 .064516 .062500
Far End Pinned
s.c.
Far End Fixed 4.16204 4.28571 4.04940 4.53333
2
2.23849 2.26200 2.28556 2.30917 "
21. 22 23 24 26 27 28 29
.050000 .047619 .045455 .043470 1 .040000 .035462 .037037 .0337l4
5.51250 5.76190 6 .OJ 136
6.26087 2 6.76000 7.00962 7.25926 7.50893
5.52632 5.77500 6.02301 6.27273 .
J
)
4
6.77083 7.02000 7.26923 7.51052'
TABLE 11.6 (CONTln) - CARRY OVER FACTORS AND STIFFNESS COEFFICIENTS FOR BEAM COLUMNS WITH AXIAL TENSION LOAD
11-96
•
STRUCTURAL DESIGN MANUAL
•
distribution (Hardy Cross method) Is used to determine LlH\ moment j n lile beam at each support point. Once determined; (1) the reactions at each supporL can be determined from statics for each span, and (2) the bending moment and deflection at any point in a span between supports can be obtained by considering each span Lo be simply supported and with the applied axial and transverse loads the internal loading can be determined.
MOlJlent
Moment distribution can be used as follows: (1)
(2)
(3)
(4)
•
(5)
(6) (7) (8) (9)
)
Assume each span to be fixed against rotation at both ends. Determine the fixed end moments, FEM, for each span due to the applied transverse loads or moments using Figures 11.66 through 11.73. Determine the fl net" moment at each joint. The net moment will be the difference in fixed end moments of the spans on each side of the joint plus any applied moment. Free a joint allowing it to rotate due to the net moment. Balance the joint by distributing this net moment to the members at the joint in proportion to their distribution factors (D.F.). The balancing moments are opposite in sign to the net moment. The sum of the moments at any joint is always zero after each balancing. Determine the moments at the far end of each span. This Is done by multiplying each distributed moment by the mEmbers carryover factor (C.O.F.). This carryover moment is assumed to be acting on the joint at the far end when that joint is freed and balanced. Repeat for all joints. Repeat the entire process. This time only carryover moments will be balanced since there will be no fixed end moments. Repeat until the carryover moments are negligible. Add up all balancing and carryover moments at each joint to obtain the final moment at the joint.
In the moment distribution process the sign of all moments (FEM, COM and balancing moments) are defined by the direction in which they act on the joint between each span. If the moment tends to rotate the joint clockwise it is positive (+). If the moment tends to rotate the joint counterclockwise it is negative (-).
•
Figure 11.74 is an example of the moment distribution as applied to a beam column with a compression load. Figure 11.75 shows the same example wi.th a Lension load.
11-97
i
100000
"I-
20
~
looooo ..
l.
u"2€
A: 50(20) 27.7
5~~~~;2
j = /~~I/p\
50
*4in
J\
1
5°~i~;2
20/10
s.c. (1'11.5) S.F.(El1.23) :LS.F.
=
D
1 .) ~-c LOOOOO
Figure 11.70
c: 4000(Z~)(~;)2(42)(l.ll.)= 10847
= 4862
:
33/17.32 = 1.905
2
.859 .859 .873 1.72ac{»;1.72Q06) 3.17(10 6 ) IX) 4.89(10 6 ) D.r'.(1~11.25) l.0 0 .351.649 C.O.F.(Tl1.5) .627 .627 .612 00
-4000(~,2(H)(42)(1.08)=-26449
-722 0
-1067 4862 o }I< 1322 .. 2463 -835 0 4778 o X1617 -3101 -lOSt 0 458 o 161 -297
2nd BAL C.O. 3rd BAL C.O. 4th BAL C.O. 5th BAL
835
Final
1392 -l392 -4459 4459
0 1051
X.:.
0 -101 0 577 101 o ~203 -374 0 55 0 -127
127 0
-19
-36
42/20
.873 .844 3.17(10 6 ) 3.22(10 6 ) 6.39(10 6 ) .497.503 .. 612 .643
o 722
C.O.
Moment
j
I
11.2
= -1067 C:
~
4000
Flgure 11.66 ) ,
LOOOOO
/10 7 /10 5\=10/3(10 7 )/10 5" = 17.32 /4(107)/105': 20
L/i
FEM 1st BAL
C
1 .67 2 = 722 B: 50(33) = 4862
- t
EI = 4(10 7 ,
~(.arrm)(.v.Jl.l unlll U,V.)(~
F·g
1l.66-11.73) B:
B
400~
,
~
EI = 3(10 7 )
•
--1---'2--i .....
if/in
w - 50
(107,~
A
(Figure
42
1M" UIII II II In lL
KI =
FE M
'1·
33
-4862 -10847 7807 1902
;;><
=
2.1
.844 3.22(10 6 )
oJ·O
.643 26449 0 Q -26449 5081 0
.1507 0 749
~
o
578
~
943
955
-182 90 -229 114
0 92 0 115
l025 -1025
•
-508l
487 0 . o -487 614 0
-1898 0
~
00
o
-614
59 0
o -59
32690 - 32690
lOOOO~~~~'.)~ IlXitrn IIIl r){.}(:~=======t~tX.~OO ~ I 1389 t , 991
3611
659
1392 100000
0000
~
126
1617
270
FIGURE ll.74 - EXAMPLE OF BEAM COLUMN WITH AXIAL COMPRESSION BY THE MOMENT DISTRIBUTION METHOD
11-98
3611
•
//0' \\) STRUCTURAL DESIGN MANUAL " \?fB~11 \\ \ V~:··~: . . .I~" ---
"
.......
'
r--
t
33
20
~ 4000
42
100000
100000 EI A
I
FEM (Figure 11.66-
11.73)
O~OO
B
• \f
2
=
J'\
621
21.3
•
20/10
=
Figure 11 66 "
0'621
-621 0 -498 498 0
-402
3rd BAL
e"o.
4th BAL 5th BAL
FINAL
402 0 28
C.O.
MOMENT
-28 0 -23
23 274 -2,74
"
Figure ii.70
10Q.POO
•
c=- 50(20)
1 D: -4000(42}{4,02)2{4 i)"{.925)=-22676
.
2
12.7
33/17.32
-939 4287 -1195 -2153 0 2703 -965 -1738
0-187 67 120 o 151
-54 -97
o -11
== .. 4287
= 1.905
30 12 2 4000(42)(42)(41) (.91) = 8911
42/20 = 2.1
1.116 1.139 6 4.06(10 6 ) 4.34(10 ) 8 4( 106 ) .483 .517 .424 .411
.
1139
4
56
20 21
-3081 308]
2184 -2184
34(10~
.
00
6 4 34(10 ) 0 1.0
411
-4287 -8911 6375 6823 913 0 -441 '-472 -737 0 356 381 51 0 -25 ... 26 -41 0
22676
0
0
-2267
2804 0 -]94
-280 4
0
157 0 -11 0
194
-L57 11
25432 -25432
400°1
10~.)~\(~~
•
D
c:
.
C.O.
*
5 ( )2 B: 0 20 = 4287 12.7 2
s.c. (T 11 .6) 1.127..6. 1.1 ~11.116 S • F • ( E J 1. • 23) 00 2.25(1()~2.25(lt.r 4.0g(106) I:S F 00 6 31(10 ) D.F.(EI1.25) 1.0 0 .357 .643 C.O.F.(Tll.6) .417 .424 .412 FEM 1st BAL C.O. 2nd BAL
4000
r
8:- 50(20)2 = -939
L/ j
C
50 #/in
fQ1It '(.)(I 111 lL 1I nU)' .)f(c:::=::::::::=====::::::::l=:::::r)(
FigU:~11.68
A: 50 (20) 32.2
= 4(10 7)
)(~~~========:::::=;l~(.) 10,2.000 . ll---;;f 85 50 II/ ig 65 • 590 3410 ~ 10~;(~UllmllUlaq 4000f i~o
~1
t
1486
t
) 25-:32
75
3410
+
FIGURE 11.75 - EXAMPLE OF BEAM COLUMN WITH AXIAL TENSION BY THE MOMENT DISTRIBUTION METHOD
STRUCTURAL DESIGN MANUAL
The slope and deflection at any point in the beam may he determined by considering each span separately and using the equations presented in Tables 11.3 and 11.4. 11.2.4
Control Rod Design
Control rods are characteristically long rods with swaged ends having rod ends lhreaded into them. Two column analyses are necessary to insure the rods are sufficient. A typical control rod is shown below.
The column must be stable as a stepped column using the analysis in Section 11.1.3. It must also satisfy the beam column analysis. Following is the procedure for control rud design: (1)
Calculate P
(2)
Determine P"u
llV
1 . ;. Pa ( Per
cr
for the stepped column using Section 11.1.3. from
1
~P a 11 / Pc r
..; Per ) Fcc A
p
cr
•
e
M
all
where P cr F cc A
Critical column load, Section 11.1.3
= Crippling stress of column = Cross sectional area
e Mall (3)
11.2.5
section
L/800 =
Allowable moment of section in plastic bending
M.S. = Pall/p
- 1
Beam Columns By The Three-Moment Procedure .~~c
The three-moment procedure can be applied to beams carrying axial con~rcssion or tcnsjon in addition to transverse loading. The procedure is descrihed in Section 9.4.
11-100
•
STRUCTURAL DESIGN MANUAL Revision ,E 11.3
Torsional Instability of Columns
The previous sections have assumed that the column was torsionally stable; i.e., the column would either fail by bending in a plane of symmetry of the cross section, by crippling or by a combination of crippling and bending. There are cases when a column will fail either by twisting or by a combination of bending and twisting. These torsional buckling failures occur when the torsional rigidity of the section is very low. Thin walled open sections, for instance, can buckle by twisting at loads well below the Euler load. Often in thin open sections the centroid and shear center do not coincide, therefore, torsion and flexure interact. In this section, it will be assumed that the plane cross sections of the column warp, but their geometric shape does not change during buckling; that is, the theories consider primary failure of columns and not secondary failures, characterized by distortion of the cross sections. There is coupl of primary and secondary failures but no method has been developed to handle them so secondary failures will be ignored. 11.3.1
Centrally Loaded Columns
Centrally loaded columns can buckle in one of three possible modes: (1) they can bend in the plane of one of the principal axes; (2) they can twist about the shear center axis; or (3) they can bend and twi~t simultaneously. Bending in the plane of one of the principal axes has been discussed previously. The latter two modes will be discussed here. 11.3.1.1
Two Axes of Symmetry
When the cross section has two axes of symmetry or is point symmetric, the shear center and centroid coincide. In this caie, the purely torsional buckling load about the shear center is given by (11.27) where: ro
polar radius of gyration of the section about its shear center
G
shear modulus of elasticity
J
torsion constant (Section 8.0)
E
= modulus of elasticity
r
wa rping
L
effective length of the member
cons tan t of the sec tion
(F iqure
11. 85)
For a cross section with two axes of sywmetry there are three critical values of the axial load. They are the flexural buckling loads about the principal
11-101
STRUCTURAL DESIGN MANUAL axes, P and P and the purely torsional buckling load, ~. One of these loads will bexminimu~ and viII determine the mode of failure. In this case there is no interaction and the column fails either in pure bending or in pure twisting. Shapes in this category include I-sections, Z-sections and cruciforms. 11.3.1.2
General Cross Section
In general a thin walled open section buckling occurs by a combination of torsion and bending. Purely flexural or purely torsional failure cannot occur. Consider a general section with the x and y axes the principal centroidal axes of the cross section and x and y the coordinates of the shear center. The cross section will undergo trRnslatign and rotation during buckling. The translation is defined by the deflections of the shear center u and v in the x and y directions. During translation of the cross section, point 0 moves to O' and point C to C· where 0 is the shear center and C is the centroid. The cross section rotates an angle ~ about the shear center. Equilibrium of a longitudin"al element yields three simultaneous equations, the solution of which results in the following cubic equation for calculating the critical value of buckling load.
r
o
2(p
cr
_p )(p
y
cr
_p )(P -p,J _ P 2 y2(p _p) .. P 2){ 2(p _p) x cr ~ cr 0 cr x cr a cr y
o
(11.28)
where P
p
x
1r2EI IL2 x
=
(11.29)
(11.30)
y
(11.31)
Solution of the cubic equation, 11.28, gives three values of the critical load, P , of which the smallest will be used. The lowest value of P will always b~rless than P , P , or P~. By use of the effective length, L,c~arious end conditions canxbe tonsidered. 11.3.1.3
Cross Sections With One Axis of Symmetry
A number of singly symmetric sections are shown in Figure 11.76. If the x-axis is considered to be the axis of symmetry, the y = 0 and the equation for a o general section reduces to
(p
cr
11-102
_ p ) y
{r
2(p 0
cr
_ p )(p _ p ) _ p 2x 2 } x cr ¢ cr 0
o
(11.32)
_.' 7{", '.'
I( / f \ \
•
...B~II ,.,"'" .. I Ij
~ ~ "
\
".>
\\
STRUCTURAL DESIGN MANUAL
':"1'--'
TIH~rp
::lH! again lhree soluUons, one of which is p. :::: P and rt~pres('nl.s pun.. ly fl('xUr'llJ buckling abouL the y-axis. The olher twu<-~n~ tKe rools of the
) Failun- of s.ingly symmelrical sections can occur either in pure bending or in silllul.lancous bending and twisting. The (,-val.uatton of the tor.~ional-flcxllral huck I ing load can neve r be as simple as the de tenninaU on of the Ell le r toad, tllt-n.-fon.:, steps have heen taken to categorIze modes of fallure. CerLai.n combinations of dimensions will prohibit torsional-flexural failures. l"or sect i.ons symmetrical about the x-axis, the critical bucklin~ load j s given by ('quation 1.l.32. The load at which the memher actually huckLeo LS (:·tth(·.r P or tht' ~llIaJl(lr root of the quadratic equation. Y
•
The buckling domain can be visualized as bei.ng composed o[ three regions. These ar(' shown In Figure 11.77 for a section whose shape is defined by th(~ raLio, bIn. Region J contains all sections for which I >1. In this region only t.orsional. f r.('xural bucJd lng can occur. Sections for Ywh12h I > I falls i.2lO Rcgi.on 2 ~r 'l. In Kegloll 2, the mode or buckling depends on the p~ram~ter LIia.. Th(' (U)a ) . clirve- [('pres'ents the boltnda~y between the two possible modes of failun~. J l jl~ln a plot of the value of tVa at which the buckling mode changes from purely [Icxural to toroionaL-flexural. 2The boundary between Regions 2 and 3 i~. localed al the jnlerSc'ction of the (tWa) . curve with the bla axIs. Sections jn '2 Region j wi I t always fail in the f~l~~ural mode regardless of the value of LUa J4' i gure 11. 78 def i nes the se curves for angles, channel s, ar:d ha L sec tl ons. .[ n Lhis (igure, lTl(;'tnoers that plot below and Lo the right of the curve faji in Lhe
lorsional-[Jexural triode, whereas those to the left and above fail in the pun: bending mode. The curves in li'igure 11.78 also give the location of the boundaries between the various buckling domains. Each of the curves approachcs a v(:rtical asymptote, indicated as a dashed line in the figure. The asymptoLt', wh.i~h is the boundary between Regions 1 and 2, is located at bla correspond.ing to secLions for which I = I . Sections with bla larger than the transition value at the asymptote ~ill Ilways fail in torsional-fJexural buckling, regardless of their other dimensions. If bla Is smaller than the value for the asymptoLe, then the section fails in Region 2 and failure can he e i th<.'r by pure flexural b~ckl~ng or in ~he to-:sional-flexural Hlo~e .. In Lhis rcg~on, LI.IC' parameter, tUa , WIll detennlne which of the two posslbl{~ modes of fal lure 1 S crilical. In the case of the plain and lipped channel section, ~here is a lower boundary Region 2. This translUon occurs wh(~re the (tL/a )1" curve jl1ll'TS~ct~ th~ b/a axis. Sections for which b/a is less than the Jf11uc at this iIlL~n;l'cli.on are locat~d in Region 3. Theoe szc.tions wilt aLways raj I in the fl('xLlraJ mode, reg~rdless of the value of tUa. For the: Upped anb1c and hat secLi.ons the (tL/a )1" curve does not int~rsect tht: b/a axis. Region 3, wlH:rc only [l(!xural bucklingllloccurs, does not exist [or these ,':wc.t.ions.
11-103
/··/T·· '..
>L:'~~b~ STRUCTURAL DESIGN MANUAL , ,1, ,.. \'
, ",
,... .'~.~.""
__
TI b
r--
__
a
T a
x-- - a
Only
~
[tL
10
I
2a
I
T
• :rorsional -Flexural
~uckling
J
,...,1 ...
r- b-1
~)nly
f
'~
x
t
I
tL
a _ ___
_~_x
FIClJHE 11.7() - SINGLY SYMMETfnCAL SECTIONS
Flexural Buckling Mode Bucklinl Depends on Value
c
I
•
2
J
I
1
I
I I
t I
,
I I
I
I I
I I
b/a FIGURE 11.77 BUCKLING REGIONS
11-104
•
•
STRUCTURAL DESIGN MANUAL Tilt' cri I iCJI Illtt'kl illl'. load ror singJy sYIIIIII('trkal S('ctiOI}S (x-axis is Llu· axis or sY"IfIll'lry) that buclde in the Lorslonal-[l~xural mode is given by the Jow<,,'!;l
rool of r
o
2(p
cr
_p )(p
x
,
\,
cr
_p) _ p 2x 2 = ¢ cr o
o.
(11.33)
2
Divldlng this equation by P P ..,{r ,and rearranging results 1n the folluwjng x .,., 0 inleraction equation:
(11 .34) in which (11.35)
•
is a ::>hapc factor that depends on geometrical properties of the cross section.
Figure L1.79 is a plot of equation 11.34. This plot provides a simple method for chccking the safety of a column against failure by torsional-flexural huckl jrlg. To
j [ a given IIlember can safely carry a certain load, P, l.t Js onLy Lo cOlllpuLe P and P for the section in question and ttwn, kllowinJ; K, use the correct cur~e to c~eck whether the point determined by the arguments PIP and PIP falls below (safe) or above (unsafe) the pertinent curve. If it is ~eslred t~ determine the critical load of a member instead of ascertaining whether i l can safely carry a given load, use deL{~rIll1n(!
n{'Ct'ssary
) which is another form of equation 11.34.
•
The interaction equation 11.34 indicates that P depends on three factors: the loads, P and P , and the shape factor, K. pr .and P are the two factors which intera~t, whire K detenmines the extent toXwhich ~hey interact. The reason bend and twisting interact is that the shear center and the centroid do nol coincide. A decrease In x , the distance between these points, therefore . \> causes a d ecrease 1n th e 'lnteraction.
11-105
-,' ;r" ".
II /. '. \ \~
'\:l:~M STRUCTURAL DESIGN MANUAL 9
b~l~~l:elll
8
7
:~
6
~I~ o 5
~
1 J
I
'/
L
lOll
~
/1 IIi
I : I
/VI) : :_1 ; fA V 000ofE-
~r/
/ t b)
~
'"
; :; ; ~.~o.y-
0.2 0.4- 0.6 Cl.81.0J O.2 0.40.6 0.8 1.0 t.2 1.40
0 .,
• 01".C'
Ii V Ilf ; ~
N Q
•
Ii
I
I
I r I
I
f
t
-0 UI
*
00 .-
VoIlJe o( • I I •
0.2 0.4 0.6 0.8 1.0 1.2
•
I ,
I.
i"
Vatu. I I f {~
°
(// ~I
"he
(el
o
I
II'0/..
0
illl/; I i
V
2
o
o
:j :I
Q~{D
a/ll. of
J
4'3
cev
l
~/I
: Ii 1~.4
1.6 1.8
" -b Raho Q
FIGURE 11.78 - BUCKLING MODE OF SINGLY SYMMETRICAL SECTIONS 1.0
~~~=t:~=;==F==t=::::r=-I-T-r---'
0.8
;----+---~-~-~
0.7
l----+----_+_
0.6
1---t----+----I----.:lIir---~.,:.I
0...
~
o 0.5 r----+----+-----,I----t---~
r-<
OC
O.l - - -
:. + :. -K (
~>.)= \--.. -
)
0.. 2 t - - - - - t - - - - ; . ,
o--------~--~~--~--~------------~--~--~ 0.2 0..8 o 0.1 0.3 0.4 0.5 0.6 0.7 0.9 1.0 RATIO P/p x FIGURE 11.79 - INTERACTION CURVES
11-106
•
j""
.:7\- . . "
;;/ /f \ "\-~'~ Ii /
•
\\
"\~~~J.' STRUCTURAL DESIGN MANUAL To ('vaJuall' Lhe' lorsLonal-rlexliraL buckling load by lIIeans or Lhl' inL(~racliOIi (·quation t il loS Ilt\c{'s~ary Lo know P and K. A conv('nienllllethod for dcLl'rlllining these Lwo parameters is therefore a~ assential part of the procedure. For any given sectIon, K is a function of certain parameters that define the shape of lhC' section. Starting with equation 11.35 and substituting for x and r , K can be reduced to an expression in terms of one or more of these p~ralOe:t('~s. If the thickness of the lIlember is uniform, the parameters will be of the fonn bla, in which a and b are the widths of two of the: flat components o[ the section. In the case of a tee section, for example, equation 11.35 can b(\ reduced to: 4
K
+ bl a} {( bl a) 3 + l}
(11.37)
in which bla is the ratio of the flange to the leg width (Fig. 11.76}.
•
In general, the number of elements of which a section 1s composed and the number of width ratios required to define its shape will determine the complexity of the relation for K. Because all equal-legged angles without lips have thc.:: same shape, K is a constant for this section. For channels and lipped angLes, K is a function of a single variable, bla, while lipped channels and hat sections require L\\lO parameters, bla and c/a~ to define K (Fig. 11.76). Curves [or d~tennjnatjon of K have been obtained for angles t channels, and hal sec Lions. These curves are shown in Figures 11.80 and 11 .. 81. A single curve Covers all equal-legged lipped angle sections. The value of K for all plain equal-legged angles, K = 0.625, is given by the point bla = 0 on this curve (Fig. lL.80). For hats and channels (Fig. 11.81), a series of curves is given. The evaluation of P follows the same scheme as that used to determine K. Slartjng with the e~uation for p~, given in equation 11.27, and substituting for r , J, and r yields: o
)
(11.38)
a general relation for P , in which, E = Young's modulus. A = cross-sectional area; t = the thickness ~f the section; L = effective length of the member; a = the width of one of the clements of the section; and C and C = functions 1 2 of bla and cia, in which band c are the widths of the remaining elements.
•
Equation 11.35 indicates the important parameters in torsional buckling and their effect on the buckling load. Similar to Euler buckling, P varies directly with E and A. The term inside the bracket consists of fwo parts, the St. Venant torsional resistance and the warping resistance. In the first of these, the parameter, t/a, indicates the decrease in torsional resistance with decreasjng relalive wall thickness; whereas, in the second the parameter aIL shows the decrease in warping resistance with increasing slenderness.
11-107
"';0-' '.' ,
/'
//.1 J(
' '.'
l
"
. \ \\
~B!ell .. ,.......... . j /......// ~'. _'1.: \ \'
\
STRUCTURAL DESIGN MANUAL 0.1 r------,~--.....--------
0.6
~
"," 0.5
•
1--~---,f-----I----+--------1
.I-----f~r..----I---.c:
....vo
~ ~
o
0."
I--------ir----+~--+-
0.3
t - - - - - t - -.........---J-~-_l__-__I
..r. (I)
O.2~--~----
o
0.2
____ ____ __ 0.4 . 0.6 1.0 0.8 --~
~
~
. 0"b R0'10
FIGURE 11.80 - SHAPE FACTOR .FOR ANGLES. ".0 r---"",--,--~-"""----,~-"""-"",----,--,,,,,--"""-------'r---.......----
•
lo-
t
0.6 t-\-~~~r+----.---+--....,----t---4--4-----i-."""
.J!
!• O.5r-~~~~~r_+_--~--~----+_~--__I_----~~~~~~~4_--~~ V)
.
O.l~--------------------~--~--~--~--~--~~--~---L--~~. 0.1 0.4 0.6 0.8 \.0
1.2
t....
Ratio
FIGURE 11.81 -
II-lOB
S~APE
0.2
0.4-
0.6
0.. 8
II. a
FACTORS FOR COMPLEX SECTIONS
\.0
1.2
1....
•
STRuc-rURAl DESIGN MANUAL The co~fficients, C and C , in the St. Venant and warping terms are [unctlons 2 1 of bla and cIa, respecliveIy. These terms therefore indicate the effecl that the shape of the section has on P • ~
Svcti.ons composed of thin rectangular elements whose middle lines intersect at have negUglblt, warp1.ng stiffness; i.c., r = O. BeCallS(l C i~ 2 proportional. Lo r, the torsIonal buckling load of these sections reduces Io:
a common point
EAC1(t/a)
2.
(11.39)
For the plaIn equal-legged angle, which falls intfr this category, P_ can be fur ther reduced to: AG(t/a)2
=
Pt7>
in which G is the shear modulus of elasticity, and a is the length of one of the legs.
•
In general, however, C and C must be evaluated. Curves for these values are 1 2 given in Figures 11.82, 11.83, and 11.84 for angles, hats, and channels • For other cross sections values of the warping constant, shc-ar Ct~nt('r are giV0t1 jn Figure 11.85.
r ,
and location of
Eccenlr ical) y Loaded Columns
11. "L 2
The previous section described the buckling of columns with centrally applied loads, i.e., at the centroid of the section. If the load, P, is applied eccentrically as shown in Figure 11.86 the general cubic equation for calculatIng Pis c.r
)
A P
3 cr
3
+ A P
2 cr
2
+
Al P
cr
+
A
o
0
(11.41)
Where:
+
A/lo. { (Ox {32
A/ J. - e
•
o
AI I o A
()
_p
(3t(P
x
{p P e
y
~
y
+ P )}- (p
x y x
P P
x y
- (e
f3
Y J
Y
(1) 2 + p (X
{p (y. x 0
y
e
y
x
0
- y ) 2 - (e 0
0
I
x
0
- e ) 2 - e {3 2 (p x x x
+ p ) y
+ Py + p ¢ )
{32 + P P e {3]} x y y
I
- X ) 2}+ I
x
+
(p P
x y
+ P p y
~
+ P P ) x
~
+ I + A(X 2 + Y 2) y
0
0
11-109
STRUCTURAL DESIGN MANUAL P P
P
EI
x
E1
y I/J
x y
1r
2/L2
rr 2/L2
.- A/I 0 (GJ +
g
r
f
3 y dA
+ f X2YdA) - 2Y 0 A
(J
3 X dA
+ J XyZdA)
"1
ill (
"2
1/1
x
~
YA
•
2X 0
A
11"2 ILl)
In the general case, buckling occurs by combined bending and torsion. In each case the three roots of the cubic can be evaluated for the lowest value.
if P acts along the shvar center axis: {'
('
x
y
X
()
y 0
and lht' buckling loads become independent of each olher, lh(, critIcal Joad wlll bp til{' lowest of the two EulE-'r Loads, P ,P and tlH' load corr('spondjng Lo pllr<·ly torsional huckling which is: x y p~ =
(1 /A) PIe
rOY
~,
+ e
x
aZ +
I
0
IA
( I I .42)
When the co I umn has one pi anc of symmetry and the J oad ae Ls in the p I ant,' of symmetry (! = 0 and buckl ing in this ptane takes' place independently and th(~ c r i U ca t I ~ad i. s thf-' same as the Eu 1 er load. HOW(~V(,T, I a U'ra I buck ling and L()'rsj onal buckl ing are coup Jed and the cri tical loads are ohtai ned [rom the following quadratic (·quation: e ) y
2
o
•
(11.43)
• 11-110
STRUCTURAL DESIGN MANUAL ~
O.4r---~----'-----r---~----~----~--~----'O.8 0.3
t---4~--+---+---I--
--+----111 0.6
u
-....o
~
0.2
1------I----3IIoor+-----I------I--
----hI''----;
"'o"'
0.4 •
!)
~
a
b
>
N
U
>
0.1 r----+----~----~~~----4_----~~_+--~~2
(a)
(b)
O~--~----------L---~--~~----~
o
0.2
0.4
0.6
Ratio
0.8.0
0....
0.. 2
__ __ ~
0.6
~O
0.8
J!. a
FIGURE 11.82 - TORSIONAL BUCKLING COEFFICIENTS C1 AND C FOR ANGLES 2
•. 01"""---....,.----.------.----------
1.0
o.e ';-0.6
....0 ~
-;
'>
0.:
0 ....
0.1 (0)
N
0 (d)
0.'
U ....
: D.4t---+----+---+---.,,~1_
'--+-----4---1
"
-;
{f 1).4
'>
....
)
0
" -: 0.2. ~
0.+
0.6
0.8
b
1.0
1.1.
I."
o. Uftr
0.3
....
I)
II
,1
8. 1
.:?
.;
Rdtioi'"
FIGURE 11.83 - TORSIONAL BUCKLING COEFFICIENTS C1 AND C FOR HAT SECTIONS. 2
FIGURE 11.84 - TORSIONAL BUCKLING COEFFICIENTS C1 AND C FOR CHANNEL 2 SECTIONS.
11-111
STRUCTURAL DESIGN MANUAL b
S &
shear center
c
~
centroid
r
= worpina constant
e= b(b))2(3b-2b,) iT [2b 1_ (b~b\)3 ]
-I
b
bI
f-
s
tfh 2h3 r = 24
h/2
~
-I e
=h
h
tf
b3
1
S
b3 .... b3 1 2
e=- 20
$1n
«-«c.OoS
c(
flC- stn (( cos or
1~2
b~
12
b3 + b3 1 2
r=--
3 b2
!
b
I"
3b 2tf e= . 6btf + htw
tf
e 0
h
s
tfb 3h2 3btf + 2htw
tw
r= - -
tf
II
12
j b'
I·
t
...... f--tw
6btf
t ~tw
tOt
-I
tf
s &0
h h/2
f
r r=
=
0
tf
I-
s
'tz
I
t
~
b
o
d
s
\,3,,2 2 [. 2tf{b 2 .... hh . . h2) 12 (2\, + h)
T
3 t bh W
]
l-
b
FIGURE 11.85 - SHEAR CENTER LOCATIONS AND WARPING CONSTANTS
11-112
•
STRUCTURAL DESIGN MANUAL Revision E
-
,
~ • b
....... ~w
t
~:
I
s..
*f ~
uI
-r--
s t--
--t
--
0
--
r
_L..-
,
3 (1,2- btl
j
l.1 < D
(w/t)h + 6 (b + 1>1)
-Values of e/h
~ 0 0.1 0.2 0.3 0.4 0.5
1.0 .430 .477 .530 .575 .610 .621
0.8 .33C
.2BC .42c .47C • 50~ .51 i
0.6 .236 .28C .325 .365 " 39~ .40~
0.4 .141 .183
.222 .258 .280 .290
0.2 .055 .087 .115 .138 .. 155 .161
b
I
r}
--,-
s
e-
l
e
h
-,.-
h
f,
n
1
0_
r-e--
.
J
...
0--- ....
....... t
Values of e/h
~ 0
-0
---
,
----
~
e
0.1 0.2 0.3 '-0.4 0.5 0.6
1.0 .430 .464 .474 .453 .410 .355 .300
0.8 .330 .367 .377 .358 .320 .275 .225
0.6 .. 236 .270 .. 280 .265 .235 .. 196 .155
0.4 .141 .173 .182 .172 .150 .123 .095
0.2 .055 .080 .090 .085 .072 .056 .040
Jo
')
Figure 11.85 (Cont'd) - Shear Center Locations and Warping Constants.
11-113
,,~ --\-'\ .'~'.' ~ '"
STRUCTURAL DESIGN MANUAL
-t;.----o-.-";7' /
P
L y
z
/Lx e
e
y
p
FIGURE 11.86 - ECCENTRICALLY APPLIED LOAD
11-114
•
STRUCTURAL DESIGN MANUAL Sl~CTION
12
FRAMES AND RINGS
This section presents thv methods of analyzing statically indeterminaLe rings and [ramps. Thc' principal analysis method discussed in thIs sectIon is that of rnoml'nt distribution. Particular solutions of bents and semicircular archE:~s under various loading cases are also given. 12.2 Analysis of Statically Indeterminate Frames by the Method of
Moment Distribution Moment distribution is a convenient method of reducing statically indeterminate strucLures to a probl~n in statics. Moment distribution does not involve the solution of simuLtaneous equations, but consists of a series of converging cycLes that may be terminated at the: degn~e of precision required by the. problem. The' th('ory of moment di stribution is not discusst,d since many pubLications are avaj lablt, on the subj(:'ct. Instead, a step-by-step procedure along with an examp It, p rob I ('m j 5 shown . 1
•
12.2. I Sign Convention Ttl(' ,o.;ign convt:nLlon for rnornenls in the method of moment distribution .is to c{}nsit\(-'r moments acting clockwise on the ends of a mcmbt:'r as posi tive. This convC'ntlon is illustrat(~d in Figure t2.1.
)
+M~
fIGURE J2.1 - SIGN CONVENTION FOR MOMENTS
• 12 .. 1
STRUCTURAL DESIGN MANUAL 12.2.2
(1)
Moment Distribution Procedure
Compute the stiffness factor, K, for each member.
K = ~I, general; K = (2)
I constant E;
1
K L' constant E1.
Compute the distribution factor, DF, of each member at each joint. .DF
= 2KK'
where the summation includes all members meeting at the joint.
(3)
Compute the fixed-end moments, FEM, for each loaded span and record. Fixed-end moments for various types of loading are given in Table 12.1.
(4)
Balance the moments at a joint by multiplying the unbalanced moment by the distribution factor, changing sign, and recording the balancing moment .below the fixed-end moment.. The unbalanced moment is the sum of the fixed-end moments of a joint.
( 5)
Draw a horizontal 1. ine below the balancing moment. The algebr
(6)
Record the carry-over moment at the opposite ends of the melllbers. Carry-over moments have the same sign as the balancing moments and are half their magnitude.
(7)
Move to a new joint and repeat the process for the balance and carryover moments for as many cycles as desired to meet the required accuracy of the problem. The unbalanced moment for each cycle will be the algebraic sum of the moments at the joint recorded below the last horizontal line.
(8)
Obtain the final moment at the end of each member as the algebraic sum of all moments tabulated at this point, The total of the final moments for all members at any joint must be zero.
(9)
Reactions, vertical shears, and bending moments of the members may be found through statics by utilizing the above-mentioned final moments.
•
It should be noted that simpler methods may be found [or the solution of rectangular, trapezoidal, and triangular frames in Section 12.3.
• 12-2
/-,.//",
/.,')
... ,
, .. \ \\
'\
II /
~B~II \' '~"'.'~""N
~
. ',,- "" \ r'"I, i"~;,./"//
STRUCTURAL DESIGN MANUAL l\evision C
6'
lJ
A
~:IIII-f---2-~-l2 2
M .. ~ 'A 96
•
M
96
8'A~ 2
)
:_5wL
B
9.
b ~ Af- a@L_h__:. . .fB
2
MA - ~ (lCr.. 2 -10a.L+3a ) 60L'" wa3 MB - - --2 (5L- 3a) 60L
M· in-lb
Mb MA - T P
~B
#~a \-'L - l~)
f
M_ -Jl-
Hel
L
(3.!!. -1'1 L ...
load (lb), w = unit load (lb/in)
M = bending moment (in-1b), positive when clockwise
•
TABLE 12.1 - FIXED END MOMENTS FOR BEAMS
12-3
STRUCTURAL DESIGN MANUAL 11.
AE
., 4 PfP
MB
L
13'~
-f!f~!.r -
M
A
!.
wa 3()
=
(10 -1*+
17. ~w
MA -
-I
.... lb/in I- a ~
a
~
B
~)
(5 L-4a)
13.52
-
wL 2 MB
S
-
wL 2
m A@P}t4
18.
=L 1.
MB
2
~ -~2
~
3wL 160
2
m~f(X)
ntfL
NA
JB
M B - -
=- -30
= - MA
~
12L
15.86
•
B
16~E¥¥l:rw MA
B
L
I
A
elliptic load
wL 2
ffffftt1tft
B
A;ntf ttftti' fttlf?:$ B _
NB = - MA
E
L
20L2
:
48
2
2
3 wa
~1B
~
•
t ; -f ~1
M - ~ (L3_a2L+4a3) ~
T
L
t
- 15PL
14. A
MB - - ~1A
~w
A
At
lA -
...ti.
a
1£
12.
M
MA
= -
~I!fm
L
wa 2 MA = - 6L (3L-2a) 15.
~B
L
MA = Pa(I - ~)
A
a
_X
.
B
•
L x(L-x)2 f (x)dx
0
J
L
2
x (L- x) f ( :<) d x
0
TABLE 12.1 (CONT'D) - FIXED END MOMENTS FOR BEAMS
• 12-4
STRUCTURAL DESIGN MANUAL 12.'2.-~
SalHpl<, Problem
20
1=3.5
~
·1·
25
FIGURE 12.2 - SAMPLE PROBLEM BY MOMENT DISTRIBUTION Find:
1~e
member end moments for the frame of Figure 12.2.
SoJution: (1) The stiffness factors of the members are:
IBA
2
25
KBJ\. - LBA
== 0.08, K BC == O. I
0.125, KCD == 0.08
KCE
( 2) The distribution factors of the members are: KBA == 0.44, DF = 0.56 DFBA BC KBA + KBC
DFcn
)
==
0.33, DFCE
DFCD
== 0.41,
==
0.26
(3) The fixed-end moments are obtained from Table- 12.1.
wL
2
FEMAB = 12
==
-2000(25)2 12
==
-104,000 ft.J.bs.
=
104,000 ft.lhs.
From case 3,
and -wI,
12
(FEM
BA
==
2
2000(25)2 12
Fixed-End Moment acting on the end of member AB labeled as B)
• 12-5
STRUCTURAL DESIGN MANUAL Frum case 2 of Table 12.1 t Pab 2 -5000(20)(i5)2 FEMBC = ~ ~ (35)2
-18,00.0 f
t. 1bs ..
and
25,000 ft.lbs. From cas e r
0
FEMC~: = PaL
)
f Tab I e L2. I ,
-IO~OO(40) =
-50,000 ft.lbs.
tlnd FEM1~C =
-PL
-8- = 50,000 ft. Lbs.
o since member CD is unloaded (4) Prepare a table similar to the one shown in Table 12.2. Enter the stiffness factors, distribution factors and fixed and moments fur each (,Iement of the structure. These numbers ar(~ shown in J inc's " 2 and 3 of Table 12.2.
BA 0.08
AB L
K
2 j 4 5 6
nF
~ o 44
FEM
-104
7 9
-1.24
o 5f
X
o
+4 -2
+3
-2
+4
+3
-18
-2
-2
+62
-62
CB O.
-48
-1 E
0.1
DC
+104 -38
-19
B
Be
0
J
33
CE
O. 125 o 4J
CD 0.08
o
26
+25 +8 -24
-50
0
+10
+7
+8
+lO
1+6 +17
EC
~ +50 +5
+6
+5 -30
+13
+60
TABLE 12.2 - SOl. UTION OF FRAME SHOWN IN FIGlfRE 12.2
(5) The unbalanced moment at joint B is
IFEM
= 104
- 18
=
86.
The rnoments at joint B may mOnll'nt by thc' dlstribution
be balanced by TTlul tiplying this unbH I.anced faeto
(6) A horizontal line may be drawn below the balancing moments. The algebraic sum of all the moments at this joint above this line is zero.
12-6
STRUCTURAL DESIGN MANUAL
•
(7)
Record the carry-over moments at the opposite ends of the members. The carry-over moments have the same sign as the corresponding balancing moments and are half their magnitude.
(8)
Steps 5, 6, and 7 may be repeated for joint C to obtain the rest of the values in rows 4 and 5 of Table 12.2. The process for the balance and carry-over moments may be repeated for as many cycles as desired to meet the required accuracy of the problem. The unbalanced moment for each cycle will be the algebraic sum of the moments at the joint recorded below the last horizontal line. Lines 6, 7, and 8 of Table 12.2 show this process.
(9)
The final moment at the end of each member may be obtained as the algebraic sum of all moments tabulated in lines 3 through 7 of Table 12.2.
12.3 FORMULAS FOR SIMPLE FRAMES This section presents formulas for determining the reaction forces and moments acting on simple frames under various simple loadings. The reaction forces and moments acting on frames under more'complicated loadings may often be obtained by the superposition of several of the simple cases. 12.3.1 Rectangular Frames Table 12.3 shows reaction forces and moments for various loadings of rectangular frames. In all cases in Table 12.3, K = 12 h/IlL. 12.3.2 Triangular Frames Table 12.4 shows reaction forces and moments for various loadings of triangular frames In all cases in Table 12.4, K = II 5 /I S o 2 2 1 0
12.3.3 Semicircular Frames and Arches Table 12.5 shows reaction forces and moments for various loadings on semicircular frames.
12-7
STRUCTURAL DESIGN MANUAL vE = Q -
1. VERT. CONCENTRATED'
LOAD
Q
--, a
b-1
fB C
1
A
3Qab
H = 2Lh(2K + 3) FOR SPECIAl. CASE:
h
V
= V
V
A
E
=
a:: b =
'2L
Q 2
II
.:
,)
3QL
H - 8h{2K + 3)
H
~2-.""'!":='~--=C~ON~lC=-=E:":'::NT=RA~T=ED=------I VA Q
= Q£Lb
+
[ 1
,
H =
M
a (b - a) ] L 2 (6K + 1) ,
VE
=Q
3Qab 2Lh(K .... 2) = Qab [ .
A
1
2(K + 2)
L
(b - .1.)
_ Qab [ 1 + (b - a) ] - L 2(K + 2) 2L(6K + 1)
~
a=b=!!
FOR SPECIAL CASE:
2
VA = VE = Q. 2 H = 3QL
-8';;::'h""(=K-+~-2-}
r-
3. HORIZ. CONCENTRATED
LOAD
TC
v
L---1 D
H E
=
~
L
=.Q![ bK(a + h) 2h - h2(2K
+
FOR SfECIAL CASE: V
+ 1]
3)
b = 0,
a = h
= Q.h L
H =H A E
=9.2
v
TABLE 12.3 - REACTIONS AND CONSTRAINING MOMENTS IN RECTANGULAR FRAMES. 12-8
l
2L(6K + 1) .
- VA
· ::'-7/'\', '"
./ ~ ;' l \ . . '.
I, /
I,
\\
STRUCTURAL DESIGN MANUAL
~Bell \\\"\- ).... ,,, ..... ~'~'"
I,.
HORlZ. CONCENTRATED
2 3Qa K
v '=
rh_
= gab
H
L
2h2
E M A
lIZ
Q~ [ 2h
Q
= Q!2h8
ME
HA
+ 1)
Lh(6K
LOAD
h
b
r
J
b + K (b - a) h(l< + 2)
b(h + b + bI<) + h h(R + 2)
3al< 1 (6K + 1) ,
+ b + bK) + h 3aK ] h(K + 2) (6K + 1)
-b(h
FOR SPECIAL CASE: _
+
b· O. a
c
h
3QhK
V, - L(6K + 1)
n
HA -- HE -- Q 2
MA 5. VERT. UNIFORM
VA
RUNNING LOAD
==
Oh(3K + 1) 2(6K + 1)
'_
ME wed
=
L
VF = we - VA
:
= w~ ( ,,+
~) ~ we (1 - ~)
H= {h r;~ ;2J .. 24Lh~;~ + 3) I 12dL-12d 2-c 21 where: Xl
c
_
X 2
==
we
we [24
24L
r
24L -
~
d3
L
d = L -
be2
L
~ + 4c 2 L
+
_ 24d 2
L
L
a, =
L
Ot
C
'
=b
= L, d =
VF a
'I
3 2 3 24!L _ 6 bc + ~ + 2c 2 - 48d 2 +
FOR SPECIAL CASE: VA
_ 6
~ 2
b
'2 - "2 H == 4h(2K
+
3)
TABLE 12.1 (CONTln) - REACTIONS AND CONSTRAINING MOMENTS IN RECTANGULAR FRAMES
12-9
"'~\ . STRUCTURAL DESIGN .. ,
MANUAL
'/
...
wed
Xl - X2
= ~ + L(6K
VA
+ 1)
Xl and X2 are given in ease 5 VF
we ... VA
=:
3(X1 + X2) H
+
2h(K
c
MA
2)
==
Xl + X2 Xl - X2 2(K + 2) - 2(6K + 1)
) FOR SPECIAL CASE: wL VA :: VF lIZ "2 wL d
D
H
L
7. VERT. TRIANGULAR RUNNING LOAD
4h(K
c:
a
=:
0, e = b == L, d ==
2'L
2
+
2)
wed VA
!:II
2L
V
.c
~ 2·
V
-
F
A
II:!:.£
(~
2L '\
+
2c ) 3
+ X4l_ 3we [ c2 2 -J H = 2h 2K + 3 J- 4Lh(2K + 3) dL - IB - d 3 [X3
WHERE: we [d
X3 :: - 2L
X4
t
t
=
L
we: [d
2L
3
3
c
= 8h(2K
c
2
L + 18
+ 3)
5Ic
3
+ '9 + BlOL
FOR SPECIAL CASE:
H
2
+ SIc
+ c 2b d 2 ] 6L -
3
810t -
c 2b 6L -
a=o, e=b=L, d
2
L
=: -
·3
·n
TABLE 12.3 (CONT'D) - REACTIONS AND CONSTRAINING MOMENTS IN RECTANGULAR FRAMES
12-10
J
2d + ciL
STRUCTURAL DESIGN MANUAL 8. VEHT. TRIANGljLAR IWNNlNG LOAn
x] .:md X4 nrc
glvcn in caf;e 7 we
VF
= '2 -
MA
= 2(K +
M
-
=
It
VA
3(X3 + X4 ) 2h(K
+
2)
c=b=L, d
==
~
X3 + X4 2)
x + X4 x) - X4 3 + --...;~_~
F - 2(K + 2)
2(6K + 1)
FOR SPECIAL CASE: VA
. wL
=~
VF =
wt
'3
r
•
==
-1. _
1
1 - 10(6K + 1)
r 1 + 20 (6K 1+ 1) 1I
H = 8h(K
MA
8=0,
wL 2 120
+ 2)
[ K +5 2 + 6K 1+ 1 J
9. HORIZ. UNIFORM RUNNING LOAD
HA
= w(a
- c) - HF
+ K [w(a2-c2)(2h2-a2_c2)
OJ
Bh 3 (2K + 3) FOR SPECIAL CASE:
)
c=o, b=o, a=d=h
v HA = wh.- HF
v
•
v
HF =
~h
r
1 + 2 (2K\
3
)1
TABLE 12.3 (CONT'O) - REACTIONS AND CONSTRAINING MOMENTS IN RECTANGULAR FRAMES
12-11
l(f\'\'~
'\iV~~" STRUCTURAL DESIGN MANUAL
•
10. HORIZ. UNIFORM RUNNING LOAD HA
= w(n
- c) - HF
Xs
_ w{a2 _ c 2) HF -
4h
+
- 2h
X6 (K - 1) 2h(K + 2)
WHERE: 3 Xs = _w_ [d (4h - 3d) ... b 3 (4h -3b)']
12h2
~
X6 ::::
12h
[a
3 (4h - 3a) - c 3 (4h -3
2
OK + 1) [ \;(a\ - c ) -X5 MA
=
2(6K
+
~[ 1 2 K +
_ OK +
2
l)r
MF -
+
+
C)]
J
1)
I+ Xs
3K
6K + 1 ..
•
2 ",(a\ - c ) - X5J
2(6K + 1) _ X6 [_1_ _ 3K I 2 K + 2 6K + 1 .
FOR SPECIAL CASE:
wh2K + 1)
HA
:=
wh(2K + 3) 8(K + 2)
M
:::: ,t/h
v = L(6K H :::: F
MA
wh2 [JOK :=
c=o) b=o) a=d=h:
24
wh - HF
2
F
+ 7
24
r
18K
+ 5
J
1]
6K + 1 + K + 2
TABLE 12. I (CONT'D) - HEACTI.ONS ANi) CONS'J'RAINJNG MOMENTS IN RECTANGULAR FRAMES
12-12
1
6K + 1 - K + 2
•
•
STRUCTURAL DESIGN MANUAL 11. HORIZ. TRIANGULAR
W (2 V = -a
RUNNING LOA.D
6L
+
ne • 2e 2)
VI. XX 7 HF = 2h + (2K +3)h
X7
=
WHERE:
f3(4d 5+b5) - lSh(3d4+b4) +
;
120h (d-b) 20h 2 (2d 3+b 3) - lSbd 2(2h-d)2 J
h
FOR SPECIAL CASE: b=c.o. a-dch: wh . wh2 HA H V =2 F 6L :I
V
V
wh IIF == 12
12. HORIZ. TRIANGULAR RUNNINC LOAD
•
n
..
]
[
7K 1 + lO(2K + 3)
w "
!""
HA·
6L (2a -I- c) (~ .. (..) w(a .. c)
- 2
- . HF
H
F
• VL
2h
+
KX 10
h(2K + 3)
WHERE: w
X10
I:
~
120h (a-c)
FOR SPECIAL CASE: wh2 V = 3L H = ~ .. H A·2 F
) HF
•
-
wh [4K + 5 ] 2K + 3
= 10
2 2 [-30h 2e (a _c 2 ) + 20h (a 3 -e 3 ) .~
-I
+ lSc(a 4 -c 4) - 12(a 5-c 5) . b=cco, a=d==h:
n
TABLE 12.3 (CONT'D) - REACTIONS AND CONSTRAINING MOMENTS IN RECTANGULAR FRAMES
12-13
STRUCTURAL DESIGN MANUAL 13. HORIZ. TRIANGULAR RUNNING LOAD
ac - 2c 2)._ HA _ MF 6L L L
v = w(a 2 + H A
=
•
w(a - c) - H
2
F
= w(a 2 + ac - 2c 2) _ X8 + X9 (K-l) HF
12h
2h
2h(K+2)
WHERE:
Xs =
X9
v
=
·v
w
[lS(htb)(d4 -b4) - 12(d 5-b S)
W
[lOd2h2(2d~3~)
60h 2 (d-b) . ] .. 20bh(d 3 -b 3 ) 2
60h (d-b)
+ lObh(4d 3+
b2h-bJ ) - d4 (JOh+15b) + 12d S + 3bS ]
["'(4 + 4~ ~~~= ~sJ 2
MA _ (3K
+
1)
-
2(6K
X9
+ '2
1)
[1 3K J K + 2 + 6K + 1 + X8
(JK + 1) MF
+
["'(4 2 + 4~
=
2(6K
X8 [
1 .
'2 K+2 -
6K
+
- 2C
2
•
)J_ Xs
1)
3K ] + 1
FOR S PEe!AL CASE:
wh 2K
v = 4L(6K +
1)
H = wh(3K + 4) F 40(K + 2)
MA
wh
= 60
2
[
271< + 1
2(6K
+
1)
+ 3K + 7 -J K
+
2
TABLE 12.3 (CONT'D) - REACTIONS AND CONSTRAINING MOMENTS IN RECTANGULAR FRAMES I
12-14
•
/;7f(~"
"'~~~~ STRUCTURAL DESIGN MANUAL l~.
HORIZ. TRIANGULAR RUNNING LOAD
v = w (2a + c) (a - c) 6L
H
HA
=
w(a - c) 2
H F
=
w(2a 2-ac-c 2) .' Xll + XI2(K - 1) I2h 2h 2h(K + 2)
~
F
where: XII
v
•
=
2W .sOh (d .. b)
.
+ x12 I-
2
K
1
+
+
2
3K
61{ + 1
" 3K+l JI" w(2a 2 6-ac-c 2 )
_t MF -
v H
]
X + 11 - X
11
2(6K + 1) 1
K
+
3K 2
6K
FOR SPECIAL CASE:
+
J
X22 - --2--
] 1
h=c=o, a=d=h
3Kwh 2 = 4L(6K
=
+
1)
wh(7K + 11)
F
40(K
wh
MF = 40
•
J
v
r )
5hd 4 -3d 5 -20hdb3 -12b 4 (d+h)
,[
2
+
MA
2)
l 21K + 6 6K
+
1
- K
!
wh
120
I:
:2
2
[87K+22 6K+1.
3
+ K+2
J
]
TABLE 12.3 (CONT'D) - REACTIONS AND CONSTRAINING MOMENTS IN RECTANGULAR FRAMES
12-15
STRUCTURAL DESIGN MANUAL 15. HOHENT ON lIORIZ. SPAN
v=!1 L
~
:=
•
3(b '- L/2)M Lh(2K + 3)
FOR SPECIAL CASE:
rr-M-
M
V
:= .-
H
:=
L 3M
2h(2K
+
3)
H
v
V
16. MOMENT ON HORIZ. SPAN
+ L2K~M
v = 6(ab
L 3 (6K + 1) H
:=
3(b - a)M 2Lh(K
+
2)
M = M '[6ab(K+2) - L [a(7K+3) - b{SK-l) 2 A 2L (K+2) (6K+l)
1 :1 _
•
ME ::: VL - M - MA
FOR SPECIAL CASE:
a=o, b=L
6KM
+ 6K)
v
:=
L(l
H
:=
2h(K + 2)
crM
3M
)
(5t< - I)M
MA • 2(K + 2)(6K + 1)
TABLE 1 L.'~ (CONT t D) - REACT IONS AND CONSTRAIN [Nt; MOMENTS
IN RECTANGULAR FRAMES
12-16
•
STRUCTURAL DESIGN MANUAL 17. MOt-mNl'
Oi~
S IDE SPAN
It
M
V=-L
n
III
3 [K ( 2a b+ a 2 ) 4- h.2] M 2h 3 (2K
FOR SPECIAL CASE:
0=0,
+ 3)
bah
M
V=L
3M + 3)
H • 2h('K
. MOMENT ON SIDE SPAN
•
+ 1)
FOR SPECIAL CASE:
..-..--II
E
ME*
H = 3bM [2a(K+l) + b] 2h 3 (K + 2)
6bKM
v = hL(6K
V ::
6KM L(6K + 1)
H ::
3M 2h(K + 2)
a=o; b=h
(?U
V
M = M!5K - I} A 2(K + 2)(6K +
1)
TABLE 12.3 (CONTtD) - REACTIONS AND CONSTRAINING MOMENTS IN RECTANGULAR FRAMES
12-17
vA = Q
- VD
n = Q.£
[:2 +
h
L
d(a+c)
2a 2 (K + 1)
J
2. VERT. CONCENTRATED LOAD
v,0
=
.
H
&:
Q£ L
[1 - d(a+d) ]
2a 2
£~b +
-----b
2Qed 6La h(K+l)
{[-' b(3K+4) ... 2L
JI~il+d~ L 11
+ 2 (2I..+b) (a+c) + 3ac
v
='
•
.Q£ L
..........- b •
D
H D
= R£ h
[.£ + L
d (h+c) 2h 2 (K + 1)
'J'
HAV}- L~ '-.
TABLE If.4 - REACTIONS AND CONSTRAINING MOMENTS IN TRIANGULAR FRAMES 12-18
•
STRUCTURAL DESIGN MANUAL
•
CONCI~N'rRATED
4. HORIZ.
v
LOAD"
I.
=.Q£ [1 _ d (h+d) L 2h2 .'
HA = Q - HI)
H : ; Q£ {b + __d__ D
Lh
r(h+d)( -b[3K+4] - 2L)
6h2(K+l) + 2{2L+b)(h7) + i:;:;Su/-rlF"
6h 2Qed (K + 1)
rl (h+d) (3K+4)
Qed
5. VERTICAL UNIFORM RUNNING LOAD
VA
= 'va
Vc
Ha 2 =2L
wa 2
6. VERTICAL UNIFORM RUNNING LOAD
[i
VA ::; wa
r 2
J
- 2{h+e) '.1
[1-;L J
wa (3K
L
C
(h + 2e + d)
H ::; 8h
24(K
..........
3aC])
b
1
+ 1+ K
3a 1 - 8L
]
"1
+ 2)
+
1)
H
C
M
.~
~
..
1"1
e
-
2 1i(K
+ 1)
TABLE 12.4 (CONT'n) - REACTIONS AND CONSTRAINING MOMENTS IN TRIANGULAR FRAMES 12-19
STRUCTURAL DESIGN MANUAL Revision E
H ..-.-
•C
::i.lle v
9. APPLIED MOMENT
:::: wh2(3K + 2) 24(K + 1)
Me = 24(K
+
M L
AT: APEX
Mr a K- +bK] 1
= hI..
I
V
2
L---~
v
3M 2L
H
3M{a - bK2 2hL(K + 1)
MA
2(K + 1)
Me
2 (K + 1)
KM M
TABLE 12.4 (CONT'D) - REACTIONS AND CONSTRAINING MOMENTS IN TRIANGULAR FRAMES 12-20
1)
STRUCTllRAL DESIGN MANUAL
.'
Revision E 1. Sinusoidal Normal Pressure
b(lb/in. )
CrrR
V
=4
H
= -4
Me
=
CR
CR
2 [ (n -28) COs
4
e -
1\'
+ 3 sin
9J
H ........ ....... .;
b
=C
sin
(Positive moment acts clockwise on section ahead.)
e
• Sinusoidal Normal Pressure
.3197 /... CR
b "C sin
~
.......H
Me
= CR 2
[ . 81 974 sin 8
.84018 +
£~se (~-
(Positive moment acts clockwise section ahead.)
O)J
on
3. Uniform Normal Pressure
M
=
0 at all points since pin poinl&
permit a uniform hoop cension. T, where: T
=V
= bR
H = 0
H
TABLE 12.5 - REACTIONS AND CONSTRAINING MOMENTS IN SEMICIRCULAR FRAMES OR ARCHES 12-21
~
~'.-\\
~~! STRUCTURAL DESIGN MA'NUAL 12.4
Analysis of Rings
Tables and figures are presented for the analysis of rings and ring-supported shells. Sections 12.4.1 and 12.4.2 show analysis methods for rings which are rigid w~th respect to the resisting structure for out-of-plane loads. The plane of the ring remains plane and the supporting structure deforms.
Only bending is considered in the deflection curves for the in-plane load cases given in Figures 12.4 through 12.29. Refer to Figures 12.30 through 12.33 to include the effects of shear and normal forces. Section 12.4.3 shows methods of analysis for circular cylindrical shells supported by "flexible" rings. 12.4.1
Analysis of Rigid Rings with In-Plane Loading
Coefficients to obtain slope, deflection, bending moment, shear, and axial force along with equations for these values are given for some of the frequently-used load cases. Figure 12.4 shows an index for the various load cases presented in Figures 12.5 through 12.29. The sign convention used throughout the rigid frame analysis in-plane load cases is shown in Figure 12.3. It basically consists of: moments which produce tension
FIGURE 12.3 - SIGN CONVENTION FOR RIGID RINGS WITH IN-PLANE LOADS on the inner fibers are positive, transverse forces which act upward to the left of the cut are positive and axial forces which produce tension in the frame are positive. Deflections in Figures 12.5 through 12.29 are based on bending only. Deflection curves for the three basic load cases due to shear and concentrated loads are shown in Figures 12.31 through 12.33. A shape factor (~) that is to be used with the curves for shear deflection of various cross sections is shown in Figure 12.30 •
12-22
•
STRuc-rURAl DESIGN MANUAL Figure 12.6 p
Figure 12.10 p
p
Figure 12.11 Figure 12.12
Figure 12.13
i@@PG p ~
p
p~
~p
Figure 12. 5
•
Figure 12.20
P
Figure 12.25
..)
Figure 12.21 Figure 12.22
Figure 12.23
Figure 12.24 P~=Pmaxcos (21'6
P¢ -Pmaxcos_
Figure 12.26
Figure 12.27
Figure 12 .. 28
Figure 12.29
Pit=- plIlSXC 0 S (1pJ)
!PSlf-Pma.x (a· +b e08~
,+c cos ¢)' p
p
•
P
p
p
max p =p
=P018X cOH¢
¢
max
P
cos¢,rnClx ' .
FIGURE 12.4 - INDEX OF IN-PLANE LOADS CASES FOR RIGID RINGS 12-23
STRUCTURAL DESIGN MANUAL 1
I
04
.oa
.03
.....
\6
\
j
1---
-
,I
0
·.03
-os
-.04 -:08
\~ -
,
~\
-,
-~
...
~~
r - -.-
.\-
'\
I
.\ - -..
'--'
\V
- -.1 I
--- .~~lL
L
/
f--
'" ,
..
f
:--.
1- -- I - -
.'
f
r~
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STRUCTURAL DESIGN MANUAL
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12-38
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12-39
STRUCTURAL DESIGN MANUAL ~"K pJl3 'll EI
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12-40
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12-41
STRUCTURAL DESIGN MANUAL
• M:K M PMAXR2 Q=K Q PMAXR NaKN PMAXR
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12-42
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STRUCTURAL DESIGN MANUAL
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12':""'43
STRUCTURAL DESIGN MANUAL pap MAX COS (2.)
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. 12-44
STRUCTURAL DESIGN MANUAL
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12-45
STRUCTURAL DESIGN MANUAL
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12-46
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12-47
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12-48
•
STRUCTURAL DESIGN MANUAL Cross -Section
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FIGURE 12.30 - SHAPE FACTORS FOR SHEAR DEFLECTIONS FOR VARIOUS CROSS SECTIONS
• 12-49
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e
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•
STRUCTURAL DESIGN MANUAL 12.4.2
Analysis of Rigid Rings with Out-or-Plnne Loads
Coefficients to obtain slope, deflection, bending moment, shear and axial force along with equations for these values are given for some of the frequently used load cases. Figure 12.35 shows an index for the various load cases presented in Figures 12.36 through 12.38. The sign convention used throughout the rigid frame analysis of out-of-plane load cases is shown in Figure 12.34. It basically consists of moments which produce ~-------------------~------------------~
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RIGID RINGS WITH OUT-OF-PLANE LOADS
tension on the inner fibers are posi tive, torque "Til and la teral shear "V· 1 are positive as shown. 12.4.3
Analysis of Frame Reinforced Cylindrical Shells
Tables are presented giving the loads and displacements in a flexible frame supported by a circular cylindrical shell and subjected to concentrated radial, tangential, and moment loads. Additional tables give the loads in the shell. The solutions are presented in terms of two basic parameters, one of which is of seoond-order importance. Procedure for modifying the important parameter to account for certain non-uniform properties of the structure are presented. Notation A
B
E
•
2.25 y4
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,-.J
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lb/in
lb/in
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2
12-53
STRUCTURAL DESIGN MANUAL
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Figure 12.36
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o Figure 12.38
Figure 12.35 - Index for Rings with Out-of-Plane Loads
.
12-54
•
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.... l
6:" ¢ "_
?r,~
~;iT\~tJr~·~-
.
ZoO.
i
P,',
A
"crf\,:--nw
P,ARCMp
- 0
~-;;;,~ .. ;= ~.;:
CTp • IS
.'.J
.04i
.1
.'l.
.04-
• OS
•I
.01.
'
en ::a Ie:
1-1 c:-)
" - ,0-51-.1
-. 1
.-1
0
1-.11-1 I Ir-T-t-t--t-t-+---t-~~-~~---l-.-1.. I ='"";= I""""" '" 1'1:
c:
::a
.0'2.
1:.:-
-.04-
I~
,..... .."
,06
'en
,as
Ie')
Z
ZIJ
40
60
80
tOO
llO
140
1 SO
180
2,ZC
260
t, de,
Figure 12.36 - Coefficients for Concentrated Out-of-Plane Load
zso
)00
3ZC
:I:
:.::z
,. c: r-
I-' t\.)
I U'1 lJ1
...... N
C~T
•
U1 0\
S'I....--!----:'-t--+---t-----r---, M t
'"
;E13-
TA CMT
T."
where CM T
L-~~~~~~-4--r-'--,
S
'9
.;,.,
-,~ (!~~l
-+---.;.
!J---...f
aI '/
'~~\,
./'
K:o
lEI ~ OJ .E:I ~ COJ
r----;..-"'--~-t
0 For any cross "ection
!111- '.--' _}'. -\!!!.~
~
;-
CI)
..... :=tJ
2-
c:
c-,
.....
D .....IIIIIi::=--+
c:
,..
:::c:J
r-
~I
c:J
~ITti~~~±H~~~~xqp~ i
I
• -I ;co
I I ~;_LJ
i ;r Ii
.-. ILJ
40
~o
so
tOO
i
12.0
I
140
I._I.
11>0
ISO ~.
i· ~OO
I: I I
I I Cl CIT) I
a~o
~40
~60
I I· I zao
I 300
i
3.!O
e
CD
:z
,..i:z c: r-
Figure 12.37 - Coefficients for Concentrated Out-of-Plane Torsion
e
en "" -
,..
}-lO
deg
e
,
-",
e
e
\-I
C
e ~
TT
.5' , T . " oTACT where:Cr ... 1--+---+-----+-- + -- .
I I
. .
A·
1 ;
I
#,
z
Zecooe
I
JEI GJ £l .. CJ
K '"
0
,
I
0
+
4, ,
I
1
. KsinO.Z lln• o
j
,~ ~
Iii'''''"......... ii''''''' '"
1 !,
I .'
R
;
I'
j
I
i\"~ ~
•
1
1
I" i"
I
I
I
,
!
I i !
'/
I
I
'\
'rJ· :
I
.:'\..
..... - ......
t
r!
j
!.
-."Z.
I i
:"
~,," ~
I
I
1
!
-,.s 20
40
60
80
100
lZO
140
I I
I
i'
I
I
i
I
I
I
I
I
i 100
180
ZOO
ZZO
Z40
ZBO
t-.J
I U1
--..J
c-)
--I f:
:::a
:r,.... ~
I: I
a)
-z:
I
300
I
~
I
' !
i 3'"
•• (leg
.....,
:=a
c::
I !
i Z60
....,•
en
Iii I
en
CJ
11
Ii
~
,---'
,
t
I!!! i I !
!
i
I ~~~~~--~~~~~~~-i_ 41
1
j'
I i i !
i'-.. ~\ .j ~~! il~~
.:'
-.4
~\
i'-. '\ '\~
I
-.3
'"
i I i
I;
I i i '" ; , \. , i '""'\'\
j
·
N:
'I '"
I
I
-----.....
~=~'
T ' '! '\ \. i , ! ! ! ! ~ I I I i i i i ! \ ~ ~,,,! i - I I : .t. ~! !.I J=3:0 i ! I! \ ,~ ~ !"l ! I ! 1~:k[Nl i i 11 il\J'\"~l' j i i ! .1 I/~ T.~ I ~ I ! :: I I i i\ '" 1-... ~: i ; ~o ~! ' \ . : i ! ! \ ! I Ii I ~ r~ ~ i ;', ~I----....~O k. o , ,~J( ... I -,......,. ~! '\ I 1"-1 I, I . ~i_ ~o I ' i r i! iii I ~I l"[ r ~1(::--="":""-'."":::=:=tJ~_""' ~~......J I ! ! ! , ~ "l ~ \J! i Ii '~I !,~ ! K~~U)~/ i ......... ~7 -.1 • ~!'i ., ~ ~.~ i I I ! " I I Ii' i I ....... """ f • , I I '- ~" I,,\",, I ~ 1<'"II#!5. ~v!
~/"
\~CI-
.
For any <';rolS sen:-'C.
I !!
-~ ~
,'---~.
•
I
'\.\N"J
I
~~J;-t -H-~ -
!
I
!!
I
!:
T ~I
Figure 12.37 (Cont'd) - Coefficients for Concentrated Out-of-Plane Torsion
HO
:r:z c: :rr-
I-' N
E1)TA
I lJ1 .:..0
CVr
!
.i."
.5
w'"'' .+
I
I
CV T '
....!..:."
0
--';-
I I
I :
?ll~\, iE;J\ ~
,..
~
,
.If - [
... ,
'~~r- ~.;
o
I
Fill" .i.l'.y
erO$.
~=~
$ .. ct;o~
en
.~
.~
r---~--~~-4--+--+--~+--+~L-~-+~~------~-;--~-~-~~~-+--~-r-4--.-~~~~-+~~~-4--+--
i7!; ('
~
t
:
I iii ilili
.V
~
:!: ! , \ ' 1i 1 I . ! ' ",1; i ...
-.1
I
L
I
I
I I:
,i'
.
"
i
t 'i!
j'
r~" J:!; I1I T I i: !i , ~,I ! I !! I I ~I . 1
!
I
I
i
I
.
I
1--
II
I
_---;,
I
I --i : I i i. i i
i!
. ,I " i i i!! • ,
l
TI
"~J
2".0
~
!
I'
I" I 'I ,
40
:
I. I' I1 I i i, i
I
I
I
1
.....L !
I
; i
so
100
I
1
t
!
1
__
i i i
120
J_ •
i 1010
,I
I
: ~
: i i 'I , ill
" I '
I
L I
i
:ec:
1:'0
Q.
;. i i i I
I
I
"
i
,
zoo
,
!
! 'Ii I
:'' 1
Ij
I1
c::
T
j
f7
1: .....iI '1
~
•
i: I Ii: ! I I / i'J I. II I I Ii II / " A VI V I :"..",1 "l ; " ', I : I I : I I ........ r"""-i I; T"
i kl
1
!
I
I I.1__ ! I!
i 111111!
60
!
.'I '1\:.: i,.
! ; . :I
c:-, -I
T
I"
j
: . , . , .
11 I
c:
1
.
!
j
iTi
I 1-+ 1- i i ; I'! ! I T : I' I! i !I i ll . ! ; :. T' ii
_.,.l 3
! : TT
I'
1'!: I I
i
2~O
I
I
I
I
1
2.40
I
,
iI
1
zt.o
280
~
laO
:>Z:J
'.....,.'
.,
.~ ",
en
-
z 31:
::Da Z
340
c:
des:
e
:::ID
::Da
CD I
i
i
1
7
::Da .-
Figure 12.37 (Cont'd) - Coefficients for Concentrated Out-of-Plane Torsion
e
.....
::a
"--""
e
e
e
e
'''"--''
~.
C MM, C TMt C"M
~ \,'-...... ../
r~/J I
.S
~\ ~
}ro-
..
MACTM
~=::::~
l.:
Z
.<4-
z
CJ') .3
-I
:::a
c::
.~
c-)
--I
.l
c: :=0
:.:.....
-.
I:J -.2.
",
.,
C i)
-z:
CI)
-.
s:
-.+
:.:-
z
-.S J.I)
40
60
so
100
12.CI
14Q
60
l30
lOa
,,0
HO
U('t
ZaC
•• d'!:g
..... rv I
Ln \.0
---.
:--.
M):t M .... C MM
Figure 12.38 - Coeff
ients for Concentrated Out-of-Plane Moment
~O!:r
no
.4 ~
c: :.:r-
(
:
STRUCTURAL DESIGN MANUAL ~
Esk
Young's modulus of skin
Ib/in
e
base of natural logarithms
F a x i a l force in loaded frame
~
•
2
Ib
~ Ib/in 2
G
shear modulus
I
moment of inertia of a typical unloaded frame'" in4 moment of inertia of an unloaded frame, distance II{II from the loaded frame ~ in4
I
o
~ in4
moment of inertia of the loaded frame
i n
K n
n
- 1 2
')
2
- '1
1 + 2
JI
+
n
2
- 1 "}
G:)
2
G:f
distance from loaded frame to undistorted shell section
L
L
c
r
characteristic length (see Glossary) =
frame
M
bending moment in loaded frame
M o
externally applied concentrated moment
p
externally applied radial load - Ib
p
in
•
-in
characteristic length (see Glossary)
~o
o
[t~r2] 1/4
J:
~
spacing~
in ~
in-lb
ax ial load per inch in the shell -
~
in-Ib
1 b/ in
shear flow in shell" lh/ln r
radius of skin line
s
transverse shear force in loaded frame
s
transverse shear per inch in shell
12-60
""'-J
in
~
~
lb
Ib/in
•
STRUCTURAL DESIGN MANUAL
•
'1'
o
(-'xlcrnaJ ly applied tangential lond'""-' Ib
l.
skin panel lhickness'""-' in
t'
effectivc skin panel thickness for axial loads
l
weighted average of all the bending material (skin and stiffeners) adjacent to the loaded frame, assumcd uniformly distributed around the perimeter ~ in.
e
u
axial displacement of shell
v
tangential displacement of shell
w
radial displacement of shell
x
axial coordinate of shell
y
"beef up" parameter I /2i L o
~
~ln
in.
~
~
in.
in.
in. c
Y for nearby heavy frame
•
rotational displacement
~
radians
polar coordinate of frame and shell In the Illethod of atta'ck with which this section is mainly concerned, a simplified structure, as shown in Figure 12.39, is used to obtain a solution for a uniform
/
/
)
I
I l
q
JOt:
•
. p
FIGURE 12.39 - SHELL WITH FLEXIBLE EXTERNALLY-LOADED FRAME 12-61
STRUCTURAL DESIGN MANUAL shell stretching to infinity on both sides of the loaded frame. Clearly the effects any frame can be propagated only a finite distance along the shell. In practice, the perturbations from the "elementary beam theory" are~ at worst, negligible ell some characteristic length "L t1 inches away from the loaded frame. Procedures for modifying the solution to acgount for discontinuities and non-uniform properties are discussed in the following sections. For the structure used, the following assumptions are made: . 0[
(1)
Concentrated loads are applied to the loaded frame and are reacted an infinite distance away on either one or both sides. The shell extends to infinity on both sides.
(2)
The loaded frame has in-plane bending flexibility. It is free to warp out of its plane and to twist. It has no axial or shearing flexibilities. Its moment of inertia for circumferential bending is constant.
(J)
The effects of the eccentricity of the skin attachment with respect to the'frame neutral axis is ignored for both the loaded and unloaded frames.
(4)
The shell consists of skin, longerons, and frames similar to the loaded frame, but possibly with different moments of inertia. The skin and longerons have no bending stiffness. All properties of the shell are uniform.
(5)
The longerons are tlsmeared out" over the circumference glvlng an equivalent constant thickness, tf t (including effective skin), for axial loads.
(6)
The shell frames, but not the loaded frame, are Hsmeared out" in the direction of the shell axis, glvlng an equivalent moment of inertia per inch, "i", for circumferential bending loads.
•
•
Characteristic Length - In this section there are two characteristic lengths, defined as follows: Lc is the distance required for the exponential envelope of the lowest order self-equi!ibrating stress system to decay to lIe (e base of natural logarithms) of its value at x = 0, provided that the skin panels are rigid in shear. Lr is the distance required for the envelope of the lowest order self-equilibrating stress system to decay to lie of its value at x = 0, provided that the frames are rigid in bending. IV
Evaluation of Parameters L , L , and Y r a Cao€ of uniform shell !"n cases where the shell happens to satisfy all the assumptions listed and, i.n pnrtlcular" i r the ski n thi(~kness, stringer area, and she 1.] - frame momen l () r inertia lll'(~ unl form In both th(;~ axial and cir.cumferential directIons, the following rt')rmuia:s I!t:ly be used:
12-62
•
STRUCTURAL DESIGN MANUAL L
c
L
r
Jb
2
::
1/4 •
•
•
•
•
.,
•
•
II
•
•
•
•
•
•
•
•
12. ,
•
12.2
.... . ... . .. . . ..... .
0
2iL
•
J\~'
r :::::--
I
Y
~t/J
r --
c
12.1
Young's modulus for skin, stiffeners ana all frames is assumed equal. Coefficients are obtained by use of these parameters (L c , L r , y) in the tables. These coefficients yield the required loads and deformations when substituted into Eqs. 12.14 through 12.21. In non-uniform shells, use the modified parameters indicated in the following equations: Case of non-uniform shell (a)
In the case that the shell properties, i, t, and tt, vary over the surface of the shell to a moderate degree, the following formulas and definitions are appropriate: L
L
c
= -r-
V
2 E
y=
0
2E
[
Ek :G
r
r
Esk t e r2] 1/4
I
-
. .. . . . . . . . . . . . . . . .. .
12.4
-
t' •
t
•
•
I
•
•
•
•
•
•
•
..
•
..
•
•
•
•
12.5
0
i L f c
The stiffness factors, Gt, Esk, t e , and Efi, must be averaged in the neighborhood of the loaded frame. The factors Gt and Eski shall be averaged over a length of shell extending approximately one-half of a characteristic length from the loaded frame in both directions.
)
(b)
When unloaded frames have unequal moment of inertia or are unequally spaced, the following weighting factor is used for computing Efi: Efi
= (Efi)fwd + 1
(Efi) fwd = L
Cfwd
(Efi)aft
• • • • • • • • • • • " • • • • • • • • • • " • • • •
12.8
(WE f If)' " • . • • • • • • . • · • • • • •
12.9
I {WEfl f }·
2.
12.7
• 12-63
STRUCTURAL DESIGN MANUAL Where W= 1 -
x
~
for x < Lc
c
=0
for x
> Lc
(x is measured forward and aft of loaded frame) The sun~ations in Eqs. 12.8 and 12.9 are to be extended over all frames except the loaded frame. The method of calculation gives greater importance to frames closest to the loaded frame and less importance to those farther away. For the case of a single, particularly heavy, neighboring frame, or for other neighboring discontinuities such as rigid bulkheads, a free end, or a plane of symmetry, the correction factors to be discussed are applicable. If those corrections are applied, the heavy frame or .other discontinuity must be ignored in applying Eqs. 12.7, 12.8, and 12.9 .. In particular, if the loaded frame is near the end of the shell, the shell must be continued beyond the end, fictitiously, in the summations of Eqs. 12.7, 12.8 and 12.9, as though the shell were symmetric about the loaded frame and extended for a length greater than L on both sides of the loaded frame. c
The method of calculation indicated in this subsection exaggerates the effect of frames which are heavier than average when compared with the more ai~Hrate method of correction given in the next section. Since Lc depends on (Efi) ,an i~itial estimate of Efl is required in order to calculate the Lc used in Eqs. 12.7, 12.8 and 12.9. Corrections to Y, the "Beef-Up parameter The general form of the modified "beef-up" parameter, y'i'c, is: y
*=
Y. f • f b . f , etc. .. • ~ • • • • • • .. . . a c
12.10
where Y is computed by the methods of the preceding section, and fa' f , and fc are b factors accounting for effects of nearby heavy frames etc. Modification for different value of L
r
IL c
The value of Lr/Lc used in the graphs are 0.2, 0.4, and 1.0. To account for values of this parameter between 0.2 and 1.0, graphical interpolation should be used. Otherwise,
1'*
= 'Y
2
l2.11
where (Lr/Lc)H is the value of the parameter for the shell, and (Lr/Lc)~'c is the value of the parameter closest to (Lr/Le)U, for which graphs are available.
12-64
/r" \ ...
,/,,;'
1(1 ".' ".
If /
\11"\
STRUCTURAL DESIGN MANUAL '~."".~"H ~ ~~"'\\
B'ell
\
Mocliri.cntion for finite rrmn(\ spacing .. The modi rication for finite frame spacing is as follows: ~o
+2LK
12.12
c 2
where ~
o
distance from loaded frame to adjacent frames
Modification for nearby heavy frames and for other similar nearby discontinuities. The corrections to "y" in a previous section are not intended to account for discontinuities in circumferential bending stiffness. The form of the correction for these effects is: 12.13
)
Fig. 12.40 shows f(2) plotted for nearby heavy frames and for nearby rigid bulkheads. Fig. 12.4l shows f plotted for a finite length of shell terminated in various ways on one s~~~ of the loaded frame. The validity of the correction is considered doubtful for f (2 ) < 0.25, due to the importance of higher order stress systems. Figures 12.40 and 12.41 are for LrlLc = 0.4, but their vari~tion with LrlLe is negligible for conventional shell-frame structures and adequate in other applications for LrlLc < 0.75. The corrections for nearby planes of symmetry and antisymmetry can be used to solve problems where two similar frames are simultaneously loaded. To illustrate the method the two following examples are given: Example 1 A frame of moment of inertia 4.0 in4 that is subjected to concentrated loads is supported in a uniform shell whose characteristic length, L , is 200 inches and moment of inertia per uni~ length, it is 0.10 in. 3 • A heavy frame having a moment of inertia 16.0 in is 50 inches to one side of this frame. The loaded frame and shell loads are required.
12-65
STRUCTURAL DESIGN MANUAL
'1-=
0.1
1, = 0.5
1J,
= 1.0
r,= r.e =
2.0 ~
Two Rigid Bulkheads Symmetrically Placed about the Loaded Frame
0.4 0.3 0.2
0.1 O&---~--~----~--~--~----~--~--~
o
0.2
0.4
0.6
0.8
1.0
1.2
1.4
____
1.6
L-~
1.8
2.0
.e I.e
Figure 12.40 - A Single Frame on One Side of Loaded Frame or Two Rigid Bulkheads Symmetrically Placed about the Loaded Frame Curves of £(2) and £(3). Lr/Lc = 0.4
. 12-66
STRuc-rURAl DESIGN MANUAL
•
4.0~--~--~--~--~---r---r--~--~---.---'--~r---r-~
f(2) To
..
00
Loaded Frame
2.5
Free End at x
== J,
1.5
1.0
Plane of Symmetry At x == l
)
Plane of Anti-Symmetry at x .: J
0.5
0.2
0.4
0.6
0.8
1.0
1.2
Figure 12.41 - Finite Length of Shell on One Side of Loaded Frame f(2) vs tiLe For Various Boundary Conditions at x = t, Lr/Le = 0.4 12-67
,~)1\.. ".-
1/ / I \ _\ \\ "
STRUCTURAL DESIGN MANUAL
.B~II \\'~"''''''''
~ ,
~,,~
1/ 1/ ~(-
The parameters needed are:
=
y
2(
.t)~200)
= 0.10
16 2(.1) 200 ~
L
50 200
c
~ and
Using
fILe
=
0.40
0.25
in Fig. 12.40 yields f(2)
.•. Y* = 0.75 (0.10)
= 0.075
= 0.75
by Eq. 12.13
Use y= 0 .. 075 instead of 0.10 in the curves to account for the presence of the heavy frame on the stresses in and near the loaded frame. Example 2: A shell whose characteristic length. L c ' is 250 inches is supported by a largE! number of identical frames whose moments of inertia are 2.0 1n4, spaced 24 inches
apart. A pair of frames 96 inches apart are subjected to concentrated loads at the same polar angle,~. The two radial loads are of equal magnitude but oppositf! sign, while the tangential loads are of the same magnitude and sign. The loads in the loaded frames and shell are to be found. i
=~ = ;4 = .0833 Q o
10 Y = 2iL
P. L
c
=
48 250
2 c
= 2( .0833)(250)
0.048
by Eq. 12.3
0.192
For the tangential loads there is a plane of s~nmetry midway between the loaded frames, while for the radial loads a plane of anti-syrnmetry exists at the same place. From Fig. 12.41 it is seen that for the radial load stress system, [(2) 0.32, while for the tangential loading E(2) = 1.75. Hence, the values of y .:c to be used in the graphs are 0.015 and 0.084, respectively. Eccentricity between skin line and neutral axis of the loaded frame: In the three types of perturbation just discussed, it is possible to account for Lhe effects by modifyi.ng Yonly, si.nce the "elernentary-beam-theory" part of the solution is always valid. In the case when the eccentricity between skin line and neutral axis of the loaded frame exists, the uelementary-bearn-theory" solution is also affected. ,
12-68
•
/ -71.' '.,
/' j 1 \ '- '\
/i I
_ \ '\
, \.B~II \'
,
,
\
. . . . . . ft· . . . .
STRUCTURAL DESIGN MANUAL
j
"-::.. 'U:' ."
Eqs. 12.14 thru 12.21 arc given later in this section, by which the effects of a concentrated load or moment on a shell-supported frame may be computed by using the tabulated coefficients given in Table 12.6. The method of computing 'Y is indicated in a previous section. These enable the shear flow and axial load at all points in the shell and the internal loads and displacements of the loaded frame to be computed. The following parts of the overall solution are omitted in the tabulated coefficients: (1) The "elementary-bearn-theory" part of skin shear flow which is calculated from beam theory. (2) The ·'elementary-beam-theory'l part of the axial load intensity in langerons which should be calculated from beam theory. (1) The rigid translations and rotation of the loaded frame.
•
As a consequence of items (1) and (2), shear flow and axial load intensity in the shell, as calculated from the tables, can be added directly to the results of an "engineers bending theorY'calculation. The shear flow and axial load distributions given in the tables are assumed to be symmetrical with respect to the loaded frame. In a shell that is unsymmetric about the loaded frame, the shear flows and axial loads are not symmetric about the loaded frame. It is not possible to derive a simple correction for this effect, but the exact solutions indicated in reference 7 are applicable. Distrihuted loads on a frame: The effect of a distributed load on one frame may be obtained· by superimposing the effects of the concentrated loads into which the distributed load can be resolved. The axial load and shear flow in the shell can be obtained for loads on several frames by a similar superposition, since "p" and "q" are tabulated in Ref. B NASA TN D402 as a ·function of x/L • c
Frames adjacent to the loaded frame:
)
At the present time it is not possible by use of tables to compute the internal forces in frames adjacent to the loaded frame. It is, however, a simple matter to tabulate the frame-bending moment per inch, urn", and the other internal forces as a function of x/L. The bending moment in an adjacent frame~ due to a force applied at the loade~ frame, is then obtained by mUltiplying "m" at the frame station by I Ii (see Appendix D of reference 6). Effect of local reinforcement of the loaded frame: It is not practical to attempt to cover, by a set of tables or charts, the many possible reinforcing patterns that can be used to locally strengthen frames in the region of applied concentrated loads. A solution is presented in Appendix A of reference 8, together with a simple example, to illustrate the numerical procedure. A loaded frame, whose moment of inertia varies around the circumference in any manner can be treated as a frame of constant moment of inertia that is reinforced to produce the actual inertia variation.
12-69
STRUCTURAL DESIGN MANUAL Tables (TABLE 12.6) The loads and displacements of the loaded frame and loads in the shell are given in terms of the non-dimensional coefficients of the tables by the formulas below. The tables contained in this section are for M, S, F, p, and q at x = o. Coefficients for displacements v. w, and 'Yare tabulated in reference 8 along wi th coefficients for "q" and'"p" as a function of x/Lco
P
T
M
q
C ~+c ~+C ~ qp r qt r qm 2 r
p =
c
M
12.14
.!£ ( Lc ~ + Cpt r I
Mo (L. .c)
+ Cpm 2" r
pp r
CPr mp 0
+ Cmt
T r + e M . • • • • • • • .. . • .. .. .. • • • • .. mm
0
0
M
P sp o
S = C
F
e
rp
+
C T st 0
0
+ e tl
0
Yr + C EI vt
p
T
o
+
vp
P
+C
0
. 3
T
. . .. .. . .. .. · . .. .. . . .. . . .
C -2. sm r
3
v = C
12.15
J.
fro
M ...2. I'
3
Yr 0
EI
0
+
. 2 Yr .. C M vro 0 EI
. · .. .. . · .. . ..
12 .. 17 12.18
.
12.19
. . . . . . . . . . . . . . . . . . . ..
12.20
0
3
· . . . . . . .. · . .. .. . . .. ..
12.16
.. .. ..
' 2
W = Cwp P0 Yr' + Cwt T0 Yr + CWID M0 EYrI EI EI
0 0 0
yr2
(J = Cf) p
t Er + CfJ o
Yr2 t To EI
+ Co
0
Yr
m Mo
EI
· . . . . . ..
0
The sign convention for loads, moments and displacements are positive in the
loaded frame as shown in Figure 12.42.
12-70
12.21
•
STRUCTURAL DESIGN MANUAL
•
FRAME LOADS COEFFICIENT
= .200
Lr IL c
= .400
Lr IL c
:::
12.7
12 .. 11
12 .. 15
C
12.19
12 .. 23
12.27
C
12.31
12 .. 35
12.39
C
] 2.8
1.2.12
1.2.16
Cst
12.20
12.24
j
C
L2 .. 32
12.36
1.2.40
G fn
12.9
12.13
12.1:7
C
12 .. 21
12.25
12.29
C
12.33
12.37
12.41
C
12.10
12.14
12. 18
Shear
QP
Flow,
C qt
12.22
12.26
12.30
C qm
12.34
12.38
12.42
Moment, M
mt
mm SD
Shear,
s
8m
Axial
Load,
F
ft
fm
)
L IL r c
emp
Bending
•
INDEX OF TABLES (FIGURE NO.)
q
At Ring
1.000
2.28
TABLE l2.6 - INDEX OF TABLES FOR CALCULATION OF FRAME LOADS
• 12-71
tf/J(\\\
"~I . -:~//
STRUCTURAL DESIGN MANUAL
)
Neutral axis
M
•
S
)
Figure 12.42 - Sign Convention for Tables 12.7 through 12.42
, 12-72
•
STRUCTURAL DESIGN MANUAL
•
..
,
,
"
On
--
.- -
Cruo .02
'7
¢
.. -"
4''''
.03
.05
.O.s36
.0646 .0296
Lr/Lc • .200 .10
.20
.30
X• 0 .50
1 00
3.00
0
0
5
.0463 .0141
10
.00011
15
- .0038 - .0047 -.0041 -.0033 -.0029 ... 0021 -.0024 -.0022 -.0022 -.0021 -.0019 -.0018
20 25 30 35
40 45
SO 55 6.0 65
70 75 ,80
85 ~O
100 110 120
130 140
150 160
170 180
- .0017
.0201 .0036 -.0032 -.0054
.009(,
.0219
... 0001
.0064 -.0029 -.0081 ... 0108 -.0120
-.0055
- .0056
-.oon
·.0050 - .0045 ~ .0041 ... 0037
-.0075 -.0072 -.0069 -.0064 -.0059 -.0056 -.0053 - .0049 -.0045 -.0042 - .0037 -.0033 -.0029 -.0020 '- .0011 -.0001 .0007 .0014 .0020 .0025 .0027 .0028
.. ,on:v.
- .0033 .0031 a
-.0029
... 0027 - .0025
- .001S - .00l} ,-.0012 -.0008 -.0005 -.0000 .0003 .0006 .0008 .0009 .0011
-.0006 -.0001 .0004 .0009 .0012 .0015 .0017
.OOII
.0017
-.0022
- .0020 - . 0017 -.0012
.0833 .0465
-.0123 -.0122
-.0111 -.0112 -.0106 -.0098 -.0091 -.0083 -.0075 -.0066 -.0058 - .0039 - .0021 - .0004 .0013 .0027
.0039 .. 0048 . .0053
.0055
.1066 .0682 .0396 .0189 .0043 -.0057 -.0122 -.0164 -.0189 -.0202 -.0206 -.0205 -.0198 -.0189 -.0177 -.0163 -.0148 -.0132 -.0115 -.0080 -.0044 -.0010 .0022 .0051 .0074 .0091 .0102
.010S
.1221 .0829 .0522 .0286 .0109 -.0022 - .0116 ... 0182 -.0226 -.0254 -.0269 -.0274 ... 0272 ... 0263 -.0250 ... 0234 -.0214 -.0193 -.0169 -.0119 .... 0068 -.0018 .0029 .0070 .0104 .0130 .0145 .0150
.1430 .1030 .0698 .0429
.1113
.2072
.1303
.1652
.09'46 .0638 .0376 .0156 ... 0028 ... 0176 -.029'; -.0367 -.045; -.0503 -.0531 ... 0543 ... 0540 -.052S -.0499 - .0464 -.0421 -.0317 -.0200 -.0078 .0040 .0147 .0237 .0305 .0347 .0361
.0214
.0043 ... 0089
-.0190 -.0204 •• 0317 ...0351 - .0371
-.0379 -.0376 -.0365 -.0347 -.0323 -.0294 -.0262 -.0190 -.0113 -.0036 .0036 .0101 .0154 .0194 .0219 .O2l]
.12(,7
.0918 .0605 .0327 .0083 -.0128 -.0301 -.0454 -.0572 -.0662 .... 0726 -.0766 -.0783 - .0779 -.0757 ... 0718
-.0665
-.0523 -.0349 - .0158 .0033 .0211 .0362 .0478 .0550 .0575
TABLE 12.7 L..r./J..~·
(;30
'1
.02
.03
.05
.200
.10
.20
.30
X• 0 .50
1.00
3.00 -.5000 -.4610
¢o
a 5 . 10 15 20 1')
]0 3')
)
/.0 45 50 5~)
60 65
70 75
aD
8S 90 100 110 120 130 140 150 160 180
-.5000 -.2464 - .0863 ... 0223 .0000 .0098 .0080 .0026 .0025 .0032 .0007 .0002 .002J .0020 .0006
.0018 .0028 .0016 ,0014 .0025
.0021 .0017 .0024 .0009 .0018 .0001. 0
-.5000 -.2738 ... 1199
- .0449 ... 0111
-.SOOO
-.5000
-.SOOO
-.5000
-.3467
-.5000 -.3813
-.SOOO
-.3066
~.3989
-.4182
... 4393
-.1637 -.0804
".2230
-.2781
-.O)I.Z
-.0802 -.0I.Z6
-.1381 -.0927
... 3423 -.2762 -.2197
- ,4204
-.1988
-.3085 ".2342 .... 1744
-.3803
... 1374
- .1269
- .1719 - .1323
-.3256 - .275; -.2301 -.1894
-.3196 .... 3390 -.2991 -.2603
- .0994 - .0721
-.1530 - .1205
-.2227
-.0494 ·.0307 -.0152 -.0023 .0084 .0171 .0244 .0303 .0349 .0385 .0431
- .0914 -.0657 -.0':'2tJ -.0226 -.0049
.01)20
-.0087 .0014 .0038 .0052 .0056 .0042
.001/1
.00)7
.0028
.0044 .0043 .0038 .0045 ,0051 .0048 .00l.S .0054 .0052 .0048 . ~Oli 7 .00J6 .0033 .0019 0
.OOi,9
.0074 .OO4S
.0042 .0041
.0026 .0018 .0017 .0035
.0027 .0026 .0034 .0032
.0028 .0Q31
.OOl9 .0022 .0009 0
-.0205 -.0082 -.0005 .0039 .0056 .00(17 .0080 .0084 •0086 .0092 .0098 .0099
-.0602 -.0374 -.0208 - .009l -.0011 .0048
... 0901 ".0623 - .0405 -.0236 -.0111 - .0013
.0093
.0065 .0125 .017L
.OQ.~O
.0124 .0146 .0166 .0180 .0189 .0197 .0205 .020) .0192 .0174 .0148 .01l7 .0080
0
a
.OLOI .01('6 .0104 .0098 .0090 .0074 .0061
.0208 .0238 .0259 .0274 .0293 .0293 .0280 ,02.')4
.0211 .0172 .0116 0
.010(•
.0240 .0355 • Olf 51
.0432
.0530 .0&41 ,0694 .0696
.0396 .0341 .0270
.0570 .04S6
.0\87 0
.0J17 0
.0445
.06')2
-.1865 -.1519 - .1190 -. Of~RO .. ,0590
-.0320 -,OO7J .0153 .0355 .0533 .0681 .0922 .1061 .1107 .1069 .09.54
I
.0774
.0545 0
TABLE 12.8
12-73
STRUCTURAL DESIGN MANUAL e,o '1 ¢o 0
5 10
~
I
15 20 25 30 35 40 45 50 55
60 I
I r
65
70 75 80 85 90 100 110
120 130 140 150 160 110 180
.02 -3.1241 -2.4792 -1.1964 -.3940 - .171S ... 0452 .0683 .0355 -.0207 .0148 .0284 -.0182 ·.0167 .0197 .0022
-.022J .OMO .0156 -.0125 .0151 -.0151 .0139 -.0071 .0044 .00)2
-.0059 .Ot40 - .OL03
.03 -2.7540 -2.2671
-1.2557
-.5499 -.2693 -.0990 .0250 .0235 -.008,) .0157
.0231 •. 0109 -. tlll4
.0127 .0005 -.0164 .0018 .0099 - .0091 .0102 -.0100 .0102 -.0038 .00 /.4 .0052 -.00l2
.05
l.r/Lc • .200 .10
, .20
.30
X• 0 ,50
1.00
-2.3243 -1.9943 • L 2662 -.6954 -.3960 -.1944 -.0532 -.0153
-1.8128
-1.3837
~1.169.5
-.9368
-1.6314
... 0690 -.0318 - .0128
-1. 1182 -.9445 -.7625 -.6121 -.4787 -.3648 - .28J.8 -.2193 ".1659 ... 1264 -.1007 -.0786 -.0591 -.0473
-.9180
- .011,9
.1. 2958 -1.0445 -.7955 -.6033 -.4412 - .3089 -.2212 -.1610
.0087 .0159 -.0050 ·.0064 .0074 -.0007
-1.1891
-.7949 -.5335 -.3340 -.1847 - .1084
-.Olt~7
-.0107 -.0014 -.0043
.. . Oll/.
-.oon
-.0004
-.0030 .0001 -.0051 .0028 ... 0014 .007l .0047 .0090 .0107 .0094 .014) .0094
.0048 -.0066 .0060 -.0056 .0076 -.0003
.0115
.0054 .0064 .0022 .0109
-.0050
.0008
- • HOI,
-.0764 - .0590 - .• 0 /,36
-.0290 -.0233 ~. 02()I• ... 0132
-.0084 - .0083 .0002 .0020 .0098 .0118 .0167
... 0386 ... 0283 -.0204 -.0162 -.0039 .0026 .0124 .D176
.0241
- .8155 -.6996 -.5966 -.4997 - .4124
-.)428 -.2858 -.2351 -.1939 -.1625 - .13'.8
... 1101 -.0912 -.0756 -.0599 -.0467 -.0365 -.0IS7 ... 0006 .0148 .0260 .0366 .0446 .0500
.0209
.0288 .0314
.02/,.)
. 03/~9
.0223
.on9
,OSt.l. .05/+6
.30
.50
.0199
-.6856 -.6942 -.6537 -.5999 -.5472 - .4932 - .4406 -.3944 ... 3527 -.3131 -.2776
-.2466 -.2173
-.1897 -.1652 .. .1/127
-.1206 -.1001 ... 0816
-.0460 -.0153 .0130 .0365 .0570 .07)0 .0845 .0920 .0939
3.00 - .4302
-.4597 -.4679 -.4669 -.4617 -.4517 -.4382 -.4231 - .4062 -.3869· -.3661 ' -.3444· ,,
-.3211 ! - .2963 t
)
-.2710 I -.2M,9
-.2177 -.1902 -.1626 -.1069 -.0528 -.0014 .0450 .0856 .1185
.1428 .1579 .1627
TABLE 12.9 (.;ao
1
.02
Lr/J..1!. •
.200
.03
.05
.10
0 5.3381 5.4895 2.5437 .9941 .8745
0 3.7739
0 2.2737
4.1517 2.3603 1. 2537 .9778 .4288 -.0862 -.0359 .0278
2.7002 1,6606
.20
X• 0
l.00
3.00
0 .3339
.1206
!!,.o
0 .)
10 15 20 25 )0
35 40 45 SO 5S 60 65
10 75 80
0
6.8999 6.6814 2.4739 .5810 .7800 .1713 -.5612
- .1377 .1709 -.2963 -.3463 .0132 -.0570 -.3732 -.1369
.2726
4.3448 - .1158 .0721 ... 2542 -.2896 -.0008 -.0879 -.)042 -.14)0
-.2486
... 1985
.. • 1497
-.1522 - .1166
-.0761
·.2211 -.2513 •. 1122
... 1982
110 120 U()
... 1401
-.1429
... 1287 - .1449
-.20 1)')
". tSf',.
-.H,l1
-.OOS~
140
".2383
-.01.20 ... 1952
- .0724 -.1589
150 160 170
.05\4
.010)
... 02/.2
-.I56J
-.12/11
.0282 0
.0107
... 0970 - .0040 0
180
0
.1134
- .US8
-.2340 - .0616
-.0236
90 100
.9864 .5741 .1887 .0709 -.0968 ... 1492 -.0832 -.12)6
... 1992
.037[. -.2538 -.2942 -.0925
85
1.2436
-.1.155
•• 1793 •• 1879 ... 1417
... 1461 - • 1'} I'; -.09% - .1301
-.0501 - .0756 -.0154 0
0
1.3216 1.6671 1;3094
1.0055 .8433 .5809 .3234 .2230 .1456 .0138 -.0509 ... 0473 ... 0899 - .1441
.8322
.7205 . • 531i .3387
.2483 .1729 .0624 - .0011 -.0162 -.0594 -.1077
0 .6150 .8224 .7232
.6217 .5574 .4381 .3111
.2420 .1800 .0965 .0417
.0171 -.0230
.4601 .4273 .3870 .3597 .3000 .2325 .1911
.1513 .0994 .0613 .0383 .0079
4.06/.5
-.0227
... 1096 - .1104 - .1406 - .1497 ... 1397 ... 1402
-.0762 -.0857
... 0377 -.0506 4.0706 -.0824 -.0925 -.0996
- • l'j(,'j
-.12~O
-.1.065
". t08)
... 1145 -.0636
-.1080 -.0671 ... 0597 -.0226
- .10/.9 -.0992
.. • 132/•
-.1223 - .1591
-.1666 -.1449 - .1447 ... 11121
... 0641
-.0210 0
TABLE 12.10
12-74
<)
.9481 1.2306 1.0231
0
- .1113
-.1226 -.1242 - .1277
- .1000 ". OtH~6
-.0817
-.067'.
-.0597
-.0547 - .0232
... 0455 -.0209 0
0
0 .1111 .1668
.1582 .1521 .1334 .1106
.0952 .0794 .6588
.0422 .0302 .0156 .0010 , -.0084 -.0169
)
-.o;ns
-.0350 -.0443
-.0507 -.OS28 -.049~
- .0453 -.0348 -.0253 ... Ol25 0
•
•
STRUCTURAL DESIGN MANUAL .03
.05
0 5
• ()Sl~6 .0206
.0626 .0274
.0744 .0380
10 15 20 25 30
.002B
.0071
.014')
-.OOlI8
- .00.13 -.0077
~.OO70
- .0071
-.00B8
-.0103
~.O060
-.0083 - .0072 -.0062 -.0052
-.0112
30
.20
X- 0 .50
1.00
3.00
.1526 .1120
.1792
.2116
.1379
.0775 .OtI S7
.1012 .0691 .0414
.1695 .1306
0
M.0073
35
-.0049
40
- .ootlo
45
~
50
5)
-.0027 -.0024
60 6S 70
-.0020 -.0018
7S
- .0017
80
-.0015 - .00l)
l(lO
170 l80
.0032
- .0022
- .0011
-.0007 -. Ooot~ .0000 .000) .0007 .0009 .UOlt .OOt2 .0012
-
.OO/~4
-.0039 -.00J4
-.OOJO - .0028 -.0025 -.0022 -.0019 -.0017
-.0011 -.0006 .0000
.000:; .0010
.0013 .0016 .0013 .0018
.000:>
-.0109 -.0100
-.OOHCJ -.0078 -.0069 -.0061 -.0054 -.0048 - .0043 -.0038 -.0033 -.0028
-.0013 -.0009 .0000 .0008 .0016 .0022 .0027 .0029
.0030
.1172 .0780 .0472 .0237 .0064 -.0060 -.0144 -.0199 -.0231 -.0247 -.0250 -.0246 -.0235 -.0220 -.0203
.0938 .0558 .0284 .0096 -.0026 - .0101 - .0142 -.0160 -.0165 -.0160 -.01:;1 -.OV~O
-.0127 -.0115 - .011)3 - .OO~H - .0080 - .00(,9 -. 00~,3 -.00)8 -.0019 -.0001 .0016 .0031 .0043 .0052 .00>7 .0059
.0926 .0600 .0341 .01.38
-.0015 - .0129 -.0209 -.0264 ~.0297
- .0315
-.0082 -.0043 -.OOOS .0028 .0057 .0081
-.0319 -.0315 -.0302 -.0285 -.0263 -.0238 -.0212 -.0184 -.0126 -.0069 -.0014 .0036 .0079 .0115
.001)9
.Olf,l
.0110 .0113
.0157 .0163
-.0184 "
.1325
-.0164 -.0144 -.OID
.0159
.0951 :0630 .0344 .0092 -.0127 -.0312 -.0466 -.0589 -.0683 -.0750 I -.0792 i -.0810 -.OSq7 -.07M -.0744 -.0689 -.0542 -.OJ61 -.0164 .0034 .0218
.01(19
.0254
.0375
.0211 .0237 .0245
.0326 .OJ70 .0385
.0495 .0570
.0250 .0059
.0178
-~0091
- .0019 -.0182 -.0312 - .0413
-.0207 -.0293 -.0354 -.0394 -.0417 -.0424 -.0420 -.0406 -.0384 -.0356 -.0323 -.0286 -.0204 -.on9 -.0035 .0043 .0112
- .0489
-.0541 - .0572 -.0585 -.0582 -.0565 -.0537 -.0498 -.0451 -.0339 -.'1213
... 0082 .0045
I
.0595
TABLE 12.11
.,
CSP
0 10 15 20 25
30 35
40 45 50 55 60 65
70 75 80 8.'}
90 100 110 120 130 140 150
160 170 180
.03
-.5000 -.2862 - .1334 -.0508 -.0096 .0100 .0141 .011) .0093 .0074 .0042 .0026 .0029 .0023 .0014 .0019 .0025 .0018 .0018 .0024 .0022 .0019 .0021 .0012 .0016 .0006 .0006
-.5000 -.3103 -.1664 -.0784 -.0279 -.0003 .0105 .0122 .0119 .0103 .0074 .0054 .0048 .0039 .0029 .00.31 .0034 .0029 .0029 .0034 .0032 .0029 .0030 .0021 .0021 .0011 .0008 0
0
X .: 0
Lr/'Lc • .400
.02
.05
.10
~o
.5
•
.400 .10
qI
90 100 110 120 130 11,0 150
)
Lr/L~ -
.02
5S
•
emn
.,
~.5000
~.5000
-.3382 -.2075
- .3715 -.2605 - .1175 . -.1744 -.0585 -.1101 -.0214 -.0636 -.0014 -.0322 .0076 -.0120 .0119 .0009 .0128 .0085 .0113
.0121
.0097
.0138 .0145 .0142 .0136
.00B7
.0075 .0063 .0060 .0059 .0054 .0053 .0055 .0053 .0050 . DOt. 7 .0038 .0032 .0020 .0011 0
.0132
.0128 .0122 .0116 .0114 .0108 .0100 .0091 .0076 .0061 .0041 .0022 0
.20
.30
.50
1.00
3.00
-.5000 -.3998
-.5000 -.4141 -.3336 -.2629 -.2023 - .1514 -.1098 -.0763 -.0495
-.5000 -.4298 -.3619 -.2997 -.2438
-.5000 -.4469 ·.3936 -.3421
-.4644
-.1944
- .2477
-.1516 - .1149 -.0836 -.0572 -.0351 - .0168 - .0015 .0109 .0210 .0291
-.2056
-.4265 - .3875 -.3479 - .3083 -.2693
- .1672
-.2311
-.1322
-.1940 -.1583 -.1243 - .0921 -.0619 -.0339 -.0081 .0154 .0364 .0550 .0710 .0955 .1099 .1147 .110 ] .0987 .0801
-.3084
- .2313 -.1680 - .1173 -.0782 -.0486 -.0264 -.0102 .0012 .0092 .0149 .Ot85
.0208 .0224 .0233 .0236 .0237 .0232 .0221
- .0284
- .0121 .0005 .0102 .0174 .0227 .0267 .0295
.0355
.0313
.01,04
.0181 ,0152 .0120 .0082 .0042
.0325 .0333 .0323 .0302 .0269 .0227 .0178 .0122 .0062
.0441 .0482 .0489 .0468 .0425 .0362 .0285 .0196 .0100
0
0
0
.0203
- .2933
- .1007 - .0724 - .0472 -.0249 - .0053 .0117 .0264 .0389 .0492 .0577 .0694 .0747 .0745 .0695 .0605 . .0483 .0336 .0172 0
-.5000
.0561,
.0291 0
..
TABLE 12.12 12-75
STRUCTURAL DESIGN M,ANUAL Lr/L~ • • 400
(;fn
1
.02
.03
.05
.10
.20
.30
.50
X- 0 1.00
¢o
0 5
-2.5833
-2.2692
-2.1796
10 15
-1. J098
-1. 97/~5 -1. 3065
-.6505
-.7536
20 25 30
-.3303
-.4325 -.2088
-.1256 .0134 .0334
-.0537 ,0080 .0293 .0330
80 85
.0152 .0322 .0Dt. .0021 -.0011 .0138 .0027 -.0113 .0019 .0078
YO
- .0068
100 110 120 130 lila
.0078
35
40 45 50 55
60 65 70
75
lS0 160
170 180
- .0013
.0115
.0062 .014) .0047 -.0062 .0011 .0052 -.0049 .0051
-1. 9132 -1. 7225 -1. 2558 -.8308 -.5401 -.J175 -.1529 -.06B9 -.0279 .0069 .0212 .0136 .0119
.0161 .0081 .0004 .0037
-1.J,981
-1.4023 -1. 129~ -.8520 -.6297 - .4404 -.2867 - .1840 -,1150 - .0607 -.0271
- .OUO - .0024 .0066 .0065 .0038 .OO5:J
- L1540 -1.1162 -.9685 - .8017 -.6514 -.5120 -.3888 -.2930 -.2180 -.1555 -.1088 -,0773
·.0525 -.0328
-.9826 -.9612 -.8695 -.1514 -.6384 -.5286 -.4273 -.3433
-.27J2 -.2123 -.1635
-.12(,5 -.09)8 -,0705 -.0523 -.0386 -.0262 -.0167
-.7962 -.8006 -.7484 -.6773 -.6039 -.5280 - .4541 -. 388/~ -.J299
-.2764 -.2J02 -.1917
-.3897 -.4238 -.4422 -.4514 -.4543 - .4513 - .4433 - .4319
-.3B11
-.4173 -.3996 -.3796 -.3578
-.3423 -.3057 -.2719
-.2397 -.2092 -.1814
... )]41 -.1088
-.1555
... 2551
-.0644 -.0485
-.1307 -.1077
-.2?6B -.1981 -.1692
.0047
.OO~2
-.0021
.002/1
-.0213 -.0139 -.0065 -.0016 -.0001
-.0] 04
-.0)56
-.O~67
.0035
.OOltb
.00~6
.0015
-.0127
-.0476 -.0141 .0157
-.Ult
- . OW,9
-.(lon
.OO)?
.0073
.oOB5
.00l12
.0059
.OOM~
-.0010
.0121 .otlll
.0)(J2
-,00)0
.0()49 .OUI5
.0191 .OJ05
.OOH
.0036
,.001.'9
.00)7 -.0020
. OOI~2
.0058
,OOI:,B .OW')
.0006 .0078 -.0007
.O()l.O
.01(lt.
.C08b
.OC)
.00)5
.0109
•• 01)/.2
-.5944 -.6156 -.6028 -.5753 -.5418 -.5028 -.4611 -.4205
-.1')77 -.1280 -.1036 -.0830
... 0077 .0077
.ooa')
..
,00'8
.Ol18
,On) . O:~()'I
. o'~no
.OlI( 1)
.1227 .1478
. (uey,
(;00
.02
.03
.05
.0219 .0243 .0235
.0)]0
.0S32
.OJ'}6
.0~72
.16)3
.0))4
.OS69 .OS75
.0993
,1684
.20
.30
X• .50
0
0
1.4049 1.8075 1.4710
.8096 1.0973 .9846 .8536 .7586 .5895 .4094 .3036 .2106 .0948 .019'. - .011, t -.0638 -.1128 -.1229 -.1291 -.1551
0 .5197 .8055 .7551 .6825 .6247 .5094 .3808
L.r/Lr. • •t...00 .10
a
1.00
3.00
~o
0 5 10
15 20 2S 30 35
40 45 50 S5 60 6S
r
70 "lS
80 85 90 100 110
0 4.5167 4.8735 2.6158 1.2698 .9842
0
).4237 3.8926 2.4071 1.4050 4.0669
.3M6
. son
-.2118
-.0112 -.0244 -.0028 ".2087 -.2/126 -.0%'.)
- .1082
.0058 -.2519 -.2785 -. ()'l16
- .ll46 -.2757 -.1503
-.0576 -.2088
•. U84 -.2460 -.1593
-.0946 -.1952 -.2084
-.2300 • . 1227 -.141,5
-.lJ'37 -.11, ',8
t2U
•• 17 SO
•• J(J?' ')
lJO 11,0
-. O()
to
-.OHO', -,1500
0 2.3731
2.8593 2.0290 1.3858 1.0863 .6291 .2055 .1032 .0/.. 41 -.1306 -.1841 - .1150
-.1505 -.2202 -.1687 - .1285 -.1878 •• 1939
-.1443 -.147/.
-.1 'i2) - • O~}()2 - .D08
150
- .1732 -.Olll
160
- .10.]6
- .0901
-.05l0 ... 0760
170
.OOH}
-.0078
-.OL5S
180
a
-.0332 0
0
1.1586 .9696
.67L7 .3780 .2464
.1448 -.0042 -.0793 -.Ol:!().'.
-.1240 -.176 )
-.161:> -.147.1
- .IRon -.lS)/) - .155i•
-.16)2
- .150<) ',H
-. DOl)
-.1/.
.. ,10811
- .1l60 -.0611 ) -.061.8 -.02 il 0
-.1545 - .1427 - .1156 - .1090 - .0716 -.0:'92 - .02/,2 0 ,
TABLE 12.14
12-76
.2973
.2197
0 .3761 .5364 .5254
.4948 .4669 .3990 .3182 .2609 .2043
.1115
.0985 .0877
.0:)/.2
-.0977
.OS50 .025] -.0026 -.0218 -.0386 -.0576
.1361
.0831 .0/.70 .0054
-.0282
.1186 .1203 .120)
.0754 .0599 .0460
.OSR5
.0211
... 0342
-. UIB -.1231
-. 0710 -.0869
-.0285
•• 1/.. /17 -,I HW ... lthO ... 10';8 - .07JU
-. I n~s
·.0962 -.0977
-.O4(»)
... 1100 - • O~g8 -.O72:!
-.08YS
- • OL. 71
-.0806 ·.0617
-.0429
-.0')72
- ,()'l',l)
- .0450
-,U2:,1
". O~l1
-.0218 0
- .0211b -.012)
0
-.0742
-.1
~%
0
)
.0081 -.0019 ,
... 0742 .0924 ... 1062 -.131) ... 1428 - . 14l,0
-.0561
•
0 .0742 .1115
.1557 .1152 .0813
.1257 .Ol99
0
.2046 .2998 .3066 .3004 .2923 .2609 .2202 .1868
)
.OI:HI7
TABLE 12.13
.,
i
-.0547 -.0014
.o:toS
.0479
J
-.2824
.0614 .0780 .0898
. Ol.otl
•
3.00
-.0111
- .0208 -.0]94 - • O/.. ""J
-,0)40
0
•
STRUCTURAL DESIGN MANUAL CmD
7
Lr /Lc • 1. 000
X • 0
.02
.03
.05
.10
.20
.0713
.0811 .0438
.0951 .0568 .0287 .0091
.117.1 .0777 .0465 .0225 .0047
.1421 .10t7 .0678
.30
1 00
.50
3 00
0 j.~
0 5 10 15 20 25 30
)S 40 45 50
55
.0117 -.0018 -.0087 -.0115 -.0117
-.0108 - .0094 -.0079 - .00(,5 - .-0053
60
-... ~0/13
65
':.0035 -.0029 - .0.024 -.0020
70 75 80 85 90
100 110
120 130 1',0 150 160
•
.0349
170 130
-.0016 - .0013 -.0007 -.0002 .000] .0007 .0011 .001.4 .0016 .0017 .0018
.01H} .0021
-.0075 -.0123 -.0141 -.0141 - .0131
-.0116 -.0100 -.0085 -.0071 -.0059 - .00·'<8 -.0040 -.0032 -.0026 -.0020 - .0011 -.OOO} .OOOS .0011 .OOL6 .0021
.0024
-.00/.0
-.0119 -.01.62 -.0181 -.0182 -.017J
-.0159 -.01.41 -.0123 -.0106 -.0090 -.0075 -.0062 -.00'50 -.0039 -.0021 -.0006 .0007 .0018
.0027 .00)5 .0040
.0026
.OM)
.0027
.00/IL,
- .0080
-.0166 -.0220 - .024') -.0260 -.02511 -.02/16
-.0229 - .020~ -.0185 -.0162 -.0139 -.0116
-.0095 -.0055
-.0021 .0008 .0r)"lJ .00S3 .006B .00]9 .0086 .0088
.0400
.0176 .0001 - .01)) -.02·~O
-.0298 - .OJlfl
- .0J04 -. OJ 71 -.0365
-.0350 - .0328 -.0300 -.0264 -.0235 -.0201 -.(lUl -.DOti.')
-.0005 • 00/.8 .0091 .0126 .0151 .0166 .0171
.1574 .111:16 .0813 .0515 .0267 .0064 -.0097 -.0222 -.0315 -.0380 -.042) -.0445 -.0451
.1763
.1988
.2222
.1350
.1569 .1187 .0834 .0538 .0269
.1799 ,1403 .1036 .0701 .0398
.09tU .0662 .0387 .015J
-.0399
- .0042 -.0200 -.0)24 -.0420 - .0488 -.OS33 -.OSS7 -.0564 -.0555 - .0533
- .O"}(}6
-.0500
-.0328 -.0287 -.0199 -.0109 -.0024 .0052 . 0118
-.0459 - .0411
-.04 /.4 -.0426
.OlH
.0209 .0233 .02t,1
.0037 -.0160
- .0323 -.0455 -.0557 - .0632 -.0682
-.0708 - .0714
-.0702 -.0674 -.0632 - .0578 -.04 /43 -.()285 - .0 US
-.OJOO - .0180 -.0060 .0053 .0152 .02-34 .0295 .0332 .0345
.OO4S .0194 .0319
.0413
.0129
-.0108 - .0311 -.0482 -.0621 - .0730
- .0809 -.0860 -.0885 -.0887
-.0866 - .0825 -.0768 -.0609 -.0411 - .0191 .0030
.0237 .0415 .0551
.0472
.0636
.0492
.0665
TABLE 12.15 Lr/Lc • 1.000 .10 .05
Csp 7
.02
.03
X II
a
.20
.30
.50
1.00
-.5000
-.5000 -.4357 - .372) - .3125 -.2574 -.2074 -.1630 -.1239 -.0900 -.0610 -.0)64 - .0158
-.5000 -.4471 -.3935 - .3413 -.2915
-.5000 -.4589 - .4159 -.3723 -.3290 -.2865 -.2454 -.2060 - .1687 -.1336 - .1010 - .0709' -.0433 -.0184 .0040 .0237 .0409 .0555 .0678 .0854 .0943 .0957 .0902 .0791
3.00
~o
)
---0- -.5000
-.SOOO
-.)359 -.20)2 -.1117 -.0517 - .0143 .0055 .0141 .0174 .0173 .014S .0122 .010) .0082 .0063 .0053 .0047 .0039 .0034 .0032 .0028 .0025 .0024 .0018 .0016 .0009 .0006 0
-.3556
5 10 15 20 25 )0 35 40 .45 50'
55 60 65
70 75 80 85 90 100 110 120 130
•
140 150 ·160 170
180
-.2338 -.1433 - .0791 -.0352 -.0082 .0069 .0151 .0183 .0182 .0167 .01S0 .0129 .0107 .0092 .• 0080 .0068 .0059 .0050 .0044 .0038 .0035 .0027 .0023 .0015 .
- .1174 - .0679 - .0)31 -.0099 .0052 .0142 .0186 .0204 .
-.5000 -.4037 -.3145 -.2374 - .1728 - .1199 - .0782 - .0461 - .0218 -.0041 .0084 .0168
.02(JS
'.0222
.0125
.0012
.0194
.0253 .0266 .0269 .0264 .0254 '.0241 .0211 .0181 .0153 .0127 .0101 .0077
.0220 .02<)0 .0340 .0372 .0391 .0399 .0392 .0364 ,0324 .0277 .0225 .0171 .0115 .0058 0
.0150 .0260 .0345 .0410 .0457 .0489 .0515 .0504 .0466 .0410 .0341 .0263 .0178 .0090 0
-.5000 - .3778 -.2700 -.18.39
.0178
.OODS
.0161 .011.5 .0129 .0115 .0095 .0079 .0067 .0058 .0047 .0037 .0025 .001.3
0
0
,
.0051
.0026 0
-.4251 -.3526 -.2867 -.2275 -.1756 - .1.311
- .0937 -.0625 -.0369 -.Ol(i4 -.0002
-.2446 - .2012 -.1615 -.1255 -.0931
-.0644 -.0391 - .0170 .0019 .0181 .0316 .0427 .0516 .0585 .0670 .0696 .0674 .0614 .0524 .0412 .0283 .0144 0
.0631.
.0442 .0227
0
-.SOOO -.4700 - .4371 -.4022 -.3657 -.3281 -,.• 2901 -.2521
-.2144 - .1774 -.1416 -.1071 -.0743 -.04)5 -.0148 .0116 .0355 .0568 .0753 .1041 .1216
~
.1281 .1244 .1114 .0908 .0640 .0331 0
TABLE 12.16 12-77
STRUCTURAL DESIGN MANUAL LfD
\.;r/Lc ..
1. 000
.lO
.20
.02
.03
.05
0
-L9400
5
-t. 7475
-1.6935 -1.5619
LO
-1. 2762 -.8448 -.5472 -.)18 ) - .1485
-1.2137 ·.8724 -.6122 -.39B3 -.2305
-1.4193 -1. 3438 -1.1129 - .8677
-1.1056 ·1.0794 -.9562 -.810l
-.8503 -.8532 -.7967 -.7190
- .6608
O. 672L~
- .6371
-.5J9St
JJ 40
-.0607
- .12G8 -.06"31
-.5517 - ./.678 -.3922
~.017i.
-.4781 -.3245 -.2147 - .1358
4S
.0185
-.0139
- .0732
50
.0329
55
.02 l18 .0221 .0251 .0164
.0129 .0180
-.0317 -.0097 .0059 . 017'~ .0195 .0179 .0190 .0180 .0133 .0115 .0062
"1 pO
15 20
25 30
60 65
70 7S
.0213
.0090 .0089
.0142 .0143 .0128
.0011
.0061
.00;1 -.OOlY
.0067
.0068
80 85 . 90
100 110 120
.0220
.0266
.0018
.0008 .0046 .0017 .0037
,
.0(4)
.0067
170
.0010 .0054
.0030 .0060
180
.0(0)
.0016
.0061 .0082 • ()Ou2
.ootl4 .0001
130
1/10 150 160
.0073 .0050
,0062
.. .1.190 -. J;~Ol ... 2:192 - .1704 -.11.68 -.O"/Bl
-.onG
-.0236 •• O(J~S
.0013 .0095 .0144 .01 ~)5
.oun •.DU,O
.0161
. olit 3 . Oll,., .0145
-.3243 -.2624 -.2090 -.1645 - .1262 -.0935 -.0675 ·.0466 -.0288 -.0147 -.0044 .0114 .0198 .0257 .0284 .030S .0)18
.01)2
.0]2) ,0 JJ:l
.OUd
.OJlB
.011.2
.30 -.7251 - .1392
-.7086 -.6594 -.6034 -.5415 - .J, 777 - .ld 7J -. 360/~
-.3065 -.2576 ~. 2146 -.1757
- .1410 - .1llS -.0860 -.0634 -.0443 -.0285 -.0028 .0147 .0277
.0363 .0426 .0/169
.50
X• 0
¥.5917 -.6HS -.6082
LOa
3.00
- .4532 - .4850
-.3238
-.4978 - .4992
-.5851
-.5553 -.5179
-.493t
-.476) -.43/.1
- .4800 -.4b16 -.4j98
-.J<.H9
... tllS3
-.3496
-.3882 -.3598 -.3307 -.3008 -.2706 •. 2408 -.2114
-.J090 -.2710
-.2347 -.2006 -.1695 -.140B -.11l.2
- .1823 ... 15/+ L
- .0900
-.0684 -.030', -.0002 .0245 .0/.36 .0586 .0696
.0/,95 .0,)J4 • CYI1 ')
.0771. .UIIl7
.30
. .s0
-.1269 -.075) -.0287 .OL28 .048J
.0779 .IOU .1177
.1279 .U12
.OR29
-.3618 -.3900 - .4107 - .4252 - .4337 - .4366 - ./d47 - .4284 - .4178
- .4034 - .3858 ·.)651 -.3416 -.3159
-.2882 -.2'589
-.2283 -.1.968
- .1]24 -.068 /1 - .0072 .Ot.89 .0977 .1377 .1672 • lI:~ 5tf .191')
•
TABLE 12.17 Lr/Lc: - 1. 000
CQP
1 ¢o 0
.02
.05
.10
.20
0 1.1976
0 .6909 .9518 .8934 .8040
0
0
5 10 15
2.3911
1.7706 2.2299
20 25 30 35 40 45
1.4211 1. 1162 .6514 .2195
2.8901 2.0653
50
~
.1338
55
-.1900
- .1289
60
- .1221
-.1085
65
-. tt.94
70
-.1581 -.227S
-.2065
75
-.1753
- .1774
80
- .1343 -.1929 ••. 1981
-.
85
90 100 110 120 130 1/.0
150 160 170
180
12-78
.0457
•. 1472
-.1493 -.1537 - .0972 - . D15 -.051:1 ... 0763 -.01l7 0
.9505 .6937 .4322
.7324 .5926
0 .3905 .5636 .5626 .5387 .5142 .4465
.4374
.3633
.2973
.3371
.3016
.1864 .0398 -.0439 -.0642 - .1142 -.1675
.2444
.2392 .1643 .10'.3 .0611 .0129 -.0328 -.0601 -.0827 -.1099 -.1268 -.1416 - .1479 -.1441 -.1263
1.5179 1. 3439
1. 7386
1. .3108 1. 0708 .7014 .3460 .2086 .1095 -.0564
.1100
X- 0
.03
1.1070
-.1631 -.1568 -.1857
152/~
-.1921
.1328
.0540 .0099 -.0 /.51 -.0973 - .1161
-.1295 -.1566 -.1676 -.1648
-.1946 -.1565 -.1527 -.1500
-.1896 -.1658 -.1585
- .L059
- .1148
-.1220 -.0607
-.115/. -.0687
-.1137 - .0773
-.O()/)O
-.0634 -.0228 0
-.0608 -.0261 0
-.01% 0
-.1497
.
.. '.1615
-.1516 -.1249
... 1122
-.0825 -.0607 -.0285 0
TABLE 12.18
0' .2770
.4076 .4195 .4130 .4028 .3605
.3050 .2612 .2148 .1584 .1107 .0735 .0325 -.0068 -.0:336
0 .1778 .2669 .2832 .2865
.2855 .2631 .2308
.2035 .1731 .1352 .1015
.0732 .0421 .0118 -.0112 -.0.318 -.0536 -.0703 -.0925
1.00 0
.0954 .1461 .1596 .1657 .1685 .. 1595
.1443 .1307 .1144 .0936 .0740 .0566 .0372
.0180 .0022 -.0124 -.0275 -.0399
3.00 0
.0340 .0529 .0593 .0628 .0650 .0629 .0583 .0539 .0482 .0407 .0334 .0265 .0188
.0109 .00/,2
-.0568 -.0825 -.1004 -.1208 - .1317 - .1320
-.1060 ... 1099
... 1196 ".1066 ... 0809 ... 0587
-.1030 -.0929 -.0125 -.0523
".0731
-.0659 -.0535 -.0386
-.0023 -.0089 -. 01~6 -.0235 -.0298 -.0330 -.0327 -.0303 -.0247 ... 0179
- .0284 0
-.0259
-.01(J5
- .0091
0
0
0
-.05B2 -.070) -.0756
~
i
I
1 ~
I
•
/
"-"0~"""" / , .' ~\ . \ \, \
"\\ ~'.,I~.,,~ ',/
// J
•
~~~j/
.,
<:mt 20
Lr/Lc :::: .200
.OJ
.05
.10
Sl 0
5 10
25
3D 35 40
0 -.0025 -.OOJO -.0028 -.0024 -.0020 -.0017 -.0015 -.0012
45 50
-.0008
S>
-.0006
60
-.0004 -.0002 -.0001 .0001 .0002 .OOO} .0005 .0006 .0007 .0008 .0007 .0007
65
70 7'j
80 85 90
100 110 120
uo Il.O
- .OOlO
0 - .0031 -.00/.0
- .0040 -.0036 - .0031 -.0026 - .0022 -.0019
-.0015 -.0012 ··.00n9 -.0006 - .0004 -.0001 .0001 .000)
.0005 .0007 .0009 .0011 .0011 .0011
.0010
150
.ooos
.0008
160
.0004 .0002
.0006
170 180
0
.0003 0
- .00!,11
0 -.0056 -.0085 -.0097 -.0098 -.0093 -.0084
-.00 J8
- • 0071~
-.0135
-.0089 -.0147 -.0182 -.0199 -.0202 -.0196 -.0183
-.00 J7 -.00:26
- .00(,4
-.0120
-.0165
-.00'))
-.0102
-.00/+3
-.0085 -.OOb7 -.0049 -.0032 -.0016 -.0001 .0012 .0025
0
-.00'.0 -.0056 -.0059 -.0056 -.00~1
-.0021 -.0015 - .0011 -.0006 -.0002 .0002 .0005 .0008 .0011
.0015 .0018 .001') .0018 .0016 .0013 .0009 .0005 0
-.0033 -.0023 -.0014 -.0006 .0002 .0009 .0015 .OO2C .0029 .0034
0 -.0076 -.0122 -.0147 -.0157 -.0156 -.01L,8
f1
.50
1.00
3.00
0
0 - .0131 -.0229 -.0298 -.0342 -.0365 -.0370 -.0361 -.0)40 -.0310 -.0273 -.0231 -.0186 -.0139 -.0092 -.0045 -.0001 .0042 .0080 .011.. 5 .0190 .Oi1.4 .0218 .0201
0 -.0162 -.0289 -.0385 -.0451 -.0491 -.0509 -.0506 -.0487 -.0454 -.0409 -.0355 -.0294 -.0229 -.0161 -.0093
0
.0035
-.0144 -.012L -.0097 -. 0074 -.0050 -.0028 -.0007 .0013 .0031 .0047
.0053 .0063 .0068
.0072 .0088 .0096
.0067
.0095 .0086 .0071
.0036 .00J5 .0032 .0026 .0018 .0010
.0061 .0050 .0035 .0018
0
0
.,
Cst .02
.03
Lr/Lc • .200 .10 .05
rf'o
10
15 20 25 30 35
40 t..5
50 5S 60
65 70 75 80 8S
90 100 110
120 130
•
.30
-.0107 -.0182 - .0231 -.02.58 -.0269 -.0267
-.0255 -.0235
-.0209 -.Ol/l,O -.OLI18
•• 0115 -.0082 - .0050
-.0019 .0010
.oo:n
.0026
.0062 .0101 .0128 .0141 .0141 .0129 .0106 .0075 .00)9
0
0
.ooso
~
I
! ! I I
,
-.0026.00.19 .0099 .020LI
.0280 .0'324
.0)]5 .0314 .0263 .0189 .0099 0
01b7
.0120 .0062 0
TABLE 12.19
--U5
.)
X •
.20
0
15 20
•
STRUCTURAL DESIGN MANUAL
140 150 160 170 180
-.0463 -.0141 -.0004 .0038 .0047 .0041 .0033 .0029 .0027 .0024 .0022
.0022 .0021 .0019 .0018 .0017 .0015
.0013 .0012
.0008 .0005 .0000 -.0003 -.0006 -.0008 -.0010 -.0011 - .0011
-.0536 - .0201 -.0036 .0032 .0054 .0056 .0050 .. 0045 .0041 .0037 .0034 .003) .0031 .0029 .0027 .0025 .0022 .0020 .0017 .0012 .0006 .0001 -.0004 -.0009 - .0012 -.0015 -.0017 ,- .00 t7
-.0646 ~.O296
-.0096 .0007 .0055 .0073 .0075 .0072 .0069 .0064 .0059 .0056 .0053 .OOt..9
.0045 .0042 .0037 .00J3
.0029 .0020 .0011 .0001 - .0007 -.00 l/~
-.0020 - .0025
- .0027 -.0028
-.0833 - .0465 -.0219 -.0064 .0029 .0081 .0108 .0020 .0123 .0122 .0117 .0112 .0106 .0098 .0091 .0083 .0075 .0066 .0058 .0039 .0021
.0004 - .0013 -.0027 -.0039 -.0048 -.0053 -.00')5
X :II 0 .20
.30
-.1066 -.0682 -.0396 -.0189 -.0043 .0057 .0122 .0164 .0189 .0202 .0206 .0205 .0198 .0189 .0l77 .0163 .0148 .0132 .0115 .0080 .0044 .0010
- .1221 -.0829 -.0522 ·,0286 -.0109 .0022 .0116 .0182 .0226 .0254
.0269 .0274 .0272 .0263 .0250 .0234 .0214 .0193
- .0051 -.0074 - .0091 -.0102
.0169 .0119 .0068 .0018 -.0029 -.0070 -.Ot04 - .0130 -.0145
-.0105
-.0150
- .0022
.50 -.1430 -.1030 -.0698 - .0429 - .0214 -.0043 .0089 .0190 .0264 .0317 .0351 .o:n1.0379 .0376 .0365 .0347
.0323 .0294 .0262 .0190 .0113 .00.36 - .00)6 -.0101 -.0154 ·.01l)4 -.0219 -.0227
3.00
1.00
.2072
- .1713
A
~.1303
-.1652
-.0946 -,0638 -.0376 -.0156 .0027 .0176 .0295 .0387 .Ol.55 .0503 .0531 .0543 .0540 .0525 .0499 .0464 .0421 .0317 .0200 .0078 -.0040 - ,0147 -.0237 -.0)05 - .0347 -.0361
-.1267 -.0918
-.0605 -.0327 -.0083 .0128 .0)07 .0454 .0572 .0662 .0726
.0766 .0783 .0779 .0757 .0718 .0665
.0523 .0349 .0158 -.0033 -.0211
·.0362 - .0478 -.0550 -.0575
TABLE 12.20 12-79
STRUCTURAL DESIGN MANUAL .,
Crt
Lr/L...... 200 .10
.02
.03
.05
0 5
-.5000
-.5000
- .2464
-.2738
10 15 20 25 30 3S 40 45 50 55 60
-.0863 ... 0223 .0000 .0098 .0080 .0026 .0025 .0032 .0007 .0002
... 1199
.0041 • 00'11
-.5000 -.3066 -.1637 - .0804 -.OJ42 -.0087 .0014 .00:38 .0052 . 0056
.0020
.0042
... 0005 ,0039 .0056
.00)7 .00'.4
,(.1067 .0080
65 70
.0020
.0014 .0028 .0026 .00l8
.0043
.C084 .C086 .0092
X• 0
20
.30
-.5000 -. )813 -.2787 ·.1988
-.5000 -.3989 -.3085 -.2342 - .1744 -.1269 - .0903
-.5000 -.4182
... 0623
-.0994 -.0721 -.049/• -.0307 - .0152 -.0023 .0084
.50
1.00
3.00
-.5000
-.5000 -.4610 -.4204 -.3796 -.3390 -.2991 -.2603 -.2227
¥'So
85
'90
.OOV.
100
.002S .0021 .0017
110 120 130 140
.0009
150
.0018
160 170
.0001. .0007 0
180
.0049 ,0074 .0045
.0023
.0006 .0018 .0028 .OOl6
75 80
-.0449 ... 0111
.0027
.0035 .0027 .0026 .0031. ,0032
.0028 .0031 .0019 .0022 .0009 .0008
.0024
0
-.sono -. J467 -.2230 ... 1374 -.0802 - .0426 -.0205
... 1381
-.0927 -.0602 -.0374 -.0208 -.0091 -.0011 .004B .0093 .0124 .0146 .0166 .0180 .0189 .0197 .0205
-.0082
.0038 .0045 .0051 .0048 .0048 .0054 .00'2 .0048
.0099 .0Wl
.00/17
.0090
.00]6,
.0074
.OO,H
.0061
.0019 .DOl2
.0,)40 .0022
.0)92 .0174 .0148 .0117 .0080 .0041
0
0
0
.O(J98
.0106 .Ol{)!..
-.OlI05
-.0238 -.0111 -. 0013
.0065 .0125 .0171 .0208 .0238 .0259 .0274
.0098
- .4393 ... 380)
-.2762 -.2197
-.2755
- .1719 - .1323
.Onl .02/.4
.030) .0349
-.3256 -.2)01 - .1894 -.15)0
-.1205 -.0914 -.0657 -.0428 M.0226 ·.0049 ,.0106 .0240 .0355 .045l
.0385 .01.31
.0530
.0293 .0280
.0'145
.0254
.0396
.02D
.0172
.0341 .0270
.OL18
:01B7
.0694 .0696 .0652 .0570 .0456 .OJ17
.0061 0
.0096
.0293
.020)
-.3/.23
.0432
0
.OMl
.0163 0
- .1865
-.1519 ",1190 -.08BO
- .0590 ' -.0320 - .0073
.0153 .0)55 .0533 .0687 .0922 .1061
.1107 .1069 .0934 .0774 .0545 ,0281 0
TABLE 12.21 L"lL,. •. 400
Cat
'1 ¢o -0 5 10 15 20
25 30
35 40 4S 50 55 60 65 70 75 80 85 90
1.3464 1.0085 .3621 -.0382 -.1457 -.2033 -.2532 -.2291 -.1925 -.2007 -.1972
-.1629 "'. 1519 - .1576
-.1360 -.1105
.05
1.1650
.9557
.9055
.7738
.3933
.4016 .1140 -.0339 ... 1303 .... 1946 -.2059 -.1975 -.1997
.0401 ".0972 ~
.1771
-.2324 ·.2239 ... 199~ -.2018 -.1952 -.1671 -.1550 - .1546 -.1356
"~1928
... 1712 - .1586
-.1530 - .1359
.10
.7093 .6008 .3692 .1674 .0362 -.0609 -.1304 -.1618 -.1732 -.1823 -.1814 -.1691 -.1591 -.1511
.20
.30
.5064 .4439
.4071 .3624 .2620 .1623 .0824 .0145 -.0408 -.0781 - .1032 - .1219
.3057 .1739 .0747 -.0061 -.0691 -.1076 ".1305 -.1470 -.1540 ·.1516 -.1473
-.1342
-.1003 -.0934 -.0848 -.0744 -.0536 -.0305 -.0092
-.1131
-.1080
",1019 -.0139
-.0994 - .0759 -,0576 ... 0204 -.0051 .02/,9 .0406 .0563
-.0777
.... 0799
... 0812
-.0559 -.0229 -.0039
-.0553 -.0255 -.0037 .0210 .0392 .0549 .0677 .0727 .0776
-.0560 -.0283 ... 0054 .0180 .0365 .0520 .0641
-.1038 -.0929 .... 0799 -.0559
.ona
.0722 .08)0 "
.0232
.M04
.0561 .0701 .0731
.0806
.0986 .0633 .0306 .0015 -.0216 -.0399 -.0549 -.0659 -.0727 ... 0771 -.0791 -,0784 -.0757 -.0719 -.0666 ".0598 -.0448 -.0275 .... 0104 • OMS
.0798 .0282 ".0157 -.0480 -.0718 .... 0904 -.1025 -.1082 -.1107 . - ,lOr) 7
-.1188
• 0261~ .0404 .0561 .07Jl. .0708 .0853
.1378
-.1229
- .1204 -.1094 -.0968
130 140 150 160 170
.2063
.1379
-.1416
-.1089 ... 0975
~.(\069
.1897 .1741
-.1312
-.un
120
.3012 .2723
... 1364
-.1139
- .0178
X- 0 .50 1.00
... 1335 -.1304
~.lO92
-.0599
-.0<)59
".0299 .... 0071 .01S/.
.0338 .0491 .0608
.0700
.0737
. I
I
TABLE 12.22
12-80
-.1321
-.1100
100 110
lao,
.03
.02
II
... 1106
.0115
.021Jl .0437
.0670
.0547 .0609
.0702
.0637
.0212
.0336 .04)0 .0485 .0507
•
3.00
•
.0799 .0743 .0610 .0461 . 0320 .0184 .0058 -.0048 ... 0137
... 0214 -.0274
-.0318 -.0349
-.0;370 -.0377 -.03J), -.0362 • ,0343 -.0315 -.0247
-.0162 -.0072
.0019 .0101 .0171 .0225
.0257 .0270
•
STRUCTURAL DESIGN MANUAL
•
Gmt 1 ¢o 0
5
10 15
20 25
JO J) 40 4S
50 55 60 6S !
70 75 80 85
90 100 110 120 130
140 150 160
•
170 180
.02 0 - .0031 - .0041 ... 0039 -.00)4 -.0027 -.0022 ... 0017 -.0013
-.0010 -.0007 -.0005 -.000) -.0001 .0000 .000l .0003 .0004 .. 0006 .0007 .0008 .0008 .0008
Lr/Lc
.03
0 -.0038 -.0052 ... 0053 ... 0048 -.0041 ... 0033 -.0027 •• 0021 -.0016 -.00l2 -.0008 -.0005 -.0002 .0000 .0003 .0005 .0007 .0008 .0011 .0012 .0013 .0012
.0007 .0006 .0004 .0002
.0011 .0009 .0006
0
0
.OOOJ
05
0 -.0046 -.0070 -.0076 - .0073 -.006; -.OOSb - .0046
-.0037 ·.0029 -.0021 ... 0015 -.0009 -.0004 .0000 .0004 .OOOS .0011
.ool)
.0011 .0020 .0021 .0020 .0018 .0014 .0010 .0005 0
II
.400
10 0 -.0065 - . 0101 -. 0117 -. 0119 -.0113
-.0103 ... 0089 -.0075 -.0061 -.0047 -.0035 -.0023 - .0012 -.000) .0006 .0013 .0020 .0025 .0034 .0039 .0040 .0039 .0035 ',0028 .0020 .0010 0
X• 0
20
0 -.0085 ... 0139 -.0169 -.0182 -.0182 -.0172 -.0157 -.0138 -.0117 -.0096 -.0074 -.0053 -.0033 -,0015 .0002 .0018 .0031 .0043 .0061 .0071 .0076 .0074 .0066 .0054 .0038 .0020 0
30
0 -.0098 ... 0164 -.0204
-.0225
-.ono -.0223 -.0208 -.0188 -.Ol63 ... 0136
- .0109 -.0081 -.0054 -.0028 -.0004 .0018 .0037 .OOSS .0082 .0099 .0106 .0104 .0094 .0077 .0054 .0028 0
1.00
50
a - .0115 -.0197 -.0252 ... 0284 -.0297 - .0295 -.0282 ... 02bO
-.0232 -,0199 -.0163 -.0127 -.0090 -.ODS/.
-.0019 .0013
.0043 .0070 .0112
.0141 .0154 .0153 .0140 .011 5
.0082 .00142 0
0
- .0138 -.0242
-.0316 -.0364 -.0389 ... 0396
-.0387 -.0365 - .03)4 - .0294 -.0249 -.0200 -.0150 -.0099 -.0048 .0000 .0045 .0087 .0156 .0204 .0230 .0233 .0215 .0179 .0128 :0066 0
3.00 0
".0166 -.0297 -.0395 ... 0464 -.0506 -.0525 ... 0523 ... 0504
-.04(,9
-.0423 -.0368 -.0.305 -.0237 -.0167 - .0096
-.0027 .0040 .0102 .0210 .0290 .0336 .0347 .0)25 .0272 .0196 .0102 0
TABLE 12.23 LrI ...e. • .400
Cst
7
.02
.03
.05
.10
.20
.30
.50
X• 0
LOO
3.00
-.1526
-.1792
-.2116
. -.1120 ... 0775 -.0487 -.0250 -.0059 .0091 .0207 .0293 .0354 .0394 .0417 .0424 .Ot.20
- .1379
- .1695
-.1012 - .0691 ... 0414 -.0178 .0019 .0182
-.1306 -.0951 -.0630 -.034 4 -.0092 ,! .0127 I .0312 i
¢o
r----T 5
l()
15 20 25
30
)
35 40 45 50 S5 60 65
70 75
80 85 90 100
•
•• 0546 ·.0206 -.0028 .0048 .0073 .0071 .0060 .0049
.0040 .00J2 .0027 .0024 .0022 ' .0020 .0018 .001] .0015 .0013 .0011 .0007
110
.0004
120
.0000 -.0003 ·.0007
130 140 150 160 170
- .0009 - .00lt "'.00 l2
180
-.0012
-.0626 ... 0274
-.0744 -.0380
-.0071 .0033
-.014S
.0019 .0017 .0011 .0006 .0000
.0100 .0089 .0078 .0069 .0061 .0054 .0048 .0043 .0038 .0033 .0028 . {l018 .0009 .0000
~.OOOS
- .OOOS
-.0938 -.0558 -.0284 -.0096 .0026 .0101 .0142 .0160 .0165 .0160 .0151 .0140 .0127 .. 0115 .0103 .0091 .0080 .0069 .0058 .. 0038 .0019 .0001 -.0016
~.OOLO
-.0016 ... 0022 - .0027 - .0029 - .0030
-.0052 -.0057 -.0059
.0077 .0088 .0083 .0072 .0062 .0052 .0044
.OO:l9 .0034 .0030 .0028
.002) .0022
- .0011
-.OOl6 -.0018 -.0018 ,
-.0005 .0070 .0103 .0112 .0109
-.oo:n
- .001+3
-.1172 -.0780 -.0472 - .0237 - .0064 .0060 .0144 .0199 .0231 .0247 .0250 .0246 .0235 .0220 .0203 .0184 . 0164 .0144 .012) .0082 .0043 .0005 -.0028 -.0057 -. 0081 -.0099 -.0110 -.Olll
... 1325
-.0926 ... 0,600 . -.0341 -.OllS
.0015 .0129 .0209 .0264 .0297 .0315 .0319
.0315 .0302 .0285 .0263
.0236 .0212 .0184 .0126 .0069 .0014 ·.0036 '" .0079 -.0115 ... 0141 -.0157 -.0163
.0/~O6
.0384 .0356 .0323 .0286 .0204 .0119
.00.35 -.0043 -.0112 -.0169 - .0211
.. ,0231 -.0245
.0312
.0413 .0489 .0541 .0572 .0585 .0582 .0565
.0537 .0498 .0451 .0339 .0213 .0082 -.0045 -.0159 -.0254 -.0326 ·'.0370 -.0385
.~ .OS89'~
.O~~~!
11'
01 ,:~,,:,;
.0801 .0784, . .0744 .0689 .0542 .0361 '"
.0164
-.0034 ~ ·.0218 1\ -.0375 ..'" -.0495 -.0570 -.0595
TABLE 12.24
12-81
STRUCTURAL DESIGN MANUAL Cft '1
.400 .10
X-
Lr/Le, -
.02
.03
.05
0 5
-.5000
1.0 15
... 1334
-.5000 -.310) -.1664 ·.0784
-.5000 -.3382 -.207,
'- .5000 - .3715 -.2605
-.1175
-.0279
-.0585 -.0214 -.0014 .0078
-.1744 - .1101
.20
.30
.50
a 1.00
3.00 -.5000 -.4644
•
~o
20
25 30 35 40 45 50
55 60 65
70
75 80 85
90 100
110 120
DO 140
150 ]60 170 180
-.2862
... 0508 -.0096 .0100 .0141 .01D .0093 .007l,
.001,2 .0026 .0029 .0021 . 00 VI.0019 .0025 .0018 .0018 .0024 .0022 .0019 .0021 .0012 .0006 . 0006 .0006 0
-.0003 .0105 .0122 .OU9 .010 !
.0074 .0054 •. 0048 .0039 .0029
.oo.n
.0034 .0029 .0029 .0034 .0032 .0029 .0030 .0021 .0021 .0011
.OOOS 0
-.06.16 - .0.l:!2 -.OUO .0009 .0ot1S
.0119
.0128 .0113 .0097 .00B7 .0075 .0063 .0060
.01:18 .0145 • OIL. 2
.0136 .0132
.0128 .0122
.0054 .OOS] .0055 .0053 .0050 .0047
.0118
.0114 .0108 .0100 .• OO~)l
.0038
.00)2 .0020 . .0011 0
-.5000
-.SOOO
~.5000
- .41l.. 1
- .4298
.0/~86
-.0264 -,0102 .0012 .0092 .0149 .0185 .0208 .0224 .02J3 .0236 .0237 .0232 .• 0221 .0203 .018L .0152 .0120 .00a2 .0042 0
-
. Ol;~ 1
.0059
... 5000 -.3998 -.3084 -.2313 ·.1680 -.1173 -.0782
.0076 .0061 .0041 .0022 0
-.3336 -.2629 -.2023 -.1514
- .3619·
-.4469 -.3936
-.2997
... 3421
-.3875
- .4265
-.2438
-,2933
-.3479
-.1944
-.2477
-.2056 -.1672
-.3083 -.2693
-.0763
-.1516 -.11'19
-.()I~95
-.0836
-.0184
-.0572 -.0351
-.1)22 -.1007
-.1098
-.0121
.0005 .0102 .0174 .0227
.0267 .0295 .0313
.0)25 .OJ33 .0323 .0302
- .0724
-.0168
-.0{~72
-.0015 .0109 .0210 .0291 .0)55 .0404 .0441 .0482 .0489
-.0249
~.Oj39
.0117 .0264
-.0031
.0492 .0'>77 .0694 .0747
. Old) 8
.0JI.S
.0269 .0227 .0178
.0425
.0695
.0362
.0605 .O/dD
.0]22
.0196 .0100
.0062 0
()
... 0921 ·.0619
-.0053
.0)89
,0285
-.2JU -.19 /.0 -.158J -. 1243
.01:>4 .0364 .0550 .0710 .0955 .1099 .1lt.7 .1107
.0987 .0801 .0564
.0336 .0172 0
.0291 0
•
TABLE 12.25 "CIt !
7
.02
0
1.0802 .8619 .4200 .0895 -.0676 a.1646
i_o 5 10 15 20 2S 30 35 40 45 50 55 60 65
70
75 80 85 9.0 100 110 120 130 140 150 160
170 180
·.2271
-.2291 -.2111 -.2098 -.2000 ... 1735
-,1594 -.1541 ".1363 - .1160 -.1089 -.0980 -.0767 -.0562 -.0215 -.0038 .0244 .0411 .0568 .0715
.0735 .0823
.03
.05
Lt-ILc • .400 .10
.9271
.7551
.5572
.7629 .4205 .1419 -.0167 -.1238 -.1947 -.2129 -.2086 -.2094 -.2006 -.1785 -.1640 -.1555
.6421 .3988 .1824
.4909
.3421 .1975
.0374
.0844
·-.0702
-.0087
~.1~65
-.0812 -.1260
-.1809
-.1925 -.2000 -.1964 - .1811 -.1681
-.1576
-.1759 -.1714 -.lM4 -.1559 -.1424
-.1408 -.1231
-.1092 -.0970 •• 0780
-.1109 -.071)9
-.0551
-.0546
-.0230 -.0029 .0235
-.0244 -.0025 .022t.
- .1141 -.1000 -.0837 -.0562 -.0270 -.0012 .0206
.0410 .0568
.M07
.0395
.056) .0693
.0552 • O(,7t~ .0734 .0771
.0705 .0742 .0809
.0743 .0793
- .1272
,.
0
.50
1.00
3.00
.3969 .3590 .2715
•.3188 .2918 .2285
.2357
.2182
.1794 .0998 .0292
.1594
.1297
.0970 .0397 -.0102 -.0488
.1481 .1386 .1157 .0889 .0625 .0365 .0122 -.0088 -.0266 -.0417 -.0535 -.0620 -.0679 -.0715 -.0724 - .0713 -.0687 -.0645 -.0588 -.0451 -.0287 -.0119 .0047 .0195 .0320
.0619 .0585 .0501 .0399 .0295 .0190 .0088 - .0004 -.0086 -.0156 -.0215 -.0281 -.0296 -.0320 ... 0333 -.0336 -.0330 -.0316 -.0294 -.0235 •• 0159 -.0075 .0010 .0089
.0414
.0208 .0240 .0251
.1766
-.0781
.0853 .0432 .0051 -.0261 -.0512
-.1261
-.1008
-.0716
-.1400 -.1445 -.1447 -.1415
... 1159
-.0865
-.1236
-.0959
-.1270
- .1015
- .126 7
-.1039 -.1025 -.0985 -.0928
- .1335
-.1221
-.1230 -.1122 -.0998
-,1146
-.0857 -.0589 -.0309 -.006) .0171 .0363 .0521 • OGI~O
.0705 .0735
TABLE 12.26
12-82
x-
.30
-.0302 -.0734 -.1040
-.1523 -.1698
- .1378 - .1189
-.0974
~
.20
-.1061 -.0957 -.0836 - .0590 -.0329 -.0088 .0140 .0331
-,08S3
-.0761
-.0558 - .0332 -.0114 .0096
.0488 .0601} .0672
.0278 .0427 .0539 .0605
.0700
.0631
.0471 .0492
.
.0156
•
STRUCTURAL DESIGN MANUAL Cmt 7
Lr /Lc • 1. 000 .10
.02
.03
.05
30 35
-. 0041~
0 -.0054 -.0080 -.0088 -.0086 - .0077 -.0065
0 -.0066 -.0102 -.0118 -.0120
25
0 -.0045 -.0065 -.0069 -.0064 -.0055 -.0034
... 00,)3
40
-.0026
45 50
- .00 18
- .0041 -.0030 -.0020
-.0085 -.0069 -.0054
¢
X
.20
t:
0
.30
.50
0 •• 0119 -.0205 -.0263 - .0296
0
0
- .0136 -.0231 - .03(J8 -.0354
-.0155 - .0275 -.0363 -.0423
- .0311
·.0377
-.0309 -.0295 -.0271 - .0241 -.0205 -.0167 -.0128 -.0089 -.0051 - .0015 .0019 .0049 .0076 .0118
-.0382 -.0371 -.0348 -.0315 - .0275 - .0231 -.0183 - .0134 -.0085 ·,0038 .0008 .0050 .0088 .0150 .0192
-.0458 - .0471 -.0466
1.00
3.00
0
0
5
10 15 20
55
60 65 70 75
... 0012 ... 0007 .0002 .0001 .0004 .0006
80 85 90 100 HO 120
~OOU
130
.00 t2
140
.0Oll .0009 .0006 .0003
1:;0 160
170 IRO
.0008
.0010 .001l .OOD
.0013
-.0113 -.0100
-.OOJ,)
-. 00 12
-.0026
-.0006 ,0000 .0005 .0009 .0012 .0014
-.00]4 -.0004 .0004 .001l .0017
.0016 .0019 .0021
.0005
.0026 .00) 1 .0033 .003) .0031 .0027 .0022 .0015 .0008
0
()
.0020 .0019 .0016 .0013
.0009
-0
.0022
0
0
-.0064 - .0138 - .0168 ... 0179 -.01.77
-.0166 -.0149 -.0129 -.0106 -.0084 -.0062 - .0041 ~.OO22
-.0005 .0011 .0024 .0035 .0044 .0057 .0064 .0065
-.0106
-.0179 -.0226 -.0251 -.0258 -.0252 -.0236 - .0213 -.0185 -.0154 -.0122 -.0090 ·.0058 -.0029 -.0001 .0024 .0046 .0065 .0094
.0061
.0053 .0043 .0030 .0015 0
.0111
.0145
.0117 .0113 .0101 .0082 .0057 .0030
.0157 .0154
0
0
.0139
.0114 .0080 .0042
-.0444 ... 0410
-.0366
-.0466
-.0314 -,0256
-.0407
1
-.OJ39
!
-.0133 ... 0071 ·.OOll
.0115
.016)
.0060
.0085 0
-j
... 0266 • ... 0190 - .0113 ... 0036 .0038 .0108 .0229 .0318 .0371 .0385 .0361 .0304 .0219 .0114 0
-.0196
.0213 .0195 .0161 0
-.0421 - .0497 -.0544 - .056 7 -.0568 -.0550 -.0514
.0046 .0099 .0188 .0252 .0287 .0293 .0272 .0227
.0213
0 w.0175 -.0315
,I
TABLE 12.27 lost
"I
.02
.03
.05
J..r/J..c • 1.000 .10
.20
.30
X• 0 .50 ' 1.00
3 00
fDo
0 .5 10 15
20 25 30 35 40
)
45 50 55
60 65 70
7S 80 85
90 100 lIO 120 130
140 ISO 160 170 180
- .0713 -.0349 ... 0117 .0018 .0087 .0115 .0117
.0108 .0094 .0079 .0065 .00S3 .004) .0035 .0029 .0024 .0020 .OOt6 ,OOl)
.0007
,0002 -.OOO} -.0007 - .0Olt
-.oott. -.OOlo
-.0017 -.0018
-.0811 -.0438 ... 0183
-.0021
.oon .0123 .0141 .0141
.0131 .0116
.00to .0085
.oon .0059 .0048 .0040
.0032 .0026 .0020 . DOll .0003 -.0005 -.OOlt
-.0016 - .0021 -.0024 - ,0026 - .0027
... 0951 -.0568 -.0287 -.0091 .0040 .0119 .0162 .0181 .0182 .0173 .0159 .0141 .0123
.0106 .0090 .0075 .0062 .0050 .0039
- .U11 -.0777 -.0465 -.0225 - .0047 .0080
.0166 .0220 .0249 .0260 .0258 .0246 .0229 .0208 .0185 .0162 .0139 .0116 ' .0095
.0021
.0055
.0006 -.0007 -.OOlS -.0027
.0021 -.0008
- .00)) - .00/10
-.0043 -.0044
-.00))
-.0051 -.0068 -.0079 - .0086 - .0088
... 1421 - .1017 -.0678 ... 0400 -.0176 -.0001 .0133 .0230 .0298 .0341 .0,364 .0371 .0365 .Ol50 .0328 .0300 .0269 .0235 .0201 .0131 .006S .0005
".1514 - .1166 - .0813 -.0515 -.0267 -.0064 .0097 .0222 .0315 .0380 .0423 .0445 .0451 .0444 .0426 .0399 .0366 .0328 .0287 .0199
-.oun
.0109 .0024 -.00';2 - .0118
- .0126
-.0171
-.Ot51
-.0209 -.0233 -.0241
-.OMS
-.0166 ... 0171
-.1763 -.1350 -.0983
-.1988 -.1569
-.2222 ",1799
".1187
-.0662
-.0843
-.0387 -.0153 .0042 .0200 .0324 .0420 .0488
-.0538 -.0269 .. ... 0037 : .0160 ' .0323
-.1403 -.1036 -.0701 -.0398 ... 0129
.0533
.0551 .0564 .0555 .0533 .0500 .0459
.0411
.0300 .01BO
.0060 -.005)
-.0152 -.0234
'
.0557 :~". .0455 .. .0632: : . 0708' .0714 .0702 .0674 .0632 .0578 .0443 .0285 .0118 ... 0011 :; -.0194 -.0319 -.0413
-.0332
-'.0472 -.0492
",t: ,
i" 9 ~:: 0860 f. • 088~
.0682.
-.0295 ~.034S
1' · ~;:0108
'.'~~1
j
:
.0887 .0866 .0825
.0768 .0609 .0411 .0191 - .0030 ' -.0237 - .0415 -.0551 -.0636 -.0665
l.V,
TABLE 12.28 12-83
STRUCTURAL DESIGN MANUAL Lr/Lf' • 1.000
eFt'
7
.02
.03
.OS
.10
.20
.)0
.50
X- 0 1.00
3.00
-.5000 -.4589
-.5000 -.4700 - .4371 -.4022 -.3656 -.3281 -.2901 -.2521 -.214,4
{;o
0 5 10 15 20 25
-.2032 -.1117 -.0517 -.014)
30 3S
.0055 .0 II.!
40 45 50 55
60 65 70 75
80 85 90
100 110 120
130 140 1')0 160 170 180
-.5000 -.3359
.017/.
.0173 .OL48 ' .0122 .0103 .0082 .0063 .00;) .0047 ,0039 .00)4 .0032
.0028 .0025 .0024 .00113
-.5000 -.3556 -.2338 - .1433 -.0791
-.0)52 -.OOB2
.0069 .0151 .0183 .0182 .0167 .0150 .0129
.0107 .0092 .0080
.0068 .0059 .0050 .0044 .0038 .003.}
.0027
.llC)) 6
.OO2'~
,OOO(J
.OOlS .0008 0
.0006 0
-.5000 - • .3 778 -.2700 -.1839 - .1174 -.0679
-.5000 - .40)7 -.3145 -.2374 - .1728 - .1199
-.5000 -.4251 -.3528 -.2867 -.2275 - .1756
-.0332 - .0099
-.0762
-.1311
-.0461
.0052 .0142 .0186 .0204 .0205 .0194' .0178 .0161 .0145 .0129
-.0218
-.0937 -.0625 -.0369 -.0164 -.0002 .0125 .0220
. Oll~)
.0095 .0079 .0067 .0058 .00'.7 .0037 .0025
- .00l.1
.0084 .0168 .0222 .0253
.0266 .0269 .0264 .0254 • 02l~ 1
.00l)
.0211 .0181 .0153 .0127 .0101 .0077 .0051 .0026
()
0
..
.0290
.0)40 .0372
.0391 .0399 .0392
.0364 .0324 .0277
.0225 .0L7l .0115
.00S8 0
-.5000 -.4357
-.5000 - .4471
-.3723
-.3935
-.4159
-.3US
-.3413 -.29p
- .3723
-.2574 -.2074 - .1630 -.1239 -.0900 -.0610 -.0364 -.01.58 .0012 .0150 .0260 .0345 .0410 .0457 .0/,89
.0515 .0504 .0466 .0/.10 .0341 .0Ul) .0178 .0090
-.3290 -.2865 -.21.54 -.2060 -.1687 -.1336 -.1010 -.0709 -.0433 -.0184 .0040
-.2446
-.2012 -.1615 -.125~
-.0931 -.0644 -.0391 -.0170
.0019 .0181 .0316
. 0237
.0516 .0585 .0670 .0696 .0674 .0614 .0524
.0409 .0555 .0678 .0854 .0943 .0957 .0902 .0791.
.0427
,01d2
.06]/.
. 02 In .01114
.0/442 .0227
0
0
0
.30
.50
- .1774
-.1416 -.1071
-.0743 ~. Ot,35 -.0148 . .0116 .0355 .0568 .0753 . 10t~ 1 .1216 .128L .1244 .1114 .0908 .06'.0 .0331 0
..
TABLE 12.29
.,
CQt .02
LrfLc • 1. 000
.03
.OS
.10
.1669 .6531 .4076
.6485 .5641 .3797
.3726
.1882
.2039
.OqOl -.0704 -.1490
.0732 - .0308 -.1092
.5185, .4622 .3345 .2051 .0993 .0093
-.1855 -.1975 -.2053 -.2016 -.1859 -'.1724
".1536 -.1765 -.1910 -.1933 -.181.1 -.1731
- .1611
- .1631 - .1471 -.1298 - .1160 -.1011
.20
x-a 1.00
3.00
.0872 .0829
.0343 .0327 .0288 .0239 .0185
~o
0 5 10 15 20 "'25 : 30 35 40 ·45 ~ t 50 ~~ 55 ~~ 60 .~ 65
!.~"<.
70 75
80 85 90 100 110
120
DO 140 150 160 110 180
-.1437 - .1254 -.1127 -.0987 -.0808 -.0549 -.0243 -,0020 .0230 .0415 .0573
-.0837 -.0558 -.0257 -.0021 .0224 .04B .0572
.0703 .0753 .0803
.0755 .0796
.0697
.3404 .2650 .1831 .1094 .• 0421
-.0633 -.1116
·.0162
-.1427 -.1642 -.1740
-.0943
- .1731
-.1683 -.1608 ".1482 - .1335 ~.U98
-.1049
-.0882
~.0609
- .1199
-.1363 -.1441 -.1468 -.1454 ".1390 -.1295 - .1189 -.1064 -.0921
-.0587
<0637
-.0285 -.0013
".0342
.0211 .0406 .0561 .0Ml') .0752 .0787
.0172 .0378 .0544
.. ,0076
.0669
.0739 .0769
.2574 .2393 .1959 .1462 .0982 .0521 .0099 -.0254 - .0543 -.0780 -.9560 - .1071 - .1143 -.1176 -.1166 - .112S -.1063 •• 0979 -.0874
-.0642 -.0383 -.0111 .0109
.0317 • 04tJ7 .061l, .0689 .0717
TABLE 12.30
12-84
.2025 .1897 .1587 .1221 .0859 .0501 .0165 -.0124 -.0370 -.0579 -.0742 -.0858 -.0939 -.0985 ~.O996
-.0977 -.0938 -.0877 -.0797 -.0605 -.0379 -.0l'H .0072 . 0269 .04)/• .0557 .0631 .0659
.1453 .1371 .1170 .0927 .0678 .0428 .0187 -.0029 -.0218 -.0383 -.0518 - .0620 -.0696 -.0747 -.0770 -.0770 -.0751 -.0714 -.0659 ... 0517 -.0341 -.0]1)1 .0036 .0207 .0),1)2 .Of162
.0529
.0554
.0719
.0585 .0443 .0296 .0152 .0020 -.0100 -.0208 -.0299 ~.0371
-.0428 -.0469 -.0493 -.0502
-.0497 -.0480 - .0450 -.0364 -.0250 ... 0121 -.0009 .0131 .0237 .0318 .0368 .0386
.0129
.0073 .0020 -.0029 -.0013 - .0112 -.0145
-.0171 -.0191 -.0203 -.0210 -.0211
-.0206 -.0196 -.0162 -.0115 ... O()(IO
-.O()02 .OOS/.. .0102 .0l:J9 .0163 .0171
STRUCTURAL DESIGN MANUAL
•
¢
O?
X -.0
Lr/Lc "" . :lOO
Cmm 7
.03
l.00
3.00 .5000 .4448 .3915 .3411 .2940
.1450 .1056
.5000 .4262 .3574 .2958 .2414 .1937 .1524
.0440
.0740
.1.169
.0240 .0094 -.0010 -.00tJ4 -.0139
.0486
.08M
.0285 .01'28 .0004 -.0093 -.0166 -.0221 -.0263 -.0293 -.0312 -.0324 -.0330 - .0317 -.0291 -.0256 -.0212 -.0164
.0604 .0384 .0197 .0040 - .0091 -.0198 -.0286 -.0356 ... 0410
.05
.10
.20
.30
.50
.5000 .3026 .1581 .0745 .0285 .0036
.soon
.5000
.3411 .214)
.3737 .2665 .1841 .1224 .0771 .MS5 .0239 ,00S9 -.0012 - .0074 - .0114 -.0142 -.0156 -.0162 -.0167
.5000 ,3900 .2937 .2160 .1545 .1067 .0707
.SOOO .4075 .3241
0
0
.5000
5
.2439
15
.0833 .0195 -.0024
to
20
2S 30 35 40
-.0119
1~5
- .00/+2
50
-.001 S -.0008
55 60 6')
70 75 80 85 90
100 110
120
DO 140 150 160 170 130
-.0098 -.001,0 - .0031
-.0027
- .0022 -.0007 -.0{)17 -.0026 -.0013
-.0010 -.0019 -.0014 -.0009 -.0016 -,0002 - .0013
.5000 .270B .1159 .0409 .0075 -.0080 -.0100 -.0067 -.0060 -.0056 -.0032 -.0023 -.0034 -.0010 -.0019 -.0026 -. DOn - .0022
-.0075 -.0083 -.0081 -.0062 -.0052 -.0055 -.00 /19
-.0041 -.0044 ".O()l,6
-.0020 -.0025
- .0040 -.0038 -.0039
-.0021
-.0035
- .OOl6
-.0030 -.0029 -.0019 -.00l9 - .0009 -.0007
-.0020
- .0009 -.0014
,0000
-.0004
-.OOOS
·.OOOS
0
-.0059
0
0
.1278 ,0704 .0334 .0121 .OOOH -,0059 -. DO9/.
- .0098 -.OlOO
... 0102 -.0098 -.0092 -.0091 -.0090 - .0OSl. -.0080 - .0077 -.0070 -.0061 -.0055 - .0043 -.0035 ... 0022 -.001~
0
-.0168
-.0165 -.OUll
-.0152 -.0140 - .0124
-.OL07 - .0087 -.0068 - .0045 -.0023 0
•• OLlS
·.0199 -.0215 -. 0225 -.0228 -.0228 -.0221 -.0205 -.0184 -.0160 - .0131 -.0101 -.0068 -.0035 0
.2532 • t.939
-.OLll
-,0056 0
-.~.so
-.0497 -.0504 -.048 L
".0435 -.0369 -.0289 -.0198 -.0101
.2500 .2094 .1721 .1378 .1065 .0781 .0525 .0296 • C09!.! -.0088 -.0245 -.0380 -.0494 -.0587. -.0718 -.0781 -.0783 -.0733 - .0640 - .0511 -.0356 -.0182 0
0
TABLE 12.31
.,
Lr/Lc - .200
C.cIm
.02
.0)
.05
.10
.20
.30
.50
X• 0 1.00
3.00
¢o
0 5
10 15 20 25 30
)
-3.1704 -2.4933 -1.1966 -.3902 - . 1669 -.0411 .0716
35
.0383
40 45 50 55
-.0180
60 65
-.0146
70 75 80 85
90 100 110 120 130 140 150 160 170 180
.Dln .OJ06
-.0160 .0216 .0040 -.0206 .0055
.0169 - .0113 .0159 - .011,7
.01:19 - .00711 .0038 .OOlf4 ... 0069
.0128 -.0114
-2.8075 -2.2873 .1. 2592 -.5468 -.2639 - .0934 .0299 .0279 -. 00'.4 .0194 .0266 -.0076 -.0082 .0156 .0032 •. 0138
.0040 .0119 -.0073 .0114 ·.0094 ,010) -. OO/~2
• DOH .00'.. 0 -.00J7 .001)9 -.{lO67
-2.3889 -2.0239 -1. 2758 -.6947 -.3905 -.1671
- .04)] -.0081 -.0080 .0150 .0218
.0006 -.0011 .0123 .0038 -.0073 .0034 .0081 - .0037
.ooso
- .0045 .0078 -.0009 .00/,0 .00114
-.000) .0081 -.0020
-1.8961 -1.6779 -1.2111 ... 8013
-.5306 -.3259 ". t 740 -.0964 -.0566 -.0196
-.0010 - .0035 -.0002
.00B5 .0048
-.0008 .0045
.0067 .0007 .0067 .0007 .0074 .0034 .0063 .0068 .0046 .0090 .0039
-1.4903 -1.3640 -1.0840 -.8144 - .6077 -.4355 -.2967 -.2048
-.1421 - .0902 -.0357' ... 0386 -.0237 -.0102 -.0056 .. ,0041 .0016 ,0048
-1. 2916 .. 1.2011 -.9967 -.7911 -.6230 -.4766 -.3532 -.2636 - .1967 -,1405 -.0995 -.0733
-.0514 -.0328 -.0222 -.0153 -.0069 -.OOll
.0032
.0007
.0082
.0080
.006l.
.0094 .0141 .0147 .0171 .0184
.0107 .0095 .0117
.0125 • OL 18
.0143 .Olll}
.0185 .0204 .0189
-1. 0199. -1. 0209 ... 8853 -.7425 -.6179
-.5041 - .4035 -.3239 -.2594 -. 20J/~
-.1588 -.1254 - .0969 -.0725 -,0548 -.0409 -.0277 -.0173 -.0103
.00]3 .0108 .0185 .0224 .0265 .0292 .0)06 .0)25 .OJ19
- .8569 - .824; -.7483 -.6638 ... 5848 -.5088 -.4379 -.3768 -.3232 -.2744 -.2320 ·.1963 -.1642 -.1354 -.1112 - .0902' -.0707 -.0537 -.0395 - ,0142 .0047 .0208 .0325 .0422 ..0493
.0540 .OS73 .0577
-.6374 -.6249 -.5946 -.5588 -.5222 -.4844 - .4464 -.4103 -.3755 -.3414 -;3089 -.2782 -.2484 -.2198 -,1927 -.1669 -.1420 - .1184 -.0961 -.0546 -.0180 .0144 .0417 .0645 .0823 .0950 .1028 .1052
TABLE 12.32 12-85
STRUCTURAL DESIGN MANUAL Lr/Lc .... 200
l:fm
.03
.02
'1 _. ¢o
0
a
0
-13 .8275 -13.4182 -5.0301 -1.2709 -L6945
5 10
15 20
25 )0
-.S018 .9398
)5 40
.0709 -.5668 .3487
45 50 55 60 65
-. 17t:. 5
70
is
.4'.n - .0336
80 85 90
-.3882 .1905 .2702
100
-.1285 -.0190 .1433 -.2J29 .2719 -. 2Q 19 .2038 -.lU6
lLO l20
DO 140 150
160 110 L80
-4.80.30
·3.8036
-2.6162 -2.0901 -1.0167 -.0102
-2.5961
... 1329
-2.1072 -1. 3014 -.5600 - .-4314
-.2806 .1546 .2072 -.1525 ... 0568
-.3669 -.0503 .0377 -.1092 - .0413
.20 0 -2.6709
-3.3895 -2.7011 -2.1198 -1.8211 -1.3210
.30 0 -1.9239
-2.5165 -2.1287 -1. 7732 -1. 5156 -1.2216
-.2714
-.8600 -.7012 -.5709 -.3686
-;1590
-.2586
- .1811 -.1087
-.2434 -.1697 -. O~J7 -.0884 -.0927 -.0)58
-.8293 -.6506 ... 5163
.50
X- 0
-1.0354 - .8049
-.6476
-.6886
-.5867 -.5276 - ./1426 -.3834
-1. 3523 -1.2492
-.5851 ... 43&8
-.3441 -.3098 -.1550 - .1421
-.2123
-.1759 -.1534
-.01/.1
-.0946 -.0732 -.0650
-.OlBB
... 04J7
- .0080
-,2663
-.1613
-.0030 -.0803
- .0llO -.0427 -.0688
.1292
.0794 .1126 -.0')6 t -.0094
.ot114
.0012
.0575 -.0101 -.0070
.011.9 -.Ol)()
-.01_88
.O~B)
.0272
-.00'.10
-.0990
-.0526 .0556
-.0097 .0086 -.0309 .0244 -.0320
-.0257 - .0340 -.0061 ... 0242 .0005 -.0090
.1"394 -.0766 0
.1111 ".1108 .08').1
-.0578 .0/123 ... 02 /15
-.0472
0
0
0
.019/.. - • () 13/..
- • .3523
-.3042 -.25)7 -.2321
-.2425 -. D02
.0979
.1858 -.1791
~.6955
-.9755 -.9369 -.8830 ... 8539 -.7591
.1982
.0974 -.1599
1.00 0
0 -1. 2578 -1. 7000 .1. 5289
.309) - .0216 .18/t l • - ,089 1 - .0134
0
-8.358]
-2.0970 -1.8835 -.7044 .5070 .0269 -.3691
.3t85
a -4.5151 -5.4556
-7.5756
-.2741 - .lt27
.. 10
0
-10.7039 -11.0342 -5.1699
. ~645
.43L9 •. 4221
.05
-.0272
.0114 -.0250 .0106 -. OWL 0
... L21.it,
-.0999 -.0756 -.0667 -.0412 -.0]98 -.0178
- .0134 0
0
3.00 0
-.2691 -.3975
-.4161 - .4253 -.4387 •. 4259 - .4038
-.3951 -.3839 -.3613 -.3/.51 -.3361 ,
-.3197 -.JOI0 -.2906 -.2796 -.2621 -.2483 -.22!19
-.197a -.1700 -.1455 - .1141
- .0895 - .0573 -.030) 0
TABLE 12.33 LrJLc •
Cf'lm
7
01
03
rl- -97.1290
°
-14.4364
.. 51. 6332
5
-41. 7926
-34.0444
-25.4518
to 15 20
40.2106 42.0065
26.5155 31. 0848
14.7833 20.2264
2.8684
5.1413
25 30 35 40
-.5785
2.1281
11. 7559
9.687,1
1.7724 -5,0669
45 50 55
1.2469 -7.9488 2.0650 5.0499 -4.3005
5.4306 3.5464 7.7161 2.3110 -2.3947
60
-2.9t:.26
65
70 75 80 85
90 100 110 ]20 1]0
11.0 l~O
160
liD 180
"
1.4656
3.4313 -3.0045 -2.0933 3.0818 .2655 -3.0739
4.5897 .4406
-4.4378 1. 2554
.tJ245 2.2)16 -1. 8694
3.29 )9 -2.6982 3.1751
2.1480
-2. 'jO/l7 1. 7';1)8 -1.2'>20
.). J2H6
2.6050 -1.7893 .4320 .6222 - L.ti L51 2. )/+6/. -2.7tl!i8
~
-30 . .3642 -16.2837
.20
-12.2413
-7.6356
-4.1889
-1.2309 .900S 3.1130
-4.7779
-2.6374 -.0799 .7197
5.9408
1.9823
10.4161 3.9087 3.3068 5.3971 2.4197 ".28)3 1.2770 1. 5479
4.9552
.5901 1. 0331 1. 0236
.2249 1.6913 .8248 1.0069 1.6068 1.0773 .5280 .7994 .7873
.1653 ,1126
.2073
.5471
.1.486
.1709 -.2426 .1218 .2394 -.2145
.2022 -.0658
1.46]) 1. 6092
3.4141.
2.4917
1.5390
-1.9354
- ~ 5619 .9697 .0524 ·1.0138
L.9284 .4982 1.1823 1.2076 -.0276 -.0555 .6422 .1224 - .4569
,1.')88
.1911
.1172
1.3Jl2 -1.1800 1. 2f:1l.9 -1.M.47
.6320 -.6454
.3181 -,3409
-1.6179 -1.3235 1. 8342 .1117
- .6911
2.2096
,6090
,2751
.1787
-.7765
- • ](n'j
. 492()
- .4'300 .2079
·,7')96 .1]62
- .41172
-.2CJ69
- .20)]
.1170
.2573"
.1)56
.1905 -.0965 .1178 -.1.879
.02!)I.
-.0266
... OMI7
• Jr:l99 .1. 2(,79
.21)]
.O()B4
- .()O7l
-.(1)2')
- • O)O{~
-.80A7
-,11512
-.2(}~O
1. 5815 -1.9078
.9]88
.4J73 - .6';10
.1790 -.3698
-.1 '>77 .0214
.. 1.2014
-.2031 .0912 -.215H
l.OMJ9 "
1.00
-9.8887
2.2199
1. 2581 2.2455
50
-17.2652
...
TABLE 12.34 12-86
.30
.0St,) - , It.S8 -.0547
.2718
'"t
X .. 0
200
10
05
- . 1cJC)6
.3489
.4945 .8383
.6052 .3514 ,4997 .4990 .2332 .2037 .3113 .1849 .0406 .D09 .1479 -.0060 .. 0826 -.Ogb4
3.00 -1.4893 -.9661 -.0984 .1872
.0828 .1494 .2800 .2152 .1408 .1.979
.2028 .1173
.1088 .11+66
,1012 .0508 .0780 .0802 .0251 .0464 -.OlS/.
.0220
.OO~8
-.Og83 -.0,}O4
... 0.11,1
- .05)7 -.1113
- . O'j"~J -.0562
-.0276
-.02.)7
-.0292
-. 135 t
-.0671
STRUCTURAL DESIGN MANUAL Cmm 7
~Oo_ 5
10 15 20
25 30 35
.SOOO
.4331 • 369ft .3105 .2569 .2088
.5000 .447S .3968 •.3480 .3015 .2517
.0231 -.0038 - .0138
.0512
.0982
.1 t,98
.1798
.0149 - .001.2
.0522
.1284
.2154 .lM7
.0875
.1220
·.0149
-.0124
.0991 .0609 .0329 .0125 - .0015 -.0108 4.0166 -.0202 -.0218 ~ .0223 - .0221 -.0215 -.0205 ... 0194 ... 0172
.0)55
• U8b 7
-.0140 -.0119 -.0085 -.0062 •. 0053
-.0041 -.0029 -.0028 -.0029 -.0023 - .0021 -.0023 -.0020 ... 0017 -.0018 - .0011 -.0012 -.0005
- .0017
120
130
-.001)
140
-.0005 -.0010 -.0002 -.0004 0
Lao
.5000
.4183
.1627
- .0021 - .00 l4 -.0012 -.0016 - .001'~ -.0010
160 170
.5000 .ft043
.1099
-.0050 - .0032 -.0032 -.0024 -.0013
150
.5000 .3914 .-2946 .2144
.O73l
.1294
50
90 100 110
3.00
.5000
-.00~4
85
1.00
.5000 .3334 .2005
.0469 .0063 -.0127
.2504
.0219 .0031
... 0156
-.0084
-.0156 - .0134 -.0112 -.0096 - .0079 -.0063 -.0056 -.0051 -.0043 -.0040
-.0 145 -.0168 -.0172 -.0167 -.0155 -.0139 -.0126 - .0115 -.0102 -.0093 -.0080 -.0069 -.0060 -.0052 -.0042 - .0033 - .0022
~.O037
-.0033 -.0029 ... 002 7 -.0020 -.0018 -.0010
~.0004
-.0006
0
0
-.0011 0
a
.50
.5000 .3065 .1612
.3651
I:
.30
.5000 .2831
1~5
70 75 80
X .20
.05
40
65
.400 .10
.03
-.0162 ·.0130 - .0107
55 60
Lr/Lc •
.02
-.0149
- .0128 - .0108 -.00B7
-.0066 - .0041.
-.0022 0
.3172
.2425
.3422 .2745
.0576
.1660 .1285 .0957
- .0276 ... 0270 - .0251 -.0225 -.0196 - .0165 - .0133 ... 0101 -.0068 -.0034 0
.0340 .0152 .0004 ... 0111 -.0199 -.0263 -.0310 -.0342 -.0361 - .0371 -.0370 -.0349 -.0314 - .0272 -.0223 - .0170 -.0115 -.0058 0
.0673 .0430 .0223 .00'.9 -.0096 -.0216 -.0312 -.0339 -.0448 - .0491 -.0538 -.0543 -.0515 -.0463 -,0391 -.0305 ... 0208 - .0106 0
.:30
.50
.0307 .0121 ~.OO16
-.0114 -.0183 -.0226 -.0255 _ .0271 -.0277
.2168 .1188 .1436 .1114 .0820 .0554 .0315 .0101 - .0087 -.0250 -.0391 -.0510 -.0608 • ,0744 -.0809 -.0812 -.0760 -.0663 -.0529 -.0368 -.0189
°
TABLE 12.35 I....s:1 L c • .400
"'!11m
7
.02
.03
,OS
.10
.20
X• 0 1.00
3.00
0
r--~-
0
5 10 15
10 25
)0 35
.0193
40
.0383 .0192
45
.0354
so 55
60 6)
70 7S 80 85 90 100 110 120 130
•
.. 2.6379 -2.2001 -1.3126 -.6457 -. )231 ... 1184
.036l .0052 .0005 .0158 .0045 -.0096 .00]1.
.0091 -.0057 .0085 - .0073 .0071 -.0034
140 150
.0025
160 170
- • ()rHO
Ino
-.00')4
-2.3318 . -2.0020 -1.3136 ... 7503
- .4248 -.2000 ... 0455 .Ob60 .0142 .0345 .0374
.Ot53 .0097 .0173
.0075 ».0037 .0039 .0072 -.00)2 .0062 -.0044
-1. 2703 ... 8313 -.5332 ... 3072 -.11.18 ... 0581 -.0179 .0158 .0290 .0205 .0180 .0215 .0135 .0046 .0074 .0080 .0007
.0053 -.0014
.0059
,00',9.
-.OOts
.0006 .0033
.0026 .0029
.0028 .007]
.. 1. 9876
.1. 7604
-.0010 .OOCJO - .0nUl
-r:r--
.OO·i6 .OOV~
.0057
. now)
-1.5919 -1.4581 -1.1581 -.8616 -.6270 -.430) -.2725 -.1679 ... 0985 ... 0447 -.0120 .0010 .0104 .0181 .0168 .0129 .01 )9
-1. 2712 -1.1942 -1.0157 -.8254 -.6578
.0131
.0127
.0082
.0122
-.5060 -.3744 -.2731 -.1949 - .1308 -.0838 -.0527 -.0290 ·.0108 -.0010 .0045 .0099
.0084
.0138
.0038 .0064 .0042 .0058 -;0062 .0053
.0116 .0126 .0113 .0120 .0124 .0121
.0076 .00')0
.0133 .0121
... 4145
-.9488 -.9126 - .8259 -.7260 -.6289 .. ,5340 -.4450
-.3223
-. )677
-.2468
-.3006 ".2410
Ml.l1S1 -1. 0598
-.9295 ':.7854
-.6523 ... 5271
-.1826
- .1320
- .1908
".0946
-.1500 - .1153 -.0860 -.0630 -.0446
-.0644 -.040) -.0239
-.012) -.002) .0045 .0080
-.0288
.0176 .017S .0185 .0189 .0189 .0199
-.0163 -.0071 .0077 .0161 .0227 .0262 .0292 .0311 .OJ21 .0332
.0191
.O·~·IO
.0142 .OL54
-.7736
-.7534
... 601) -.5933 -.5728 -.5465
".7040 -.6444 .... 5173 -.5832 ,,',4S.s6 -.5206 ;. ...41.$24 - .4592 ,,~l("~92 -.4024 :"·...:·3861 -.3499 ~ ':'::'3530 -.3009 -.2568 -.3207 -.2178 -.2895 -.1825 .b·....; •• 2591 , ... 2296 - .1508 -.2014 -.1232 -.0989 -.1744 ·.1484 -.0771 -.1237 -.0579 -.1003 -.0416 -.0569 - .0137 -.0186 .0072 .0239 .01S0 .04)5 .0361 .0670 .0457 .08S) .0526 .0984 .0572 .0601 .1064 y:...Q~18
.l089
t
TABLE 12.36
12-87
STRUCTURAL DESIGN MANUAL Lr/Lc '" .400
(;fm
.02
'1 0 0 5
.03
.05
.20
.30
.50
0 -1.6469
0 -1.1811 -1. 6663 -1. 5926 -1.4739 -1. 3838 -1.1780 -.9441
0 -.7199
-1.1281 -1.13J2 -L0985 -1.0684 -.9571 -.8189
-.7992
... 7263
-.664')
-.6337 -.5161 -.4269 -.3696 -.2993
.10
X ., 0 1.00
3.00
0
0
0
-6.B7S1
0
-9.0612 -9.8022
;'7.8404
·4.7739 -5.77JB
15 20 25
-S.3140
-4.8966
-4.1404
-2.6485 -2.1029
-2.9189
30
-.8884 .25.31
-2.26B2 -1.1654 -.1602
-2.8804 -2.3070 -1.4174
.0117 -.2367
-.2195
10
3S 40 45
SO 55 60 65
- .1559
.2599 .2962 -.1686
.1735
.2245 -.0826
-.0594
0
-2.8376 -3.670]
-3.024] -2.4261 -2.0737 -1. 5025
-.9)8b -.6975
•• ,5935 - .4110 -.3144 .0174 .l07S - .0458
·.0118 .1928
.0125
.0112
-.S14b -.2354
-2.2499 -2.0515 -1.8161
-1.6516 -1.3381 -1. 0014 -.8119 -.6462 - .4334
- .1022 - .1148 -.0405
-.2996
.0539 .0154 .018~
-.0736 -.0617 -.0552
-.2474 - .1009
70
.2S23
75 80
·.0068 -.1984 .1005
-.1243
.0298 -.0566
.0734
.0585
.0429
... 0069
.lt~ 16
.0985
.0694
.0t.88
.0082 ... 001.4 .0027 .0097
85 90 100 110 120 130 140 150
Ibn 1.70 l80
.1413
-
-.06B1
-.0462
-.0248
... Q()26
-. OLOl
... 0075
.0028
.01112 -.1218
,0494
- .0044 .0290 - .0514
-.OB29 -.O()1.9
.0569 ... 0;71
.0717 •. 0396 0
-.OZ/d 0
.0954
.141.7 -. 1370 .1O(}4 -.O'it\4 0
.0159 -.0266 .0273 -.0302. .0206
.0431
-.0126 .0134 ... Ot(lO
.0095
-.0070 0
-.0128 0
-.4952 -.3717 -.3154 -.2321 - .1507
-.1226 - .1011 ·.0545 -.0327 -.0243
-.0097 .0004 ... 0118
.0069 -.OlLO
0
-.1762 -.2783 -.3195 -.3495
-.6548 - .6955
-.7097 -.7191 ... 6809 -.6230
-.5822 -.5365
-.3752
-.3822 -.3796 -.379!J
-.4233
-.3158 -.36)6 -.3527
-.4741
-.2308 - .1953
- .3857
-.3441
-.3400 -.2940
-.3307
4.2639
-.3037 -.2913
-.3154
-.1652 -.1216
-.2362 -.2019
- .09'.7 - .0673 -. Ol~20
- .li6) -. LJ96
-.2)48
- .1067
-.2062
-.0244 -.0238 - .0()70
... 0803
-.1771
- .Or,t.8
-.1497 ... 1187 -.0911 - .0596 -.0 }O6
0
- .OL4b -.0012 -.00')2 0
.30
.50
.00';5 ... 00')1
0
- .4370
... 0/.34 - .0 }J8
....0190 -.Ollb 0
-.2755 -.2613
0
TABLE 12.37
.,
~Qm
.02
.03
.05
Lr/Lc - .400 .10
.20
X•
a
1.00
3.00
-2.4849 -1. 6754 -.3128 .1873 .0958 .2483
.. ,8858 -.6134
~o
10
19.1109
-46.3842 -23.7304 11. 5482
15
25.0324
17.4571
10.5683
20
6.1706 3.6888
5'.6958
4.3340 3.8175 5.8716
0
5
25 30
-62.1528 -30.0038
4.2100 7.5251
35
8.9015 2.2920
40
-3.4110
-1.5650
45
1.2677 2.6000 -2.3734 -1.7017
1.2710 1. 9523 -1. 5494 -1.1773
2.25145
1.4671
.DbO
-2.3966 .5871 1.6735 ·1.4478
1.0854 -1.0169
1.6180 -1. 77b) 1. 3239
·1.2274 .8695
50 S5
60 65 70 75 80 85
90 100 110 120 130
2.689~
-.7206
.0268
- .6211 .9048 -.OUO
.0)62 -.0429 .6225 .0807
-1. 6838
-1. 0745
-.5123
.3399
.1448 .5987 ... 6725 .5998
.06)4 .2682
1:0(3)
170
.2857 ·.9856 1. 18n
.1670 -.6922 .7766
180
-1.4738
-1.022lJ
lbO
.7956
1.3967 1,3423
-.6848
150
2.7776 -.0631
-18.2282 -10.6842 1.5118 4.9268 2.4420 2.5915 3.8263 2.3028
1.3908 1. 5795
-.9143 • l89 L
140
-31. 5480 -17.2080 5.5868
.1023
-
-.7822 .4944 -.4476 "
.0308 .0697
-10.2682 -6.3692 .0654 2.0886 1.1474 1.4597
2.2428 1. 5709 .8480 1.1408 1. 0762
.3662 .2650 .5377
.2086 ~
.1398 .1067
-4.6437 -3.0428
-.2490 1.2067 .6816 .9794 1.5726
-.3649 .5730 .3233 .5626 .9705
1.1743 .7225 ,.9323
.7718 .5297
,8906 .4083 .3257
.6746 .6592
.3715 .J180·
.4S89 .2485 -.OOSO .1396
.2520 .0867
.1670
.1620
-.0041 .0693 -.1076 -.0007
.4077
.1596
- • .1869
.1751 - .1809
.2455 - .4/.89
.1013
·.0869 .0156
-.2648
... lH61
.0544 .. ,18)5
.0162 -. tMifl
-.0600 -.0481
-. OCJ9l
".O(,lJl
- .0(,24
... 0697
- .1784
-.1111)8
- .1248
.0014 -,1829
-.035J
.2001 -.2109 -.0268 ... 0059
... 4515 .4396
-.2(J81
.1a 15
.0481
•. 6521 -
-.3690
-,2280
TABLE 12.38
12-88
-7.2619
-4.6271
-.1)01
- .1/,6,)
-.1526 .0233 .0031 . OM 7
.4823
.1530
.4086 .3081 .3963
.1379 .1128 .1492
.3995 ,2621
.1559 .114)
.2380
.Lun
.2824 .2015
.1256
.1136 .L435 .1372 .0/.60 .0658
-.0)88
.0004 -.0663 -.05).1
-.0625 -.0950 -.0530 -.1096
.0992 .0694
.0781 .0739
.0409 .0411 -.0010 .00l,6 •. O~46 w. n271 -.0)46 ... 04')1
- .0374 ... 0571
•
STRUCTURAL DESIGN MANUAL Cmm
Lr/'Le • 1.000 .10 .05
.02
.03
0
.5000
5 10
.5000 .3502 .2258
30
.3314 .1968 .1049 .0454 .0089 -,0099
35
·.0 l71)
40 4S 50 55 60 65
-.0199 •. 0191 ·.0160 -.0129 - .0105 -.OOBI -.0059 -.0048 ' - .0040 -.0029 -.0023 -.0019 -.0015 - .0012 -.0012 - .000 7 -.0008
'1 ~.,o
15 20
25
70
15 80 85 90 100
.0216
.0014
~.0191
-.0121, - .0196 -.0225 -.0229 -.0219 •• 0199 - .0173
- .0213 ~.0202
-.0180 -.0156 -.0129 -.0102 -.0033
180
0
0
140 150 160
.1054
-.0]22
- .0003 -.0003
130
.2598 .1721
.00t?
170
120
.3713
.0566 .0231
-.0068 -.0053 -.0043 -.0031 -.0024 - .001,8 -.00l6 - .0011 -.0010 -.0006 -.0004
no
•
.1345 .0705
.5000
•. 0150 - .0128
-.OLD7 -.0089 -.0064 -.0046 -.0034 ... 0027 -.0020 -.0016 ~ .0010 -.0005 0
.5000 .3953 .3007 .2207 .1549 .1022
.0616 .0312 .0090 -.0066 -.0167 •• 0229 -.0263 -.0274 - .0271 -.0259 -.0241 - .0219 - .0196 - .0154
... 011B -.0088 ·.0066 -.0048 -.0034 - .0021 - .0011 0
X •
a
.30
.50
1.00
3.00
.soon
.SOOO
.4145 .3349 .2640 .2024 .1497 .10S9 .0701 .0412 .0184 .0010 - .0120 -.0214 ·.0278 -.0319 -.0341 -.0)49 -.0345 -.0334 -.0298 ... 0253
.4238 .3518 .2863
.5000 .4335 .3698 .3105
.5000 .4lt34 .3864 .3360
.2277
.2561
.2867
.5000 .4525 .4057 .3601 .3160
.1764
.2069
.1321
.1631
.0945 .0629 .0369 .0159 -.0009 -.0140 -.0239 ... 0310 -.0360 -.0392 ... 0408 -.0413 -.0397
.1244 .0907 .0616 .0369 .0160
.2407 .1983 .1595 .1243 .0926 .0644 .0395 .0177 -.0012
.20
-.0207 -.0164 -.0124 -.0089 -.0057 -.0028 0
-.0359 - .0309 -.0256 -.0202 -.0149 -.0098 -.0048 0
- .0013
- .0153 -.0266 -.0353 -.0419 -.0466 - .0497 -.0520 -.0504 -.0461 -.0400 -.0329 - .0251 -.0168 -.0085 0
- .0173
-.0)08 -.0420 -.0509 - .0519 -.0665 -.0692 ... 0670 -.0609 -.0519 -.0407 -.0280 -.0142 0
.2737 .2334 .1953 .1595 .1260 .0950 .0664 .0404
.0168 -.0042 -.0229 - .0391 -.0530 -.0645 -.0812 -.0898 -.0910 -.0859 -.0753 -.0604 -.0421 - .0216 0
TABLE 12.39 Ll"/L(' •
t;sm
,02
J
.03
.05
l. 000
.10
.20
.30
.50
X- 0 1.00
3.00
-.6519
".5460 -.5417 -.5302 -.5143 -.4953
0
_£0 5
10 15 20 25
)
-1. 7745 -1. 60S7 -1.2320
-1. 5144 -1.4006
-1.2227
-1.1416
- .874 /"
.0228 .026,>
.0044
.0291
.0183 .0280 .02B1i
.0100
-.0348
.0115
.0175 .015)
.0024
.. omn
.0254 .0251 .0229
.0058 - .0017 .0040 -.0006
.0077
.0172 .0136
-.0166 -.0019 .0088 .0157
.0018
.0021 .0022 .0006
30
- .D68
-.2164
35 40 '15
-.0499 -.0079 .0264 .0394 .0)01 .0264 .0286
- .1127 -.9500
.OL9)
.0262 .OUH
SO
85
90 100
no
120 130 1'10
150 160
-.9924 -.9549 -.861,5 -.7589 -.6547 -.5518 ... 4545
-1.0027 -.8328 -.6711 -.5318 -.4023 -.2961 -.2143 -.1444 -.0911 -.0535 -.0247 -.0028
-.3068
55
~ 1.1571
-.8768 -.6569 -.4662 -.30B3 -.1966 - .1176 -.0)59 -.0158
... 6047 -.3860
60 65 70 75 80
•
-2.0112 .. 1. 7824 -1. 2879 -,6430 - .5J85
.0092 .0109 .0105
.ooty
170
-.0006 .On37
180
-.OUlS
-.0023
.0325
.0011 ,0041 ,0006
.0067 .0065 .0032 .0035
.0234
.0260 .0250 .0236 .0181 .0152 .0111
,0034
.0021 .OOJS
.0092 .0077 .0063 .0066
.0000
.0017
.0054
. DO)/.
-.8826 -.8557 -.7900 -.7109
-.7680 -.7504 -.7065 -.6520
-.6300
... 5940
~.5468
.... 5479
-.5069 -.4653
-.3691
-.3951
-.2945 -.2283
... 3289
-.5332 ... 4722 - .4142 -.3594
.... 2684
... 3077
-.1726
-.2153 -.1701
-.2602 -.2177 ·,1790
-.1275
-.0897 -.0585
.0245
.0263 .0262 .0236 .0214 .0192 .0172 .Ot6()
.0157
-.4680
- .1306
.... 0966 -.06Mry - .0461 -.0268 - .0114 .0002 .0170 .0256 .0301 .0310 .0)08 .0298 .0286 .0281 .0275
- .1443 -.111~0
-.0876 -.OM2
-. 0441 -.0273 -.0004 .0179 .0305 .0)83 .0433
.0461 .0476 ,0484 .OM~4
... 6420 -,6165 -.5835
-.4238
- .3829 -.3427 -.3041
-.2675 -.2327 -.1998 -.1694 - . 1412 -.1149 -.0909 -.0691 -.0310 -.0002 .0246 .0438 .0585
.0692 .0764 .0807 .DRlO
-.4735
- .4495 -.4240 ... 3973 -.3696 -.3413
-.)128 -.2842 -.2556 -.2273 - .1996 -.1723" -.1451 -.1201
:
- .071 5 -.0273 .0120
.0458 .0740 .0962 .1121 .1218
,12';0
TABLE 12.40
12-89
STRUCTURAL DESIGN MANUAL Cfm .02
"1
.03
.05
Lr/Lr .. 1.000 . 10
X = 0
.20
. .10
.50
l.00
3.00
0
0 -.2186 -.3474
0 -.0957 - .1611 -.2009 -.2345 -.2645 -.2849 -.2992
..) 0
'--t-5 10
15 20 25 )0 35 40 45
50 5S
-4.2130 -2.9511
0 -3.5690 -4.5150 -3.5595 -2.7305
-2.3889
-2.2761
-1.4619 -.6217 - .4247 -.3L65 .0238 .lt92
-1. 5620 -.87 /,6
-1. ot~ 71
-.6219 - .4t141
-.7992 -.5978
-.1310
-.3234
-.0029 -.0587 .010) .1139
-.1729,
0 -4.8099 .) .8355
60 65
-.0314
.0276
-.0142
.0432
. olin
80
M.0448
- .0086
85
90
.0686 .0779
.0670 .0709
100
-.0191
-.0005
lLO
-.OOO'} .0)17
.0064 -
- .1473
.0·~2t
.1560
-.0111)5
-2.7701 -2. )228 -2.0355 -1. 5465
.02/,)
70
uo
-3.2111
-.0601 .0358 .01B8 .0001 .0544 .0610 .0182 .01.80 .0237
75
120
0
-2.4230
lf~O
.0583
150 160 170 180
-.0)61
.0393 -.0377
.0262 -.0217
.0417
.0292
- .02.>9 0
... 0161
.0[79 ... OO~H
U
0
0
0
• L4096 -1. 9708 -1.8701
-.8087 -1. t825 .. 1. 2076
-1.71b9
-1.1863
-1. 59~4 -1. J4£..4
-1.1629 -1.0521 -.9092 - .8078 -.7035
-L057) -.8787 -.7140
-.5094 -.3687 -.2955 -.1983 - .104:;
-.0752 -.0545
-.5725
- .4692 -.3979 ·.311,3 -.2335 ... 1873
0
·.5818 -.8704 -.9213 -.9349 -.9400 -.8801 ... 792 5 -.7270 -.6')47 -.5fl07 -.4822 -.4227
.0168
-.0647
.Ol60
-.OJO) -.0033
- .07t9 -.0)58
.0125 .OOM8 .0198 .0058 .0125 .0016
-.Oll7 ... 00'+ 7 .0086
.0060 .0228 M. OOli 5 .0127 -. DOll 0
0
-.4715
-.4781 -.471.3
-.6116 -.5712
-.5143 -.4637 -.4221 -.2851
-.1482 -.0973
- .4016 - .4402
-.6819 -.7056 - .6854 -.M41
-.3728 -.3226
•. 0039
.0275
-.5890
-.6487
... 3536 -.2856
-.2403 -.1999 -.1523 - .1176
.023:1
•• 3834
-.4660
- ..)123
- .4539
-.3215 -.3253 -.J275 -.3286 -.3261 -.3210 -.3158 -.3089 -.2993 -.8891 -.2665 -.2395 ... 2097 -.1784 - .1440 -.1098
-.4310
-.4mn
-.3888 -.3630
-.3)50 -.J119
-.2499
-.21:l86
- .2100 - .1778
-.2&20 ... 2]85
- .1285
- .1971 -.158l,
-.0871 -.0558
-.1245 -.0976 ... 0709
- .0318
-.0189
.0026
-.otld,
.om~)
.00l5 0
-.ootI2 -.O()'J4 0
.30
.50
-.0522 ..·.0]17
- .0732
-.Ol63 0
... OJ71
1.00
3,00
a
. TABLE 12.41
.,
L. r I LC •
(,;Qm
.02
.03
.OS
1. 000
.10
.20
x '"
0
~o
0
-31.7536
5 10
-17 .1813
IS 20 25 30 35
40 45
50 55 60 65 70
5.4898 10.55l9 4.)776
-23.1673 -13.2664
3.8961
2.7088 6.8034 3.1981 3.1934
5.9630
4.7197
2.8636 .0015 1.11429 1.6132 -.7033 -.61&1
2.6501
.9018
.6S~0
1.4980 1.4836 -.1903 -.21/.9 .7029 .0260
-8.6690 ·5.5108 -.264] 1.4750 .8392 1.1908
-4.7950
-3.3512
-3.1804
-2.2751 -.4507 .2270
1.8966
.9885
1.4126 .8644
.8125 .S8M
1. 1099
.7410 .7298
.6127
1.0S42 .4707 .3677 .5607
.1382
.2707
.3036
-lS.3880 -9.2513 .7776 J.7189 1. 9501 2.2354
3.3307 2.1608 .9600 1.4191 1.3352 .2395 .1258
... 4778
.4932
.2789
.lIn
.5517
.3318 .6535 .5626 .4335 .5534
.4416 .3841
170
.O:,HH .06% -.4)16 ./.:\98
IBn
-.6521
150 160
.26U4 -./dd6
~.O/t31
-.0238 -.2323 .1210 -.3120
-.0549 -,0753 -.0631
-.0341
-.0872
-.0401
.0127 .0727 -.1131•
-.O63l -.1654
-.0828
.0175
... ()IIb~)
-.20n
... 1>14
- .1369
TABLE 12.42
12-90
.0646
.00)2
-.0754
-.50'Jf, ,3512 -.5078
.n'
.0777
M
-.6825
• :l')'))
.2772
.0304 .0704
-.O'IL7
90
"')') 1~ -.00/1\ . 0\95 ·.3294
.1481 .1992 .2130 .1671 .16J5
~.()(d2
.1932 .1869
.492'] .. ,MllU
.1692
-.Db ) j
.129)
.1403 .1747 - .1275 .0736 -.2333 • () 11,1 - ,17~H - .CJ7S0
1)0 14()
.0109 .0534 .0543
- • ()l)t~~
-.ons
.0656
-.3917 2 ".21.21
-.0210
-.<1772 ... 0837 - .1243
~.3703
.0683 .3)48
.5932
-.0137
-.0264 .0593 .1769
-.08.H
-.7279
• L333 .5879
-.786'5
-.2345
.0611 ' .0155 .1654 .3788 .3428 .2815 .3670 .3804
-.3945 -.2841 -.0958 -.0197
- ,lIOI11
75
100
... 3"635
-.7945
-.01114 -.12,Ul
.4674
60 85
110 12n
.H80 .39)4 .2812 .1598 .1955
-1.1238
.2612 .2971 .2)01 .1'i5b .1738 .1620 .081) .0818 -.0170 .001 J
-.0213 -1.08')6
.2110 -.3270 . 1670
.5579 .3731
-2.1254 -1.4713
.1813
.0584 .0804 -.0(J09
- .066() -.11.14
- .lOS7 ... 01'16 ". l106
.0736 .0693
. tS43 .1524 .11 J6
.0)21 .0550
.L240 . Ll64
,0522
.0736 .0667 .0086 .UUH2
.0)73
•
(J'l'j)
.0329
.0l04 .llOb7 ~.OllO «
.0174
-.0249
-.OJlb
•
STRUCTURAL DESIGN MANUAL
• 12.4.4
Frame Analysis by the Dummy Load Method
The previous sections have dealt with ttcookbooktl methods for determining loads and moments in rings and frames. This section presents a method for analyzing any type of frame with any type of external loading and reaction system. Generally, a frame will be redundant by more than one degree. This is especially true when a symmetrical frame is cut along its axis of symmetry and one-half is analyzed. This situation has three degrees of redundancy. The degree of redundancy can be said to be the number of external reactions which must be removed to make the structure statically determinant.
A redundant structure cannot be analyzed by the simple equations of statics, i.e., :EF = 0 and EM;:: O. Additional equations are necessary and these generally involve the deformation of the structure. The expression for the deflection at any point of a structure may be expressed as:
•
<5 :::-
' Mmdx J EI
12.22
where M is the bending moment in the structure caused by the external loads. The EI is the elastic modulus and the moment of inertia of the structure and m is the bending moment at any section of the -structure caused by a "dummy" load of unity acting at the point of desired deflection and in the direction of desired deflection. Another deformation equation, for rotation, may be expressed as: Q
=
f
MIn'dx
EI
12.23
where M and EI are the same as for equation 12.22 and ro' is the bending moment at any section of the beam caused by a dummy moment of unity (1 in-lb) applied at any section where the change in slope is desired. When using equations 12.22 and 12.23, it should be noted, that though mmay be considered to be a bending moment, it is in fact a length in equation 12.22 and dimensionless in equation 12.23.
•
The dummy load method is described in S-ection 9 .. 3. 3 so only the procedure for solving a redundant frame will be presented here. Figure 12.43 shows a frame with symmetrical external loads reacted by shear in the side skins. The frame is symmetrical about the vertical center line. The procedure is as follows: 12-91
~
\~~~]~l' STRUCTURAL DESIGN MANUAL p
p
Figure 12.43 - Symmetrically Loaded Frame
1). Cut the frame about the centerline of symmetry. Apply the external loads and the balancing redundants as shown in Figure 12.43.
2). Cut the structure at approximately each 20° making a cut at each point of reaction or load application or change in cross section. Number the cuts.
•
3). Calculate the cross sectional area, moment of inertia and neutral axis location at each cut. 4). Calculate the shear, axial load and bending moments at each cut for the applied load acting separately and reacted at the bottom centerline. 5). Calculate the shear, axial load and bending moments at each cut for each of the redundants applied separately and reacted at the bottom centerline. The total strain energy, U, is determined from 2 2 2 U= M dx/2EI + P dx/2AE + V dx/2AG
.f
J
J
12.24
This expression contains the terms for bending, axial and shear strain energy_ Generally bending is considerably larger than axial and shear combined so the last two terms in the previous equation are ignored. If, however, the combined effects of axial and shear are in ~xcess of 10% of the total strain energy,
12-92
•
STRUCTURAL DESIGN MANUAL Revision A
consideration should be given to including them in the solution. In this procedure axial and shear will be ignored. It should be remembered that they can be included in a manner similar to bending. The term for the bending moment at any point is M
Where MO
Ml
=
M2 M3 =
=
moment moment moment moment
MO + x1M1 t x2~I2 + xSM3
due due due due
12 .. 25
to applied loads to xl to x2 to x3
By expanding the equation for strain energy and applying Castiglianots Theorem, au/ax 0, three equations with three unknowns are
obtained:
12.26
•
12.27
au
12.28
--=
6). Calculate the values for the previous three equations using the moments and section properties previously determined. 7). Solve equations 12.26, 12.27 and 12.28 for xl )
t
x2 and
xa-
8). The moments at each cut can be determined using equation 12025. A tabular form for the solution of a three-redundant frame is shown in Table 12.43.
12-93
::0
co
...... tv
I \D If,lo.
J-3
~
Beam
J,
E1
MO MI
~
M2
M3 k
ro
IkMII2J
i k
~lMI
k
EI
E1
rt
k
~lMJ E1
k
kM 2
2.
EI
k
kM~M.j k ~2MJ EI
EI
<:
. ~:
..
::l
)I
.
~
N
~i:-:::>\ J
r
~
~"~':',
/
...
,,-~. ; ~ --
.~
'
- ;/
.J" / '
"!:'~
.j::--
;-
W
I
t-j-
p:,
I
I
I
I.
I
CI)
I
......
~
::a c:
....... \J)
ti
c.n 0
#
~
!kM
C
rt
.
1-'-
g
r..
k
3
n
......
z,
kM3MOt
k
F~T
RT
'\lMl
X?M2
c:
MO + X1M1 + XZM Z + X3 M;
X~Ml
:=cJ
:J>
r-
Hl
0 Ii
1-3
ei
Ii
.."
::r'
I-
(I)
ro
I
-
CI)
I ~
en
I'D
z
0..
§
NOTE:
0.. p:,
::;
(1) Use equations 12.26, 12.27 and 12.28 to solve for xl, x2 and x3-
31:
(2) Reference section 9.3.3 for values of k.
rt
:J> Z
I-:rj
Ii Ol
3
c:
•
:J>
(D
r-
e
e
e
STRUCTURAL DESIGN MANUAL
•
SECTION 13
SANDWlCH ANALYSlS 13.0
GENERAL
Structural sandwich is a layered composite, formed by bonding two thin facings to a thick core. It is a type of stressed-skin construction in which the facings .resist nearly all of the applied edgewise (in-plane) loads and flatwise bending moments. The thin spaced facings provjde nearly a11 of the bending rigidity to the construction. The core spaces the facings and transmits shear between them so that they are effective about a corranon neutral axis. The core also provides most of the shear rigidity of the sandwich construction. By proper choice of materials for facings and core J constructions with high ratios of stiffness to weight can be achieved.
A basic design concept is to space strong, thin facings far enough apart to achieve a high ratio of stiffness to weight; the lightweight core that does this also provides the required resistance to shear and is strong enough to stabilize the facings to their desired configuration through a bonding media such as an adhesive layer, braze or weld. The sandwich is analogous to an I-beam in which the flanges carry direct compression and tension loads, as do the sandwich facings, and the web carries shear loads, as does the sandwich core.
•
In order that sandwich cores be lightweightt they are usually made of low density material, some type of cellular construction (honeycomb.like core formed of thin sheet material) or of corrugated sheet material. As a consequence of employing a lightweight core, design methods account for core shear deformation because of the low effective shear modulus of the core. The maln difference in design procedures for sandwich structural elements as compared to design procedures for homogeneous material is the inclusion of the effects of core shear properties on deflection, buckling and stress for the sandwich. Because thin facings can be used to carry loads in a sandwich, prevention of local failure under edgewise direct or flatwise bending loads is necessary just as prevention of local crippling of stringers is necessary in the design of sheet stringer construction. Modes of failure that may occur in sandwich under edge load are shown in Figure 13.1.
)
•
Shear crimping failure, as shown in Figure 13.1, appears to be a local mode of failure, but is actually a form of general overall buckling in which the wavelength of the buckles is very small because of low core shear modulus. The crimping of the sandwich occurs suddenly and usually causes the core to fail in shear at the crimp; it may also cause shear failure in the bond between the facing and core. Crimping may also occur in cases where the overall buckle begins to appear and then the crimp occurs suddenly because of severe local shear stresses at the ends of the overall buckle. As soon as the crimp appears~ the overall buckle may disappear. Therefore. although examination of the failed sandwich indicates crimping or shear instability, failure may have begun by overall buckling that finally caused crimping . If the core is of cellular (honeycomb) or corrugated material, it is possible for the facings to buckle or dimple into the spaces between core walls or corrugations 13-1
,,(~~
STRUCTURAL DESIGN MANUAL
• ~FACING ~
FACING-...... CORE
IJ
J
tttt
t ttt
A. - GENERAL BUCKLING
8. - SHEAR CRIMPING
• J
HONEYCOMB
~~
CORE
SEPARATION FROM
fI ..... ~.,.
CORE
~
I~
CORE
~ CRUSHING
tttt c, - DIMPLING
f ttt
't t t t
D, - WRINKL ING OF FACINGS
OF FACINGS
FIGURE 13.1 - POSSIBLE MODES OF FAILURE OF SANDWICH UNDER EDGEWISE COMPRESSION 13-2
STRUCTURAL DESIGN MANUAL
•
as shown in Figure 13.1. Dimpling may be severe enough so that permanent dimples remain afler removal of load and the amplitude of the dimples may be large ('nough Lo cause the dimples Lo grow across the cell walls and resuJ t In a wrLnkLillg of lhe facings. Wrinkling, as shown in Figure 13.1, may occur if a sandwich fac'ing sUbjected to edgewise compression buckles as a plate on an elastic foundation. The facing fnay buckle inward or outward depending on the flatwise compressive strength of the core relative to the flatwise tensile strength of the bond between the facing and core. If the bond between facing and core is strong, facings can wrinkle and cause tension failure in the core. Thus, the wrinkling load depends upon the elasticity and strength of the foundation system; namely~ the core and the bond between facing and core. Since the facing is never perfectly flat, the wrinkling load will also depend upon the initial eccentricity of the facing or original waviness. The local modes of failure may occur in sandwich panels under edgewise loads or normal loads. In addition to overall buckling and local modes of failure, sandwich is designed so that facings do not fail in tension, compression, shear or combined stresses due to edgewise loads or normal loads and cores and bonds do not fail in shear, flatwise tension or flatwise compression due to normal loads. The basic design principles can be summarized into four conditions as follows:
• )
(1)
Sandwich facings shall be at least thick enough to withstand chosen design slr~sses under design loads .
(2)
The core shall be thick enough and have sufficient shear rigidity and strength so that overall sandwich buckling, excessive deflection and shear failure will not occur under design loads.
(3)
The core shall have high enough modulus of elasticity and the sandwich shall have great enough flatwise tensile and compressive strength so that wrinkling of either facing will not occur under design loads.
(4)
If the core is cellular (honeycomb) or of corrugated material and dimpling of the faces is not permissible, the cell size or corrugation spacing shall be small enough so that dimpling of either facing into the core spaces will not occur under design loads.
The facing to core bond shall have sufficient flatwise tensile and shear strength to develop the core strength in the expected environment. The environment includes effects of temperature, water or moisture, corrosive atmosphere and fluids, fatigue, creep and any other condition that affects material properties. 13.1 13.1.1
•
Materials Facing Materials
The facings of a sandwich part serve many purposes, depending upon the application, but in all cases they carry the major applied loads. The stiffness, stabilitYt configuration and, to a large extent t the strength of the part are determined by the characteristics of the facings as stabilized by the core. To perform these functions the facings must be adequately bonded to a core of acceptable quality_ Facings sometimes have additional functions, such as providing a profile of proper 13-3
STRUCTURAL DESIGN MANUAL at!rodynamic smoothness, a Tough nonskid surface, or a tough wear-resistant floor covering. To better fulfill these special functions, one facing of a sandwich is sometimes made thicker or of slightly different construction than the other. Any,thin sheet material can serve as a sandwich facing. usually used are shown in Table 13.1. 13.1.2
A few of the materials
•
Core Materials
To permit an airframe sandwich construction to perform satisfactorily, the -core of the sandwich must have certain mechanical properties, thermal characteristics and dielectric properties under conditions of use and still conform to weight limitations. Cores of densities ranging from 1.6 to 23 pounds per cubic foot have found use in airframe sandwich, but the usual density ranges from 3 to 10 pounds per cubic foot.
)
A wide variety of core materials can be constructed of thin sheet materials or ribbons formed to honeycomb-like configurations. By varying the sheet material, sheet thickness, cell size and cell shape cores of a wide range in density and properties can be produced. Most honeycomb cores can be formed to moderate amounts of single curvature, but some can easily be formed to fairly severe single curvature and to moderate compound curvature. Honeycomb core cell size is determined by the diameter of a circle which can be inscribed in a cell. Two types of honeycomb core showing the cell size tis" are shown in Figure 13.2. Honeycomb core cell sizes used in airframes vary from about 1/16 to 7/16 inch, usually in multiples of 1/16 inch. Not all sheet matt!rials are formed to all of these cell sizes because some sheet materials are so- thick and stiff that they cannot be formed to core of cells less than 3/16 inch in size. For special. usc, such as an insert, honeycomb cores can be densified locally by underexpanding, crushing the core locally or by inserting a higher density core locally. Cores for airframe sandwich construction are presently being made 'of thin sheets of aluminum alloys, resin-treated glass fabric, resin-treated asbestos,_resin-lreated paper, stainless steel alloys, titanium alloys and refractory metals.
•
Honeycomb cores fabricated from nonmetallic materials have better thermal insulating characteristics than metallic honeycomb cores, even though both allow transmission of heat by radiation in the open cells. In considering thermal effects on sandwich slructure, it should be understood that the sandwich can act as a reflective thermal insulator. The effeclive thermal conductivity of a honeycomb core depends upon conduction of the material of which the core is made, radiation between facings and connection within the core cell and can be computed approximately as 13.1
where
13-4
Ke = effective
conduct ivity conductivity of core ribbon material Ko core solidity Ac = = Wc/Wo We = core density Wo = core ri bbon, material density (J = Stefan-Boltzmann constant
•
•
STRUCTURAL DESIGN MANUAL Revision A FACING
YIELD STRENGTH Ffrvpsi
MODULUS OF ELASTICITY EflVpsi
ALUMINUM: 1100-H;14 2024 .. T4 3003-H16 5052-H38 6061-T4 7075-16
6
17,000 47,000 25,000 37,000 21,000 73,000
10x10 6 10x10 6 10x10 6 10x10 6 lOx10 6 10x10
Af =1
-j;!
2
.89 .89 .89 .89 .89 .89
WEIGHT PER MIL THICKNESS 2 LBS/FT
.014 .014 .014 .014 .014 .014
\
•
.040
6 30x10 6 30x10
.94 .94
.040 .040
6 15xlO 6 16.8x10
.94 .94
.0235 .0230
50,000
30xlO
STAINLESS STEEL: 316 17-7
60,000 200,000
TITANIUM: Ann.Ti-75A H. T. 6A1·-4V
80,000 143,000
FIBERGLASS PRE-PREG: EPOXY F155 EPOXY F16l PHENOLIC FI20 POLYESTER F141
~
6
.91
MILD CARBON STEEL
6
63,000 62,000 48,000 48,000
3.5x10 6 3.5x10 6 J .. 5xlO 6 3.5xlO
.98 .98 .98 .98
.0095 .0088 .0094 .010
2,650 300
6 1.8x10 6 0.4xlO
.99 .99
.003 .004
)
PLYWOOl>: DOUGLAS FIR HARDBOARD
•
TABLE 13.1 - TYPICAL SANDWICH FACING MATERIALS
13-5
((1\:\\
"~~. STRUCTURAL DESIGN MANUAL to
•
AXES
• ~~~ AXES
FIGURE 13.2 - HONEYCOMB CORE NOTATION
13-6
•
STRUCTURAL DESIGN MANUAL
•
Revision A
L4 --r-
-
-
-
1--'
- -
1--
-- - - --
-
--f--
r
l2
~~ GI
H 0 -I-J
U
m r.....
LO
C 0
-..-I
.u
-
H
r----
0,0
!
-
I
I
•
II
,
.
I
..
I
I
H/ - + - C I •
I _~'ON·t.rl:~"
I
I
1.5
i-
!
I . ,'-
; I ------,--r--t---r.-.---!
1.0
=q=-.- -
- r~--t rr--- - -w-"-= --8-- -1-J----At,LIC It/\:
I-- "-r--
--
I-
-+ -1---1-- -- ---1-$
-.........t ~'lilJ' -~,..--,- ,\1
!
I
0.5
i
I
!
---o
I
!
1-
0.6
I
I
~c.-1t= -- -
I
I
I
r I "'~i ~l::~ ~--~-- - ~-o:::...:::'- 4-- - ---~-- - --
-
0
U
r
--
U aJ I-i
--+-j--'
I I I
f-
i
--t+ '.
I
2.0
Core Thickness - Inches
FIGURE 13.2a - SHEAR STRENGTH CORRECTION FACTORS
)
13-6a/13-6b
"., I,
•
••
,
,
\
\
\, \
~. :a;;Bell \'
,
•
.... u.~· .. "
STRUCTURAL DESIGN MANUAL
~.--
Revision F tc
core thickness 'I'm fTl<:an absolule temperalure of the two facings .£1 :: emissivity of inside of facing 1 -E.:2 = emissivity of inside of facing 2 F12 = geometric view factor between facings = connective heat transfer coefficient in:ide core cell 11 Sheets of corrugated metal foil are usually assembled with the corrugations parallel to form honeycomb cores. The foil may be perforated for use in core where solvents or gasses must be vented. Perforated foil in sandwich panels that are not sealed or are poorly sealed will allow penetration of moisture, etc., which may cause severe deterioration of the core. If the sheets are assembled with the corrugations in adjacent sheets perpendicular to each other, a well vented crossbanded core is produced. Crossbanded cores may be cut so that the corrugation flutes are at an angle 0 of 45 to the sandwich facings, giving the core a trussed appearance. Crossbanded cores are not as strong in compressions in the T direction or in shear in the TL or TW planes as honeycomb cores of the same density. Honeycomb cores, however, have negligible compressive strength in the Wand L directions and shear strength in lhe WL plane, whereas crossbanded cores have considerable strength in these directions. Crossbanded core is not readily formed to curved surfaces because of its relatively high stiffness in all directions .
•
Many core materials and core configurations are available, but the aluminum honeycomb with a hexagonal cell is the most commonly used at Bell Helicopter. Table 13.2 shows· the mechanical and physical properties for many of the available core materials. The metallics used at Bell are SOS2-H39 and 5056-H39 aluminum. They should be procured in accordance with Bell Specification 299-947-059. The nonmetallic honeycomb materials are as specified in Bell Specifications 299-947-103 and 299-947-337. Regardless of the core material, the final bonded panel must meet Bell Specification 299-947-091. Figu:=e 13.2a shows correction factors for core shear strength at various core thicknesses. 13.1.3
Adhesives
III thl- fahric:.ttjon of :~:jncJwich, adh<:si.v(;s arc used for hondin~ fact..'s to cor<' and br)ndin;,; r;JCinA.!,s and fillings, rei.nrorcin~ plates, edge sLrips and oLht,C insLrL~.
The adhes lV.{:S us<.:d are resin formulations especially developed to give high strE:ngth bonds over a wide range of exposure and stressing conditions. Adhesives can be used to bond many types of metal in highly stressed applications. They can also be formulated to have resistance to moderately elevated temperatures. The intrinsic elastic properties and strength of adhesives have not been evaluated to any 1arbE extent. Instead, adhesive bonded lap joints are used to evaluate the strength of an adhesive. The standard test is .063 aluminum sheets bonded with an overlap of 1/2 inch. The lap joint strength is the load required to shear the bond divided by the bonded area. Table 13.3 shows typical lap joint strengths of several adhesives. Lap joint strength is not considered of prime use for determining adequacy of adheThe need of an adhesive to form strong fillets at the ends of the core cells to produce satisfactory sandwich has prorrlpted the evaluation of peel strength and sandwich flatwise tensile strength.
~ives for bonding sandwich facings to honeycomb cores.
13-7
STRUCTURAL DESIGN MANUAL 5052 ALLOY HEXAGONAL ALUMINUM HONEYCOMB AEROSPACE GRADE '+1
f
0.. ....-I
C
-r-!
Ce Il-Material* 8
o
Gage
CI}
~ (IJ
:z.~
1/8-5052-.0001 1/8-5052-.001 1/~-j052 ... 0015 1/3-5052-.002 5/32-5052 .0007 5/32-5052 .001 5/32-5052 .0015 5/32-5052 .002 ~/32-5052 .0025 3/16-5052 .. 0007 3/16-5052 .001 3/16-5052 .0015 ~/16-5052 .002 3i16 ... 5052 .0025 ~/16-5052 .003 1/4--5052-.0007 1/4-5052-.001 1/4-5052- .0015 1/4-5052-,002 V4-5052-.0025 V4- 5052-.003 111- 5052-.004 ~/8-5052 .0007 ~/8-5052 -.001 3/t)- 5052 .. 0015 3/8-5052 -.002 3/8-5052 .0025 3/8-5052 -.003 ~/8-5052 -.004 ~/8-5052 - .005
-
t3-8
>-.-1 .j.J
3.1 4,.5 6 .. 1 3.1
PLATE SHEAR
COMPRESSIVE
u
iHONEYCOMB lDESIGNATION
..c: JJ
Stabilized bJ) ~ Modu- ..c: CI} OJ ::l !-I.r-! Strength Strength Ius I-IJJCI} psi psi u~ ksi typ min typ min typical 270 200' 290 215 75 520 375 545 26 405 150 450 870 650 910 680 240 1400 1000 1470 1100 350 750 Bare
~
"til Direction ModuF.s
Strength psi typ
Ius ksi
"w" Direction ModuStrength lus psi ksi
ty~ ~ I'~£S \ .'l}{) '.lJ'n;
455
168 272 400
typ 22.0 31.0 41.0 54.0
r~'
.2.l.(1
4..5 -_
340 505 725
285 70 .. 0 455 98.0 670 135.
~
220
320
n}_~!1/
90 ~
2.6
200
150
215
160
55
90
165
120 37.0
100
70
19.0
3.8
395
285
410
300
110
185
270
215 56.0
175
125
26.4
5.3
690
490
720
535
195
340
420
370 84.0
270
215
36.0
6.9 1080
770
1130
800
285
575
590
540 114.
375
328
46.4
8.4 1530 1070
1600
1180
370
800
760
G90 140.
475
420
5t.0
-
2.0
130
90
135
100
34
60
120
80 27.0
70
46
14.3
3.1
270
200
290
215
75
130
210
155 45.0
130
90
22.0
4.4
500
360
525
385
145
250
330
280 68.0
215
160
30.0
5.7
770
560
810
600
220
390
460
410 90.0
300
244
38.5
6.9 1060
770
1130
BOO
285
575
590
540 114.
375
328
46.4
8.1 1400 1000 S5 60 1.6 2.3 165 120 3.4 320 240 4.3 480 350 5.2 670 500 6.0 850 630 7.9 1360 970
1470 95 175 340 505 690 880 1420
1100 70 130 250 370 510 660 1050
350 20 45 90 140
725 85 140 235 320 410 495 700
670 60 100 180 265 360 445 650
455
235 340
750 40 75 150 230 335 430 725
400 54.0 32 : 11.0 57 16 .. 2 105 24.0 155 29.8 200 35.4 265 40.5 390 52.8
190
135 21.0 32.0 50.0 66.0 82.0 96.0 130.
85 150 210 265 315 440
50
1.0 1.6
30 85
20 60
45 95
20 70
20
25 40
45 85
32 12.0 60 21.0
30 50
20 32
7.0 11.0
2.3 3.0
165 260
120 190
175 270
130 200
45 70
75 120
140 200
100 ~2.0 145 43.0
85 125
57 85
16.2 21.2
3.7 4.2 5.4
370 460 720
390 485 745
180 220 360
260 310 430
200 55.0 255 65.0 380 86.0
170 200 280
115 150 228
26.0 29.0 36.8
970
285 355 535 750
105 135 200
6.5
270 335 500 700
265
505
545
500 105.
350
300
43.5
1020
10
TABLE 13.1 - PROPERTIES OF TYPICAL SANDWICH CORES
•
•
STRUCTURAL DESIGN MANUAL
•
5056 HEXAGONAL ALUMINUM HONEYCOMB AEROSPACE GRADE
~
PLATE SHEAR
COMPRESSIVE
li 1"\
HONEYCOMB
DESIGNATION ~r; Bare ~..-I Cell-Material.. ..-IS s::fIJ Strength o Q) psi Gage ZO 1/8-5056-.0007 3.1 4.5 [1/8-5056 ... 0015 6.1 [1/8-5056 .... 002 8.1 ~/8 ... 5056 .... 001
..c:
~
Stabilized
b.O
ttL U Direction
.c: OJr:: Strength Strength Modu- fIJ=' 1-1-.-1 psi H ~ ~ psi Uti) ~~f
"Wit Direction
Modu- Strength psi lus
Modu-
~~f
typ min 340 250 630 475 1000 760 1520 1200
typ 360 670 1100 1700
min 260 500 825 1300
typ 97 185 295 435
typ 170 320 535 810
typ 250 425. 640 900
min 200 350 525 740
typ 45.0 70.0 102. 143.
typ 155 255 370 520
min 110 205 305 440
typ 20.0 38.0 38.0 51.0
180 360
260 500
185 375
70 140
120 235
200 335
152 36.0 272 57.0
120 205
80 155
17.0 24.0
615 920
865 1340
650 1000
240 350
420 650
530 760
435 85.0 610 118
310 430
250 360
33.0 43.0
110
160
120
45
75
140
105 27.0
85
50
13.0
250
360
260
97
170
255
200 45.0
155
110
20.0
460 685 75 145 300 440 600 25 75 155 240
650 980 110 210 420 620 820
490 735 80 155 315 465 645 35 80 155 260
180 270 30 58 115 172
310 480 50 100 200 300 410 35 50 100 160
410 585 90 170 290 400 500 60 90 170 245
340 480 78 130 230 325 425 45
68.0 94.0 20.0 32.0 50.0 67.0 84.0 l5.0 78 20.0 130 32.0 190 43.0
245 340 60 105
175 240 300 35 60 105 145
198 280 38 62 130 190 245 25 38 62 100
27.5 36.0 12.0 15.0 22.0 27.0 32.0 9.0 12.0 15.0 19.0
min 400 600 770 950
typ typ 82.0 315 470 118 590 148 650 170
min 250 375 470 585
typ 33.0 45.0 54.0 64.0
180 120
143 88
23.0 19.0
~/32-5056-
2.6 255 ~/32-5056-.001 3.8 475 :>/32-50565.3 820 .0015 5/32-5056-.002 6.9 1220 3/16-50562.0 155 .0007 ~/16 .. 5056.001 3.1 340 3/16-5056.0015 4.4 600 3/16-5056-.002 5.7 910 1/4-5056 - .. 000 1.6 100 /4-5056-.001 2.3 205 .. /4-5056-.0015 3.4 395 fL/4-5056-.002 4.3 580 /4-5056-.0025 5.2 790 3/8-5056-.0007 1.0 35 3/8-5056-.001 1.6 100 ~/8-5056-.0015 2.3 205 ~/B-5056-.002 3.0 320 .0007
• )
50
110 210 340
230 15 30 58 92
2024 HEXAGONAL ALUMINUM HONEYCOMB
•
1/8-2024-.001: 1/8-2024-.002 1/8-2024-.002: 1/8-2024 ... 003 3/16- 2024-. .0015 1/4-2024.... 001~
5.2 6.7 8.0 9.5 3.5 2.8
min typ typ 700 525 780 1100 825 1225 1480 1100 1650 1970 1475 2300
min 620. 980 1320 1725
370 250
290 165
330 220
250 165
typ typ typ 500 200 425 760 300 640 960 380 840 480 1120 1150 86 40
200 110
290 200
230 55.0 140 42.0
TABLE 13.2 (CONTINUED) .. PROPERTIES OF TYPICAL SANDWICH CORES 13-9
•
5052 ALUMINUM FLEX-CORE - AEROSPACE GRADE
......t
Ma terial-CellGage Count
to
{:: or4
g
'"
Bare
~
Stahlli
.... ..j.J
Strength ~ psi
(f)
typ
3.1 340 4.1 540 5.7 900 4.3 575 6.0 ,..000 B.O 1'-570
5056/F40-.0014 5056/F40-.0020 5.056/F40-.0026 p656/F80-.0014 5656/F80-.0020 5056/F80-.0026
typ 2.1 215 3.1 405 4.1 645 4.'3 680 6.0 1150 8.0 1730
min typ 260 470 690 740 1300 1800
~CG
5.2
3.6 1.8
-
--
-
typical 595 325 95
~
min 157 280 420 700
-
min
-
--
typical 610 340 110
liT
II
n.f""A,.f-~,..,n
~ IlJ ~
~
Strength psi
typ 65 125 185 290 195 310 400
typ typ 80 90 165 180 250 260 380 400 280 440 620
min 63 126 182 280
typ 65 125 185 195 310 410
typ typ 105 215 310 335 520 740
min
typ 148 92 24
"W" Di reC'_ t _nn
b(J
ucn
~052/F40-.0019 ~052/F40-.0025 ~052/F40-.0037 ~052/F80-.0013 ~052/F80-.0019 ~052/F80-.0025
-
..c::
~:ecl
lModu- ..c= Strength lus Ul ::::I psi I-l psi
min typ 126 225 238 400 378 600 630 1000 650 1050 1600
~052/F40-.,OO13 2.1 180
1/4-.003 3/8-.003 3/4-.003
PLATE SHEAR
to)
0.
zp
~CG ~CG
I.H
COMPRESSIVE
~
HONEYCOMB DESIGNATION
-
-
-
-
typ 245 175 50
-
-
typical
Strength psi typ typ 18 50 32 100 45 150 68 230 45 160 72 240 98 345
min 37 75 115 170
typ typ 18 55 32 120 45 .175 47 185 73 285 100 410
min
210
typ 63 40
95
16
345
-
-
typical 215 130 55
ModuIus psi typ 10 13 17 23 18 24
31
typ 10 13 17 18 24 32
•
typ
31 20
8
~
TABLE 13.2 (CONTINUED) - PROPERTIES OF TYPICAL SANDWICH CORES
13-10
•
STRUCTURAL DESIGN MANUAL
•
HRP GLASS REINFORCED PHENOLIC HONEYCOMB
PLATE SHEAR
COMPRESSIVE
HONEYCOMB DESIGNATION Ma t I 1- Ce 11Dens! ty
Bare
liT
Stabilize~
Strength
Strength
ModuIus k.s i
•
at
ModuStrength lus Iksi Dsi typ min typ
Hexagonal
typ
min
typ
min
typ
HRP-3/16-4.0 HRP-3/16-5.5 HRP-3/16-7 .. 0 HRP-3/16-B.0 HRP-3/16-12.0 HRP-l/4-3.5 HRP-l/4-4.5 HRP-l/4-5.0 HRP-l/4-6.5 HRP-3/B-2.2 HRP- 3/8- 3.2 HRP-3/B .. 4.5 HRP-3/B-6.0 HRP-3/B-B.0
500 800 1150 1400 2280 350 630
350 600 900 1100 1600 260 450 510 850 105 245 450 750 920
600 940 1230 1600 2300 500 700 820 1180 200 440 690 1000 1200
480 750
57 95 136 164 260 46 70 84 120 13 38 65 100 150
260 425 500 660 940 230 300 340 450 105 200 300 400 520
350 600
625 950 1230 425
43 65 84 32 60
210 270 395 '240
25 37 49 37 49 61
125 200 280 195 265 390
700
1025 150 320 610 900 1060
-
1280
-
400 560 660 900 145 350 550 750
-
"w"
Di rp(, ti on
210 370
-
600
-
170 250
-
75 160 260 340
-
D; rf=
Stren~th
ps
typ
11.5 19.5 2B.O 34.0 55.0 9.0 14.0 17.0 25.0 5.0 8.0 14.0 22.5 31.0
140 220 290 400 570 120 170 200 260 60 105 170 260 320
-8.0 10.5 14.0 4 .. 5 10.0
250 330 450 150 300
12.5 15.0 22.0 20.0 25.0 31.5
70 105 140 100 140 205
l'_
ti on
Modulus ksi
min
typ
110 190
B.5
100
5.0
-
12.5 15.0 25.0 3.5 6.0 7.5 11.0 2 .. 0 3.0 6.0 10.0 13.0
--
1B.O
--
20.0 9.0 17.0
370
140
-45-
85 150 210
OX-CORE HRP/OX-l/4-4.5 HRP/OX-l/4-5 .. 5 HRP/OX";1/4-7.0 HRP/OX-3/8-3.2 HRPjOX-3/8 .. 5.5
520 810 1150 340 700
-
260 580
820
-
--
140
--
-..
..
15.2
FLEX-CORE HRP/F35-2.5 HRP/F35-3.5 HRP/F35-4.5 HRP/F50-3.5 IHRP/F50-4.5 HRP/F50-5.5
•
180 320 440 300 400 600
-
-
-
-
240 400 600 425 600
8BO
300 300 500
-
.. 140
-
14"0 200
..
-75
... 75 100
-
7.0 10.0 12.0 10.0 13 . 0 16.0
TABLE 13.2 (CONTINUED) - PROPERTIES OF TYPICAL SANDWICH CORES 13-11
STRUCTURAL DESIGN MANUAL
•
NP GLASS REINFORCED POLYESTER HONEYCOMB COMPRESSIVE
HONEYCOMB DESIGNATION Mat'l-CellFabric-Density Hexagonal NP 3/16-4.5 NP 3/16-6.0
NP 3/16-9.0 NP 1/4-4.0 NP 1/4-6.0 NP 1/4-8.0 NP 3/8-2.5 NP 3/8-4.5
Bare Strength psi min typ 520 470 880 615 1700 1.200 420 295 880 615 1400 9BO 200 140 520 365
PLATE SHEAR
Stabilized "Ltt Direction ModulModuStrength Strength Ius l~uS psi psi ksi ,si min typ typ typ min typ 670 470 80 280 195 13.5 1050 735 116 330 230 15.0 1260 180 1800 460 320 20.0 560 390 68 260 180 13.0 1050 736 116 330 230 15.0 lOBO 160 1540 410 290 18.0 2BO 195 34 170 120 10.0 470 670 80 280 195 13.5
"Wit Direction
Strength psi min typ 130 90 155 110 230 160 120 85 155 110 205 145 10070 13.5 90
~odulus
kai -typ 5.2 5.8 7.5 5.0
5.B 7.0 4.0 5.2
OX-CORE NP/OX 1/4-4.0
NP/OX 1/4-6.0 NP/OX 3/8-4.5
350 700 420
-
-
...
-..
... ... -
...
160 275 190
5.0 7.5 5.5
-
1C)O 375 285
... ...
12.0 19.5 15.0
HRH-327 GLASS REINFORCED POLYIMIDE HONEYCOMB
HONEYCOMB DESIGNATION Ma t I 1- Ce 11Density
PLATE SHEAR
COMPRESSIVE Stabilized Modulus Strength psi ks!
typ HRH 327-3/16-4.0 440 HRH 327-3/16-4.5 520 HRH 327-3/16-5.0 600 HRH 327-3/16-6.0 780 HRH 327-3/16-8.0 1300 HRH 327-1/4-4.0 440 HRH 327 ... 1/4-5.0 600 HRH 327-3/8-4.0 440 HRH 327-3/8-5.5 680 HRH 327-3/8-7.0 1.000
min
-
400
-
625 1000
-..
325 540
-
typ 50 58 68 87 126 50 68 50 78 106
ilL" Direction
Modulus ksi
Strength psi typ 280 320 370 460 650 280 370 280 420 550
min
-
220
-
345 500
-...
195 300
...
typ 29 33 37 45 62 29 37 29
41 53
"Wit Direction Strength Modulus psi ksi
typ 130 150 180 230 410 130 180 150 210 310
TABLE 13.2 (CONTINUED) - PROPERTIES OF TYPICAL SANDWICH CORES 13-12
•
min
... 110
-
170 330 ... ... 100 160
-
typ 10 11 25.5 15 22 10 12.5 12 13.5 18.5
•
•
STRUCTURAL DESIGN MANUAL AVERAGE ALUMINUM TO ALUMINUM LAP SHEAR BOND STRENGTH AT ROOM TEMPERATURE - (PSI)
ADHESIVE TYPE
(A)
NITRILE PHENOLIC
3500
(B)
VINYL PHENOLIC
4200
(C)
EPOXY PHENOLIC
3400
(D)
UNMODIFIED EPOXY
3100
(E)
MODIFIED EPOXY - 250 CURE
4500
(F)
MODIFIED EPOXY - 350 CURE
3300
{G)
EPOXY POLYAMIDE
5500
(H)
POLYAMIDE
3300
STRUCTURAL ADHESIVES USEFUL TEMPERATURE RANGE, STRENGTH PROPERTIES 192 HR. EXPOSURE
• )
•
ADHESIVE TYPE
TYPICAL VALUES USEFUL TEMP. RANGE LAP SHEAR (OF) (PST)
PEEL STRENGTH
(A)
NITRILE PHENOLIC
-67 350
3500-4500 300-1700
GOOD TO EXCELLENT
(B)
VINYL PHENOLIC
-67 225
2000-3000 100-1800
FAIR TO GOOD
(C)
EPOXY PHENOLIC
-70 500
1300-5000 200-1900
POOR TO MEDIUM
(D)
UNMODIFIED EPOXY
-67 300
1300 ... 3000 800-3000
POOR TO MEDIUM
(E)
MODIFIED EPOXY250 CURE
-250 180
1500-2500 1000-1900
GOOD
(F)
MODIFIED EPOXY350 CURE
-67 250
3000-3500 1500.. 2500
(G)
EPOXY POLYAMIDE
-300 250
4000-5000 2800-3300
GOOD
KH)
POLYAMIDE
UP TO 600
3300
POOR
GOOD
TABLE 13.3 - ADHESIVE PROPERTIES
13-13
STRUCTURAL DESIGN MANUAL Peel strength is determined as the torque necessary to peel a facing from a sandwich core. Table 13.4 shows values of peel strength and fillet strength for several adhesives. Adhesives are available in the form of liquid t paste, powders and supported or unsupported films and can be applied by spray, roller, spatula or hand lay-up. The form of the adhesive (liquid, paste or film) is chosen to suit the lay-up operation and glue line thickness requirement. Structural adhesives have good shear and tensile strength, but resistance to peel stresses is relatively poor. Bonded joints should be designed to take advantage of the high shear and tensile strengths of the adhesive and avoid peel stresses in the bond where possible.
FILLET STRENGTH
PEEL STRENGTU
I""
ADHESIVE 73°F lb/in
180°F lb/in
-67°F Ib/in
180°F 73°F -67°F in-lb/in in-lb/in in-lb/ir
NITRILE ELASTOMER-PHENOLIC PLUS MODIFIED EPOXY FILM
73
51
84
66
32
21
POLYVINYL-PHENOLIC
42
35
49
25
22
14
EPOXY-POLYAMIDE
71
39
66
18
24
15
MODIFIED EPOXY3500F CURE
62-86
27-65
61-87
16-97
13- 82
1.2-49
MODIFIED EPOXY250°F CURE
46-76
34-38
47-91
1.5-40
L8- 23
19-26
33
89
27
NITRILE EPOXIDE
61
TABLE 13.4 - .sTRENGTH OF ADHESIVES IN SANDWiCH WITH HONEYCOMB CORE
13-14
18
28
•
STRUCTURAL DESIGN MANUAL
13.2
Methods of Analysis
The analysis procedures described in this section apply to sandwich structures having isotropic facings and either orthotropic or isotropic cores. The isotropic ITlaterials are those having essentially constant properties in all directions. The ortholropic materials are those whose strength properties are nat constant in all directions, such as honeycomb cares. The assumption is made that adhesive failure does not occur, a reasonable assumption if proper care is taken in the selection of the adhesive system. This requires that the adhesive shear and flatwise tensile strength be greater than the respective core strength. This can be assured by specifying that the finished panel Ineet the requirements of Bell Specification 299-947-091.
If the sandwich has thin facings on a core of negligible bending stiffness, as is usually the case, and after assuming is given by the formula:
Al = A2 = A, the bending stiffness
EitlE2t2h2
D
•
= ~~----~--~ (Ei tl + Ei t )A
(for unequal facings)
2
E'th
2
D=-
(for equal facings)
2i\
Figure 13.3 shows the notation used for the analysis of sandwich panels in this section.
Facing 2
Facing 1
FIGURE 13.3
NOTATION FOR SANDWICH COMPOSITE
The notation used throughout this section is shown below: 1 2
- subscript denoting facing 1 denoting facing 2 - subscript ,
13-15
STRUCTURAL DESIGN MANUAL a, b ... length of panel edge; subscripts denoting parallel to a or b c subscript denoting core or compression cr ... subscript denoting critical D - bending stiffness d - total sandwich depth or thickness E - modulus of elasticity E' - effective modulus F - allowable stress f - applied stress G ... modulus of rigidity h - distance between facings centroids J ... polar moment of inertia K - a constant L - length M - bending moment N - load per unit length of edge P load p - distributed load r - radius; subscript denoting reduced ... ratio R S - shear load normal to surface of panel ... core cell size; subscript denoting shear s T - torque - thickness; facing without subscript t ... transverse shear stiffness U - parameter relating shear and bending stiffness V ... weight W w - density ... axis x axis perpendicular to x axis y z - axis normal to surface of sandwich
a
- jE'a/Etb
{3
... ... ...
y
o €
A
,.,.
13.2.1
•
a' '-'ab + 21" shear strain; elastic property parameter deflection strain 2 (1 ... JJ ) - Poisson's ratio
Wrinkling of Facings Under Edgewise Load
Wrinkling of sandwich facings, as shown in Figure 13.4, may occur if a sandwich facing buckles as a plate on an elastic foundation. It may buckle into the core or Adhes'ive Bond Fai lure
1.3-16
FIGURE 13.4 FACE WRINKLING
•
STRUCTURAL DESIGN MANUAL away from the core depending on the relative strengths of core in compression and adhesive in flatwise tension. The facings of a sandwich shall not wrinkle under design load. The wrinkling stress formulas are given for two types of sandwich; sandwich with continuous cores and sandwich wi th honeycomb co'res for which elastic-moduli in the plane of the core are very small compared with the elastic modulus in a direction normal to the core plane. 13.2.1.1 (1)
Continuous Core
Determine the 'parameter q:
(
where
)1/3
A,'
q
EfE c Gc
13.2
E = core flatwise modulus of elasticity C G = core modulus of rigidity c
(2)
Determine parameter K: 13.3
•
where
~~
total amplitude of initial facing waviness usually between .0005 - .005 inch Fc = flatwise strength of core or bond, whichever is less
K and determine value of parameter Q.
(3)
Enter Figure 13.5 with q and
(4)
Find the stress Fer at which face wrinkling occurs: E E G
fcc Fer = Q ( A (5)
•
Honeycomb Core
(1)
Enter Figure 13.6 with values of T1/t(... and Fc;/EG. to obtain K.
(2)
Enter Figure 13.7 with value for K and (Ect/Eftc)1/2 to obtain Fcr/E f .
.(3)
Solve for Fer.
(4)
Computed compressive stress (ff) must not exceed Fer.
13.2.2
13.4
Computed compressive stress (ff) must not exceed Fer.
13.2.1.2
)
)1/3
Dimpling of Facings Under Edgewise Load
If the core of a sandwich construction is of cellular (honeycomb) material, it is possible for the facings to buckle or dimple into the spaces between core walls as shown in Figure 13.8. Dimpling of the facings may not lead to failure unless the amplitude of the di'mples becomes large and causes the dimples or buckles to grow across core cell walls and result in wrinkling of the faces. Dimpling that does not
13-17
........ W I
........ (XI
~} 11~~
;-
'C'
0.
"
1',0
~
'.
---r-----r--- r---)'f!:--r---to""
f
o. ....
~~ r:--:...
~
b--.
I--... ~ !'-....
............
~
I nl
~ r--~
.........
r--- r-t--r-
-.
r---;--t--- I---
r---- I--
~~O
""
.........
c::
I
.,
c-)
~
--I
r - - I---
- r--- r---- r - -
c:
I
.!
I----
i
0.4
08 •
i I_t?
16
----r-==::::::-- ---
20
I
r---~_
~
r---
-r---
-
.t!.;.!...OO
-'-----2.
~.8
-
.16
4.0
~
r--- [-:00-........
--- ---
~
r---- r--- r--_
r-
-I--
c::I ~
~ :
:::::::1 I I
--:
-
KO.t?OO
32
I
4:4
q
4.8
5.t?
~_6
6.0
::a
r-
I--
r--
~
t-_ _
~
t---r---- r--r-- ~r--- t - - - r--- t--- t----
~ ~
.~
I'--.
--.
~~;
r::",
~ f".... .........
--
~
~
I
r--:--
r--- r---
I"--.
t-..,
a 1'~ c. 16
r---- ~
'(1)
..... ::::a
r---I---r--- t--r--t--~~ r--- t--- ~ r----r--- r--r--- r---- r--- I---' " r--~ , J ~ ~~ r--- r---r---r--r--;--r-- r--- I--r--~ ~ L:-:---., t---t--r--- ---... r--- .. r--r--A~ ~ l/ \ r--t-...
ID
I--...
--
r--- ~ r---
r---- ---
r---.
r-
-:-
.............
c. !2
,)
r--~ t----
Q. "-
Q ."
I
":0.01
~~
o. 6
J
I--
6A
72
6.8
U
110
a.
BR
J2
en
-
CD
:z ~ I~
:z: ~
~
r-
FIGURE l3.5 - PARAMETERS FOR DETERMINING WRINKLING OF FACINGS OF SANDWICH WITH CONTINUOUS CORES
•
•
.--..-
•
STRUCTURAL D'ESIGN MANUAL 2.0~~--~--~--~--~--~·--~------~--~------~----~
7)
tc ·0.0100 0.0090 ~2~~--+4----~-+--~--~~----~~--~~----~+---~~
0.0080
0.0070
f( a8r---~~~----_+--~~-r----~~----~~---------~~----4
0.0060 0.0050 0,0040
0.4 r-------+~~--_+--...;;:-.".~__+_------~----~"""+_=__----_+__----____I 0.0030
-T----+------it ----~=-~C_:-:-=-~j
0.0020 0.0010 0.0005
o~----~------~------~------~------~------~----~
a
0.001
0.002
0.003
0.004
0.005
"\_.")
0.005
0.007
./
(I /,.. . --I - -
FIGURE 13.6 - RELATIONSHIP OF K TO CORE PROPERTIES (F c IE c ) AND FACING WAVINESS ( 6/tc)
13-19
t--'
W
•
r-...l
o
'1-'
0.010 I II
I·
I
::a
i
~ I ,/I >'fi/-
0.0061
I
0 \ 0 'Z.
'1-. . '1-. .
~
5 .0.
,,.~
--
,
""iCD ~
J1«,.0
.
en
.....
:::a c:
>"
c-)
~l./""
1t<_Z.O
~
c:
:::ell
Fer
Ef
'}~ \'!.-~~"
,~
;-
!
0.0081
o
:.:0.0041
>"7'7' !
7',.LC
r-
::::::;oao ...........
CJ ...,
,-,~-",
'.1
0.002 I
r
hy:'L
71.......
en
-:z
::::aa>'' ' ' ' ' -I
CD
.,.)
''''J. O~~--------~------~--------~------~--------~--------L--------L
o
0.002
0.004
0.006
;~~)8J
0.010
0.012
:I:
:.:-
______~
0.014
0.016
z:
~
:.:-
\t;fc
r-
FIGURE 13.7 - GRAPH FOR THE WRINKLING STRESS OF FACINGS OF SANDWICH WITH HONEYCOMB CORES ," ~
e
e
e
STRUCTURAL DESIGN MANUAL Revision C cause total structural failure may, of course, be severe enough so that permanent dimples remain after removal of load.
Faces Buckle Into Core Cells
FIGURE 13.8
INTRACELL BUCKLING (FACE DIMPLING)
If dimpling of the facings is not permissible, the core cell size shall be small enough so that dimpling will not occur under design loads. It is assumed that faIlure in the facing-to-core bond cannot occur prior to dimpling. Figure 13.9 can be used to determine the critical facing stress (stress at which dimpling will occur). The curves in Figure 13.9 represent a plot of equation 13.5 which can be used instead of Figure 13.9. 2
13.5
A
13.2.3
Flat Rectangular Panels with Edgewise Compression
The method presented here is used in design of a flat, rectangular sandwich panel subjected to edge compression. The panel is simply supported at the four edges and the load is applied equally and uniformly to the facings at two opposite as shown in Figure 13.10.
)
r---
a
--1
..
3-..--....r--~------,.~---
i
N=Edge Load, lb/in b=Loaded Edge
_=j ___'--------t:=--N--"r'--a 1
=Un oade d Edge
FIGURE 13.10
COMPRESSION PANELS
Overall buckling of the sandwich or dimpling or wrinkling of the facings cannot occur. without possible total collapse of the panel. Detailed procedures follow ing theoretical formulas and graphs for determining dimensions of the facings and core, as well as necessary core properties. Facing modulus of elasticity, E, and stress values, F, shall be compression values at the conditions of use; that is, if 13-21
STRUCTURAL DESIGN MANUAL l
I
I
II
I. 00 0.80 0.60 ~
~
0.40
::t (,.) 0.30 ~ ........ ........
.. 020 .
~
...
la.J f\j
.........
V)
.....,J
0.10
~ 0.08
(,.)
Lv 0.06
~
~
0.04 003 0.02
0.0/ ........
C)
CJ CS
C\J CJ
h)
"'\l'C)
C)
C) C)
CJ
c::s
C)
C)
~
~
CJ CJ
CJ CJ
c::i
c::i
C) ........
CJ C:)
CJ
~
<:::)
CJ
CJ CJ
C)
(J
CJ
(J ........
Cj
C)
Cj
c::i
CJ
h)
"t
AFcr Ef FIGURE 13.9 - CHART FOR DETERMINING CELLULAR CORE CELL SIZE SUCH THAT DIMPLING (INTRACELL BUCKLING) OF SANDWICH FACING WILL NOT OCCUR
13-22
Q:)
•
s-rRUCTURAL DESIGN MANUAL Revision C application is at elevated temperature, then facing properties at elevated lemperature shall be used in design. The facing modulus of elasticity is the effective value at the facing stress. If this stress is beyond the proportional limit value, an appropriate tangent, reduced or modified compression modulus shall be used.
(1)
Choose an allowable design compressive stress (F > and determine the required f facing thickness from
+ t2Ff2 = N; unequal faces t
=
N/2F ; equal faces
13.6
13.
f
When the elastic modulus of one face is different from the elastic modulus of the other face, equation 13.6 must be satisfied, but also the stresses Ff! and Ff2 must be chosen so that 13.8
The lower of the ratios in equation 13.8 must be used for design, otherwise the face with the lower ratio will be overstressed. (2)
The critical facing stress (Fer) at which buckling of the panel will occur is
13.9
Fer
where E and A are values for the facing with least FflEf ratio as determined f from equation 13.8. If the facings are of equal thickness and of the same material, equation 13.9 becomes
13.10
In equations 13.9 and 13.10 13.11 K is determined first by going through the following steps 3 to 8.
•
(3)
Determine the value of parameters: alb or bfa, whichever is
<1
13.12
1.3-23
STRUCTURAL DESIGN MANUAL 13.13
13.14 (4)
Enter the appropriate figure (13.11, 13.12, 13.13 or 13.14) with equation 13.12 at V = .01 (Choosing a low finite value of V to start with since V = 0 gives h as a minimum and Gc as infinite). Project laterally to parameter 13.13 then vertically down to parameter 13.14 and horizontally to h/a. Evaluate h.
(5)
Determine core thickness from
tc
=h
t1 -
+ t2 2
13.15
; for unequal faces
13.16
h - t; for equal faces (6)
Determine constant K' in the equation V
Kt
2 Kt
(8)
K'/G C from
for unequal faces
7r
(7)
=
13.17
tcEft
= ---_
for equal faces 13.18 2 2Aa Determine tentative core modulus of rigidity (G c ) from Gc Kt/V for V = .01. If this value of Gc is not within the range available in the desired core material and type, enter Figure 13.15. Project diagonally along the line V = K'/G c until a practical value is reached. For the new value of V, repeat steps (4), (5) and (6). From the appropriate Figure (13.16 through 13.27) find the value Figure 13.28 obtain the value of ~O. Find KF
of~.
From
13.19
for equal faces
For values of b/.a greater than those in Figure 13.28 assume KF
13-24
13.20
=
O.
e
STRUCTURAL DESIGN MANUAL
• 4 -"-'- -
5 ..
~ -r
6 -
~ ~
~
1 1
ENO'"
10
)---b-+i
_J_ _
15
20 30-
o
Q2
0.4
%
0.6
0.8
1.0
0.8
0.61.. 0.4
0.2
o
'7"0
0.001
--l-----.l0.03
)
--1---1-----10.05
~I..Q
-----1--··---10.06
LEGEND: --80TH FACINGS ISOTROPIC - - -80TH FACINGS ORTHOTROP/C
For dissimilar faces see 13 2.3.
'.
0.07
0.08 ---1---10.09 ---'-_......I 0.10
FIGURE i3.11 - CHART FOR DETERMINING h/b RATIO (V=D) SUCH THAT A SANDWICH PANEL WILL NOT BUCKLE UNDER EDGEWISE COMPRESSION.LOAD
13 ... 25
STRUCTURAL DESIGN MANUAL
tt t
,
rI I
!
~~t
1:1
Gc
....---tIoo
!
.t
l.···b-·..J
0.2
0
o
ii
0.0 0 / ~-==F".--1!-----I---I----t---t---b"....".q,.+--.J..~-b~~~-I-----.j 0.02
+:r----:lol'--+---I---I----I---IO.O 5
.c:: I~
LEGEND: - - 80TH FACINGS ISOTROPIC - - - 80TH FACINGS ORTHOTROP!C
~-I-----I~rG-+--,II~---J.-For
dissimilar faces see 13.2.l-
.09
O.Ol,
FIGURE 13.12 - CHART FOR DETERMINING h/b RATIO SUCH THAT A SIMPLY SUPPORTED SANDWICH PANEL WITH ISOTROPIC (Geb = Gca ) WILL NOT BUCKLE UNDER EDGEWISE COMPRESSION LOAD 13-26
tt
STRUCTURAL DESIGN MANUAL
• ~--+--+--+--r--r-~
-'--r I
-.--~---
•
o
0.2
0.4
0.6 a
F
0.8
1.0
a8
0.6
h
0.4
0.2
o
7i
0.02
0.001
0.03
)
---+'---10.04
!l,j1
~~ O.005~~r-~_.~~-; __~__~__-+.~~~-+__-+__- r__- r__~ 0.05 .. .....
-.c:::l~
~~~~--~--~~~--~~~~-+---+---+---+---+--~0.06
0.0 1
r---r-~---~~-4--~--~~-+---+---+---+---+---+--~0.0?
LEGEND: --BOTH FACINGS ISOTROPIC - - -80Tl-l FACINGS ORTHOTROPJC
C.OB
~----'r-----+---::1',",-+_-7f-_-+-_F 0 r dis simi 1 a r fa c i ng sse e 13" 2" 3 0.09
. 0.02
•
~~~~--~~~--~--~--~--~--~--~--~--~--~o.IO
FIGURE 13.13 - CHART FOR DETERMINING h/b RATIO SUCH THAT A SIMPLY SUPPORTED SANDWICH PANEL WITH ORTHOTROPIC CORE (Gcb = 0.4 Gca ) WILL NOT BUCKLE UNDER EDGEWISE COMPRESSION LOAD.
13 ... 27
1f11V\\\ STRUCTURAL DESIGN MANUAL ~~~.~ ~j;
'"~ ,~~
... - ~<: ~ r-:~ ~ .
-...
v::o.~
-
/
/
-
//
,-~
:& V::Oj
V
~ ~ ~
l
i
-~"
~1 T '" ~ ~ ~~ ---- l ~ iff t _.-. ~ -i ~ b
-
~- ~
f--~
-
~
-
~-- b
........
~
t 2
L
V= ff 0 ·b 2 U
I
0.2
-
-------- ---
2.5Gc
1
/ t V t o
-- - .. --- --
-"'...:.~
'" ""
1/
~ /1
-
""-' "
~~
v;o.2/ i''l.1 1 , / ~; / / V ~'/ I'
--
""", ~ ""s- '--~~ ---
./ • .L. f-.""".
•
- r----r--
~ ~
J,
0.4
a6
a
as
1.0
0.8
1!
-
----
0.6 : Q4
Q2
1i
~~,
•
o
[=C~~=+=f=+=f~35~~~~O
O.OOO/~_~ __~__~__~__r-~~~~-r~~~~~~~~~ 0.01 r:::~~~~---+---+---r-~--~~~-+~--~~~~--~002
~ ~~005~~__~__~~~__r-~~~~~__~__~__~~__~O~ ~I~ ~Ilil
-t------i
O.06
0·0 I ~-------+--+---~:""'-+-~f---7I'~---I----+----I-----I---I----+----JO.07 . LEGEND:
F----+~.L-f---t-----,....~~of___+ --80TH
- - -80TH
ISOTROPIC
FACINGS FACINGS OR THO TROPIC
~-+_-+-~+---::lol'---t_JFor dissi.milar faces s~e 13.2.3
02 0.
.LU_~ I
I
I
0.08 0.09 .0.10
FIGURE 13.14 - CHART FOR DETERMINING h/b RATIO SUCH THAT A SIMPLY SUPPORTED SANDWICH PANEL WITH ORTHOTROPIC CORE (Gcb = 2.5 Gca ) WILL NOT BUCKLE UNDER EDGEWISE COMPRESSION LOAD. 13-28
•
STRUCTURAL DESIGN MANUAL
•
~
~
•
~
\:) ~
....: c::s
<:5
~
""
c::;
~
~
~
<:S
\()
C)
C)
~
C)
cs
~ ~
........
~
,
FIGURE 13.15 - CHART FOR DETERMINING V OR WAND G FOR SANDWICH IN EDGEWISE COMPRESSION c 13-29
·;;;7!\~-~"
'~~I~~ \\
STRUCTURAL DESIGN MANUAL
\~\ '~'"""'."
~/
14
~--~--~--'--~--~--~I~-----------~--------~
I
I
I-
,
12
v=o I [ 10
,
I\V:O., \\
l\~--"-
\
8 1-----6+---,-\
LEGEND: - - Borff FACINGS ISOTROPIC - - 80TH FACINGS OR THO TROPIC For dis s i mil a L fa c e sse c 1 '3. 2 . .]
,~ l~
__._-+,,-1\
6
1\
I'
1----I----l\~1\~\\
,\. \\
-!-+-"- - - I - - + - - - - + - - . - - - I - - - - + - - - - l - - - - - I
\.
J=j::~ \~~
n:/
•
.••••••••••••• •n=2 .:" .-.n-:l.·-·- + - _. -
~___t=v::o I '\~~~_=_.- -~-~e-=' ~
'---
2
4 . -~~
1---
'... ..... ........
~-=--
'-
-
-r- -
- - - _ .. -
----.f---.. ' -_.
----
-
---
~--""-----
-.-
.
..
..
.. _ ... -.
-
"""- - -- ---r---- - - .I----~- - - - ... ---- ... --...
,
--_._--
o~--~--~----~--~--~----~--~----~--~--~
o
0.2
0.4
0.6
~
b
0.8
1.0
0.8
0.6
0.4
0.2
0
b
a
FIGURE 13.16 - KM FOR SANDWICH PANEL WITH ENDS AND SIDES SIMPLY SUPPORTED AND ISOTROPIC CORE. (G cb ; Gca ). 13-30
•
I/:-~;rr:' '\
'~~~~~~ STRUCTURAL DESIGN MANUAL 1
1I
:I
. l--~~-~---l
12 I------.r--.---
LEGENO: - - 80TH FACINGS ISOTROPIC - - 80TH FACINGS OR THO TROPIC
For dissimilar faces see 13.2.3
_.
,
-----
..
0
0
0.2
0.6
0.4
0.8
1.0
0.8
0.6
a
0.2
0
b a
b
•
0.4
FIGURE 13.17 - KM FOR SANDWICH PANEL WITH ENDS AND SIDES SIMPLY SUPPORTED 'AND ORTHOTROPIC CORE~ (G b = 0.4 G ). c
ca
13-31
STRUCTURAL DESIGN MANUAL 14~----~tj~----~--~I-----I--------------------~
•
12 ----.-- --- -- +-----1-
,l
I----+---H---i---+_
l V:O 10
tr------+----'-'+-+---.....j....
~:O.l \\ _.__ ~
81------1---
\ -- -_._,\~
.-----1-
-
~
4
2
'"
,
80TH FACINGS ORTH.oT(?PPIC For dissimilar faces see 13.2.1
'\-+---+----+------.- -1---1--\~
\
~\..'~'I~\ ~r-",~ --
t
-
'\
61----------1-'\ V=O.2
LEGENO: _ _ 80TH FACINGS ISOTROPIC
. '~-',
I,~~
v=O.4
,
...........
--'-..
•
~=2 n=~
n=1 ••••••...•.••••
••• •
... , ~ ~~~~--~----~--~--~----~--~
""----- - - -
-1----'"---
-.r------"*-- -- r-- - '-- . ~ -- -
-
~ -I-
-
-_._+--__.._+-_-_....;-~-==--=-=+'l-===-::-===-=-"'F-=--==-:-~----=--~"--~--...-~----__t"'-I----.--.....,. -
- - - - - .--~I---_l_---l---+__-_l_-_+--_+---I_-___l
0.2
O.~
0.6
a
b
0.8
1.0
0.8
0.6
0.4
0.2
b 0
FIGURE 13.18 - KM FOR SANDWICH PANEL WITH ENDS AND SIDES SIMPLY · SUPPORTED AND ORTHOTROPIC CORE (G c b = 2.5 Gca ). 13-32
0
•
STRUCTURAL DESIGN MANUAL
•
I~
t \
7J2 Ncr: K /)2 0
V=O
1
12
I
I
."I
"20 V: IJ2LJ
1
"
\
"-
\
\~
, \\ .. .. \ -V=O.I
\
~
•
\
~~
t t
I
..
I •
... •••• n=I 2 •••... n:3
.. .-..
'-
~-.
-- - - ---
1 - - - - ~--
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-- - -
- ---
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-
r--
. ~-
~-
-- ---
--- --
I-- - - ~--
-- -
-~--
V=O.4
-
_
roo-
---
--
LEGENO: - - 80TH FACINGS ISOTROPIC - - 80TH FACINGS ORTHOTROPIC For dissimilar faces see 13.2.3
o
o
0.2
0.4
0.6
a b
•
I
''-...... ........
.)
.:
~
~ r---.
2
ENO
.
n=/ .... ••'
V=O.2 ,
--
V)
.. t.
\~
'--
~1 ~l
~ t., ...... /" t.l
\.
\
6
v
r-b-i
' 1
\
' 'l
f.O
'\
I
\
t
" .. Gc . "
t
8
"'\ '\ '\
I
10
t t
-...:
I
I
I
0.8
1.0
0.8
I
I
0.6· 0.4
I
0.2
0
b
a
FIGURE 13.19 - KM FOR SANDWICH PANEL WITH ENDS SIMPLY SUPPORTED AND SIDES CLAMPED AND ISOTROPIC CORE, (Gcb=G ca ). , 13-33
STRUCTURAL DESIGN MANUAL 14
I
I I I
,, l
i ,
_
V=O
TJ2D 1J 2 U
\.
\
~
\
\
.,",,~
,
'.~
,
~~l ~rT
END
tt
a,
n=1 ..... ..'
V~O.2 ""'~ ~
l/
~b---+{
'.
~-
~~l
0.4Gc 1/ .. .. 1/
j
..•..•
V
+ +
'\:
~
\\ '[\\\\
8
6
V=
\\
~V:o.,
112
Ncr - K b 2 D
It
10
•
I
LEGEND: - - 80TH FACINGS ISOTROPIC - - 80TH FACINGS OR THO TROPIC , For dissimilar faces see 13.2.3
I
12
I
.......n=2 ... .' ....'.n:3.-. I . --..
..
.. .... _-
- --
~-
- - -
~
-- --
\
-----.~'"
' .......
-- --
"- ... .. _- V=O.4 - -. - -- --2 ~ ...... - - - , -IV=O.8 .-.
-
........
I
o
o
0.2
I-- -
-
"'-- -
~-
._--
1---
-
_0
----
-
-
0.2
0
'
0.4
0.6
a b
0.8
1.0
0.8
0.6
i<- -
0.4
b a
FIGURE 13.20 - KM FOR SANDWICH PANEL WITH ENDS SIMPLY SUPPORTED AND SIDES CLAMPED AND ORTHOTROPIC CORE (G c b=0.4 Gca )' 13-34
•
~
;/1/\\\\
,,~~,~
•
STRUCTURAL DESIGN MANUAL
14
r I
, I
I
I
12
I
V
k\
=
\\
i·5G~
1-
1\~=O.I \~V=O \
\
6
\
,
'
...
V=O.2 ....... ~-
2
j
~h~
"-
\
4
l
t
n=1 .,'
.
. ..
..
I
I .. I"'· ". '. _•. n=2 ... · ....n=J" ..
-\
\
~~
~
1
..........
\\ --
~ 1--.-
-- - _. --- -- - - -
-
--
- -- -- - -- --- "" -- - -- _. -- - - --- -- - - - --.._------ - -~
~
~
.,---
_ ..
LEGEND: --80TH FACINGS ISOTROPIC ---BOTH FACINGS OR THO TROPIC For dissimilar faces see 13.2.3
o
o
"
[/
END
j
\\
,
8
~
., ~l ~~'T ,"~~ ;:"-
~T
~
.
,"-"
1I 2 0 h2y
1
I--
10
ff2 = Kp 0
Ncr
0.2
I
I
I
I
0.4
0.6
~8
£0
~
I
I
0.8
0.6
JL
0.4
0.2
o~
b
b
a ~"
~
FIGURE 13.21 - KM FOR SANDWICH PANEL WITH ENDS SIMPLY SUPPORTED AND SIDES CLAMPED AND ORTHOTROPIC CORE (G c b=2.5 Gca ).
13-35
STRUCTURAL DESIGN MANUAL
/2~--~---+----~--~
I0
~--t-----+--+----:-t--+--+--':"'--I-
8~--t---~r---~--~-+--4---+
I----t---+~;--t---_i_~-~--+_-__+_---.f._.....---.-
-
2~-~---+----~---~--
LEGENO: 80TH FACINGS ISOTROPIC - - 80TH FACING$ OnTHOTROPIC For dissimilar faces see 13.2.3 o~--~--~----~-~--~--~---~-~--~-~
o
0.2
0.4
0.6
a b
FIGURE 13.22
,
13-36
~
0.8
1.0
0.8
0.6
0.4
0.2
b
a
FOR SANDWICH PANEL WITH ENDS CLAMPED AND SIDES tSIMPLY SUPPORTED AND ISOTROPIC CORE (Gcb=G ca ). ~
0
STRUCTURAL DESIGN MANUAL 14
LEGEND:
80TH FACINGS ISOTROPIC ---BOTH FACINGS ORTHOTROPIC For dissimilar faces see 13.2.3 12 ,,2
Nc:., =KI7D 2
V
= .If;f2; b U
~~1 t.J~ p4G~
8
~
END
~
1 1 I:l
I---h-., 6
) V=O.8 ----
--
.~t--
. _ - _ .. _
~
_ _ _ .__ . . . ___ _ _ _ ~
O~--~--~----~--_L
o
0.2
0.4
0.6
a b
~._.
__________
~
____ _
____
O.B
L __ _~_ __ L_ _ _ _L __ _._L_ _~
1.0
0.8
0.4
0.6
0.2
0
b
a
FIGURE 13.23 - KM FOR SANDWICH PANEL WITH ENDS CLAMPED AND SIDES SIMPLY SUPPORTED AND ORTHOTROPIC CORE (G cb = 0.4 Gca > 13-37
STRUCTURAL DESIGN MANUAL 14r----.----r---nr----r-----.----r----r------------~
".2
12
Nc.r: K b2 D
-
v= rr2 0
~~l ~
b 2U
2.5Gc
.
10
M-
END
l
tl
J
,
, ~b~ 8 ~~
-
---
6
41----~
2
LEGEND: - - 80TH FACINGS ISOTROPIC - - 80TH FACINGS ORTHOTROPIC For dissimilar faces see 13.2.3
0.2
0.4
0.6
~
b
0.8
1.0
0.8
0.6
0.4
0.2
0
b a
FIGURE 13.24 - KM FOR SANDWICH PANEL WITH ENDS CL:.MPED AND SIDES SIMPLY SUPPORTED AND ORTHOTROPIC CORE (G cb=2.5 Gca ) 13-38
)
STRUCTURAL DESIGN MANUAL -'--r-
14
\\V:O
12
N.
\\
\
\
10
~
- K rr2D
cr -
\\~ \
I
\
V=
"
"
" '\,
~
'""
~
-:::
I
I
~:;,
'-
w*
~~
"
"",
I"---.......... ...-..,'--
~ ........ ~
................ ~ r---.
--
~
ENO
f/t"/l
~ '~:2
~,~
~ - ......
~vl
~b----i
-.
,~ ..
"
(,!S
.. Gc .. V
A
'III
6
/r "t~/ ~V ~ 'V
"
, "'\'"
~v_._ /
'\
'\'.. \
V=O.I
. ~'" t " l
'\,
" "-
\.
8
b2 rr2D b 2U.
'\
'\
,
I
f
--
........... """'-
~ ~:0.2 ....
~ ..... 1"""-
"""'- ...........
----
- ---- -1""-------~-=
""---
~
V=O.4
- -
2
--
~--
LEGEND: - - . 80TH FACINGS ISOTROPIC - - 80TH FACINGS ORTHOTROPIC
-
For dissimilar faces see 13.2.3
o
o
0.2
0.4
0.6
o
b
•
t
I
I
I
I
I
0.8
1.0
0.8
0.6
0.4
0.2
0
b 0
FIGURE 13.25 - KM FOR SANDWICH PANEL WITH ENDS AND SIDES CLAMPED AND ISOTROPIC CORE (Gcb=G ca )' 13-39
STRUCTURAL DES-IGN MANUAL
,
14
,
I
\\ v:1o /2
\\\
\
\
\
\ \ /0
TT2
Ncr::KpD
".. ,~~ \
""\
--
e
\
,,
\.
'.
\
•..•.
~-b-" ~=2
~.
, ..
,{=O.l
...
, , ~::3. ,
~
,'~ "-
"'I"-
~ '-,
-~
~ """-
,. ~
)
-
I.
I
-----.....
---
...... ~ l"" _____ ...... ......
~~ -...
-.---..:..---..
~...... -...-
V=0.4
2
-
~
~
4
-
-
......
......
-- ~~
I
'" ,
", ~
--~
1
f /f/A ~<_.
END'
/l/
~ 6
~;
~.46; V
"
t,
,,
~l ~t lIJ;
"
""-
'-1V V
~
~ ..
\
., "
. n::' . .,,.,
~"~,, t
-
-... - -'---:
I
V=0.8 I
LeGEND: -80TH FACINGS - -.80TH FACINGS
to
.. 0
0.2
0.4
0.6 .Jl
b
"-r--'------
IsaTROPIC ORTIfOTROPIC
I
For rlissimilar. faces see 13.2 .. 3 I 0,8 /.0 0.8 0.6 0.4 0.2
b
a
FIGURE 13.26 - KM FOR SANDWICH PANEL WITH ENDS AND SIDES CLAMPED AND ORTHOTROPIC CORE (G cb= 0.4 Gca ) 13-40
0
STRUCTURAL DESIGN MANUAL
• /21----::l----+-
8
' 6
.... ~"-'-
!
'1-
--........
'"
.........
,~
or--
--
r-"""1-~~~~!:- --- --_-_-11_~ -~I:-;:==:-~~ --~---!
j - - ..... -
"'f' /.
,
f---·-·-t---+-----!"--
r
---
-
_
.... _ _
1----+------1,----- r - - - --------/------/----+---------+----I
)
2 I----j----+----+.---j.-------,..----+-------t----I---- - r - - - - -
~---.
-
-----._,.
------
LEGENO: - - BOTH FACINGS ISOTROPIC - - 80TH FACINGS ORTHOTROPIC
-
For dissimilar faces see 13.2.3 O~--~--~-~---~I--~I---~I-~I----~J----I~~
o
0.2
0 .. 4
0.6
0.8
1.0
0.8
0.6
~
0.2
0
b
b
•
0.4 0
FIGURE 13.27 - KM FOR SANDWICH PANEL WITH ENDS AND SIDES CLAMPED AND ORTHOTROPIC CORE (G b= 2.5 G ) c
ca
13-41
STRUCTURAL DESIGN MANUAL IOO~------------~------~----·-r----------------~
90
•
80 701----%.60~---~~-------~-----4--4------+
50~----~~----~------~-----+
40r--------~,--~--------~~~---+
---
30
-l I
20 ~
~~
10 f - - - - - - - - - - - - - - - + - -
9-------8
1---------------+-----
7 1---------------16
-------------- __ 0----
5J-----4
LE,jENO: --BOTH FACINGS ISOTROPIC - --BOTH FACINGS OR THO TROPIC For dissimilar faces see 13.2.3
3 '------------------'0.1 0.2
0.3
0.4
05
0.6
a b
13-42
FIGURE 13.28 - VALUES OF KMO FOR SANDWICH PANELS IN EDGEWISE COMPRESSION
0.8
.'.0
•
STRUCTURAL DESIGN MANUAL
•
(9)
Evaluate Fer by using the relationships in step (2). If the applied stress exceeds Fer, repeat steps (3) through (9) for a stronger panel.
(10)
Analyze [or face wrinkling, Seclion 13.2.1.
(11)
Analyze for intracell buckling, Section 13.2.2.
13.2.4
Flat Rectangular Panels Under Edgewise Shear
The following method is used in the design of flat sandwich shear panels. It is assumed that the shear load is equally and uniformly distributed over the edges of the panel as shown in Figure 13.29.
I~
a
..... N
I ----
•
----..
......
FIGURE 13.29
~I
....
~I
III
SHEAR PANELS
Overall buckling of the sandwich or dimpling or wrinkling of the facings cannot occur wilhout possible tolal collapse of the panel. Detailed procedures follow, giving theoretical formulas and graphs for determining dimensions of the facings and core, as well as necessary core properties. Facing modulus of elasticity, E, shear modulus, Gt and stress values, F, shall be values at the conditions of use; for example, if application is at elevated temperature, then facing'properties at elevated temperature shall be used in design. The facing shear modulus or modulus of elasticity is the effective value at the facing stress. If this stress is beyond the proportional limit value, an appropriate tangent, reduced or modified value shall be used. (1)
Choose an allowable shear stress (F ). sf using
Determine the facing thickness (t)
13.21 t
= N/2F sf ;
equal faces
13.22
When the shear modulus of one face is different from the shear modulus of the other face, the face stresses are balanced by the ratio
•
13.23 The lower of ,the ratios in equation 13.23 must be used for design, otherwise the face with the lower ratio will be overstressed.
13-43
STRUCTURAL DESIGN MANUAL (2)
The critical facing stress (F Scr ) at which panel buckling will occur is given by
13.24
where E and A are values for the facing with least Fsf/Gs ratio as deterf mined from equation 13.23. If the facings are of equal thickness and of the same material, equat,ion 13.24 becomes
2K (l!b) 2(~'"f ) ,
1[' Fscr = -4-
13.25
In equations 13.24 and 13.25 13.26 (3)
Evaluate the following parameters
b/a
13 .. 27 13.28 13.29
•
where equation 13.29 uses the values of the facing with the minimum ratio from equation 13.23. (4)
Enter the appropriate chart (Figure 13.30, 13 •.31 or 13.32) with parameter bla (13.27) to V = .01. (Choose a low finite value to start since V = 0 gives h as a minimum and Gc as infinite). Move laterally to parameter, equation 13.28, and then downward to equation 13.29. Project laterally and read value of h/b. Determine h.
(5)
Evaluate core thickness from tc
=h tc
(6)
-
t1 -: t2
2
=h
; unequal facings
- t; equal facings
13.31
Delermine the value of K' from
unequal facings
13-44
13.30
13.32
•
STRUCTURAL DESIGN MANUAL
•
'SO'"
u"-
~
~9p~
-;,
~
:,
Tl I~
~ ~
111.--.._Gc_"---I ~.- h ---4 ISOTROPIC CORE
o
•
Q2
0.4
1.0
0.6
.lL a
0.001 .
'-"'- 0.04
0.005 0.06
)
--i~--t-----+--I-----+---+---+-----j 0.08 cr---+---+---'!--
----
--+-'-+---"-'-IO~ 10
LEGEND: --BOTH FACINGS IS07ROPIC -:-BOrH FACINGS OR rHO TROPIC
0.12
For dissimilar. faces see 13.2.4
•
--------------------------.~O'I4
FIGURE 13.30 - CHART FOR DETERMINING h/b RATIO SUCH THAT A SIMPLY SUPPORTED SANDWICH PANEL WITH ISOTROPIC CORE WILL NOT BUCKLE UNDER EDGEWISE SHEAR LOAD 13-45
STRUCTURAL DESIGN MANUAL
•
----
T~ ~l
-
v
lil~ -----
~
I
~'-b-~
)
ORTHOTROPIC
CORE
l-----I---f--__+_
o
Q2
V ~ If.t. 0 - bJU
0.6
Q4·
b
as
•
1.0.
(1
.. ··0.02
0.001
i .-+---t-~, ~-.-.-- 0.04 I
0.005
I
--- .-- -.. . 't--.-
0.06
-c:1-Q
.J.
----L --
... ---. - ....---
0.08
·0.10
LEGEND: --BOTH FACINGS ISOTROPIC - -BOTH FACINGS ORTHOTROPIC
0.12
For dissimilar faces see 13.2.4 I L - - - - - l _ - - L _ . - - L _ - - - - - L - _ - - - L - _ - ' - -_ _ _ _._ _ _ _ _ _ _ _ _-------'
13-46
0.14
FIGURE 13.31 - CHART FOR DETERMINING h/b RATIO SUCH THAT A SIMPLY SUPPORTED SANDWICH PANEL WITH ORTHOTROPIC CORE WILL NOT BUCKLE UNDER.EDGEWISE SHEAR LOAD (Gcb = 0.4 Gca ).
•
ST'RUCTURAL DESIGN MANUAL
•
~
~ -;:;;:
Revision C
!ft!L
~
~
-;;>
Ej, "
-
v
Tj ~! ~ ~ lJ ----2.5Gc
1--0-1
ORTHOTROPIC . CORE
•
o
a2
as
1.0
1----t----+-,=--~=---I---1--::;;;I!'~-_t_-~:.........,~~~-+--+-_t_---_jO.04
0.005
)
~J
r:=:---J-~~-+---+-----+-::~-I----::~-----::;,.r--t--t---j--1---t---
0.06
r: -~j-J--i-jOO8 ~I"
0.010
1 :L-~----L-- --- -jo./o L~~ENO --BOTN FACINGS ISOTROPIC - -BOTH FACINGS ORTHOTROPIC
1°.12
For dissimilar faces see 13.2.4 ~~-~--L--L-~--~-----
•
_ _ _ _ _ _ _ _ _ _ _ _ _ _~o.I4
FIGURE 13.32 - CHART FOR DETERMINING h/b RATIO SUCH THAT A SIMPLY SUPPORTED SANDWICH PANEL WITH ORTHOTROPIC CORE WILL NOT BUCKLE UNDER EDGEWISE SHEAR LOAD (Gcb = 2.5 Gca ). l3-47
STRUCTURAL DESIGN MANUAL K' ::;
(7)
equal facings
Determine tentative core modulus of rigidity (G c ) from 13.34 for V this value
13.33
.01.
•
If
13.34 of G c is not within the range available in the desired core material and type, enter chart in Figure 13.33 along line V ::; K'/G c until a practical value is reached. For the new value of V repeat steps (4), (5) and (6). (8)
From appropriate charts in Figures 13.34 through 13.39, read directly the values of ~ and ~o. For detennining ~o' assume V = O. Evaluate ~ by using
13.35
KF
t
2
K . . equal facings 3h2 -Mo'
= ---
13.36
Detennine the value for K from 13.37 (9)
Substitute the value of Kinta (2) and solve for FScr> This stress must be greater than the allowable stresses Fsfl and Faf2 determined by step (1).
(10)
Check for face wrinkling as outlined in Section 13.2.1.
(11)
Check the panel for intercell buckling, Section 13.2.2.
13.2.5
•
Flat Panels Under Uniformly Distributed Normal Load
This section gives procedures for detennining sandwich facing and core thickness and core shear modulus so that design facing stresses and allowable panel deflections will not be exceeded. This procedure is used in the design of a flat sandwich panel with equal facings, simply supported at the four edges and subjected to uniform normal loading. Facings are isotropic; core may be isotropic or orthotropic. In the case of an orthotropic core, Gca is the modulus of rigidity associated with the shear distortion observed in a cross section parallel to side a. Correspondingly, Gcb is associated with side h.
• 13-48
~(,{j
/i
/
~o\
//---tr-//
~f-r--~I(iJ/ I
~\9
OO~/
~" -~o~)'
--~~ y
W- v
1.)1
.;
0
~i'
/
7
./
,I'
"/
V
7 v /
/
/
/
\O~
/
/
/
v
V
OlL
V
//
v~
~
/
V
/
/
/
)
/
/
l/
1I
/
./
v
'/
/ v lI
/
/
7
7
V
v
V
V1
V
7
v
/
/
V
/
~
cs
/
II
V
uv 7
/
~~Y"
./
,Iv
t.,oO/
/
~/ I;Y ./ . - 'V0°/
vV
V
/
,U;!
~
7
/
V~
V
V •
/
/
/
.7
l/
,/
/
)'
7
V
,uJ'/
/
./ V
7
Vl7 oo~ / />/ i/
/7
I/
/
/
/
,91
.;
/
/
7
/ ;/. /V bt9~ r:h/ /V i//' )" ~/ 1/1V /~. oy v , 7,~/ ,(;0/
~
L
/ 1// 1/
/
/ !.;4
0"/
j"
/
v/
YV
. ~v
7
/
1/
V
(;(, ~7
~~ Y
/7
/'/
\(;(,
oo~ qO'-/
l/ /
p~,/
rPJ
/
/
/
/
~\
-/
~v
/
~~ 1/
"
/
/
1
/ ~.jI
/
V
jzt
7
. U(,J
.~/I 0 }? oor!}
J
/
/
\2
00 ~
/ /
/
/I
/
/
/ [7 (;(,
7
7
/
V \ooy ~~/
V
V
v /
--
1I
II
~ ~ ~
C)
ti
CS
~
C)
M YO 11
FIGURE 13.33 - CHART FOR DETERMINING V OR W AND G FOR SANDWICH c IN EDGEWISE SHEAR 13-49
!/1ff\\ STRUCTURAL DESIGN MANUAL "~-q9.~.~ ...\. '/~~ 10 --a...
9
--IIIIt...
IT
~
-
1 ~ J
8 -
~l
..
V
-.:.r.---:.....
Gc
".2
t b
..
.....-.....- .....-- ...---
V - .,,20 - J)Ti)
j..-' b ------t4
1
/ I 'i-.y'
~
~'
~
5
4
/
)(111°/
- ..--. .
~
~
l...o--:::;::;"
~ r-:::-~
~
/
/'
,//
~//
V
~
i-""'"
Y //
"""",,1-
. ~ ..........
0.05
/
,//
CLJ~~
- - -.--::::. -::-:::::~
// /
FO~~
/ /
/ /
7
0
/
U~~
6
----.--
/1
/
I
~
--~-'
L...--- ~
i-- ---
0.10
0.10
l.---- 0.20
------- .-....... - -- -- -pooo-"
0.05
"'"
- .".,..
---,..",..
O
.-
/~
[1
--I--
T-
Ncr:: K /;TD
0.20
I-- ..-
3
2
--~---
~ r-----I--~it--=.
1--"---'
1--'---
........ --.---+----1----..-
~-----
LEGEND: --BOTH -
-80TH
... --.. -
---
_ _ '- _ _ OAO
- - - '---+---4---1
-------.-~-.-
-
"·1-'-'-'---1 -f--
_._ ..
FACINGS ISOTROPIC FACINGS ORTHOTROPIC
.!L o
FIGURE 13.34 - KM FOR SANDWICH PANEL WITH ALL EDGES SIMPLY SUPPORTED, AND ISOTROPIC CORE. 13-50
.
•
STRUCTURAL DESIGN MANUAL 10 .. --...
9 -
~ ,~ J
8
------
~
~
-
_........
t
~I
.
t
o.4Gc
..
."..- . . . - ......-. -..c--
~-h
f
---+i
I
Ncr::
t)
V:
~l
,,2
KbT D
l~o/ 7-
I~
,,€.
l¢T I
~
---
i==" ...........
-----
-
-.::::'~
4
p.--.-"::-::-.
~
r----" I-- ' f-
3
)
~-
~I"
~ ..
~....,,~
- -
/
V
/
~ ,..,... ... ...,....
~
' .. ~
~
.... 10-
0.04
~
T
~
~
".-
~
I-i..- - -
~
0.08 0.08
_._--
._-- ----- - - -
--------- -----
~
-;;:; ---..."....,/
- " " - "'"
-- I---
0.04
0.02
,,/
.. "" .".""
'----.-
--
~
.-.---- ~ -----
.- .. ~-
---
J'
0.02
o
./
.-~ ~-------
f----
.. ~
.."....,
:;...-'
.. I- -_::: t----.
I-- -
~ ~~
C
~~---
0.16
~ 0./6
..~~
I-
f - - - - - - 1------ 1--'
1------- f - - - - -
I
~
~D~\IE
/
"'"--
-------r-- -
2
-- ...
~
/
AfO~
~ P'
-'--- -
-~---.--
~
~tI'o'
V
-.::;:rt~
~
o
/
'/
J
I
6
/
,,20 ITU
/
7
•
---- -.-- v
•
~.---
--
--- .. --
LEGEND: --BOTH FACINGS .-
ISOTROPIC -80TH FACIN6"S OR THO TROPIC
For dissimilar faces see 13.2.4
_. ' - - -
..
._-._---
a
o
0.2
0.4
0.6-
1.0
O.B
b
•
a
FIGURE 13.35 - KM FOR SANDWICH PANEL WITH ALL EDGES SIMPLY SUPPORTED, AND ORTHOTROPIC CORE. (G b= 0 4 G ) c
•
ca·
13-S1
STRUCTURAL DESIGN MANUAL
•
/o~--------------------------------~.---,----,
9
+--------+-
8 -
~/
---t
/V
f---b ._-..1 -
0
+----r~~~----+---~
/V
.
//
1~---+---~---4----+---~---+~~r---1----t~~
~~--
/
1----------+----+---t---t..-;rfOIlr----6
..
.
~4~f, L---.-v-- /-
1------
~
--
~--- -/r..L.· .-----~ _. ---. ~----.--
"-.
~~
......--- _ .. - - - . . . . . . . - - -
-
-
_I--"'
-,
...
...._____V
-
/0.125
P-'""-.. ----
,-'
.
--....---+----- - - - ...- I
------ ---------=-=r--------#- --- -- ---- ---- ---- --
~L-. _ _~
4
0.125
.. V I fO~~/ ~. ~L----·--cC-U~£ ~ .,/
5
v /
___
-
....
~
__
---~
___
____~
~~------.-
0.25
-.--------- 0.25
-----1-.........,.......-+---=------1--+-----+-------+---- - ----
-
3
~
1-----+----.-+------+-----+----+-----. r------I------ - - --._.---:0.50 L-L--J--t--+--+--t=-=±=-=1=-=-r-~
21-----+---0-----+-------+---+-·-
- - - f - . - . -.. ----- -
I-_-----+---:--.+:I---~d--=_=-=._d-;_=-_=_:;""I_1-:: ..=-._::-:-=_:;; __ .--:_= ___:::-_~ _-,t-_~__~_==-:__ ~___ -:-::._~ -:f-:-:::=: __ ...-=.
/ ------_._-
._._- - - - - -
f - - - - - - ---.--- ----.. ---
__ .==__
-=-.::::' . .:==:1_
1.00
LEGEND: --80TH FACINGS iSOTROPIC - - BOTH rACINGS OR THO TROPIC
For dissimilar faces see 13.2.4 -
01..-----.1...---- _______ J ___1______.J _____ --..: _______ ~ ..___ .___ !___ .J o 0.2 0.4 0.6 0.8 J.O· b (1
FIGURE 13.36 - KM FOR SANDWICH PANEL WITH ALL EDGES SIMPLY SUPPORTED, AND ORTHOTROPIC CORE. (G cb = 2.5 Gca ). 13-52
•
-----.:....:.-.--~-
16
-ll T
II~ 14
~! t 1"
i~
r
.(5
I . Gc... I Il -=- ~.;::! t
12
''
L
r---
=K f! 0
V·
,,'0 b'iT
b---.j
-' 1
~
1=::'-~
1
'-
-
2
14 r
_
-c--
1 ~t :" I
j
j j ..a4Gc~......
, 10
-
~
~t""
aos
_
--
.... ~
"'
_
J
J--+-t----t-T
,- lI
-
---~~-~
!
-I-r-
,-----
---r-
1
1
Q2
-r--
_- ;
~L--l~t:""""L 0.4
b/a..
~ a6
1
-
.
~h---1
---
~
0.10
/~/-
[7
~ '"
V
/
/
cj§lV
6
\~ ~fD--
~I
I
2
(I)
......
=a c: --I
~
-
~
-
l,-.---
~
~
~
c: ::=a 0.051_ rCI
t--- ~
I..-:--
~ 0.10 I
.=
0.20
FIGURE 13.37 - KM FOR SANDWICH PANEL WITH ALL EDGES CLAMPED, ISOTROPIC FACES AND ISOTROPIC CORE
Z
iI: Z
I
1.0
JYI
(I)
CD
c:
L-l
0.8
___
(II)
0.20
I
o
o •
Q2
~ --~
Q4
b/o.
a6
0.8
r
~=~
~
----
4
"'- ~
(~!II
fO~
I
KM
_
==::::::
7)TiI
1
V ~, o~
/
Voir TrIO
8
--~-
-
---r-
:
v
TT2 Ncr:: KI7D
.-...
~.......-..---
/2 ! -
_
~
'
I
"L..
~--~~I-~~r--~ 1 _
.-
IJl W
e
-.I11III.. - - - .
-r--= r---1""--, ..L-+-+-1 - ,___
I--,----r- - - ' I -
ol o
--JIiIoo...
o
V
___ ~
~=r~--I--'
4-
~
-.
-
I~r-- r - - r - - ~
W
L
/V
~
6 --+----.,--r-
.......
_
16
'if,
~
.---
L
iv
+:-fO
r-
8
I
~~
.. T 'f..~O It 1
)~
Ncr
J
I
-10
e
e
e
1.0
FIGURE 13.38 - KM FOR SANDWICH PANEL WITH ALL EDGES CLAMPED, ISOTROPIC FACES AND ORTHOTROPIC CORE (G cb = 0.4 Gca )
r-
:
STRUCTURAL DESIGN MANUAL 16 ---......-....IIIIIro.~
14 -
Ncr
'1 ~1 :
~ ~
f
.2.5~ ..... ~
~
~
,..-
12 '-
........
........ ....-
1 b
I
V'~
-
~\lE.
-
~
--
V ~ ~
V
/
V ~
-
/
/
/
•
fO~
~
l.---- ~
v o
TT'
K /iTO
2 V.I' 71' 0 /J2LJ
J.-b--.j
10
z
~~
~
-
0.05
V V ~
l.----
0.10
6
0.20
4
•
2 ----- ----
o
o
0.2
0.6
0.4
0.8
1.0
b
a
FIGURE 13.39 -
13-54
KM FOR SANDWICH PANEL WITH ALL EDGES CLAMPED, . ISOTROPIC FACES AND ORTHOTROPIC CORE (G cb = 2.5 Gca )
•
STRUCTURAL DESIGN MANUAL
1....
T
ca G
...
b
Gcb
.I FIGURE 13.40
a
-1
1
uniform normal load t lb/in 2 long side a Gc= core modulus of rigidi ty p
II::
SIMPLY SUPPORTED FLAT PANEL WITH UNIFORMLY DISTRIBUTED NORMAL LOAD
(1)
Evaluate the maximum bending moment per inch using the equation given in Figure 13.41.
(2)
Tentatively select panel materials and establish allowable stresses.
(3)
Determine facing thickness in the following weight minimizing expression.
13.38
•
where F
allowable facing stress, psi density of core, {;finS We = W = facing density, #/in 3 f
f
Increase t to nearest standard gage. (4)
Determine core thickness (tc ) from
13.39
)
If practical considerations require unequal facings or different t c , make the necessary changes at this point. (5)
For a panel configuration thus determined, evaluate the parameter V
unequal faces
v=
equal faces
13.40
13.41
13-55
STRUCTURAL DESIGN MANUAL Enter the appropriate charts in Figures 13.42 through 13.46 with b/a and V to determine the value for constants C and C 2 3 (6)
The maximum bending moments occur at the panel center and are determined by the following expression 13.42
13.43 Moments obtained are per unit width and length of ,panel respectively. (7)
Calculate the resulting facing stress from f
f
=
2M a t(d + tc>
13.44
t(d + to)
13.45
The equations 13.44 and 13.45 are based on faces made of the same material and of .equal thickness. If materials or thicknesses are different, the stresses must be calculated using Me/I. If the facing stress is greater than the chosen allowable design stress or if considerably below, iterate the previous procedure to obtain a more nearly optimum design. (8)
The maximum shear loads occur at the mid-length of the panel edges and are determined from S
a
Sb
= 16pb 3
C . shear on side a 4'
13.46
= 16pb 3
C . shear on side b
13.47
rr
5'
1'('
Enter chart in Figures 13.47 through 13.50 to determine C and Cst 4 (9)
Evaluate shear stresses 2S
f
sa
= _--.;..a_ d
+ tc
2S f
sb
b
=--d
+
to
13.48
13.49
Choose an available core to meet the stress requirement of 13.48 and 13.49. (10)
If panel deflecti?n is limited by the design criteria, it may be determined by
13-56
•
STRUCTURAL DESIGN MANUAL
•
Revision A
f3 pb
M max
2
=
moment @ middle of pane 1
I--b a/
b
1
1.2
1.4
1.6
1.8
063
.075
.086
.095
2
3
4
.119
.102
.124
5
.125
.125
FIGURE 13.41 - MOMENT AND DEFLECTION IN THE CENTER OF A RECTANGULAR PANEL WITH UNIFORMLY DISTRIBUTED NORMAL LOAD 0.8
-----..
•
0.7
0.6
CL
anti C3
..
Gc
'" " ~
Gc
'""
C2
"-
0.3
~
0.2
,.....
C3
0.1
•
Isotropic Core
~
0.4
for l--- b~
"
0.5
....
{}
0
---
V
/'
~
.....
~
~
~
0.25
0.5 b/a
0.75
1 • 00
FIGURE 1 ~L 42 - MOMENT CONSTANTS FOR FLAT PANEL WITH ISOTROPIC CORE UNDER NORMAL LOAD
13-57
I-'
VJ I lJI ():)
1.0
1.0
r~
.9
.9
,
'" \
\: Itt
::;or,
01
£
,~~
~~ .. .~
._~,
,ell
.. . '\
"
;:~y/
"~'~"~7 "'i=::::;::;::V
.8
.8
(I)
.....
•7
.7
I j
V=l 5
.6
.6
V=2JO
b/a
::a
c: n -I
b/a
.5
c:
.4
.4
:c-
.3
.3
.5
::0
......e
~
Gca .1
•3
.4
.5
.6
•7
.8
0
~
--'
•1
.2
.3
.4
.5
.6
.8
•7
FIGURE 13.44 - MOMENT CONSTANT, C2' FOR FLAT PANEL
FIGURE 13.43 - MOMENT CONSTANT, C2' FOR FLAT PANEL WITH ORTHOTROPIC CORE UNDER NORMAL LOAD
e
101
Z
.
0
.2
CD
~b--t
o •1
-
-
•
~b~
o
m_
(I)
Gc b=2.5 Gca
.2 \-
GcatDI
•1
c:::::. I
ORTHOTROPIC CORE
ORTHOTROPIC CORE Gc b:;.4 Gca
.2
....
WITH ORTHOTROPIC CORE UNDER LOAD
e
NO~~L
e
:cz e: :cr-
STRUCTURAL DESIGN MANUAL .6
OHTl10TROPIC CORE
Gcb = 0.4 (;ca
...
...
.4
C 3
.3 .2 1.0
.1 0 0
0
•1
.2
.3
.4
.5
.7
.6
.8
.9
l.0
FIGURE 13.45 - MOMENT CONSTANT, C3J FOR FLAT PANEL WITH ORTHOTROPIC CORE UNDER NORMAL LOAD
•
0.25
ORTHOTROPIC CORE
- -
Gcb = 2.5 Gca
0.20
Gca
IOI j-.:-b~
O. 15
')
C3 O. lO
v=
0 .. 05
v
1. 5
= 2.0
o o
.1
.2
.3
.4
.5
.6
•7
.8
.9
1.0
b/a
FIGURE- 13.46 .. MOMENT CONSTANT, C3, FOR FLAT PANEL WITH ORTHOTROPIC CORE UNDER NORMAL LOAD 13-59
STRUCTURAL DESIGN MANUAL ~ 0.9
ISOTROPIC BORE
... ... c
!--
Gc 0.8
-
~
~
'"~
101 t--b--i
Cs
0.7
0.6
o
•1
.2
.3
.4
.5 bJa
.6
-
~
~ "" ---
..........
r---- ~
.7
1.0
.8
FIGURE 1.3.47 ... SHEAR CONSTANTS, C4 AND CSt FOR FLAT PANELS WITH ISOTROPIC CORE AND NORMAL LOAD
1.0r---~----~----~----~----~----~----~----~----
v=
__--~
0
•
O.9r----4-----4----~----~~~~~~_r----_r----~----~--~
ORTHOTROPIC CORE Gcb = .4 Gca
0.8
... .....
C 4
Gca 0.7
2.0
to} t--b--t
0.6~---4-----4-----+-----+----_r-----r-----r----~~~*---~
o. I)
L -_ _- - - J . - -_ _- - - 1 - - -_ _- - ' -_ _- - ' -_ _ _--"--_ _ _ _- - I -_ _ _ _- - ' - -_ _ _~_ _ _ __ ' _ _ - - _
o
.1
.2
.3
.4
.5
.6
•7
.8
b/a
FIGURE 13:48 - SHEAR CONSTANT, C4, FOR FLAT PANELS WITH ORTHOTROPIC CORE UNDER NORMAL LOAD 13-60
.9
1.0
•
STRUCTURAL DESIGN MANUAL
•
I
I
..
Gel>= 0.4 GCll
ca t
1.0
0.8
V ==
lop
V =
o.p
1- _ _ _
V
=
...., -L
------
r-----r-----.
~ I--. ~ ~
~~
---'---------------r-----.
0
~
r-
0.6
o
•1
.2
.4
•3
-
a
~
0.7
• .5
.6
•7
.8
b/a
•
T
I--b
--- r----- ~
r---
....
G
---r--=:
V == 2. ~ V ::: 1.5
0.9
--...
.9
1.0
FIGURE L3.49 - SHEAR CONSTANT, Cs, FOR FLAT PANELS WITH ORTHOTROPIC CORE UNDER NORMAL LOAD 1.0
v ::
2.0
I.S 0.0
ORTHOTROPIC CORE Gcb = 2.S Gca
0.8
Gca~Dl
...........--..
t--
C4
)
I
ORTHOTROPIC CORE
b~
and
Cs
0.7
0.6
o.s
•
o
•1
.2
.3
.4
.5 b/a
.6
•7
.8
.9
1.0
FIGURE 13.50 - SHEAR CONSTANTS, C4 AND CSt FOR FLAT PANELS WITH ORTHOTROPIC CORE UNDER NORMAL LOADS 13-61
STRUCTURAL DESIGN MANUAL 13.50 where
•
13.51
D
equal faces
13.52
and C is determin~d from charts in Figures 13.51 through 13.53, interpolat1 ing between values when necessary. If 6 exceeds the design limit, increase the core thicknc"ss, and if necessary, the facing thickness until the deflection is acceptable. Repeat steps (5) through (9) to determine new, lower stresses. 13.2.6
Sandwich Cylinders Under Torsion
This section gives the procedure for determining core thickness and coce shear modulus so that overall buckling of the sandwich walls of the cylinder will not occur . Buckling of the sandwich walls, dimpling or wrinkling 6f the fairings or sidewise buckling of the cylinder cannot occur "Wi thout posslbl(' tolal collapse or Llw cyl ind~r.. Detailed procedures giving theoretical formulas and graphs for deLcrmining dimensions of lhe facings and corc, as well as necessary core propt!rties follow.
•
I-L--1 T
j
t1{f---B T
r = radius of outer surface in inches
FIGURE 13.54 (1)
CYLINDERS IN TORSION
As a first approximation in determining the required facing thickness assume each face carries half of the shear load. Then t
where T t
r 13-62
torque
= thickness = radius ~f
of either face outside surface
13.53
•
STRUCTURAL DESIGN MANUAL ISOTROPIC CORE Gc
2.0
1.5 C 1
v=
1 0
=
0 5
V
\
1.0
0.5
o
o
.2
•1
.4
.3
.5
•7
.6
.9
.8
L.O
b/a
'FIGURE 13.51 - DEFLECTION CONSTANT, Cl' FOR FLAT PANELS
•
5.0
tv =
4.5
2 .0
'"" ~
4.0 3•
V = 1 .5
Jc:
.......
3.0 C
1
2.5 V
=
lJ
{J
--
S
~
1.0
-
V = 0
.5 0
•
0
- ..
~
1 .0
~
=
.1
-
.2
.3
t01
Gea
t-b--l
~
" "-.. ~ ~~
,
-
Geb = 0.4 Gca
~ '-
2.0
1.5
I
I
I
ORTHOTROPIC CORE
~
.......
~
'---..
~
~
"
~
~
"'-
'" ~
~
~
~
------------- ------..............
-..............
~
.4
.5
.6
.7
.8
.9
1.0
b/a
FIGURE i3.52 - DEFLECTION CONSTANT, CI , FOR FLAT PANELS WITH ISOTROPIC CORE UNDER NORMAL LOAD 13-63
"~
'" ;7f'\~ --.; ." 1
/;//
,\\\
"\;V~", STRUCTURAL DESIGN MANUAL
• 2.0~--~--__--~----~--~--~--~
1.5~--+---~---+
ORTHOTROPIC CORE Gcb :: 2.5 Gca
-
____~__4 -_ _~_ _~
0.5 r---~---r--~----+----r~-d----~~~--~--~
o
•1
.2
•3
.4
.5
.6
•7
.8
.9
b/a
1. 0
•
FIGURE ]3.53 - DEFLECTION CONSTANT, Cl, FOR FLAT PANELS WITH ORTHOTROPIC CORE UNDER NORMAL LOAD
13-64
•
STRUCTURAL DESIGN MANUAL (2)
Choose a practical core depth and density.
en
For
tht· previous cOl1rij.;.urnLion determine thf' [acing .sln'sses hy
Tr f
so:;
0
J
i
where
13.54
inner facing
13.55
radius to midline of outer facing r. == radius to midline of inner facing Jl :::: polar ~oment30f inertia of cylinder r
:;::
0
= 21rt ( r.
l
( 4)
outer facing
+
r
0
)
Calculate the shear load on the cylinder by T
=
N
5
217'r
13.56 c
where
+
r
r (5)
c
=
0
r.
1
2
13.57
Delermine the critLcal buckling load [rom 2KE t tc N
f
:;::
cr
r
13.58
c
where K is determined by entering the appropriate chart in Figures 13.55, 13.56 or 13.57 with parameters 2
J' = L /dr
c
13.59
ltcEf V
:::
2ArcdGc
13.60
where GC is the circumferential core shear modulus. (6)
If Ncr is smaller than the shear load Ns calculated in step (4), increase the. sandwich strength and repeat steps (1) through (6).
(7)
Analyze the des
13.2.7
for intercell buckling per Section 13.2.2.
Sandwich Cylinders Under Axial Compression
This section s the procedures for determining core thickness and core shear. modulus so that overall buckling of the sandwich walls of the cylinder will not occur. 13-65
STRUCTURAL DESIGN MANUAL
2.2
~----+---+--+-4-+~~+-----~--~
2 .. 0
I-------+----+---+----il--l-t-+-IH---~
)
I.• 6 tt----+---+---+--+--+~I-++-----
L.. 4 f-+----y'f---+---+--+-+~~+------
K 1.2
~~--~--+--+-+-+~~+------
ISOTROPIC FACINGS ISOTROPIC CORE
o 10
20
40
100
1000
l"IGURE 13.55 - BUCKLING CONSTANT, K, FOR CYLINDERS WITH ISOTROPIC FACINGS AND ISOTROPIC CORE AND TORSIONAL LOADING
13-66
10000
)
STRUCTURAL DESIGN MANUAL 2.0
It
1.8
lo6
l.4
~
\
\
1.2 K
1.0
.8 •6
i'-...
IG c
VV = (
"< /
/
-
-I.-
r-
--
"-- ~I-
( .2
/V
"--'" -=0
V'
~V
.4
.-""
~
r--
4 I
~ .......
~
r-
/
( .6
~
......
----- --
'~
---
r---
.2
rrr-- r-- r-
c:?
- --
ISOTROPIC FACINGS ORTHOTROPIC CORE
- ....... -r-
:--
i
0 10
e
40
20
1000
100
FIGURE 13.56 - BUCKLING CONSTANT, K, FOR CYLINDERS WITH ISOTROPIC FACINGS AND ORTHOTROPIC CORE AND TORSIONAL LOADING
2.0 1.8 1.6 L.4
1.2 1.0
.6
'--
\
~
IG
\
\
V
,,\
V ~
/V
~ 2.5G
" ---- ----
0.2
.4
c
.....
- -
V
( .4
-
-
I--
-
I
~ ~ ~ I'--. ~ '<:
l-
-C - -
0
'"
~
K .8
10000
---
-H-
0.6
\
~ :::::: :::: r-.. C2 r--:::: ::: ~~ .......
---=:::::
.2
-
0
10
20
40
100
1000
10000
FIGURE 13.57 - BUCKLING CONSTANT, K, FOR CYLINDERS WITH ISOTROPIC FACINGS AND ORTHOGRAPHIC CORE AND TORSIONAL LOADING
13.. 67
STRUCTURAL DESIGN MANUAL Theoretical formulas are based on buckling loads for classical sine-wav~ buckling. The theory defines the parameters involved rather than determining exact coefficients for computing buckling "loads. Large discrepancies exist between theory and tests and, unfortunately, the test values for buckling of thin walled cylinders in axial compression are much lower than expected by theory. Design information based on large deflection theory and diamond shaped buckles give results less than onehalf the buckling loads given by classical theory. Until sufficient test data are available, the two methods of analysis, large deflection, theory, and small deflection or classical theory must be used. The two methods are presented in this section. The designer may take his choice, but this choice should be dictated somewhat by the application of the structure.
A.
Large Deflection Theory
The following method is used in the design of a sandwich cylinder subjected to axial compression loading. Assume" the load is applied uniformly to both facings. Either the outside or the inside diameter is given. The sandwich has isotropic faces and isotropic or orthotropic core.
* I-- L--I N
t-1-" --~ N
N
uniform axial load in Ib/in of circumference"
FIGURE 13.58 - CYLINDERS UNDER AXIAL COMPRESSION (1)
Choose an allowable compressive stress {F } for the facings and determine f the approximate required thicknesses by 13.61 t =
N/2F ; equal faces f
13.62
For facings of different materials, maintain the ratio 13.63 (2)
Determine the following parameters. assigning subscripts in such a manner that equation 13.64 is ~ 1. 13.64
13.65 13 .. 68
STRUCTURAL DESIGN MANUAL
•
(3)
I
Enler chart in Figure 13.59 with V c 0.1. Project vertically upward to parameter determined by equation 13.64. Proceed horizontally to appropriate cone shear modulus (G ) curve, then downward to parameter of equation 13.64 c and across to equation 13.65. Project upward to hlr and read the value; use it to determine a tentative sandwich configuration (1' ~ l' ). o
h = r (hII')
13.66
Determine core thickness (tc) from tc
=h
- (t
te = h (4)
1
+ t 2 )/2; unequal faces
13.67
~
t; equal faces
13.68
Estimate the value of r, the radius to the centroid of the cylinder wall, from 13.69 This is true for equal facings only but is sufficiently accurate for most practical cases involving unequal faces.
(5)
I
I
Determine constant K , relating V
K'
=
to G by c
13.70
2 EfIt 1 tc' 3 A I'd I
For unequal facings evaluate K for each facing; use lower value. (6)
Enter chart in Figure 13.63 with VI = 0.1. Project horizontally to approximate VL = KIIG diagonal and read the value for G. If this value c c of G is impractical, move diagonally to a desired value. Read the new Vi • e
For the new value of V' repeat Steps (3) through (6), iterating until a satisfactory solution is reached.
)
NOTE: (7)
J
For values of V 13.62.
1.0 use charts in Figures 13.60, 13.61, and
For sandwich buckling analysis t evaluate the parameters
V
•
~
V
=
2 3A Eftt c
3AdrGc
unequal faces
equal faces
13.71
.13.72 13.73 13-69
~
ISOTROPIC CORE
EJ
fG c
c:-:;> .4 Gc
ORTHOTROPIC CORE
o
.1
.. 2
.3
.4
.5
.6
.7
.8
.9
1.0
.12
.14
.16
.18
.20
Vi
hi r .1
1
2 .1
4
.001
.005
.010
.02
FIGURE ll.59 - APPROXIMATE DESIGN CURVE FOR CYLINDERS UNDER AXIAL COMPRESSION 13-70
.. 03
.04
•
STRUCTURAL DESIGN MANUAL
'\11'17\'\\\ \ffi-~.~.~., '~, ,-!/
.9
.0-··
ti
\
.8
.7
4 3
Ef2t~ IV; / /V / \.... - ...-.. --.~ --///
~\
\
1
\
\
\
\
\
r\.
/;1/
\ \S
IV/I / //; I / V/ V II V
-
\'
I
jJ
\ \~ -\
f/ /
~
JV '\V 1
2
3
0
~
Ef1t}
1 2 3 4
~ /
8
.12
.14
10
9
hI r
ISOTROPIC CORE
Ef2t2
7
V'
~ c
~
6
5
4
~ Gc
)
/
V
\ \\
\,
~ V
'& v
, ~L ~~
r'
.02
.04
~ ~ 1\\
.06
\ \ \."
.08 .10
.16 • 18
........
~,,~ '-.~
,"'-
Ff J)'"
~ ".
" \ \\ \:"'-'"~ \ \ \ \: ~ '" \ \ \ ~ '" \ \ 1\ ~'"" Ef
~
~
~
~
"",
\
.001
•
1
2
.002
~003
.004
~ .01
'~ ~
"""- .007
.005
FIGURE 13.60 - DESIGN CURVE FOR CYLINDERS WITH ISOTROPIC CORE UNDER AXIAL COMPRESSION
13-71
STRuc-rURAL DESIGN MANUAL .8
.7
2
4 3
Ef2t;2
____~__~_r+_--4---+---~--~E~~tl --~--~~++~~--~--~1
1
EJ~Gc
2
3
4
5
6
7
8
9
10
.10
.12
.14
.16
.1B
Vi
hI r
~ .4 Gc ORTHOTROPIC CORE
0
.02
.04
.06
.08
.01 1 2 3 4
.007
I
FIGURE 13.61 - DESIGN CURVE FOR CYLINDERS WITH ORTHOTROPIC CORE UNDER AXIAL COMPRESSION 13-72
STRUCTURAL DESIGN MANUAL .9 .8.7 432 1 , E t, f2
\
le
d
1.0
\\
\
\
----
--
-
'----
\
\
\\
\
/
.# \
\
II J
\
1\ \ \ \ -
\~
~,
Lf / J / V '/; 1/ / V V> V ~-
// / / II / /
\
\ \\ \
1
If/ V
/
J/
r
1
3
2
4
5
()
7
8
q
.10
.12
.14
.16
10
Vi
, C
hi r ORTHOTROPIC CORE
0
I
/
/~
)
/, 1 2
3 4
??; /
& v
~
"A
~
~
.02
.04
.08
~ ~ r---- I t \'\ 0 "~ J r---- r----
\ \\ ~ \ \\" ~~
~
\ '\ \ \ \ .001
•
.06
.002
r-
~ "-
--
~
~ ~, \1\ ~ ~ .003
.004
.1 8
F£' rx' E,..
~
~
----
,
...........
"
.005
FIGURE 13.62 - DESIGN CURVE FOR CYLINDERS WITH ORTHOTROPIC CORE UNDER AXIAL COMPRESSION
13-73
.01
.007
STRUCTURAL DESIGN MANUAL 2.00
.~ ,
1.00 I ' 0.80
0.60 0.40
0.20
V' 0.10
0.08
l'\
~
'"
~
"I~'"I'" '",
'\..
"-
"-
,
"-
c·
0.01 1
2
.'-..
~.
'"
"
"
)'
6
....
"-
""-
1"\
"\ ~
L
I~ ~
~
"'.." :.'\.. :-'
-'-
·V/'J.:'\.. ~
f"
,
)..,'\ " ",~
'.:'\
°h
/P.d\ '"
~
.
-r.,. '
..
'-
Ou"':\.
'Q~ V~0
c" , -'
ftt;"',-.,. "' r\. .~.
~~
I'\.
!'..
"-
"-
" "\
"~
t>r
'-
~
If) ~
~
"'0;,
-'K'" r\
" "-
~~,
r-:0
~~
VI)
~['~ ,;) .
~
"
" Ji0 '!.h. £
IO~ ~ ~
,'"
~,
"'.r"-...... ,
(f)" .. ~.
~ _"7:
~~Li., ~.,. '"
~
"-
I\~ i\
'"f\ ~~ r\.
1" "I
",
"-
'""-f'\. "'1"\ I", '" "' " K " " ~ '1\ ~~I'"~ eo ~.~I"" "I\K
I\~
f'1\
""\
~~
~
L:..~
v
·vOO~ !\. .-(
.~
'"
8 10
i\
"
~
~
~
1\1\
~~
20
\..
~
1\
~
(
'[\
"
~
'\
-r ._
fa t\ ':1(; t'
"
~
'/~
~rl~
.1. .....' \
4
1\
~,
~
"t ~
~
"~.~ .
~
~
"'"-
"-
-'.
",,-"-
IL
i'
"'\.. " .", "
I"~
1'"
L~
,"" ,
,
" [\
/0.
0.02
~
~
''""
~I"" '-
I"
, " ~
~
~
" , ~
"-
"
~~
~
""
'~
f' [\
I"\.
"'"' " " '\.. " "i\
0.06
0.04
'"
I'"
40
60
"
I"
l"-
~
100
~
1\1\
1\
1"\
"
200
~
400 600
1,000
G c ' ksi FIGURE 13.63 - CHART FOR DETERMINING Gc FOR CYLINDERS UNDER AXIAL COMPRESSION 1.0
r\,
Ut
~
r
,\
l'
0.8
Gt ISOTROPIC CORE
~\ ~\
.\\'
~
0.4
G,
C~
,,\\'\\
FIGURE 13.64 - CHART FOR DETERMINING K FOR CYLINDERS 0.6 UNDER AXIAL COMPRESSION WITH ISOTROPIC CORE
I--
t
\\ ~
1\' "- ~
~ \ '~ ~
'- ~
~
0.2
~
"~ t' ,
."
r--..... i'o..
j"--...
"'" ~
~
~
tit *
.. !"fl_' d
..
-r""'"'---. r--
,
.
0.7
r--Ir- r-- r--
r-.. r- r--
0.8
0.9
~
1.0
o 13 .. 74
•
o
2
v
3
4
5
•
STRUCTURAL DESIGN MANUAL
•
Enter the appropriate chart in Figures 13.64, 13.65, or 13.66. value of K. (8)
Obtain the
Determine the ratio of the facing stiffness to the sandwich stiffness
13.74
unequal faces
RF (9)
2
/3h
2
=4
5'
ccr
FCCl'
=
K
2 K ~ ~l+RF 5 r -A-
unequal faces
13.76
13.77
; equal faces
Unequal faces must both be checked to insure that F (10)
13.75
equal faces
The value of facing stress (F ) at which buckling of sandwich wall will cr occur is
F
•
=t
ccr
> ff.
Check for overall column buckling using
N
13.78
eer
)
equal faces
(11) B.
13.79
Cheek the cylinder faces for intracell buckling per Section 13.2.2. Small Deflection or Classical Theory Proceed through step (6) in the previously described large deflection method. This will give a satisfactory first approximation.
(7)
For sandwich buckling analysis evaluate the parameters
v 13.80 13-75
STRUCTURAL DESIGN MANUAL FIGURE 13.65 CHART FOR DETERMINING K FOR CYLINDERS UNDER AXIAL COMPRESSION WITH ORTHOTROPIC CORE
,
1.0
U
\
\
•
G,
C-~
\,
0.8
t
O.4G"
1\
ORTHOTROPIC CORE
\ 'l
0.6
i\
1\' \ \:
K
\~ ~
0.4
\
~~
~ "'r-.;,.. ~ ~ "'- '- ~ r-.
",
\
o
I'-.
r--. r- r-
~ ............
o
1
d
........ ....... ~ ~ r- ~ to-- !-- i--
......... 1-0.,.
0.2
,iL
:-- r-
---I-.
3
2
-
0.7
!-- i--
r-- !--
0.8
0.9 1.0
•
5
4
V FIGURE 13.66 CHART FOR DETERMINING K FOR CYLINDERS UNDER AXIAL COMPRESSION WITH ORTHOTROPIC CORE
1.0
\.
0.8
~
U1
G~
,
~
IT
\
2. 5GcORTHOTr.OPIC CORE
\
,
\
I
~ \
0.6
\\ \. \\
"" " " "",
K
1\ \.
I~
~
0.4
~ '\
'- "- ~
',-
~,
0.2
o
.......
~
i'. "-
"
"- 1"00.... "
~
"' r-...... '0
,
2
~
......
r-3
V 13-76
d !
---- -
r-- .......
........ ......... ~
- r-~
..........
t
...:E..
~
-
""'-
ctt
"'"'-
--
0.7 0.8 0.9
1.0 4
s
•
STRUCTURAL DESIGN MANUAL
•
v
Ef tc t
=
equal faces
13.81
2 ~hrGxz
(8)
Enter chart in Figure 13.67 and obtain a value for K.
(9)
The value of F at which buckling of the sandwich will occur is cr F
_ 4 K Eft Ef2 hfi1t2' cr ~a(tl t 2 )
unequal faces
KE h
F cr =
(10) and (11)
•
~; equal
13.82
13.83
faces
Same as steps (10) and (11) for large deflection.
(12)
Check for face wrinkling per section 13.2.1.
13.2.8
Cylinders Under Uniform External Pressure
This section presents formulas, theoretical equations, and a design procedure for determining the sandwich facing thickness, core thickness, and core shear modulus such that overall buckling of a sandwich cylinder will not occur at the facing design stresses. The following method is used in the analysis and design of sandwich cylinders subjected to uniform external pressure. The facings 'are isotropic, but may be of different materials and different thicknesses. The core may be either isotropic or orthotropic. The outside diameter and cylinder length are given as part of the design criteria.
p
.>
uniform external pressure, lb/in 2
p
FIGURE 13.68 - CYLINDER UNDER UNIFORM EXTERNAL PRESSURE (1)
•
Select a tentative sandwich configuration given cylinder length (L), the outside diameter (Do) and external pressure (p). Choose: a.
Facing materials and thicknesses.
b.
Core material: must satisfy F
The flatwise compressive strength of the core, (F ), c
~ 1.5
c 13 • 84 13 ... 77
STRUCTURAL DESIGN MANUAL 1.8
,
~
1.6
~
\
,
1.4
\
G,,:
~
\
" --"
,~ .Q ' '/
...
.)'~
·0
0.6
.".I
1\ \ t!i>'\
0.8
./
"CSl
K
-
G"t
1.2
1.0
•
""-""",,,,-
~
~ \. ~ """"
~
0.4
0.2
o
0.2
0.4
" '" 0.6
~
0.8
......
............
---- -...
1.0
1.2
•
FIGURE 13.67 - CLASSICAL BUCKLING COEFFICIENT ISOTROPIC FACES AND ORTHOTROPIC CORE
13-78
•
STRUCTURAL DESIGN MANUAL
•
c. (2)
A tentative value for the centroida! distance between facings (h).
Calculate the total depth of the sandwich (d) and the mean radius (r ). c d
h
a
+ (t! + t 2 ) ; unequal faces
13.85
2 r
c
= D0
-
d
13.86
2
d ( 3)
h+ t; equal faces
13.87
Calculate the parameter (R) • R =
unequal faces
E1 tl
13.88
E2t2
R
•
(4)
1; equal faces
=
13.89
Calculate the parameter (V).
v
=
13.90
V
=
Et
equal faces
13.91
3r A G
c
where G re
)
(5)
re
Core modulus of rigidity in the radial and tangential direction. 2
Calculate the parameters ( a ) and (Lire ).
Llr c
the ratio of cylinder length to the mean radius
13.92
13.93
h/r
•
c
the ratio of the centroidial distance between faces to the mean radius, where the mean radius is the distance to the center of the core.
13-79
STRUCTURAL DESIGN MANUAL (6)
Enter charts in Figures 13.69 through 13.84 with the values of a 2t LIT c and R. Determine k. If there is no exact chart for the given values, interpolate between adjacent charts.
•
A simpler but more conservative method based on the assumption of a very long cylinder (LIre > ~OO) may also be used to determine the value for k directly. . l2R a 2
k
; unequal faces
13.94
(R + 1)2 (1 - 12 Va )
k
::.II
3 a. 2
equal faces
13.95
1 - 12 Va (7)
Calculate the critical buckling pressure (q
cr.
). 13.96
2Etk r
(8)
13.2.9
c
equal faces
13.97
A
If q
is less than the external pressure, p, select a new sandwich cr configuration and repeat steps (1) through (7). Beams
This section contains the procedure for the design of sandwich members used as beams. The.load is applied nonnal to the face of the sandwich and the member has reaction points at the ends. The edge of the beam is assumed to have no support. (1)
Determine the maximum bending moment (M) and maximum beam shear (S) for the design loading and end support conditions. Figure 13.85 shows some commonly used beams with maximum moments and shears.
(2)
Choose an
(3)
Calculate the required section modulus per unit width (Z) from
allowab~e
design facing stress (F ) which does not exceed either f the tensile or compressive yield stress of the face material.
13.98
where b 13-80
::.II
beam width.
'.
STRUCTURAL DESIGN MANUAL
•
T
0.006
0,001
~'-~.--------~r=---I------------
a:.l.:.-
,.:,/..
0.0001
1------:.---lo...--_-lI"._ _ _~--::lo_~I____4..___--------___I
0.00001 I--_ _ _ _ _ _ _ _ _ _
~I__~....::..._--..l......_-----~
10
100
Ljre . Values of k for V;;;::. O. and for J~I ~~E~12
•
(),OO() I
I.
FIGURE 13.69 - BUCKLING COEFFICIENT FOR CYLINDERS UNDER UNIFORM EXTERNAL PRESSURE
13-81
0.01
I I
0.001 ~-~..--,------3~---t-------'-----;
tV~·
,';oJ.
0.0001
{tOOOI
I-----l'c"---~--~~-~_+__..lr--------~
0.00001
_---5',000006
0.00001 L-----------+---~~-~--==:::::::::====i
FIGURE 13.70 - BUCKLING COEFFICIENT FOR CYLINDERS UNDER UNIFORM EXTERNAL PRESSURE
13-B2
•
STRUCTURAL DESIGN MANUAL 0.01
.
----r---,-- I I
I I I I
q. ,=_~:.!.~. t_~~~·,~::. r ~ ( 1- Ii:.')
k
0.001 t-.~----","" ~-~---+-------------I
0.000011--------
0.000001 __._ _'__--I.-LL1...LJ....l-L...--..J.------I-J.--.IL.-L--'-"-'-' 100 10 1 Ljrr Vlducs of ~ for V := 0, and for
•
=3 .
FIGURE 13.71 - BUCKLING COEFFICIENT FOR CYLINDERS UNDER UNIFORM EXTERNAL PRESSUHE
13-83
STRUCTURAL DESIGN MANUAL
• q..,:;:.:. E,t!±f:'~_'::
k
1C'(JMI-'~)
ll':.!
0.0001
~~_~_~_--l~~.--+-4-
·0.0001
_ _ _ _ _ _ _ _ _-.I
0.00002
O.ooooJ
0.00001
I------------I-..:s.--~k_..l~----....j
)
0.000001
'0
L/'~ V"luc~wf k
for J' .. 0, ami for
El
_.:.1_1_.
4.
/':::1,
FIGURE 13.72 - BUCKLING COEFFICIENT FOR CYLINDERS UNDER UNIFORM EXTERNAL PRESSURE 13-84
•
.'.
STRUCTURAL DESIGN MANUAL 0.01
r------r-···
I
I
I I
0.001 r-~:-----~~---+-------
•
0.00002
0.00001
o.OOOOO(
0.00001 j - - - - - - - - - - - - + - - - . 3 I r r r - - 4 - - - - - - - ' - - - - 1
)
•
FIGURE 13.73 - BUCKLING COEFFICIENT FOR CYLINDERS UNDER UNIFORM EXTERNAL PRESSURE
13-85
STRUCTllRAL DESIGN MANUAL
0.001 l--~-------JIi,r------+-----------I
)
~::::O.OOOI
~
0.0001 ~J.A3l.r---::"--~~r---~-+-~.----------l
• 0.00001 l----------+-~Ior_-+--======1
)
0.000001 '--_ _-I-----JI-....I..-.L.-JL...L...............l-_ _-'--_-L--L-.....&._. 10 1
100
LIre Values of Ii. for V ~-:: 0.5. and for !!.!!L :;:-: 2. E:!(.
FIGURE 13.74 - BUCKLING COEFFICIENT FOR CYLINDERS UNDER UNIFORM EXTERNAL PRESSURE 13-86
•
STRUCTURAL DESIGN MANUAL
• E,I. +E,t!l qr_' _ ----
k
'I'\:(1-p!l)
"2::-..:0.0001
•
0.00002
0.00001
r----------~~---\~~-------J
0.00000 1 ~_~I-----L._1......J~...L.Jl-L:-L---L-1-LU-L.L1J , 10 100
Ljre Value!> of k for Y
•
= 0.5, and for
_.!i.~~~I . . £'lt~
: .:.: 3.
FIGURE 13.75 - BUCKLING COEFFICIENT FOR CYLINDERS UNDER UNIFORM EXTERNAL PRESSURE . 13-87
STRUCTURAL DESIGN MANUAL
• O.OOl~
______
~~
______
~~
________________
~
•
100
10
Lire Values or k for V~' 0.5. ittld for
4.
FIGURE 13.76 - BUCKLING COEFFICIENT FOR CYLINDERS UNDER UNIFORM EXTERNAL PRESSURE 13-88
•
STRUCTURAL DESIGN MANUAL
Ell. +E;!l;! qf ; : : : ; - - - - - fc (I.p.!l)
I<
0.00001 t-----------I-~~--.J~-------I
)
0.000001 : -_ _~--L-L_JI..._1....J.....I_l..L-_ _L____'_L__L'_L..L1J 100 I 10 L/rt V"lucsof k for V:c.: 1, and for
~..!!!..
I.
E'J1';l
•
FIGURE 13.77 - BUCKLING COEFFICIENT FOR CYLINDERS UNDER UNIFORM EXTERNAL PRESSURE 13-89
STRUCTURAL DESIGN MANUAL
• qcr::=
!:~'-±_~3-'~ re (J
0.001
0.0001
I-------~
It
-Jl~)
____ + _ - - - - - - - - - _ _ _ ;
1--11oor--~_---J'r-~-~-t-----lIr------------t
• 0.00001 I----------+-~~-+_--======t
)
0.000001
r_ _ _+-_'--'-...&-""--J.-I-&....L..--......I..-----.-40.................
1
100
10
LIre Elll Valuetl of k for V :;,: 1, ilOd flU -----
;.c-'-
2
•
E::/~
FIGURE 13.78 - BUCKLING COEFFICIENT FOR CYLINDERS UNDER UNIFORM EXTERNAL PRESSURE
13-90
•
STRUCTURAL DESIGN MANUAL
•
~
0.0001 ~::-----'~-~;--.::tr----1~+--\----------I
•
0.00002
0.00001
0.00001 f:"""-----·---Ir-~---1~~~~m~
100
10
1./,,-
E Values of k for V.- I, and for _ ,t_1 E2 r2
•
=3.
FIGURE 13.79 - BUCKLING COEFFICIENT FOR CYLINDERS UNDER UNIFORM EXTERNAL PRESSURE 13-91
STRUCTURAL DESIGN MANUAL
• E,II +E:t /:.> qu=-----
k
Tc O-I':!)
0.001 1 - - - - - - - 3 0 o r - - - - - - - I r - - - - - - - - - - - - - t
.:.t
0.0001
0.0000]
•
0.000006 0.00001
to
Villucs of k for V
LIfe I, and for
FIGURE 13.80 - BUCKLING COEFFICIENT FOR CYLINDERS UNDER UNIFORM EXTERNAL PRESSURE
13-92
•
STRUCTURAL DESIGN MANUAL
~
0.0001
~~
__~__~~~~~__~______________~
• 0.00001 1-------------------1~_*--_4~--------~
)
100
10
Lire
Values of k for V
•
= 1.5. and for
EI/I
E'J I,
= I.
FIGURE 13.81 - BUCKLING COEFFICIENT FOR CYLINDERS UNDER UNIFORM EXTERNAL PRESSURE
13-93
~~
"~-c;~ STRUCTURAL DESIGN MANUAL ~ Elll qr,= _ _+£~/:! __
•
k
Tc (J -p:!)
...
0.001 .I------~"...___--_+_---------
...lIi£
0.0001 ~:---.:r...r------I.r--~-.lor--I----\O---------i
• 0.00001 1----------+-~tr__-~--=======:::f
100
10
Ljre
E, I,
Values or k for Y:::: 1.5. and for - - - ::::: 2.
E:Jh
FIGURE 13.82 - BUCKLING COEFFICIENT FOR CYLINDERS UNDER UNIFORM EXTERNAL PRESSURE
13-94
•
STRUCTURAL DESIGN MANUAL
a 2 =OJ)001
0.00001 1'----------+-~-__4.___::~-------I
)
lOU
10
L/r( Va1ues of k for JI = 1.5. and for
=3.
FIGURE 13.83 - BUCKLING COEFFICIENT FOR CYLINDERS UNDER UNIFORM EXTERNAL PRESSURE 13-95
STRUCTURAL DESIGN MANUAL
0.001 j----...;:a...,.r-~----_t_---------.....
i\'~::~:O.OOOJ
.:./.
0.0001
1--~----"";:liIo&-~t--'4r--...::!io.k-----------f
•
0.00002
o.oonol
0.00001
0.000001
I-----------~..._-~~o:__----_I
I...--_ _+----L_"--I........I-L-..........L----'---"'--'--..L-.!......I.-L..I-"
1
Values of k. for V
10 LIft" 1.5, and (or
100
FIGURE 13.84 - BUCKLING COEFFICIENT FOR CYLINDERS UNDER UNIFORM EXTERNAL PRESSURE
13 ... 96
STRUCTURAL DESIGN MANUAL ..
'
TYPE OF LOADING AND END SUPPORTS
P:'-.1
q
+ t +
TIJ~
~
a
p -==-q
a
--I
POINT OF DEFLECTION
UNIFORM LOAD SIMPJ.JY SUPPORTED
MIJ.)SPAN
UNIFORM LOAD FIXED ENDS
MIDSPAN
M
Pa 8
V
.£. 2
q
f"~~'~ jIll
p
-=q
a
d~"lIl I-- /zl a
tllllll~a~
j:qn II f 'L J...-
POINT LOAD AT MIDSPAN FIXED ENDS
MIDSPAN
p
12
2
Pa
R
4
MIDSPAN
2
Pa
P
8
2
--
P/~
P/2
--4 n/ 4
~
t
t
P/2
POINT LOAD AT MIDSPAN SIMPLY SUPPORTED
Pa
a/ 4, .
P/ Z
p:pcUlJITl1 -l ad-- ~ a/ 1-4
POINT LOADS AT QUARTER SPAN SIMPLY SUPPORTED POINT LOADS AT QUARTER SPAN SIMPLY SUPPORTED
MIDSPAN
Pa
P
8
2
LOAD
Pa
P 2
FREE END
Pa 2
p
FREE END
Pa
p
8
q
~Jll~' ~
P/=q -[J[f]['[[ a UNIFORM AND ~il n ~ CANTILEVER
~ ~
"
a
-=-I
~mrlP a
~
POINT LOAD AT FREE END CANTILEVER
F'IGURE 13.85(a)- BEAM
MOMENTS
13-97
STRuc-rURAl DESIGN MANUAL TYPE OF LOADING AND END SUPPORTS
t • ~ q. 1 ~ II
q =f a
I
II
.
A
~~
---I
a
f!~~I~I~
P -:::q a
Illlld;lIIl
IIIIIIIIG~~ ~ I
jll:HII:IL
-1 a/
--f a/ 41-P/ 2
~
,
4
P/2
q --{ ad- --l a/ 1-jll=IIOII
4
q
~ ~ j
"'~
MIDSPAN
POINT LOADS AT QUARTER SPAN SIMPLY SUPPORTED POINT LOADS AT QUARTER SPAN SIMPLY SUPPORTED
CANTILEVER
II U B
a
---l ,
~ 1I11l111ITn :\
~-
UNIFORM LOAD FIXED ENDS
t 1 P/ a =q UNIFORM AND
i
~ II II II ~
MIDSPAN
POINT LOAD AT MIDSPAN FIXED ENDS
P/ 2
P/2
UNIFORM LOAD SIMPLY SUPPORTED
POINT LOAD AT MIDSPAN SIMPLY SUPPORTED
J-a/-,.4
a--1
POINT OF DEFLECTION
MIDSPAN
MIDSPAN
MIDSPAN
5
1
8
1
1 8
1
1
48
4
1
1
192
4
11
1 8
768
LOAD
FREE END
KS
384
384
%
1
1 8
1. 8
1 2
1
1
P
POINT LOAD AT FREE END CANTILEVER
FREE END
FIGURE 13.85(b) - DEFLECTION CONSTANTS
13-98
~
3
)
)
STRUCTURAL DESIGN MANUAL -- ..--
.6
--,--.----r---..---r-----t--~..__-__.--....,.__-__;
.1:15
.6
.100 ,41--+-
.071 .063 ,050
.032
FACING THICKNESS
.0:15
inches
.010
•
z
.006~~flft~~--4_-_4---4_-~~--~----r_--~--~
)
.004~~~-~--4_-_4--4--~~-~--r_-_+-~
.002
.'
t-1'+--+--,--t---+---_4---+--~""-----~----.j__-__+_-__I
.001
o
.5
1.0
1.5
2.0
2.S
3.0
3.S
4.0
4.5
S.O
d-inchcs
FIGURE 13.86 - DESIGN CHART FOR BEAMS WITH EQUAL THICKNESS FACES 13-99
s-rRUCTURAL DESIGN MANUAL (4)
Calculate a starting value of dlt from the weight minimizing relation 13.99
where W and Wc are the densities of chosen facing and core materials f
respectively.
(5)
From Figure 13.86 find the value of d associated with the value of Z determined in step (3), and move horizontally (Z = constant) to the nearest standard sheet gage. Read the corresponding panel thickness d.
(6)
With the values of d and t thus determined, check the facing stress, 13.100
M
This value of ff should equal (7)
Ff (step (2».
Determine core thickness tc = d - 2t
(8)
13.101
Solve for core shear stress from fs =
13.102
2S
+
b(d
t c)
This value of fs should not exceed the allowable shear strength of the chosen core. (9)
When stiffness is an important consideration, determine the two stiffness parameters, 0 and U
3 3 D. :2E~f /d - c (1 - :; ) t
where Et
c
Ef
= Elastic
13.103
modulus of the core in the spanwise direction
Elastic
modulus of face material
A = 1 _ #1 2
p.
= Poisson's
ratio for face material
For beams with cellular cores, E'c is often very low in comparison to E , f and the ratio Ele/Ef is then assumed to be equal zero. Calculate shear stiffness 13.102 t
13-100
c
•
STRUCTURAL DESIGN MANUAL
•
where h = d- t G c
( 10)
=
= core
tc
+
t
modulus of ridigity
Compu te the def lee tion ( ()
from
13.103 where P L
applied load length of beam
The coefficients
~
and Kg are given in Figure 13.85 for various beam
loadings and end support conditions. If the computed value of 0 is greater than that compatible with the design criteria or good design practices, the beam's stiffness may be increased by increasing core thickness, or by using a core with a higher modulus of rigidity, or both. Any of the above calculations affected by the change should be repeated. (11)
Determine the flexure induced core compressive stress
f
(12)
13.3
)
13.104
c
The core should also be analyzed for local crushing due to concentrated loadings, either applied or at reaction points. Attachment Details
All sandwich parts must be attached to the framework of the airframe and often to other similar parts; therefore, means for transferring the concentrated loads imposed at these attachments must be provided. Occasionally, on very lightly loaded parts, unreinforced bolt holes or subsequently inserted reinforcements will suffice, but in most structural applications, local reinforcements must be incorporated during fabrication. 13.3.1
Edge Design
Sandwich parts are normally joined over a framing member. The edge configuration is often dictated by the loads to be transferred, core, smoothness requirement, fasteners, facings, panel usage, etc. Figure 13.87 shows some commonly used edge configurations. Care should be used in selecting the edge design. If the methods of Section 13.2 are used in the design of the panel, both faces are capable of reacting load. Then, in order to fully utilize the sandwich concept, the edges must be designed to be compatible. 0
Some of the edge configurations have beveled edges, such as a 45 chamfer with fiberglass closure. This is a commonly used configuration at Bell. The load that is introduced into the inner face at the edges is only what can be transferred through the fibergfass edging or shear lagged through the core. 13-101
STRUCTURAL DESIGN MANUAL \ IlllI111!!!
J
,
t 'REINFORCEMENT
[111111111111111
r~
EDGE CD LS FlU CD
HIGH-STRENGTH INSERT,
~IIIIIIIIIIIIIJDF METAL
MZ'
FIGURE 13.87 - SOME TYPICAL EDGE DESIGNS
13-102
STRUCTURAL DESIGN MANUAL
•
Regardless of the type of configuration, the edge design should be such as to keep moisture out of the core. This can be accomplished by the use of potting compounds or fiberglass closures. The facings which have been sized for the type of failure modes discussed in Section 13.2 will not necessarily be thick enough to develop the fasteners. An edge reinforcement must be installed with a thickness sufficient to develop the fasteners. The reinforcement is a doubler, either internally or externally. In some cases, it can be chern-etched integral with one of the faces. When the loads on a sandwich panel are normal to the panel. the edging doubler may have to react these loads. In this case, it must be thick enough and wide enough to develop the bending moment in the edge. 13.3.2
•
Doublers and Inserts
The design of a sandwich structure may be such that loads must be transferred to or from individual parts at points other than at their edges. Inserts in the part are required at these attachment points if the loads are of appreciable nagnitude. Typical methods of introducing high loads into a sandwich panel are shown in Figure 13.88. These may be in the form of strips (metal, wood, foam), inserted continuously across the panel or as local reinforcements under individual bolt patterns. Shear loads on attachment bolts may require additional reinforcement to provide adequate attachment bearing area. This can be in the form of a doubler which can be installed internally or externally. One method of densification (increasing the density of the core so that concentrated loads can be introduced) is to cut out an area of the core and insert a piece of denser core. Another method of densification is to compress the core in a local area so that the cell size is smaller than the main body of the core. 13.3.3
Attachment Fittings
Accessories, such as shelves, fittings, mounting brackets, are often fastened to the sandwich panels. Figure 13.89 shows some examples of how fittings can be attached to sandwich panels.
• 13-103
STRUCTURAL DESIGN MANUAL
A
-
/ HIGH STRENGTH
I
zz
,
)
INSERT
1
'/
RE/NFI~wc. "MENT FOR ADOl TIONA L BOL T BEARIN6 AREA
8
i
1//11/1/ [
IC?III IIIII ~
C
METAL EXrRUSION
o
FOAMEO
•
CORE REINFORCEMENT
FIGURE 13.88 - SOME TYPICAL HIGH-STRENGTH INSERT DESIGNS
•
•
~
,,:W~
STRUCTURAL DESIGN MANUAL
'-:.. ~lJ.:' .-.
.1._
~·t
{
,
!
I _._--
.[fIi·tfllll
L~ ~
1
•
~r
{r~ ~'
I
~ I·' i q~:J
~ Ii [)ift III~
•
FIGURE 13.89 - SOME TYPICAL ATTACHMENT FITTINGS
13-105/13-106
•
•
•
STRUCTURAL DESIGN MANUAL SECTION 14 SPRINGS 14.0
GENERAL
The proper design of springs requires an understanding of spring materials, design formulas, stress analysis and manufacturing procedures. Various aids are available to the analyst including special spring slide rules, tables of constants t curves, charts and nornographs. All are helpful, but an understanding of the basic fundamental formulas and experience in their use is essential to good design. The purpose of this section is to describe the design methods used for each type of spring commonly used. 14.1
Abbreviations and Symbols
The following abbreviations and symbols are used throughout this section unless otherwise specified.
A
)
•
constant for rectangular wire B = constant for rectangular wire b width, in C Spring Index = DId CL = compressed length, in D = mean coil diameter, in d ; diameter of wire or side of square, in E = modulus of elasticity in tension, psi F deflection, for N coils with load P,in deflection, for N coils, rotary, degrees FL = free length, unloaded spring, in f = deflection, for one active coil, at load P, in G = modulus of elasticity in torsion, psi ID = inside diameter, in in. = inch K stress correction factor for curvature L = active length subject to deflection, in 1 = 1eng th , in lb = pound M = bending moment, in-Ib N = total active coils n J vibratIon per minute OD = outside diameter p = load, lb Pr::: applied load~ Ib, (also P2' etc. ) p= pitch, in psi = pounds per sq. in. R ::: dis t.ance from load to central axis, in r spring rate load per inch, Ib/in rt = torsional spring rate, in lb/deg Sb= bending stress, psi St= torsional stress, psi 14-1
STRUCTURAL DESIGN MANUAL Sit= torsional stress due to initial tension, psi SG= squared and ground SH= solid height (or SL = solid length), in s = height load is dropped, in T = torque = P x R, in lb TC = total. coils t = thickness, in O U = number of revolutions = F 360 0 W = weight, (also applied dynamic load), Ib x = multiplied. by Y = constant for Belleville springs Zl= constant for Belleville springs Z2= constant for Belleville springs ~= angle of movement, deg. TT = 3.1416 #J. = Poisson IS ratio
14.2
Compression Springs'
Most compression springs are open coil, helical springs which offer resistance to loads acting to reduce the length of the spring. The longitudional de[l~c tion of the springs produce shearing stresses in the spring wirc. Where particular load deflection characteristics are desired, springs with varying pitch diameters may be used. These springs may have any number of configurations, including cone, barrel and hourglass. They may be made from wire of round, square or rectangular cross section.
•
Figure 14.1 shows a typical compression spring with the nomenclature and a description of the four types of ends which can be made. The ends can be finished into 1) open not ground; 2) closed not ground; 3) open ground and 4) closed ground. Open ends not ground; sometimes called plain ends t has the largest eccentricity of loading. These are used only when accuracy of loads is not important. This type is seldom used because such springs tangle severely during shipping. Closed ends not ground; also called squared ends, cost approximately the same as open end types and have less eccentricity. This type is often used on light wire springs under 1/32 in. dia. wire and for heavier wire where the index exceeds 13. Open ends ground; also called plain ends ground, are seldom used because they cost about the same as the closed ends ground, but have high eccentricities of lOlullug unci tungl c cluTI.,ng shtppl.l1g.. They arc sOUlctilllt' .... used wh(~rc lhl' ~olld height is very lilnited and it is necessary to have as many active coils as possible in the least space.
14-2
•
STRUCTURAL DESIGN MANUAL
• Frp.f' fpnp;th
...
{~rnu nrl
~
'-
Qi
c.u
Sec t ion
~
mQi Qi E
~m
SIze of
Ret\'J~en
11atpri al
rolls
TYPES OF Ftff"J FIniSHES
a-Ea•
Plain Fnrls - ro;lpn
~ight
r-1anrf
Total r.oils
)
= Active roil
Squared
nrounrl rnrls roilerl left PClnrf Tot rofls= Act roils
+
2
_e_e
SQuarerl or t'o t ~ rOllnrf,
Tnt rolls
r,loserl Fnrls
ro i 1en
= Act
Rt. !'rl.
Coils
FIGURE 14.1
+
2
Pl aT n Fnrls - t'ot r,roJnrl I ef t "ann Total rofls = ~ctlve roils
ro j 1en
COMPRESSION SPRINGS
14-3
STRUCTURAL DESIGN MANUAL
Closed ends ground; sometimes called squared and ground, is the most popular type as it provides a level seat and reduces the tendency to buckle. This is the most expensive type and should be avoided for springs made of very light wire. Each end coil is ground for 270 0 plus or minus 30 0 • 14.2.1
•
Design Formulas
Formulas for design of helical compression and extension springs without initial tension are given in Table 14.1. Dimensional characteristics for the four end finishes of compression springs are given in Table 14.2. 14.2.1.1 Diameter Changes in Compression Springs When a helical compression spring is compressed, an increase in the outside diameter occurs because the angularity of the coils changes so that it is nearl.y at a right angle to the axis. The outside diameter when the spring is compressed solid, can be obtained from the following formula: 2 11
14.2.2
J
+
14.1
d
Buckling
Compression springs having a free length greater than [our (4) times their mean diame~ become critical in lateral stability. When deflected beyond a certain percentage of the free length, a spring will buckle. Figure 14.2 shows the maximum deflection which may be expected without buckling if the ends of the spring are closed and ground. Buckling can be reduced, space permitting, by redesign using a heavier size wire and increasing the diameter of the coil. Buckling caus~s an undesirable reduction of the load and may cause early spring failure. If properly guided in a cylinder or over a rod, buckling can be reduced, although friction against the guiding member will affect the load and shorten the spring life •
...
.J:.
100
~
c
&!
G.I G.I
....
0 0
"
....
G.I 0
70
50
0 .~
lU
to GO
~
u
1)0
)0(
Ij...
.= c .-....0
r~D .... Q;
c; G.I
'- .... '"-
•
30 25 20 0
1
2
3
Sl~n~p.rness
6
Il
r.atlo·
FIGtllH 14.2
7
r.r~~ I~nrth f1p;'ln
14 -4
•
!'1"nAt,.r
f1AXUIUI1 I'EFlECTIOfl UITHOtJT JllICKUtlt. OF SPRHIC'iS tIlTH nos EI' AMO r.nnllfll' fT'lIlS
8
e·
STRUCTURAL DESIGN MANUAL
•
TARtE 14.1 r- "
,
~.---
. . . . . ,....
FORMULAS FOR COt1PRESSION "NO EXTFt'SIOtJ SPRItH:;S HITUOOT t~tlTIAl. TENSION
----
.~
.... - - - - .. - - - .
Propl"rt,
"
....... ,....
-~
..
~--
---.-~-
- - r - - - - ' _ .. _-......o.- _ _ _ _ _ _ •
Round wJre ..
Square wire
PD
PD
_._---------_ ... ------ _----- ---_._-_._-- - --
Torsional strt"ss. psi
-_. __
"'.~~.-
_
......... __ . . __
~_
...... ~ _ _ _ _ _ _ _ ... _ _ _ _ . _ _ _. _
Ueet&ntrul.r wlr.-
..
---.-~---.-
PD
- ;rf 1)'2-2.32 N D ~ -'N"D2---0'"-------.-- ... _----_._..... -- ......--. ----,---_. -,_ .•.. _._--_.. ----,-,-._-'-'"
Deflection, in.
-"-"1f"tii-----.--
"---'~---.-.
",St N DZ
F
St N D2
5.68 P N D:I
8 P N Da.
-:A G 't- - - - - - -.. - .. - . ---------.1 -G~
2.32 8 t N D2
-Gd--
--~
Change itl load lb P2-Pl
LI-L2 ~-F--
-p-
Compression springs only
Change in load Ib P2-Pl Extension sprlngg only
L2-Ll --F--
p-
Stress due to initial
•
St
St
tension. psi Sit
-XIT
Rate Ib/in.
P -F-
- X IT
p.
P
-----·---~---.I---=_---J---~---!-----""""-_I
r
P
~
* See FIgure 14.3 cOt~pnESS I ON SPR I ~IG FORMULAS- FOR PIMENSIONAL CHARACTERISTICS
TABLE 14. 2 1"'"'------
TJ'P' of flDtU Dimen.lonal ebaraeu.orlstlea
.-. Pitch (p)
)
---------So1id Height
Open or plain
OJleD 01' plaia with
Square or dOlled
(not.lfround)
eadalrrGliDd
(Dot IrroUDd)
FL-d --N--
FL -TO-
FL-3d N
FL-2d N
TCXd
(TC + 1) d
TC X d
(TO
+ 1) d
(SH)
j
-----.---N-TC
Active Coils
or
(N)
FL-d
N .... TC-l
or FL
~
p _.
•
CIOMd .l1d l'J"QuDd
--.-.. -1 p
N .... TC-2 or FL-Sd
N-TC-2
p
p
FL-:-3d
p
-FL p
(pX Te) + d
pXTC
(pX N) +3d
Total Coila
FL-d
(TC)
Free Length
p
+2
Qr FL-2d
FL-2d p
(p X N)
+2
+ 2d
(FL)
14-5
STRUCTURAL DESIGN MANUAL
.05
.GO .55 )
.50
round .45
Rectangular wtre with sharp corners .40
.35 3
2
1
n{ltio
4
7
5
hIt = (\lirfth = lonp;f'r sitt".) (thicknpss = srortp.r sirlp.)
flfllnr 11~.3 r.ntl~T~JITS 1\ Atln ~ fnp. PFrTfQ-!rlll..t\p
p,n I. '- Fr
Slow.,. appUtd JM4
Buddftbt appUed iouI
W F-r
p--
AppW kJ.d (1IPIiq'
~-
W-PI +
TABLE 14.3
•
IW r
boPped ~
IDttIaD:r
~).
V (W_Pt}1 + IWn
ApP.... load with drilr.IDIr ~eIocl~ of V la/INC.
(aPrbt.
ID I.orl.... taI plaDe)
-Pl+
p-
r
• ., ...... II 110& 1aJ1:Wtr·...
14-6
'.11 nr:
nr
V
WrVI
P11+-aie
r
' . . . . . . Pi
LOAD-OEFLECTION FORMULAS FOR COMPRESSION ANP EXTENSION SPRINGS
•
•
STRUCTURAL DESIGN MANUAL 14 .. 2.
'~
Unless functional requirements dictate a definite direction, the helix of compression and extension sprIngs should be specifJed optional. To' prevent intermeshing of coils when springs operate one inside the other, the helixes should be specified as opposite hand. For the same reason, springs which operate to slide freely overscrew threads should have the helix specified opposite to that of the screw threads.
14.2.4
•
lIclix Direction
Natural Frequency, Vibration and Surge
The use of springs for loads which are applied dynamically; i.e., with impact or rapidly repeated, will be in error if the spring is designed on the basis of static or slow loading. The load, stress, deflection, etc., will have been calculated for applications where the load is applied and held, or the rate of load application is below the natural frequency of the spring. Because of the inertia effect of the coils in instances where the load is suddenly applied, the load on the spring does not have time to distribute itself uniformly throughout the mass of the spring. This non-uniform loading causes deflection or. a surge wave in a few coils of the spring which results in a high stress in this area and a lower stress in the remainder of the spring. In applications of high rate of repeated loading~ non-uniform load distribution occurs in the same manner as suddenly applied loads and the natural frequency of vibration of the spring may be excited. The excitation of the natural frequency of vibration, in some instances t may be of such magnitude as to cause the spring coils to clash causing the spring to destroy its constraint on the mechanism .. This is known as spring surge. The following methods may be used to prevent spring surge:
1. Stiffen the spring a. Increase wire diameter b. Decrease mean diameter c. Decrease number of coils d. Use square or rectangular wire 2. Use nested springs
)
3. Use conical spring 4. Reduce or vary the pitch of the coils near the end of the spring. 5. Use stranded wire The formulas for calculating the natural frequency of steel springs are:
nt
•
= 761,SOOd/ND 2 ; 187.6
UNLOADED SPRING
)TIi- ; LOADED
SPRING
14.2 14 .. 3
14-7
STRUCTURAL DESIGN MANUAL If the frequency of the spring and its harmonics are too low, the spring will surge causing the coils to clash. In general, if the natural frequency of the spring is at least thirteen (13) times that of the maximum frequency of the applied load, the design should be satisfactory. 14.2.5
Impact
If the load is suddenly applied, dropped vertically from a known height or strike the spring with a known velocity, the deflection can be calculated using the equations in Table 14.3. The applied stress can then be determined from Table 14.1. 14.2.6
Spring Nests
The nesting (one inside the other) of helical compression springs is a method of obtaining maximum energy storage in a limited space. An example is shown in Figure 14.4. It is desirable to design the springs for equal life with 60 to 70 percent of the load on the outer spring. Maximum energy storage Is obtained in a spring nest when the value of the spring indexe~ are between ') and 7, when solid lengths of all the springs are approximately the same and when the working stroke (L2-L1) is of constant magnitude.
• FIGURE 14.4 - SPRING NEST 14.2.7
Spring Index (DId)
The spring index is the ratio of the mean coil diameter of the spring to the wire diameter (DId). This ratio is one of the most important considerations in spring design inasmuch as the deflection, stress, number of coils and selection of material depends upon this ratio. The best proportioned springs have an index of 7 to 9. Ratios of 4 to 7 and 9 to 16 require more than standard tolerances for manufacturing; those with values less than 5 are difficult to coil on automatic coiling machines.
14.2.8
Stress Correction Factors for Curvature
The equatl.ons for stress in Table 14.1 are based on the assumption that the ,nagnitudc of the stress varies directly with the distance from the center of the wire. Actually, the stress is greater on the inside of the cross section due to the curvature of the spring coil. A correction factor has been determined to account for the increase in stress level due to curvature. This
14-8
•
STRUCTURAL DESIGN MANUAL correction factor gives the effect of both torsion and direct shear. For helical compression and extension springs, the curvature stress correction factor (K) is determined from the following equation: K
:=
4C-1 4C-4
.615
+
C
14.4
The total stress becomes S
max
= St X
K
14.5
where St is determined from Table 14.1. This is the stress which should be compared to the allm.Jable stress to determine whether or not the spring is safely designed and is the sole use of K. The stress determined in equation 14.5 should not be used in calculating the deflection or number of coils. The stress determined from Table 14.1 is used without correction for these purposes. 14.2.9
Keys tone Effec t
~len
square and rectangular wire are coiled into springs~ a change in shape occurs. This change takes place because some of the material on the outside diameter is drawn into the spring and the material on the inside diameter upsets, thereby changing the wire into a trapezoidal This is known as Keystone effect. The original thickness of the curve is maintained at or near the mean diameter of the coil. It is necessary to take into account this upsetting of the material in determining the solid height of the spring. The dimensional depends upon the sprIng index and the thickness of the material and may be determined from the following tl
)
= 0.48t (OD + 1) D
14.6
where t' is the new thickness of the inner edge after coiling and t is the thickness before coil Equation 14.6 can be used for both rectangular and square wire. 14.2.10
Design Guidelines for Compression Springs (a) Compression springs ordinarily should not be permitted to go solid.
(b) Whenever practicable, springs should be designed so that if they were compressed to solid height, the corrected stress still would not exceed the minimum elastic limit. (c) The length of a compression spring at maximum working deflection must not be too close to the solid length. As a minimum, a clearance of 10;0 of the wire diameter should exist between coil s. (d) The selection of springs for continuous cycling should be made on the basis of the fatigue allowables given in Section 14.7.
14-9
.I
j
STRUCTURAL DESIGN MANUAL Revision A (e) The outside diameter of a compression spring when compressed must be less than the minimum hole diameter, if the spring operates in a hole. When operating over a guide, the minimum inside diameter of the spring must be larger than the maximum diameter of the guide. (f) The possibility of buckling must be investigated and guides used~ if necessary. (g) Use compression springs in preference to other types since they are easier to produce, less expensive and have a deflection limiting feature in the solid height. (h) The best proportioned springs from the standpoint of manufacture and design have a spring index between 7 and 9, although indexes of 5 to 16 are commonly used. (i) For indexes less than 5 in the larger diameter wires, it may be necessary to use annealed material and harden after forming. (j) Specify baking immediately after plating to relieve hydrogen embrittlement. (k) Springs operating in parallel have a total spring rate equal the summation of individual spring rates; i.e., r t :; ~ri (1) Springs operating in series have a total spring rate equal the reciprocal of the summation of the reciprocals of the indivi.dual spring rates; i.e., l/r t = l;l/r.1 ' _ . 14.3
Extension Springs
Helical extension (sometimes called tension) springs differ from helical compression springs only in that they are usually closely coiled helixes with ends formed to permit their use in applications requiring resistance to tensile forces. It is also possible for the spring to be wound so that it is preloaded; that is, the spring is capable of resisting an initial tensile load before the coils separate. This load does not affect the spring rate. 14.3.1
Design Formulas
The same procedure as described in Section 14.2 is used for extension springs. The difference is in the end design and reload. 14.3.2
End Design
Various types of ends which can be obtained on a tension spring are shown in Figure 14.5. Loading an extension spring having hook ends causes the hooks to deflect. The amount of this deflection depends on the type of hook used. For a half hook the deflection per hook is equivalent to .1 of a full coil and the total number of active coils for design purposes will be N + .2. When a full hook is turned up from a full coil, the deflection per hook is equivalent to .5 of a full coil and the total number of active coils for design purposes will be N + 1.
14-10
STRuc-rURAl DESIGN MANUAL
,ft)01(
WinE 011\MI:11:11
0,£,.'"0
ttHan.
TYPES
M.~hI ...
Halt
II~
•••• C.,,'e.
or ENDS
~
011 ..... 1I..1t .1
OU III I.,.: O''''Mr:tt:ft
DI:UOt 1I0on
USED ON EXTENSION SPRINGS
".1" '. . . Cui ["".
'I'"
nl •
[~I lIon411.Ul_..
•••ie .....'
'_p .....
hub'. twh'." filii eenl..
I..,."
to .. , "ound , .. "
H ••k ••• ,
c ...,.,
S...."H
r. ..d
.1••11 .... Co"l.,
".nl,,,, t ..1f
'0.......'0
t:........" .:,. h ....
• '11'1 •• c;.al., a' lihht
A.... al.d
Co .... d tn" wUb IbM' 5wlul Ey.
c.... ct
CIOII.d tnd te 1t.ltA 10,", '.Inl t,•
.tlow
9
) .,......... p.t Iidi.
,,,,.11 1,••t .'d.
Im_"C...fTle,•• nr
Mach'''. l ...,. lIII"d Mftchln. U....h :1110." in Lin.
MGn" lo.,p ond tloo" o' II.,hl ""ql ••
FIGURE Ifl.5
.w....
C.".d E..d
£nil ",llh a",lnl'"o'''
w""
lolt
Mnchh•• t.op .... " M.,., .. h ... Hoolo 510.0....... "''1ltl A ....,I ...
• ...... loo ..... Sid . . ."d 1 ..... 1t a:'t hom C.II'.'
EXTENSION SPRINGS
14-11
STRUCTURAL DESIGN MANUAL
The hooks at the ends of extension springs are subjected to both tension (bending) and torsional stresses. These combined stresses are frequently the limiting factor which determines the characteristics of the spring. These stresses occur at the base of the hooks and their magnitude is higher than the stress in the body. This, then, is the weakest point in an extension spring lind the stresses should be calculated. The allowable working stresses should no t ('xc(:eo those shown in Sec tion 14.8. Figure 14.6 shows a typical hook end.
The bending stress at Section A is
•
p
FIGURE 14.6. TYPICAL HOOK END A
calculated using .
Sb = 32PR --3 "d
14.7
x
The torsional stress at section A'is calculated using:
st = Where
x
14.8
mean radius of hook, in. 2 = mean radius of bend, in. 3 inside radius of hook, in. 4 inside radius of bend, in. 1
For best results, the inside radius should be at least twice the wire diameter. Special ends can be used when high stresses occur in the hooks. By using a smaller diameter for the last few coils before the hook, the magnitude of PR is reduced. Thus, the stress is reduced in direct proportion to the decrease in PRe By using as large radii for ra and r as the design will permit, the stress is 4 further reduced.
14.3.3
Initial Tension (Preload)
Initial tension is a load in pounds which opposes the opening of the coils by an external force. It is wound into the springs during the coiling operation. Extension springs \I"i11 have a uniform rate after the applied load overcomes tbe load due to Initial tension. The number of coi.ls do not affect the amount or initial tension except when the weight of the coils is heavier than the initial tension. The amount of initial tension is dependent on the spring index (D/d);
14-12
STRUCTURAL DESIGN MANUAL
•
the smaller the index the larger the initial tension. Initial tension does not incf('ase the ultimate load or capacity of the spring, but causes a larger porUon thereof to be exerted during the initial deflection. For exanlple; if the injtial tension is 4 Ibs. and the spring rate is 9 lbs/in. then, at 1 inch deflection the load is (1 x 0)
+ 4
=
13 Lbs.
3 inches of deflection gives a load of (3 x 9)
+
4 = 31 Ibs.
In computing the total torsional stress, add the torsional stress caused by initial tension to the torsional stress caused by deflection. Figure 14.7 shows the amount of initial tension in terms of torsional stress (without application of curvature stress correction factor) which can be coiled into extension springs made of music wire, oil tempered, corrosion resisting steel and hard drawn spring steel. Reduce these values 20 percent for springs made from ldckelbase alloys such as Monel and Inconel. Hot rolled springs and those made of annealed materials cannot be wound with initial tension. Springs which require stress relieving will lose 25 to 50 percent of their initial tension. This loss can be compensated for during the coiling operation by winding more initial tension into the spring and, thus, obtain the required initial tension after stress relieving. 14.3.4
Design Guidelines for Extension Springs (a) Avoid using enlarged, extended or specially shaped hooks or loops; they may double the cost of the spring and have high stress concentrations. (b) If a plug must screw into the end of a spring, the spring should be coiled right hand. (c) Nearly all extension springs are wound with enough initial tension to keep the spring together. Always figure at least 5 to 10 percent of the final load as ini tial "tension, unless otherwise specified. (d) Electroplating does not deposit a good coating on the inside of, or between, the coils of extension springs. (e) Hooks on extension springs deflect under load. Each half hook t made by bending one half of a coil~ deflects an amount equivalent to 0.1 of an active coil. Each full hook is equivalent to 0.5 of an active coil. Allowance for this deflection must be considered. (f) If the relative position of the ends is not important, note this fact on the drawing. (g) For standard hooks keep the OD of the hook the same as the 00 of the spring and the distance from the end of the body, or from the last coil, to the inside of the hook about 75 to 85 percent of the ID of the spring. (h) The body length or closed portion of an extension spring equals the number of coils in the body plus one, mUltiplied by the wire diameter. 14-13
~ STRUCTURAL DESIGN MANUAL
";~..
\
\ , \ \ \ \ .. \rl1~'6·~ \ ' \ 1.. lb~. ~ 1\ \ \ ~-:~eo::::,.\:h~~,.hI' i\. '\'" " ..... '"1,..~ -"""illl., ..-:-'-
\
40
38
36
\.
\
/
'II/I.
'eli lId -"'"°0
~ c.~
••cw ...o~
J)f;.. "at
:L
I.II
.'Woid IAle ~
U·
.~
---""-'\ r(
'\
i\.
••••
~\1)..
. P ..\.\lP
~
b'~~_~ ~ ~
!
..
\
'~iD\lliI ~~~-~ \ itiP \e~ b\e-' "
\
J' " " ~
"
16
.
:0\.' 'fa' "...tIfIII' "~.-o..~ ~
~
j
~~.
"-
l,fI'
.....
\..
V1'/ ".. . . . , ~
\4'J'~"'~Z~ ~,~ ..~\~.~
..,.
10
8
L... "q'oJ'/
""',".'"l" ,
~
~
. " IfII
'"
" '"' ~ " " • -., ~" ~
~
~
-
6
~
.....
""'"- .... ......... -............... ......
~
3
4
5
6
7
8
9
o
s,n... ..... - Ci -
10
11
12
13
14
15
16
MeaRDi. W,.. Di.
PERHISSIRLE TORSIONAL STRESS RESULTING FROM '~'T'AL TENSION IN COlLEn rXTENSION SPRt~JGS FOR OIFFEREt.'T £lId RATIOS
14-14
~
~ ~~\.'&P~ ~"'Z·~ .~ ~1"';-4~ / > . . ~
12
FIGURE 14.7
'\
~
'V "- " " " "V~~;.~\ "-
.,.' \It'l
., 14
.
""
J "-
"
~
I'" " ,"'\
" " ~~~ '-,- "
~~\o"'~1 .... tp)'G' ~ ,at; ~
)
~'"
\
A 'II"'.J .r.•,io.Ilatt.bl.
J
\
)
.
STRUCTURAL DESIGN MANUAL
1~ times the maximum deflection as assembled, the total stress should be less than the minimum elastic limit shown by the curves in Section 14.7 as modified by their multiplying constants.
(i) When deflected
14.4
Torsion Springs
A helical spring can be loaded by a torque about the axis of the helix.
Such loading is similar to the torsional loading of a shaft. The torque about the axis of the helix acts as a bending moment on each section of t~e wire. Ordinarily, round wire is used, but where added elastic resistance is needed in a limited amount of space, square or rectangular wire is frequently used.
The design theory for helical torsion springs is the same as beam theory. The wire in the torsion spring, as in the beam, is essentially in a state of bending. The analysis is simplified by assuming a constant moment throughout the wire cross section and a moment equal to the product of the load and the distance from its point of application to the central axis of the spring coil. 14.4.1
Design Formulas
The stress in helical torsion springs is a bending or tensile stress. The stress caused by a load should be compared with the elastic limi.t in tension of the material to determine the allowable stress. Comparison should also be made with the curves of allowable stresses, corrected for torsion springs, as shown in Section 14.7. Table 14.4 shows the various formulas for designing helical torsion springs. 14.4.2
)
End Design
Frequently, the limiting stress value in helical torsion springs is the stress value in the ends. When a helical torsion spring has an eye or bent off end as shown in Figure 14.8, the stress at the inside of the bend is a tensile stress. The sharp curvature causes the neutral axis to move inward toward the center of the curve and the tensile stress becomes that of a cantilever multiplied by a constant (K). The formula for determining the stress in the bend of the eye in Figure 14.8 is
Sb
32PRK TT
where
•
R
d
14.9
3
mean radius of eye, in
ID of eye + d 2
OD of eye - d
14.10
,2 K
2
4C -C-1 4C(C-1)
14-15
STRUCTURAL DESI,GN MANUAL Property
Torque, Ib in. T (&lao. PR)
Bendlne .trells. pal Sit
Roulld wlr.
Square wire
E d'F o
.. E d4. FO
Reet.l'urulal'
_-
4,OOO-ND
-2~376
--
..,ire •
-_ ..
E b t 3 FII
Nif
'2,af6ND
Sb dl -r~
-6-
--"6'-
lO.2PR --(1';-
6 I) R ----d 3' · -
GPR
-,
EdF·
-S92N 6-
Sbd:l
Sbbt2
..-
-bii -
EdFo
EtF·
-S92N'D
-3!J2rf"D --~
4.000PRND
DeBection,
F·
--Ed-'-392 Sb ND
-tti"" Change in
F1I 2 -Fol
2,376PRN
-'-Edt:
.......
2,376 PRN D - - E b't3 ,.. _.-
392 Sb N D
392S.,N D
-Ed---FII2 - Fat -F"i-
-F"o-
-T
T
Et F"2~F·l
moment T~ - Tl
--F e-
1D after
N (ID free)
~J!I?{rceJ
N (ID free)
In. IDa
N+ FO -360
N+-
N+-
deflection
T
F·
360 ~-~
nate r t Ib.in./Deg
TABLE 14.4
T
.---
T
FO
F·
-
---~Fo-
_u
S60
T
FO
FORMULAS FOR HELICAL TORSION SPRINGS
When a sprIng has (makes) several complete revolutions, FO multlpl'led by the number of revolutions.
= 360 0
* Rectangular wire may be cotled
on edge or on flat, but h is always parallel to the axfs of the spring and t is always perpendicular to the axis.
) .
F t CURE 14.8
14-16
TORS ION S PR I tH~ F~1f') nFS I CHI
•
STRUCTURAL DESIGN MANUAL
For bends of the coil as shown in Figure 14.8 the stress value in the bend is
14.11 Where 11
= distance
from center of bend to load
curvature correction factor as defined by Equation 14.10
K
When the length of the material in the arms of a helical torsion spring approaches the length of material in one coil, the deflection of the arms will cause the deflection under applied loads to be in error. As shown in Figure 14.9, such ends deflect as a cantilever and may be calculated as such or the PI NEGLECT! NG ARt~
DEFLECTION
INClUOJNG ARt~
•
DEFLECT! ON
FIGURE 14.9 - CANTILEVER ENDS formula [or sp'ring rate including arms may be used. The formula for spring rate when the deflection of the arms should be included is : rt
where 11 12 L
.)
4
Ed 1170(L+ll/3+1 2/3)
14.12
length of arms from center of coil to point of load (Pi), in length of arm from center of coil to the point of load (P2)' in active length of material = 1TDN
In springs with a large number of coils and short arms, the deflection of the arms is neglected. However, short arms should be avoided since this causes difficulty in coiling and forming.
14.4.3
•
=
Change in Diameter and Length
When a helical torsion spring is deflected, a reduction in diameter and an increase in length occurs. In order to prevent binding or scuffing, which reduces spring life, sufficient space must be provided when operating over a rod or in a cylinder. The new inside diameter ID, is obtained from Table 14.4. The change in length is due to the increase in the number of coils at the deflected position. If the spring makes one complete revolution, the increase in length is equal to one thickness of wire~ plus an allowance for the space between coils, if'any. 14-17
STRUCTURAL DESIGN MANUAL 14.4.4
Helix of Torsion Springs
The direction of coiling (helix) should always be specified for torsion springs. A torsion spring should be so designed that the applied load tends to wind up the spring and increase its length. In springs operating under high stress it is desirable to design the springs with open coils. A slight space of about 1/64 inch or.20 to 25 percent of the wire diameter will eliminate friction between coils and reduce stress concentration which will lengthen the spring life. When long helical torsion springs are used there exists the possibility of buckling. Since buckling will cause abrasion between coils, erratic loads and early spring failure, it should be avoided. Buckling may be reduced in varying amounts by providing some means of lateral support such as:
•
1. Mounting the spring over a rod or guide. 2. Mounting the spring in a tube. 3. Clamping the ends. 4. Winding the spring with a small amount of initial tension. 14.4.5
Torsional Moment Estimation
Table 14.5 is an aid to quickly determine the torque (T at PR) that can be applied to a wire diameter at the suggested basic stress listed. For example, what wire diameter is required to $upport a torque of 10.5 in lbs1 .From the table it will be observed at 0.090 diameter music wire or corrosion resisting steel; 0.0915 diameter carbon or alloy steel and 0.125 diameter copper and nickel alloys could be considered. The final determination must be made by use of formulas,but Table 14.5 gives a good starting point. The basic stress indicated is a bending stress Sb caused by a torque T or PRJ corrected for curvature. 14.4.6
Design Guidelines for Torsion Springs a. Always try to support a torsion spring by a rod running through the center of the spring. Torsion springs unsupported or held by clamps or lugs alone are unsteady, will buckle and cause additional stresses in the wire. b. Torsion springs should be designed and installed so that the deflection increases the number of coils. This increase should be allowed for in' the design of space requirements. c. The inside diameter reduces during deflection and should be computed to determine the clearance over the supporting rod. d. Use as few bends in the ends as possible. They are often formed in separate operations, are expensive and cause concentrations of stress and frequent breakage. e. Consider tolerances on diameters when determining clearances over rods. f. Always specify the direction of coiling as either right-hand or left-hand on drawings. g. Springs may be closely or loosely wound, but they should be wound tightly except when frictional resistance between the coils is desired.
14-18
•
•
STRUCTURAL DESIGN MANUAL
• TABLE 14.5
MOMENT VS. WIRE SIZE CHART (AIBON I ALLOY'
MUSIC Will Cln,ct.1I
WI,.
1•• le
Momen.
r:..
S" ...
...... 11.
Diem
It,..
,.,
Cartu'," WI" Mom"'l Dlefln. Ib.·llt,
1,OU 1.ISS 2.27 1.47 5.24 7.'2 10.49
201,000 200.000 .01'6 .010 199,500 .OU' .011 19'.500 .0337 .012 .o42S ,013 196,100 .On7 .014 '9S,500 .07'0 .016 "-4,000 .1101 .0" 193.000 .ISOS .020 "2,000'
,D'D' .", ..,
"',000
..., .na .441
.Ot. 1'5,000
.stl
.011 .OU .035 .031 .039
17f,Joo 17'.000 171,000
•
.0"
1",500
.041 .045 ,047 .049 .051
173,000 1",500 170,000 169,000 "7,500 165.000 162,000 161;000 15'.'00 157.000
.631 .755
..., 1.'JO I,t .. 1."0 I.UI '.73
I.'S 2.11 2.70 J.26 J.tS ....70
)
'.JO
"a. 1."
'.04 '".10 D.•'
.....
'4.11
1t.IO
.0.55 .OS'
.0'3 .0" .071 .075 .010 .01S
.... .Otl .100 •106 .U2
n.os.
Ito,OOO 11',100 117,500
.022 .024 .026
.256
1 '.21 15.72 25.'
31.'
I",
nuu
C."ulld
p.1
1.... 111•
.125 .1,.
.,,:as
13:1,000
13t,0 163
.2253 •2437
.75
.250 .2,2S
"',000
6 •••
79.1
IS.'
'"
'01
.2112 .306.1
315
.1125
'67
.331 .3431 .3625
229
40S 464
154,000 S06 113.000 17S 110,000 62a 147,100 7JJ '4',500 191 ....,000 1060 145,000 "10
... ' ' ,600 ,
Wit.
D'..... I".
.OO~U .001 .00.s09 .009 .00"4 ,010 .000U .011 .01) .0152 .0119 .014 .0111 .01' .0" .019' .0146 .020 .0127 .021
104.6 '16.S
112,000
Mom ....
'.9,500 .04' .047' 14.,000 146,700 .054 .0625 145,000 ,072 143,000 .010 '''1,500 ,O"S 139,100 ,0937 n9,OOO .10SS 136,600 .120J 134.000 "'.700 •• 4U '2',200 .1562 121,100 121,100 .162 .117 124,'00 .U1S 123,100 122,400 .192 120,200 .207 .21'7 1",'00
4'.4 "'.0 53.1
CO" .. " NlCleU ALLOYS
80.le Strll.
.375 .3931 •.eN2
117,000 115,000 114,000 112.200
'0',600
.1056
.2" .310 .421 .591
...6 l.lt4 1.6'" 2.U
"'1.•
97,'00 95,'00
•4~15
'",'''
.46.7 .500 ,J62J .6iS
",500 12,700 7.,100
tI,IOO
",.00
.a.'
'1.7 122 IA•
'"
a•
441
".100
6'.100 "~JOO
69,100
.04' ••,toO .051 66*500 .057 61,600 .064 '4,600 .072 '4,100 .010
'0',700 31 .... 99.000
70.600 70.. 500 70.200
",'00
.Ot • •102 .114 .125 .12' .144
11.'
7.,.00
.036 .040 67,200
a.1'
'06,400 '1.2 10J,JOO 17.2 103,000 21.9
71,200 11,000
.021 ",000 .029 ••,200 .032 6',toO
4.61
'.4' 1.12
••• 1. Sf, ... "t
63,200 62,300
'I,... 60.600 '9,100
.'.1.500 ,IOG
.162 .7,200 .112 56,200 .204 11,100
.221 11..100 .211 12.600 11,100 .2"
.12' so,. 00 .161 .4'. 0',300 .460 46,_
.,,100.
)lOft: The values for MUllic Wire may atao be used t.or Corrollon Resisting Stee1a.
• 14-19
STRUCTURAL DESIGN MANUAL
h. Avoid using double torsion springs. Two single torsion springs, one coiled left-hand and one coiled right-hand, usually can perform the same action as a double torsion spring, at less than half the cost. i. When deflected 1-1/4 times the maximum deflection as assembled, the total stress should be less than the Minimum Elastic Limit shown in Section 14.7 modified by their multiplying constants.
14.5
•
Coned Disc (Belleville) Springs
The coned disc (Belleville) spring or washer is a plain dished washer of a particular diameter, sectional profile. and height suited for an intended purpose. It is used in a variety of applications, all having the cornmon characteristic of necessity for short range of motion and attendant high loads. In order to calculate the free spring height and required thickness of stock in a relatively simple manner, it is necessary to know the outside diameter (OD), inside diameter (10) and the load (p) for a specific deflection. Design Formulas
14.5.1
By obtaining the value for constant ( Y ) from the proper curve, Figure 14.10, the following formula can b~ used to calculate the load-deflection characteristics: p =
Ef
(1- 112) Y
where a h
8
2
14.13
•
one half the outside diameter, in free height minus thickness, in
By obtaining the values for constants Zl and Z2 from the proper curves, Figure 14.10, the following formula can be used to calculate stress
14 .. 14 It is possible for the term (h~f/2) to become negative if f is large. When this occurs, the terms inside the bracket should be changed to read Zl(h-f/2)-Z2t. This means, in this instance, the maximum stress is a tensile stress. For a spring life of less than one-half million (500,000) cycles, a stress of 200,000 psi can be substituted for Sb' even though this limit might be slightly beyond the elastic limit of the steel. This is because the stress is calculated at the poInt of greatest Intensity, which is on an extremely small part of the disc. Immediately surrounding this area is a much lower stressed portion which so supports the Ilighcr stressed point that very little yielding results at llt11lospherLc temperatures. For higher than atmospheric temperatures and rong spring life, lower stresses must be employed.
14-20
•
•
'8:.
STRUCTURAL DESIGN MANUAL ZI r Z2 2.3 v
1. .8
2.0 1.9 1 .. B
•1 .6
1.7 l.G
.5
1.5
• II
1 . l~
.Q
.3 .2
•1
•
2.2 2.1
n
1.3 1.2 1.1 1.0 1 1.f1
2.4
P.<'tin
r- If~ n~ E
14. 1 0
"f-
3.2
~aO
s.
nn/lf'
~ T r F. S~ A Nf' !1 [F I. r: CT , nt! r, nt f c) TA~t T S FOP. RF'lLr:VIt.l.f HA~fH:R~ OF n~HFOr.t1 TltlCKNF.C;5
)
FlnURE 11,.11
'·1F.THOf'S OF STACKIt·l(l cOttEn DISC (RELLEVILLF) SPRI~'GS
• 14-21
STRUCTURAL DESIGN MANUAL
Belleville springs can be stacked to obtain various load/deflection relationships. Figure 14.11 shows typ1cal arrangements of these stacks. When fineconed disc (Belleville) springs are stacked in series as in Figure 14.11 J they have a spring· rate only one-fifth that of one disc and the solid load will be the same as for one disc.
•
When six discs are stacked in parallel as in Figure 14.11, they will have a spring rate and a solid load six times that of one disc, disregarding friction. When six discs are stacked in parallel-series as in Figure 14.11, they will have a spring rate only two-thirds that of one disc and the solid load will be twice that of one disc, disregarding friction.
14.6
Flat Springs
Load requirements are intimately connected with spring dimensioning and the space available for the spring. The point of load application, deflection, length, width and thickness should be clearly specified. Formulas for designing various flat spring characteristics are given in Table 14.6. The stress in flat springs is in bending and should be compared with the elastic limit in tension which is shown in Section 14.7.
14.7
Material Properties
Various materials are available for springs. The selection of material is based on wire size, temperature of operation, load application (i.e. impact or slowly applied), cycles of load, corrosion, environment, etc. Table 14.7 shows a list of commonly used spring materials and the recommended usage.
14.7.1
•
Fatigue Strength
The fatigue strength curves shown in Figures 14.12 through 14.18 are for the most popular spring materials. These are for compression springs, based on the minimum torsional elastic limit of each material. The values may be increased 25 percent for springs that have been properly stress relieved, cold set and shot peened. Table 14.8 shows the proper usage of the allowable curves. Figures 14.12 through 14.18 show the torsional minimum elastic limit and maximum solid stress at which a spring can be stressed. In addition, fatigue curves are shown for three different ser.vice conditions; light service, average service and severe service. These service conditions are defined as:
)
14.7.1.1. Light Service - This includes springs subjected to static loads or small deflections and seldom used springs such as those in bomb fuzes, pr.oject1les uncI safety devices.. This service is for 1,000 to 10,000 cycles. 14.7.1.2 Average Service - This includes springs in general use in machine tools, mechanical products and electrical components. Normal frequency of deflections not exceeding 3600 pet hour permit such springs to withstand 100,000 to 1~OOO,OOO cycles. 14-22
•
STRUCTURAL DESIGN MANUAL
•
TABLE
PROPERTY
.
1~.6
6 [
FL~ ~Ld ~::I ~""~F ")--F
-.e:c::... __ ~,
L-
d •
Ib.1
PLAN,
Deflection
P LJ
F
4 E b t
Inches
Load
3
Pounds
4 E b t3 F E b t
L:l
4 L3
3 P L 2 b t2
Stress
s"
Bending
2 L2
(' L2 ub 6 E F
Thickness t
2 Sb J.J 2 3 E F
W. 14
Inches
l
E t F
:~
L2
4 E b F
.87 S
EDt
Sil b t 2 oL
6 P L b t2
6 E t F
psi
3
E b t3
E t
6 L
P L'J
E b F
~ b
t
3
T4
•
5.22 P LJ
Sb L2
~ E t
:::J~~
13
I.::J
3 fi E 1. E b t3
b t 2 SI;} b t 2
12
>
t
- ~I
1$ --
LJ E b t3 2 S L2 12
3 L
p
PlRN
4 P
Sb L2 6 E t 2 8
•
FORMULAS FOR FLAT SPRINGS·
Sb b t
L2 2
5 L
F_
6 L3
6 P L
Jt:;
2 t3
EL 2
6
b t2
b t
E t F
[~
L2
F
5.22 L3
t
.87
F
LZ
Sb L2
.87 Sb L2
E F
E F
3 6 P L3 Rib F
1
5 • 22 P L'J E b F
* Based on standard neam formulas where the rleflectlon Is small.
Mol.riot
)
u,.
Speciftcotion
Ineonel Wire
OQ·W·390 Cond.C
This mot.rloll, elCcelt.nl for appliCotion, requiring good corto'ion 'elistonce one on oblli.y to wlll\,lond operation ot temperotures from sub-uro to 650·F.
'ntonel Sheet & Strip Spring
Mlt-N·6840 Condition S
Incone' pouesus high el.clrlc:ol,.slstonce and ,hould nol b. u •• d os a conductor; " Is non-mogn,fic.
Incon.' X Wlr. Sptlng temper
Jon-W·562 Clota2
The applicoble divislonol stoff unit mull b. con.ulled betol. r.taasl' of 'pring designs ,allinQ for this moter· 101.
Incon.r X WI,. No. I Temp.r
Jon-W·562 Closs I
Th's mo •• rlolls like Incon.lelCcetptfhot it h.al It.ofobi •• The spring temper ... Ir. 1& to be used when''1.r moximum mechonicol properti.l or. desired ond moxlmum lemperoture or operation will not ekceed 800·F. Th.s moteriolls also .lIcell.n' 'or sub-zero appli·
Temper
•
'S
,h.
calions.
The Clou I t.mper, .,hen ptoperly heot treoled. mov be used up to 9,SO·F. If. r",i,tonc. to r.lolColion is luperlor 10 the Cfon 2 temper and should b. given preference for opplicatlons when this choroc:teristic.l. desired.
TABLE 14.7
APPLICATION OF COMMONLY USEn SPRING MATERIALS
14-23
STRUCTURAL DESIGN MANUAL /i.JJ
200 '90
I~O
ItO
140
VI
0 C 0
.....
liD
U')
I~O
0..
140
110
100
"C
90
... OJ
I)
.... ....
0 0
0
4U
....
...
U)
0
to
,-.. '"t::
60
0 0
U
.. 0
0 0
0
0
11\
IIJ
0
0
0
0
o
0 0
~
0
o
0
0
0 0
'"
CD
0 0
0
~
d
0
If)
Gi
0
'0"'
! l! 0 0 0
;;
c:
~O
40 JO o 0
0 o 0 0 0 0 0 .. 4D «I 0 N 'II' CD o 0 q 0 00 0 0 0 0 6 6
0
-
N N
.. 0
0 IIJ
0
N
N
N
0
0
0
0 ci 000
a
N
CJ
CD
~~S 1"1 1"11")
0 0
1"'\
)
WI re otameter, In.
U')
VI Q)
1'0
I-
OlL. T[MP( RED
140
150
140
(/)
130
n:I
120 110
C
100
0
90
VI
80
0'"'
70 6O
.....
,.,
':>°0
0 0
0 &n
'0"' ..... o
0
."
0
-
0
0 0
~~e
00 000
-
000 0
- 0- 0'" 0 0 0 0 0 0 '" WIre DIameter, In.
~
....
o 0 0.0
.n
~
00
o
n
d
/11
-
N
N
/11
N
N
~
1"\ ' "
If)
0
150
.....
120
10
70
I-
WI re Dtameter, In.
(.)
VI VI
...
0 N 0 0
0
110
80
Gi
0
I~O
0 ....
.....
10 70
U
•
302
100
0 0
'130 120
,-..
".1-."
UO
170
0..
COli 05010 .. IlstStINO
100
'0
80
10 60
$O~
...0 ,...~ 0
0 ....
0
on
~ 0
1"'\
0
0
'"00
0
d
~
0 1"'\
0 ci
0
-.
~
0
0
,..0
(h
N
N
N
0
....- ,.,1"'1
0
0
0
A.
0 0
2
70
60 ~o
-
l.
cr-1·-- .... _ J"'~/tt ,,- -.!-,,~
IQIlI' .JC'f' lIl
.......
- A"'Itf'O."
~
-'-"- - -- -_.. -J-t--I--I-- -. -. -
'--,
__
, _
•••
i
S~"l!'"
_
If)
0 ....
1"'1"" 00
0
0
,.,'" 0
..
H.RO DRAWN SPRING STEEL
IlO
.00 90 80 _
a
0
.J_.I_
INCONt'-
tiD
..
0
WI re ntameter, In.
no 120
0
1t"I/,,.,,
.7:"
..
._ _
FIG. 14.
IlO
r.
\a
.+
gllo 2
::. . :;. '0 c
.o~
I-
~O
..
0
0
§
~
CD
0
o 0 0 6
00
ON
dd
.,0 ci WHt
0
0
ID
\I)
-0 0
N N
00
..
0
N
IIJ NN
N
0
0
00
ci 6 0 0
0
0
0
orol'Pl ttC't. In.
It)
oON 0 .... ~
0 ,.,.. .., 0
\0
6
"0
0
0
.,0
~ 0 0 0
.
0 0
0
~
N
N
0
0
0
0
0
..
0
0
6
WUf'
CD N
0
,., N
0
0
\I)
o~
~
'It
0 0 0 Diomtte r.ll'I.
0
.. .,.
0
.. 6 0
ID
0
0
III
III
\I)
6
0
0
N
\I)
FIGURES 14.12 14.18 FATIGUE STRENGTH CURVES - RECOMMENOEO MAXIMUM WORKING STRESSES FOR COMPRESSION SPRINGS 14-24
0 0
•
STRUCTURAL DESIGN MANUAL
•
•
•
TABLE 14.8 CRITICAL STRESS DATA* COMPRESSION SPRING TORSION SPRING EXTENSION SPRING 1. Torsion Stress - Compare 1. Torsion Stress(Coils) 1. Bending Stress (Coils) Compare calculated calculated stress in coils Compare calculated design stress in coils design stress in coils with service curve of with service curve of with service curve of Figures 14.12-14.18. Figures 14.12-14.18 2, Solid Stress - Compare torFigures 14.12-14.18 multiplied by 1.5. sion stress in coils when multiplied by .85. compressed solid with minimum 2. Torsion Stress (Hooks) 2.Bending Stress (Ends) Compare calculated Compare calculated elastic limit curve. design stress in ends design stress in hooks with service curve with service curve multiplied by .• 85 .. multiplied by 1.5. 3. Bending Stress (Hooks) 3.Bending Stress in Coils at Maximum Compare calculated Deflection - Compare design stress in hooks with service curve mul- calculated stress in coils at this deflectiplied by 1.5. tion with min elastic 4. Torsion Stress (Coils) at Max Extended Length- limit of Figures 14.1214.18 multiplied by 1.5. Compare calculated stress at this length 4.Bending Stress in Ends with min elastic limit at Maximum Deflection curve multiplied by B.S. Compare calculated stress in ends at 5. Torsion Stress (Hooks) at Max Extended Length- this deflection with min elastic limit of Compare calculated stress at this length Figures 14.12-14.18 with min elastic limit multiplied by 1.5. curve multiplied by .85. 6. Bending stress (Hooks) at Max Extended LengthCompare calculated stress at this length with min elastic limit curve multiplied by 1.5. ,'(Note 1:
After tentative spring configuration has been determined, use data in above table in association with Figures 14.12-14.18, to ascertain that allowable stresses are not exceeded.
Note 2:
The above referenced "calculated design stresses" are TOTAL STRESSES. They include curvature stress-correction factors, except for extension spring hook stresses which include the correction factor in the basic formulas •
14-25
STRUCTURAL DESIGN MANUAL 14.7.1.3 Severe Service - This includes springs subjected to rapid deflections over long periods of time and to shock loading such as in pneumatic tpols, hydraulic controls and valves. This service is for 1,000,000 cycles and above. Lowering the values 10 percent permits 10,000,000 cycles. 14.7.2
•
Other Materials
For materials not shown on the curves of Figures 14.12 through 14.18, the following multiplying factors may be applied: a. Beryllium Coppet - multiply the values of the phosphor bronze curves by 1.20. b. Spring Brass - multiply the values of the phosphor bronze curves by 0.75. c. Monel - multiply the values of the inconel curves by 0.82. d. K-Monel - multiply the values of the inconel curves by 0.90. e. Duranicke1 - 'use the same values as for inconel. f. Incone1-X, (Drawn to spring temper and precipitation hardened) multiply the values of inconel curves by 1.25. g. Silica-Manganese - multiply the values of the Chrome·Vanadium curves by 0.90. h. Chrome~Silicon - mUltiply the values of the Chrome-Vanadium curves by 1.20. i. Value Spring Quality Wire - Use the same values as for Chrome Vanadium. j. Corrosion Resisting Steels Type FS304 and FS420 - multiply the values of the Corrosion Resisting Steel curves by 0.95. k. Corrosion Resisting Steel Type FS316 - multiply the values of the Corrosion Resisting Steel curves by 0.90. 1. Corrosion Resisting Steel Type AISI431 and 17-7PH - mUltiply the values of the Music Wire curves by 0.90. 14.7'.3
•
Elevated Temperature Operation
Springs used at elevated temperatures exert less load and have larger deflections under load than at room temperature. Compression and extension springs subjected to the temperatures and stresses shown in Table 14.9 will have a loss of 5 percent or less (or if the load remains constant they will deflect an additional 5 percent) in 48 hours. Elastic limits and modulus values are also reduced. thus necessitating these lower allowable working stresses.
• 14-26
STRUCTURAL DESIGN MANUAL
•
Permissible Elevated temperature F deg
Spring Material
•
Brass Spring Wire Phosphor Bronze Music Wire Beryllium-Copper Hard Drawn Steel Wire Carbon Spring Steels Alloy Spring Steels Monel K-Monel Duranickel Corrosion Resisting FS-302 Corrosion Resisting AISI 431 Inconel High Speed Steel Cobeniurn. Elgiloy Inconel X Chrome-Moly-Vanadium
150 225 250 300 325 375 400 425 450 500 550 600 700 775 800 850 900
Maximum Hec onunended
working stress St PSI 30,000 35,000 75,000 40,000 50,000 55,000 65;000 40,000 45,000 50,000 55,000 50,000 50,000 70,000 75,000 55,000 55,000
TABLE 14.9 - Permissible Elevated Temperatures for Compression and Extension Springs (Loss of load at these temperatures is less than 5% in 48 hours.) 14.7.4
Exact Fatigue Calculation
The curves shown in Figures 14.12 through 14.18 show allowable stresses for three types of service conditions. The exact life of a spring cannot be determined from these curves. If for some reason the exact life is necessary, a Goodman diagram can be combined with an S-N curve for the spring material and an exact life predicted. First, an S-N curve is drawn from the data in Tables 14.10 and 14.11 and the strength properties of the materials. There will be a separate S-N curve for Structures in Bending 10 10
•
4 5
cycles - 80"/0 of F
tu
cycles - 537.. of Ftu 6 10 cycles - 50io of F tu 7 10 cycles - 48io of F tu
Structures in Torsion 4 cycles 5 10 cycles 6 10 cycles 7 10 cycles 10
.. 45%-,'c' of F
tu - 3570 of F tu .. 33% of F tu - 3070 of F tu
*35% for Phosphor Bronze or AISI302 TABLE 14.10 - Design Stresses for Cyclic Service - Springs Not Shot Peened
14-27
STRUCTURAL DESIGN MANUAL
Structures in Bending 4
10 105 6 10 7 10
cycles - 80'70 of F tu cycles - 62'7,. of F tu cycles - 607.. of F tu cycles - 58'7,. of F tu
Structures 4 10 cycles 5 10 cycles 6 10 cycles 7 10 cycles
in Torsion - 45io of F tu - 4270 of F tu - 40io of F .. 36°1.. of F
tu tu )
TABLE 14.11 - Design Stres'ses for Cyclic Service - Springs Shot Peened each material by size and also by stress type (torsion or bending). The abscissa is also used as a double scale - a log scale for the number of cycles, N, and a linear scale labeled "minor stress" of the same rate as the ordinate or "major stress" scale. Figure 14.19 shows the diagram. A 45 degree line is drawn from the origin of the plot. On this line, a point tlN r is marked corresponding to the ultimate strength of the material. This is the tensile strength for structures in bending and the torsional strength for structures in torsion. Torsional strength can be taken as two-thirds of the tensile strength. These two lines constitute the combined S-N and Goodman diagrams for the given material and the pertinent stress type. If a minimum service life must be met, draw a vertical line from the appropriate value on the "N" scale to its intersection with the S-N, curve ItBtt .. At the intersection,' draw a horizontal line to the intersection with the ordinate or "major stress" scale tiC'. From that point, draw a straight line to the tensile strength point HAft. Along this line "AC" lie the combinations of major and minor stress that will meet the desired life. 14.8
•
Spring Manufacture
Certain processes may be employed during manufacture of the spring to greatly en h ance the performance of the spring. Information on a few essential operations is given here. 14.8.1
Stress Relieving
The usual types of hardening and tempering ovens are used for stress relieving. Springs made from prehardened wire such as Music Wire, Oil Tempered, Hard Drawu t Corrosion Resisting 18-8 and similar materials are stress relieved by heating at low temperatures from 400 to 650°F to reduce the residual stresses trapped in the wire during the coiling operation. Springs made from annealed wire are hardened and tempered in a manner somewhat similar to tool steel. Precipitation hardening materials such as Beryllium-Copper, K-Monel, Inionel-X, 17-7PH and others are heated at varying temperatures, depending upon composition, for extended times from 1 to 16 hours.
14-28
•
STRUCTURAL DESIGN MANUAL
rH no r S t res S I 100 0 psi
o
200
100
300
U')
a. a
o o M
t...
o
-,
-... ttl
~!umhe
r=IGunr 14.19
r of eye 1es, r!
GOO"~4A~r
f" AGRAt~ COt,1P I ~'[r Stt CURVE
\'/ I Tit
-I
14-29
STRUCTURAL DESIGN MANUAL 14.8.2
Cold Set to Solid
This process is used to stabilize the free length of a compression spring, so that subsequent inadvertant or intentional compression to solid height will not change the loads at working deflections. a. If a compression spring is designed so that the elastic- 1 imi t is not exceeded when the spring is compressed to solid height, no appreciable permanent set will occur, other than removal of small kinks in the wire. The note "Cold Set to Solid" should be specified on the drawing of such springs. b. If a spring is designed so that the elastic limit is exceeded when the spring is closed solid, permanent set will occur and the free length will be decreased. Residual stresses of opposite sign will be set up in the wire when the load is released so that if the spring is again closed solid, it will withstand a higher calculated stress than the stress corresponding with the elastic limit. If the initial free length of the spring is made greater than the calculated free length by the proper amount, overstressing the spring beyond the elastic limit by compressing it to solid hei,ght will stabil ize the characteristics and produce the desired loads at working deflections. Additional cycles of compressing the spring to solid height and releasing the load will not further change the free length. However, there is a limit to this process. After a certain initial free length has been reached for a particular spring, the final free length after compression to a solid height will remain constant no matter what 'increases are made in ini tisl free length. c. When a spring is designed so that the stress at solid height is so far above the elastic limit that the spring will not have the desired loads at working deflections if cold set to solid; the note ttShall compress to ••••••• in. without permanent set" should be placed on the drawing. The computed stress at the specified length (equal to or less than the final assembled length) must be less than the stress at the elastic limit. Whenever practicable, this design of spring should be discarded in favor of a spring having a solid stress within limits that will permit closing solid without ,permanent set. 14.8.3
Grinding
End coils of compression springs are ground whenever it is necessary for the springs, (1) to stand upright, (2) to obtain a good seat against a contacting part, (3) to reduce buckling, and (4) to cause the springs to exert more uniform pressures under a diaphram or against a mating part. This operation is expensive and should be avoided whenever it is practicable to do so - especially on light springs with wire diameters under 1/32 inch and where a large spring index ratio prevails, such as 13 or larger. 14.8.4
Shot Peening
Spring life can be increased at least 30 percent and has often been increased from two to ten times by shot peening. This process may be applied to all highly stressed springs made from steel and non-ferrous materials usually over ,14-30
)
STRUCTURAL DESIGN MANUAL !
1/16 inch wire diameter. Extension springs and closely wound torsion springs are difficult to shot peen because the tiny steel shot is frequently trapped between the coils a'nd is difficult to remove. The large increase in fatigue life of helical springs due to shot peening is accomplished by a combination of effects: a. Small surface irregularities seen only by the microscope are hammered smooth. b, The surface of the wire is thoroughly cleaned and sharp burrs are made dull. c, This additional cold work hardens the surface of the wire and raises the physical properties where the stress is highest. d. Cold forging traps beneficial compression stresses near the wire surface which must be overcome by the destructive tensile stresses that cause fatigue failure before breakage can occur. All heat-treating of springs and all stress-relieving processes should be done prior to shot peening, except in those instances where electroplating is used; it is then necessary to reheat 0 after plating. Heating the springs above 500 F after shot peening counteracts much of the beneficial effects of the trapped compression stresses produced by shot peening. Springs made from annealed, oil tempered or alloy steels that must be electroplated can be shot peened principally to clean the surface. thus avoiding the necessity of soaking them in acid solutions to remove scale. The slightly roughened surface of shot peened springs does not produce a bright glossy electroplated coating. Protective Coatings
\
\
Uncoated or oil dipped springs are satisfactory where corrosive conditions are not a factor. Black japanning is often used as it is a flexible, inexpensive finish suitable for many applications. Enamels, lacquers and paint are occasionally used. Cadmium with supplementary chromate treatment provides one of the best electro-deposited coatings because it is both flexible and corrosion resistant. 14.8.6
Hydrogen Embrittlement
Steel. particularly hardened steel, is susceptible to embrittlement resulting from hydrogen introduced by acid pickling, electroplating or cathodic electrocleaning operations. Absorbed hydrogen results in brittle behavior, particularly under sustained loading in the presence of stress concentrations. Baking to relieve hydrogen embrittled springs should be accomplished.
14-31/14-32
)
STRUCTURAL DESIGN MANUAL
•
SECTION 15
THERMAL STRESS ANALYSIS 15.0 GENERAL When a structural element is subjected to a change in temperature, it will either expand or contract depending on whether the temperature change is an increase or decrease. If the element is restrained, such as is common in airframes, the attempt at expansion or contraction will induce stresses into the structure. Not only will the individual element be affected but the surrounding structure will have induced loads from the temperature change. In the absence of constraints at boundaries, thermal stresses in a body are self equilibrating.
•
Except for a few simple cases, the solution of the thermoelasticity problem becomes intractable. Therefore, for thermal stress analysis, approximations leading to the strength of materials and finite element methods are used extensively. Depending on its geometry, a structural element is classified as; rad t beam, curved beam, plate or shell. If a structure consists of one of these elements or some simple combination of them, the method of strength of materials will yield good results. If a structure has a complex geometry, the finite clement method is easier to use and the results are satisfactory. The finite el~ment method used at Bell Helicopter is NASTRAN. It should be used on an idealized structure whi.ch consists of a large number of smaller simpler el.ements to provide approximately the configur~tion of the actual structure. In a constrained structure, compressive stresses resulting from thermal, or thermal and mechanical, loading may produce instability of the structure. The linear thermoelastic solution of the problem excludes the question of large deflections. Thus, for buckling t or for structures where loads depend on deformations, nonlinearity that is due to large deformations must be incorporated in the problem formulation. The extreme difficulty involved in solving the nonlinear thermoelasticity problem has led to the approximate methods of strength of materials and finite elements. The strength of materials solutions for simple structural shapes are presented in this section. The finite element solutions must be obtained by the use of NASTRAN and is not within the scope of this manual.
15.] Strength of Materials Solutions
•
Thp assumption that a plane section normal to the reference axis before thermal loading remains normal to the deformed reference axis and plane after thermal loading, along with neglecting the effect on stress distribution of lateral contraction, lays the foundation 6f the approximate methods of strength of materials. Materials solution improves with the reduction of depth-to-span ratio, if the variation of temperature along the length of the beam is smooth. As in the case of mechanical loads t a considerable error results in the vicinity of abrupt changes in the cross sections. If the temperature is either uniform or linear along the length of· the beam, the assumption of a plane section is valid and the strength of materials ITlethod gives the same results as those by the plane stress thermoelastic method.
15-1
STRUCTURAL DESIGN MANUAL Thermal stresses are induced in structures as a result of a. Heating or cooling of an element which has some restraints (elements with no restraints have self equilibrating stresses). h. Heating or cooling of a structure composed of elements with different coefficients of thermal expansion. c. Unequal heating or cooling causing a non-linear or non-uniforITl temperature distribution within a beam or plate. d. Unsymmetrical heating or cooling through the thickness of a plate or beam producing bending moments, with or without external restraints. Thermal stresses can be added linearly to mechanical stresses if the total is below the proportional limit of the material. Above the proportional limit, the sum of the thermal and mechanical stresses can be obtained using a strain analysis.
i5.L.1 General Stresses and Strains The following equations will give the strains in the x, y and
z
directions.
lIE [ ax - lJ.(ay
+
az)J
+ a(T-T 0)
15. ]
€y = I/E[Uy - 1J.({1x
+
(]z)] '+Q(T-T ) O
LS.2
EX =
15.3
where:
coefficient of thermal expansion Poisson's ratio E ::; modulus of elasticity TO == reference . temperature (zero thermal stress) temperature at point in question T a I-'
The equations for stress in the x, y and z directions are Ux
= ( 1 .+11)Er1
(]y = O'z ;:
- 21J,)
EI:!;,
(1
+ 1-') ( 1
- 2p,)
EIJ,
(1 +1-')(1 - 21J.)
(Ex
+ Ey +
(Ex + Ey (Ex +
+
Ez ) +
E'z)
aE(T-To) 1 ... 2 I-'
15.4
E"y ..
O!E(T-TO) 1 - 2~
15.5
E frz ( 1 """ J.L)
aE(T-Tql 1 - 2p.
15.6
E Ex _ ( 1 +Jl)
E
+ ( 1 +/J)
E'y + Ez ) +
In the pLane stress case (a z = 0), these equations become Ea ~T-Tol Ux = 1E.p2(€x + lJ€y) 1 -
Uy = _E_ (E y + P,E x )
1 -
/J 2
- EO!1 -(T-TO) Il
For uniaxial conditions (O'y =
(1z
O"x = E Ex ... E a (T-T 0) ::; E [Ex -
15-2
15.7
IJ.
J5 .. 8
0)
a (T-To)l
15.9
STRUCTURAL DESIGN MANUAL
•
15.2 Uniform Heating Following are some typical beam elements using the previous equations to determine the effects of uniform heating.
15.2.1 Bar Restrained Against Lengthwise Expansion
~~L_He_ated_Bar----f'1 ~
L
~
FIGURE 15.1 - FULL RESTRAINT, UNIFORM
•
Hf~AT
P = -AE 0 (T-TO) = -E 0 (T-TO)
15. L0 15. 11
(j
15.2.2 Restrained Bar With a Gap at One End
p~--II--p ~L~~g
FIGURE 15.2 - FULL RESTRAINT WITH GAP, UNIFORM HEAT If gIL ~ O(T-TO), P = 0 If giL < 0 (T-TO), P = -EAa
and
(j
=
(T-TO) + gAE/L -E a (T .. TO) + gE/L
15.12 15.13
• 15-3
STRUCTURAL DESIGN MANUAL 15.2.3 Partial Restraint
)
P
=
wl)(;~re:
~
FIGURE 15.3 - RESTRAINT WITH SPRING, UNIFORM HEAT a1L1(T-T O)1 + a 2L2(T-TO)2
lIe + L1/A1El C = spring ratc for L2 at its final temperature T. be real or represent another structure.
This spring may
15.2.4 Two Bars at Different Temperatures The bars are attached such that the cold bar restrains the expansion of the hot bar. The bars remain straight with no bending.
(1) Hot Bar
(2) Cold Bar
FIGURE 15.4 - TWO BARS AT DIFFERENT TEMPERATURES O"t ;: -E
1 0!1(T1-TO)C 1
0'2 .::; ... Al ' A2 0'1
C =l:::~ + E2A2Jl'
1
15-4
15. j 4
15. ] 5
~17i:~i::O) I
l5.16
•
STRUCTURAL DESIGN MANUAL
•
15.2.5 Three BaTS at Different Temperatures
FIGURE 15.5 - THREE BARS AT DIFFERENT TEMPERATURES (J'
1
0!1(T -T )C 1 O 1
15.17
- E2 Q'2 ( T2- TO) c 2
15.18
-E
1
~ = 1/A 3 (0'1A l + (J2 A2)
/
c2 =
15.19
_A_2E_2_+_A_3_E_3_ _ _ 1 1 - A20!2 E2(T 2 -T O) + A3Q'3 E 3(T 3-TO) AlEl + AZE2 + A3 E3 0!1(T I -TO)(A 2E2 + A3 E3)
AlEl + A3 E3
15.20
1 - A10!1E1(T1-T O) + A30!3E3(T3-TO) 15.21
15.2.6 General Equations for Bars at Different Temperatures
(J'
i
15.22
where i refers to the bar in question and P is the externally applied axial load. 15.3 Non-Uniform Temperatures The following are equations which can be used to determine the stress in beams with temperatures varying through the depth. Figure 15.6 shows beams with uniform and non-uniform thickness •
• 15-5
STRUCTURAL DESIGN MANUAL
o
y-
y
y
y •
+c
-+
cent.
x
cent.
z
-c
x
•
~
-.f To ~
FIGURE 15.6 - TYPICAL BEAMS WITH VARYING TEMPERATURES 15.3.1 Uniform Thickness
f
~c
I1x
= - or E(T-TO) +
1/2C
orE(T-TO)dy
+
3y/2C
-c
3
+C
j orE(T-TO)ydy
15.23
•
-c
Notes: (1) If the beam is restrained frorn expanding and hending, drop the last two terms. (2) If the beam is restrained from expansion but is free to bend, drop the middle term. (3) If the beam is restrained from bending, drop the last term.
15.3.2 Varying Thickness If the beam has a varying width through its depth and is symmetrical about its vertical centerline, the previous equation becomes 11
x
= - orE(T-TO) +
I/Af E (T-T )td Y O
A
+
Y/lzjorE(T-TO)Ytd Y
15.24
A
where I is moment of inertia about the centroidal axis and t is a function of y aszshown in Figure L5.6. 15.4 Linear Temperature Variations Th(' foil ow'i ng are equB t ions wh i ch c an be used to de terml n(~ the s trcs ~ in bC'ams wr.th Ilnear temperature varIations between the two faces.
• 15-6
STRUCTURAL DESIGN MANUAL 15.4.1 Restrained Rectangular Beam t Uniform Face Temperatures
\ I
"
FIGURE 15.7 - RESTRAINED, DIFFERENT FACE TEMPERATURES M O"b
=±Ea(T -T )/2 1 O
max
e·
15.25
Ela(T1-TO)/t
15.26
15.4.2 Pin-Ended Beam The foJlowing equations are for a pin-ended beam with rectangular cross section and different uniform face temperatures.
p
-" I
_/
FIGURE 15.8 - PIN-ENDED COLUMN, DIFFERENT FACE TEMPERATURES e
T
~
Q
2 (T ('oTO)L /8t
p. = 1J'2EI/L2
•
15.28
(e T _+ e1,)/(1- PIp·)
e final
=
M
PCe f lna 1) ·
max
15.27
15.29 15.30
eccentricity due to temperature
where: c.1
= any
initial eccentricity
15-7
STRUCTURAL DESIGN MANUAL
•
L5.5 Combined Mechanical and Thermal Stresses If a member is subjected to external loads and moments with temperature variations in two planes as shown in Figure 15.9, the stresses can be calculated as follows. y
y
~My FIGURE 1.5.9 - COMBINING MECHANICAL AND-THERMAL STRESSES
= E.
Mz +
Y.
i l l
[ (
.ttY. E.A.
1-
1
1
1
2
n
a.1 (T.1 -To)} + Z.
,,",Yo E.A.
~
1
1
1
1
i=l
(MY +
~z. E.A. a. (T. -To)}
1-
1
1
1
1
•
1
2
n
,",Z. E.A.
~
1
1
1
i=1
n
+ -i~lE.A. a.1 (T 1'_TO)~ _ 1 1
f
+
E. a. (T. -T ) n E A l l 1 0 ,L... i i
15.31
.~.
i=l
where:
M = moment about z-axis z moment about y-axis M Y
If no bending in the x-z plane is assumed the quantity in the second parenthesis is eliminated. 15.6 Flat Plates The following equation is the general expression for flat plates with the temperature varying through the thickness and independent of the length or width. Figure 15.10 shows the nomenclature for this ~quation.
• 15-8
STRUCTURAL DESIGN MANUAL
• z
~~----------~------~~~----.-x
FIGURE 15.10 - FLAT PLATE, NON-UNIFORM TEMPERATURE VARIATION
S C
(Jy =
-aE(T-T O)
(1 _ p.)
1
+
2C(1-P.)
aE(T-To)dz
-c
•
+
~
c
Z
2C (l-P.)
Sa
E(T-TO) zdz
15.32
-c l5.6.1 Plate of General Shape The following equation is for a flat plate of any shape, rotationally restrained at th(~ edges, with linear temperature gradient between the two faces, both at different uniform temperatures. r~a(TI-TO) (Jrnax =
+
15.33
2(1-~)
15.6.2 Square Plates The following equations are for a square plate, rotationally restrained at the edges, with linear temperature variation between the two faces, both at uniforrn temperature. Near the edges:
C1
bmax
15.34 15.35
•
15.6.3 Flat Plates with Uniform Heating The following equations assume that there is uniform heating, no bending and that the edges remain straight and parallel.
15-9
STRUCTURAL DESIGN MANUAL 1) Uniformly heated rectangular plate restrained in the x and y directions only.
•
y
FIGURE 15.11 - RESTRAINED PLATE, UNIFORMLY HEATED
= (]y x
(J
=
-EO! (T-TO)
15.36
(1 -IJ.)
2) Partial restraint of a uniformly heated square plate.
•
2)
(2
(2)
x-x
FIGURE 15.12 - PARTIALLY RESTRAINED SQUARE PLATE, UNIFORMLY HEATED -E10!1(T1-TO)K
=
( l-IJ.)
15.37 15.38
where:
15.39
15-10
•
STRUCTURAL DESIGN MANUAL :n
Pllrt,lal restraint of an uniformly heated rectangular plate .
.... 1
J
y
AS 2
FIGURE 15.13 - PARTIALLY RESTRAINED RECTANGULAR PLATE, UNIFORMLY HEATED 2 E1/(1-1J) [€
x
+ P€y
15.40 15.41
•
(J
2
=
15.42
l5.43 1:F :EF
15.44
x
15.45
Y
where: A \
bt
xl
A
= 2wt2 = at
A3
=
A2
Yl
2vt
3
Equations 15.44 and 15.45 can be solved simultaneously for E and E. Substituting back into equations 15.40, 15.4l, l5.42 and 15.43, th~ stres~ at the edge of the members can be obtained •
• 15-11
STRUCTURAL DESIGN MANUAL 15.7 Temperature Effects on Joints Temperature affects the preload in a fastener and also induces loads into the clamped sheets. These effects are obtained using the following equations. 15.7. I Preload Effec ts' Due to Temperature Joints with bolts, or other threaded fasteners, which are exposed to a temperature change after installation will have a change in preload. The change can either be an increase or decrease. Figure 15.14 shows a typical joint.
f
- -.,.- -
.1--
\
)
::::t-:~'"=""!!!!!!"'!I!I~
• FIGURE 15.14 - JOINT NOMENCLATURE The following equations assume no washer deformation and no gap exists.
C
s
eM
Q!BLB (T-T O)
+
~B (~ss + ;:r +
= (T-TO)(aata + abtb
a)
15 .. 46
r
+ actc) - P(A::a +
~~b + A:~~
15.48
e B = eM
(T - TO) ( QIa t a +
(Y
b t b + act c .. a BLB )
P t wiJ J bf.\ tension if T >TO and aM >~.
(+) is tension in bolr J 15-12
15.47
(-)
would unload a preloaded bolt.
15.49
•
•
STRUCTURAL DESIGN MANUAL If al
three materials are the same, equation 15.49 becomes
J
15.50
If a gap exlsts, no preload in bolt, equations 15.49 and 15.50 become l5.5l
(+) is tension in bolt, (-) would unload a preloaded bolt
-g(T-TO)( aMt M - aBL B) _1 (.!a + !r + + tM ES
•
As
Ar
-1&) 2Ar
15.52
~EM
(+) is tension in bolt, (-) would unload a preloaded bolt where:
~
Ar As DB
DM
= Area = Area
=
of root of bolt of shank of bolt BoLt diameter Effective diameter of material exerting bolt; assumed to be 2 DB
the~mal
eB
= Deformation
of bolt over length, LB
eM
= Deformation
of material over thickness, LM
= Gap
g
LB
=
Ls + Lr + Lg/2
Effective length of bolt
= Final
T
temperature Initial temperature
To tM E
load on
=
Total material thickness
=
ta + tb + t c.
= Modulus of eiasticity
a • Coefficient of thermal expansion 15.7.2 Thermally Induced Loads in Material
•
When same into ment
dissimilar materials are subjected to a uniform temperature change from the initial temperature, TO' to a final temperature TI and T , the loads induced 2 the clamped materials are as follows. Figure 15. 5 shows the general arrangeof the joint. ,The equations assume that the bolts are concentric in the
15... 13
STRUCTURAL DESIGN MANUAL
r-
L
L
-r
L
(1) ~2)
Single Bol t
Two or Three Bolts
FIGURE 15.15 - JOINTS WITH DISSIMILAR MATERIALS haleSt i.e., the gap is equal all the way around the bolt. For heating of sheet 1 or cooling of sheet 2;
a 1(T-TO) - Q2(T 2-TO) - (&1 + g2)/L P = l/AIEI +' 1/A2E2
15.53
The force P will.be compressive in 1 and tensile in 2. For heating of sheet 2 or cooling of sheet 1;
a 2(T 2-TO) - Ql(T1-T O) - (g1 + g2)/L p =
l/AIEI + 1/A2E2
•
15.54
The force P will be compressive in 2 and tensile in 1. If the joint is a
singl~
lap joint as shown in Figure 15.16, the equations for
(1 )
:J..l . FIGURE 15.16 - SINGLE LAP JOINT loads in the materials are as follows. 15-14
•
•
STRUCTURAL DESIGN MANUAL p = ClJL (T1-T O) 1
Q!
ZL 2 (T 2 ,,"T O)
+
&1 + g2
15.55
L}/A,El + LZ/AZE Z
For riveted joints, g1 and gz are set equal to zero in equations 15.53, 15.54 and 15.55. 15.8 Thermal Buckling Thermally induced strains can induce buckling in beams and plates. It is assumed that this buckling occurs in the elastic range. The following general equation can he used to detennine the temperature differential which would initiate. buckling of a column.
p2
C 11'2 L2C
[aCT-TO) ) CR =
15.56 1
where:
T-To = Temperature change C = Column fixity coefficient
C]
•
C 2
C2/(C 2 + AE/L)
Stiffness of restraining structure
If the structure being heated is a flat plate with uniform heating the temperature differential which would initiate buckling is
Fully restrained in one direction: K 71'2
t 2
(T 1-T O)CR = 12(l +~)a(b)
15.57
where K is the non-dimensional buckling constant shown in Section 10.
15.58
Fully restrained in two directions, clamped edges:
=
(TJ-T)
o
CR
1f2
l2(l
('1)2 4(1-J.l.) 112) b
301(1
+ !~)
(3b 2+ 3a22 + 2)
/
15.59
li
• 15-15/15-16
• ) )
•
-/
•
'.
Revision D
STRUCTURAL DESIGN MANUAL VOLUME II
PROPRIETARY RIGHTS NOTICE ,THESE DATA ARE PROPRIETARY TO BELL HELICOPTER COM-
PANY. DISCLOSURE, REPRODUCTION, OR USE OF rHESE DATA FOR ANY PURPOSE IS FORBIDDEN WITHOUT WRITTEN AUTHORIZATION FROM BELl. HELICOPTER COMPANY.
I. ')
Bell Helicopter i i l:i
itt·]:I
STRUCTURAL DESIGN MANUAL This Structural Design Manual is the property of Bell Helicopter Textron and no pages are to be added or withdrawn except as directed by revision notices.
Information contained herein which is specifically identified may
not be reproduced or"further disseminated without the approval of the Chief of Structural Technology, Bell Helicopter Textron.
) Jj
• i1
•
® cl c s
=
c
c
9.2
9.2
c
c
8)
s
s
5.8
Ic
Sl
J4
13
\
\
~-.~
12
,
L
-[ I
~ \
\ ' r\ \ , \ " "'"
~3J
10
KS
---4)
9
-
--
~
\
\
\\
-L
6
..
I
'" " ~ ~
I
=
I
-...-.
\
_
,,~
~
"'"
"
f - - - 1---
I
\\.
'\
7
TN 37 Bl
I
t2) 1\
1I
IS
a long side b = short side
Ref . Nl4.CA
11'1' \.V
~
5.5 s
s
, i
8
Cl
clamped simple supported
J5
•
Is
®
"" i'... ~ ............
\
~ r--
---
...
r-- r-. r---
....... I
5
o
1
2
3
4
5
6
PANEL ASPECT RATIO, alb F1C;URE 10.18 - SHEAR BUCKLING COEFFICIENT, K s
10-19
STRUCTURA,L DESIGN MANUAL ,
,
•
Revision A 10.4
SHEAR BUCKLING
The critical- shear stress at which a plate first buckles is given by the equation:
T
cr
where Ks (Fig. 10.18) is the non-dimensional shear buckling coefficient and is a function of the plate geometry and edge restraints. The values of Ks and ~ are always the elastic' values since the plasticity correction factor, ~ t contains all changes in those terms resul ting from inelastic behavior. The term- b is the smaller dimension of the panel.
)
A great deal of work has been done relative to the value of the plasticity correction factor. the expression for ij must involve a measure of the stiffness of the material in the-elastic and inelastic ranges. A simple means of obtaining a value of ij is to take the ratio of the shear secant modulus to the shear modulus. shear secant modulus shear modulus
10.4.1 CRITICAL BUCKLING STRESS WITH AXIAL LOADS When axial loads are present the actual shear buckling stress 10.4 will be different. The presence of compressive stresses stresses causes the panel to buc'kle at a lower value of shear were present. Tension causes the panel to buckle at a higher
defined in paragraph together with shear than if no compression shear stress.
•
When shear and compression are present the panel buckles according to the interaction f
c
IF c
+ cr
(f
s
IF
s
)2
1.0
cr
where FCcr and FScr are the critical panel buckling stresses for pure compression and pure shear. From chapter 7, section 7.3 the buckling stress for a panel under compression is F
c
cr
For any particular panel
FC
/Fs - A, (a constant) cr cr From conventional means the applied compressive stress, fc' and the applied shear stress, fs, can be calculated. These stresses will have a constant relationship • with each other until the panel buckles, after which the compressive stress no longer increases. Thus
10-20
)
•
STRUCTURAL DESIGN MANUAL The shear stresses are calculated from the shear flow equation:
f
s
v
=
q/t
(h )(t) c
(k)(f ); fs
s
~
s
=
(l-k)(f ) s
As the load increases beyond the initial buckling load, a higher percentage of the total shear is carried by tension field. This causes the ratio f Ifs to become s cr an important parameter. Methods of analysis for three specific types of tension field beams are given: 1.
Flat tension field beams with single uprights. Flat tension field beams with single uprights and access holes. Curved tension field beams.
2. 3.
•
The curves given for use in these analyses yield results with a reasonable assurance of conservative strength predictions t provided that normal design practices and proportions are used •
10.5.1
Effective Area of the Uprights
In order to make the design curves apply to both single and double uprights, it is necessary to define an effective upright area A ue For double uprights, which are symmetric with 'respect to the web:
Aue
= Au =
total cross-sectional area of the uprights.
For single uprights: .
)
A A
u
ue 1
+( ~y
where p= radius of gyration of the stiffener and e = distance from the centroid of the stiffener to the center of the web.
If the upright has a very deep web, Aue should be taken to be the sum of the crosssectional area of the attached leg and an area 12 tUt where tu is the upright thickness. 10.S.2
Moment of Inertia of the Uprights
The uprights must have a sufficient moment of inertia to prevent buckling of the web system as a whole before formation of the tension field, in addition to preventing column failure due to the loads imposed upon the upright by' the tension field. Forced crippling failure, caused by the waves of the buckled web and possibly most critical, must also be prevented by the upright. The required moment of inertia of the upright may be determined by iterating through the appropriate Table 10.1, 10.2, 10.3, or 10.4.
10-21
STRUCTURAL DESIGN MANUAL
•
Revision B 10.5.3
Effective Column Length
The effective column (upright) length is calculated by the equations: If d < 1.5 h , u c If d > 1..5 h , c u
10.5.4
L
L
h
e
e
..Jl
+ k
2
(3 - 2d)
h
h
')
Discussion of the End Panel of a Beam
The following analyses are concerned with the ftinterior" bays of a beam. The uprights in these areas are subjected, primarily, only to axial compressive loads. The end panel, however, is a special case. Since the diagonal tension effect results in an inward pUllan the end upright, bending, in addition to the usual compressive axial load, is also produced. There are three general ways of dealing with the edge member subjected to bending. 1.
Sufficiently strengthen the edge member so it can carryall of the loads (which is inefficient, weight-wise, for long unsupported lengths).
2.
Increase the thickness of the end panel either to make it nonbuckling or to reduce k, which would reduce the running load producing bending in the edge member. (This is usually inefficient for large panels.)
3.
Additional uprights may be provided to support the edge member and thus reduce its bending moment.
10.5.5
•
Analysis of a Flat Tension Field Beam with Single Uprights
Table 10.1 is a step-by-step procedure which yields the stresses in the flanges, webs, rivets, and uprights of a flat tension field beam with single uprights (Figure 10.19) .. Table 10.1 is based on a single web with parallel flanges and parallel uprights. Most beams consist of more than one;web. At various locations in the following table adjacent panels must be considered. Such a situation occurs for rivet load, stringer axial stress, upright stress and moment in stringer.
• 10-?' 2
•
STRUCTURAL DESIGN MANUAL 4 .. BORIZ. CONCENTRATED
LOAn"
v
I.
=.Q£ [ 1 _ d (h+d) L 2h2 .'
HA
=
Q - lIn
H = Q£ {b + __d__ r(h+~)( -b[ 3K+4] D
Lh
- 21,)
6h2(K+l)
+ 2 (2L+b)(h+e)1-
M
A
=
MD --
Qed
6h 2 (K
•
[ (h+d) (3K+4) - 2 (h+e)
1) "
Qed
-1
(h + 2c + d)
2
6h (K + 1)
5. VERTICAL UNIFORM RUNNING 1,OAD
+
3aC.i}
VA = wa
[1 -;L J
l>,a 2
Vc =2Lwa 2 H = 8h
[ 4b
L
6. VERTICAL UNIFORM RUNNING LOAD
+
_1
J
1+K _
3a "1 8L
2 = wa (3K + 2) A 24(K + 1)
M
Me
•
",B 2
= 24(K
+ 1)
TABLE 12.4 (CONTln) - REACTIONS AND CONSTRAINING MOMENTS IN TRIANGULAR FRAMES 12-19
STRUCTURAL DESIGN MANUAL 7. HORIZ. UNIFORH RUNNING LOAD
wh2 V =2L
•
w
J 8. BORIZ. UNIFORM
V ;: 3wh
RUNNING l.OAD
2
8L
f
wh
== 8L(K
H ---
c
~"'c v
9.
+ 1) [b(3K + 4) + a]
= wh2(3K + A
24(K
2)
+ 1)
Me
II:
24(K
+
M L
1)
•
:::-
;: !Lr hL
bKJ
a -
K+ 1
)
I
v
.
= 3M
2 _ H -
----~
Ii
V
2L
3M(a - bK) 2hL(K + 1)
KM MA ::; 2(K + 1) L----IIIII.
v
Me
M == 2(K
+
1)
TABLE 12.4 (CONT'D) - REACTIONS AND CONSTRAINING MOMENTS IN TRIANGULAR FRAMES 12-20
•
•
STRUCTURAL DESIGN MANUAL 1. Sinusoidal Normal Pressure
v
b{lb/in.)
CnR
:=
4 CR
'4
H ==
2 CR Me = ~
[ (n-2B)cos
a-
n +
3
sill
oJ
H '--. ....
b ::: C
5
in
a
• Sinusoidal Normal Pressure
CrcR V:::: -4-
~R [3:
H =
•
M
= CR
2
4
H
b
=C
sin
~
(Positive moment acts clockwise· on section ahead.) .~
~
.......H
MO
r
2
.·,Y J = .
n3
= CR 2
lon.l
8 _ 1f2
J
:=
[.81974 sin
31974CR
.054 78CR
e
- .84018 + (Positive moment
section
ahead.)
2
£~so
acts clockwise on
3. Uniform Normal Pressure
H
M
= 0 at
T
= V = bR
all points since pin points permit a uniform hoop tension. T, where:
H = 0
TABLE 12.5 - REACTIONS AND CONSTRAINING MOMENTS IN SEMICIRCULAR FRAMES OR ARCHES 12-21
/,11\,'\\
Ii , \ \\
~
'\ ............ a_II. ", V ~ ,1/ .....
12.4
Ito
STRUCTURAL DESIGN MANUAL
:~",
AnaLysis of Rings
Tables and figures are presented for the analysis of rings and ring-supported shelts. Sections 12.4.1 and 12.4.2 show analysis methods for rings which are rigid with respect to the resisting structure for out-of-plane loads. The plane of the ring remains plane and the supporting structure deforms.
•
O~ly
bending is considered in the deflection curves for the in-plane load cases gIven in Figures 12.4 through 12.29. Refer to Figures 12.30 through 12.33 to include the effects of shear and normal forces.
Section 12.4.3 'shows methods of analysis for circular cylindrical shells supported by "flexible u rings. 12.4.1
)
Analysis of Rigid Rings with In-Plane Loading
Coefficients to obtain slope, deflection, bending moment, shear, and axi.a1 force along with equations for these values are given for some of the frequently-used load cases. Figure 12.4 shows an index for the various load cases presented in Figures 12.5 through 12.29. Tile sign convention used throughout the rigid frame analysis in-plane load cases i.s shown in Figure 12.3.
It basically consists of:
moments which produce tension
• FIGURE 12.3 - SIGN CONVENTION FOR RIGID RINGS WITH IN-PLANE LOADS en the inner fibers are positive f transverse forces which act upward to the left of the cut are positive and axial forces which produce tension in the frame' are positive. Deflections in Figures 12.5 through 12.29 are based on bending only. Deflection curves for the three basic load cases due to shear and concentrated loads are shown in Figures 12.31 through 12.33. A shape factor (~) that is to be used with the cllrves [or shear deflection of various cross sections 1s shown .In ~',igUT(! 12. 'HL
12-22
•
STRUCTURAL DESIGN MANUAL Revision A
e
t t- t·
+ + ++ + +f-
1-
he
1e. T
+
+
+
+-
+ +
+
+
+
+
. . . + + +- + +
+ :+
WEB2
d
1-
:1
he
+ +
1
+ +
++++++
+
ItL
n.L
IlL. k
UPRIGHT-z........
+
+
+-+++ 1- r
I
.~
FIGURE 10. 19 - FLAT TENSION FIELD BEAM WITH SINGLE UPRIGHTS Description
Variable and Equation
CD E 1as tic
mod u 1u s E e
Q) Upright
spacing, (NA to NA)
(}) C.) ear web between uprights
Numerical Value
d dc
(rivet to rivet)
•
C0 Distance
from median plane of web to centroid of upright
e
@Clear web between flanges
(rivet to rivet)
® Di stance
be tween f lange
he
centroids
(1) Leng th
hu
® Web
t
() f uprigh t be tween upright to flange rivets
thickness
@ Upright
@Flange 1])
Lhickness thi.ckness
Uprigh t area
) U> Flange
a rea
~ Radius of gyration of upright ~ Moment of inertia of upright ~ Moment of inertia of flange ~ Applied load - upright
~ ~ ~
Iu
IF
Applied Load - flange
Pu Pf
Applied web shear flow
q
Web shear stress
T
q/t =
@ /®
e~------~--------~-TABLE 10.1 - ANALYSIS OF FLAT TENSION FIELD BEAM WITH SINGLE UPRIGHT 10-23
STRUCTURAL DESIGN MANUAL Parameter
@ Il+(®21 0 Aue/det = @ Ie})@)
Parametet'
het
~ Effective area of upright
@ @
Aue =
=@®
© Parameter
dc/hu =
@
tf/t =
Parameter
© Parameter ® Parameter © Parameter
01 (J) ® /@
=@I@
tult
he/de
=0/0
dclh e = 1 I
@ )
® Parameter
=@ 10 t/h e =®/0)
Q9
Upright restraint coefficient
Rh, Figure lO.20(b)
Q»
Flange restraint coefficient
Rd, Figure IO.20(b)
@
Parameter
tIde
~ Theoretical buckling coefficient
@
kss' Figure 10.20(a)
Elastic buckling stress: de
@(D@21®
de> he Tcre
@CD®21®
Q9 Initial
@ Stress
buckling stress ratio
~ Diagonal tension factor
@
QY Ratio
of upright to shear stresses tension angle
Stress in median plane upright/
web
T/Ter =
1@
Uu =
-
avg max
@@@ /@ = @@ / @ = f41\ @ ~
(§l ( 3- 2 @) I~
Le= (j) I /1+ Le=hu=0
CD
~ Slenderness ratio
Le/ 2p
~ Column allowable
(Teo
€)
Proportional limit
Fpl' Section 5
@
Strain, if ®>@
O"u/ E =
=@
12
= 1T2@/
@
@2 or Section 11
leD
TABLE 10.1 (cont'd) - ANALYSIS OF FLAT TENSION FIELD BEAM WITH SINGLE UPRIGHT
10-24
•
Tana, Figure 10.25(a)
(Tu
0>1.5
(1-@) 2
luu, Figure 10.23 max , (TuIT, Figure 10.24
€) Upright
~ Effective column length: If @< 1.5
2
=0
(Tu
O"u
If
@
k, Figure 10.22 @ 300tdc /12h e
~ Upright average stress
maximum stress
_®) @31 _@) ®3!
Figure 10.21 (See Note 2)
det
~ Ratio of upright stresses
€y
Ter ,
~(® + ~ (<19 +
~+l (l-k) =@+
Parameter
€9 Diagonal
2)
) .
STRUCTURAL DESIGN MANUA'L
o
®
Revision A From stress strain curve
F c ) use ~ to determine all.
Ma'rgin of Safety: column yield
~:~
® MS - CoLumn ® Pararne te r .
MS=@I@-l MS = I @ - 1
e
MS=<@I@-l
~ Upright allowable (forced
®
/t)1/3 =
k2/3( t
u Figure 10.26
0"0'
213
® 1/3
crippling)
~ MS - Forced crippling
®
Parameter
c
® Pararne te r. @ @ @ @ @
o
o @
C , Figure 10.27 1 C ' Figure 10.28 2
Parameter Max imurn web
= @ I @-1 Wd = .7Q)(®/2@®)1/4
MS
stress
OJ)
i' = max
(1 +~2 ®)(l +'36'@) ~ ~
Web aLlowable
":11'
MS - Web
MS
Parameter
C ' Figure 10.28 3 MSB = (1/12) (@
Secondary bending in flange
® /®
= 45°
-1
®
Di.stance from NA to extreme fiber of flange
C
Distance - NA to ne.ar fiber of flange
D f
Flange ;lppli.eu stress
(J'
Diagonal tension stress-flange (camp)
=
Figure 10.29 @ a pDT
f
=@ I
©
U:T = -<@QJ/@."12 @ I @+ 1
Secondary bending stress-flange uSB = (comp) Secondary bending s tress-flange uSB =
@0 I
.5(1-®>]
©
@@I @
(tension) Flange stress-inside fiber
U
Flange stress-extrc.:::me fiber
O"tot
Allowable crippling stressflange
Fcc
Allowable tension stress-flange F MS - Flange (tension) Flange (compression)
tu
or F
S= @ @
Rive t f ac to r Rivet load-w0b to flange
&> + @ + © = c© + ~ + @
tot =
1
"
+ qR
ty I @-1
10- 1 @ = @@ 0.414
TABLE 10.1 (conttd) - ANALYSIS OF FLAT TENSION FIELD BEAM WITH
~INGLE
UPRIGHT
10-25
STR'UCTURAL DESIGN MANUAL QJ Allowable
rivet shear load
P
af
~ MS - Flange rivets
MS =
~ Rivet load-upright to flange
P
@9
p
Allowable rivet load-~pright
@ /
@-1
=@@
u
au
~ MS - Upright rivets
MS =
@}
Fir' Section 10.6
Interrivet buckling allowable
~ MS - Interrivet buckling
MS =
~ Ultimate tensile stress of web
F
~ Rivet tensil~ stress-upright/
(fR
tu
@ I
®-1
(@ / @
-1
' Section 5 =
.22@@
web
~ Rivet allowable tensile stress ~ MS' - Rivet tension
F , Section 6 RT MS = (@ I @ -1
NOTES: (1)
(2)
If any of the margins of sa ety are less than zero, the design is inadequate. The (eficient area must be corrected and this table repeated. If the web is subjected to iension or compression as well as shear, the initial buckling stress of the web must be modified according to the method described in Section 10_4.1.
TABLE 10.1 (cont'd) - ANALYSIS OF FLAT TENSION FIELD BEAM WITH SINGLE UPRIGHT
10-26
)
•
•
THIS CURVE IS SHOWN FOR d < h . c
c
IF d c > h c THE ABSCISSA SHOULD BE READ AS }
'-'
1 I
~
9
L
d
J~
'--
~V' L t
1
~~
,a
~
kss 6
c(1_ r--
I
.1
~
--J -
.....
N
5
--
J
h
2lS-
c
~~.~~ -
1.6
'If-l-
I
~d
1.8
1.4 1.2
-
c
/'
1-
1.0
Rh , Rd
/
/
.8
.
.4
-
c
=00
I I /'
•2 r
rr-
1
... ...L
1
2
3 h d
(A)
I
4
a 5
CJ ,."
-:z en
/
CD
i'
1
2
I
FIGL~E
t
(B) EHPIRICAL
RESTRAi.~T
CALCULATI~G
BUCKLING STRESS OF
~EBS
4
:a:
::.:.
t
c
10.20 - GRAPHS FOR
3
-u ,
...... N .....&
r-
I
c
THEORETICAL COEFFICIENTS FOR PLATES WITH SIMPLY SUPPORTED EDGES •
c-)
::.:.
~
~
a
:::.:::.
c::
:::a
I
I
I
/
-I
c:
/ \ . . J.
/
(I)
-I
---~
/
I f
h
J
/
.6
r--
/'
/
------
t
COEFFle [F;"NTS
~
-: ;
h -
1
7
..........
;=::.::::-"'
~-
c
."""~'
:'~r-
c
8
o
/.,
c
10
4
/4~~',. K.-; ,
UPRIGHT COEFFICIENT COEFFICIENT
FL~~GE
Rd -
2:
c::
::.:. r-
I--'
o
•
N
(Xl
~
>~ ~'.
,
",\
~.
~_-
~
' .." , _ J /
~!:L:~:::~>" ;-
....::a
40
VV
30
T
eI'
, KSI 20
// ~
~
?- ...-----
v---
/
~
-
(I)
ALCLAD 707 5~ ..--1~
c:
c-,
......
c: :::a ):II r-
2024-T3 ~
CJ rrI
en
-z
l7
10
o
V.
o
/
W
a)
v
31:
):II
z
10
20
30
40
50
c: :.tar-
60
eI', ELASTIC' ~SI
T
FIGURE 10.21 -
•
G~4PHS
FOR CALCULATING
BUCKLI~G
•
STRESS OF WEBS
•
fI
/
/, '.~ 1\' <.. \ \ /: i ' , \
•
"
"~!t~~~ STRUCTURAL DESIGN MANUAL
• -- -
.'-----___
iii
,--.,-----'- 0 _ _ _
_
~
- -__
r---.
-.................
1--.. . -I-- ............... i'--I--.r-........_
---
-r--_ ----~- ... --
- -. .
-
--1--
-~ -
-I--
-_
~~~~~_4~-+~~~~~~~~~+_~0
o .-I
•
(1'\
.
OJ
,......
\.0
iii
..::t
M
N
.......
0
FIGURE 10.24 - DIAGONAL TENSION ANALYSIS
10-31
;f:1f~'
"~'~ 1.0
STRUCTURAL DESIGN MANUAL
~~
"~ ~ '"
'"
.9
~
~ ~ '" ~
'"" '" " '" '" .........
.........
......
.8
...........
'-... ~
'",,-
tan a
~
..........
.7
.6
-.......
~
'-....
'-..
k
~
~ ............
-........
,
----
..............
.............
o
r-... ............. .............
------ -
~ "'-,
I-- ............. 1 """"'".5
------
1.5
.5
)
,
o
2.0
INCOMPLETE DIAGONAL TENSION
(iJ)
48
2Af lit"
44
-
_I--""
~
/"
40
/
/
/ '/ " /
36
II 'h /
V
1/
V
l~
r--
8
/ " co'
/"
-
---
-
2~
./
~~
-
~
f.-- ~ i--!----
.-- 1----
I
IV/, j/
32
1,111
tlV fj
28
V
I
24
20
o
.4
.2
.6
.8
1.0
1.2
1.4
Aue dt
(b)
PURE DIAGONAL TENSION
FIGURE 10.25 - ANGLE OF DIAGONAL TENSION 10-32
•
. (//1\' \'\\
•
"~,~,~
STRUCTURAL DESIGN MANUAL .12
\ \
.10
~
.08
CL
\
\
\
.06
\
.. 04
\
I
i"
.02
" ~r-.. ~
o .. 6
.5
.7
.8
---
LO
.9
tan a
FIGURE 10.27 - ANGLE FACTOR C 1 1/4
•
wd
~
O.7d
((IC+~T)he )
1.2
--
1.0
C 2
C
3
I---
C 3
t-
""""-
.t)
l---.
.6
/
Lt ~
/C 2
V .4 .2
-~
0
o
1
2
/
/
-
/
3
4
wd FIGURE 10.28 - STRESS CONCENTRATION FACTORS, C AND C 2 3
• 10-35
STRuc-rURAl DESIGN MANUAL
•
30 "-
~
25
>--
............
~ ~ ~~ ~
I-
-- ,'" ~
-
::---.;: ~ ::::,....
..... ~
20
.......... ~
-.....
---r~
.......... :-....
~
-k
Tall' ksi
----.- ::--
....... .........
-
45 40
-r--- r--
""'"'-
-r--. I-- . -....... r...... ..............
........
15
-.......
-
--- ----..
-
35 30
)
25
20
10
o
.6
.4
.2
.. 8
1.0
k
2024 ALUMINUM ALLOY..
F
tu
= 62000
•
psi
DASHED LINE IS ALLOWABLE YIELD STRESS 35
,
I\..
~
~~
'0: ~~
30
;-....~
~~
~ ~ ~ ~~
'" '"
25
-r-- r--
" -- ---- -- - - -- ---. . . r-... .......... ........... r--.. ..........
.............
r--
r-- -.
r--.....
r---~
r--~
............
........ ~
~
'k
T a 11' ks i
~
....................
~
....................
---..
I'-....
20
r----.....
()
• "2
.4
.H
k
7075 ALUMINUM ALLOY.
F
tu
72000 psi
FIGURE 10.29 - BASIC ALLOWABLE VALUES OF TMAX 10-36
--
45 40
15 30
25
20
I • ()
)
;;:/T\'\' ~
•
"'''-~~II STRUCTURAL DESIGN MANUAL '\\ \ ... ....... . ,
\
'\
f"
~:-:,;L'
.-,/
Revision A
",.
®
EFFECTIVE AREA OF UPRIGHT
A
@
PARAMETER
A
@
PARAMI~T.ER
he t
©
PARAMETER
de/hu =
@
PARAMETER
tf/t=@/@
®
PARAMETER
PARAMETER
tu/t =@I@ h Id ,=0)/(1) c c dc/h e = 1 I @ tIde =@IG)
PARAMETER
t/he
®
UPRIGHT RESTRAINT COEFFICIENT
R , FIGURE 10.20 (b) h
@ @
FLANGE RESTRAINT COEFFICIENT
R , d
THEORETICAL BUCKLING COEFFICIENT
k
@ @ @ @
@
PARAMETER
ELASTIC BUCKLLNG STRESS: C
<
>
h he
c
~
V:.:;;JI
=@®
0) 10
=0/0) FIGURE 10.20 (b)
55
,FIGURE 10.20 (a)
=@~2r® + ~ (@ - ®)@~
TCTe
=@CD@2r@ + ~ <@ - @)@l
INITIAL BUCKLING STRESS
T
STRESS RATIO
TITcr ::: @ I @
@
DIAGONAL TENSION FACTOR
k, FIGURE 10.22 @ 300td 112h
@
PARAMKrr~R
cr
, FIGURE 10.21
~(1-k)
+
Aue t C
=' A
~
RATIO OJ." UPRIGHT STRESSES
~
RATIO OF UPRIGHT TO SHEAR
@
D1 AGONAL TENSION ANGLE
~
STRESS IN MEDIAN PLANE UPRIGHT / q, = WEB
@
UPRIGHT AVERAGE STRESS
~AVG
=
@@ I @
Q
UPRIGHT MAXIHlJM STRESS
(J"uMAX
=
@)@
au MAXI (]"u ,
+ (1-
c
=0
@)
2
FIGURE 10.23
(J" IT , FIGURE 10.24
STRESSES
EFFECTIVE COLUMN LENGTH: IF
u TAN c¥, FIGURE 10.25 (a)
>J.5
@@@ I @
Le
=G) I r1 +
Le
= h u =0" \.!../
(i}<1.5
@
(See Note 2)
c
@
@
•
= ~ 1f3V'8\
'Tere
d
)
u
Id t ue C
PARAMETER
d C d
A
=
ue
®2 (3 -
2
0) I~
j
TABLE 10.2,(CONT t D) - ANALYSIS OF FLAT TENSION FIELD BEAM WITH DOUBLE UPRIGHTS 10-39
STRUCTURAL DESIGN· MANUAL @
€9
SLENDERNESS RATIO
e CT
COLUMN ALLOWABLE
STRAIN, IF
@>@
~
FROM STRESS STRAIN CURVE
@
®
PARAMETER
~
~
1f
tp I \.!.J
©2
or SECTION 11
Fp1 ' SECTION 5
@
MARGIN OF SAFETY: CO~UMN YIELD MS..; COLUMN
2
=
co
~ PROPORTIONAL LIMIT
®
=@/@
LIp
CTu/E
@>Q)
@)<@
=
® leD
Fe' USE ~ TO DETERMINE ALLOWABLE
:~:~~~=i MS=(@/@-l k 2/3 ( tu /t ) 113 = ®2/3 @1/3
UPRIGHT ALLOWABLE (FORCED CRIPPLING)
FIGURE 10.26
PLASTICITY CORRECTION: IF
®>@
.
~ MS - FORCED CRIPPLING
® © (§)
PARAMETER PARAMETER
C , FIGURE 10.27 1
PARAMETER
C , FIGURE 10.28 2
~ MAXIMUM WEB STRESS
6
*all'
(1 +@2@)(1 +@)®)
WEB ALLOWABLE
T
MS - WEB
MS=@/®-l
PARAMETER
C , FIGURE 10.28 3 MSB = (1/12)(
SECONDARY BENDING IN FLANGE q
=@
T'MAX
FIGURE 10.29 @ Q pDT
C
DISTANCE - N.A. TO NEAR FIBER OF FLANGE
D
FLANGE APPLIED STRESS
© onT=-<@@1®>j12 @;@+ (T =-@@ I © SB
8
SECONDARY BENDING STRESS FLANGE (COMP)
9
SECONDARY BENDING STRESS FLANGE (TENSION)
45 0
®©@Gi @)
DISTANCE FROM N.A. TO EXTREME FIBER OF FLANGE
DIAGONAL TENSION STRESS FLANGE (COMP)
=
f
f
ua
=@
erSB =
I
.5(l-@)i
@@ I @)
TABLE 10.2 (CONTtD) - ANALYSIS OF FLAT TENSION FIELD BEAM WITH DOUBLE UPRIGHTS 10-40
•
STRUCTURAL DESIGN MANUAL Revision A lT
tot ::;
@+© +®
FLANGE STRESS - EXTREME FIBER
(Ttot ==
® + © + @)
ALLOWABLE CRIPPLING STRESS FLANGE
F
@
ALLOWAllLE TENSION STRESS FLANGE
F tu or F ty
@
MS - FLANGE (TENSION)
MS=@I@-l
~
MS - FLANGE (COMP)
MS=@I®-l
®
RIVET FACTOR
R=1+O.414@
(jJ
RI VET LOAD - WEB TO FLANGE
Rtl
~
ALLOWABLE RIVET SHEAR LOAD
Paf
®
I"LANGF: STRESS -
@ @
MS -
)
([9
INSID~~
FIBER
FLANGE RIVETS
cc
=qR=@@
MS=@/@-l
@@
RIVY.:T LOAD - UPRIGHT TO FLANGE
P
ALLOWABLE RIVET LOAD
p
MS - UPRIGHT RIVETS
MS=@)/@-l
STATIC MOMENT OF CROSS SECT. OF ONE UPRIGHT" ABOUT MEDIAN PLANE OF WEB
Q
WIDTH OF OUTSTANDING LEG OF UPRIGHT
b
UPRIGHT COLUMN YIELD STRESS
F
RIVET LOAD - UPRtGHT TO WEB
~
RIVET ALLOWABLE LOAD
P
MS - RIVET, UPRIGHT TO WEB
MS=@/@-
ULTIMATE TENSILE STRESS OF WEB
F
RIVET TENSILE STRESS J UPRIGHTI WEB
(F
RIVET ALLOWABLE TENSILE STRESS
F , SECTION 6 RT
o
u
au
coy
,SECTION 11 2
@@Q)I @@
ar
tu'
R
::;
1
SECTION 5
.15@@
®
MS - RIVET TENSION MS = I @- 1 NOTES: (1) If any of the margins of saCety are less than zero, the design
•
( 2)
is inadequate. The deficie~t area must be corrected and this' table repeated. If the web is subjected to ~ension or compression as well as shear, the initial buckling stress of the web must be modified according to the method des~ribed in Section 10.4.1.
TABLE 10.2 (CONTln) - ANALYSIS OF FLAT TENSION FIELD BEAMS WITH DOUBLE UPRIGHTS 10-41
STRUCTURAL DESIGN MANUAL 10.5.7
Analysis of a Flat Tension Field Beam with Single Uprights and Access Holes
The following step-by-step procedure (in Table 10.3) is an analysis of a flat tension field beam with single uprights and access holes (Figure 10.31).
FLANGE
T h
e
1 e
i
T
T
+
+++++ t ++ + + +
..J..
0
+
+ -t
+ ..,.
, + + 1-
+ t
++++ +
~-
f-+-
d
c
+
ILL ~
0 + + +
1
D
WEB
+
+ +
I h
c
1
l!...
FIGURE 10.31 - FLAT TENSION FIELD BEAM WITH SINGLE UPRIGHTS AND ACCESS HOLES Table 10.3 is based on a single web with parallel flanges and parallel uprights. Most beams consist of more than one web. At various locations in the following table adjacent panels must be considered. Such a situation occurs for rivet load, stringer axial stress, upright stress and moment in stringer.
• 10-42
STRUCTURAL DESIGN MANUAL Revision A Diagonal Tension Coefficient
k, Figure 10.22 @
Parameter
@
Ratio of Upright stresses Ratio of Upright to Shear Stress
u
Diagonal Tension Angle
tan a, Figure 10.25 (a)
Stress in Median PlaneUpright to Web
0 @ @I@ u uavg = O@ I © uumax = 0 @ 2 Le =CD/[l + 0 (3 - 2 @)I~
Upright Average stress Upright Maximum Stress Effective Column Length If~<1.5
• )
•
+ ~(1 -
umax
U
u
lu, Figure 10.23 u
IT,
o*u
0)
Figure 10.24
=-
=CD
If~>1 .. 5
Le == hu
Slenderness Ratio
Le/ 2p = @/2
Column Allowable
0"
Proportional Limit
F p1 ' Section 5
Strain, If
@ @ @) @ @ @ @ ~
300 td c/12h c == 0
<3
or section 11 co = 1J'2rp~2 \..:!;;f~
0> 9
CTu/E ==
@
leD
From Stress-Strain Curve
Fe' use ~ to determine allowable
Margin of Safety: Column Yield
MS
MS - Column
MS=@/@-l
Parameter
k2/3(tu/t)1/3 == @)2/3 ~/3
Parameter
C , Figure 10.32, 4
Parameter
CS ' Figure 10.32
Parameter
O@@
=@
MS ==
/0-
@) /
Parameter
@+@
Parameter
@/Q)[)
Upright Allowable (Without Access Hole)
TABLE 10.3
(CONTI'D)
-
0"0 '
©-
1 1
F igu reI 0 • 26
ANALYSIS OF FLAT TENSION FIELD BEAM WITH ACCESS HOLE AND SINGLE UPRIGHTS 10-45
STRUCTURAL DESIGN MANUAL =0 1 @
~
Upright Allowable (With Access Hole)
ero
~
MS Forced Crippling
MS=@/@-l
@
Parameter
wd c
Parameter
C , Figure 10.27 I
Parameter
c2 '
Maximum Web Stress
T~ax = @
t
= .7~/23®1~
Figure 10.28
(1
/(1
+ @~l
+@ @)-I
@
Web Allowable (Without Access Hole)
r* all, Figure 10.29 @ a pOT =45 )
@
Web Allowable (With Access Hole)
If"
@ @ @
MS - Web
MS
@
Distance From N.A. to Extreme Fibe~ of Flange
C
(i)
Distance - N.A. to Near Fiber of Flange
Of
@ @
Flange Applied Stress
O"a
Diagonal Tension Stress Flange (Camp)
(TOT
= -(~G
<@
Secondary Bending Stress - Flange (Comp)
er SB
= - @@ I@
(@
Secondary Bending Stress - Flange (Tension)
lTSB
=<@Q)I@
Flange Stress - Inside Fibers
er
tot
=@+@+<@
U
tot
=@+@+@
=@ 1
®-l C3 ' Figure 10.28
Parameter Secondary Bending Flange
=@@I@
s
in
Flange Stress - Extreme Fibers
Msa =
(1/12)
(0 ~@Gi
@>
•
f
=@I@
+ .5
Allowable Crippling Stress - Flange
F
Allowable Tensile Stress - Flange
F tu or F ty
/
(1 -
~~2 I (i)/ ~ it
cc
TABLE 10.3 (CONT 1.0) - ANALYSIS OF FLAT TENSION FIELD BEAM WITH ACCESS HOLE AND SINGLE UPRIGHTS
10-46
•
STRUCTURAL DESIGN MANUAL
•
Revision A MS - Flange (Tension)
MS=@/(@-l
MS - Flange (Comp)
MS=@/<@-l
Rivet Factor
R = 1 + 0.414
Rivet Load - Web to Flange
RIP = qR =
@) Q]) @
Allowable Rivet Shear Load MS - Flange Rivets
MS=@/@-l
Rivet Load - Upright to Flange
Pu
Allowable Rivet Upright Load
p
MS - Upright Rivets
MS=@/O-l
Inter Rivet Buckling Allowable
FIR' Section 10.6
MS - Inter Rivet Buck-
MS --
au
ling
•
Ultimate Tensile Stress of Web
=@@
§ I @ -
1
F tu ' Section 5
Rivet Tensile Stress Upright to Web Rivet Allowable Tensile Stress
F
MS - Rivet Tension
MS=@/@-l
RT
, section 6
NOTES:
)
•
(1) If any of the margins )f safety are less than zero, the design is inadequa-e. The deficient area must be corrected and this _able repeated. (2) If the web is subjected to tension or compression as well as shear, the nitial buckling stress of the web must be modified aecording to the method describ~d in Section 10.4.1.
TABLE 10.3 (CONT'D)
- ANALYSIS OF FLAT TENSION FIELD BEAM WITH ACCESS HOLE AND SINGLE UPRIGHTS
10-47
If'fl\\\
"~~~,,~ STRUCTURAL DESIGN MANUAL
1.0
/'
.8
/
.6
~
~
/ •
I If
.4
I
.2
V
/ o
V
V
o
/
- 'V
1.0
2.0
.3.0
4.0
~ d/h
1.5
..,,- v
1.4
/
~!-
V
/
1.3
I
C 5
V
1.2
/
1. 1
L.O
o
---
V
/
r/ .2
.6
.4
.B
nih FIGURE 10.32 - ACCESS HOLE REDUCTION FACTORS 10-48
1.0
•
STRUCTURAL DESIGN MANUAL 10.7 COMPRESSIVE CRIPPLING Introduction Compressive crippling or local buckling is defined as an inelastic distortion of the cross-section of a structural element in its own plane (rather than along the longitudinal axis, as in column buckling). The crippling stress, which is the maximum average stress developed by a structural shape, is a function of the cross-sectional area rather than the length. The crippling stress for a given cross-section is calculated by assuming that a uniform stress is acting over the entire section, Pcc = Fcc' A. In reality, however, the stress is not uniform over the entire cross-section. Parts of the section will buckle at a stress below the gross area crippling stress, while the more stable areas, such as intersections and corners, reach a higher stress than the buckled elements. Method of Analysis The allowable crippling stress may be obtained from the procedure outlined below.
L. Divide the section into individual segments as shown in Figures 10.43 and 10.44. Define for each segment a width b and a thickness t. ment will have either zero or one edge free.
Each seg-
2. The allowable crippling stress, Fcc, for each segment is obtained from the compressive crippling curves of Figures 10.43 or 10.44. 3. The allowable crippling stress for the entire section is found by taking a weighted average of the allowable stresses for each segment: hI tl F ccl + b 2 t2F ec2 + ... E bntnF ccn F =-------cc bit l + b 2 t 2 + ... I:bntn
)
The same procedure is used to analyze formed and extruded sections. Care must be taken in segmenting an unbalanced extruded section. When the thicknesses of the segments differ by a factor of 3.0 or more, the excess thickness should be discounted in calculating both the crippling stress of the segment and the effective load carrying area of the section. Also note that the bend radii of formed sections are ignored. For formed sections with lips, Figure 10.45 may be used to determine whether the lip provides sufficient stability to the adjacent segment.
10-69
~
~ ~
o ~ o
<
~
:!~"
FIGURE 10.43 - COMPRESSIVE CRIPPLING FORMED SECTIONS, GENERAL SOLUTION
1 EDGE FREE t
1 EDGE FREE
o ;;::: ':", ::s y ~ ~ '\\
I/'-
; __
tIJ--..
-\>,~.
t
~--"h" ~-
~ bN tN FCCN
t 2
Fcc =
Eb
N
'{"
tN
--
I
~ _~----
A,/ !
(I)
~b~
-I o EDGES
.10 I'
,j::
I ..
1 ..... - -
!
1-'
! .,
! -- 1
:::;~ --l :- :
F
I --.
, ... - _ ,-
.
~ :.
1
'- .;
,,,:.,'---
:- ;_.
.:: :: 'l~ : -:
I- .
..
.
c:
c-)
- -I
I' ~ T .08". ..,:- ~.-. ... >:_:-i::-::;cutoff at Fey.:: -~.; __ -- ~ .• 1:~_. , - --- '1 .-. I
:::a
FREE
..
-I
~I
c: :::a :Dr-
.:.'--"'::'-::_:~". ~.~1 . --I
c::t rn
cc
\/F~y
E
03
1--
.,
---
.... _--.. --.
~ ~
,.
·-1-..·•
..
~~
..........
'"
-
(I)
.,,-
· :~:=:~~:: :~~ :~: J;~~--i2:~I~~2l;~ ;~~~~t; :~c~_ ~~~~ 02
:.
I
_
-1-
.,
. . . ,..- _.-
... 1-
• 01 J 5
Z
~~_ ::~, -~~:_~ ::=" :=--=- _:~: :tf:~":: :: I: _--':'~r"I--'-~:-'-,_.-
.... ----- .'1'- ,-,---.- .. -. -- _._-
•.•
t :)
-
1
-- I
7
-,
.-.-. 1---- - .
f--
---
--.
I
1--"-
"
.
~
--- _ .•. \-, ,
I
I ... - . . j'-' i 10
20
:DZ
r 1"" -- .... K" . __. -----T-- --.. [--" 1--
\
I.
s:
30
40
--.
K.
50 60
=
--1,,-
~
80
I
100
:D-
r-
150
b t
e
e
"-/
e
•
e
(~
FIGURE 10.44 - COMPRESSIVE CRIPPLING OF EXTRUSIONS, GENERAL SOLUTION
y, -
1 EDGE FREE
t2 1 EDGE FREE
L, -' _rZ77ZA J I--- b---l
t
.:: ___ -
"-.
t3
t2
t3
2
2
.10
'J
.Og
I'
j
!
I
-I: -:
,
Iii
Cutoff at ' I
~-rT
I:b
N
tN
: I
'i
.!' ~~~'l I ~ __
Fcc
,~
.. _
.06 .05
-
.04 7:-. :.:- ~~
~~:
'.:'
':
.03
.02
I
r
!
I I L. -,
; J ' "",I
01 •
It
Ii····
.-
5
.II!
.. "'I,'-'j'--, ---
---f·-
7
-
10
_ ow.".
i , t
I 20
•_ .
!
w
••• _
F·
50 b
I-'
o • ......
I-'
t
I:: -
_.
_~
ISklll·,·,-----,---,
1----I-T11t _
_
_
_
_
~
~
.,.
,
__
_
,
1'· ..--,"·-, ~
I I I ... ~ ..~~:-
40
50 60
80
100
•
_~~~~\ .,~
'~=~'
:Eb N tN FCCN
EDGES FREE
I
--1
'~ ,~.- ~.;/ ',in ~,/
t2
o
~ .,: 't\
I
150
• "
t-' -
o I
...... N
~r~I~.; .-
FIGURE 10.45 - DESIGN RANGE FOR LIPS ON FORMED SECTIONS
Lr C
bL
l
1 In
- - - --::,
I
t
b
I
((1)< \~CI ' i_
ABOVE DESIGN RANGE:
BELOW DESIGN RANGE: F WITHIN DESIGN RANGE:
___J~U
I'
LIP LIP LIP AND
'BUCKLES BEFORE FLANGE TOO SMALL TO PROVIDE SIMPLE SUPPORT TO FLANGE PROVIDES SIMPLE SUPPORT TO ADJACENT ELEMENT IS TREATED AS A FLANGE IN CRIPPLING ANALYSIS
en
..... :::a
c:
n
..... c::
::a
"' Vv~
.3 b
b
/
L
F
~
~
V .2
'"
/ MAXIMUM LIP SIZE
V-
"
CJ
.o
~,t,
~
94-
~
en
-
4J-t
'
~ t--
MINIMUM LIP SIZE
•1
>
r-
~
C i) ,
r----. r----
z
~ r--.. .
3:
> z
c:
::.:-
0 0
10
20
30
40
50 b
60
70
80
r-
90
F
t
e
e
-
e
e
\-......'
2.8
2
1.2
3 5 S
1.]
10 15 20
/~ -~/
i
2.6
"""
;-
2
! I
n
3
2.21\
.9
L
I
CI)
5
II
--I
8
2.0
:::a
c:
FO.7 .08
F
.8
1.8
.07
FO.7 .06
.. 05
.......
--~"
.... ....-
-, ..
c;-)
--I
c::
1.6
,
:::a :D-
.6
r-
.5
...,
-"1
~
,04 .4
.1
1-- '.
.2
.3
.4 B
.5
..6
.7
.... ~ ~ ~ .. - '..
~ .;'._,
.01 .1
'-
...
---..
I~. __A ~ ______ ~~ __ ~~~~ I --.2 .6 1.0 1.2 1..4 1.6 1 ..8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 .~~----:-~~~~~~- ..--
L
..8
....... I to-'
......
-
:z
.02 .2
.......
en
G ')
.. 03 .. 3
II
.~
"'~!-~ ~CI-~
n
2.411
~ ~\.
~ -""---!-- ./.:{'
\
I
"'-
1.0
e
4.
FIGuRE 11.8 -
6.
8..
COLut~ BUCKLI~G
10.
B=
12.
LbJ-rn
F O.7'
14.
16.
18..
20.
s:
:D-
:z c: :D-
r-
STRUCTURAL DESIGN MANUAL
11.1.3
Columns With Varying Cross Section
The conventional Euler critical column stress equation: 1T2 E
Fcrit =
(LIp
t
is only valid for a straight column under compression with constant bending rigidity (EI) and a constant area along its length. When the bending rigidity varies along the length of the column, determination of the Euler load becomes more difficult. In this section column buckling coefficient charts and the appropriate formulas for the Euler loads are given for numerous columns of varying cross section. CPS Program SC5001 is a computer ana1ysis of stepped columns.
)
The critical buckling l.oad for variable section columns In the elastic range is given by general equations of the form
P = MEI/L2 cr
(11.J3)
where M it; the column buckling coefficient and is a function of Lhe cotllmn geometry, bending rigidity and end restraint. Values of the column buckling coefficient, M, for various stepped columns shown in Figure 11.10 are given 1n Figures 11.11 through 11.29. For tapered columns with the moment of inertia varying at the ends according to
Ix = 12 (X/b)
n
•
(11.14)
where b, x, Ix and 12 are defined in Figure 11.9, the values of the coeffici~nt, M, to be used in Equation 11.13 are obtained from Figures 11.11 through 11.29 for the ca~es given in Figure 11.10.
p
.....--p......-----~-,...--p _......
---
Figure .11.9 - Column with Varying Cross-Sections
11-18
•
e »o:r r-'
t:::
C 1
LOADING
~
.2
P~P
t::! ........
Uniform Decreasing Load
('j
o Z
'0::1 ~
~ ('j
o
t""
c:
2 ('jj
~ H
1-3
:::t:
~
1--1
>r-'
1-3
C
L
P~P
MOME~T
Occurs at: Cosh(x/j)=(j/L)Sinh( Solve for j and x, Substitute into Equation 11.19
CoshCL:x)=(j/L)Sinh(L/j _wj 2(1- xtL) Solve J for x/j and x, Substitute into Equation 11.19
2
-WJ
Tanh(L/j)
Z
Ul
o Z
Partial Uniformly Distrl'buted$ad
-+f--;1
F+ nljw, ed
..-{ p ~x
~
.
f _
••. ~. 1(1:,·\,...· vdU
a <:
x <: b:
?
~1
.3
~ Sinh(L/2j L
- 2W) x
)
I
L 3 ~ L -2wj 20-x/L) L Tanh(Z'j) - wj 2
b L
F
...... I
co ......
r~
~
I-'
P
L
_ wj 2
2~,'j2Si:-.h(dl2;)Si:l~(C/j~) Tanh\L/ j)
-1
P
,2S' ... · \ . 1 r. I,.·' I \ a! .!. 1 !
- \.;> 1
< x < L·a; 'i
-wj~Co5h(
L-a <:
X
i
T:"
(i:/
't)l
,J
Wj2Cosh(~/j)
-2wj2Sinh(d/j)Sinh(e/f)
o "wj2Cosh(a/j)
)Tanh(I!2j)
< L:
wj2Sinh(J/~j)Cosh(L!~)
0
~-----------+-----1f-------------I
Ccs:-:tL/:':j)
Cosh(L/2j)
c::
c-)
......
-2wjLSinh(d!2j)Sinh(L/2j),
o
CJ ..... en
s:: ):II z
c:
0 .2 -wJ
r-
Z
b
a
-I
::a
a) ISee Note 6
,ZGCOSh(a/ j ) wJ Co S h ( L / 2 j )
-0J
.~
'~ ~ =:::::;
-
o
,1
_ '"
Symmetrical Partial Un' fonn Distributed Load I x < a:
~~n1I/~ a~ t t t t t t w, # Ii r _ r--
o
tr/')
If/in
f--.~ P
,
):II
.2
o
~~ :~t~( ;)~!;:'
-\\'1
~,',,'-
:=cI
x < a~
-
-, '"
c::
x < L/2: 2wj 3 LCosh(L/2j) x> L/2: -2wj 3Cosh(L/j) LCosh(L/2j)
trl
1-1
,::-:~,
~;;,_.-, ..
f /--
CI)
Occurs at:
~P Loa~fi
-WJ x
Sinh(L/j)
.2
Symmetrical Triangular
.2
o
WJ
MAXIMUM
f(w)
2
Uniform Increasing Load
1-3
.c-...........
e
e
\"-",,
):II
~
:
,
~....;
cor
NI:">1 ;.-
..c-..-... ()
o z
~
C1
LOADING
.....I > tp
x
Two Syrnmetrical Concentrated Loads w W
+-
r- a
~x~
b
o
C 2
-~jCosh(b/2j)
o
Cosh(L/2j)
-+a1
L-\P
a
-t-ijSinh(a/j) -WjSinh(L/j)Cosh(b/2j) Cosh(L/2j)
MAXIMUM MOMENT
f(w)
~~
o .Sinh(a/j) "WJCosh(L/2j)
o
/'
tp
r
rr:I
»::s:: (')
.-
~ ~!!.::::::/
o
en
o
t""'
~
C/)
~
f-I ~
:::c
S< f-I » t""' H
P--px~
Le~
See Note 6
x>a: -MCosh( al
j) Tanh(L/j)
MCosh(a/j)
L
?;
o
-I
c: ::a :r-
-wLj
wLj 2Tanh[L!2j)
2
, .2
• 2 [L::.!./...;;,2otL.i----.,~
-wJ
wJ
ITanh( Lf 2j )
r-
-~
CJ
P
Fixed End neam.;concen ... trated Load at Center
P~
c-,
At x -= 0
ffHt4}
P
::a c:
SinhILfj)
;tb;
4JJ g;in§j
~
~
-I
o
Fixed End Beam ... Unifonn Load
Z
C/) H
o
,r i+p
,."
en
x< L/2: -Wj
!:ifl -
x> L/2:
.!2 [2Cosh( L/2j) 2
-z ,..z CD
o
2
COSh(L/2Pj
2 lSi nh ( L12 j )
o
it:
-1J
Cantilever - Concentrated End Load
c:
JW : ~L=l-P e
):II
r-
WjTanh(L/j)
'----
e
:::::l
~~.- ;::~i:
x
a
'~
'---......~
r -
'-'"
Concentrated Moment
;,,-
...........'
e
STRUCTURAL DESIGN MANUAL Revision C 11.3
~
Torsional Instability of Columns
The previous sections have assumed that the column was torsionally stable; i.e., the column would either fail by bending in a plane of symmetry of the cross section, by crippling or by a combination of crippling and bending. There arc cases when a column will fail either by twisting or by a combination of bending and twisting. These torsional buckling failures occur when the torsional rigidity of the section is very low. Thin walled open sections, for instance, can buckle by twisting at loads well below the Euler load. Often in thin open sections the centroid and shear center do not coincide, therefore, torsion and flexure interact. In this section, it will be assumed that the plane cross sections of the column warp, but their geometric shape does not change during buckling; that is, the theories consider primary failure of columns and not secondary failures, characterized by distortion of the cross sections. There is coupling of primary and secondary failures but no method has been developed to handle them so secondary failures will be ignored. 11.3.1
•
Centrally Loaded Columns
Centrally loaded columns can buckle in one of three possible modes: (1) they can bend in the plane of one of the principal axes; (2) they can twist about the shear center axis; or (3) they can bend and twist simultaneously. Bending in the plane of one of the principal axes has been discussed previously. The latter two modes will be discussed here. 11.3.1.1
Two Axes of Symmetry
When the cross section has two axes of symmetry or is point symmetric, the shear center and centroid coincide. In this case, the purely torsional buckling load about the shear center is given by
P
~
= 1 / r2 {GJ + E r 7r 2 / 1 2 } 0
(11.27)
-
where:
ro
polar radius of gyration of the section about its shear center
G
shear modulus of elasticity
J
torsion constant (Section 8.0)
E = modulus of elasticity
•
r
warping
1
effective length of the member
constant of the section
(Fiqure 11.85)
For a cross section with two axes of symmetry there are three critical values of the axial load. They are the flexural buckling loads about the princtpal
11-101
STRUCTURAL DESIG.N MANUAL axes, P
and P
and the purely torsional buckling load,
~.
will bexminimu~ and ~ill determine the mode of failure.
One of these loads
In this case there is no
interaction and the column fails either in pure bending or in pure twisting. Shapes in this category include I-sections~ Z-sections and cruciforms. 11.3.1.2
General Cross Section
In general a thin walled open section buckling occurs by a combination of torsion and bending. Purely flexural or purely torsional failure cannot occur. Consider a general section with the x and y axes the principal centroidal axes of the cross section and x and y the coordinates of the shear center. The cross section will undergo tr~nslatign and rotation during buckl • The translation is defined by the deflections of the shear center u and v in the x and y directions. During translation of the cross section, point 0 moves to 0' and point C to C' where 0 is the shear center and C is the centroid. The cross s~ction rotates an angle ~ about the shear center. Equilibrium of a longitudinal element yields three simultaneous equations, the solution of which results in the following cubic equation for calculating the critical value of buckling load.
2(p
r o
cr
_p) (p
y
cr
_P) (P
x
cr
_p) _ p
¢
cr
2 Y 2(p 0
cr
_p) _ p
x
cr
2.x 2(p 0
cr
_p)
y
o
(11.28)
'."here P
p
(11.29)
x
.)
•
(11.30)
y
(11.31)
Solution of the cubic equation, 11.28, gives three values of the critical load, P , of which the smallest will be used. The lowest value of P will always b~rless than P , P , or P~. By use of the effective length, L,c~arious end conditions canxbe tonsidered. 11.3.1.3
)
Cross Sections With One Axis of Symmetry
A number of singly symmetric sections are shown in Figure 11.76. If the x-axis is considered to be the axis of symmetry, the y = 0 and the equation for a o general section reduces to
(p
cr
11-102
_ p )
y
{r
2(p 0
cr
_ P )(P
x
cr
_ P ) _ p
•
cr
2x 2 } 0
o
(11.32)
•
;/;1\' \'\~
.r.~~~.. STRUCTURAL DESIGN MANUAL 0....
0.3
0.8
--+----,1 0.6
t------"IIrl---f-----f----i--
u ~
o .. 0.2
J----+--...lIIop~-....f.._-___.~
-+-.iL--I
:.-
0.4 ..
!)
'0
o
>
N
U .....o
>
O. \ t----+--+----+-~I__-_l_--h;oIE~_l_-___I 0.2
to)
(b)
o~-~----~--~----~~~----~--~--~ 0.2 0.4 0.6 0.8,.0 0.2 0.40.6
o
Ratio
.2 Q
FIGURE 11.82 - TORSIONAL BUCKLING COEFFICIENTS C AND C FOR ANGLES 1 2
1.0
1.0.....--....----,...--..,.-----.......- - -
.---...,..--...,.---.,....--.,..-----1..__-__ "
~C<
u o., "i r :: 0.4
0.1. 0 (d)
0.'
0 ...
til.
~O.4
D.3
....
I)
,2-
II
S·l
~
0.+
0.1
0.6 D
•
b
1.0
1.2
I .• '
:
tll
... r 0
~
",dho d
4)
0.1
FIGURE 11.83 - TORSIONAL BUCKLING COEFFICIENTS C1 AND C FOR HAT SECTIONS. 2
0.4-
0.6
0.8 b Rdtl ..
a
1.0
1.1-
'
.....
FIGURE 11.84 - TORSIONAL BUCKLING COEFFICIENTS C AND 1 C FOR CHANNEl. 2 SECTIONS.
11-111
STRUCTURAL DESI,GN MANUAL b
S ::
shenr center
o :: control d
r
= woroina conston t
o
t(h 2bl r: 24
n/2
tf ~
e=
h (b ,)2(3b-2b\)
,
{f [2b 3- (b.bl)3 ]
b3
tf
e
t
= h ---=..Ll J
" e-::,
b +b 1 2
h tf
'tfh 2
b~ b~
12
b3 + b3 1 2
r=-
2
a
c.o..s
C(
SIn "C'o.s
a:
Sln.- ft. ft.-
210 S" ~ 6 (•• n r.;::-
.
3
•
tf
.
e=-
3b 2tf
6btf + htw
3 (b 3 + b~ I 2 3b
r: -1
h 0
tw
r=
tf
tfb 3h2 3btf .. 2htw -12- 6btf + hrw t
I·
2
-I h
-
hl2 ~.
r=
,...tz d
c
b
t}
3 2 __ b _h_-2[ Ztf
+ h2)
~ 3t W
bh]
• I_
t 3 b-t-t3d3
D
T
b
"I
r=::l
144
FIGURE 11.85 - SHEAR CENTER LOCATIONS AND WARPING CONSTANTS
11-112
c)~
" - Sin II( cos I(
b
e
I
J
.,
hl
I-
\
"I
b
t-
0
..L 36
-.' -0" '.. ;',
Ii
oJ
.l
(
\
,
'\
'\
STRUCTURAL DESIGN MANUAL
\ \
\\~ '.~'"
-.
~'\/'~'. -.~1L
q:
b
j
I
-+
~w
*f
J -,-.
t
0
h
~ l
1 (b2 _ br)
j b,
Qa
< b
tw/t)h + 6 (b + bl)
-I--
~t
uI
-,----
~
....
~t
n
~ 0
0
..---
Values of e/h
.
f--
0.1 0.2 0.3
I
~~
1
I'"
h
.....--
.33C
0.6 .236
0.4 .141
0.2 .055
.183 .222 .258
.087 .115 .138
• 39~
.280 .290
.155 .161
.477 .530 .575
.47C
.280 .321:\ .365
.610 .621
.50 .517
.40~
.28C .421:
b
I"}
-,---
0
0.8
L
e
•
0.4 0.5
1.0 .430
l-
r.-t
Values of c/h
~
0.5
.430 .464 .474 .453 .410 .355
0.6
.300
0
f--- - _ . - -
~
0.1 0.2 0.3 I-" 0.4
_F--
I
e
1.0
0.8 .. 330 .367 .377 .358
.. 320 .275 .225
0.6
0.4
0.2
.236
.141
.055
.270
.1 7 j
.080
.280 .265 .235 .196
.182 .172 • J 50 .123
.155
.095
.090 .085 .072
.056 .040
)0
Figure 11.85 (Cont'd) - Shear Center Locations and Warping Constants .
• 11-113
• p
)
•
L y
z
~x e
y
---- . .---f...-.-.------
X
p
FIGURE 11.86 - ECCENTRICALLY APPLIED LOAD
• 11-114
FACSIMILE MESSAGE Reference. ~1.41.~' Aotion: ~
Sender ~ Leth and Assooiates
Phone: (206) 622-4546 Fax No: (206)392-4482 Name:
--
Infol
I Date
~,: 'LI;.IJ.I
Operator Instructions:
I page
&'/OU7iJ,
I
1 of' - "
Please forward XMMEDIATBLY.
To:
,
\e
P.i
OCT 05 193 17:49 LETH & ASSOC (336)392-4482
[{!-I7e
Messaget
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"To
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Hn5
•
•
OCT 05 '93 17: 50 LETH 8. ASSCC (206) 392-4482
•. .
"
'
.
.
'
Analysis. of Stiffened Cu:r;ved Panels Under .Shear and Comp~ession.
1
M. A. MELCON· AND A. F. ENSRunt L~cklteuJ d
lrcr4li CorpDrdllon
I"
'~ t.t MUlti. . IIIed 011 a study of the available !lteratu~ giving' );; tlteentical apd explril1umtal data perWnlq to thI.lub-. I· , ject... method has been deveioped tor the prediCitJon of . tM ultimate 'atreDlth of Iheet anet loagitutUnal stiffeners \I a earved panel. "-
totAt AfCU. oIlld..
-
i• . ·
S' . .. lbell parameter
t
t.
it; .. •
J
\
,'.',
:-:
-' Utlc:lcAea of Il;.gen~ ~ ~ Il4ttor ret1ectlnl thf! w.r1ouI d'WI of 'tilt dlaaOO&1 . tension a.ld on th,- stiff,tt&' . '.
factor lt1dJeatin, Ih. inttIQlltJf of d~tW ·terillon In ' web _ • IKQiOI' redu:mnl the tritltlal shear atreIiI £Om..
. '" l
'.
).
.,
- n.dJul ot QurvfthU't a, panel dtl~U.CM ~r:"Qb:, . .
R
_tilt
blncd loadl .. ,
.... - . . ,
- ItUhner .,a.clur .. . , Cp .. rlwt factor, wig 'of ,JI..c tG ltON·ara of web 1l '. - allowable com~ "ft:U,1or ttllener alone" ·F~I la , the t6W. of eicpur til. COll.U1\ll' .Uowlbh~,(uae ... "sit,. of a lor .dlfmcrl condauOUI acmea Mia) or tke cripplml QutoJl' of the ltilenet. . I ,~, • alIGw&ble COR\.PNallon stn:aa for l'dSea.. piu. efteatl'ft , t'
: *111
~ ~. tile Stn.&Otutca Seilfoa. Tw.dJCo'lb. MeetJar. 1.1.1.) J_uar,v't8-P.btuary 1. 19G~, N.w York. RWiIl4ld aud
........ ~'.,.loa.
,
'. • GtouP ........... StruOi.qr~ M~tbods and DPliHlt&. . t Itt&IfIItMIU Bnslnm'. . .. ..
•
I
I
...
•
'!
,
of,
j'
"
•
to
,da... At8aI. pkll . .t.ur. . ara l)iul
. A. • atila. area.· 4
.
- AI10wabie CfOIIIII a.t~.. abtar Itretl ~ web failua. /1ft - bAsic allowQb\c lfO'IIo a.fllL tbtar Iib'l!::d (for bo.ttOPIUi:~ dlaCOftll ten lion field at 4,5-) P,' - ~rilkul buddin, tbeci.......... for aheJl . F," - criUr:,d buetlin,lheat Ittei.l fer &c puel F... .. 1'.' + '11M . . critical budtlm,lbeu It... fotCW'VN pAneS 11ft • Wtim_'te sheAr alrefS Dr the ma.,.riaJ ill IIIpplIl!d IfDII:i .&'eO .hear' atreea . hDf' - .,~ ot Ippttcd .bClU' atnii anitd 111 4htpJrual 'UlPalott File - ultimate lea.u,e rtr:1I& of ..b. tnat.rh&1 In - moml!nt of mert .. rsf iltifrtnf:1" about centJ't.dd&l adl pAru.llcl to till! tanretlf of tbe 111£111 ecmt4»r J" - tN'lioua' stiffrtw f~or: fat open teetiotJI, J ~ ..4.,;.,'/3, lor elated aeotlou.l J - 4At'"/~, where A j, ..ht euclOied flftll ,\ltd P is tlte perftntter
.tm
as,
.
1'.
tNra OP aRBAT ADVANCiMBNTS in the design of air· , eraft IItntCtureal the analysis of ltiffaned IhcU. , ~tI .. c:haJl!Dginr probJem; The difficulty arises I MiDty 'from, the {act thAt no practical m"thematic:a1
t_
.
. break ".., ,han hUG PllLtlell. . ' apJ)1I1\1d COlllprcutOil lAra. buaed OQ deeUvl'l wldtb of skin - awlied l,!Outpt~liol1 ItnJl baled on
"
! NTII.ODUCTtON
t..-tlMnt of the Btresl pattern in the poetbuckUn, .tate hal tlt(fl1' develoJ)ed ye~. ,Therefol;e,' fQf; tha, beirii. the detiper haa to be. satisfied. with iCtJlli.· .-piricat tolutioDI that show lufflclf!nt agreemeat with
lenllon.
N' '. a men,,.rc of IItUTell9!' "141111 .dd'uul Aquikd
,'., ..... _. ttd eompudDC teclullflUQ. A an,mplc Int.."I'I.ctJgn dillf&f" " ~d Ihowin. ht,w the rt~1t. of tbll mctbod ....)' be pre. . . .('far fIt't.t1~~ appUc..dotJ.
,
- . tlreatlve QOm~~J1' atftM In .dft'ener d.ue to dIqoo.1
Jl
. SVMlLi\kV
.' Tw. PaPS' prllUlUa. ftlOth.od of cmalyti.lor wrvfid PIUtel. mo- . IM'otd by laqltudiaal Kift'~Qna .ubJl!l!tt!d t~ ebMr at1.d tamnru" . . . . . ~IW 'ermulal have bClClu derived lor the eritiOid ~1iIl' :··f ..... of cqrvtd.:paetsln .bcar and tOl'the varlOWl .ktlol thG . '~ lillUlIi~1 IMId. 2U the J1Ojtbuekltn. etate of the .~. .:. f t . kttl!l' eflt.I!tI are: CI'pI'Clleci by :6ctltiolu compnBlvlltre8let. ..... llll Itlffaal!' wbiob IU'9 t.hen «:ombln~ by an llltuaetiou cqua· ,'," with ike cmti~ alrella dlle to tbe exten:Iat toadlnr. ' ... acNltion. a formula .af,Yefi' for the ultJ;mate.lIeu Itren.rth or ':. . . . . . . Tbe rn**hod ptOpoaed In this paper hal beeri. com... ,.... with a.YD!lable QXp.ime.r,uAl ·data. aud. iAtlafutory &Jl'fIC" : . . . k.fGua.d. The metbocl ru....bt:m, put bt a form tbt.t requirel :, dWJ -tudoa 01 certain mu.them.&tleol ral&tioU aod & mllIUn\Ull 01 : .:, ·......t _4iAg ..~d, heaee, iI readil, adaptable to IBM or other
, "IeU,,1 I• ,T't'PICAL $HILL ,STRUCTURf
•
•
OCT es '93 17:52 LETH & ASSOC (206)392-4482
,
.
. T
112
J 0 URN A 1. O.P ."I' H E A B A ON AUT '.
teA-I..
I
M£TlfoD Ol":AHALYS11lt
. '.
8 C] ENe H S --:'F B B oR U A R V.
. .
loft".
.
•
"
•• ;
.,'
!~!
.~? I.
.·... 1 '.
•
"J.
this C!4Ie K 1:: a.9& + (1'/')(~/a)iS"(. • . .S - (l,t/RJ. deaotel t.be thell patAinetu. Thll fomnl~::. for the critical bu~kling atrels of a ml'V~ panel 'may ~:.
..
wh~.rtn
J.9lH'J:
,~
ft •
I.'
A aemlfilollocaqUf! theD relnloreed. by rings Bnd tudinaJ stiffeners (Fi" 1) may· f.ul in tbrce ways; (1) . trlllttlorrned to the form failure of ItHfenerl, ($) failure at sheet. and (3) failure of rings. If several ~mponents fail Ilmult&t1eaully. the F#tp. - P/ F/ type of I~ure ia WluaUy reter:rett 1:0 as general mlta- where bllity. ·It is assumed that th4l atiB(!hments bef:'rfeen a1dn and supporting.mem~ra willl16)t,i1ear or·pop and 11.' - C../4-t/,s)(B'/R) thus rc
+
.
:.
( . 2) .. /(
.,
.. '. "
structure. The followillg delibttliLtlon.a are Limited to an investi-. Pcntbudrll.. , IHlUltJ10r 0/_ ShMr PM., lation of the Jint two type, of iailure. The results of. With the introduc&n pi thin sheet constntdiot'l. ter::~~ this investigation are then cheeked against the te.t data alrcra.ft Itruc:ture. e.nginCCfI bogan to accept ~ new.; of I~irnefls that failed by buckUng of stiffeners Cf idea that the buckling ot structw-w componen(s.d1d 1101::. tearing of tlu:: skin. Pomtulu derlvM in this paper necessnrily .indicate faUur~. Sinc:e that time.: a grttt .. apply specifieaUy only to thln-walled shell•. of circular dc:al haa been wri Uan ott tile poat.buckling behav.ior Qf. \ ' crOll se~tiol1 With COfl.8tant values for sheai' 60\", atitT- 'structureK. Of foremost importance in. the Study' of:, ener stiffener spacing. and lheet tbiekne••• Adap- tid. subject, It the theory of the incomplete diasqnaJ.: tationlS for variation in tlte~ items are easily ma.de. tension Jield. . ,.: The proposed tncthoc:l tna.y be outttl1ed ns follows! ~rhf. stress pattern in a ~at sheet 9pbjected to shw,~ : 'rhe eriticDJ bUc::klinlltrtlll of a curvedpaDel is given forces beyond ita buCklit;g sti'cnrth ia know as di.·.~ as the sum of the atrengUt of the ahen and tJwt ot .th~ agonal tension. A rigorous. ~onnu1ation o~ th~ ~. :! .. .panel. If the applied shear SlUe,. it below the crit1~3.1 tion from the unbuckled sute to Wagner.' IdeAl dl~ '; shen~buckJing .stress, no 10ngitudinal Qtiilellerl !U'e ;e- '. ago~al. ten'ion field hat not 'yet been acco~plish~< quite-d. On the other hand. it Is obvioul t,hat. the ltift'- . Setlliempiric:al formula. developed by t;:uhns art!! tt)G~t'::l en.:". mUlt have a certain amount of bendlllK stiffness widely "sect. < in order to lubdivide the! melt Into panels. ~ loon as Whert the .tRIJ in it plane web ata.rtsto exceed its ': the nppUed melt ex.eeeds the criti.cat b.uc~inlltrell for initial b1u:lding strength. the appUed shear foreel .~" a c::urvec1 pal1el, a rather ~mplex intemctton' between gradu.tty taken by a combined truu aetlon of the web ..:~ sheet and supporting ml!mbers is started. The math~ atld It£ffeners.· The sheet acts more a~d more likl! . matica,l trea.tment be~tilel extremely involvtd, if not . we.gonal_ while the atiffelJe.rs talc:e..~ place .of uprlghti ,;. impoasib!c. There is a teridellcy' tOt .. bue1cle to form from oorber to :"; At the start. the shell between the rings boWJ out~lud corner of the panel, proviclec.t tms pattern is eompattble .; , like a barrel. With incnalling shear fiow I the radial with the deformation of' the stlft"enen supporting the .:::. cor:npone~tl5 of the. diago~~1 tension ~eld will ~ul1 ~e edsci of the ·panel. . Itritlgett Inward. _ In aclcl,tlon, 8 nOldtntar relationship . The various methclds propolled for the an",1ytis of flat between the appllc,d torque and the co~np;resslve Itr~ss panels in the po.tbucJdjJ1C ltate di1fer in the usurnption .... in, the Iti.ft'ener!!l c:an .be Obtet~~1 FaJlure ultta.11y of magnitude ot compression stull the wet ia:ab1e tc .:. occurs from an individual buckle In the &hell f~t(!lng the su.tain in its buek1ed Ibap~. Additioual complieatiorus :: ~tlffene!' out of its .or'~nall?cation. thul btl~,IlJ'Ig about EIrlsc when the pa.nel is curved. The diagonal tension In' ': ' IU.collapse by an mtrlC:~te beam-column RC'tlOtl. . the Meat tends to reduCe Its curvature in the dIrection " , . of tM wrlttkles. Thit action induct8 nonbnitofln radial . 'Iouis on the longitudinalltilfeDers. A simplified e:rprCSlion has bun derived for the crit.i-
area,
cat budding Stl'etls of u curved pnnel. For 11 fiat panel tile eriticlll bncldb\g stress may upr!IRd by
pt
'F
j '
..
KE(t/b)1
O}
wbere the marnitude of the coeBlcle.ttt Ie depeuds on tbe material, tho degree of edge testra!nt. lUld the IUIpcet ratio of the panel. For aluminum alloy. and ttandarcl construction. this vdue Is taken equal to 0.25 irre5pCc:tift ot Illpel'!t ratio. The variation in panel length is taken care of in the term tba.t reflects tho efi'eet of ~urva* ture. The c:riti~nl hucldlng stress of the cDtvr!d p~nel may be expteMed by P,••. - KJ;Ut/6)1
1."he tonlitudiaal ItiftenerS t11 a reinforced flhell per-. fonn three functions: .(l) They subdivide the IheU; hlto panels: (2) they IUltaln axial and radial loads, in~ dueed by the teos10il field; and (3) they austaln directly applied axts.l loads. AI long as the shc11 Itiifened by nfl,P 0IIly il able to IUltain. the applied Ihear .flow without buddinS'. 110 Itift"enm are required for the purpose of reduclng the panel Bi~. With Increasing shear flow, it UIWIU,. become.s mote de!lirable to raise the initial' buclding stre.. by the addition ollon.g;tudina.l stiifenen than by thickening of th~ shell. The raqulted i?eltdiul' . sti:ffnea of tl,e
stUr~ner!l
whiclJ will, naise the initial
:
"
• ..
•
wbere F/ - 1ft EJ,,/Q.IA., fd.eCu tbe coJumtl Itrengtb ·of the atUiener. Combining and solving these equatiOns
gives
.
• J
J/ -
FIGURe:
(3)
Tltf. equation lndicatci that 11e3vy akin add wide riDI Ipadtll Rquirc ItroDI Itifftneta "".order to avoid excesaive values or 1. •. Howev~, thi. portion of Ule total eff'ecth·'c comprcaalOlt !ttelS in the ttHtener in mOlt Itrudut8a is usual1y Irtuilt Tb~ ~ft'ett!l of the diagoual ~~Blon field on the ltiE. enert in ... buckled tbeU Are mther complex. The axial load bw1d-up iu the lonsitudillu.l stifi'enen cu.used by tbe dlngonul temdon. (Fig. ~) 41 aJven by
...
•
F/(M/A,,) ~O.053Z./G .
t.
.P -
if, -: Jt,,~..)bt cut
«
. . . . . .4
tu
In tLdditiOl1, there arc radial components of diagonal . tension .field which produce bending *uomel~ts in the stiffenerl. The pun ~r runnina inch (Fig_ 3) exerted by the tension field along this chord line is
T
:=t=
(f. -
F,c..)~ tim a
Bud tbc md"iulload per ruulling inch i&
Pi --, T~/R FlGURI'S,
Ut -
F;erJbl(ta~;a/~)
The unknown angle a may be found by trial and error.
However, thi. standard approach to the a.Daly•• 01 lonsitudinal edtr"nen does not agree tnt. for mAny reasons, nnd. hence. a differCJlt approach to the
..nth
valu~ nf shell IlJu5 Ilanel nlo.y
ucltJiJ1r ttre•• to the fuil
detennined IlCcordiJlg to Seydel'sl' tomtulll, fOT tbe trfUca1 shear ttre.. of flat orthotropi~ plates.
,!
F••. - (~:2/1a~),~AAa
: D,·. ·'t
.D.
I,..
,D
~
=
,problt!1U ",... ehcaeu. .' llig. 4 pic lures " stiffened sheJl lubjected to torsion. Al8ulnlng a diagonal ~nsion field of 45 ~ an.d no bending in the stiffeners. the induced cm'npreas1on streS$ in t.he stilft'nera Is given by . ." i
// - (f, - F...:)(~/Ad) ,
D '
Uti/b. 1~1/12(1 -
IJ.I)
.~ IIttiar thilS·expreaion equal to F/ and eliminatit1g 011 .:, lied!. tktea. the~ follows'the expression for. the portion 01 .;'." moment 01 'inertia J 4.1"t or the sti.:ffener'required to : .iivWe the .hell into pa.nels,
41.1 - (bt l /5)(a/6)'I•
.1.
mentioned 'previously. ·thil requireinent will .Pe ......,.,m.d into & fictitious «lllrpn..asion stress indi, the portion 01 the 'stJ'ener luenrth neMed for lltetive tubcUvi.ion i~ to pmell. It ,il aSlUll1ed tQw.t lie ratio of the stiffener mOinent of iDe-rtiQ required for PlU'poae to Its tot81 tnom~nt at inertia may be taken to the ratio ot an additional &.rea to the total area ~ stiffener. The fonowiol' re1.ati.o~ for the fic. aonlJ*etsiOlt 8trelft If II ia eata.blilhe~:
f."A jl
-
F/AA ..
ua \,
!
'<'J
•
J
114
u: RNA LOP
T H,E A 8 RON AUT t, CAL SCI B NCR 8 -
fl"rfB It U A.R V;
1,98 tl
,!
,'The 'act.~al anl~ of the '~ulion,ftekt I. uluaUy'attta.Ucr" panels ullder wmblned 'loadlngl. HoweVe~. lor the "; tJwi 41J·,.ttl &rldltiGt•• th....... pun l'.t,,1I1 the tlt'.1!00lion of _the w.tInk1es whk:b f~ the Itt~ toward. tbe,J:en~ of ~cyllnde~. ,I.t,caa be lien front Pig... that the pUll P.hua a greater e«~ Qil stringers of smalteJ"
f1.ex1blJlty, be4:auae the.
"t a tater .tate,
een fiUppcrt in un. ,caR atarts
Furthennare, it is obvioU6 that this , effect is increased with sharper ,curvature. Theae , In additJon to other ,corlIide.ra.tionl. led ·to the detetmillation of the empirical factor •• which was introduced into Sq. (4) to ,give a fletitioul c:ompref$sion Itrca ~hl\t is a meuure of required stlffener strength to sustain the eWeds of diagonal tension. Bq. (4) tben'translornll
into
purl.OM 8t ha.nd, theae t!qUatlotll are Dot se.tl1llsl!tory, ' ; lint';e they wiU ~at pntper1y account tor t1ae amOuDt. ot the alleRT belli. c:amed ill difl.F.tW tetl8ioR nor witl the , angle! 0( dfajoaal tension be the .me al when the Iheet : i:' buc:k1ed uDder .bear alODO. In o,tiLer words, it would be ,',: ~~ ' coJieerw.tive to usume' that. after buekHng ocCUrs! In :" 'r eombined Ih~a:r and eomptesslon. any additionallfhell, ' j~~ is riarried i'n the 'same mRn~ as, tf tbe J;tuckling were due ," to shear alone. If ti"~.r interaction were ruJlutnl!d for . .~: ~niti&1 buckling in'compreieion and shear, the rotlowing " :,;. equation Jnay be written: : ;~ ' (IIF. ) (I IF ) _ 1 J,
; f/ ..
F'd/r.)(bl/A,~ ,
.".(ft -
(6)
"here II -
1
+
: ,
;.
..
I:'
+ J"
ft'_
.,;
1
comes. then, '
,.1 + 1.~ -
'II.
Then the thear streM at which the panel buck'~s becomes
(G/R)~(I•.IJtl)('lb)·
Thl, IQrintda give. latisfactory mults for ibt!l1s with R'~ 100 In. The total effective'or tlctitious Mtnpn!Sa Mon ,trese for whtcl?, the stiffel1er must be designed be, '."
r
.
J, -1
+
I
(f,.IJ,)(F••,.1 F ..J P••,
': : ,)
Based on this. retlflOnlni, n fl1ctor A was' a.rbitrarllyr .J ; .. ulected to refteet the reduction in tht! critical -a.Iteaf ",:. ,::. buckling stress due to oompmsi~n. :. ':~
,
to~ ~dfect,iv~ ',~;p~sslon ~tress in, ~tifl'.
(6), ;
~(
;'~
f;
,
The atlowable atre!" tn be used -fOr deterlDining the strength of the .ueener isF/, which is the lower of the co1umn aUowable Culling a fixity 'Of 2 for stlttener8 that
are continuou8 acrOll rings) or the cnpplib, cutoff for ,the stiffener aione. Allowablell deterntined from testa on the stift'ener by Jtaelf' may often· be unsuitable fot thil analysil. linee the !ltifl'.eoe.r may loll in a. tnode not pOisi· ble for a .tift'ener that Is attaehed to the abell. Table A .ummari2elii the declive compreaion streiB to be used at different v,dues of l* when the panel is subjected to
shear a1one. f,'
TA.r..A IlHd , / Whe2l Panel is BtiDJececd ta 8ht!ltr 01111'
I,ll
. N 1/ Jt
~
0, whllH /. < F,.r,
flU, '-
'1/
-
' '.t 1M Ad"
when ,. it 11*....
/.' - O. when I,
"'-II-, ~ IOJ)S3b
t._
IP I
w~n r~
...
...
X
i. < II'H.
!i!!! ~ /0.063;
/I /. t
< I,'
(I' ;./.')(~)
It' -
-
A" "'-a~' when J.
w1iere ! if is the appUed ca~pns!lion Iml! baSet;t on ~: stiffener &.rea plus total of akin. ': The predictiOn 'of stiff:ener lailure due to the com- ' bined action of wa.r and dIrect comprelsion I.' bast!d· . " upon the follo,..111, inter~e~OD equation: ' .
area
.f t,'·"(" + (~t'" wbere
-'J ,.
(1) ..
ie Is the dlrett compl'Hllon atrelS hued on
.tIl:Yen~ &rCa plu. effective width ols1cin a.t1d Ft is tbe allowable cotnprasion .Uell based on the _me 8f'ea. In other word&, tbe ratlo I J F. II 'the ratio of the a.pplied (!O~preuloii load'to the a,l1owab1e compression loa.d of the atltretter plus sJdn. Table B summa.rlul the efFr:e~ tive c:ompreaslon atress to be u~d at different vaJue. of, f, wllen the panel illUbJteted to shear and compression. For ~ - J, Table n ts ldel'.ltlcal 9o'ith Table A. '
1/
Jl'
;at Filet,
the
~ut1ined
,
for determIning the .
e.ft'ects of Mf.ltU' on the stUfetJet fs that Ute critical s11ta.r Little. teet ,1ata could be rrmnd for shells relblorced by bucklin, Itre~I, bl this case, must be reduced because , atift'eners with clo"ed seCtiOll. Whether the factor ., of the effects of compreslioJ3. There are. wcl1-1mOWII etnt applfes iu this cue could therefore not be sub.; lnt;ersctlon equatioD. that give the initial bu&litlg of : .tautiated .by comparison,. with· test data. Th~ only.
" '; ,
.
;~ :i,,.
.:: ; I,;' 'f
,
. ':
,~----------------~--------~--~----" TABLII D Ii .lldla' \\"hd Part,1 It Subjeetcd to Shear and Cotn~..:.
If, itl acldition t.o sheW'. panel is also subjected tf' direct compnsslon. the e1feetB of sbear 0!1 the stifrener Gfe eomb1l1ed. by flO interi1eUol1 equation with tho cd· feetl of tlle direct compreMion. The only exeeption to
the method as previoualy
:
~
•
I
I,
A N'A.t;
'
v S l SOP
B '.r
r v VR
N B D ,C U R V HI)
i' AN R.L ~
11G
jEtlr I· .p, e r---t--t--t--t--,+-j'.;-:.,f-+ 11-+++-11-'--+--+--t--+---+---l--+-~ • !b-'tqr .' I~, .1\. I '
F. ._It
~ ; II ",I
t:' c. ~.. ~p~) ,,1.,I,J.,0,
Ift.'tM••
l , un 3. '1 It IO~----r---+-~--~~~·--~~~~·~_~I----~I--+--+~+---~~~4-~~~
e.Ln.
'r-~~~~f~I~I~I~I~(~I~I~~~~.~~ I· I 1 I I I I I
FGr aiMr ",ot'llab. modi', b)' ratio of Fa •
o~~~I~.I~I~jl~lt~I~IU~~~ I e , .
4
~
G ,. e I 10
'. 1t"t
·10
30·
~O.9
40
70 80 t~ 100
FIGURE S - UL.TIMATE ;\I.I.CWA81.E. Gftose AREA SHiAft STRESS.
<
,:. two I~h.tu~nl thu.t hl1d ,closed stitFenert '",iled by ,tear- . if alb 1~ use 1:1 if alb -> a. uje 3.: ultimate ClJ!PW" iat of the web. ': ttble graN area web sben iI, thClrifore. '
The
.'
,
F t ~ '/~.,.
',:....,.flJ8hMr Wllb .
,+ [(f~ -
"
F.,,')/,]
(0)
Tile formula tor the anQwA~le gross lU'eo. sbear stress where F~. is obtained from Fig; 5• ......ate
rS
IJ
P"~' - 1 ~he basic allowablo : ' " of itt 11'011 area. In tbe unbu~kled condition af modified 80 thu.t for 'a. webt the ntaxitdlllQ tension sttell itL the aheet 'is, ahen.r stress becomoa FH - O.75F'I/;' Thtfi c:orrectJon Is . ·"~,....l to the .belL[' stress. For aluminum ~UOYlI the mu.do so 'th~t net a.rea loollr stress win ne,ver exCeed ,...... t.e tensile strength it approximately 1.67 tilDH F,,,,. In ox.eeptioual eaRS wbctC the net ahta of the sheet ill eIII! ultimq.te shear Itre-ngth Qf the olaterU~l. "rhis in· ,. 4iM-. a DUlIlina! tentian capacity ot 67 per aent~ In' less dian 75 l)er ~ellt "of the gr~ ~11t an addhiona1 , tM 'deal tension field, where the coutp!'CtIi.c,U Itrength cJiCcL:: it reeommended. The circ..'wu tntetaeti01t tor ' .• dIe Ih~et il'aaumed equal to zero, the maxhnum ten- cq.ubblt!d tel1aion Bnd .bo~r il liven by tha ~orIJ1Ulf& "JIon of tho , ....t ia equal to twiee the .!&bear streM,' This F~]I (10) :~teoli that in the homogeneous tenaion field the alleet ellF"" CliP. '~I!U:"0t develop ·tbo ultimate shear Itre~ of, ~h~ ~. '}lfial. The, curve of Fig, 0, giving the butc allowable ' wllore C, dell~t.e8 the rivet'tactoE'• ..... atreJs' Pi.. takeS! aA.."eOUnt of this. ,However, a ,If coml~realon 4s aupcrhnpo,,:d Oft tfle ~t' ~tre. bt , ~tiO.n il '&110 r&eedClid to provide for angle lilt dll- . the panel. the initial bucldil'llltre,. il _in multiplied , .....t iroIq 46° and fOE' wrinkles forming from corner to by the factor). hi both Rqa. (9) BJld (10). of tJie panel. This r:om:ction ie i'eftected by the r-_ _i&l/actar oj A'i~lrinfJftr' At either end ('if tbe '• •t pr at spUees, the lpad in the : Iheet must 'be transferred ~y ~vets or las~n~ other
[---l!-]t + ['!J1...
_1
an
."""9'"
I.
, (
..
:...
:',
','
1_
*0
'j' "
JOUllN'AL .Ott l'H8 AIlRONAt1TtC.A~ SCIJ1NCB8-P;BBR:UA.RV, 1963
110
,
IIUIAINCE I
a
.'
t
I.
~,
..........- - JII-
.. .~
f
~---.. -I!!!!!iE---1
I • .
.
r:* .~. (.~
'. \
.1'I~
FleUR!. 6 - TYPES OF SPECIMENS TESTE.D parts 01 the structure. Tha shear toad for whieh each f.,tener Ihoulcl ~ c:b~ked is given by p _ w.hf!~
J]'
ql>~1 + [!C!. ~.F...
I! .. pJtc:h nE
(11)
fut~m"rl
a.ild q - sItell' ftow in _sheet. A.Jain, reduce F,u•. by the J!J,ctor A if ~o.mp~ .. _sian. £8 present. .
·rb~ BLtbstanli~ti()u fo!' the method. Pl'OI)uacd herein i'Lte baaed upon the test work perfonned by the N.A.C.A,"'t the A1t1mlnum Company of America, ',I Do",I11.1 Aircraft COltrpnny. InCH· arid the National Btfft.'\u c.f Standardl.' A sum tDtal of 54 test specimens Ito been fUtRIYled in t1t11&Investigatlon. Fig. 6 shan Ute gct1crol oonf1~lu·at.ious of the typea of .pecitnot1s tested. As Cl1ti be observed. IOUlt of the si)Ccimens lire complete cylinders stiffelled with tongitudinru attn'ener. and rinp, whUe othera are portions of cylindcn. The radius of specimens varied froID 15 ta 00 in.; skin pre, frotn 0.020 to O.OBl: and the factor If had a ntnle 011 the apechUf!111 exattlined frofll approxDllatcly 1.10 . to S.
Tuble 1 gives" ."mpanson 01 Ute. ctitie:n1 bu~kJiJtg' ::. mear I9treBS romputed by the method of this ruper, by the 1ttethod. of reference 9. and ob~ test valuet.. ! Twenty-ltix po.nC'ls WC~ checked. Test valUt'f.l uv~ra.ged. ~.O
per cent. higher tbnn predicted by the propoled method and 5.8 pc!!' (cnt hlgber than by tlte method of .. tef~rence 9. Figure 7 shows gru;phic::-ally the- reJatiolt.eb1p between. test ond pred{cted values. There. l.--no -:.... IirniticAut c1iffottmee between tho prediction~ o{ 'the t~ meijtods. The advantage of the proposoQ method lies "', in the fnet that t1!(~--bueklin8-·&trcSl·-ef-·tht -f!urvc-ej:- . i.:~ pattel: II exprel!lCd expUdtly in ternlS of paran1eterIJ . of .the panel and that Meteac:e to ehartl is not ft!qutred. "
A tota1 of 24 specimens tbat were subjected to meat .' Pig. 8 mows the..relation· ship between the ahea.t IItfet1gtb predided by the tnethod 01 thill)Cl.per and teat &becv strertgth. rn~lud;tlr an s~meul.· the aYe1't1ge con.ervAtl!m of the .predietion ii 7.8 ~r cent. The resulu frons aperonei! 2 tJt tefE!fenCf! 2 and s~imen 10 of reference 41vere,averly . mnaerv8;tlve. In tlle firs.t CRIC th~ stiffener appil.rently: .
Gilly railed In the 9tul'ener.
OCT
es
'93 18:"'0 LETH & ASSOC (a215J392-4482
A N A L V S r SO"" S 'A' 1 }r If KNIt D
j', ~ ,
t ,.. , ,
'1".\01.111
1
~
• • 1..1
C LT RV B [) PAN E t.. S
117
.~- ..--:-:- , - - -
.., Campa~ of lnitiaJ B\lClktin, Stl'1..'116 CDmputed by P.. .., • P/ ... F/' by Nt'tbod at RtJ~tK:e' 0 a.ud Obmvecl Test ViJut'!•• Pa.nQ~~_8uhJl!fWd as tlu.l.l' 0 ..1, ., " tabial BlU!kiinl StrHl Ratio of Test to 1-'red1cted ObHrved PropOHd. Method of
b,
method
tl!~t
1.* 1,001)
2.960 9,870 4.120 8,400 4.l:ttl
2
1.199 1.]46 l. 000 0.080
~.860
. rei.\)
1.039 1.0« 1.21~
1.00& 1.107
1.040
3.100
0.048 1.112
0.086
6,040 6,840 "t .160
1.082 . 1.016' 1.008 0.964
t.OBO 1.086 1. UJI 1.088
~,12()
a.iUO
A.elm
LOll
:1,'100
0.0&8
1.012
4.100
0,918
1.0U)
1.003
8.11IJ()
1,013 LLU04
O,91Ja 1.01'1 .
6,400 6,700 8.800
0.IM2 1.034 0,114 l,023 1.046 I.lgS
1,040
2,38C) ., .900
1.210
1.~3,·
1.020
1.011 0.'73 l.aU 1.111 1.179
------------------, --."-, -"lOped u fixity of 4 iustefld ¢ 2 ila determiMd from
" .. eomprcaion tttc.sea re"d in the strail1 gages. find ilt , .... lKond calK! the test is qUestiOlled l since this sped'. Wlil one. of Q lamlly of specinlem and the teat ft· , ... were incompat.iblo with the rest. It these two .......... are excluded, the average COllgerVs.tism be-: . . . . L'j.~ per cent.
; ·eM
~,'" "'J.t:hItl' ~ SlNf.Jr O"~ (No &tlJlener
,,,'hue,) ,
. ' ',:.... a ftoptlvt clteek of the method' for predicting ' ....... failures, 16 specill1eUI that did .not laU in the , , were .flnaJyzed for lltiffenel' fAilure. The pur, .. ,.. of thi. wat to deter..1lliDO to what extent ltiffenet ;'....... '!"Ould be predicted by the proposed. method ~ ,~tht!y did not a.(!tuu.l1y oceUr ill the teat panel. 9 i. a graph ot predicted sheru- stress VS. test shear I
....
:, . . GBd tndic.u.t~ that lor moat of the &~tn1en8 the il lower than the predIcted atrass. This is as It -.ould M, since. these l)a.nets did nQt lail1n the .tUI.
_1treII
'.r.
'nu..
.
~c!McI
~fle1A that bad a test st:rea& dose to the
Itreu·bued on stUfener failure faDed by pop-
W of the rivetI, and it il ql1cttionable whether the
, . . wqu14 h4ve aarried mu~h morc It~1I had the ~
nat f a i . , . . ·
hn,', SubJ~t«l to
Combined SIHta,. .,.d .C1omlWtllllon ",.
(StlJ/tm4lr '"IIUt"') ,
.
A total of 11; spcc:htten.. aubjeeted 'to shea.r and compre.Sion were analyzed. Fig. 10 shows the interacdon between elfective colnpreasioll in stiffener due to Ihear and the direct CDlnprc.ui.on tor the test .pedmcnJ. All"· lYzed. . The results hld~(!tltc an DV(..'rQgc conservatism tor the lllCtl104 of predietion 016.0 per eent. The teat ftlUlt f.rolll one specimen from retcnnee 4 with a high COIl&erVp.tiBm wall con'sidered tD be questionable, aineeft waa Ilf,)t ',compatible\Vith othertest&in~besameaeriesoflpecimen~• If this specimen is neglected, the average c;onservatiJm drops to 3.8 per eent.. The aet.na.l ~Oillerv.tJlm. in an interactioD d;u.gram b~tween shear flow inakib and compreilLon load on stUfener would be le•• than tbe IlQove. YQ.IUG, Anco the conservu.U.1n euten into. only that portion of the shear flow supported by tlle &tifT~ etters. PflM" SUIi}.u;t"d ,.., ShHr Oftly (W,,, I'oUau,;)
Nine specimens 'ailed til the web. Fig. 11 altows R ~aph of t~t I~ VI. pred~et~d atren for web failure. The resultalndiC'o.te thut tile meU,od is conservative by ... average ~f 0.:1 pcir cent. There it :realOn to questioD
an
•
J 0 U R. N' A t:. 0 1! T H:
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til r---...,...-.....;~~..--.-...,.---_--
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Tv,ICAL·ApPUCATIQN
A t1Pi~I.ppU('!~ti.Oft to,..h1ch tb1& 1Uetho~ of I.ueiytll : :' may be .p.pned is the detetm~na.tlon 'of allowable· Itrengthl fot a tuaelap shell 8ttuMure. The lolt0tMr . b as,. outline of the pro~dure lot' COl1.truetmr AQ il1tef.. \ ,
n.etion: eurve of corubiaed shear.' and C01npte8!ilon 011 I:: :, curved panei. An eJeample of such curve. i. shown.1rs :.. Pig_ l~t
"
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, I !3
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.- RIf'. I
+'. ",,.1 .- •. 1 I~~~--+---+---*---~
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'''OPo.lO METHOD· ILIt
FaIUftC 7. - IMTIAL .•HEAR IUOKLtNG PANaLS .. '
7
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.
~
~
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."tAII If"lIl '''lEO'' tTl""" '.t.1L.UII£ .... :.
'10 ... iHiAR STMiS AT F'AILUM .1 PfI£DICTID IHOa 8Tf11!1iS , . .... PAHD.S wrrHOUT 8TI"E"!" PMUJIIU. ~cttD ' TO SH!A" ONLY.
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""ID!GTlD IHIAII IoTMlI IAKO 014 1T"''''teft '~It.UR! - ..
f FIG • - SHEAR STe. AT 'AIWftI! 1/1 PA!DICTED SHEM ITRI!IS FOft
,.,..., wmt STII=P'£NER F'AlUJAII. FMil.t SUI"I!C1'ID TD
\
IH!Aft ,ONLY.
.I the re5ultg (roltl three of the lpectmeus. Speebttens 20a and 3001 ot refereDce I) iaUed by tearing the aiteet in tbe end panels QtLd wus P(Ccil)itated by shearing ot nveta
.I LO
\ oonnecttng itrJ~gerll to jig. In testing specimen 1 of', Ti reference 3, the loading' ji, bound, and the applied to.rque was actually about 10 p-=r cent greater tilan the. Ft8Uft! 10 -IHTWCTltIH CW COMP....OH DU! TO SftEM : teat gu.ie indica.ted. II theBe t~ lpecltttens wer~ AND otRECT COMPR!8SIOttt ,aft PNm.S WITH . OIt1itted, the /lVc:nrc wnlCrvati~n becoanes 4.0 per BTI"1fHE1II FAtLURII. ~ IUIJIDTID 19" cent.
ItEAFl AND OOMPRE88tOH.
j
r
1.19
ANALVStS OF STIFPBNED CURVED PANBLS
''''lure
''';''''';lM 1M $tlJlVruw (1) The ring spsclnll ItUfenu llpacinl•• tift'ta"; teo. .It1 pge. and the radiul of the parlel ~ knDwn; • the factor , may be ~It',uhlted. " . "(2) DetertnJne' }t~ I &Ild p.. note may be
.lCI
~
la aDd. obta.in corresponding values of ), from
~/ ~/f,~l'"
_ 1
I
+
(I
/
Q
~ ~)I'H
I'
(1) " . value ~f fr time. i it then the shear flow th .. t . ~ .'My iii ~ppJle
V V·
;10
Sq. (6). .~ u.volvel}"i whleh I. dewmined by P/(A.ri bl).) (I) From Table B,· detarminc}/ and/: for BS$umcd ..... of J. and COn'eBpondlng valUe8 ~. (I) n.iettuine value ot 1. whicll. ma.ke.
V
/
I • f ,.
The web failure curve with the etJ'ect of com~ included may be ob·tained as Follows: . ., (1)' C4.pu~ 'I, . (2) hrlorm steps 3 llDd 4 un.der ~tion above. . : (,). Dtternilne the value of J~ Ittch that I, - F•• .... P, iI given by Eq. (9). It should be noted that .,k tMN &tetenninatioDII F.fII'. Dlust be multiplied by the . ...... l. If the rivet factor CI: < 0.76. thenaa o.dditio!2a1 be rru;u:!e in aeoorduC'e with Eq. (10).. The . . . /, tilM•• t is then the altear flow tn&t wiD produce ~ ·"fli&wre. . . 'nit: etVl'tope of the .titTener fallure curV'~ _ad web .~ ewrve form. the c!'lmplete i.nteta.edao curve for ... ,..... In plotting the c:orve..s, it i.e recommended til; - 0 be the fIrIt I) oint in~ettip.ted, lince in thil _ ). il. alwaya flqual to one. aDd Ole trial.ctd-errol' . . . . . iftvolved in lieJ)8 4; li, a~d «1 of the section. above II . . . .atec.i. Once the value af It at p ~ 0 hal been ~I it is jx,uible to make a close first approlCima~ . . M to wbat /, wiU be at ather' vflluea of P. nnd "-Ra.in ...... of trial and ttrOr :witt be grfttly redu~ed. ).,
Kf.•
• tau. I
I ~ • m Pl'ltDtG11.D ,,-t.JM.MIt.E ~ ITRIM - ..
m
FIOURE II. - IHUft I'TRIESS AT '''willi va ,,"ECIC'rEO .....~ ... IHOR $TREtS 'OR MHW WrTH WU- FaI....... 'I, NN£LI 1UI.lICTID TO IHEAR ONLY.:' • 1000
.l!ffIUIl....
~ •
el'eet 'of com prell Ion on th.e u.timate shear .........t1t. ot the wctb illman,lin~e ita only influence is to : ...... the ~rit~ buckling shear ~rellj F bY' the
a
A ~r
•
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~
V
, f
It
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I., I
,II .
ruo • 10··
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). • .00
I
_
1II·fi au; .
• • 10.1 , 40' pII~
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I
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.1111 U,·T.I .ALCLA.6 A". ,lOT 111.1
11.4 11'1.
400
I 0-
IGO.
... n..t
CONCLUSIONS
'. \,.... Ihnplifled formulu for the ealcu.latlon ot critical __ bttcklinr ttreu of curved panels have been ........ Telt values averaged 3.0 per ~nt higher than .~valUeil. '. ' ..
oL-...I---JI....--I........_J.U.....L_..........................
o
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..
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(.OAl)
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10
F.oW.1 12." EX.AMIt.E OF INTERACTION CU.RVa· 0" ~~ aE'AIt AND COMPRESSION ON CUR\IIO PANELS. '
(2) A method haa been determined £OJ' predicting the ultimate atnmath of. a Itift'ened ~ed panel subj~cted to shear. The averap predlc:rlDD II approximately 6 pet cent connrv&.tivo for loplit"di~l .tijJ~er failure ,1'1'1 4: per ceat ~serV&tive for ,web failure. . :1a) A nletbod haa been detemlined for predietiag the w;mate Itten8'~ 01 a ltiBened ~rved pattelaubjected to'!!Qmblned shear aDd compression. The aVl\rap pre· cUction JI .lIghtly less than 3,S per C!snt coftlerVatiV! for lonaitudiftal .tiffeuor failuR. eCofldfJdnl tH& ~" 1. . )
•
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'Jeadilll h, this result. an adjoint variational prindpli, •
or
!f f
",dS -
~ff
'1'110 the C'Ouftterput (J10)
hilS
er' one
,~"
already knowu ,...
steady fto,YI. has been elltablisbed. This may be use,ul
in the approxitnate fIOtution of liftJnl-~ probtem ' in nmudea.dy flow. &vera~ appticatloDa of th" geam! . theorem te probttml in nonstationary wiag theory luift', been given. Thue included determination o£ rela~ . . between eertai~ a.er:adynQ.mic c:oefficlents for plan f~·.: CO~,C~V8toNS in 'tlirect and revene Bow ·wtd establll:1hment or infi~cc . A reneral rtIation ,bet.ween linearized 101utiona of function. lor total lilt, pitching moment, and rollitl, ' lifting-,urtac:e problems in direct and reverse flow haa moment lor, winp oscillating witb arbitrary' motlon , ':~, beeh esi:Bbliihed for ~ompf'f!SSib1e nonsteildy 110m. :'fro. , a.nd det0rtt:J4tion or the 'plan form. The htBut:nce fune·· relatlon JI a. dirut extet1.ion of that alRady known for tionA were fctund to be ~tQ.in sinlple solut~ons'/or th! ' .', ateady·flow lolutione~ ,On the bUI of the analysis plan tonn in rtv~r&e flc)w. The integral 011 the leIt ,Side. il the roUitJg moment that gi"l\'tS the required ruult. •
:,j
I
'
:'
I
'~
i
t, R,.,IUUfHCBB I
vou IC6rm(m; Til •• S"/m'HfttC A."Iid~ftflllli'I-PriflriJt/" d"rt
.4. ;llittlHtml. J tJUrilal of th~ 'Aerot\Qutical Sei\Hleea. Val. 14, N". 6. pp. 873--.00. Jul,. IA·n. t liayes. W. LLHf(Jri~. 'sU;,tJ';Hi, North Atuft'kan.
D.•
P""".
Ayhdlou. [Ile .. Rupert Na. ALpUi. Jl:lnc 18. 1047. 8 FIIlX. A. t.f •• , R~HDMi' 1JftrlJHfJ "", CII4i'rlUtri,ticl
ttt 4 Wt"" FlO'W. 'ourlud af tbe AL1'Otlautieu.l Saleneelil t Vol. 16. No.8, p,. 490-604. AUII1't , .. ).funk, M.. M., TiHt Rsw"td TlltlJ~rlfl 01 LintaN•• &~p"""jt: Air/Dil Tilt"". JOIfrnAi of At)pUtd Pbylb, Vol. ~n. No. I. pp. IS-lfn. Febl'Ud.t'Yt 1960. 'HltrntU1l 1 S.. ' rlll'D'~"t4~ Rt:kJ,Urtn. 11dt/.M:Il" ,l, Sla.61Hly D"_~Wu 6/ c Witt, i. Djr~c' iu,d ttl. R.'II Sll;tlrU1lk 1'16111, N.A.C.A. T.N. No. 1943, Septesnber, 1949, • '.'Brown. C. H" Th R,"r;rlbllfJ~ TMQI'f'IH 1M 7'hltl A irj()m i • . Sub."" ..c4"d Su;,rJ(llfie ltIttW, N.A:C.A. T.N. No. 19U, September tUld
JII R~rlf ,. SHPtr6(H1tl.l
.18"'.
1949.
r Ul'Sf!IL P.,. Qud Ward. G. N'l em Smlt. C,Mt't'JJ lIttf)tm,I Ht 1M LI,.,,,,rltt>s r"tlJ" ".f Camp"m"bk FlotJ.i, Qua.rletb· ] out111l1 of
MKbanlet & AppJiqd 'MAthema.tiH. Vol. UI, Pa.t't 3, pp. 326-'1" SeptetuberJ 1930. . • Flax, .A. H •• G.Mrai RlU,;rt" PlUfIJ a"tI VdHs'iCJfIJl,t TMDfltliU in i.ifti"l.St""~ r/"iI'Y.,I0u.tu~1 lOr 'hil Aol'OllQuticord Selerl~. Vol. 1~. No. 0, pp. 361-!J14, JUf.1e JOM. . , t \\'eb$tcr. A. (,1., PtJyl1ttl DU'rtnlitll BttlitsliOrt)DllI.bdA""tlt.i.trtJ1 P).7Ii~l. p. 2UI: O. B. StClOkerl &: Campaqy. 1983. II RI.y1el,h. Lord. Tlleo,., oj S~1Intl. Vol. U. pp. 14&-147: DoVer' Fubl~tloh!l. Ine .. IP46• U R.eluner. n.. 0" lilt Gtftl&ral n,ol'10f niH .A. ;'r/oil.'f I'" .VOft-. U"ilM'fff AfOllicm. N.A.C.A. l\N. No, ftO. AUIUlt. 1944. II St!hwartl. 1.., }ge,.tr'b"~, d" cir.fI' 1r",,,,,,~,,,,Jt lfll fll!r/mrInJ4en Tratftllf.~r; in #be#~,. &~fl"g, LuftfRhrtfw·· tehul1l. Vol. ti, pp. 379-386, 1940. . -, S"hnpn. H '. &hwMrt~. I..• lInd Diet..., F.• Three PBptr. (1'011' Col1rert~ DtI ·'W;n.r Md Tan Surface ~llto.tlatta. n Muuleb. Mltreb 6-8. 1941. A111o. N.A.C.M, T.M. ~o. lerm, Aurullt,
D,.,.cht",Uu.,
1951.
Analysis of Stiffened Curved
Pa.nel~
Under Shear and
Compr~ssion
(Co"c.lrtde4. fro", PORI tID) bFR.INcaa
C.. r"r,.w" TIIII ~J S'II'II«J Cfrtfd"f' Cylirrflt.rJ. N.A.C.A. A(lvn.ltt'Cd Jtclluit'ttd R~Qrt No.4 B31.19+4. , 1 C'.l1J.rJl'j 1. W., and Moore, R. L.. r",imt Tutt ttl Al""u'u,mr AU", Stifflnm C~'fif'I4p' C,ti"tltf'. Udpubtishtd Alr:oa Raport. No. 12-ftO 15-A, 10tit. .I XubfJ. hul, lind PClet'lOH, ]l1tt1Q' P•• A. SU"",u'r:I f,lJ Dla,~ Mtdl Ttfls-ltm, N.A.C.A. T.N. Na. 2062, lOBS. (N.A.C.A. T.N. No. 148l. UJ4'1. Ditlft'fUll r,.,ltJH I .. 0fI.rw4 WtMj,) • AnilcrMll1, P. N .• CqflfliiuM Comprtt,nnn Afttl Slrttr; in .'1;11 llIuf SUjftN., Ctmt6f.f.UJli""'" J:)oullu Attemft Company l{epon No. aMI. 1041. • G{J(I~hnan, Stanley, T"J. 0/ Rfl14/fJ'I't«l C~rtled Sltl:tl i" SJaa.r. National BUrNt/. of Stan.dardl Lab. No. 04180 PRI, 1949. • ).JtDOt'e, R.. L'J cmd WI!tICOQt.
a
IlleterROn, Jdmll8 1'.,
Circsmr C,u,.4,1"
lHw4U,t'lJirHC nf SH.!..ttt~ Combirmf T(tfIit'lH atrd Cp",p,rINon.
.B.~pmm'nlo.l
SIlI})fJtI~r III
N.A.C.A. T.N. No. 2188, 1960. 7 WCl.l11L!!r, H~rben, Fkd S'/ttd lIItt1aJ (;irdl1'l CiIItIJ t··fr), TltiJl M"tttli Wt6. N.A.C.A. 'r.M. No. 006. 100L • Kuhn. PR.ut. ,nud PcttfMm, JlUJle. P., SU'rtl.lla A "n.l~.sil ,,} . SlII'HM. BCQu, Wtb.t, N.A..C.A. T.N; No. 13434.. 1947. , 'Batdorf. S, D., A Si",plljuJ. M,IW ,,1 EltJ.lHc $I6"#lI1 A.u;,.;. lor TlJi" C"UiUlriwJ SAt-III, N.A.C.A. Repcrt No. AN, J041.··
'1.1del. R.•
,T"~
o-mMl
SlJt41'
'
Ltlll.d
pi R"_""11dll'
Pktltl.
N.A..C.A. T.M. No. 70lJ. 1(»33. ' Jl !.wIn. L. It.• sad Snhd,-", C. W•• Ir.) A8oJ,Ji, &tiftlfli niti 8;0_ W~,. N.A.C.A. T.N. No. 1800. lS".Y.
Sl,,,,,,,.
(It
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.'
Bell Helicopter' i ~:; i.t.]: I INTER-OFFICE MEMO
2 May 1991 81:KET:rlc/42-005
MEMO TO:
Airframe and landing Gear Design
COpy TO:
B. Alapic, R. Alsmil1er, J.O. Clark, J. Fila, T. Fox (BHTC), L. Graff, G. Grimes, J. lang, T. Meyers, G. Moore, D. Newland, T. Pekurney, R. Seago, W. Taylor, W. Thomas
SUBJECT:
MIL-S-81733 vs. MIL-S-8802 Sealant
In a meeting held on 5-1-91 between representatives of Airframe Design, Airframe Structures, Chern. Lab, Materials and Processes, and Methods and Materials lab it was decided that, for all future airframe applications, corrosion inhibiting sealant per MIL-S-81733 will be used in preference to sealant per MIL-S-8802, unless fuel resistance is the primary consideration. MIL-S-8802 will continue to be used to seal fuel cell areas. When new assembly or installation dash numbers containing sealant are created on existing drawings, the sealant should be changed to MIL-S-81733 at that time.
Kurt E. Tessnow Group Engineer Airframe &Landing Gear Design
MIL-S-6}733C
ao ..
9999906 011Q3a3 5
r---:-:----_ __
..
H.... 109-3(J
MlL-S-81733C 13 March 1980 SUPERSEDING HIL-S-81733B(AS) 7 June 1976
MILITARY SPECIFICATION SEALING AND COATING COMPomm, CORROSION INllIBITIVI
This specification is approved for use by all Department. and Agencies of the Department of Defense.
1.
SCOPE
1.1 Scope. This specificatioD cover. acceler~ted, room temperature curing synthetic rubber cOIIIP0UD.Cls used in the aealing and coating of metal components on we.polUl anel aircraft _,st._ for: protection agatost corrosion. The 8ealant is effective over a continuous operating te.perature range of -540 to +93°C (-65° to +20QoF). 1.2 Classification. The sealing compound shall be of the following type. as specified (see 6.2): Type Type Type Type
I
- For brush or dip applications
II - For extrusion application, gUD or spatula III - For spray gun application IV - For iaying 8urfac~ application, gun or spatula
1.2.1 Dash numbers. The following daah numbers shall be used to deaignate the mintmu. .pplication time in hours.
Type I Type II Type III Type IV
-
Dash Dash Dash Dasb
numbers shall be -1/2 and -2 numbers shall be -1/2, -2 and -4 number shall be -1 numbers shall be -12. -24, -40 and -48
Bxurple - Type 1-\ shall designate a bruahable material having aD. application tme of ~ llour.. Type 1-2 shall designate an applica.tion time of 2 hours. All other type. and daah Dumbers ahall be designated in • sillilar' IIaIlD.er.
Beneficial CDCmencs (recoIlllleD..dAtionat~additioDl, -deletioDJI) au any pertinent data which . ..,. be of use .in improviq this dO~UIDI:Ilt should be addre •• ed to: Engineering Specifications-and Stao.dard& Department (Code 93). Naval Air hgineering .Center, Lakehurst, llJ 08733, by using the self-addre•• ed StandardizatioD Docu.ent Improv~ent Proposal (DD For. 1426) appearins at the end of this docuaent, or by letter.
' .. ESC 8030 r~----------------------------------~'
THIS OOCUMENT CONTAINS
Ul
PAGES.
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Rectangular frames with hinges supports-verticallodlHng. 2
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Vi
At" = AI,
~'
= Pa (I - ~I) + /tI, ~' M. * AI.
(I -
n
+ M, ~
Rectangular frames with fixed supports
Rectangular frames ' with fixed supports-simple loading. 21- (1
IJ ---j
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,
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11.
L
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----t
HI 1\1. = ~a~ [~ ='= L:-2~] 2L D LE ' ',. For :i: signs: ,1I. == -; lilt == Pah [ 1 1.1 - 2a ] 'AI:. AI3 = --y;- -6 "*: 2L/~"-
+
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+;
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[1 _
When
J:
~ a,
At. - (Pb+At.)
When
%
>",
!of.
1'. == P - V',
1+.1(, ( 1- -f:)
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=
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--
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+
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=
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== -2iU t wb%
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JlI.t. =
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BELL HELICOPTSR POMPANY Engineering Department June 15, 1972 Page 1 of 2 STRUCTURES
e-
INFORMATION MEMO NO.1
TO:
Mr. N. J. Mackenzie
COPIES TO:
SIM Distribution
SUBJECT:
PROCEDURS FOR STRUCTURSS 1~FORMATION MEMO (S1M)
.REFER8NCSS:
(a) ( b)
As As
ENCLOSURES:
(a) (b)
SIM Index SIM Distribution
required requ ired
This memo is written to establish a procedure for making new and unique structural design information available to members of the Structures groups and appropriate design groups. Much useful information is either generated or collected_ by members of the Struct~res groups during the nor~al performance of their duties. This information is usually available to a limited number of persons and is often filed away and forgotten. In order to prevent valuable information from becoming useless and forgotten, the Structures Information Memo is hereby established as the vehicle for conveying this information. The Methods and Materials Structures "Group Engineer witl be thecoordinator for all SIM's and will assist in determining what information is valuable enough to publish. He will retain all originals, will assign S1M index number, and update the index and distribution list as required. Each SIM shall include a cover memo, addressed to the Chief of Structural Design, giving a brief synopsis of the material. The memo shall be signed by the originator and approved by the SIM coordinator. The originator of each SIM shall be responsible for establishing the credibility and accuracy of his information and for preparing the SIM for distribution. Each SIM shall "stand on its own ll and be thoroughly checked ana referenced_ Format of the material is left to the discretion of the originator. however, it should be remembered that all SIM's will be considered for
e
June 15, 1972 rage 2 of 2
incorporation in a Struc~ures ~anual to be issued at a later date. Similar significant structural information originating in any design group will be welcomed and handled in the same manner. Any additions or deletions to the distribution list should be directed to the SIM coordinator.
M. J. McGuigan Chief of Structural Design Bxt. 3147
•
.'
-
..,.
....
t'~
8Y
e.
Your Name Here
BELL. HEUCOPTER
COM~
CHECKED ____--------__~
MOO E L
PA GE
_4_o--.;.f--.;.4---11
Leave Blank on Comm. HELl.
RPT
"
-----------1.
Add dwg. number part being analyzed in this box. Leave space for revision letter later.
TITLE HERE
I
DEtAIL PART NOMENCLATURE HERE
I.
State geometry, loads, detail and location or reference Sect. A, Pg. _ __ where this is shown.
See #3.
compute the actual ultimate stress level,
l
gener~lly
from limit loads,
referencing a report -or page number for the loads. It may be necessary to compute section properties, loads on the section
being analyzed or to determine and show a static balance prior to computation of the stress level. compute an allowable or state a
r~ference
for the allowables being used.
State limit loads and yield allowables when. these are uSed to
pro~e
structure is non-yielding at limit load. State the margin of safety. This is the purpose and conclusion of the analysis.
Be sure to include
fitting factors, casting factors in the M.S. and so state, i.e., "using
1.15 fitting factor". M.S.
·• > w a:
.. ·.. n
In
State which formula is used such as,
= (R + R 1)(Factor) 1 2
-1
=
+.xx
L Confine the analysis within these limits and thereby preserve the neatness of the report.
..,. f.
BELL HELICOPTER COMPANY Engineering Department August 28, 1973 Page 1 of 2 STRUCTURES INFORMATION MEMO NO.3 TO:
Mr. R. Lynn
COPIES TO:
SIM Distribution
SUBJECT:
LOAD SHEETS FOR STATIC TEST
OF
CASTINGS
This memo is written to standardize the data furnished on the'Load Sheets prepared for the Mechanical Laboratory and used by them iR the static tes.ts--··· - ... of castings. The following information, as a minimum, should be included on all Load Sheets. 1.
Title, part no., and name, thus: CASTING LOAD SHEET Part No. 204-XXX-XXX-X, Bellcrank, Cyclic. Control
2.
Indicate loads as ttlimit" or "ultimate""
3.
Dra't~
4.
When available the report number from which loads and reactions were obtained shall be referenced, thus:
sketches of part showing 'the external load applica tion, direction and magnitude, apd the reactions (usually designated by ;' ~ , / ~J and ~~ ). Sufficient views shall be used to completely define the cri tical loading condition. Each view shall show the--" reactions necessary to place the part, with its applied external load(s), in a state of static equilibrium. The loads and reactions shall be the same as those used in the structural analysis to insure that the part will be tested in the same manner as it was analyzed. Where moment vec.tors ( "",.) are used a note shall be included to indicate whether the right or left hand rule is applicable.
Ref. 5.
Ultimate preferred.
~eport
20S-XXX-XXX
A note shall give a brief description of· the loading condition. thus: Loading Condition:
8G Forward Crash
.,..'
- ,
~
Page 2 of 2
•
6.
A note shall indicate the casting factor, thus: Loads include a 1.33 casting factor Where the casting factor is unity, so indicate, thus: Casting factor of 1.00 is applicable
7.
Any other special information necessary to assure that the casting will be tested as it was analyzed.
8.
All Load Sheets shall be
prepa~ed o~
An example Load Sheet is attached. was omitted.
stress pad paper.
Note that the rule for the moment vectors
M. J. McGuigani
Chief of Structural Design Ext. 3147
•
BELL HELICOPTER CO~~k~ Engineering Department
27 February 1974 Page 1 of I
StRUCTURES INFORMATION MEMO NO.4
e·
TO:
Mr. R. Lynn
COPIES TO:
SIM Distribution, B. Alapic, J. Duppstadt, J. Garrison, J. Gilday, W. Rollings, C. Sloan, K. Wernicke
SUBJECT:
BOLTS IN MOVABLE CONTROL SYSTEM JOINTS
In order to avoid the possibility of in8ta11ing provide increased resistance to repeated loads, implemented on the Model 409, Model D306, Model of the ~lodel 214 and Model 206L; and all future
an understrength bolt and to the following p~licy shall be 391; the production series designs.
NAS quality bolts shall be installed in the movable portion of all control system joints.
M. J. McGuigan
:::='f
Chief of Structural Design Ext. 3147
•
•
BELL HELICOPTER COMPANY Engineering Department
12 March 1974 STRUCTURES INFORMATION MEMO NO.5
•
TO:
Mr. R. Lynn
COPIES TO:
SIM Distrib\ltion
SUBJEct:
JUMP
TAKEOFF LOADS
Recently, it has come to my attention that we are not add~essing the rotor tilt for the jump takeoff conditions in a consistent manner. In order to provide a uniform approach, the following procedure shall be followed: - Assume the helicopter has landed on a slope Qf specified magnitude in any direction (normally 6 0 ) and executes a vertical takeoff at max~um load factor for this condition. The rotor tilt will be that which is necessary to execute this maneuver •
.
--
O~~·Lt< ~~
O. Baker Senior Group Engineer Airframe Structures
•
•
BELL HELICOPTER COMPANY Engineering Department 2 August 1974
STRUCTURES
INFOR}~TION
MEMO NO.6
TO:
Mr. R. Lynn
COPIES TO:
SIM Distribution
SUBJECT:
DETERMINATION OF FAILURE MODES
ENCLOSURE:
Suggested Form for Recording Failure Modes
Beginning with the Model 222, and effective for all future design activity;the Airframe and Dyn~ic Structures Groups will establish and maintain a notebook which shows the first and second predicted failure modes for all structural elements. The maintenance of these notebooks "will be the responsibility of the lead structures engineer for each project. The determination of these failure modes will consider static and dynamic loads along with other contributing factors, such-as temperature, corroslon t and fabrication effects. The primary control will be maintaine~ at the subassembly level "(i.e., engine mount, bulkhead, main beam, etc.). Primary and secondary failure modes for static and fatigue loadings will be deteroined for each subassembly. For those elements which are subjected to static or fatigue testing, the results of those tests will be entered in the notebook. In addition, any service problems encountered in the production cycle of the element will be entered. A suggested form for these r~cords is -.. -enclosed. To aid the designer in his determination of these failure modes, the structural design groups will 'supply the designer wi th. the cri tical loads for the structural element under consideration. These will be supplied in the form of a sketch or free body of the element with the applied loads and reactions. These loads will be updated as the mathematical model is refined during the design process. The establishment and maintenance of these records can mean much in establishing the rationale for a particular design, tracking its performance and guiding similar designs in the future. Your cooperation in implementing the procedure is essential to its success •
.
"
•
M. J.I McGuigan --, , Chief of Structural Design Ext. 3147
••
e DWG. NO.
DESCRIPTION
PREDICTED FAILURE MODE PRIMARY ~ SECQN"pf'.~Y
••••
•
TEST RESULTS
SERVICE HISTORY
•
BELL HELICOPTER COMPANY Engineering Department 8 August 1974 Pa.ge 1 of 2
STRUCTURES INFORMATION
•
M~10
NO.7
TO:
Mr .. R. Lynn
COPIES TO:
SIM
SUBJECT:
'STRUCTURES APPROVAL OF ENVELOPE, SOURCE CONTROL, AND SPECIFICATION CONTROL DRAWINGS
Distribution
It has recently come to my attention that some of the subject type drawings do not always contain adequate information to allow us to properly validate the item to the government or the FAA. For example, castings may be purchased from a Source Control Drawing without proper inspection or test requirements being fully met within the company_ The Source Control .Drawing may make no reference to x-ray requirements, static test requirements or any other special inspections required on castings • Therefore, all Structural Design personnel who have occasion to sign Envelope, Source Control, or Specification Control Drawings shall, as a minimum, establish that the drawing adequately defines the fo110'ving: o o o o o o o o o o a o o
Configuration Mounting and mating dimensions Dimensional limitations (interferences) Performance (loads, environment, life, etc.) Weight limitations Reliability requirements Interchangeability requirements Test requirements Verification requirements (analysis or test) Material limitations (example, no castings allowed, etc.) Casting classification if allowed (also casting factor) Primary Part designation Reference to applicable specification
Also, if special inspections and tests such as x-rays and static tests are required t Project should be alerted so that plans can be made to procure parts for the required tests. •
•
"Approved Sources of Supply" or USuggested Sources of Supply" shall not be approved by Structures until we are completely satisfied that the proposed vendor item does meet all structural requirements. This may mean vendors must submit stress analyses of their design or test data ~s a part of their proposal.
8 August 1974 Page 2 of 2 On all Primary Parts or other items with significant structural requirements, . the Structures Engineer shall retain a copy of the approved design, vendor stress analysis and test data and file this information in the proper Drawing Check Notebook. It is hoped that other design groups will use this or some other check list for processing these type drawings ..
(
•
~, ~t" .. -:,~;k""v -"I ..
"'-.1"4-/,,~'
M.. J. McGuigan Chief of Structural Design
Bell Helicopter i i ~:, i it.] : I INTER·OFFICE MEMO
MEMO TO:
Airframe Designers, Stress Analysts, Checkers, E. Ryba
COpy TO
B. Alapic,
SUBJECT:
BONDED PANEL INSERTS
o.
Baker, T. Eidson, W. Fontain, K. Tessnow
REFERENCE: (a) IOM 81:JMS/DET:jhb-460 Reference (a) specified three types of potted inserts, NAS1832C, NASl834C and NASl835C, which are acceptable for use with GRIEP composite panels. It is the opinion of Airframe Structures that the use of these inserts be limited to non-structural and structural shear applications only. These inserts must be considered non-structural for tension applications (unless modified with an enclosing doubler), because when installed there is no bearing surface on the composite facings of the panels.
•
A vendor is investigating the manufacture of the 80-007 insert using 300 series CRES material. This insert shoul~ be available for callout on drawings presently in work. Also under consideration is a flat ,head 80-013 plug and sleeve type in$ert that is domed on one end for use in fuel cell panels. These inserts are considered to be structural for tension and shear applications. Information regarding these inserts will be forwarded to the cognizant personnel as soon as it becomes available.
G. R. Grimes Group Engineer Airframe Structures
BELL HELICOPTER TEXTRON INC.
Engineering Department 24 January 1984 Page 1 of 1
STRUCTURES INFORMATION MEMO NO. 17 TO:
SUBJECT:
SDM Distribution LATERAL LOAD CRITERIA FOR COLLECTIVE
CONTROL
To preclude inadvertent damage from handling , the following additional criteria will be met on all future collective control systems:
•
170 pound limit load applied separately in a horizontal plane inboard or outboard at the center of the cOllective·
handgrip.
o.
for
K. McCaskl.lI O. Baker Manager of Structural Analysis
("
F. W ner DiretOr
Of~le
Design
•
STRUCTURES INFORMATION HEMO NO. 14
.
To:
SIM DISTRIBUTION
Subject:
STRUCTURAL APPROVAL POLICY
Reference:
Structures Information Memo No. 7 . "structures Approval of Envelope, Source Control and Specification "Control Drawings"
Structures Group approval of any drawing is defined as structural approval of all parts called out on that drawing regardless of whether or not they are Bell designed parts.
• r
(. \,
It is therefore the responsibility of the Structures-Engineer who signs a drawing to satisfy himself that all components of that drawing, including vendor part numbers" standard parts and specification controlled items, meet Bellis structural requirements for that particular installation. For components that are defined by Bell Procurement Specification, Specification Control Drawing or Source Control Drawiug, the guidelines of SIM No.7, as amplified here, are to be followed. The Structures Engineer must be assured that the controlling Bell specification or drawing contains adequate requirements for vendor stress analysis _ and/or structural test proposal and results report to assure that scrength requirements are met. Provision should be made for FAA conformity and for Bell witness of testing, if required. In the case of a product de£ined entirely by vendor's' drawings and procured by their part number, the Structures Engineer must "notify the Project Engineer in writi~g of the extent -of structural substantiation by analysis or testing required from tne vendor;--Provision should be made for FAA conformity and for Bell witness of testing, if required. It must be made clear that drawing approval is contingent upon successful completion of analysis or testing and submittal of these data for structures approval. If Bell testing is indicated, ~~As and schedules must be written to establish these tests.
Dave Poster Director of Design Engineering
•
/J4~L
OrVirl-e Baker Manager of Structural Analysis
•
BELL HELICOPTER TEXTRON Engineering Department 31 August 1981 .Page 1 of 2 STRUCTURES INFORMATION MEMO NO. 13 ,SIM Distribution
To:
Subject:
FITTING FACTORS, THEIR DEFINITION AND
Reference:
APPLICABILITY
FAR 29.623 1 29.619
A fitting factor is a 1.15 load factor, applied to limit loads, and is in addition to the 1.50 factor of safety. It accounts for uncertanties such as deterioration in service, manufacturing process variables and unaccountability in the inspection processes.
• ( \
•
For design considerations, a fitting shall be defined as partes) used in a primary structural load path whose principal function is to provide a load. path th~ough the joint of one member to another. The connecting means is generally a single fastener • A fitting factor is applicable to ~~e fitting, the fastener bearing on the joined members, as well as the attachments joining the fitting(s) to the structure. It is particularly considered when failure of such fitting would not allow load redistribution in a manner that would provide continued safe flight and that load redistribution cannot be verified by -analysis or test. Obviously then, a fitting factor is applied to non-redundant connecting" members in primary load path applications the failure of which may affect safety of the aircraft and its occupants. It is applied until the load is distributed into the surrounding back-up ~tructure to which the fitting is attached. A fitting factor is
~
applicable to:
a)
Crash load factors that are the only design condition and/or crash load factors that exceed limit load factors x 1.5 x 1.15.
b)
A continuous riveted joint(s) in basic structure when section properties remain consistent throughout the joint and the joint consists of approved practices and methods such as splices of main beam caps - riveted door post caps to bulkheads, riveted skin splice
•
31 August 1981 Page 2 of 2
SxM No. 13
doublers, continuous riveted skins to longerons, continuous riveted structure such as bulkheads to beams or intercostals, or frames, etc. c)
An integral fitting beyond the point where section
properties become typical of the part. Example, integrally fabricated lug on a forging, or machining. d)
Welded joints.
e)
To a member when a larger load factor is used such as a larger special bearing factor, a 1.25 casting factor, a 1.33 fatigue factor, a 1.33 retention factor of seats and safety belts.
f)
Systems or structure when they are verified by limit or ultimate load tests. The fixed control system is an example of this exception.
q)
Bonded inserts and/or fittings in sandwich· panels.
h)
A
-
fitting in redundant connecting members •
•
Orv1l1e Baker . Manager of Structural Analysis Ext. 3147
£J.P~
D. Poster Director of Design Engineering
.( \
BELL HELICOPTER TEXTRON ENGINEERING DEPART~mNT
7 Auqust 1981 Page 1 of 1 STRUCTURES INFORMATION MEMO NO. 12
To:
Mr. R. Lynn
Copies to:
SIM Distribution, Engineering Design Groups, Check Group, D. May
Subject:
7050-T73 RIVETS IN LIEU OF 2024-T3l (ICE BOX) RIVETS
-
7050-T73 rivets will be utilized in lieu of 2024-T3l "DOli (ice box) rivets as of 20 July 1981. The 7050 rivets can be'stored at room temperature, thereby eliminating numerous problems that exist with the 2024 rivets. The followipg policy will be implemented. 1.
•
Manufacturing will utilize the 70S0-T73 rivets to supersede the MS 2042600 and MS 2047000 rivets (Reference, the SUPERC~ESSION LIST, BHT Standard 170-001, Revision ftG tt ) , effective the target date of 20 July 1981 •
2.
The 100° ,flush and protruding head 7050 aluminum alloy rivets are delineated in BHT Standards 110-174 and 110-175, . resp'ective1y.
3.
All out for and
new,drawings initiated after ,this date will call the 7050 rivets for 3/16 and 1/4 inch diameters. Approval other diameters l-1UST be obtained from applicable Structures Design Group Engineers prior to utilization •
..
4.
The 7050 rivets "will not" be utilized to replace nAD n rivets (generally used in 5/32 inch diameters and smaller) at this time (not cost effective) •
5.
The driven shear strengths for both the 7050 and 2024 rivets are established for an Fsu = 41 ksi. Until MIL-HDBK-5 al10wables are available, 2024-T31 MIL-HDBK-S data in tha 3/16 and 1/4 inch diameters, for both protruding and 100 flush heads, are acceptable for 7050-T73 installations and should be so identified for report referencing. It is anticipated that 7050-T73 MIL-HDBK-5 allowables will be available during the 1981-1982 time frame •
( \
•
1).
ilL
D. Poster Director of Design Engineering
o.
Baker
~~nager
of Structural Analysis
e·
BELL HELICOPTER TEXTRON DEPARTMENT
ENGINEERING
16 January 1980 Page 1 of 1 STRUCTURES INFORMATION
~~~O
To:
SIM Dist=ibution .
Subject:
Emergency Float Kit Loads
NO. 11
In addition to the existing design conditions, emergency float kit loads must be developed for the following cO!ld:i:tions: 1. Floats in the water at 0.8 bag buoyancy and combined with salt water drag for 20 knots for~ard speed. These loads will be treated as limit loads. These loads will be applied at angles corresponding to the righting moments, but not to exceed 20°.
•
2. For skid mounted floats; a) A computer drop will be done in a tail down attitude' for limit sink speed. Skids will be checked for a positive M.S. at yield. b) Crosstubes will not yield with the helicopter in the water, floats inflated and no rotor lift.
::Z?L/47'A~_~
M.. J. /McGul,gan, Jr. /
Manager o£ Structures Technology
e
BELL HELICOPTER TEXTRON ENGINEERIN~ DEPARTMENT 9 March 1978 Page 1 of 2 STRUCTURES INFORMATION MEMO NO. 10 To:
SIM Distribution
Subject:
Design Criteria for Doors and Hatches
Unless otherwise specified in a Detail Specification or Structural Design Criteria Report, the structural Design Criteria presented herein should be used on new designs for the following: 1. 2. 3. 4. 5. 6. 7. 8.
Access doors . Hinged or sliding canopi~s Sliding doors Passenge~ doors Crew doors Cargo compartment doors Emergency doors Escape hatches
All loads associated with the use and operation of doors and '. hatches terminate in the latches and hinges and their ,attachment to the airframe. The sources of these loads are: 1. Open canopy during approach or taxi operati.on 2.-Gusts " 3.-0utward push from personnel 4. Air loads 5. Rough handling
1. Open canopy during approach or taxi operation If a sliding or hinged canopy is used, it should be designed to withstand an air load from taxi operations of up to 60 kt. 2. Gusts All doors that are subject to damage by ground gusts and wind loads from other helicopters b~ing run up or taxied nearby or flown close overhead, should be provided with a means to absorb the energy resulting from a 40 kt ground gust occurring during opening or closing. Doors and access doors or panels that have a positive hold-open feature should be capable of withstanding gust loads to 65 kt when the door or panel is in the open position and unatte"nded.
I
9 March '1978
•
Page 2
.
3. Outward push from personnel
f
t
2
Due to possible inadvertent loading by personnel, passenger d90rs should be capable of withstanding an outward load of 200 lb. without opening. Also, doors between occupied compartments shall be capable of withstanding a load of 200 lb. in either direction without opening. These loads are assumed to be applied upon a 10 sq. in. area at any point on the surface of the door. Yielding and excessive deflections are permitted but the door must not open. 4. Air loads'
I
The air loads on doors and hatches for helicopters piobably are minimal when compared to the many personnel-oriented ,jloads. The air loads, however, should be investigated, including the appli~ cation of the appropriate gust criteria. All doors should be capable of withstanding air loads up to Vn in the closed position. All sliding cargo and passenger doors should be capable of withstanding air loads up to 120 knots in the full open position and up to s.o knots in any partially open position. 5. Rough handling
All doors and hatches that are likely to receive rough handling during their lifetime should be capable of.withstanding loads "they are expected to receive in operation. Passenger·and crew doors should withstand a 150 pound load applied downwaxd at the most critical location without permanent deformation. All other doors that are unlikely to be stepped ,on or used as a handhold or which are marked with a UNO STEp·' or "NO HANDHOLD u decal should withstand a SO pound load parallel to the hinge pin axes and a 50 pound load perpendicular to the surface without permanent deformation. -
McGul.gan, Jr. of Structures Technology
•
Bell Helicopter' i ~:, i t(· J: I JNTER-OFF1CE MEMO
16 February 1979 81:GRA:jo-83l Memo To:
Production Airframe Stress Group
Copies To:
Messrs.
Subject:
Analysis of Castings Where Foundry Weld Repair is Allowed
Reference:
S1M No. 9
o.
.
Baker, M. Glass, W. Kuipers, O.K. McCaskill, J. McGuigan, G. McLeod, R. Scoma .
(
The referenced SIM specifies the amount of reduction to be applied to the allowables of five (5) commonly used casting alloys when-·· the foundry is allowed to make weld repair of casting 'flaws per BPS 4470. Our Final stress analyses should take these factors into account in the following manner •. 1. List basic allowab1es at the beginning of the analysis
of the part in question. 2. If weld repair is allowed in area'being analyzed, state "Weld Repair Allowed," and reduce ~ allowable by the amount shown in SIM No.9. Note that the reduction factor is for weld repair. This memo should be attached to your copy of SIM No.9.
(. Group Enqineer Production Airframe Stress
1
.
BELL HELICOPTER TEXTRON Engineering Department
• (
4 May 1976 Page 1 of 1 STRUCTURES INFORMATION MEMO NO.9
To:
SIM Distribution
Subject:
Mechanical Prooerties Reduction Factors for Castin&s with Foundry Weld Repair,
References:
a) BHC Report 599'~233-909, "The Effect of Weld Repair on the Static and Fatigue Strengths of Various Cast Alloys" b) BPS FW 4470 - In Process Welding of Castings c) ASM Technical Report W 6-6.3, "Static and Fatigue Properties of Repair Welded Aluminum and. Magnesium .Premium Quality Castings"
• •
,(
•
Future casting drawings should have a note that permits the in process welding of castings per BPS 4470. To allow for this l0,7eld repair, parts should be analyz.ed using the following reductions in allowables • Reduction Factors for Foundry Weld Repair Material 356-T6 A356-T6 AZ9l-T6 ZE4l-TS 17-4· PH
Ultimate Tensile
Yield Tensile
10% 10% 25% 10%
5%' 13% 22%
0
2%
0
.Endurance Limit
Elongation 0
10% 10%
0 ,.
50% . 50% 30%
-
-
10%---"-10% 10%
In those circumstances where the part cannot"be sized to allow for weld repair throughout the part, a weld map should be provided on the drawing to indicate those areas which may receive weld repair. If the entire part is so critical that no weld repair can be permitted and the part cannot be redesigned, the drawing and all analysis should be clearly marked uNo ~~eld Repair Allowed." All 201 Aluminum Alloy castings shall be marked uNo Weld Repair Allowed".
;w .-~r}"_4'- . ..,./.....
_~
M. J/ McGuigan·,/Jr • .Chief of Structures Technology
.s
.. .
BELL
•
I """-
DIS,...;:', i$.u;-7 c·....)
HELUCC::>PTE~~ COMPANY
INTER-OFFICE MEMO ~ngineering
TO:
Design
& Group Engineers
2 June 1976
Oate
Chie~s,
81:WCF:bb-092
cOPIes TO:
Messrs: A. Green, L. Hochreiter t -J-..:"{:Mc:Gtrigan'1 D. Poster, M. P. Smith, Jr. 1 J.·Weathers
Reference:
(a) (b)
Subject:
IN-PROCESS REPAIR
S.I.M. No.9 dtd 4 May 1976 BPS 4470 - In-Process Welding of Casting ~~LDING
--.
OF ROUGH CASTING
In accordance with Reference (a), future casting drawings will be analyzed to allotV' for in-process repair 'velding of rough castings. New casting drawings shall have one of the following notes. -'
.
.
-~:
If the entire casting can be repair welded, the following general note shall be added: . In-process repair ,welding permissible per BPS 4470.
•
In those circumstances where the casting cannot ,be slzed'to '~llow for repair weld throughQut the part, the dra~ving shall indicate those areas which may receive repair weld. This area will be flagged with the following ~ote: In-process repair ~.;elding permissible per 'SPS 4470 in this area only.
Example of callout
on FlO. "-
r- - -
-+--1---
I , I
\
\
"-.:-~~~)
If the entire casting is so critical that no repair weld can be permitted, the following note shall be added: No repair welding allowed. Please circulate to all Design Personnel. This added to the DRM at the next revision •
•
......
procedu~e
will be
If/ ~ ~.?f?= W. C. Fountain Chief Draftsman
..
•
BELL HELICOPTER COMPANY Engineering Department 26 February 1975 Page 1 of 2
STRUCTURES INFORMATION MEMO NO.8
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l
TO:
Mr. R. Lynn
COPIES TO:
SIM
SUBJEcr:
EOOE DISTANCE REQUIREI-IENTS FOR NAS 1738 AND NAS 1739 BLIND RIVET INSTALLATION
Distribution, Engineering Design Groups 1 Check Gro\lp- .
As stated in MIL-HDBK-5B, paragraph 8.1.4, Blind Fasteners', "The strength values were established from test data and are applicable to joints having values of e/D equal to or greater than 2.0. Where e/D values less than 2.0 are used, tests to substantiate yield and ultimate strengths must be made." On page 1-11 of MIL-HDBK.. 5B, e is defined as the distance from a hole centerline to the edge of the sheet and D is the hole diameter.
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The ultimate and yield strength values for NAS 1738 locked spindle blind rivets are based on a hole diameter of 0.144 for a 1/8 rivet, ,0.177 for a . 5/32 rivet, and 0.2055 for a 3/16 rivet. reference-MIL-HDBK-SB, Table 8.1.4.l.2(d). The shank diameter for the NAS1738 and NAS 1739 rivets are 0.140 for a 1/8 rivet. 0.173 for,a 5/32 rivet, and 0.201 for a 3/16 rivet. Loft and somet~es Engineering Design will dimension edge distances and parts for the NAs 1738 blind rivet based on two times the 5/32 value (.31) rather than two times the 0.177 MIL-HDBK-5B value (.36), for example. This practice results in a rivet edge distance of less than 2.0; th~reforet the MIL-HDBK-5B strength values in Table 8.1.4.1.2(1) f01: NAS 1738B rivets a"r'enot applicable. -
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In conclusion, to ensure the correct edge distance is used wnen planned patterns of NAS 1738 and NAS 1739 rivets are installed, Structures Group recommends that the correct edge distance dtmension be specified on the face of the drawing for rivet patterns rather than using the drawing note that states rivet elD is equal to two times the rivet shank diameter. Also. special attention must be given to skin overlaps, and bulkhead and stiffener flange dimensioning. The edge distance for the countersunk NAS 1739 rivet of 2.5 times the rivet shank diameter is valid because MIL-HDBK-SB values for the NAS 1739 rivet are based on two ttmes the hole diameter. The table below summarizes the recommended minimum nominal edge distance values for NAS 1738 and NAS 1739 blind spindle locked rivets •
26, February 1975 Page 2 of 2
• Rivet Size
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1/8 5/32 3/16
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EDGE DISTANCE NAS 1739 NAS 1738 ~29
.36'
.41
.32 .39
.47
OJ. McGuigan
D. p~
D. Poster 'Manager of Design Engineering
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. Chief of Struc,tures Technology °
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