NASA Astronautic Structural Manual Volume 1

S

NASA

TECHNICAL

MEMORANDUM NASA TM X- 73305

ASTRONAUTIC STRUCTURESMANUAL VOLUME I

(NASA-T_-X-733C5) MANUAL,

AS_EONAUTIC

VOLUME

I

(NASA)

8_6

N76-76166

STRUCTURES p

_/98 Structures

August

and

Propulsion

Unclas _a_05

Laboratory

197 5

NASA

George C. Marshall 5pace Flight Center Marshall Space Flight Center, Alabama

MSFC

- For_"

JlgO

(l_ev

June

1971)

APPROVAL

ASTRONAUTIC STRUCTURES MANUAL VOLUME I

The cation.

information

Review

Atomic

Energy

in this

Commission

Classification

Officer.

This

report

of any information

document

programs

This

has

has

report,

also

been

concerning

been

has

reviewed

for

Department been

in its

made

entirety,

reviewtd

security

classifi-

of Defense

by the has

and approved

been

MSFC

or Security

determined

to

for technical

accuracy.

A.

A.

Director,

McCOOL Structures

and

"_

Propulsion

" LI.S.

GOVERNMENT

Laboratory

PRINTING

OFFICE

1976-641-255/446

REGION

NO.4

.

TECHNICAL I

REPORT

NASA 4

NO.

12.

AND

AUTHOR(S)

9.

PERFORMING

George

REPORT

NO,

3.

SUBTITLE

STRUCTURES

ORGANIZATION

MANUAL

NAME

C. Marshall Space

TITLE

CATALOG

PAGE

NO.

REPORT

Space

Flight

AND

6

ADDRESS

Flight

Center,

DATE

August 1975 ,_ERFORMING

, 8.

Marshall

STANDARD

RECIPIENTJS

,5.

ASTRONAUTIC VOLUME I -7.

ACCESSION

I

TM X-73305

TITLE

GOVERNMENT

PERFORMING

10.

Center

ORGANIZATION

WORK

CODE

ORGANIZATION

UNIT

REPC)R

r

NO.

I. CONTRACTOR GRANT NO.

Alabama

35812 13.TYPE OF REPORT& PERIODCOVERED

12

SPONSORING

AGENCY

NAME

AND

ADDRESS

Teclmical

Memorandum

National Aeronautics and Space Administration Washington, 15

SUPPLEMENTARY

Prepared _ IG,

D.C. NOTES

by Structures

and

Propulsion

Laboratory,

Science

and

aerospace

document strength

cover

most

of the

actual

ranges.

analysis

for the

background of the

devoted

to methods

Section

D is on thermal

These

KE_'

three

and that

as a catalog

are

sophisticated not usually

Section

and

C is devoted

E is on fatigue machinery; NASA

and

TM

Section

X-60041

WORDS

SECURITY

18,

CLASSIF,(of

thl=

¢epart_

Form

3292

(Rev

December

1972)

enough

to give

the

but

in scope

accurate

elastic

available,

and also

SECURITY

CLASSIF,

to the

curves;

topic

to stres_

as a reference

and

DISTRIBUTION

(of thl=

page)

For

sale

National

Technical

B is stability;

Section

F is

H is on statistics. NASA

TM X-60042o

STATEMENT

-- Unlimited

21,

NO.

OF

PAGES

839 by

of methods Section

of structural

mechanics;

Informatlnn

_ervice,_pringfleld,

22.

in

estimates

inelastic

introduction

interaction

Unclassified

Unclassified MSFC-

20.

for

and fracture

Unclassified

19.

general

A is a general

stresses,

Section

Section

supersede

are

methods

themselves.

combined

G is on rotating

that

of industry-wide

enough

techniques

is as follows:

analysis;

a compilation

out by hand,

analysis

methods

manual

stresses;

presents

of methods

on loads,

volumes

III)

It provides

of strength

Section

and

can be carried

of the

sections

on composites;

that

expected.

not only

and includes

I, II,

encountered,

strength

An overview used

(Volumes

structures

It serves

source

:L

Engineering

ABSTRACT

This

17.

14. SPONSORINGAGENCYCODE

20546

PRICE

NTIS Virginia

221_1

STRUCTURES

MANUAL

FOREWORD

fThis Branch vide in

manual

to a

ready

this

is

and

use

of

sidered

It oped

in

recognized the

This to

body

and is Table

as

as

Many zation

on

cataloged Utilization

the

the

included

of

are all

the

manual;

is requested to:

Chief, Strength Analytical

Marshall

15,

1970

pub-

to

are

have

of

con-

the

range

of

Contents to

of

content

added. as the a

New demand

completed and

been in

Univac in

the

comments

are be

not

develmaterial

topics not arises. avail-

supplements

are

for

1108,

computerized

Language or

IBM

concerning

this

manual

Section

Laboratory and Flight

Space

Center,

Administration Alabama

and

Computer

Division

Aeronautics

for

7094

Analysis

Branch

Mechanics

Space

or

section

adapted Fortran

Structural

Requirements

Analysis

is

revisions

VIII,

Structural

are

remain index

written

problems

any

analysis

necessary.

included

Executive

Table

make

addition,

become

that

the sections

new material will be treated

In

and

of they

possible.

alphabetical

ii

August

pro-

universities, book

method

wherever

in

utilized

they

the

procedures

some an

been

Astronautics National

to

contained

by text

wherever

subjects

programs

MSFC

the

indicated

that

with example Manual.

It

either

are

methods

These

Analysis and

information

industries,

clarify

possible. as

of

the

to

updated as of Contents has

soon

utilization.

directed

of

incorporated

Strength

published

tables

However,

arrangement be

data

future.

is provided listed in the

the

analysis

material

aircraft

Limitations

in

of

structural

Generally,

and

and

the

is the

of

agencies.

curves of

personnel

of

missile

necessary.

present

the

data.

problems

the

applicability

able

for

government

Illustrative the

to

methods

a condensation

journals,

lishers,

issued

uniform

reference

manual

scientific

is

provide

35812

be

utiliare

SECTIONAI STRESSAND STRAIN

TABLE

OF

CONTENTS

Page At.0.0

f

Stress

i.I.0

and

Strain

Mechanical

Properties

I.I. 1

Stress-Strain

1.1. 2

Other

I.I. 3

Strain-Time

1.1. 4 i.i. 5

Temperature Hardness

1.2.0

Specification

1.3. 3

Equations

1.3. 4

Distribution

1.3. 5

Conditions

1.3. 6

Stress

1.3. 7 Use Theories 1.4. i

Elastic Interaction

Mechanics

at

of Strains

Curves

of Materials

a Point

7 12 ......... ........

...................

in a Body

21

...................

23

......................

25

............................. from the Theory ............................. .............................. ............................

A1

iii

17 18 i8 19

........................

of Compatibility

Failure

3 5

of the Theory of Elasticity and Stresses ..................

of Stress

of Equations o¢ Failure

1.4. 2

of the

of Equilibrium

Functions

i

.......................

........................... Tables .....................

Applications for Forces

1.3. 2

1

..........................

Effects Conversion Theory

Elementary 1.3. 1 Notations

...................

.........................

Properties Diagram

Elementary

1

of Materials

Diagram

Material

1.3.0

1.4.0

.................................

27 of Elasticity

.........

28 34 35 36

Section A I March

i, 1965

Page I AI. 0.0

Stress and Strain

The relationship between stress and strain and other material properties, are used throughout this manual, are presented in this section. A brief

which

introduction to the theory sented in this section. AI. I. 0

Mechanical

of elasticity

for

important detailed

mechanical discussion

ber of well mechanical

known texts on the subject. properties of most aerospace

(reference

1).

of these

values

sults of one type or another. is the stress-strain diagram. the next subsection. AI. I. i

is also

pre-

properties may be found

of materials in any one

The numerical values materials are given

are

One of the A typical

obtained

from

most common stress-strain

is given of a num-

of the various in MIL-HDBK-5

a plotted

set

of test

sets of these plotted diagram is discussed

resets in

Stress-Strain Diagram

Some strain Figure

applications

Properties of Materials

A brief account of the in this subsection; a more

Many

elementary

of the

more

useful

diagram. A typical AI. I. i-I.

properties stress-strain

of materials curve

for

are

obtained

aerospace

from

metals

a stress-

is shown

The curve in Figure A1.1.1-1 is composed of two regions; the straight line portion up to the proportional limit where the stress varies linearly with strain, and the remaining part where the stress is not proportional to strain. In this manual, elastic. employed stress.

stresses below the ultimate However, a correction (or in certain types of analysis

Commonly briefly

in the E

used following

properties

shown

tensile stress (Ftu) plasticity reduction) for stresses above

on a stress-strain

are considered to be factor is sometimes the proportional limit

curve

are

described

paragraphs: Modulus of elasticity; average stresses below the proportional E = tan 0

.ratio of stress to strain for limit. In Figure A1.1.1-1

in

Section

A1

March 1, Page 2 A1. I. 1

Stress-Strain

_--Elastic,

_'_

(Cont'd)

_

Plastic,

ep

|

ee

(pfsi)

Diagram

_/

i965

Ftu

_eld

Point

I

I

eu

-4

e Fracture

e (inches/inch)

Figure

E

S

AI. 1.1-1

A Typical

Secant the

Et

Stress-Strain

modulus;

proportional range.

Tangent

modulus;

range.

ratio

of stress

limit;

portional tan 01

at any point;

Diagram

In Figure

reduces

In FigureAl.

to strain

reduces

slope

AI.

of the

above

to E in the prot. 1-1

Es =

stress-strain

l.i-i

proportional df E t - de - tan

curve

to E in the

02

Section

-f..

A1

March Page AI. i. I

Stress-Strain Diagram

Fry or

1965

(Cont'd) Tensile materials

Fcy

1, 3

or compressive do not exhibit

the yield method.

stress This

yield stress; since many a definite yield point,

is determined by the entails the construction

. 2% offset of a

straight line with a slope E passing through a point of zero stress and a strain of. 002 in./in. The intersection of the stress-strain curve and

Ftp or

the

constructed

straight

tude

of the

stress.

yield

Proportional

Fcp

Ftu

F

limit

sion;

the

stress

vary

linearly

line

stress

Ultimate

tensile

reached

in tensile

the

E

The

U

strain

magni-

or compres-

stress

ceases

to

strain.

stress; tests

the

maximum

of standard

Ultimate compressive stress; less governed by instability.

CU

the

in tension

at which with

defines

corresponding

stress specimens.

taken

as

Ftu

un-

to Ftu.

Elastic strain; see Figure AI. I. I-i.

E e

plastic strain; see Figure Ai. I. i-i.

E

P efracture

(% elongation)

Fracture determined

strain; gage

percent length

elongation associated

failures, and is a relative of the material. Ai. i.2

Other Material Properties

The in stress

definition analysis

of various other material properties work is given in this subsection.

in a prewith tensile

indication

and

terminology

of ductility

used

SectionAi March i, 1965 Page 4 A1.1.2

Other

Fbry'

Material

Fbru

Properties

(Cont'd) Yield and ultimate bearing stress; in a manner similar to those for compression. plotted

A load-deformation

where

the

deformation

the hole diameter. fined by an offset bearing stress F F

shear

Proportional

sp

is the

For

is

change

in

actual

failing

stress. limit

in shear;

usually

to 0. 577 times the proportional for ductile materials. Poisson's to axial

curve is the

Bearing yield (Fbry) is deof 2% of the hole diameter;

ultimate (Fbru} divided by 1. t5.

Ultimate

SU

determined tension and

taken

limit

equal

in tension

ratio; the ratio of transverse strain strain in a tension or compression test.

materials

stressed

in the

elastic

range,

v

may be taken as a constant but for inelastic strains v becomes a function of axial strain. V

P E

G-

2(I + v)

Plastic

Poisson's

Vp may

be taken

Modulus elasticity

ratio; as

of rigidity for

pure

unless

otherwise

or shearing

modulus

0.5.

shear

in isotropic

Isotropic

Elastic

properties

are

Anisotropic

Elastic

properties

differ

Orthotropic

Distinct

material

properties

pendicular

planes.

stated,

the

same

materials.

in all

in different

of

directions. directions.

in mutually

per-

Section Ai March I, 1965 Page5 A1.1.3

Strain-Time

The

behavior

Diagram of a structural

material

is dependent

on the

duration

of loading.

This behavior is exhibited with the aid of a strain-time diagram such as that shown in Figure A1. i. 3-1. This diagram consists of regions that are dependent

Strain

/,/Fracture///_J_

Creep

I

Strain Elastic

I

Limit

Elastic

(no

fracture)

Recovery

Curve

= Elastic

Strain

f

Constant

v

Loading Loading

A1.1.3-t

upon the four loading conditions as loading conditions are as follows: Loading

2.

Constant

Strain

P_rmanent _ Time Set

Recovery

Unloading

Figure

1.

_

loading

Strain-Time

indicated

Diagram

on the time

coordinate.

These

SectionA 1 March

1, 1965

Page AI.

1.3

Strain-Time

Diagram

3.

Unloading

4.

Recovery

The weeks

(no

interval

tively short time curve

(Cont'd)

load)

of time

or months.

6

when

Whereas

the

the

(usually seconds can be represented

load

time

is held involved

constant

is usually

in loading

or minutes) such that by a straight vertical

and

measured

unloading

in

is rela-

the corresponding line.

strain-

The following discussion of the diagram will be confined to generalities to the complexity of the phenomena of creep and fracture. A more detailed cussion on this subject is presented in reference 5.

due dis-

The condition referred to as "loading" represents the strain due to a load which is applied over a short interval of time.

This strain may

vary from zero

to the strain at fracture (_fracture - See Figure AI. I.l-l) depending upon the material and loading. During strain-time possible below.

the second loading condition, curve depends on the initial

strain-time

a. experienced b. becomes

This

action

(Figure

t,

initial

In curve for

load

curves

the

entire

In curve

and

is indicative

In curve

inelastic

deformation

complex

deformations

A1.1.3-1)

strain

initial

then

This

strain

remains which

that

is elastic

interval.

of slip

sulting from the shifting most favorably oriented stress. c.

time

2, the

constant

slip until a steady which is generally

the

where the load is held constant, strain for a particular material.

curve

typifies for

for

is characterized

(slip) of adjacent crystalline with respect to the direction

3, there

is a continuous

increase

state condition is attained. the result of a combined within by slip

the

unordered

and

result

are

discussed

and no additional

increases constant

could

strain

elastic

a short

the

intercrystalline of the

after

of the

by a permanent

set

the period. re-

structures along planes of the principal shearing

in strain

after

This curve is indicative effect of the predominantly

fragmentation

is

action.

period

remainder

the The

boundaries ordered

crystalline

the

initial

of creep viscous and

the domains.

Section A 1 March

I, 1965

Page 7 Ai. i.3

Strain-Time d.

ference

Curve

from

period

4 is also

curve

in fracture.

fracture

is indicated

During

the elastic strain "elastic recovery.

the

creep

may

by the

unloading,

(Cont'd) a combination

3 is that

This

and

Diagram

the

reduction

recovery

last

condition

period.

The

_f

after

In this

plastics)

effect.

dicated family

at any

shaded

area

period,

is true that

of the The

to its curves

to be discussed

ThisI

height

creep. until

time

of curves

only

dif-

material

the

fails

constant

load

A 1.1.3-1. l,

2 and

3 is equal

upper

bound

strain-time

diagram

some

of the

strain

indicated

for

many

viscoelastic

shaded

by the solid horizontal of possible strain-time

configuration immediately 3 as there will be some

particularly

do not show

lower

initial 2 and

on the

real area

creep,

only

in Figure

to

strain

materials

(such

is called

the set

elastic

maximum

possible

The lower confined

bound could be any one of the within the lower shaded area.

by a line

permanent

strains.

is the

the permanent If slip action

that

mechanical properties of a material are usually This effect will be discussed in general terms

specific

information,

and

is in-

set curve is negligible,

approaches

The perature.

zero

asymp-

Effects

see

the applicable

temperatures

below

Ductility

chapter room

is usually

example

of aluminum

steels behave in a similar ture magnitudes.

alloys

for

the

is given

manner

effect

affected in this

in reference

temperature and The

in Figures

but generally

A1.1.4-1 are

less

the

the

notch

opposite

of temperature

by its temsection. For

1.

increase

decreased

of the metal may become of primary importance. true for temperatures above room temperature. A representative

the

recoverable

A 1.1.3-1

Temperature

of metals.

concerns

as inelastic

delayed

A1.1.4

In general,

after residual

line. curves

this limiting curve would be represented totically with increasing time.

properties

the

during

in Figure

The limiting curve of the lower bound would approach due to slip as indicated by the horizontal dashed line.

properties

The

incurred during loading. This reduction is referred to as the " It can be seen in Figure Ai. I. 3-I that in the case of curve

is recoverable. as flexible

and

continues

place

in strain

I the structural member will return unloading. This is not the case for strain. The

action

take

upper

of slip

sensitivity

is generally

on the through sensitive

strength

mechanical 4.

Most

to tempera-

Section A I March

i, 1965

Page 8 A1.1.4

Temperature

Effects

(Cont'd)

120 \ \\\\

100 Q) \',,

2 cD

QJ

8O /_

hr

O O

/100 _10,000

60.

hr hr

¢J

;h

40 _

2O

0 -400

-200

0

200

Temperature, Figure AI. I.4-I

400

600

°F

Effects of Temperature

on the Ultimate Tensile

Strength (Ftu) of 7079 Aluminum Ref. I)

Alloy (from

800

Section AI March

i, 1965

Page 9 AI. 1.4

Temperature

Effects (Cont'd)

140

120

\ \

100

f_

-_

hr

8O I 100 hr

8 _.10,000

hr

>_ 60

,t0

20

0

-400

-200

0

200 Temperature,

Figure A1.1.4-2

Effects Strength Ref. 1)

400

800

°F

of Temperature (Fty)

600

of 7079

on the Tensile Aluminum

Alloy

Yield (from

Section

A1

March Page At.i.4

Temperature

Effects

1,

1965

t0

(Cont'd)

140

120

100

O O

a_

I

\

80

60

2O

0 -400

0

-200

200 Temperature,

Figure

AI.

I. 4-3

Effect Modulus Ref. i}

and

600

800

°F

of Temperature (E

400

Ec}

on the of 7079

Tensile Aluminum

and

Compressive Alloy

(from

Section

A 1

March Page

AI.

1.4

Temperature

Effects

1,

1965

11

(Cont'd)

100

8O

/

6O O

O

40

2O

,

0

I00

200

300

Temperature,

Figure

AI. i. 4-4

Effect 7079-T6

of Temperature Aluminum

400

5C

°F on Alloy

the (from

Elongation Ref.

of 1)

600

Section March Page Al.

I. 5

Hardness

A table

for

Conversion converting

AI 1, 1965 12

Table

hardness

numbers

to ultimate

tensile

strength

values

is presented in this section. In this table, the ultimate strength values are the range, 50 to 304 ksi. The corresponding hardness number is given for of three hardness machines; namely, the Vtckers, Brinell and the applicable scale(s) This

of the

Rockwell

table

is given

materials-property whenever necessary.

in each

machine. In the

haaktbook

remainder should

Tensile

Vickers-

Brinell

Strength

Firth Diamond

3000 kg 10ram Stl

of this

be consulted

section. for

The

appropriate

additional

information

Rockwell A Scale

B Scale

C Scale

6O kg

100 kg Dia Stl

150 kg 120 deg Diamond

Ball

Cone

Ball

ksi

Hardness Number

Hardness Number

120 deg Diamond Cone

1/16

in.

50

104

92

58

mm

52

108

96

61

1B

54

112

I00

64

_W

56

116

104

66

58

120

108

68

60

125

I13

70

62

129

ii7

72

64

135

122

74

Table AI_'I.5-1

,&l

Hardness

Conversion

Table

_m

Section A I March

i, 1965

Page 13

AI. I. 5

Hardness

Conversion

Table (Cont'd)

Tensile

Vickers-

Brinell

Strength

Firth

3000 kg I0m m Stl Ball

D Jam ond

ksi

Hardness Num be r

Hardness Number

Rockwell A Scale 60 kg 120 deg D Jam ond Cone

B Scale

C Scale

I00 kg 1/16 in. Dia Stl Ball

i50 kg 120 deg Dmmond

66

139

127

76

68

143

i31

77.5

70

i49

136

79

72

153

140

80.5

74

157

145

82

76

162

150

83

78

167

154

51

84.5

8O

171

158

52

85.5

82

177

162

53

87

83

179

165

53.5

87.5

85

186

171

54

89

87

189

174

55

90

89

196

180

56

91

Table AI. I. 5-i

Hardness

Conversion

Table (Cont'd)

Cone

Section March Page AI. 1.5

Hardness

Conversion

Table

Tensile

Vickers-

Brinell

Strength

Firth D Jam ond

3000 kg 10m m Stl

At 1,

1965

14

(Cont'd)

Rockwell [

A Scale

B Scale

C Scale

60 kg 120 deg Diamond

100 kg

150 kg 120 deg D iam ond

Ball

ksi

Hardness

Hardness

Number

Number

Cone

1/16 in. Dia

Stl

Ball

Cone

9t

203

186

56.5

92. 5

93

207

190

57

93.5

w--

95

211

193

57

94

--m

97

215

t97

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

2O

110

245

230

61

99, 5

21

t12

250

235

61.5

100

22

i15

255

241

62

101

23

118

261

247

62.5

i01.5

24

120

267

253

63

102

25

Table

A 1.1.5-1

Hardness

Conversion

Table

(Cont'd)

A1

Section

1,

March Page

AI. 1.5

Tensile Strength

Hardness

Conversion

Table

Vickers-

Brinell

F irth

3000 kg 10ram Stl

D iam

ond

1965

15

(Cont'd)

Rockwell A Scale

B Scale

C Scale

Ball 60 kg ksi

Hardness

Hardness

Num

Number

be r

120

deg"

Diamond C one

100

kg

1/16 Dia

in. Stl

Ball

150

kg

120 deg Diamond Cone 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

33

150

339

318

67.5

34

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

7O

39

Table

A I. i. 5-i

Hardness

Conversion

103

Table

(Cont'd)

Section

Ai

March Page Ai.

i.5

Tensile Strength

Hardness

VickersFirth Diamond

Conversion

Table

l,

i965

16

(Cont'd)

Rockwell

Brinell 3000 kg 10ram Stl

A Scale

B Scale

C Scale

60 kg i20 deg Diamond

i00 kg I/t6 in, Dia Stl

120 deg Diamond

Ball

ksi

Hardness Number

Hardness Number

Cone

Ball

150 kg

Cone

181

396

377

70.5

40

188

406

387

71

41

194

417

398

71.5

42

201

428

408

72

43

208

440

419

72.5

44

215

452

430

73

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

5O

264

539

5OO

76.5

51

273

556

512

77

52

283

573

524

77.5

53

Table Ai. i.5-i

Hardness

Conversion

,45

Table (Cont'd)

Section

A1

March

1, 1965

Page AI.

1.5

Hardness

Conversion

Table

Tensile

Vickers-

Brinell

Strength

Firth D ia m ond

3OOO kg 10mm Stl Ball

Hardness Num be r

Hardness

(Cont'd)

Rockwell .m

A Scale

B Scale

6O kg ksi

17

100 kg 1/16 in. Dia Stl

120 deg Diamond

Num be r

C Scale

Cone

150 kg 120 deg Diamond Cone

Ball

294

592

536

78

54

304

611

548

78.5

55

Table

A1.2.0

Elementary In the

strain

A1.1.5-1

Theory

elementary

is generally

Hardness

of the

theory

Conversion

Mechanics

of mechanics

assumed.

This

state

Table

{Concluded)

of Materials of materials,

of strain

a uni-axial

state

is characterized

by the

of simpli-

fied form of Hooke's law; namely f = E _, where • is the unit strain in the direction of the unit stress f, and E is the Modulus of Elasticity. The strains in the perpendicular directions { Poisson's ratio effect) are neglected. This is generally justified in most elementary of mechanics of materials. generally placements magnitude independent

and practical applications In these applications, the

subjected to a uni-axial state are of secondary importance. of each of the

Frequently

of a set Poisson's

in design,

of stress Also,

of bi-axial stresses ratio effect. there

are

this

in which

occurs)

(or tri-axial) the magnitude

mary theory

This type of application must be generally A brief account on the use of the theory

elementary

applications

is given

in the

next

dependent upon the and displacements

subsection.

and

disthe

is generally

the magnitude

of a set of bi-axial ratio effect; and/or importance. of elasticity.

are strains

in the theory members are

and/or the strains in these applications, (when

applications

stresses of the

considered structural

of each

Poisson's are of pri-

analyzed by the of elasticity for

Section A 1 March Page AI.

3.0

Elementary

The

difference

between

is that

rio simplifying

elasticity latter.

Because

distribution of Hooke's noted that

of this,

Some

the

l The

in the

of the

body

following

for

stresses

mechanics

is made

necessary

deviates

subsections

Forces

acting

of stress,

shearing

indicating indicating

stresses

the direction the direction

subscripts for

fll =f

the

and

on the namely

In Figure AI. 3. l-i parallel to the coordinate

notation

of Elasticity

of ordinary

assumption

it becomes

physical

Notation

components three

Theory

method

and

concerning

to take

from

are

but are applicable to problems containing the third dimension.

A1.3.

like

of the

1965

the

into account

the

theory

of

strains the

in the

complete

of the strains in the body and to assume a more general statement law in expressing the relation between stresses and strains. It is the stresses calculated by both methods are only approximate since

the material both methods.

field terms

Applications

1, 18

the

written

ideal

for

a three

in two dimension

assumed

by

dimensional

simply

stress

by neglecting

all

Stresses side

the

of a cubic three

normal

element

can

stresses

be described fll,

by six

f22, f33, and

the

fl2 = f21, f13,= f3t, f23 = f32. shearing axis.

stresses are Two subscript

resolved numbers

into two components are used, the first

normal to the plane under consideration of the component of the stress. Normal

and

positive

x-y

coordinate

directions system

are

as shown

in the

figure.

and the stresses

second have

An analogous

is: xa

X

f22

f22 = fy f12 = f

material

!

S

fs3f"'-

xj

I" Figure

AL. 3. 1-1

Representation an Element

of Stresses of a Body

on J

Section

A1

March

F_

1, 1965

Page A1.3. f

1

Surface

Notation

for

Forces

and

Stresses

19

(Cont'd)

forces

Forces

distributed

body

on another,

Body

forces Body

forces

as gravitational in motion. A1.3.2

over

the

or hydrostatic

are

forces

forces,

Specification

that

are

magnetic

of Stress

equations of statics. be neglected since

of the

body,

such

are

called

surface

distributed

forces,

over

or inertia

the

as pressure

volume

forces

of one

forces.

of a body,

in the

case

in Figure A1.3. 1-2 are known for any given inclined plane through this point can be calculated

Body forces, such as weight of the they are of higher order than surface

element, forces.

X2

C N

x_

x_ Figure

AI.

3. I-2

such

of a body

at a Point

If the components of stress point, the stress acting on any from the generally

surface pressure,

An Element

Used

in Specifying

Stress

at a Point

can

SectionA I March I, 1965 Page Ai. 3.2

20

Specification of Stress at a Point (Cont'd)

If A denotes

the

area

of the

inclined

face

BCD of the tetrahedron

in Figure

AI, 3. t-2, then the areas of the three faces are obtained by projectin_A on the three coordinate planes. Letting N be the stress normal to the plane BCD, the three components of stress acting parallel to the coordinate axes, are denoted by NI,

N 2, and N 3.

The components

ordinates X|, Xz, X 3 are AN_, AN2, relationship can be written as:

cos (NI) = k,

cos (N2) = m,

of force and

acting

in the direction

AN 3 respectively.

of the co-

Another

useful

(1)

cos (N3) = n

and the areas of the other faces are Ak, Am,

An.

The equations of equilibrium of the tetrahedron can then be written as:

NI = fil k + f12 m + f13 n

(2)

N2 = fi2 k + f22 m + f32 n Na -_ fl3 k + f23 m + f33 n

mined

The principal stresses for a given set of stress by the solution of the following cubic equation:

components

can be deter-

fp3 _ (fli+ f22+ f33)fp2 + (fllf22÷ f22f33+ fllf33- f232

(3) - f132 - f122) fp - (fli

The

three

roots

of this

f22 f33 + 2f23 f13 f12 - fll f232 - f22 f132 - f33 f122) = 0

equation

The three corresponding sets can be obtained by substituting stress) into Equations 3 and

give

the values

of the

three

principal

of direction cosines for the three principal each of these stresses (one set for each using the relation k 2 + m 2 + n 2 = i.

stresses. plan_s principal

Section

A1

March Page

A1.3.2

Specification

(fp - fit)

k

f12 k + (f f13k-

The obtained

of Stress

fl2 m

-

at a Point

1,

1965

21

(Cont'd)

ft3 n = 0

-

(4)

- f22) m - f23 n = 0

f23m

+ (fp-

shearing by:

t fl2 = + _-(fp!

f33) n=

stresses

0

associated

! _ fp2) , fl3 = + 2-(fpl

with

the

three

principal

stresses

can

be

- fp3),

(5) ! f23 = + _- (fp2 - fp3)

where

the

stresses fp2' and

superscript and the fP3"

notation

stresses

associated

The maximum shearing the largest and the smallest between these two principal AI. 3.3

and the

to distinguish with

the

between

principal

normal

the

applied

stresses

shearing fpl,

stress acts on the plane bisecting the angle between principal stresses and is equal to half the difference stresses.

Equations of Equilibrium

Since in the within

is used

no simplifying

assumption

is permitted

as to the distribution

of strain

theory of elasticity, the equilibrium and the continuity of each element the body must be considered. These considerations are discussed in this subsequent

subsections.

Let the components of the specific body force be denoted by X1, X2, X3, then the equation of equilibrium in a given direction is obtained by summing all the forces in that direction and proceeding to the limit. The resulting differential equations of equilibrium for three dimensions are:

Section

A1

March Page

AI. 3.3

afli 8x i

Equations

+_

afl2 8x2

_+ 8x 2

axl

8f33 --+ 8x3

_

afi3

+

+

of Equilibrium

8f13

8x 3

+Xi

1965

(Cont'd)

= 0

+X2=

8f23 +--+X3= _)x2

0

(6)

0

These equations must be satisfied internal stresses must be in equilibrium of the body. considering

i, 22

These conditions the stresses acting

at all points throughout the body. with the external forces on the

of equilibrium at the on Figure AI. 3.3-1.

boundary

are

----_

The surface

obtained

_x1

%

Figure

Ai.

3.3-1

An Element

Used

in Deriving

the

Equations

of Equilibrium

by

Section A i March

I, 1965

Page 23 AI.

3. 3

Equations

By use

Xl

of Equilibrium

of Equations

1 and

(Cont'd) summing

forces

the

boundary

equations

are:

= fll k + f12 m +f13 n (7)

X2 = f22 m + f23 n + fl2 k X3 = f33 n+ft3k+f23

in which of the of the

k,

m,

n are

m

the

direction

cosines

body at the point under consideration surface forces per unit area.

of the and

external

normal

X1, X2, X 3 are

to the the

surface

components

The Equations 6 and 7 in terms of the six components of stress, fll, f22, f33, f12, f13, f23 are statically indeterminate. Consideration of the elastic deformations is necessary to complete the description of the stressed body. This is done

by considering

A1.3.4

have

the

Distribution

elastic

deformations

of Strains

in a Body

of the

The relations between the components of stress been established experimentally and are known

deformations normal strain

where superposition is written as:

1 el = _ [fll

applies,

Hooke's

body.

and the components of strain as Hooke's law. For small law

in three

dimensions

for

- v (f22 ÷ f33) ]

1 £2 = E- [f22 - v (fll

+ f33) ]

1 e3 = E- [f33 - v (fil

+ f22) ]

(8)

Section

A1

March Page

A1, 3.4 and

for

Distribution shearing

of Strains

2(I+ v)

fi2 = G

Tts =

2(t E + v)

ft3 = G

These

(9)

f23 G

f23 -

of strains

of displacements.

ment dxl, placement point

+ v)

six components

components

(Cont'd)

+_.,v,

E

E

i965

strain

• l_ =

T_-a = 2(i

in a Body

1, 24

can be expressed

By considering

in terms

the deformation

of the

three

of a small

ele-

dx2, dx 3 of an elastic body with u, v, w as the components of the disof the point 0. The displacement in the x 1 - direction of an adjacent

A on the x 1 axis

is

au

u + _xl

due

to the

dxl

increase

(au/axl)dx

x l, It follows that In the same manner directions The AI. 3.4-i x I x 3 and similarly. The

are

the unit elongation it can be shown

given

distortion

by av/ax2 of the

to be av/ax x 2 x 3.

six

The

ax 1 ,

angle

1 + au/ax shearing

components

au el -

1 of the

from 2.

u with increase

of the

coordinate

at poiqt 0 in the x 1 direction is au/ax 1. that the unit elongations in the x 2 - and x 3 -

and aw/ax

3 respectively.

AOB

This

strains

of strains

to A'O' B' can be seen

is the between

in terms

shearing the

of the

strain

other

three

from

between

two planes

displacements

are

Figure the

planes

obtained

are:

aw

av _2 - ax2

function

'

_s =_.. (10)

au

av

Ylg- = 2ax-- + ax t

au "/i3

Dw

ax 3 + ax I

av _23

aw

ax 3 + ax 2

Section

Al

March Page

At.

3.4

Distribution

of Strains

in a Body

X_

_

I,

1965

25

(Cont'd)

u +iL_. 8x2

dx 2

i I

T-

At

1 0 dx 2

+ a_v dxt Ox!

v

0

_

.

_ Xl

4- J _x 1 dxt

Figure

A1.3.5

A1.3.4-1

Conditions

Distortions

Due

to Normal

to Define

Strains

in Terms

and

Shearing

Stresses

Used

of Displacements

of Compatibility

can

The conditions of compatibility, that assure continuity of the structure, be satisfied by obtaining the relationship between the strains in Equations

The

relationship

can

be obtained

by purely

Differentiating Q twice with respect Ti2 once with respect to x t and once with of

(1 and

ax]

e2 is found

+ ax_

to be identical

= axtax2

to the

mathematical

manipulation

10.

as follows:

to. x2; e2 twice with respect to xt; and respect x 2. The sum of the derivatives derivative

of Tt2.

Therefore,

Section A1 March 1, i965 Page 26 AI. 3.5

Conditions of .Compatibility (Cont'd)

Two of the

more

relationships

subscripts

Another as follows:

1,

set

of the

of equations

Differentiate

same

kind

can

be obtained

by cyclic

interchange

2, 3.

e 1 once

can be found

with

respect

by further

to x I and

mathematical

once

with

manipulation

respect

to xs;

_/12

once with respect to x t and once with respect to x3; _/13 once with respect to x l and once with respect to x2; and _'23 twice with respect to x I. It then follows that

8x_0x 3

axl0x3

Two additional scripts

the

as

Oxiax2

relationships

0x l"

can

be found

by the

cyclic

interchange

of sub-

before.

The six differential relations equations of compatibility and

8x_

between the components are given below.

= axlax 2 '

8x28x 3

ox2ax3 '

axlax 3

8x i\

ax 3

ax2

of strain

are

called

8x i j'

(II) 8x{

the

These strains

ax{

equations of compatibility may in Equations 11 are expressed

law (Equations for substitution,

8 and 9). we have

Differentiating

be stated in terms each

in terms of the stresses if of the stresses by Hooke's

of Equations

8 and

9 as required

Section

A 1

March

i,

Page A1.3.5

Conditions

of Compatibility

(Cont'd)

a20

a20

(i

+,)

_72fil

(1

+ v)

V 2 f22 + 0x--_ = 0

+ _x I

=

( I + p)

0

-

V 2 f23 +

020 ,

(1

+ v)

V 2 fl3

+ 0xlDx3

(1

+ v)

V 2 fl2

020 + -OxlOx

020 + v)

0

Dx20x 3

_20

(1

1965

27

V 2 f33 + --Ox2

-- 0

- 0

-

(t2)

0

2

where:

V2

D2

_)2

_2

and

0 = fll

For

+ f22 + f33

most

system

cases

components

equations

A1.3.6

Stress

It has the

satisfied

and

are

11 or

linear

12 are

The

discussed

use

and

superposition

sufficient

applies,

to determine

of stress

functions

the

the

to aid

stress

in the

solution

below.

Functions

shown

in the

(Equations not

(Equations

1t)

must

the

body.

a distribution

also The

element mean

stresses

throughout

sections

ensure

of every

necessarily

boundary

previous

6)

equilibrium

does the

strains

7,

ambiguity. are

been

of equilibrium serves

6,

without

of these

since

where

of Equations

must be

satisfied

problem

in the

that also

that

the

the

body.

The

distribution

be satisfied. to ensure

is then

differential

of stress

to find

an

fact

that

are

compatibility

proper expression

strain

that

pre-

these

of stresses The

the

equations

in a body

are correct equations

distribution that

satisfies

all

Section

A1,

March Page A1.3.6

Stress

Functions

these conditions. function that meets will deal

only

1965

(Conttdl

The usual procedure this requirement.

with problems

of the body will also

1, 28

is to introduce a function For the sake of simplicity,

in two dimensions.

The stresses

called tliis

a stress section

due to the

weight

be neglected.

In 1862,G. B. Airy introduced a stress function (_b (xl, x2) ) which is an expression that satisfies both Equations 6 and II (in two dimension) when the stresses are described by:

fll

-

,

f22 = ax I

,

f12 = -

_xl_x2

By operating on Equations 13 and substitutinginto Equations il, we find that the stress function _b must satisfy the equation

+ 2

Of the (7)

+

= V4qb = 0

Thus the solution of a two-dimensional biharmonic equation (Equation t4)

of the

At. 3.7

problem reduces which satisfies the

to finding a solution boundary conditions

problem. Use

Proficiency

of Eqtmtions in the use

from

the

Theory

of stress

It is not unusual to find an expression to determine what problem it solves.

use

(I4)

The following problem of stress functions.

is presented

of Elasticity

functions that

is gained

satisfies

to illustrate

mainly

Equation

the

basic

by experience. i4 first

and

procedure

then

in the

try

Section Ai March 1, 1965 Page 29 At. 3.7

Use

Statement

for

of Equations

of the

Determine a cantilever

shown

the

Theory

of Elasticity

(Cont'd)

problem: the stress function beam of rectangular

in Figure

and compare mechanics.

from

A1, 3.7-1.

with the

that corresponds cross section

From

maximum

this

stress

flexure

p/unit

to the boundary of unit width and

function

stresses

as

determine obtained

the

by the

conditions loaded as stresses method

of

length V ° = -p L _

_

ii

_

_

_

_

.__._._.p,

V-

Mo=-

Xl

2

L-

X2

Figure

At.

3, 7-i

Sample

Problem

Solution: Assume

that

the

stress

function

is

_b = ax2 s + bx23xt 2 + cx23 + dx2x 2 + ex 2

Operate

on

_ to satisfy

V4_b = (5-4"3.2)

Equation

ax 2 + 2( 3.2.2

24x 2 (5a+b) from

which

a

= - b/5

14

bx2)

= 0

= 0

(a)

Section March Page A1.3.7

Use Since

condition

of Equations

Equation

Figure

the

Theory

of Elasticity

14 can now be satisfied

to satisfy

From

from

is the boundary Ai.

3.7-1

by letting

A1 i,

1965

30

(Cont'd) a = -

b/5,

the only

other

conditions.

the boundary

i.

f22 = -P

at

x 2 = - h/2

2.

f22 = 0

at

x 2 = h/2

conditions

are

from

ZF=0

as follows:

h/2 3.

f

fl2dx2

= -pL

at

x1 = L

-h/2

h/2 4.

f

fltx2dx2

= -pL2/2

at

x1 = L

from

ZM

= 0

-h/2

5.

fl2 = 0

From

fli

Equation

=

x 2 = h/2

i3

= 20ax3

f22 -

+ 6bxl2x2 + 6cx2

= 2bx_ + 2hx 2 + 2e

f12 = - OxiOx2

Using

at

boundary

f22 = -P

= -

6bx, , condition 2bh 3 8

2dh 2

(b)

2hx, I

+ 2e

(c)

Section

-f-

March Page AI.

3.7

_Use of Equations

from

boundary

f22 = 0 -

adding

(c)

and

condition

2bh 3 2dh 8 + 2

or

of Elasticity

+ 2e

(Cont'd)

(d)

condition

(e)

3

h/2 f12 dx2 = _hf/2 [- 6bx22xl - 2hxl} dx2

=2 [ -_ 6 bLx23-2hLx2]_/2

(f)

= -pL

bh 3 or

from

1965

2

e = -p/4

boundary

h/2 -h/2

Theory

1, 31

(d)

4e = -p

from

fro m the

A1

+ 2dh -_p

2

boundary

condition

4

h/2 - hf/ 2 [20ax24 + 6bx}x22 + 6cx22] dx 2

=2 [

2___a x_ + _xl2x3+ 56 cx 31 _ 6b

h/2 0

ah 5 4

bL2h 3 +_+

ch 3 _ _ pL2/2 2

Section A 1 March Page Al.

3.7

Use

of Ec_uations

substituting

c = -pL2

from

fi2

Equation

the

Theory

a and solving

of Elasticity

bh_xi

condition

- 2dxi

1965

(Cont'd)

for c

- b ( L2h a - hs/iO) ha

boundary

= _

from

f, 32

(g)

5

= 0

3 =-x i ( _ bh 2 # 2d )

or

(h)

Solving

d = 3p

Equations

and

f and h simultaneously

b = -p/h 3

we get

(i)

Substituting b = - p/h 3 into Equation g

(J)

Section

A1

March

1,

Page Ai.

3.7 The

Use

of Equations

stress

from

function

¢) = -px 2 (x_/h

can

3-

the

Theory

of Elasticity

now be written

3x2/4h

1965

33

(Cont'd)

as

+ 1/4) (k)

+(ph2/5)

and

(x25/h 5 - x23/2h 3)

the

stresses

as

fll

P = - 2I (X2

(see

X 2 +

f22 = - -P---(x23/3 2I

fi2

where

= -P-21 (x22x!

11 felasticity elementary

fmiechanics

The

h2 xz/10

b)

-

h2xy/4

(i)

2x_/3)

+ h3/12)

(m)

- h2xl/4)

(n)

I = h3/12

Comparison x 2 = - h/2

from

-

Equations

of maximum

= ph 4I

flexure

stresses

from

Equation

1 with

x 1 = L,

(o)

/L 2 - h_l

mechanics

Me I

_ pL 2 h 4I

difference

is then

felasticity 11

- f_i echanics

= - ph 60I3

(P)

p 5

(q)

Section A 1 March Page A1.4.0

Theories Several

load

theories

discussion

The

to aid

member.

in the

Each

of stresses

prediction

theory

of the

is based

or strains

normal

critical

on the

constitutes

the

assump-

limiting

stated

theories

books

such

in this

subsection.

of failure

can

as references

A more

be found 2 and

in most

3.

Theory

stress

begins

are

other

text

Stress

in a material

theories and

analysis

Normal

maximum

point

useful

on these

strength

Maximum

any

advanced

combination

of the more

elementary The

been

The margin of safety of a member is then predicted by comparing the strain, or combination of stress and strain with the correspondas determined from tests on the material.

Three detailed

have

on a structural

a specific

condition. the stress, ing factors

1965

of Failure

combination

tion that

1, 34

theory

of failure

only when

the

states

maximum

that

inelastic

principal

stress

action

at

at the

point reaches a value equal to the tensile (or compressive) yield strength of the material as found in a simple tension (or compression) test. The normal or shearing stresses that occur on other planes through the point are neglected. The Maximum

Shearing

Stress

Theory

The maximum shearing stress theory is based on the assumption that yielding begins when the maximum shear stress in the material becomes equal to the maximum shear stress at the yield point in a simple tension specimen. To apply it,

the

principal

stresses

are

first

determined,

with

the

then,

according

to Equation

5,

fimJax = I2( fpi _ fpj)

where i and respectively.

j are

associated

The

Maximum

point

The maximum energy of distortion in a body under any combination

energy

Energy

of distortion

of Distortion

per

unit

volume

maximum

and

minimum

principal

stresses

Theory theory states that inelastic action at any of stresses begins only when the strain absorbed

at the

point

is equal

to the

strain

Section March Page A1.4.

/

0

Theories

of Failure

and

l+v 3E

wl

-

the

strain

W

35

per unit volume at any point of uniaxial stress as occurs

The value of this uniaxial test is

maximum

in a bar stressed to in a simple tension

strain

energy

of distortion

F 2 YP

energy

--

[(fpl

.6E

of distortion

in the

general

case

is

- fp2 )2 + (fp2 - fP3)2 + (fpl - fp3)2]

where fpl, fP2' fp3 are the principal stresses and Fyp (For th_ case of a biaxial state of stress, fP3 = 0.) The

i965

(Cont'd)

energy of distortion absorbed the elastic limit under a state (or compression) test. as determined from the

A1 1,

condition

for yielding

is then,

is the

yield

point

stress.

w = w 1 or

(fPl - fp2)2+ (fp2 - fp3)2 + (fpi - fp3)2 = 2 Fy/

AI. 4. I Elastic Failure The the

choice

material.

of the proper It is suggested

for brittle materials maximum-shearing-stress The considering catastrophic since the

choice

and

between

theory that

of failure the

maximum

either the maximum theory for ductile the

two methods

is dependent principal

on the stress

behavior theory

energy of distortion materials.

for ductile

the particular application. When failure results, the maximum-shearing-stress resuits are on the safe side.

materials of the theory

theory

may

of be used or the

be made

by

component leads should be used

to

Section

A J.

March Page Ai.

4° 2

Interaction

No general conditions

1965

Curves

theory

in which

1, 36

exists

failure

whichapplies is caused

instability case or other critical load or substantiated by structural tests. tions are discussed in Section A3.

in all cases

by instability.

for combined

Interaction

curyes

loading for the

conditions are usually determined from The analysis of various loading combina-

Section

A1

March Page AI.

0.0

Stress

1,

1965

37

and Strain REFERENCES

1.

MIL-HDBK-5,

"Metallic

Structures," 2.

Murphy,

Glenn,

Company, 3.

Materials

Department

Inc.,

Seely,

Fred

Second

Edition,

Advanced New

B. and John

and Elements

of Defense,

York, James Wiley

Mechanics

Washington, of Materials,

for

Flight

D.

C.,

Vehicle August,

McGraw-Hill

1962. Book

1946. O. Smith, and

Sons,

4.

Timoshenko, McGraw-Hill

S. and J. N. Goodier, Book Company, Inc.,

5.

Freudenthal, and Structures,

Alfred M., The Inelastic John Wiley and Sons,

Advanced Inc.,

New

Mechanics York,

Theory of Elasticity, New York, 1951. Behavior Inc., New

of Materials, 1957. Second

of Engineering York, 1950.

Edition,

Materials

SECTIONA2 LOADS

TABLE

OF CONTENTS Page

A2.0.0

Space 2.1.0 2.2.0 2.3.0

Vehicle General Loading

Loads

..............................

1

................................... Curves

1

.............................

3

Flight Loads ............................... 2.3.1 General ................................

4 4

2.3.2 Dynamic and Acoustic Loads .................. 2.3.3 Other Flight Loads ........................ 2.4.0 Launch Pad Loads ............................ 2.5.0 Static Test Loads .................... 2.6.0

Transportation

2.7.0

Recovery

and Handling Loads

.............................

A2-iii

Loads

................

........

5 5 6 7 7 7

v

Section

A2

April

15,

1973

Page A2

SPACE

A.2.1

COORDINATE The

and

directions

aircraft

Figure

A2.1-2.

center

of gravity

outboard

left, Any

are left

applied when

or up; section

moments

acting

positive

the

right

under

rear,

left,

or above.

the

rear,

left,

and upper

any

section. The

external

1.

Flight

2.

Launch

Pad

fibers.

which

Loads

rockets,

axes

are

Moments

used

applied

loads

the

left

missiles,

X

axis

taken are

or upper tends bending Positive

may

act

acting

in the

is

in

positive

part,

upper

tends

to rotate

Z

to move

clockwise

moments

produce

axial

produces

load

on a space

vehicle

X

airplane direction,

direction. are

rule).

part

in

at the

of gravity

(left-hand or

shown

aft in the

center

A2.1-2

outboard,

are

directed

airplane

in Figure

torsion

Loads

Z

conventions

and upward

about

as follows.

for

longitudinal

system.

direction,

Positive

loads

and

when

outboard,

the

Y

as positive

rear,

positive

sign

externally

as shown

shear the

Y

used

rule.

figure

in the

been The

The

a right-handed

the

defined

have

A2.1-1.

direction.

right-hand

In this

to the

Externally

flight

analysis

which

in Figure

to form

by the

For

axes

shown

in the

as determined

under

are

as positive

positive

coordinate

vehicles

appropriate

LOADS. SYSTEMS.

standard

launch

taken

VEHICLE

defined

At ,any section

tends aft, when

to move right, viewed

tension

aft, or up. from

compression

are

as

in across

categorized

Section

A2

April

15,

Page

Z. 0

1973

(L, d>, p, u)

+X

IN. _, r, w) +Z

+Y (M, (_, q. v) -Z

i

i,

FORCE SYMBOL

MOMENT SYMBOL

LINEAR VELOCITY

LONGITUDINAL

X

L

u

LATERAL

Y

M

v

YAW

Z

W

ANGLE

SYMBOL

ROLL

¢

Y to Z

PITCH

e

ZtoX

POSITIVE DIRECTION

YAW i

NOTE:

Figure

ii

A2.1-1.

ANGULAR VELOCITY

q

Xto Y i

4

r !

Sign convention follows right-hand rule.

Coordinate

axes

and

symbols

for

a space

vehicle.

--:

Section

A2

April

15,

Page

2. 1

1973

f-

0 _:::u ,-.4

_'_ f

I-d 0

\ bD _I

"_

o

F

._ _._ o

°_

,,k

NASA--MSFC

Section

3.

Transportation

4.

Static

5.

Recovery

Since analyst the

to obtain

that

qualitative the

"Loads

quantities

assumed

coordinate

the

will these

Page

2. 2

1973

Loads

Loads. practice

magnitudes Group"

in the

loads

are

in his

stress

and

industry

loads

organization, in this

furnished

is required. for

airframe

of external

not be presented

description axes

Handling

15,

Loads

it is universal

cognizant

these

Test

and

A2

April

for

the

the

methods

manual.

to the stress These

loads

aerodynamic

are

for

space

Rather, analyst generally

the

stress

vehicle

from

of calculating it will so that

be only

resolved

their along

analysis.

w/

MS

FC_I_A,

A_

Section

A2

March Page A2.2.0

Loading The

of the

1965

2._

3

Curves

loads

are

station curves, where to as vehicle stations. flight

1,

vehicle.

usually locations These

At each

presented along curves of these

in the

the are

form

of load

versus

vehicle

longitudinal coordinate are referred plotted for various times during the

times,

the

longitudinal

force,

the shear

and the bending moment are plotted as a function of the vehicle station. curves showing the bending moment and longitudinal force distribution a vehicle can be seen in Figure A2.2.0-I.

Typical along

.2 I

0

.2

Bending

Moment

l

Longitudinal

i

2800

L - S -_

2400

__

2000 Vehicle

Fig.

A2.2.0-1

1200 Station

Force(

!

I

I

1600

800

400

~ Inches

Typical Bending Moment Distribution Curves.

and

Longitudinal

Force

Section A2 March 1, 1965 Page 2._ A2.2.0

Loading

Curves

It is necessary along the is applied

vehicle to the

(Cont'd) to know

at times structure

the

circumferential

pressure

of critical loading. This circumferential pressure along with the critical loads during strength analysis

of the vehicle. Typical distribution of this circumferential ular vehicle station may appear as in Figure A2.2.0-2.

ax

Figure

A2.3.0 A2.3.

A2.2.0-2

Flight I

Typical Curves

at a partic-

P

Circumferential Pressure at a Vehicle Station

Distribution

General vehicle

is subjected

to flight

loads

its flight. These flight loads must be investigated loads on the vehicle. Although it is not possible times

pressure

Loads

A space

loads

distribution

will

occur

during

up of critical as follows:

the

without flight

loads.

considering where These

the entire

conditions times

exist

and the

loads

of varying

magnitudes

to determine to know when flight

which which

history, are

the these there

favorable occur

may

during

critical critical are for

certain the build-

be summarized

r

Section

A2

March

1,

Page 1.

Liftoff

application

and

- As

the vehicle

redistribution

which

may

be critical.

2.

Maximum

nation result.

of vehicle

lifts

off the

of loads

Dynamic

velocity

on the

Pressure

and

air

launch

(Maximum

density

pad

vehicle.

is such

there

This

is a sudden

causes

q) - At this that

the

1965

5

dynamic

time

maximum

the

4.

Engine

A2.3.2

Dynamic

Cutoff

- Engine

thrust

and

cutoff. During cutoff, of these loads. and Acoustic

high

inertia loads

may

air to air-

loads

are

result

because

maxi-

Loads

Dynamic loads are loads which are characterized by an intensity that with time. These loads may be analyzed by one of two methods. One is to replace the dynamic load by an equivalent static load, and it is the

varies method preferred

method

for most

cases.

The

other

is justified only in those cases where the good and the design is felt to be marginal.

from

longitudinal dynamic

combi-

airloads

3. Maximum qo_ - At this time the combination of vehicle velocity, density and vehicle angle of attack is such that high bending moments due loads and vehicle acceleration result.

mum just before of the redistribution

loads

Acoustic loads extraneous disturbances

are determined static pressure

are

only in shell structure. A2.3.3

ferentials,

loads induced such as engine

by using an equivalent static acts in both the positive and

sure fluctuates about a zero the design inflight pressure or panel

Other

Flight

Other

flight

must

stress

mean value. to obtain the analysis,

method

confidence

is a fatigue in the

by pressure noise. The pressure negative

in the

,analysis

and

time-history

it is

fluctuations resulting effects of these loads

load. This equivalent directions, since the pres-

This pressure total pressure,

not

load

analysis

should be combined with and should be considered of primary

or supporting

Loads loads,

be considered

which

are

in the

caused stress

by pressure analysis.

and temperature In addition

to the

dif-

Section A2 March 1, 1965 Page6 A2. 3. 3

Other

longitudinal

Flight

loads

a longitudinal

and the pressure

internal

pressure

resulting

be known

tudinal

the difference

on vehicle

location

the

at the

the

local

hoop

external

desired

of the point

analysis. curve.

A2.4.0

The pad. These categorized 1.

Pad

vehicle loads are as follows: Holddown

a holddown mechanism this time are referred

down settles

loads The

range

onto the

external

effects.

vehicle

the vehicle

external

and on the

of values

These

station,

internal

pressure

pres-

which either in the

the

pressure

is a function

range

results

of values

in a maximum

used and

of longi-

in the a mini-

and temperature differentials caused by aerorocket heating and cryogenic propellants result must be considered. The effects of these

properties

must

also

be investigated.

Loads may

be subjected

referred

Loads

to as

- The

to various launch

vehicle

pad

loads loads

is usually

during engine ignition. The to as the holddown loads.

2. Rebound Loads - During engine the engines due to some malfunction. back

and

is

flight. The ambient while the vehicle

internal pressures longitudinal load

at a particular

local

ambient

there

and ambient pressure. The pressure depending on the circumferential and

in question

This

on material

Launch

the

A2.1.1,

during only,

and venting

pressure

time.

Temperature magnitudes dynamic heating, retro or ullage in additional vehicle loads which temperatures

in Section

between

trajectory

of attack, dynamic pressure may be positive or negative

aerodynamic mum design

diagrams

usually produce positive net or decreases the compressive

to determine

between

the angle difference

loading

from

depends

In order must

in the

vehicle internal pressure at any time is a function of the vehicle's altitude

sures in combination increases the tensile vehicle.

difference

(Cont'd)

presented

load

pressure external

Loads

launch

pad

are

referred

while and

held loads

it is on the

are

launch

generally

onto the

launch

on the

vehicle

pad

by

during

ignition it may be necessary to shut The loads on the vehicle as it to as rebound

loads.

/

Section A2 March

l, 1965

Page 7 A2.4.0

Pad

Launch 3.

Surface

Loads Wind

pad, i.c. , unsupported surface wind loads. ical

location

from effect

and

Loads

The

should

(Cont'd) - While

except for magnitude

the

vehicle

is freestanding

launch

the holddown mechanism, it is exposed to of these loads will depend oil the geograph-

be specified

in the

design

specifications.

4. Air-blast Loads - The vehicle may be subjected an accidental explosion at an adjacent vehicle launch of this air-blast on the vehicle must be determined.

A2.5.0

on the

to an air-blast loa, i site. The potential

Static Test Loads

The statictest loads are the loads on the vehicle during static testing of the vehicle. These loads are summarized as follows: 1. 2. holddown

Engine

Longitudinal and rebound

3.

Wind

loads

The

dynamic

investigated A2.6.0

since

and

they

are

these loads ments.

and are

Recovery The

and

transportation

transportation

loads

loads due conditions

Transportation The

A2.7.0

gimbaling

to various

acoustic higher

loads during

Handling

handling

static

static

test

loads

of the space primarily

for

loadings

firing than

during

tests

the

should

in flight,

also

in many

be cases.

Loads

,and handling

required

propellant

for

arc

vehicle. the design

the

loads

which

In the dcsigm of ticdown

occur

of the

during

vehicle,

and handling

attach-

Loads

recovery

particular structural also include the loads

loads

are

the

loads

which

occur

component or stage of the vehicle. which may occur during descent

during and

the

recovery

These recovery impact.

of a loads

SECTIONA GENERAL

._J

ASTRONAUTICS STRUCTURES MANUAL SECTION

SUBJECT INDEX

GENERAL SECTION Ai

STRESS AND STRAIN

SECT I ON A2

L(_DS

SECTION A3

COMBINED

SECTION A4

METRIC

STRESSES

SYSTEM

STRENGTH SECTION

BI

JOINTS AND FASTENERS

SECTION

B2

LUGS AND SHEAR

PINS

SECTION B3

SPRINGS

SECTION B4

BEAMS

SECTION B4.5

PLASTIC

SECTION B4.6

BEAMS UNDER AXIAL

SECTION B4.7

LATERAL

SECTION

SHEAR

B4.8

BENDING

BUCKLING

BEAMS

SECT ION B5

FRAMES

SECT ION B6

RINGS

SECTION

THIN SHELLS

B7

SECT ION B8

TORS ION

SECTI ON B9

PLATES

SECTION

HOLES AND CUTOUTS

BlO

STAB IL ITY SECTION

Cl

COLUMNS

SECT ION C2

PLATES

SECT ION C3

SHELLS

SECTION

LOCAL

C4

INSTABILITY

A-tlJ.

LOADS OF BEAMS

SECTIONSUBJECTINDEX (CONTINUED)

SECTION

D

THERMAL

STRESSES

SECTION E1

FATIGUE

SECTION E2

FRACTURE

SECTION FI

COMPOSITES

SECTION F2

LAMINATED

SECTION G

ROTATING MACHINERY

SECTION H

STAT I ST I CAL METHODS

MECHANICS

CONCEPTS COMPOSITES

A -iv

i

SECTION A3 COMBINED STRESSES

-._J

TABLE

OF

CONTENTS Page

A3.0.0

Combined

Stress

and

Stress

Ratio

....................

i

P

3.1.0

Combined

3.2.0

Stress

Stresses Ratios,

...............................

Interaction

Curves,

and

Factor

of

Safety ..................................... 3.2.1 A Theoretical Approach to Interaction ....... 3.3.0 Interaction for Beam-Columns .................... 3.3.1 3.4.0 3.5.0 3.6.0

3.7.0

Interaction for Eccentrically Loaded and Crooked Columns ...........................

General Interaction Relationships ............... Buckling of Rectangular Flat Plates Under Combined Loading ....................................... Buckling of Circular Cylinders, Elliptical Cylinders, and Curved Plates Under Combined Loading ............................ _ .......... Modified Stress-Strain Curves Due to Combined Loading

Effects

................................

A3-iii

1 8 I0 12 14 18 22

27 31

Section

A

i0

1961

3

---F July

Page

A

3.0.0

Combined

Stresses

A

3.1.0

Combined

Stresses

When such

as

an

determine pal

element

tension,

of

and

structure

compression

resultant

Stress

maximum

Ratio

is

and

subjected

shear,

stress

1

it

to

is

combined

oftentimes

values

and

their

through

the

use

stresses necessary

respective

to

princi-

axes.

The

solution

graphical

may

be

construction

Relative

Orientation

and

fx

and fy are applied normal stresses.

fs

is applied stress,

fmax

and

Mohr's

circle.

Equations

of

Combined

of

equations

or

the

Stresses

shear fy

fmin

resulting normal

attained

of

are

the

principal stresses.

fSmax is the resulting principal shear

f8

stress.

0

is

the

angle

principal

L

of

axes.

e Sign

Convention: _

Tensile

stress

fs

is

positive.

45 °

fy

Compressive

stress

is

negative. Shear as

stress

is

Fig.

positive

A

3. I. 0-I

shown.

Positive

e is

clockwise

as

countershown.

Note: This

convention

this

work

only.

of

signs

for

shearing

stress

is

adopted

for

Section

A

I0 July

1961

Page A

3.1.0

Combined

Stresses

Distributed

3

2

(Cont'd)

Stresses

on a 45 ° Element

t t'yt f-"_2

Y

fx

fx -III-------

fs =._

j-

_ "

/

%

fs

I

v

Fig. A 3.1.0-2 Pure Tension

Fig. A 3.1.0-3 Equal Biaxlal Tensi_',

L v

fs fx

_em,,.

Ffg. A 3.1.0-4 Equal Tension & Compression

Fig. A 3.1.0-5 Pure Shear

A

3.1.0

Combined

Stresses

fx fmax

+

-

fy

(Cont'd) / ,_

2

+

fy _

2

+

/ fmin

-

fx + fy 2

V\

2f s TAN

f

- fy

fx

= Sma x

fy

+

2

Constructing Mohr's Fig. A 3.1.0-6a)

___

f2s

............

(1)

_s

............

(2)

fx Y

two angles representing the principal axes of inl The solution results fmax and fmin:

28 fx

Circle

1964

9, 3

2

\// -

A3

July Page

2

fx

VI

Section

(for

f2 s

(Disregard

the

stress

Stress + Shear

f

Sign)

condition

.......

...........

shown

(3)

(4)

in

fs

fx ht

fmin A

(a)

hand face

T

---fn

0

+ Normal Stress

fx +

fy

fmin

(c)

(b)

fmax

Fig.

A

3.1.0-6

A

3.1.0

I.

Make

stresses

Combined

Stresses

a sketch

of

are

known

an

and

Locate

the

center

element indicate

a distance of (fx tive, compressive

of

A 3

I0 July Page 4

1961

(Cont'd) for which

the

on

proper

it the

2. Set up a rectangular co-ordinate axis is the normal stress axis, and stress axis. Directions of positive and to the right. 3.

Section

the

+ fy)/2 stresses

and

sense

shearing

of

these

stresses.

sy_em of axes where the horizontal the vertical axis is the shearing axes are taken as usual, upward

circle,

from are

normal

which

is

the origin. negative.

on the

Tensile

horizontal stresses

axis

are

at

posi-

4. From the right-hand face of the element prepared in step (I), read off the values for fx and fs and plot the controlling point "A". The co-ordlnate distances to this point are measured from the origin. The sign of fx is positive if tensile, negative if compressive; that of fs is positive if upward, negative if downward. 5. Draw the circle with center found point "A" found in step (4). The two circle with the normal-stress two principal stresses. If principal stress is tensile,

in step (3) through controlling points of intersection of the

axis give the magnitudes and sign of the an intercept is found to be positive, the and conversely.

6. To find the direction of the principal stresses, connect point "A" located in step (4) with the intercepts found in step (5). The principal stress given by the particular intercept found in step (5) acts normal to the line connecting this intercept point with the point "A" found in step (4). 7. The solution of the problem may then be reached element with the sides parallel to the lines found indicating the principal stresses on this element. To determine associated normal I. per

Determine previous

the maximum stress:

the principal procedure.

or

the

stresses

principal

and

the

by orienting an in step (6) and by

shearing

planes

stress

on which

and

they

the

act

2. Prepare a sketch of an element with its corners located on the principal axes. The diagonals of this element will thus coincide with the directions of the principal stresses. (See Fig. A 3.1.0-7). 3. The magnitude of the maximum (principal) shearing stresses acting on mutually perpendicular planes is equal to the radius of the circle. These shearing stresses act along the faces of the element prepared in step (2) toward the diagonal, which coincides with the direction of the algebraically greater normal stress.

A

i0 July

1961

Page

"7

A

F

Section

3.1.0

Combined

Stresses

3

5

(Cont'd)

4. The normal stresses acting on all faces of the element are equal to the average of the principal stresses, considered algebraically. The magnitude and sign of these stresses are also given by the distance from the

origin

of

the

co-ordinate

system

\

f'

to

the

fmax

=

\

center

+ 2

of

fmin

=

fmax

\ \ fmin Fig.

A

3.1.0-7

Mohr's

fx + 2

fy

" fmin

circle.

Section

A

I0 July

1961

Page

A 3.1.0

Combined

Stresses

Mohr's

(Cont'd_

Circle

for

Various

Loadin$

Conditions

+ fs

fx_

fx

_

fs_

g _-

O_fx_

Fig.

A 3.1.0-8

+

Simple

+

f._x

fn

Tension

fs

9

_-Fig.

A

fx -_

3.1.0-9

+

Simple

fs

_

" fn

Compression

fSmax

0

+

fn

_y

Fig.

A

3.1.0-I0

Biaxlal

Tension

"

6

3

Section

A

I0

1961

Page

A

3.1.0

Combined

Stresses

Mohr's +

Circle

July 7

(Cont'd)

for

Various

Loadin$

Conditions

fs

Point

0

+ fx_fS

=

fn

0

fy

Fig. A 3.1.0-II

Equal

Blaxlal

Tension

"

fn

f +

fs

fSma x

"

fs

Fig. Equal

0

" fn

+ f n

fnmin_

x = fs

Fig. Pure

A

3.1.0-13 Shear

Tension

A

3.1.0-12 and

Compression

3

Section A3 July 9, 1964 Page A

3.2.0 A

Stress

means

without method.

Ratios,

of predicting

determining

The

basis The

I°

under

bending,

The combined stress ratios,

Failing

can

effect

of

represented curve or

by

by

test,

the

interaction

condition

is determined

by

is represented

(tension, test or

by

either

theory. load

or

STRESS

R1 or

R2.

been

STRESS

rupture,

loading

and

or

OR

yield,

equation

have

schematic

shown material

in

on

by

a

buckling,

another

etc.

simultaneous

loading

Ra

is

interaction The

equation

determined

by

combination

interaction

Fig. A or size

influence and

as

loading

1.0

A

all

one

R1

may

LOAD

mean

an

involving curve

theory, of both.

loading

etc.)

OR

combined

follows:

simple

LOAD

of Safety

__

FAILING

The

each

Factor

under

is known

loading condition "R" where

APPLIED e

is as

buckling,

and

failure

stresses

this method

strength

Curves,

structural

principal

for

shear, .

Interaction

8

it.

the

3. Z.0-1. effects

This

possible

Rz

1.

the

Rx

4.

value

0

of

represents

of

point and

Rz

of

Rl

R1

at

If R1

remains

point

c.

If Rz

remains

point

The

factor

and

the

R1

and

R2 0

a. can

/11

\

point

/

1o1

\

R2 increase until

occurs

at

Type not

failure.

proportionately

3.

is

curve:

locate 2.

curve cause

the Let

will

combinations

thatwill

Using

curve

Fig,

1.0 A 3.2.0-1

failure

b. constant,

Ra

can

increase

until

failure

occurs

constant,

R 1

can

increase

until

failure

occurs

(2)

is

at

d. of factor

safety of

for safety

for

(3)

F. is

S. = (ob+oa)_or(oh+oe),(or F.S.

= (fc+

fa).

og-of)

Section

A

i0 July

1961

Page A 3.2.0 In cally (one

Stress

Ratiosj

general, the for interaction

term

may

Interaction

formula for equations

be missing)

is

as

Curves_

and

Factor

...................

where

R''

designates

the the

sum sum

of of

9 (Cont'd)

analytii or 2

follows:

IR+J_2+_21 designates

Safety

the factor of safety stated where the exponents are only

FoS,

R'

of

all all

first-power second-power

ratios. ratios.

3

(1)

Section A 3 i0 July 1961 Page I0 A 3.2.1

A Theoretical

For

combining

equations

or

are

Let F rupture.

from

Approach

normal

and

F s be

Maximum

=

f/F;

Normal

shear

of

as

most

the

The

F;

replace

resulting

fs by

equation

Maximum

_Rf_

this

equation

Shear

Stress

Theory

Divide

The

by

Fig.

plot

of

A

3.2.1-1.

for

+

Fs;

replace

f by

equation

when

Rf

i-f this

stress

such

show

this

as

yielding

ratio

to vary

2

Eq.

(I)

Sec.

Rf

and

Fs/F

by

by

A

3. I. 0

k.

= F is

(kR s )

k

...................

= 0.50

f2

and

k = 0.70

Ref

Eq.

is

(I)

shown

(4) Sec.

A

in

3.1.0

s

R2

+ s

equation

Ref

2

_(2)

1

f/F

2

=

resulting

A

fmax

+

plot of 3.2.1-i.

fSmax

will

f2

RsFs,

when

I:T-+ A A

stress,

s

Rf

Fig.

principal

Theory

-- 7 +

by

the

failing

materials

+

Divide

stresses,

R s = fs/Fs

Stress

fmax

Interaction

to use.

defined

Let k=Fs/F; tests 0.50 to 0.75. Rf

and

convenient

to

for

RfF,fs/F

fSmax

s by

= Fs

R s and

F/F s by

I/k.

is

................................. (2) k = 0.50

and

k

= 0.70

is

shown

in

Section A 3 i0 July 1961 Page I 1

f--

A

3.2.1

A Theoretical

Approach

to

Interaction

(Cont'd)

Conclusion

From the foregoing analysis, only Equation (2) with k = 0.5 is valid for all values of Rf and Rs. It is conservatively safe to use the resulting Equation (3) for values of k ranging from 0.5 to 0.7, since all values within curve (_ must also be within the other curves. The use results.

of

other 2 Rf +

and

the

Factor F.S.

curves

2 Rs = 1

graphical A 3.4.0-1

Fig.

A

3.2.1-1

may

lead

to unconservative

.......................................

(3)

of Safety 1

= VR2f

For the of Fig.

of

.............................

(4)

+R2s

solution may be

for used.

Factor

of

Max.

Safety,

the

Shear

2 2 R1 + R2 = 1

curve

Stress

Theory

k -- .5; Rf 2 + Rs 2 =

k =

1.6

.5Rf 2 +

Rs 2 = 1

O Max.

Normal k =

Rs

.7;

1

1.=1-

\

4@k

=

Stress

Theory

.5;

Rf +_f2

.7;

Rf

+ Rs 2 = 2

+_/Rf2 +

(1.4

1.

0

-

,,,,1 .2

I .4

r.O

Fig.

A

3.2.1-I

I 1.2

L

@

Valid

@

Partly

@

Invalid

@

Partly

Valid

Valid

Rs )2 :

2

Section A 3 I0 July 1961 Page 12 A 3.3.0

Interaction

for

Beam-Columns

P Fig. A 3.3.O-1 Sinusoidal Moment

P

Fig. A 3.3.0-2 Constant Moment Curve

Curve

= applied

load.

2 E1 Pe

=

L2

(Euler

load).

(Reference

_2E Po

= buckling

load

Section

C

1.0.0)..

(I)

I

=

t

.........................

(2)

L2

or

applicable

short

column

formula.

M

= maximum

applied

Mo

= ultimate Section

bending B 4.0.0)

Ra

= p-_

Rb

M M

(Reference

bending moment

moment as

Section as

a beam

C 1.0.0)

a beam

only.

only.

(Reference

P (column

(beam

u

stress

stress

ratio)

ratio)

.........................

(3)

..........................

(4)

o

f

from

which

=P+k A the

M-c I

interaction

R a + kR b = i Po Let For

_

sinusoidal

bending

is:

............................................

(5)

Et E moment

(plasticity

coefficient)

...............

(6)

curves

i

k= I

Rb =

= Pe

equation

- P/Pe

(i - Ra)

(I -

_ Ra)

................................

(7)

A 3.3.0

Interaction

Interaction Fig. For

A

curves

Beam-Columns for

various

A

i0 July Page 13

1961

(cont'd) va]ues

of

values

of

_

are

shown

in

are

shown

in

3.3.1-5.

constant k=

A 3.3.

for

Section

Interaction I-6.

bending

moment

curves

1

curves

for

various

_

Fig.

Conclusion Comparison of Figs. A 3.3.1-5 and A 3.3.1-6 show changes in shape of the primary bending moment diagram influence the interaction curves. Therefore, Figs. A A 3.3.1-6 should be adequate for many types of simple

that significant do not greatly 3.3.1-5 and beam columns.

3

A3.3.1

Interaction

for

Eccentrically

Loaded

and

Crooked

Section

A

3

I0 July Page 14

1961

Columns

M=Pe

LIIIIIIII i_ e

I

P P Eccentric Fig.

A

Reference Re

e

Column

Crooked

3,3.1-1

Section

Fig.

A

3.3.0

= e__ (eccentricity

for

beam-column

ratio)

Column

A

3.3.1-2

terms

........................

(I)

eo

M o

e°

=-Po

(base eccentricity, for Po to induce

which a moment

is that Mo)

required

r

...

(2)

For a particular e, M would be a linear function of P as shown Fig. A 3.3.1-3. A family of such lines could be drawn which would represent all eccentric columns.

same P, M,

To obtain Fig. A 3.3.1-4 form as the interaction and

e

of

Fig.

A

3.3.1-3

(a nondimensional one-one curves of Figs. A 3.3.1-5 may

be

divided

by

Po,

diagram of the and A 3.3.1-6),

Mo

and

e o respectively.

e e o

P Ra

=_o

M

M Rb Fig.

A

3.3.1-3

Fig.

in

= M-_ A

3.3.1-4

A 3.3.1

for

Interaction

In using Fig. crooked columns

for A

Eccentrically

3.3.1-6 for the following

Loaded

Determine Po, the buckling short column formula.

2.

Calculate

3.

Determine Mo, using Section

4.

Calculate

e o = Mo/Po,

5.

Calculate

R e = e/eo.

6.

Calculate

_

7.

Knowing R e and _ appropriate curve. of Safety of 1.0.

' Ra = P/Po may This value of

8.

The

ultimate

is

9.

The

Factor

=

load

of Safety

P

load

the

the ultimate B 4.0.0.

Pu F.S.=--

_2EI/L2,

= Po/Pe,

by

Euler

moment

base

the

plasticity

for

an

A

i0 July Page 15

1961

Columns

Fig.

or

A

as

3.3.1-5

applicable

a beam

only

eccentricity.

coefficient.

be determined R a corresponds

R a.

applied

load

P

is

3

(Cont'd_

load.

bending

= Pox

and

_ 2Etl/L2

the

Pu

Crooked

eccentric columns steps are taken:

I.

Pe

and

Section

from the to a Factor

Section A 3 i0 July 1961 Page 16 A 3.3.1

Interaction

0

for

Eccentrically

Loaded

and

Crooked

Columns

(Cont'd_

0.2

1.0

/ !

/

/ I

0.8

P

Po

=0.0

=Y

_ = 0.2

0.6

=0.4

o Ow

_=0.6

mw

"q = 0.8 _3 = 1

II o_

0.4

0.2

i0.0

0

0.2

0.4

0.6

0.8

R b = M/M ° Interaction with

Sinusoidal

Curves

for

Primary

Straight Bending

Fig.

A

or Crooked Moment

3.3.1-5

and

Columns Compression

1.0

Section A 3 I0 July 1961 Page 17 A3.3.1

Interaction

for

Eccentrically

R

e

= e/e

Loaded

and

Crooked

Columns

(Cont'd)

o

1.0

2 0.8

0.6 O

0

1 8 II

2.0

0.4

0.2

i0.0

'i|l*|lll_

0

0.2

0.4

0.6

0.8

R b = M/M °

Interaction Curves for Columns Bending Moment and Axial or Fig.

with Constant Primary Eccentric Compression

A 3.3. i-6

1.0

,

Section I0 Page

A3.4.0

General

Interaction

A

July

3

1961

18

Relationships

0 ._ V

0

co o _

o_

,_ ,-_ _

00

0 0 o

ml _

_

0

co

_._ o _ [z_oo

= _ o_

_J

eq 0 [--i + _J

Om

+ oq_

! !

.0 co

+

i+ o

,._

cN o !

o I ,.-4

II II

II

II

4J c,q _

r-_ co

II

II

Jr

o_

.I..I

co +

II

Jr

+

II

+

+

,--4 I

I

I

o

o

c_

I

o

_J

!

oo

..4 .<

o

_0 o

co co

o co

r_

0.1

I_

o r.j

m

o

co

G

0

o

cJ -,_ ;_

_

o

_J.J=

I=

m

•

o ,.c_ _J _o0oo

(2 co r_

"o

= m'_

.IJ I

"_ co

co

._ _ i_1 o

!

o

"o

"o

co "o .,-i I=

!

o

!

.1_ I_

co •

c3

_

CO

cO 4-J

0

_ =_ _-,_ _

,-_ 0

_pf

A

3.4.0

General

Interaction

Table NOTE:

Relationships

A

3.4.0-1

For

round

(b)

See

Section

tubes A

in

3.2.1

compression for

see

discussion

A

3

I0 July Page 19

1961

(Cont'd)

(Cont'd)

Care must be exercised in determining Safety for limit or ultimate loads.

(a)

Section

whether

Section of

range.

to

C.3.0.0.

check

Factor

of

A 3.4.0

General

Interaction

Relationships

Section

A

i0 July Page 20

1961

(Cont'd)

1.0

0.9

3+

=I

0.6

;

0_

,

\\

:

1 5

R 1"

/

- /

_

2

+ R2 = + I R2

= I!

0.3

RI

0.2

I R1 + R2 =

1

/ /_/

I i

_//

0.2

0.3

_

0.i

0

"1

I

I

I

0.1

0.4

0.5

0.6

R1 Interaction Fig.

A

Curves 3.4.0-1

0.7

0.8

0.9

1.0

3

V

Section A 3 I0 July 1961 Page 21 A 3.4.0

General

Interaction

Relationships

(Cont'd)

I I NOTE:

AN-IO

30

Formula

Interaction

R _

+R

S

t

2

\

= 1

Curves

not

applicable where shear nuts are used. Curves are based on the results of combined load tests of bolts with

\ \

25

\

-

nuts fingertight.

\

\ i

\

20 AN-8

\ \

O,

\ , 15

\

\

m o

\

\

\

\

\ •

\

,AN-6 I0 ---_

_

\ \ \

\ \

AN-5

\

\

5

\ t

_-4

\

I 5

I0 Shear

AN

Steel

Bolts

15 Load

20

-Klps

Flg.

A 3.4.0-2

(125,000

H.T.)

Interaction

Curves*

25

A

3.5.0

Buckling

of

Rectangular

NOTE:

See

Flat

discus=ton

Plates

Under

in Sec.

C

Section

A

19 July Page 22

1961

Combined

3

Loading

2.1.4

iIIEIH tttttttttt Combined Fig.

Table

THEORY

LOADING COMBINATION Biaxial pression

Loading

A

3.5.0-1

A

3.5.0-I

INTERACTION FIGURE

ComA

3.5.0-2

For plates that buckle in sq. waves, R x + Ry

Longitudinal Compression Shear Elastic

Longitudinal Compression and

3.5.0-2

A

3.4.0-I

Rx+Ry

R c +¢R

2 + 4R 2 s

Bending

Bending Shear

and

Bending, Shear, Transverse

Compression Longitudinal Compression and Shear

R2 + R2 = I b s

_/R 2 + R 2 s

&

Compress ion Longitudinal Compression, Bending and Transverse Inelastic

A

FOR FACTOR OF SAFETY

= I

For Long Plates. = I Rc + R 2 s

and

EQ

EQUAT ION

A

3.5.0-3

A

3.5.0-2

A

3.4.0-1

i

2 R2

+R

=

I

i v

c

s

_/

2 R2 Rc + s

/+--_

A 3.5.0

Bucklin_

of Rectangular

Flat

Plates

Under

Section

A

19 July Page 23

1961

Combined

3

Loadin$

(Cont 'd) (a)

(b) 1

l.O_

a/b

=

_--,>,.

O. 8 0

o.__

Rx=

\

,a/b - l. o

_

0 06

_o_

\

R

¢\

04

o_ 0.2 \

(

\ \\\

, 0._ o. 9\ \

\

0.2

0.6

\\\\,,\

\"\\\\ ,\i _.o._ ,\,\ \ o_\\ \\\\,

0 2

• _ \ \ \\\\

\

\ \

0.8

1 o ....

1.0

_ _\\

0.2

0.4

0.6

0.8

\

\ 1.0

(c) 1.0 alb

-- 1.20 f

0.8

0.6

Y

__\_\

\_-o

Rb 0.4

0.2

0

\\ \\\o_ \_,\ \\ \ _o._, \,\\_\

:

a

_IIIIIIIIIIIII'

\ \ o..,\ \ \ \\\

o::\ \ \\ \\\\

.... I 0.2

I

0.4

0.6

0.8

1.0

Ry Plates

Interaction Curves for Simply Supported Flat Rectangular Under Combined Biaxial-Compression and Longitudinal Bending Fig.

A 3.5.0-2

Loadings

Section

A

i0 July

1961

Page

A 3.5.0

Buckling

of

Rectangular

Flat

Plates

under

Combined

3

24

Loading

(Cont'd)

(d) 1.0

_

a/b

= 1.60

--- L....

_

:°

o.8"-""'m!.. 0.6

-_

Rb

--_ _-

_

0.4

"--

0.ii0

.

0.80 _

0

O.

_o ,','

RI

_

i

_,,I\

' :

0.8

r

_'-_

Rb

_

o.4

3.

"

L

3.0

=

a/b

(g)

__-

__

0

--

8

_

I_

=

oo

-_!_ \

:

II!o

O'_n<_'_ I_

I

--

)

0.6

0._

1.0

\

alll

0

0.2

\ 0.4

0.6 R

Y

Y Fig.

A

3.5.0-2

(Cont'd)

\

\

%.0.90i \ \

Z

-I 1 R

-'_

' I I!1°

|lit

0.4

o

_1 I!1

•

oo_

R__._..L=C

-....,_ _

_

0.2

_

"illl o

\o_o\

"0.2

a/b

I

_

0

2.0

3.

_

0.6

=

'I

\\:I

"-0.90_

_._

\\

.-;

(f) 1.0

_

a/b

O.

r",_z). 2o

\

0.2

(e) l,

1.0

Section A 3 I0 July 1961 Page 25 Buckling

of

Rectangular

Flat

Plates

under

Combined

Loading

I 1.0

0.8

0.6 R b

0.4 1.0

1

0.2

0

_nallllln 0

0.2

0.4

0.6

0.8

1.0

Rs

Under

Interaction Curves Various Combinations

for Simply Supported Long Flat Plates of Shear, Bending, and Transverse Compression Fig.

A 3.5.0-3

Section A 3 I0 July 1961 Page 26 A 3.5.0

Bucklin$

of Rectansular

Flat Plates

under

Combined

Loadin$

(Cont'd_

0.5

__

___

10.8

f f

0._f

/

0.6 Rb

Rs

0.4

0.2 m

l"

1.0

o o.''"1"

0.8

o,.2

0.6 Rc

0.4

0.4

0.6

Rs

0.2

//

0.2

0.4 R c

0.6

0 0

//

Rb

0.8

1.0 Interaction

Curves

Fig. A 3.5.0-3

(Cont'd)

_J

Sect

ion

February Page

A3.6.0

Buckling

Curved

Plates

of Under

Circular

Cylindersj

Combined

Loadin$

Elliptical

Cylindersj

27

and

kl E_ 0 2_ 0 I

kl

I +

+ O ¢x.1

,.-( +

_>

cy

+

C_O

+

O

0

.SZ

_z

_

r-4

-4©

II

II

II

LJ

o

,.Q

II •,_ II

_

_

_

II c_

+ u_

,x:l

+

_J

c_

+

I

T

T

O

O

II

4

T

.

T

T

0

o

o

oi

o

4

4

$

41

4

0 <

<

<

Z g_

_

_o

°0'1

_o_

o o _

o o

._ _

_ _

,.,

z H

C

m

(1)

_)

g_

o o _

I

o_ OLj

_

c _

A3 15,

1976

Section

A

i0

1961

July

Page

A

3.6.0

Bucklin$

Curved

Plates

of under

Circular

Cylinders

Combined

t

Loadin$

Elliptical

Cylinders

3

28

t

and

(Cont'd)

[I3 0 4-I eq 03 + ¢q oq

o

03

o +

+ r/3

,m

.o o

o' ,-4

,-4

II II

= 0 O

II

¢o#m

,--4

II

%.s

¢_ 03 03

,-4 ! O

M

e4 03

+ +

ZO z _0

+ .m

v

¢q

II

¢_1 03 O_

.U ¢_1 03 O_

03 +

+

+

II + 03

_CY

+

+

I

I

.4

..4

<

<

+ 0

o

o

,m

Z 0

i0

m = •,4 _ :3

c _ 0.,4 .,4 _ 03 = 03 0

• > 03 = _l

= = •,4 _1 ¢1 '13 03 = _

.,4 _

[-_ •

O > •,4 03 m C m

03 _

•,4 _

• 03 _

C .,4

_ C

C C o-,_

C o

"o D

03 C 03 •

03 _

•,4 _ _0

[-_

C

i1/

C > •,4 m -o C

C

_ _

C O .,4 .,4 -O m

CD _

=eft1,

o

o

_,_

o

D

_.c

o

o

o

_

C

o

_

_

O

"o I 0

I

I

-

I

I

•,4 m-,4 .,4 _

>_ _rD-O

m

f

Section

A

I0

1961

Page

A

3.6.0

Curved

Bucklin_ Plates

of

Circular

under

Cylinders_

Combined

Loading

Elliptical

Cylinders_

(Cont'd)

o

4-1

=° •

d

_ Vl

_

I_

o _

4-J

_: Ii _

_1--_ +

_ln

II

II

V o o °_ I

o

,g ,-.-4 121t_

.,-4 0

_

4-J

,x:l N

,._

m

_

o

n_

_

flJ ¢.1

t.J

.0_

0 .,-I 01

.,II

,-4

oo

Vl

_J

g

u_l_ _ o

o

e, O_

July 29

and

3

A 3.6.0 Buckling of Circular Curved Plates under Combined

Cylinders_ Elliptical Loading (Cont'd)

Section

A

I0 July Page 30

1961

Cylinders

3

a and

1.0

RI3 - R 2 = 1 0.8

//

0.6

/j/

2 RI• - R 2 = 1 R2 0.4

o2

j'

....

"|

0

I 11

0.2

0.4

0.6

0.8

1.0

RI

Interaction Fig.

A

Curves

3.6.0-1

1.2

1.4

Section A3 July 9, 1964 Page 31

A3.7.0

Modified Stress-Strain

Curves

Due

to Combined

Loading

Effects

An

analysis

properties

that

derived

thermal

effects,

Plastic

uses

such

plastic

Energy

curve

combined

loading

the

and

of or

is

other

member

reaches resisting

the ultimate combined

tensile loading

reaches

Ftu

Several been hedral

Shear

Assumptions

,

of

the Stress

&

fl, i.e.,

piastic

when

the

Ftu of before

Section

in

A1).

one

plane

when affect

effect.

stress,

For

(P/A),

material, maximum When

loading

curve

but a member principal

buckling

or

effects,

other

modified

required.

modifying

method

the the

a modified

Poisson

average

Elastic-

require

a modified

the

material columns,

buckling,

stresses to

or

beams,

may

from

combined

not

etc)

due

stress may fail

are

methods

developed;

fails

include

curves

and 5.7, or

planes

(Reference

parameters

stress-strain

Loads

in

of

derived

involved.

stresses

a tension

stress

B4.

curve

(analysis

elastic

Section

properties

example,

empirical

stress-strain

a curve

bending,

Theory

stress-strain

loads

a uniaxial

from

uniaxial

presented

here

stress-strain is

derived

curves from

the

have Octa-

Theory.

Conditions:

f2

and

f3,

the

three

principal

stresses,

are

in

proportion;

fz = El fl

(_)

f3 = K2 fl

(2)

K 1 _ K 2 See

Fig.

A3.7.0-I

for

direction

of principal

stresses.

A3.7.0

Modified

Stress-Strain

Loadin$

Effect

Curves

Due

Section

A3

July Page

1964

9, 32

to Combined

_Cont'd)

3

3

foct

f2-

.._ ---'-2

Figure Directions

2.

Prime

(') denotes

c'

3.

and

of Principal

a modified

=

modified

strain

V,' =

modified

modulus

In this method, modulus

for

any

principal

Stresses

value:

of elasticity.

principal

of elasticity

of the other

A3.7.0-I

are

stress

modified

fi' the total strains to include

the

effects

stresses.

Procedure:

I.

2.

Calculate

the

principal

(Reference

Section

Determine

the

=

for

a given

load

condition

A3.1.0).

effective

-

stresses

uniaxial

- (f2 - f3

stress:

+ (f3 - fl

(3)

Section A3 July 9, 1964 Page 33 A3.7.0

Modified Effect

Stress-Strain

Curves

Due

to Combined

Loading

(Cont'd) I

and

calculate

EI '

o

Enter

=

an

effective

modulus

of elasticity,

E l, by:

(f_ll) El

the plastic

(4)

stress-strain

diagram

for

simple

tension

of

--

the (See

material, Figure

stress-strain A3.7.0-2a

if available, (A3.7.0_2b)

curve ) by:

at

fl

at the Otherwise, and

value

f

of fl and determine Esp. enter the simple tension

determine

E'

(see

Figure

sp

f i E' sp

-

(5) _i - _I O

°

Use this known,

[p

value of E'sp 1 find e I from P 1 -

E,sp

and

a value

of

gp

= 0.

5,

if

not

accurately

(6)

(fl

-

gp

E

f2

- _p

f3 )

Esi __

EsPi

_-

pt.

I

_--

I

--

i

Plastic

Secant

modulus

'¢ip :

F (a)

Engineering

_ le

¢I stress-strain

P curve Figure

(b) A3.7.0-2

Plastic

stress-strain

curve

A3 °7.0

Modified Effect

5.

Stress-Strain

Curves

Due

to

Combined

Loadin 8

can

be

determined

for

Section

A3

July Page

1,564

(Cont'd_

Once

E' has

been

found,

c'

of fl by!

any

value

le

(7)

fl Ie

N_

I

6.

Determine

°

Repeat

the

all

a plot of stress-strain

fl

total

effect

e

p

steps

until

vs

strain,

el,

for

each

of

fl

sufficient

cl (see curve.

points

Figure

A3.7.0-3

are

obtained

) which

is

t

E 1

/ /

to the

modified

E

El s

construct

t

it

/

l

Any Point

e l t

I

Ip O

¢'1 Figure

A3.7.0-3

Modified Loading

by:

(g)

t

f

value

Stress

Strain

Diagram

Due

to

Combined

_, 34

Re ferences

:

Popov, 1954.

P.,

E.

Structures Fort Worth.

Mechanics

Manual,

of

Convair

Materials,

Division

Prentice-Hall,

of

General

Inc.,

Dynamics

Section

A

July Page

1964

New

9, 35

York,

Corporation,

3

SECTIONA4 METRIC SYSTEM

v

TABLE

OF

CONTENTS

Page METRIC

A4.0.0

P /

A4.

SYSTEM

Introduction

1.0

A4.2.0

The

A4.2.

Basic

A4.4.0

International 1

A4. _F

5.0 A4.

System

5.

A4.6.0

1

A4.6.

1

A4.6.2 A4.7.0

SI A4.7.

1

Units

Dual

.........

Notation

Drawings

Units

3 4 4

Quantities

.......

................

5

................

5

and

in

Analyses

........

A4.7.

3

Tabular

of Units

Data

A4. 7.4

Collateral

A4.7.

Temperature

8

..............

8

...................

Use,

SI and

Non-SI

7 7

....................

Identification

4 5

/

A4.7.2

2 3

................... Units

for

SI

.................

Physical

Photometric

on

2

..................

SI Symbols

Units

Units,

1 1

..................

Quantities

Rules

........

...............

for

System

Dimensionless

Other

(SI)

...................

Incoherent

Physical

of Units

.....................

Symbols

Giorgi 3

of SI

SI Units

CGS

A4.4.2 A4.4.

System

Advantages

A4.3.0

A4.4.

1

......................

International

1

1

.....................

Units

.......

8

f

5

Scales

f

A4-iii

..............

9

TABLE

OF

(Continued)

CONTENTS

Page A4.8.0

Transitional A4.8.

1

A4.8.2 A4.9.0 A4.

A4.

A4.

Mass

vs

Indices

.................

9

Force

.................

9

Examples

of Nomenclature

Measurement

10.0

Preferred

A4.

10. 1

Volume

A4.

10.2

Time

A4.

10.3

Energy

A4.

10.4

Tempe

A4.

10.5

Prefixes

11.0

of Angles Style

12

.....................

12

......................

12

.....................

12

.................

14

....................

Conversion

Factors

Basic

A4.

11.2

Noncritical

Conversion

A4.

11.3

Conversion

to Other

Linear

Tables

14

................

11. 1

Conversion

11

...................

rature

10 11

...............

A4.

12.0

...........

Unit

15

............ SI Units

.................

A4-iv

15

...............

.........

15 15

Section

A4

1 February Page METRIC

1970

1

SYSTEM

Introduction

The

metric

purpose

system

also

and

presents

of

System

and

to

length,

the

foot;

and

A4.2.0

(SI),

the

based

System

is

its

length,

the

the

Metric

of now

"--

f

involving

and

time,

system,

is

System

of

Units,

referred

to,

definitive

purpose

less

of of

to

both

System

the

English

these

second.

are:

Note

common

terms

or

precise

to

both

Syst_me

that the

the English

Internationale

terms, The

system,

previous

SI,

as

the

Meter-

therefore,

although

it

is

should much

be

broader

system.

SI significant

relating

eliminate

varied

any

SI has

work to

section

(SI)

system.

metric

than

Advantages

tend

in

(MKSA)

use

This

the

System.

International

the

with

tables. basic

the

The

and

will

English

Units

development SI

the

of

The and

are

System

as

1

time

reader

system.

conversion

International

considered

A4.2.

pound;

the

English

The

sometimes

scope

In

sexagesimal

Kilogram-Second-Ampere

in

and

System.

the

the

and

mass,

mass,

is to acquaint

over

symbols,

Metric

on

section

advantages

definitions, Units

second,

of this

to wasted derived

advantages

space time from

technology. and

costly a multiplicity

in

all

phases

For

of

instance,

errors

in of

research the

use

computations

sources.

The

Section

A4

1 February Page

utilization

of

a uniform

fies

the

exchange

and

will

do

of

so,

organizations

A4.3.0

Basic

the

basic

units

des

Poids

of

et

Units,

installations

and

has

Mesures

degree candela

the

of

atomic

a basic

tool.

The of

pound

the

atomic

space-ori-

in

been

1960

recommended

for

the

following

to

reach

mole substance

in

pound

weights

of

based

SI

agreement

may

for be

in

on

the

the

a unit

of

grams

{gram

the

amount

atoms

Units,

of

basic

quantity

in

mole;

weight)

Carbon

fact

OK cd

recommended

molecular all

upon

Kelvin

that

The (tool),

Symbols that

A

determined

quantity.

or

are

International order

a

also

mole,

weights

In

was

as

amount or

sum

it

addition,

treated

weight;

necessary

G_n_rale

System

kg s

as

A4.4.0

International

kilogram second

symbol:

These

simpli-

world.

ampere

mole,

the

SI thus

and

contractors

m

be

to

the

the

centers

associated

meter

would

ular

among

NASA

as

:

In

defined

among

such

v

Units

name,

ConfErence

measurement

data

throughout SI

The

of

in-house

eventually,

ented

by

system

1970

2

unit

is

the

chemistry, gram

which

constituting

substance

is

moleccorresponds

the

molecule.

1Z.

SI an

international

symbols,

names,

system, and

it

abbreviations.

was

_

Section

A4

1 February Page

1970

3

_F

A4.4.

1

CGS In

have

System the

special

field

of

names

Conference

on

and

Weights

mechanics,

the

symbols

and

which

following have

units

been

of

this

approved

by

1, b, h

centimeter

cm

second

s

m f, v

gram hertz

F

dyne

E,U,W,A

erg

p

microbar

The of

units

for

quantities

( = g. cm/s (=

The the

MKSA

name

or

m-kg-s-A

dyn

2)

erg _t bar 2)

p

time,

and

ampere

A

MKS

system.

The

MKSA

used

is

magnetism,

electric

kg

a

coherent

based

current

on

system four

basic

intensity.

S

based the

system,

system

and

kilogram second

by

commonly

2)

s/cm

m

system"

most

dyn.

meter

system

four-dimensional

(=

electricity, mass,

The

the

cm2/s

( = dyn/cm2)

system

mechanics,

mechanical

g.

System

: length,

"Giorgi

General

g Hz

( = s- 1)

poise

Giorgi

the

Measures:

t

A4.4.2

system

on

these

four

International

which

was

Electrotechnical

is

based

system

of units

system

of

together

units

on

forms

equations

with

these

given

the

name

Committee

the

first

a

coherent

previously

equations.

three

units

system mentioned,

in

1958.

only,

has

of

units and

in is

Section

A4

1 February Page

A4.4.3

Incoherent

_ngstr_m

A

barn

( = 10-Z4

V

liter

(=

t, T

minute

min

t, T

hour

h

t, T

day

d

t, T

year

p

atmos

p

cruZ)

b

1 dm 3)

1

a pher e

kilowatt-

Q

atm

hour

kWh

. calorie

cal

Q

kilocalorie

E, Q

electronvolt

eV

m

ton

t

m

p

bar

Physical The

German:

is

number)

A4.5.

or

and

1

symbol

Examples:

for

'physikalische

to

atomic

mass

(=

10 6 dyn/cm

(=

10 5 N/m

unit

u

Z)

z)

bar

Quantities

symbol

equivalent

kcal

( = 1000kg)

(unified)

A4.5.0

4

Units

1

Ma,

1970

the

a unit,

of

i.e.,

physical

the

quantity English,

dimensionless

and

is

explicitly

E

= 200

erg

F

=

N

'grandeur

(French:

value

quantity

physique';

'phys ical magnitude

sometimes:

numerical

Physical

For

27

Grosse';

product

Dimensionless

not

a physical

(or

the

= numerical

measure, value

a pure

x unit.

Quantities

physical

quantities

the

indicated.

nqu v

= 1.55 = 3 x 108

s-1

unit

often

has

no

name

')

Section

A4

1 February Page

r" A4.6.0

Other

SI Symbols

The and

symbols

Weights

following which

and

have

units

of

been

approved

the

ampere

Q

coulomb

C

farad

General

special

m

kilogram

1, b, h

meter

F

newton

R

ohm tesla volt

P

watt

$

weber

corresponding

to

mZ/s2)

J kg

( =kg. (=

B

the

Vs/A)

m

V

m/s2)

N

V/A)

(= (=

Wb/m2)

T

W/A)

V

(= J/s) (=

W

V.

s)

Wb

Units

field

of

the

basic

symbol:

an

quantity,

additional

luminous

unit

intensity.

(candle)

cd lm

E

lux

( = lm / m2)

lx

upright b.

period),

and

in

this

This

candela lumen

for

units

is

I

Rules

for

basic

names

Roman

cd.

photometry

Special

a.

on

C H

(= ( =kg.

Photometric

s)

F

henry

A4.6.2

names

Conference

( = C/V)

joule

candela,

the

have

A

L

In

system

by

( = A.

E

1

MKSA

Measures:

I

A4.6.

1970

5

field

introduced unit

is

the

are:

Notation

Symbols

for

units

of

physical

for

units

shall

quantities

8hall

be

printed

in

type. Symbols

shall

remain

unaltered

not

in

the

contain

plural,

a final

e.g.:

full

7cm,

stop

(a

not

7 cms.

Section

A4

1 February Page 6 Symbols

Co

upright shall (weber);

type. start

However, with

Hz d.

fractions

or

e. prefixes

are

a capital

for the

units

shall

symbol

Roman

be printed for

in lower

a unit

derived

e.g.:

m (meter);

letter,

case

from

1970

Roman

a proper

name

A (ampere);

Wb

(hertz). The

following

multiples

prefixes

shall

be used

to indicate

decimal

of a unit.

Prefix

Equiv

deci

(10 -1)

d

centi

(10 -2 )

c

milli

(10- 3)

m

micro

(i0-6)

nano

(10- 9)

n

pico

(i0 -IZ)

p

feint.

(10 -15 )

f

atto

(10"18)

a

deka

(I01)

da

hecto

(10 z)

h

kilo

(10 3 )

k

mega

(106 )

M

giga

(109 )

G

tera

(1012)

T

The available.

use

of double

prefixes

S)rmbol

v

shall

be avoided

when

single

Section

A4

I February Page

f.

combination or

Not:

m_ts,

but:

ns

Not:

kMW,

but:

GW

Not:

_

but:

pF

When

a prefix

shall

cubed

be

without cm

A numerical

prefix thus,

always

No

symbols,

or

following

example

as

mA

shall is

never

be

never

periods

SI

units

A4.7.

the

on 1

non-SI

before

symbol,

a unit which

symbol,

can

be

the

squared

_s 2

2,

used

before

a unit

written,

and

never

or

hyphens

shall

be

Prefixes

symbol

means,

is

which

0.01

(m 2)

but

are

joined

used

with

directly

to

SI

abbreviations,

units,

as

in

the

s :

mN

kV

kHz

MV

mA

GHz

cm

Units

The

(picofarad)

2

prefixes.

SI

(gigawatt)

placed

a new

MN

A4.7.0

(nanosecond)

brackets.

(0.01m)

go

is

2,

crn2

means

symbol

considered

using

Examples:

squared,

F,

1970

7

on

DrawinGs

following

drawings

and

in

paragraphs and

Dual

Units

When

SI

units

units

of

measure

in

are

Analyses

describe

general

techniques

for

using

an

analysis,

analyses.

specified shall

for be

used

use

on

a drawing

parenthetically

or to

in

facilitate

Section

A4

1 February Page comprehension omitted

on

A4.7.

Z

.of the the

assumption

identified

by

the

A4.7.

3

a note

shall

be

on For

Non-SI

are

familiar

units with

shall the

never

be

SI units.

the

used

drawing

frequently

to avoid

on a drawing

repetition

of unit

shall names

be through-

example,

ALL

Tabular

Data

provide

placed

users

of measure

NOTE:

To

analysis.

of Units

units

drawing.

or that

Identification Basic

out

drawing

1970

8

DIMENSIONS

maximum

in separate

ARE

clarity

columns

IN mm

(in.).

of presentation, or

SI and

in separate

tables

non-SI

if the

need

units is

indicated. A4.7.4

Collateral Place

the

alents

in parentheses.

as

one

unit

or

column.

present shows

in a row In

the

some

equivalent

a drawing

with

Use,

SI and

metric

units

or

Non-SI first,

Intables,

other

column,

followed

complex

tables

units

in separate

both

units

given.

and

Units followed

immediately

formats

may

by the

other

drawings tables

be

desirable,

unit

it may and

by the

drawings.

equivsuch

in another be desirable Figure

row to A4-1

Section

A4

1 February Page

"--r_

6 MM 4.236

/ M' 4.236

-

UPRIGHT

(BRASS)

2 REQD

IN.)DRILL-(2) OMM

FRAME t'-'- 5-

0.51

BASE

C394

IN)

--..

300

7. 5

as

the

non-

SI unit,

MMC

DIA

ROD

,SMMC.S ,N..)--,.i ,

11.8 IN,)

..j.--,

I REQD

Collateral

Us_

of

Units

Scales the

with

Kelvin the

or

the

Fahrenheit

Celsius scale

temperature being

scale

optional

as

may

be

used

a parenthetical

SI unit.

A4.8.0

Transitional

The

and SI

A4-1.

Temperature Either

ROD

iBRA_,S)- ---I _

ROD (ALUMINUM')

Figure

SO

MF (2) ,_.-6.35MM(.251N)

A4.

IN)

300 MM( ,,.8 IN)

$

FRAME. MM

1970

9

preferred systems

A4.8.

1

following

styles to Mass The

Indices

explanations

which

are

indicate

to

be

used

nomenclatures,

methods,

the

from

during

transition

non-

SI. vs term

Force "mass"

(and

not

weight)

shall

be

used

to

specify

the

Section

A4

1 February Page quantity

of matter The

acting

on

of the

The kilogram

by

defined

as

pound

thrust,

weight

to be

located.

pound

mass

being

S.

National

exactly

4.448

defined

as

at the

earliest

abbreviated

Ibm,

the

pound

thrust

be

of force,

A4.8.2

The vehicle

abbreviated

is

48 The

(kg),

weight,

or

being

exactly

of Standards;

the

5 newtons

4.448

221

shall

of the

object

615

was

0. 453 pound

by the 260

gravweighed

592

force

NBS;

37 (lbf),

and

5 newtons,

the shall

date. to SI units,

the

pound

be abbreviated

lbf,

mass and

the

shall

be

pound

the

SI unit

of

mass,

shall

not

be used

as

a

thrust.

of Nomenclature

dry 600

force

by a statement

by a statement

260

period

the

to lbf.

kilogram

Examples

exactly

Accordingly,

the

as

615

force

where

practicable

transition

unit

or

defined

221

gravitational

be accompanied

location

Bureau

being

the

The

location.

object

at the

(Ibm),

as the

should

of the

During

shall

be defined

of an object

in m/s2

U.

objects.

at a specified

location

the

be abandoned

shall

object

acceleration

is assumed

in material

"weight"

corresponding

itational or

term

a material

statement of the

contained

1970

10

mass kg

weight

(107

of the

S-I

(first)

stage

of the

Saturn

I launch

139 lbm).

of a man

of

70.0

kg

(154

Ibm)

mass,

standing

on the

Section

A4

1 February Page

surface

I13

of the moon

newtons

(25.4

The

is 6. 689

MN

(i 504

(tad),

there

ever,

the

for the

radian

000

being

stitute

a

ades

it

A4.

10.0

is 1.62

m/s

of the Saturn

I launch

2,

is

vehicle

weight,

and

thrust

is the newton.

be divided

into a rational

number

of radians

2_

radians

arc

degree,

(approximately

arc

of plane

6. 283

minute,

angles.

and

Decimal

tad)

arc

in a circle.

second

multiples

may

How-

all be used

of the degree

or

preferred.

"grad"

right

will

is

angle.

be

found

useful

to a.

measurement

to

wherever Spell

symbol

of

is

angular

not

for

an

SI

many

measure unit,

but,

wherein since

100 it

is

grads

based

con-

on

dec-

purposes.

Style

order

adhered

a unit

This

Preferred In

related

S-I (first) stage

unit of force,

cannot

measurement

are

acceleration

of Angles

circle

radian,

gravitational

Ibf).

preferred

The

be

of the

Measurement A

the

il

Ibf).

thrust

The

A4.9.0

where

1970

in

ensure

maximum

accuracy,

practicable out

a term

parentheses.

in in

full

the

engineering when

Thereafter,

following

analysis

first

used, use

the

style

shall

documentation.

followed related

by symbol

the for

applications. b.

In

general,

state

the

measurement

in

terms

of

the

system

Section

A4

1 February Page of units for

used,

followed

example,

48 c.

place

to the

126

306.

A4.

10.1

kg

(107

between

each

and

359

applicable 000

numerical

right

204

000

In using

a space

used

by the

translated

lbm),

and

values

group

25.4

of decimal

meter

(m3)

lbf

involving

of three

left

value

digits.

points.

Commas

newtons). than

Such

12

in parentheses:

(113 more

1970

three

spaces are

digits,

shall

not

be

used:

60.

Volume The

cubic

The

liter

is now

A4.

10.2

Time The

defined

as

preferred

should

exactly

be used

in preference

to the

liter.

1 dm 3.

unit

of time

associated

unit

of energy

with

time

rates

is

the

second. A4.10.3

Energy The

and

all

other

preferred forms)

although

listed

should

be avoided.

A4.

10.4

in this

International

atures

joule

(J).

document

for

The

Btu,

electrical, calorie,

information,

are

and

thermal, kilocalorie,

poorly

defined

and

Temperature Either

Practical

is the

(mechanical,

the

Practical Celsius

in degrees

Thermodynamic Kelvin

Temperature Rankine,

Kelvin

Temperature Scale Fahrenheit,

Temperature Scale,

may etc.

or

Scale, the

the

International

be used.

Equivalent

, may

be included

temperin

v"

Section

A4

1 February Page

1970

13

-W" parentheses.

Note

degrees that

(OK)

and

degrees

ature

in

Nomograph,

the

defined atures

and

in

a

degrees

Table

and

Celsius

interpolation

are

equations

in

numerically

are

Table

Practical

Practical of

(°Cels)

expressed

Centigrade

A4-2,

International

set

differences

degrees

Figure

International by

temperature

Celsius

Celsius

The and

that

Kelvin

equal

identical.

See

and Temper-

A4-15.

Kelvin

Temperature

Temperature based

Scale Scale

on

the

of

reference

1960

of

1960, are

temper-

A4-1.

Table

A4-1.

International

Temperature

Reference Practical

Temperatures, Temperature

Scale

oK

of

oC

Oxygen:

liquid-gas

equilibrium

90. 18

Water:

solid-liquid

equilibrium

273. 15

0.00

Water:

solid-liquid-gas

273. 16

0.01

Water:

liquid-gas

equilibrium

373. 15

100.00

solid-liquid

equilibrium

692. 655

419. 505

717. 75

444.6

Zinc:

Sulphur: Silver:

Gold:

equilibrium

liquid-gas

equilibrium

solid-liquid

equilibrium

solid-liquid

equilibrium

1233.95

1336.

15

-182.97

960.8

1063.0

Section

A4

1 February Page

A4. 10.5

the

coherent tities

14

Prefixes "Coherent

without

1970

application

units

'_ are

the

entire

desirable.

system

of coherent

science

and

units

be

can

full

range

the

the

rationally

and

the

of all

of numerical

use

of

of physical

SI is

needs

in equations

exclusive

values

stated,

to meet

directly

The

of numerical

previously

represented

be used

coefficients.

available The

that

range

As

technology.

can

units

of numerical

over

is highly

quantities

units

only

complete

branches

values

conveniently

quan-

of

of physical

by utilizing

SI

units. To facilitate approved (or

prefix

before

fractions

A4.

combination

Only

previously

multiples

The to which

the

able

with

sion

factors

listed

prefixes

in units

application

other

of approved

of decimal

information

have

of ten

is

employed,

or

before

an SI unit

is placed

an

shall

be used

to indicate

decimal

Factors

number

applicable

a power

of ten

of SI units).

Measurements by the

a power

of an SI unit.

Conversion

converted

either

representing

any

or

11.0

this,

is to be measuring

been

tabulated

than

those

numerical

places

should

put and

by the

instruments according

of the

and

SI are

conversion be

methods.

to physical

factors.

governed

degree

preferably

by the

of accuracy These quantity.

purpose attain-

conver-

Section

A4

l February Page

A4.

II. I

Basic

The

used

in

prefix

basic

accordance

be

as

exactly

11.2

used

unit of measurement

A4.6.2(d).

on

a

2.54

Noncritical

in

be

11.

herein

3

to

a prefix

in

analysis.

an

is

chosen,

The

no

inch

be

other

(in.)

is

mm

Conversion

to

are

critical,

not

convenient

non-SI

numbers

set

Conversion

to

Other

Conversion

to

SI

and

data

converted

to

the

word

"nominal"

appended

mm

shall

12.0

the units

SI Units

units

other

than

follow

the

rules

as

forth.

Conversion The

itate

or

Once

shall

parentheses.

A4.

A4.

rounded

drawing

Prefixes

is the meter.

cm.

If dimensions shall

15

Unit

with

shall

defined A4.

Linear

1970

conversion

conversion of

the

Tables

Metric

of

factors

most System

given

commonly (or

in

used

conversion

the

following

units of

of non-SI

tables

the

will

English units

to

facil-

system SI

units).

Section

A4

1 February Page

Table

To

Convert

foot/second (gal)

inch/second

Acceleration

To

squared

galileo

A4-2.

squared

Symbol

meter/second

squared

m/s

meter/second

squared

m/s$

meter/second

squared

m/s

Table

To

Convert

1970

16

A4-3.

Multiply *3.048

x

I0 "l

*I.

x

I0 "z

z

000

*2.54

z

by

x

I0 -z

Area

To

Symbol

Multiply

by

sq

foot

sq

meter

m z

*9.

290

304

x

sq

inch

sq

meter

mZ

*6.

451

6 x

10 .4

sq

meter

m z

074

8 x

circular

rail

Table

To

Convert

gram/cu

A4-4.

10 "1_

Density

To

centimeter

5. 067

10 "z

Symbol

Multiply

kilogram/cu

meter

kg/m

3

*1.00

x

by I_

pound

mass/cu

inch

kiiogram/cu

meter

kg/m

3

2.767

990

5 x

I¢

pound

mass/cu

foot

kilogram/cu

meter

kg/m

3

1.601

846

3 x

I0 l

slug/cu

foot

kilogram/cu

meter

kg/m

3

5.153

79 x I0 z

Table

To

Convert

ampere

(Int

ampere

hour

coulomb

(Int

faraday

of

A4-5.

Electrical

To 1948)

of

1948)

(physical)

Symbol

ampere

A

coulomb

C =

coulomb

C = A"

coulomb

C = A.s

Multiply 9. 998

A" s s

farad

(Int

of

1948)

farad

F = A"

henry

(Int

of

1948)

henry

H

ohm

_=

V/A

tesla

T =

Wb/m

ohm

(Int

of

1948)

gamma

*Exact,

as

defined

by

the

National

Bureau

of

Standards.

,'3.60

s/V

= V-s/A

l

by 35

x

x

I0 "I

I03

9. 998

35

x

i0 "l

9. 652

19

x

104

9. 995

05

x

10 "l

1.000

495

1.000

495

* 1.00

x

10-9

Section

A4

1 February Page

Table

To

Convert

[

(Int

of

1948)

maxwell

Btu

(Cont'd)

Symbol

tesla

T

= Wb/m

volt

V

= W/A

weber

Wb

Table

To

Electrical

To

gauss volt

A4-5.

Convert

A4-6.

= V" 8

m

joule

J=N.m

joule

J=N.m

joule

J=N.

joule

J=N.m

joule

5=N.

joule

J=N.m

joule

J=N.

m

joule

J=N.

m

{Int

kilowatt ton

{nuclear

watt

hour

of

1948)

hour

(Int

of

equiv

1948)

of

TNT)

Table

To

A4-7.

Convert

Energy/Area:

To

*1.00 m

0 x

I. 000

165

3. 600

59

m

4.20 "3,

foot.rain

watt/aq

meter

W/m

z

*$Btu/sq

inch.sec

watt/sq

meter

W/m

2 2

erg/sq

centimeter.sec

watt/sq

meter

W/m

watt/sq

centimeter

watt/sq

meter

W/mZ

*Exact,

as

ochemical

)

109

x

l0:

l06

Multiply

$*Btu/sq

rm

60

x

x

l0 "z

Time

I.

**{the

10 "?

011

2

Standards.

x

4.214

Symbol

of

10 "19

9

W/m

Bureau

x

10

817

meter

National

I0 _

1. 355

watt/sq

the

x

184

1,602

foot.sec

by

by

190

*4.

*$Btu/sq

defined

10 "a

02

J=N.

joule

x

4.

joule

poundal

*1.00

87

(thermochemical)

foot

330

i. 055

calorie

force

1.000

10 .4

m

m

pound

x

J=N.

J=N.

foot

_'1.00

joule

joule

erg

by

Multiply

{mean)

volt

z

Symbol

calorie

electron

Multiply

Energy

To

(mean)

1970

17

134

893

1 x

104

1.891

488

5 x

I0 z

1.634

246

2 x

106

*i.00

"I.

by

00

x

10 .3

x

10 4

Section

A4

1 February Page

Table

To

Convert

force

Force

To

dyne

kilogram

A4-8.

(kgf}

Symbol m/sZ

newton

N=kg.

m/p

z

*9.

806

65

m/s

z

*4.

448

221

615

260

138

5 x

10 "l

(avoirdupois)

newton

N=kg.

ounce

force

(avoirdupois)

newton

N=kg.m/s

Table

A4-9.

astronomical

unit

foot

z

10 -5

2.780

Symbol

Multiply

meter

m

*1.00

meter

m

meter

m

*3.

x

1. 495 048

by

10 "1° x

10 Ix

x

10 "i

foot

(U.

S.

survey)

meter

m

'1200/3937

foot

(U.

S.

survey)

meter

m

3. 048

meter

m

"2.

meter

m

9.

460

55

m

*1.

650

763

inch

light

year

wavelengths

micron

meter

m

*1.00

rail

meter

m

*2.

meter

m

meter

m

(U.

S.

statute)

yard

Table

To

Convert

force,

kilogram

mass

pound *Exact,

mass as

secZ/meter

(mass)

(avoirdupois) defined

A4-10.

To

gram kilogram

Kr 86

54

meter

mile

by

the

National

006 x

x

10 73x

10

-5

*1.609

344

x

*9.

x

144

x

I0 "|

-z

x

54

,s 106

10 3

10 "l

Mass

Multiply

kg

*I.

O0 x

kilogram

kg

*9.

806

kilogram

kg

*I.

O0

kilogram

kg

*4.

535

Standards.

10

10 -6

Symbol

of

096

x

kilogram

Bureau

5

Length

To

angstrom

x

*1.00

by

N=kg.

force

Convert

Multiply

newton

pound

To

1970

18

by

I0 "3 65

923

7 x

10 "l

Section 1

February

Page

Table

To

Convert

mass

{avoirdupois)

ounce

mass

{troy

or

pound

mass

{troy

or

2000

(Cont'd)

Symbol

Multiply

b/

kg

_;°2. 834

952

3i2

apothecary)

kilogram

kg

*3.

347

68 x

apothecary)

kilogram

kg

*3.732

417

216

kilogram

kg

390

29 x

kilogram

kg

847

4 x

pound)

Table

To

1970

19

kilogram

slug

{short,

Mass

To

ounce

ton

A4-10.

_\4

A4-11.

Convert

II0

1.459

::'9.

071

5 x lO -z

I0 -z

x

10 l

10 z

Miscellaneous

To

Symbol

Multiply

by

degree

{angle)

radian

rad

1. 745

329

251

994

minute

{angle)

radian

rad

2. 908

882

086

66

second

{angle)

radian

rad

4. 848

136

811

x

684

659

2 x

474

4 x

10 -4

cu

foot/second

cu

meter/second

m 3 /s

cu

foot/minute

cu

meter/second

m3/s

_:"Btu/pound

mass

°F

:'.:2.831 4. 719

joule/kilogram°C

J/kg°C

*4.

184

x

103

joule/kilogram°C

ff/kg°C

*4.

184

x

10 3

joule/kilogram

J/kg

2.

324

444

joule/kilogram

J/kg

::,1.

O0 x

roentgen

coulomb/kilogram

A.

*2.

579

curie

disintegration/second

*3.

70

l:_::Kilocalorie/kg ,_*Btu/pound Rad

°C

dose

absorbed)

Table

To

3 x x

Convert

A4-1Z.

To

l/s

s/kg

10 -z

10

10 -z 76

x

4 x

x

10-4

10 l°

Power

Symbol

Multiply

by

watt

W=

J/s

1. 054

350

264

488

**Btu/minute

watt

W=

J/s

1.757

250

4 x

10t

**calorie/second

watt

W=

J/s

':*calorie/minute

watt

W=

J/s

6.973

333

3 x

10 -z

W=

J/s

1.355

817

9

pound

10 -4

10 -6

**Btu/second

foot

[0-

3

mass

(radiation

I0 "j

force/second

:_'Exact, as defined ::_':: (thernlochemical)

watt by the

National

Bureau

of Standards

*4.

184

888

x

10 t

Section

A4

1 February Page

Table

To

Convert

A4-12.

Power

To

20

(Cont'd)

Symbol

Multiply

by

foot

pound

force/minute

watt

_'=

5/s

2. 259

696

6 x

10 "z

foot

pound

force/hour

watt

W=

J/s

3. 766

161

0 x

10 -4

watt

W=

5/s

7.456

998

7 x

l0 z

watt

W=

J/s

*7.46

_;_kilocalorie/sec

watt

W=

5/s

*4.

_kilocalorie/min

watt

W = 3/s

6. 973

333

watt

W=

1. 000

165

horsepower

(550

horsepower

(electric)

watt

(Int

of

ft Ib force/sec)

1948)

Table

To

Convert

To

atmosphere centimeter

of

centimeter

of water

dyne

/ sq

foot

of

inch

of

mercury

inch

of

water

mercury

(0*C) (4°C)

centimeter water

A4-13.

(39.2°F)

(60°F) (600F)

Symbol N/m

z

newton/sq

meter

N/m

z

22 x

103

newton/sq

meter

N/m

z

9. 806

38 x

I01

newton/sq

meter

N/m

z

:',,I.00x

newton/sq

meter

N/m

z

2. 988

98 x

103

newton/sq

meter

N/m

z

3. 376

85 x

103

newton/sq

meter

N/m

z

2. 488

4 x

N/m|

kilogram

force/sq

meter

nev, ton/sq

meter

N/m

newton/sq

meter

N/mZ

newton/sq

meter

N/m

newton/sq

meter

N/m

newton/sq

meter

N/m

force/sq

foot

millimeter

torr

mercury

(0°c)

(O'C)

*Exact, **(

of

the

as

defined

rmochemical)

by

the

National

Bureau

of

Standards

101

z

*1.013

by

I. 333

meter,

pound

3 x

105

newton/sq

(psi)

103

25 x

centimeter

inch

x

Multiply

newton/sqmeter

force/eq

force/sq

184

10 z

Pressure

kilogram

pound

J/s

x

I0 =|

*9.

806

65

*9.

806

65

I0 z x

104

6. 894

757

2 x

103

4. 788

025

8 x

101

z

1.333

224

x

2

1.333

22

z

x

10 z

10 z

1970

Section

A4

1 February Page

Table

To

Convert

A4-

14.

21

Speed

To

Symbol

Multiply

by

foot/second

meter/second

m/s

'::3. 048

foot/minute

meter/second

m/s

:_5. 08

foot/hour

meter/second

m/s

8.466

inch/second

meter/second

m/s

:,_2. 54

meter/second

m/s

2.777

777

8 x

x

kilometer

/hour

x x

10"i 10 -3

666 x

6 x

(U.S.

statute)

meter/second

m/s

;:'1.609

344

mile/minute

(U.S.

statute)

meter/second

m/s

'_2.682

24

meter/second

m/s

:,_4. 470

4 x

(U.

S.

statute)

Table

To

Convert

A4-15.

x

101 10 "t

Temperature

To

*Celsius

*Cels.

*Centigrade

°C

*Cels.

*Fahrenheit

*F

*Centigrade

°C

*C

=

5/9

(*F-32)

°Rankine

°R

*Centigrade

°C

°C

=

5/9

(*R-491.

*Reaumur

*Re

°Centigrade

*C

°C

=

5/4

*Re

*Fahrenheit

*F

*Celsius

*Cels.

*Cels.

*Fahrenheit

°F

*Reaumur

*Re

*Re

*Fahrenheit

*F

*Rankine

°R

*R

*Rankine

°R

*Celsius

°Cels.

*Cels.=

*Rankine

*R

*Reaumur

*Re

*Re

*Reaumur

*Re

*Celsius

*Cels.

*Cels.

*Centigrade

*C

*Kelvin

*K

*K

To Btu.

*Exact,

inch/sq

as

Convert foot.

defined

Symbol

A4-16.

Thermal

To second.

by

*F

the

National

joule/meter,

Bureau

second-*Kelvin

of

Standards.

10 -l

103

Symbol

Table

10 "s

10 -z

mile/second

mile/hour

1970

Computation =

*C

=

= =

5/9

4/9 *F

(0F-32)

(*F-32) +

459.

5/9 =

=

4/9

69

(0R-491. (*R-491,

=

*C

69)

5/4

+

69)

*Re

273.

16

Conductivity

Symbol J/m-

Multiply s.

°K

5.

188

731

by 5 x

69)

10 z

Section

A4

1 February Page

g_

m

,i,l,l,l,i,l,l,l,i,

"

g

N N

"_....

N

°

"

In

g

N

S

I1 I'-

1970

22

,s:l

In N

:ii'li'lililiili'lilllllllllllllilllll,liiillilillllll bg 0 0

w_

_o

X,,,,I,,,,J,,,,I,,,,

,,,,I,,,,I,,,,I,,,

,,,, ,,li Ji,J ,,,ll,,,,I,,,,I,,,

I

0

ii)

0 I

l)

,

,,,,I,,,,1,,,,1,,,,

T

T

T

_

?

!

lillllililililliliiliilililliliiiililiililliliililili I

C

0

il

I o N

,i,l,i,l,,l,I

,

I,

Ill

o

iiii Ill

o ill

o ill

ill

o

iT7iii ' Ill

Ill

g Ill

o ill

ill

0

0

0

_

_

s ,l,l,l,l,l,l,l,l,i, _1'

,i,l,i,l,i,l,i,l,i

ill

!

W li: h

i. 0

!!

s N

g

s --

ill

Section

A4

1 February Page

Table

To

Convert

day

(mean

day

{sidereal)

hour

(mean

hour

{sidereal)

solar)

(mean

minute

(sidereal)

solar)

month

{mean

second

(mean

second

(sidereal)

tropical

year

Jan,

calendar) solar)

day

Time

To

solar)

minute

A4- 17.

23

Multiply *8.64

x

second

(mean

solar)

second

(mean

solar)

second

(mean

solar)

second

(mean

solar)

second

{mean

solar)

*6. 00

second

{mean

solar)

5.983

second

{mean

solar)

second

{ephemeris)

second

(mean

second

(ephemeris)

104

8. 616 *3.

60

409 x

Use 9.

0 x

170 x

4 x

103

10'

617

628

104

103

3. 590

"2.

by

x

4x

10

106

equation

of time

972

695

7 x

10 -I

*3.

155

692

597

47 x

*3.

153

6 x

IO T

solar)

1900, 0,

hour

12

year

{calendar)

second

(mean

solar)

year

{sidereal)

second

(mean

solar)

3. 155 815

0 x

IO T

year

(tropical)

second

(mean

solar)

3. 155

6 x

iO

IO T

?

Table

To 8q

Convert sq

centipoise

mass/foot,

pound

force'

second second/sq

foot

poise

pounds1,

second/sq

slug/foot,

second

:_ Exact,

as

defined

Viscosity

To

foot/second

pound

A4-18.

foot

by

the

National

Symbol

meter/second

Multiply

m'/s

newton,

second/sq

meter

N's/m

z

newton,

second/sq

meter

N.

s/m

z

newton,

second/sq

meter

N.

s/mZ

newton,

second/sq

meter

N.

s/m

newton,

second/sq

meter

N.

s/mZ

newton,

second/sq

meter

N.

s/mZ

Bureau

of

692

Standards.

z

*9.

290

':'1.

00

by

304 x

x

10

10 "J

1. 488

163

9

4. 788

025

8 x

':'1.

00

1. 488 4,

788

x

-2

10*

10"*

163

9

025

8 x

I0 i

1970

Section

A4

1 February Page

1970

24

TableA4-19.Volume To fluid cu

Convert

ounce

To

(U.S,)

foot

gallon cu

(U.

S.

liquid)

inch

liter

pint quart

ton

(U,

S.

liquid)

(U.S.

liquid)

(register)

Table

To

Symbol

Multiply

cu

meter

m 3

*2.

cu

meter

m 3

"2.831

684

659

cu

meter

m 3

*3.

411

784

x

cu

meter

m 3

*1.638

706

4 x

10 "s

cu

meter

m 3

000

x

cu

meter

m 3

764

73 x

cu

meter

m 3

529

5 x

10 -4

cu

meter

m3

684

659

2

A4-20.

Alphabetical

Convert

Listing

352

785

1,000

.4,731

9. 463

*2.831

of Conversion

To

957

by

ampere

A

abcoulomb

coulomb

C=

abfarad

farad

abhenry

henry

abmho

mho

abohm

ohm

abvolt

acre

25

10 -3

I0 -4

Multiply

by

10 l

A- s

* 1. 00

x

10 l

F=

A.

s/V

*l.

x

10 #

H=

V.

s/A

*1,00

x

10 -9

* I. 00

x

109

f_ = V/A

*I.00

x

I0 -_

volt

V=

*I.00

x

I0 -s

sq meter

m

ampere

A

angstrom

meter

m

*1.

are

sq

m z

*1.00

ampere

{Int

astronomical

of

1948)

unit

meter

meter

atmosphere

newton/sq

meter

N/mZ

bar

newton/sq

meter

N/m

barn

sq meter

* Exact,

as

defined

by

the

National

Bureau

m z

of

Standards.

00

*4. 046

856

9. 998

m

00

35

422

x

x

*1.013

4 x

10 "i

x 10 -l°

1.495

z

10 -z

10 "3

x

z

10 -s

2 x

*1.00

W/A

x

Factors

Symbol

abampere

956

10 z 98

x

10 It

25

x

105

*1.00

x

l0

s

*I.00

x

I0 "z8

10 s

Section

A4

i February Page

1970

25

f_

Table

To

A4-Z0.

Alphabetical

Listing

Convert

To

barye

newton/sq

Btu

(Int

Btu

Steam

of Conversion

Table)

Factors

(Cont'd)

Symbol meter

N/m

Multiply z

::.'I.00

x

I0-

by

L

joule

3=

N.

m

I. 055

04

x

103

(mean)

joule

J=

N. m

1. 055

87

x

I0

Btu

(thermochemical)

joule

J=

N.

m

I. 054

350

Btu

(39°F)

joule

J=

N. m

I. 059

67

x

103

Btu

(60°F)

joule

J=

N. m

I. 054

68

x

10

3

264

488

888

688

x

x

3

bushel

(U.S.)

cu

meter

m 3

,x3.523

907 56

cable

meter

m

"2.

caliber

meter

m

,._2. 54

joule

J=

N.

m

4.

186

8

4.

190

02

calorie

(Int

calorie

(mean)

joule

3=

N. m

calorie

(thermochemical)

joule

J=

N,

calorie

(15°C)

joule

J=

N.m

calorie

(20°C)

joule

J=

N.

calorie

(kilogram,

Int

joule

I=

calorie

(kilogram,

mean)

joule

calorie

(kilogram,

thermochemical)

carat

Steam

Table)

Steam

Table)

(metric)

*Celsius

(temperature)

centimeter

of

mercury

centimeter

of

water

(4°C)

_4.

x

x

184 185

80

4.

181

90

N.m

4.

186

8 x

J=

N.m

4.

190

02

joule

I=

N.m

kilogram

kg

meter

N/m

newton/sq

meter

N/mZ

_._4. 184

_2.

*K

newton/sq

m

00

x

x

*K=

z

I0 z

10 -4

4.

*Kelvin

(0*C)

m

194

016

I03 x

103

103

10 -4

°C

+ 273.

1. 333

22

x

103

9. 806

38

x

101 I0

16

1

chain

(surveyor

chain

(engineer

circular

or

or

gunter)

ramden)

mil

cord

--

*Exact,

as

defined

by

the

National

Bureau

meter

m

_:,2.011 68 x

meter

m

_:¢3. 048

sq meter

mZ

5. 067

074

8 x

cu meter

m 3

3.624

556

3

of

Standards.

x

i0 l

I0 -l°

I0 "z

I03

Section 1

A4

February

Page

Table

To

A4-20.

Alphabetical

Listing

Convert

of

Conversion

To

Factors

1970

26

(Cont'd)

Symbol

Multiply

by -!

coulomb

(Int of 1948)

coulomb

C=

cubit

meter

m

_4.57Z

x

cup

cu

m _

_2.

882

curie

disintegration/second

I/s

*3.70

x

1010

*8.64

x

104

day

(mean

day

(sidereal)

degree

solar)

(angle)

denier

(International)

meter

second

(mean

solar)

second

(mean

solar)

A" s

9- 998

35 x

365

I0

I0 "l

365

x

10 -4

8. 616

409

0 x

104

1. 745

329

251

994

3 x

I0 _z

5 x

I0 "3

5 x

l0 "_

radian

rad

kilogram/meter

kg/m

*I.00

x

I0 "7

dram

(avoirdupois)

kilogram

kg

:sl.771

845

195

312

dram

(troy

kilogram

kg

_3. 887

934

6 x

10 -3

dram

(U.S.

cumeter

m3

*3. 696

691

195

312

newton

N-- kg.m/s

joule

J=

N.m

joule

J=

N. m

or apothecary)

fluid)

dyne

electron

volt

erg

z

_I.00

x I0 "5

1.602

I0 x

I0 "19

,',_ I. 00 x l0 -7

"Fahrenheit

(temperature)

"Celsius

°C

°C

=

5/9

(°F

- 32)

°Fahrenheit

(temperature)

°Kelvin

°K

°K =

5/9

(°F

+

farad

(Int of 1948)

on carbon

12)

farad

F=

A.s/V

9. 995

05 x I0 "I

coulomb

C=

A, s

9. 648

70 x

A. s

9. 649

57 x 104

9. 652

19 x

faraday

(based

faraday

(chemical)

coulomb

C=

faraday

(physical)

coulomb

C-- A.s

fathom

meter

m

_ I. 828

¢" rmi

meter

m

_I. 00 x

cu meter

m 3

*2. 957

352

meter

m

'_3.048

x

meter

m

_ 1200/3937

459, 69)

104

104

8 -15

0

j fluid ounce

(U. S. )

I foot foot

(U.

*Exact,

S. survey)

as

defined

by the

National

Bureau

of

Standards.

I0

956

I0 "i

25 x I0 -5

Section l

A4

February

Page

1970

27

v

Table

To

A4-Z0.

Alphabetical

Convert

Listing

of Conversion

To

Factors

(Cont'd)

Symbol

Multiply

by -!

foot

(U.S.

foot

of

survey)

water

(39.

meter Z°F)

newton/sq

foot-candle

lumen/sq

furlong

meter

gal

meter/second

m

3. 048

006

meter

N/mZ

2. 988

98

meter

lm/m

I. 076

391

squared

z

m

_:_2. 011

m/sZ

*I. 00 x

096 x

x

10

10 3 0 x

68 x

l0 |

I0 z

I0 -z

gallon

(British)

cu meter

m3

4. 546

087

x

gallon

(U.S.

dry)

cu meter

mS

_4. 404

883

770

86 x

gallon

(U.S.

liquid)

cu meter

m s

_3. 785

411

784

x

gamma

tesla

T=

Wb/m

z

::_i. 00 x

i0 "9

gauss

tesla

T=

Wb/rn

z

*i. 00 x

I0 "4

gilbert

ampere

gill (British)

cu

meter

gill

cu meter

(U.S.)

turn

(angular)

10 -s

7. 957

747

2 x

m s

I. 420

652

x

m s

I.

941

2 x

grad

degree

1°

grad

radian

rad

grain

kilogram

kg

182

10 -s

10 -s

i0 -|

10 -4

10 -4

::_9.00 x 10 "!

I. 570

796

:,_6. 479

3 x

891

x

i0 "z

10 -5

-s gram

kilogram

kg

* 1. 00

hand

meter

m

'_1.

hectare

sq

mZ

',_ 1.00

henry

(Int

of

hogshead

1948)

(U.S.)

henry

H=

eu meter

mS

watt

V.

998

7 x 10'

9. 809

50 x

W=

horsepower

(electric)

watt

W = J/s

horsepower

(metric)

watt

W = .I/s

National

Bureau

of

Standards.

495

7. 456

watt

the

1. 000

104

W = J/s

(boiler)

by

x

10 "!

423

horsepower

defined

x

809

(550

as

s/A

016

10

*2. 384

horsepo_ver

_Exact,

foot Ibf/second)

meter

x

J/s

*7. 46 x 7. 354

10 s

I0 z 99

x

102

92 x

10 1

Section

A4

1 February Page 28

Table

To

,6,4-20.

Alphabetical

Listing

Convert

of

Conversion

To

Factors

(Cont'd)

Symbol

horsepower

(water)

watt

W=

hour

(mean

solar)

second

(mean

solar)

hour

(sidereal)

second

(mean

solar)

Multiply

J/s

7. 460

by

43 x

*3.60

x

I0 z

I0 z

3. 590

170

4 x

103

hundredweight

(long)

kilogram

kg

*5,080

234

544

x

hundredweight

{short)

kilogram

kg

_4.

923

7 x

101

meter

m

*2.54

inch

inch

of

mercury

(32°F)

newton/sq

meter

N/m

inch

of

mercury

(60°F)

newton/sq

meter

N/m2

inch

of

water

(39.2°F)

newton/sq

meter

N/m

inch

of

water

{60°F)

newton/sq

meter

(Int of

joule

1948)

joule

kayser

°Kelvin

(temperature)

kilocalorie

(Int

Steam

Table)

kilocalorie

(mean)

kilocalorie

(thermochemical)

2

535

x

I0 "b

3. 386

389

3. 376

85

z

2. 490

82 x l0 z

N/m

z

2. 488

4 x

J=

N- m

I. 000

165

I/meter

I/m

*I.

°Celsius

°C

°C=

00

x

x x

10 3 10 a

10 b

10 b

°K

- 273.

16

joule

3= N. m

4. 186

74 x

103

joule

J= N. rn

4. 190

02 x

I03

joule

J= N. m

:_4. 184

* I. 00

I0 |

x

103

kilogram

mass

kilogram

kg

kilogram

force

newton

N-- kg.m/s

b

*9. 806

65

newton

N=

kg.m/s

z

*9. 806

65

newton

N=

kg. m/sb

*4. 448

221

615

260

meter/second

m/s

444

444

x

8

103

kilopond

force

kip

knot

(lnternational)

5. 144

lambert

candela/sq

meter

cd/mZ

lambert

candela/sq

meter

cd/m

langley

joule/sq

Ibf(pound

*

Exact,

force,

as

defined

avoirdupois)

by

the

National

meter

newton

Bureau

Standards.

3. *4.

J/m2

N=

of

* I/pi

z

kg.m/sZ

1970

x

I0 "i

10 4

183

098

184

x

_4.448

5 x

x

104

221

615

260

5

103

Section

A4

1 February Page

1970

29

_f.Table

To

A4-Z0.

Alphabetical

Listing

Convert

To

of

Conversion

Factors

(Cont'd)

Symbol

Multiply

by

kilogram

kg

_4.

535

923

7 x

meter

m

'_5.

559

552

x

meter

m

"5. 556

x

meter

m

:._4. 828

032

meter

m

9. 460

gunter)

meter

m

ramden)

meter

m

liter

cu

m 3

lux

lumen/sq

maxwell

weber

Ibm

(pound

mass,

league

(British

league

(Int

league

(statute)

avoirdupois) nautical)

nautical)

light-year

link

(surveyor

link

(engineer

mete

or or

r

meter

meter

wavelengths

2

Wb=

V.

Kr 86

103

103

x

103

55

x

10Is

:',_2.011

68

x

I0 "i

#3.

x

048

1. 000

lm/m

I0 -L

I0 °i

000

x

10 -_

1. 00 s

*1.00

::,1.

x

650

10 "a

763

73

x

l06

micron

meter

m

_ 1.00

x

l0 -6

rail

meter

m

;"2.

x

l0 °5

meter

m

:_ 1. 609

344

x

10 _

meter

m

,*1.853

184

x

l03

meter

m

*1.852

x

10

meter

m

* 1. 852

x

10 _

mile

(U.S.

statute)

mile

(British

mile

(Int

mile

(U.

nautical)

54

3

nautical) S.

millimeter

nautical) of

mercury

(O°C)

millibar

(angle)

minute

(mean

minute

(sidereal) (mean

solar)

calendar)

oersted

ohm

SExact,

(Int

of

as

meter

N/mZ

newton/aq

meter

N/mZ

radian

minute

month

newton/sq

1948)

defined

by

the

National

rad

1. 333

-'*1.00

224

x

Z. 908

second

(mean

solar)

'_6.00

second

(mean

solar)

5. 983

second

(mean

solar)

,,_2. 628

x

l0 z 882

x

617 x

7. 957

747

ohm

i'l = V/A

I.

495

of

Standards.

66

4 x

I01

106

A/m

Bureau

086

i0 1

ampere/meter

000

10 z

2 x

10 i

x

10 .4

Section

A4

1 February Page

Table

A4-Z0.

Alphabetical

To Convert

Listing of Conversion To

Factors

30

(Cont'd)

Symbol

Multiply by *2. 834 952

ounce

mass

(avoirdupois)

kilogram

kg

ounce

force

(avoirdupois)

newton

N= kg. mls z

ounce

mass

(troy

kilogram

kg

*3.

ounce

(U. S.

fluid)

cu mete r

m3

*2. 957 352

pace

mete r

m

"7.62

parsec

meter

m

pascal

newton/sq

or

apothecary)

meter

N/m

1970

2. 780

312 5 x 10 -z

138 5 x I0 -i

110 347

68 x 10 "z 956 25 x 10 "5

x 10 -t

3. 083

74 x 1016

*I. O0

z

cu meter

m3

*8. 809 767 541 72 x I0-'L

pennyweight

kilogram

kg

*1.555

173 84 x 10 -3

perch

meter

m

*5. 029

2

phot

lumen/sq

peck

(U. S. )

meter

Im/m

z

1. O0 x 104

pica

(printers')

meter

m

'_4. 217

517

pint

(U. S. dry)

cu meter

m_

*5.

104 713

cu meter

m3

.4.731

754

73 x lO "4

meter

m

*3.

598

x lO -4

poise

newton.second/sqmeter

N. slm z

*1.00

pole

meter

m

*5.

*4. 535

923

7 x 10 "t

z *4. 448

221

615

260

*3.732

417

216

x 10 "1

.1.382

549

543

76x

*1.

101

220

942

715x

9. 463

529

5 x 10 .4

pint (U. S. point

liquid)

(printers')

pound

mass

(1bin0 avoirdupois)

kilogram

kg

pound

force

(lbf,

avoirdupois)

newton

N =

pound

mass

(troy

or

kilogram

kg

newton

N =

apothecary)

poundal

kg.m/s

kg.m/sZ

505

514

6 x lO "3

x 10 -l 029

Z

quart

(U.S.

dry)

cu meter

m3

quart

(U. S.

liquid)

cu meter

m 5

Rad

(radiation

joule/kilogram

J/kg

* I.00 x I0-z

*Centigrade

*C

"C = 5/4

° Reaumur

*Exact,

dose

absorbed)

(temperature)

as defined

by

the National

Bureau

of Standards.

575 x lO "4

5

10 "t 10 "a

"Re

_"

Section

A4

1 February Page

Table

A4-20.

Alphabetical

Listing

of Conversion

Factors

1970

31

(Cont'd) ]

To

Convert

To

rhe

sq

rod

meter

roentgen second

(angle)

second

(mean

second

solar)

(sidereal)

section

Symbol

meter/newton,

second

'_1.

00

m

'x5.

029

coulomb/kilogram

C/kg

_2.579

radian

rad

second

(ephemeris)

second

(mean

m z /N.

Multiply s

x

solar)

10 l

2

76

4. 848

Use

by

x

10 -4

136

811

equation

x I0 "6

of time.

9. 972

695

7 x

I0"*

sq meter

mZ

,2. 589

988

II0

336

kilogram

kg

,_I.Z95

978

2 x

10-3

shake

second

s

skein

meter

m

slug

kilogram

kg

1.459

span

meter

m

,wZ. 286

statampere

ampere

A

3. 335

640

x

I0 "_

statcoulomb

coulomb

C = A" s

3. 335

640

x

10 "l°

statfarad

farad

F=

A. s/V

I. llZ

650

x

stathenry

henry

H=

V.s/A

8. 987

554

x

I0 Li

statmho

mho

I. 112

650

x

i0 "tz

statohm

ohm

f/

8._)87

554

x

I011

statvolt

volt

V = W/A

Z. 997

925

x

I0 z

stere

cu meter

m3

stilb

candela/sq

stoke

sqmeter/second

m z /s

'>1.00

tablespoon

cu meter

m 3

*I. 478

676

478

teaspoon

cu

m 3

*4.

921

593

scruple

*Exact,

(apothecary)

as defined

by the

National

meter

meter

Bureau

of Standards.

I. 00 x

,xl.

= V/A

cd/mZ

097

x

106

x

l0 "s

10 "8

Z8

x

l0 z

390 x

Z9

x

l0 t

10 "l

I0 "*z

*I. 00

I. 00

9Z8

x

l04

x

10 -4 125

75

x

10 -6

Section 1

A4

February

Page

Table

To

A4-20.

Alphabetical

Listing

Convert

of Conversion

To

Factors

1970 32

(Cont'd)

Symbol

Multiply

by -2

ton (assay)

kilogram

kg

2. 916

666

6 x

I0

kilogram

kg

*9. 071

847

4x

I0 z

ton (long)

kilogram

kg

$1. 016

046

908

8 x

ton (metric)

kilogram

kg

,_I. 00 x

joule

J=

659

2

ton (short,

ton

2000

(nuclear

pound)

equiv,

of

TNT)

ton (register)

cu

tort

newton/sq

(0°C)

meter

N.m

m3

meter

-*2.831

z

x

109

684

I. 333

22 x

9. 323

957

2 x

1.256

637

x

sq

unit pole

weber

Wb=

volt

V=

W/A

I. 000

330

watt

W=

J/s

1.000

165

meter

m

watt

(Int

of

(Int

of

1948) 1948)

yard

m z

103

township

volt

meter

N/m

4. 20

V.s

I0 z

I0 v

10 -7

*9.

144

x

10 -l

*3.

153

6 x l0 T

year

(calendar)

second

(mean

solar)

year

{sidereal)

second

(mean

solar)

3. 155

815

0 x

107

year

(tropical)

second

(mean

solar)

3. 155

692

6 x

107

second

(ephemeris)

592

597

47 x

year

1900, day

_Exact,

as

tropical, 0, hour

defined

Jan,

s

_3. 155

103

107

12

by

the

National

Bureau

of

Standards.

_:

Section

A4

1 February Page

Table

A4-Zl.

Inch

Inch

Decimal

Fraction

Decimal

and

Metric

Equivalents

of Fractions

1970

33

of an Inch

Millimeter

Centimeter

Meter

(mm)

(cm)

(m)

0.015

625

1/64

0.396

87

0. 039

687

0. 000

396

87

0.031

25

1/32

0.793

74

0.079

374

0. 000

793

74

0.046

875

3/64

1. 190

61

O. 119

061

0.001

190

61

O. 062

5

1/16

1. 587

48

O. 158

748

O. 001

587

48

0.078

125

5/64

1.984

35

O. 198

435

0.001

984

35

0.093

75

3/32

2. 381

23

0.238

123

O. 002

381

2-3

O. 109

375

7/64

2-. 778

09

0.277

809

O. 002-

778

09

1/8

3. 174

97

O. 317

497

O. 003

174

97

O. 125 O. 140

625

9/64

3.571

83

O. 357

183

0.003

571

83

O. 156

25

5/32

3.968

71

0.396

871

0.003

968

71

O. 171

875

11/64

4.

57

0.436

557

0.004

365

57

O. 187

5

3/16

4.762

45

0.476

245

0.004

762-

45

0.2-03

125

13/64

5. 159

31

O. 515

931

0.005

159

31

0.2-18

75

7/3Z

5. 556

2-0

0. 555

620

0.005

556

20

0.2-34

375

15/64

5. 953

05

0. 595

305

0.005

953

05

I/4

6. 349

94

0. 634

994

0. 006

349

94

17/64

6. 746

79

0. 674

679

0. 006

746

79

0. 25

365

0. 265

625

0.2-81

2-5

9/32

7. 143

68

0. 714

368

0. 007

143

68

0. 296

875

19/64

7. 540

53

0.754

053

0.007

540

53

0.312

5

5/16

7.937

43

0.793

743

0.007

937

43

0.328

125

21/64

8. 334

27

0.833

42-7

0. 008

334

Z7 17

0.343

75

11/32-

8.731

17

0.873

117

0.008

731

0. 359

375

23/64

9. 128

01

0.912

801

0.009

12-8 01

3/8

9. 5Z4

91

0.952- 491

0.009

524

91

25/64

9.921

75

0.992

0.009

921

75

0.375

0.390

625

175

Section

A4

1 F-ebruary Page

1970

34 v

Table

A4-21.

Decimal

and

Metric

Equivalents

Inch

Inch

Millimeter

Decimal

Fraction

(ram)

of

Fractions

of

an

Inch

(Cont'd)

Centimeter

Meter

(cm)

(m)

0.406

25

13/32

10.318

65

1.031

865

0.010

318

65

0.421

875

27/64

10.715

49

1.071

549

0.010

715

49

0.437

5

7/16

11.

40

1. 111

240

0.011

112

40

0.453

125

29/64

II.509

23

I.

150

923

0.011

509

23

0.468

75

15/32

11.906

14

1. 190

614

0.011

906

14

0.484

375

31/64

12.302

97

1.230

297

0.012

302

97

1/2

12.

699

88

1. 269

988

0. 012

699

88

096

71

1. 309

671

0. 013

096

71

0.5

112

0. 515

625

33/64

13.

0. 531

25

17/32

13.493

62

1. 349

362

0. 013

493

62

0. 546

875

35/64

13.

45

1. 389

045

0. 013

890

45

0.562

5

9/16

14.287

37

1.428

737

0.014

287

37

0.578

125

37/64

14.

19

1.468

419

0.014

684

19

19/32

15.081

11

1.508

111

0.015

081

11

15.477

93

1.547

793

0.015

477

93

5/8

15.

85

1. 587

485

0. 015

874

85

0.593

75

0.609

375

0. 625

39/64

890

684

874

0.640

625

41/64

16. 271

67

1.627

167

0.016

271

67

O. 656

25

21/32

16.

668

59

1. 666

859

O. 016

668

59

O. 671

875

43/64

17.

065

41

1. 706

541

O. 017

065

41

0.687

5

I1/16

17.462

34

1.746

234

0.017

462

34

0.703

125

45/64

17.

15

1.785

915

0.017

859

15

0.718

75

23/32

18.256

08

1.825

608

0.018

256

08

O. 734

375

47/64

18.

652

89

1.865

289

O. 018

652

89

3/4

19.

049

82

I. 904

982

O. 019

049

82

O. 75

859

O. 765

625

49/64

19.446

63

1.944

663

O. 019

446

63

0.781

25

25/32

19.843

56

1.984

356

0.019

843

56

SECTION B STRENGTHANALYS I S

SECTION B l

JOINTS AND FASTENERS

TABLE

OF

CONTENTS

Page BI.0.O

Joints

1.1.0

and

Mechanical

Joints

i.i.i

Riveted

1.1.2

Bolted

Joints Joints

Fasteners

.................

I

..............................

i

...............................

2

Flush

Rivets

................................

19

1.1.5

Flush

Screws

................................

24

1.1.6

Blind

Rivets

................................

27

1.1.7

Hollow-End

1.1.8

Hi-Shear

1.1.9

Lockbolts

...................................

39

Jo-Bolts

...................................

41

Welded

Rivets

Rivets

Fusion

Welding

1.2.2

Effect

on

1.2.3

Weld-Metal Welded

1.2.5

Flash

1.2.6 1.2.7

Brazing

Adjacent

Welding Welding

Due

in

to

39

Spot

46 46

.............................. Parent

Metal

Due

to 46

............................

Allowable

Reduction

39

.............................

Welding Cluster

Spot

............

...................................

1.2.1

1.2.4

Bolts

...........................

Rivets

Joints

and

2

Protruding-Head

1.1.4

Fusion

1.3.0

and

1.1.3

i.i.i0 1.2.0

Fasteners

Strength

...............

47

..............................

49

...............................

49

................................

5O

Tensile Welding

Strength

of

Parent

.......................

.........................................

Metal 56 59

1.3.1

Copper

Brazing

..............................

59

1.3.2

Silver

Brazing

..............................

59

Bl-iii

Section 25 Page

B 1.0.0

Joints

B

I.i.0

Mechanical

B

i.i.i

Riveted

and

Although it

is

at

the

and

and

Fasteners

state

ignore

rivet

the

holes,

made,

stress

unequal

applied

load

the

rivets,

friction

riveted

is

as

rivet

assumed

be

the

is

complex,

concentration among

the

and

as

to

between

load

across

summarized

joint

stress

o£

stress

between

are

The

a

division

shear

stress

which

in

considerations

of

bearing

are

of

such

distribution

of

(l)

and

actual

to

of

assumptions

1961

i

Fasteners

Joints

the

nonuniform

rivet

i

Joints

customary edge

B

September

fasteners,

section

plate.

of

the

Simplifying

follows:

transmitted

entirely

connected

plates

by

being

ignored. (2)

When is

the on

center

the

the

total

are

assumed

and

to

to

carry

be

The the

shear rivet

(4)

The

bearing

loaded

on

equal

is

stress

or

this

line,

the

parts

assumed

distributed

times

plate

in

compression

the

of

or

long

riveted

between rivets,

joints

out

are

summarized

as

follows:

The

distance

from

1 3/4

diameters,

than

on

of

against

in

a

the of

is to

rivet or

assumed

is

The

minimum

the

same

size;

otherwise.

across

assumed

to

rivet

rivet

spacing

be

assumed

basis them

of is

be

diameter

uniformly

to

due

a plate

between

bending

along

a

standard

to to

a a

these

be

uniformly

shall

assumptions

strictly

to

rivet

or zigzag

causes, and

sheared

edge or

be

3 diameters.

rivets by

shall

rolled

such of

upsetting

when

specifications

planed

edge

insufficient line

is

correct.

secondary

r-

(2)

of joint

areas

the

to

diameters.

-

of

the

distributed

rivet

failure

the

failure

staggered,

(1)

if

rivets

area.

none

rivets,

tensile

guarded

of

section

equal

member

gross

secondary

tearing

adjacent or

the

although

of

or

a

over

possibility

shearing

and

is

The

practice,

uniformly

the

centroid

thickness.

(6)

design

load

be

area

of the

rivets

their

an

tension member the net area.

accepted

the

each

when

to

plate

over

in a over

stress

of

to

between

uniformly

of

load,

The stress distributed

plate

iA

is

area

the

proportionally

stress section.

the

of

(5)

The

of

cross-sectional action

area

(3)

The

as

of

rivet

distributed

the

of

line

are

provisions

not

edge,

be I

less

1/2

Section

B

1

25 September Page 2 BI.I.I

Riveted

(3)

Joints

The maximum 7 diameters,

(Cont'd_ rivet pitch and at the

be 4 diameters of the member.

(4)

1961

for

in the direction of stress shall be ends of a compression member it shall

a distance

equal

to

1 1/2

times

the

width

In the case of a diagonal or zigzag chain of holes extending across a part, the net width of the part shall be obtained by deducting from the gross width the sum of the diameters of all the holes in the chain, and adding, for each gauge space in the chain, the quantity $2/4g, where S = longitudinal spacing of any two successive holes in the chain and g _ the spacing transverse to the direction of stress of the same two holes. The critical net section of the part is obtained from that chain which gives the least net width.

(5)

The shear and bearing basis of the nominal the hole diameter.

stresses shall rivet diameter,

If the rivets of a joint are so arranged of the load does not pass through the centroid the effect of eccentricity must be taken into B 1.1.2

Bolted

be calculated on the the tensile stresses

that the line of the rivet account.

on

of action areas then

Joints

Bolted joints that are designed on the basis of shear and bearing are analyzed in the same way as riveted joints. The simplifying assumptions listed in Section B i.i.i are valid for short bolts where bending of the shank is negligible. In general when bolts are designed by tension, the Factor of Safety should be at least 1.5 based on design load to take care of eccentricities which are impossible to eliminate in practicaldesign. Avoid the use of aluminum bolts in tension. Hole-filling fasteners (such as not be combined with non-hole-filling bolt or screw installation).

conventional fasteners

solid rivets) should (such as conventional

0

B

1.1.3

Protruding-Head

Rivets

and

Bolts

The load per rivet or bolt, at which the shear or bearing type of failure occurs, is separately calculated and the lower of the two governs the design. The ultimate shear and tension stress, and the ultimate loads for steel AN bolts and pins are given in Table B 1.1.3.1 and B 1.1.3.2. Interaction curves for combined shear and tension loading on AN bolts are given in Fig. B 1.1.3-1. Shear loads for MS internal wrenching bolts are specified

and tension ultimate in Table B 1.1.3.3.

Section 25

1.1.3

Protruding-Head

In given in

computing

in

Table

rivet

rivet for

D/t

room

yield elevated for

bolted

Where

D/t

_

performed. rigid

parts

room

temperature;

For

convenience,

B

of

1.1.3.7.

strength

to

contains

unit

riveting, B

1.1.3.5,

it

in

I00

These

without temperatures

however,

sheet

Factors

representing

i00

are

ksi

bearing is

which

strength

unnecessary account

in

bearing

of to

for

will

use

high

B

sheets the

and

and

1.5

no

be

higher

the

data

correction stresses

relative

Yield

those

for

rivets,

given sheet

Table

in

based Table

bearing B

1.1.3.9

magnesium-alloy

factors the

of

Table

rivet.

and

specified

available.

on

only

of

parts.

For

5.5.

applicable

are

actual

bolts.

<

bearing

than

is

at

riveted

D/t

possibility

of

of

to

where

are

is

1.1.3.8.

materials properties

ultimate

strength

on

bearing

the

and

applicable

diameters,

ratio

Table

on

protruding-head

various

stresses

there

the

given

are

yield

hole

for

used

quantitative

nominal

reductions

joints,

strength

deformation

no

"unit" and

are

bearing

where

for the

and

holes

low

ksi

in

alloys,

factors

the

1.1.3.6.

stresses

joined at

stresses

strength

substantiate

joints

bearing

shear B

correction for

single-shear

given

of

the

compensate

high

Table

are

group

of

for

stress

given

to

to

for

bearing

be

stresses

a

basic

cylindrical

tests

ultimate

on

or

where

5.5,

the

The

ultimate

alloy

design of

is

strength,

3.0

1961

3

(Cont'd)

used from

of

temperatures

each

must

the

motion

and

joints

strengths for

rivets

and

stated

excess

joints.

The

be

resulting

in

Bolts

shear

should

strength

double-shear

and

rivet

1.1.3.5

ratios

aluminum-alloy

or

aluminum B

shear

at

Rivets

I

September

Page B

B

Section

B

i

25 September Page 4 B

1.1.3

Protrudlng-Head

Rivets

and

Bolts

_Cont'd)

_0 o• 0 _D

= : : = 0

0000

0

:i

e

•

.

:

,

i"

:

:i

"

°

•

•

=

00000

O0

0

000 _0

00000 0_0_0

000 000

00000 00000

_

_00.

• : : : : : : : : : : :

_0

_

_-..t _..I

F...4 ¢',4

_0

00000 00000

O 0 r._

O0 O0

I.I

0

= .,-I

el

00 _0

_

0 _0 •_ 0

_

_

_

°

°

.e _0000 __

O0 O0

,.-I ,--I

Omm

Omm::

ggX

ggg'! :

00000 _000 _0_

O

,4 ,.4

0 0

0 "_

0

_

_1

O0 O0 _

__0 _0_ __0 0000_ °.°.°

_" _ _ __ °°0,°

_0_ _0_ •

°

°

.

.

moo

_m_::

d_d

d£d':

0_ 0_0

__ _0_ _0_ °°°0°

0__ _0_ __ __ 0°°°°

__ __ 0__ _0_ °°°°°

__

_0_

_00 _0_0 000_ °°°.°

z

__

°°°°.

°°0°.

'_'_

-*__

0 _

N

_1_1

0__

__

I

.

1961

Section

B i 1961

25 September Page 5

BI.I.3

Protruding-Head ,._ 0

_ • ,-I

Rivets

and

Bolts

(Cont'd)

0 _

,-q 0 !

•,_ .,_

I

I

I

I

I

I

I

eq

-_"

_D

O00

I

I

I

I

00

o'_

t

_J

co

O

_

,,-.4

o

000000000000 _00000000000

_J

0_

[13

_00_

Oa

L_ ,--4

O 4J

;J

_ u_

0_0_0_000 ____ ___0_

U'3

_J I

o

0 00000000 _000_0_ _ _0_0_0

_ __

_O 4J _

4_ Q;

_0_

__0_00_

_

n_ 0

'.D

_

-,-4

_____

_

__0_

_J OO

__

____

__0_

Oq __t_O_

L_ <

U_ 0

U-_.
L_O_ r_ r_-

-_' _

0000 00000

¢xlO0'--_ t_'O_

_00_ Cq
O0

',_0 C_ ',.00_

O_

t"_'

O0

¢xl

,'--_ 000

,--_


0

cO

.= _;

4J _3

-_

0 000

00

0

o°°o°°°°o °°°°ooo

r_ ,--_

o ¢'_

o'_ _-

..._ r._

u_

oooS o

qP _J GO

.-I 0-_

_0____0___

(D

.c=

b9 0

CO

z

<1"

_D

O0

0

_

_

_

O_

¢q

U ,-4 .,-I

r-q

0.., 0

_

0 r-_

.a E_

_J

-,-4 _ _

.,.-4 JJ ,--4

N CO

_ 0

_ _0_0

._

•

_ • u_O_O0_

_

•

• ____ _

___

I

Section

B

1

25 September Page 6 BI.I.3

Protruding-Head

Rivets

and

Bolts

!AN-10

3O

(Cont'd)

Interaction

r--_._.

x3

Formula 2

25

=

20

Where : x =

\

,

_

y = a = b =

shear

load

tens ion load shear allowable tension allowable

\

_-5

i

0

5

I0 Shear

15

Load,

20

25

Kips

Note: Curves not applicable where shear nuts are used. Curves are based on the results of combined load tests of bolts with

nuts

Fig.

fingertight.

B i.I.3-I

Combined

Shear

and

Tension

on

AN

Steel

Bolts.

1961

Section B 1 25 September Page 7 B i. 1

3

Protruding-Head

Rivets

0 0 0 u_ !

r-_ q_

and

Bolts

(Cont'd

OOOOOOO OOOOOOO

I

o o OO ,-q I O

o

O u'_ OO

"_

-_ O ,.D

OOOOOOO OOOOOOO

v

,-I

I

I c_ Q;

¢1

__O_ OOOOOOO OOOOOOO

0

O r-_

U

-rq

u_

b'3 q_

-U 0.CI .,-.4

r_ C_

)-I J_b CO

f--I

.X3 ,-4

!4 r-q

O O O

o]

_D

OOOOOOO OOOOOOO

O

! O3 ,--4

O '_ r--_ O_

_

-,-4

{a

.,-.4

°,.-.I

_

m

1.4

0

o O ¢1

_OO

CO

Q; • ,-I

O_

.I-I I

,--I

_-_

_"

O OO

,--I

0

_

-_ _

_

_J

0.0 _

,.-I

0000000 0000000 0__0 '-' m

q.I ,--I

4J ,-q

,.D

_

o.x::

t:_O.X:l ._ _.._

'_

0

000000_ 0000000 0000000

c_ 0

_ .l-I

_m

0

_

_

O

m

OO -,-I

t_ _

u_ o.

_

_

_

O r_ _3

oq

0

1961

Section

B

1

25 September Page 8 1.1.3

4-I

>

r=4 0 _o 4-J 0_=

•M ¢'_ m[--_

0 4.1 °_ 4.1

-*g

r-t [--i

Protruding-Head

Rivets

and

Bolts

(Cont' d)

1961

Section 25

B

Page B

1.1.3

Head

Protruding-

Rivets

and

Bolts

(Cont'

1961

9

d)

eq o r_ o 0_ o

00000

00000

_J

1

September

__0 __0 __0 .°.°.

>

o _o_o _o

00000

<

_0 •

!

,

,

:

:

.

0 el

E

_o

_00 0__ _0_

,--4

:

_0.0. _0..0°0

<

:

_:::

• | ,_o

• I

_0°oO _0,o,

_0

_

.

•

.

•

?

•

•

•

• °

• ._

...... .

_0o

_.._:

OO

•

|

•

•

"

o

o 0'_

o

(_

0

°

•

•

•

•

° •

; : : • • : : : i : :

O

: : : : : , . , _J _J oO

•

i

•

: :

oJ GJ

:

,• :

•

: : : :

.c -_._

•

:

:

:

:

• ,

• •

• •

• •

• : : :

L_ •° • .. _

_0 _

_

_ II

_

_ II

_

:

__

•_

.

0000000000000

_ooo_oooooSooooS_SS rl/

(J3

. i : •

: : : : : : ,

: : : : : : : : : : : : : : :

m

0

e

. . • . : ,--I

,--_

,--_

,--_

cq

Section

B

i

25 September Page I0 BI.I.3

Protruding-Head

Rivets

and

Bolts

1961

(Cont'd)

__0 __0 __0 °...° 0

g __0 __0 °°°...° 0

o o0 O eq ,-_ O u_ r-00 ..T O'_ e'_ t-.. ¢"_ 00 '.O r-- r-. o0 oS O'_ O., • ° , , , , , CD

I

__O"

_

O O O • ,-4

• * : • •

I

O

,°°..°°.•

Sa O _J o

oO oO _O

O -.T I_.

¢q O'_ e,_ oO r--. oO

O_ ,-_ oO ..J" ...Jr-. O O o% o% o'_ O

o

,-4

i *

: !

4_

,

C oO _D

Sa _J

•

-.T O_ _'_ r-. _ r-. 0 _---r-- oO oO O_ O_ 0 • • • ° o • .

o

, •

, •

,-4::

• •

:

"i

• . : : : •

>

:

i i

.,4

i

.......

O

.=

_o r-. r-.o0 •

.

.

• • •

:

: : : : : • :

oo_o,_.._-r-.o

I

: :

_-4

:

:

:

:

:

: :

. • ,-.-4•

• •

• •

o_ o_o_o ,

,

.

.

CD

:

• •

:

:

.

• •

: •

• •

• • :'ii!!.

O

i

!

!i!iiiiiiiiiii! i!i iii i:iiii! c m

•

. ........

(]3 •

•

•

•

•

•

•

•

•

•

..........

_

0

0

0

0

. .

. :

:

•

•

•

•

. :

:

•

. .

0

0

0

0

0

0

0

0

0

,_

0 _D ,-_ ,-_ ,-_ eq

.,4 0 "'I

.= m I

E 0

Section 25

Protrudin$-Head

Rivets

and

Bolts

•

u_ o

oO c-4

JJ

c-_ t_ ',D

01

F_ ! rq

•,_

_

OE-_

0

_4_ 0

..4 o_

_ _

_ ,.c:

o

4J

0

0

0

c_

0

01

0

._--I

"_

0

01 0 • ,-_ ,Z_ _-_ _ 01 ,-_ 000 _1

_ cco

O

_

u_

O_

_

_

0 4J

00 CO 0

_

I I

0 to

-_

i

01 4J

"0 0

._

r-c-4

¢-4 o'I

1961

i i

(Cont'd)

• _1

1

September

Page BI.I.3

B

01

I 0

,M _

I 0 0

r--

P-_

0 ¢-q

"_ co

cO I

0

o_

oO ¢xl

_o c_j

,-q cO c'q ! 4..1

I-

0

•

>

•

co

_-4

i/3 ,---t

ol

0

0 >

._

bO._ 0

(D

E--t 4J CO

0

0

_J

4J 0_ 0

oO

0 [o

[--t

,-4 <

0 CO

0

0

(LI '..../ _./ 4J

0

rz_

_ 0

0 ,-_

_

,-.4 _

Section

B

1

25 September Page 12 BI.I.3

Protruding-Head

Rivets

and

Bolts

(Cont'd)

1.0

0

\

.9

o

\

.8

.7

\

"illllllll

.6

0

i00

200

300

400

O

Temperature Fig.

B

F

1 1.3-2 Reduction Factor for Allowables of Protruding Head, AN470-AD (2117-T3), Rivets at Elevated Temperature for Five Minutes

1961

Section 25 Page BI.I.3

Protruding-Head

Rivets

and

Bolts

B

1

September

1961

13

(Cont'd)

I

Rive

t

Dia.

/

/

318 /

//

_

/

/

L

/

/

/

.,4

/

/ /

/

/ /, / / ///' /

oO O_ ._4

0_

/

7 / /

/

/

3/16/

#

/

/

-r4

/// / ///_->

/

/

.,

_

S_ _i, II ,f_ll

i

/

/-

//j /

lillllih

.02

.04

.06

,08

.i0

.12

Sheet

Fig.

B

I.i.3-3

Unit

Thk.,

Bc,aring Fbr

• 16

.14

i00

.20

.22

.24

iu.

Strengths =

.18

ksi

of

Sheets

on

Rivets

.26

Section 25

September

Page B

1.1.3

Protruding-Head

Rivets

and

Bolts

B

14

(Cont'd)

= .,-4 .,4

,-4

.,4

O O ,.-4 II

U .,4 ,=

.,4

.,4 ¢J 4.J .,4

>

.,4 ,-4 O

am

4_

o=

> .,4 .,4

am In W4 O

o

=

am

am &J .,4

=

&a

,-4

&a .r4

O4 o_ .,4 o'I _J

.,4

_oo

.,4

o=

_o ,-4 ,-4

r-4

.,-4 v

i!!iiiiii iii ii :

:

: :

•

.

:

: :

:

•

i:i

: : :

: :

:

i

• :

ca

_n _J

_J .,4

.= &a

am

iii!, _-4 _ O (D

,4 O

e4 O

,,i iii!iiiii N O

tn OOOOOOOOOOOOOOOOOOO

eO O

eO-,T O O

-.T u_ (D O

_) O

r'- OO O C,

O_ O

O eq _O O ,-_ ,'4 ,-_ o4

_ O4

i 1961

Section 25

P_otruding-Head

Rivets

_-_

•

•

•

and

•

•

•

Bolts

I 1961

September

Page BI.I.3

B

15

(Cont'd)

•

•

•

°

•

,-q

> _0

o ,,.,.., {J

,.°.°•°°

__

__

_

,...

o

•

oo°•oo

,o••,.°•

o,o°

_0

o o •°l°,lo•°°•°,o••o.°°Ol°O°••°o•

r-_

o .,'4

c}

v 4-1

Cq °•••••°°••.°°.°.••

....

•°.•0•°•

Q.I

_

,,4 _

_ >

,rl Q;

,--4 .._

,--4

&

,--4

.•.•.•

.......

•

.......

•

....

••°•

<,-4 0

_

u v

c-J

E _ <

&

........

•

0 -r-I (_

l.J

,--4

II

•

....

•.•.•}•

& [_

C) •

O9

C) •

u_ °

,

,.-4 M m

.........

O

0 •

V

0 _ _r_ 0 ¢"q ,_

L}

0

0

°

, 0 WhO C',I

cr_ <0

•

•

•

T _ VI

V

_

._"

0

<0

_J-

O_

C)

<0

0

',/

<0 °

, , <1" _O 0

tr/

Ch

.........

_

0 0 ,r_ 0 c',,I t,"l

, ,., 0 _-_ 0


....

,

_ °

, , ,

_

o ,r_ C-I .

<0 °

o 0 i_ •

o--, _-I,-_ _0 0 0 ° °

_D .

V/I

<0 °


<0 •

V AI

<0 •

V

_ •

AI

_ •

0 °

0 wh C',I

,--4

:

°

,

" :

:

: A • -o

°

.ID

4J ..q

c'l

-d(D

OO q.}

0 !

I


,N eq

4.}

,'_ _'1

._

o'5

-n'-

I

I

I

I

C) cxJ

08

_.l ILl

I

_T nJ

-.1_I

0 ¢-.]

0 r_!

b-_ I

_ I

_

[--4 I

[-.4 I

I

-.1 0 r'l

0 C_I

_-_ .,-I ©

0 CNI

0 ¢xl

c'-I O cxI

"El r',l

cq

r_)

_

c-J

c'I

c'_]

c

¢1

¢'xI

I

,--4 U

r--I U

_ U

,-q r...}

r-_ U

E-I I

,--4

tr/ f-.. O

CO

r---

v

•

, ,., ,.D 0 _-_ _I" 0 0

r',l _-_ 0 °

0 •

,--_ 0 _rl

Section

B i

25 September Page 16 BI.I.3

Protruding-Head

Rivets

and

Bolts

(Cont 'd)

u_ ,-i

o

,--_ o.I i-_ 00 _

i_. II

,.a

_A o 0

O O. JoloJe

JJ.

_8 00 '_ .e# u_ _oo

___g_.o

o r-4 O

1.1

r-4 O_

o _ i

oo

o

__o___ _o___

_

_

Ill

rd

u_ > oD

__ oooo

_o _o

_

__ __

_

o

o

o

.,-I

,-4

o ,--I

I

_

g_

_o

_

_

_

_

gg

_

_

_

: : :

:

: :

:

: :

: : :

"

"

"

.,--I o _

o_o',o_oo_oo

o •

¢_ -,T O_ O

o_._-oooo

-r,I

""

¢O O

_OOOO _-_ _ 00

, , . OO • ._- ._- . 0 _ .

u_ 0 o,l u_

,o [..-t

tg

_o

"

: : : :

"_vi_i

, , , , 0 o,-I .=

O

"

: :

: :

: :

:

•

•

•

•

•

.

•

•

.

,

: : : :

. . : :

:

• .

: :

i

::::::: : " : :

: : "

•

• :2_ii"

•

:

_

Y

Y i_="

0

0

_

0

• . _, u_ o_ •

_._ ,._

o o. I|

1_':"

:

:_®_

__

IIIIIII ffl

_

0000000

o

1961

Section B 1 25 September Page 17

BIol.3

Protruding-Head

Rivets

and

Bolts

1961

(Cont'd)

28

I Bolt

I or

'

Pin i

/

24

Dia I i

/

20

/

/3/_

. / 16

/51_--

/

/

.i.I

f

12 ,,,4

/ "_

.-/// /,/"

8

/,-_

4

_31s

//_5/16' /

/ /

_

I .--5/32 .....3/16

_

--3132 --1/16

0

.04

.08

.12

.16

Sheet Fig.

B 1.1.3-4

Unit: Bearing Fbr

Thk,

.24

.28

in

Strengths =

.20

100 ksi

of

Sheets

on

Bolts

and

Pins

Section 25 Page BI.I.3

Protruding-Head

Rivets

and

Bolts

(cont'd)

"_

0

0

O0

O O

u_ I_.

O

eq

0 0 000 OO_ O0_

II .Q

00

,-._ O

--

= = "_

_

co _J

OOOOOO 0 Om _O_OO_

_'_

O

O0

0

= 0

__ _

>

_O_O_OO_ _O_OO_ _ O _

"_

O

_

_

O

O

_ _

OOO OO

_

_O_OO_ _O_OO_ __O_ _ u_

u_ 0

°,.-I

,--I

.= =

_J o3

O _ _OO _ _O _O_OO_ O

,--4 .,-.I

O.0 _Z

_D

_O

_O _OOOOO _ __O_O_ ____O_

_

I

_OOO_

= °4

OD

O__ O_O__O_O_

O_ .,-I _J °4 =

_OO_O_

°,-I u_

_OOO__O_O_OO_ _O_O__O_OO_ ___O_O_

_4

,.-I

>

-,--I

_J _O__O _O____O_

_O _O _O__O_O_ ____O_

_O_OO_

__O_

_OO_

.,-I 0 _-_ _ O ,.--I

N o3

°,-I

_ O __ OOOOOO

_O_ _

_O _O OOO__

O

O

_OOO _O_ 4-1 0 Z

B

September 18

1 1961

Section

B

1

25 September Page 19 B 1.1.4

Flush

1961

Rivets

Table B 1.1.4.1 through B 1.1.4.3 contain ultimate and yield allowable single-shear strength values for both machine-countersunk and dimpled flush riveted joints employing solid rivets with a head angle of i00 °. These strength values are applicable when the edge distance is equal to or greater than two times the nominal rivet diameter. Other strength values and edge distances may be used if substantiated

by

tests.

The allowable ultimate loads were established from test data using the average failing load divided by a factor of 1.15. The yield loads were established from test data wherein the yield load was defined as the average test load at which the following permanent set across the joint

is developed: (i)

0.005

inch,

up

(2)

2.5 percent of than 3/16 inch

to and

including

the rivet diameter.

diameter

3/16 for

inch rivet

diameter sizes

rivets. larger

Section 25 Page Flush

1.1.4

Rivets

(Cont

'd )

_0 _ _ ___00_

[,_

_o _J

> ¢'4 0

.,4

r"-. ,0 r",-

_

_ _

_0 _0

_ _

0 _

UOU_O

:

:o

:

:

,,.-4 __00_

=

,.._

: :_ ,,o_-

= =

o -.1"

8

:CCC2CCCCC

I

:

I

l •

u

o

O%

U

U

_

t_

O_

O

u%-.T

• •

O

_ _00% _ _'_ _,S_0% o

o O

_

O

O

•

|

_

.t.1 %0

0%

:o •

O_OO_'.: :_-

.-4

uu

I

o

o4

m

_

O

_D

: :

_ :

00

u% o'_ _o__

.u

o •

U

•

%0 P,,, r--- r-. 0

r-..

oO oo

• •

0%

('_

O

u_

•

•

r_

Cxl ..T

_.D

r_

°

•

•

¢'4

•

0%

• •

O0 O0

m I

ul

..1- c,4

0o0o0 eq e% o o

r_

,-4

_

(xl

-_4

,.O ,--i .r4

..T u

¢q 0 eq

c',l e.I

r.-. o4

0 e")

¢_eO¢_

--

.

•

•

l

•

•

e

"':!i"

eq

_o

•

: :

: "

•

¢0 1-) .M

e•

.g .,-4

,-4

v

J

co co

i:::

i:::::::

•

.

c o •,-I ,._

4J

,n _.J

>

..j

-_4

c0

.c

0 C'q 0 ° 0

' iiiii

: : ' : • : L,'_ C',I 0 . 0

C'-I C_ 0 o "-_

0 0 C"_ ..J" LP, '4D 0 0 0 . • • 0 0 0

: ,--_ r-0 ° 0

0 0 _ Ub 00 0", 0 04 0 0 ,_ ,--_ . . , ° _ 0 0 0

0 <0 ,-_ • 0

: :

: :

0 0'_ _1 .,-_ • ._ 0 '.,0

B

September 20

1 1961

Section

B

1

25 September Page 21 B 1.1.4

Flush

Rivets

(Cont'd) _0__

I

C O CD

p,_ 0

m

0

.

_0 C

•

:

.

.

r_

Lm

> --

o_ 0o

--

:

C_

cq 0 Cq

•-_ 0

CO [-_

_J C 0 _D I

.C

•.

,--I 0

oo

,.0

,-I

,-I

,--4

_D r-q

: •:o_ _ _._o-_

: : oo : :

•

_

CO

C 0 Cq

.C U

o,1 C

• • •

u_

:

0 0 O 0 o,1 .,.4 ,-4 0

0 I_. ¢q

L_ -d" c_

,--I 0 _T

,-.-I (_10'_ oo _0 ¢_ ._" u'h _D

• •

:

: '

:

: : : :

_0 •

m

,a

cO _ u_

u_

•

cq

cO

.d-

u_

_

e"_-d-.._--d-

_ID ,.0

r_

oo

i

I

0 ffl

I r_ ,--4 ,--4 o.I

cq CD c_

[-_ ,

oo _

• : • ,-_

._ >_

o4

m

i :

_0__

cq

:

cq

•

• • : •

c_ .,-I

: • • .

: : ¢ : .

• . ; : I : • : : .

;

:

•

•

.... : : : i i! : : : .... .

.: : •

•

,

•

iii!i

•

•

•

°

•

•

•

•

•

•

•

•

,

.

•

•

•

•

•

•

•

•

•

•

•

•

•

.

•

.... .... • • . .

.

..4

J

.,-4 c_

i:''"

•

:

: • : : : • :

: : : : : : : : : : : :

: •

: : : . : :

•

: : : : : : • :

: O

,1

.u ..Q > -,-i

_J

l

: : . . : . : : : : •

8 000000000__ ••.o°•°°•••_° 00 O0000

i

• :

:

• ! : "

cq c_

.,-4

i

: : ! . • :

O000

O0

O

1961

Section

B 1.1.4

Flush Rivets

25 September Page 22

(Cont 'd_ 0_ooo_ oo__ _0__

.4"

t i I "_ t .,_r .,,? -._- _u'r_ 000 ,-4 eq eq eq ¢O

0 f-_

I

¢q O eq eq 0 eq

_0_0_: ___.

_eq 0 ¢q

_CCC:

4J

• 0ooo

_4

¢q _D eq oo eq oo _ m- _D 0 [-_ 0 [-_ 0 [-_ _ 0 [-_ _4 ¢q ¢.4 P-

°__0_

I -4"

I "_-._

°__0

('4

eq

_

O..T

u% eq

"

" i

: __g_

,-4

o7 oO_

%0

_D

! 'T_ I 4J 0 ¢'4

0 r.-

._

t_ I r-r-4

:

u

:_o_. ._o_o_o. :_o__:

o eq ,-4

I

I

I

I

C',l C4 _1 I:: C'4 000 _ 0 ¢q C_l _'%1 C',l

m

_o_o_

::

• __oo_

•

m IIII

0000

: :

¢q

• _0_ "___. : _0000

_D

J

O_

,4"

I

I

• ' _ oO "

,O

•

_

,-4 _1 >

,-4

::!

-.? "D -.T eq _eq o mo ¢q ('4 I

_ .1.1

I

,-4

,-4 ('4 %0 r-4 0 O0 eq eq

i**•

o

o_ 0

_o_: __. __" Ill

.,4

I

_q I!11

000

_O

,-4

_0_0

_n

: : : : F-4

,-4

0

J

O,

U

B i

• _ _._

_ 000000 _00000000000

00_

1961

Section 25 f

Page Flush

Rivets

(Cont'd) .q-

00 I

".D _

eq

a •.d- cxl o_ 0

co _J q_ >

i _1 i_

_

i

_o,i

cO

c'q 0 o-1

q_ r-q

_D ,-_

0

-..1"

_

eq 0 c'.l e_

.r4 _D CO

O O O

0

0 t'M

O

t_ 1'

0

i

O cq 0

_ ,.D O0

_J

[-_00 I C'q Oq .q,-

_J

"O

L¢-5

qJ

,,D

¢q

._"

!_ l _ILq

c-,l

0 r_

_

,--4

p-4

t-¢3

_q a

i

I

I

_

i t'M

CM C,,I 0'4

_,_

I

I

_ 0

oO

--d" b'lO

I I
.,-I

t¢'3

C'-I

..j-oq _0 ¢q

<0 eq _0_ ¢q

I .._ eq E.-_ 0 cq

oo

_ ¢-_ co _0 [_ 04

! <0 _-_

_ _

¢q 0 oJ

O0 _-_

c_

1-1

•_1 -,-I

._1

1

September

k

B 1.1.4

B

23

1961

Section

B 1

25 September Page 24

B i.i.5 values

Flush

Table for

Screws

B 1.1.5.1 contains ultimate i00 ° flush-head screws with

and yield allowable strength recessed heads installed in

machlne-countersunk clad 2024 and 7075 sheet. These strength are applicable when the edge distance is equal to or greater times the nominal screw diameter. Other strength values and distances may be used if substantiated by tests.

Higher

1961

These strength values may be used for the design values may be used for dimpled joints if based

of on

values than two edge

dimpled joints. test results.

The allowable ultimate loads were established from test data using the average failing load divided by a factor of 1.15. The yield loads were established from test data, wherein the yield load was defined as the average test load at which the following permanent set across the joint is developed: (i)

0.012

(2)

4.0 1/4

inch,

up

to and

percent of the inch diameter.

including

screw

diameter

1/4

inch

for

diameter

screw

sizes

screws. larger

than

The test specimens used were made up of two equal-gage sheets lap jointed and machine countersunk with washers to build up thickness to minimum grip. All joints had 2D nominal edge distance in the direction of the load and were either of the three-screws-across or the twoscrews-ln-tandem type. For the latter type, the on opposite sides of the joint to assure 2D edge

flush heads distances.

were

placed

Section 25 Page B

1.1.5

Flush

Screws

(Cont'd}

cq

00 °r4 O

_OO___

:

CO

ii 0,--_ r'_

0'_

O_O_O_

_o__% _ 0

m

_

_

_

_

_

_

co

co

CO

_

! o-, ,--_ O '..D O'_ 00 Lrh r"-- O

,._

Lr_ ,_ c'q cq ,,.00 c_ ,..0 c',_ u-'_ ,.D r---

c'q Ch 00

,,.C: _D c'q _'q ,--_ ;._

• • °

: I

:

:

:

:

:

•

:

•

:

CO

L) _

• :

:i

o,._ I ._

o

o

,r-I

0_ _:

.._

_

_

: _ : : : : : : %___

._

"';L::

0

co

|

_

u_

m

:

og_o

_

: _

_

co co _

_

_ _

_

:

_

_

c)

::: , :

:

CO

CO

:

.

•

• : • •

.

, : : :

|

_ .r-I

(D _---. ¢'4 tc'_

• ,-_-._

o_ _

_

_o.._

:_°_°__i..1-,o_,_o •

E

--

,-4 .-4 ,-q ,-4 ('q ¢q

CO

,. _m_o_o o_ _ o _ o_ _o c_ _- c_ c_ ._o o_ e_ _ _ _- oo _ _-_ •

_.4

" •

: :

O'3 _D

_4

u

•..-4

°

_:

._

_

[--t

_

_

-_

4-1

.I--)

_

[-.-t Crl

_

_0

.

• •

_O

_

o_

.

.....

:..:::: . .... !ii.!::.. • : • . ,-_

" • :

:

:

:

"

: .: : : :

.

• : : : : . : • : : . : O O 6 u_

_ co -.1- u'_ ',O r-- 0o 0", • ,-I O O O O O O O i--! . ° . ° .......... 4J O O O O O O O J.J

_

:

: _ . : ' : _'_ : : • : • O O oh ,__ eq

q--I QJ

i

:

:

:

:

: • : • : : .' . : • O O• _ eq m

O ,-_

c-,I ,..o ,--_ ,-_

o'_ ,-_

u'_ ¢q

,-_ co

r-co

C)

O

O

O

O

O

O

B

September 25

1 1961

Section 25 Page

BI.I.5

Fldsh

Screws

(Cont'

B

September

1 1961

26

d)

tN

=o eqQO

0_o00_

:i

0 r_o

_0

,._ u'_ 0

_-_ r_

QO

_-_ O0

eq

•

•

Q;

: : •

u

_%

o__

.

m _J

_o

_,__,_-,_

o

:.

!

I =

)

C;

,=

• : :

: :

"

: .

•

:

: :

•

•

:

. ,-_o0,_o

o_

4-1

o o

=

• . :

: coco._m_r_o0o_

0

ii"

q_ 0

: : : ,_r_,_O..q-o_o_,-_r--e_

•

: :_o___" : : __=__"

• :

4J

C_ 0

:"

cO J--

_J Ore

e

•

•

:; t_

•

:'

•

e'

• __0_

:

•

ee

:

:

•

:

>4 =

G; _J

_D

• ____:i

:!!_ ":

_J T-4

ee

P-4

d .H

: : •• .

: :

°

!i

: : ....

: :

_ __ 0000000 ,)e°e,•elelleO 000000000000 q-I

•

e

l l

; ! • :

: .

. :

'

!ii!!!iiiiil •

_J

: :

i_

0__ __m O0

Section 25 Page B

1.1.6

Blind

Tables

These

rivet-hole and

may

be

The

Where

e/D

values

joint

was

is

the

0.005

(2)

2.5 3/16

For

tables

and

grip

or

blind lengths

respective

holes

yield

flush-head

the

the

oversize

of

2.0

must

are

be

divided

by load

to

tests

and

manufacturers,

improper

grip

Yield

following

were

yield

set

pro-

failing

values

the

permanent

of

average

strength

wherein

2.0. yield

values

the

from

than

substantiate

from

data

obtained

greater

strength

obtained 1.15.

data

or

to

Ultimate

test

the

test

equal

used,

were

which

from

e/D

made.

rivets

yield

inch,

B

the

in

table were

load

across

is the

up

to

and

1.1.6.2

rivet

and

comparable

B

diameter

data

obtained

inch

for

the

shear

Test

were

3/16

1.1.6.3

rivet

B 1.1.6.3. based

including

diameter

rivet

sizes

ultimate of

on

the

which

larger

rivet

strength

using

rivets.

2117

standard

than

shear solid

strength rivets,

strength

values

degreased

clad

of

specimens.

In

view

of

the

and

ables

are

recommended.

blind

rivets

in

should

absence

of

sheet

may

Since be

used will

applicable Drawing

when

by

established

values

than

at

aluminum

rivets

were

blind

load

tables

2024-T4

not

if

ultimate and

only

recommended

percent of the inch diameter.

on

noted

tion

contain

protruding

developed:

based

for

less

average

(i)

these

for

reduced

specimens

from as

1.1.6.6

applicable

as

having

flush

test

defined

are

strengths

and of

are

values

specimens

ultimate

load

1961

27

used.

strength

obtained

as

strengths

of

truding

B

strengths

substantially

are

tests

through

tolerances

lengths

and

1.1.6.1

single-shear

rivets.

1

Rivets

B

allowable

B

September

be such be

wide

variance

magnesium

in

alloys, Allowables

double-dimpled established data,

on

the

or

ultimate

dimpled, basis for

methods

standard

for

or

allowables

dimpling

no

and

uniform

and

tolerances load

shear

machine-countersunk of

blind

specific

tests.

rivets

in

allow-

strengths

of

applicaIn

the

machine-countersunk

used.

blind

rivets

in

applications

exist. use

MS33522.

of

are

primarily where

Reference blind

rivets,

shear-type

appreciable

should such

be

made

as

the

fasteners, tensile

to

the

loads

they on

requirements

limitations

of

usage

should the of on

the

Section 25 Page

B'I,I.6

Blind

Rivets,

(Cont'd_

0_00

O

>

eq eO

D O _0___ 0 0_0 ,-4 u_

0

=

0_ 0

,,,I" eq

_0

i!:

OO

_.j

_J oJ

W 4_

_o

o;

,,M 1.I O

: :

o

: . ." : :

_00__

eq

•

r.. _o c.4 e,I _-,-_._.

e,I o4 .4..._ u,% %o

0 u_ r-.

o-_ 0.._ e,_ oq ,_ _Do._ oo o_ oxch

eq ¢,4 u-_ 0

u'-, ,t.i ,-I 0o

_ ..J-

u-_ _0 oo o,,i

'_-t

J

i

om_mooom

! _J o_ oO_ C _n

:: N

• •

• °

: :

:

,

.....

t

11

:

: :: : ::

: :

: ::

:

i!!i

:::: _

Be

• • : :

o W

=

,• _oo_oooo • __

__

: : : :_o__

0

_

IJ 1.1 ,,M

,r.l ld

u'3

0

"_'i

¢Xl ¢_I _0

u'_ O0

: : _.• ":

•_= : :

0 •,_ _J

¢xl 0

:

•

,--I

.'ii A'

_ _

_ _J

0

_ .,_

._

000

_0000000000000

O0

O00000_

_

_ _

B

September 28

1 1961

Section 25 Page B

1.1.6

Blind

Rivets

(Cont

B

i 1961

September 29

'd)

0

.,4 4J

O

0 O O

>

"0

= .,4 _J U I. 0 u_ m 0 .,4 ,4.1 •,4

_J

I_

0

_v O q4 4...I

,--4 0

>, o IJ

0

,..c; ,.= ,,_ r..,t3 0_3

4J o'3

4,,.I o0 .;-I oO

o

0

ffl 4..i

.,-I

0

_

0 -,4

_

m

nJ

o

4_1

0

0

4.1 _ _

o rj

•,-i _

_ ._

o ,'l:J _ 4..1 .,.4

ill 4_ ,.--4

m _

"tJ

,-'4

0

m

_-I

4-1

0

,"4

m

_

.,4

0

ffl o r_ b-t

>

0

= ,.c:

,-4,.=:

O

Section

B 1.1.6

Blind

Rivets

B

1

25 September Page 30

(Cont'd)

1961

_0_00_0 0_00_

_ _

_ oooo oo • •

•

•

_O_, •

:

,_

,-.

u

_

>_

_

._

:

• . : : : • . : : .

: :

_!:::: _

_-_

: : : : :

:

:

•

:

:

_00000

: :

:.

0000

O000

co H

> •_ od

_ _ co

>i • =_I

co

O00

•

:

_ O000

o

•

: :

i" :

_,°,....°,°.

!

:

_

_J

Section 25

September

Page B

1.1.6

Blind

31

(Cont'd)

Rivets

m

O4 O _O O Oq I

B

_

_0_0000

_

,-4

_0__ _0_

_

m

.,-I

O O _D O

•_I

¢q

_'

I

j_

v r---t

"o

!

"_

.

0 U

D_.J

4_1

O_

_

m

_ 1_o c,l •_

..c:

__0_ _0_

_ bO t¢'5

m "_ {¢) •lJ

-_.4

_

,-.-4

_ m

:> U'?

,-.-I

• ,.-I

_

._I

0

_ r--

0

_

[-4 _

ocq

o

Cxl _..J

¢q

__

r_ 0__ _0_

_._ 0

,.-4

! -,-4

:

:

:

:

:

.

_J CO

!!ii ,-.-4

•

v ,--4 ,-_

_

._I

O

_

_

:

:

:

!

i

:

:

:

ii! ::!!" 0_00_000_ ___

"

.= u_

o O_

000000000_

, I-_ ,-,K

_

U

1 1961

Section 25 Page B

1.1.6

Blind

Rivets

(Cont'd)

O %0 O eq I

.=

•-4

%0

%O

14

,

4J

o_ =

eq

,=

o

4_

o

_J W O 4-1

O ,--I Neq I _.._ m

m

_

: :

: :

O0

•

•

: : : :

u I,.4 O

_

4J

m ..4 4J _=_

u'% o_

¢q

%0

00

_o_o

.-4

• :

: •

. .

: • : :

" " : : • : • •

" : i : : . .

: : : : : : .

: : • :

: : : :

: . : :

" : • : :

:

:

: : .

:

•

: : : • : :

: •

: .

:

:

• i iil

.4

• • • : : : : :

: :

: • : •

• • : : : . : : :

i . : : :

ii:: • : : : o iii.:::_

-H .61 _

°

c_

.M

_

i

: : • : i

"

. :

:

_.!!:..:: _ : " • • • ,-_

0

;_ ._000000_ _00000000

m

_

B

September 32

i 1961

Section 25 Page B

1.1.6

o

Blind

Rivets

(Cont'dl

B

September 33

I 196.1

Section

B I

25 September Page 34 B

Blind

1.1.6

Rivets

(Cont'd)

L

4J

_O en O

c_

CO

_

o

m o,o

_ _

_-

-

_ 0

.4" _

o

g

_

_

-_

_-_-

o

_

4J

4=1

CO

"_

_

_J IJ

_D

_

_,

,

u3

: :

O_oh

•

_

: : !

•

1961

Section 25 Page BI.I.6

Blind

Rivets

(Cont

B

September 35

'd)

_J 03 03 CO

_D 0

o_ 03

eq c_

U3

"_-

c OO

03

c

0 0

_

03

N

_J 03

,._

>

0

¢'_,

c _

=-

03

_J

< C _D u_ O u_

_-4 o'3

_J O'3

',0 _ '

"_ •_,4

0 C

,_

i

o(3

_J

c O t_

O L) v •

_0

_0_

_J C 03

D_ "O Lr_

03

OO C

03 _J

E 4J ,-q

d v

03 ,-4

0

_

(D u) 03

03 >

m

> 03 >

i 1961

Section

B

1.1.6

Blind

Rivets

(Cont'd) _o ,-4

oo

"o

>

z=

(-4 o_ Ou"_

u_ _o oo

,--t

,-4

|

oo I

o"_ p._ c.q

< "o

o_ :

,-4 u3 0 ¢'_ u% " O_ 0"_ -,T 0 ',.0* r-. oo 0 ¢q C.4 • oO

i "

:

,m

o2

_=

"O

,'--_,--_ ,,-_

•

: : . •

:

•

•

, •

oO

i

eq r'... C'q O_ 0

,-4

,,

,,c,r-ooo_ " :

oo ¢q o

o

.!!

• : : : : • •• : :Q

,--I

o2

: : • : : • . •• • : :

,.-4 .= ,-4

0

bo

0

.w

¢q

: : • : : •

: : : : : :

:

:. • • : • : :_ 1_

•

ooo _

u% eq o_, ,-4 ¢q c'q ¢0 c,'_

oo

ooo

:

,-I_D O,-'4

•

•

.

•

,,

: :

: : : _,-_=_ . : :_.o

• O

u'_u% O

•

•

|

:

I

!

O

..

a.;

: : _ _ooo_

,_i

: :#

oo0

w

0

.=

:

::i

I

: : :

IlJ W W

,.-4

_.

• •

00 =

O ¢O u,-i ,-4 O _"

'O

O0

_

O0 u'% •. 00

.-.I 11

•.

. : " : • : i " : • " "

O

O2

_

_

O2

o2

eo

.

: .

o_oooo _o0000o_o_

: : : : : . : : . • : : : " : . : .

:

• • : : :!'"

: : : : :

"i'ii''''" : : : .... v

:

i: : ! :

Ne4eq

: : • _ooo_o _o_
u'3

•

WD

::! . _0 u_o r-,,(%1 O0 0 •

:i •

¢O

o

: : _ooooo : : _o_,_n • • ,--II_, ,--Ii"_ u_ : " N__

:

m .,4

•

4

•

•

•

•

. :

•

•

•

•

• •

• •

• .

• •

o: i i i i ....:.. ... m • o2 •

:

:

: :

•

:

•

: : :

oo,A_...

i": •

: •

• •

....

G,-',,_ooo_,.o

•,.-4 C',le4 cq .4- u"_ xo r.,. 00 0'_0 e,l _.-'_ .=: 000000000 ,--4,...-4 ,.-..4 4J

•

0000 _

_J

_J

o2

B

i

25 September Page 36

_J o2 X_ u_

•

•

•

•

O0

•

•

•

O00000

•

•

•

.

1961

Section 25

September

Page B

1.1.6

Blind

Rivets

37

(Cont'd)

O L3 v

Ce_

.._ _n,.o,.o

r... r-.

: !

> =

cg

CO

0

_O cO ¢q

_

,._ u_ r_

._u_u_. or--00o_ r-.r--cooo m m _ _,-,

: : : : • • : : .'.

: : • : :

' : : . :

•

: :

:

: :

_o_

: .

: : .

: : • . : :

: • :

: . : : .

m

•

•

: •

:

• •

_

°_ "_

_

_ooo_c

OO

_m_m O

•

,,-I _0

B

ffl

_ _

" i

.: " : • •

: ,

"1D

• .,-I

.

: : i : i ! ooooo .. : •

: :

• :

• :

._

:

:

: oooooum

0

• •

• "

•" _0 00 *.I_ ,_

¢'_ ao CO

. :

--

_

0

0

.iJ

:

0 ,--I

cq ,--I

._ ,.._

_ ,--I

0

0

0

0

" :

.

_J O

co O

• o_._-_ :x_ _e4_

_

i

.

•

•

:_

,.--I

•

°

._

:

:

:

: :

u'_ 0

0

_

: _o

u_o

.

mo

:

:

i

,

.

•

,--I

,,-p !

.

•

0 "
tJ

_

.t..1

_0

u--i

m

,

:

•

:

......

•

•

•

•

e

_/3

•

• •

• i

{_0

0

•

,--4

_

j

cq

0

_

u_

oo r_ _J J

•

•

•

•

• ....

ce_ °r-.I

,

_O p-q

.,-1

..

•

•

•

•

i

i+

•

•

•

•

•

•

.

.

•

•

i

,

-

•

•

|

-

-

•

•

•

•

I

•

•

•

•

•

+

•

:

:

• ;

:

: • : •

:

:

•

.

•

.

.

.

:

.

.

.

.

•

:

.

•

.

.

.

.

: :

:

:

0 0

_

"_

i i

.I...I

,---4 I

+l r-_

.

• .

:

: :

_ O

m

co

v

t.M 0

tC_

0

_

c_

._

,t_

_

qJ

c_

J

_J

.,_

: : C

:

0

: : 0

' : : 0

0

" :

: 0

0

• . i : ' 0

0

.

. :

oo

,--_ ,-_

,--_

_n

>

oooodo_ooSo_ r-q

•,..4 t_

.M ,._

1 1961

Section 25

September

Page B 1.1.6

Blind

Table

Rivets

B

1961

38

(Cont'd)

1.1.6.6

Explosive

Rivets,

Ultimate

Rivet

.025

.032

.'o4o

.o51

5/32

320

410

513

3/16

--

495

620

Rivet Size

B i

Sheet

Load,

DuPont

Extended

Cavity

Lb/Rivet

Gauge .064

.072

610

610

610

796

880

880

.081

880

Section 25 Page B

1.1.7

Hollow-End

If the equal

to

that

the

the

meter

B

provided

1.1.8

B

to

specified

1.1.9

the

by

and

in

an

stumps

For shear to the

lockbolt

least

25

portion may

material,

percent

locations

of

of

be

taken

provided

the

rivet

the

hollow

end,

where

they

toward

"Hi-Shear"

aircraft

rivets bolts

is

heat

diaand

will

not

the

same

as

treated

to

125

lockbolts

and

tension the

following

ksi

R t and and

shear,

- Rt +

are

the

when

type

in

material

strength

Rs I0

ratios

respectively.

= of

1.0, applied

tensile

lockbolt

values

were

by

the

ultimate

and

combined

loading

available.

(blind)

equations

and

guaranteed and

be

manufacturer

and

strength

strengths

BL

cutoff

interaction

lockbolts Rs,

the

installed

added

shear

values

shall

the

lockbolts

These

conformance and

by

allowable

minimum

yield

be

in

practices,

Huck

1.1.9,1. are

tensile will

shear

ultimate

B

and

but

installed

recommended

flush-head

table

and

all

The

data

be

procedures

and in

test

shall recommended

with

strengths

make

stumps

method.

Shear

Steel

tension

same

a

stresses.

for

standard

protruding

from

bearing

where

the

in

tensile

manufacturer's

contained

manufacturer.

ing,

of

measured

used

load

accordance

for are

enough

are

for

rivets

these

1.1.3.2.

equivalent

established

of

the B

lockbolt

strengths

yield

at

as

sections of

Lockbolts

inspected or

shear

table

Lockbolts with

1961

39

Rivets

for

in

is

shear,

they

cross

strength

rivets

cavity

appreciable

allowable

given

the

solid

of

that

Hi-Shear

The that

the

plane

solid

used,

of

of

the

with

are

strength

subjected

and

rivets

450)

bottom

from

further be

(AN

i

Rivets

hollow-end

length

B

September

under

having

a

critical are

thickness

for

the

large

shear

load-

applicable:

7075-T6 load

lockbolts to

allowable

- Rt + load

Rs 5 in

=

1.0,

Section 25

B 1.1.9

Lockbolts (Cont' d) .0

Page

B

September

i 1961

40

_O

W

C 0__

•,4 _

O_

C _o 0_ c

c

,-4

C

_0000

C_g 0

0

_

I_

m

CO

_

r_

0 W u,-i 0

> cn

"0 _J

.M

rn

m W I

:C_gg k_

r

"00_0 0 O ,-4

0

,-_

;>_

,-_ ",4 m 0_

.c

g

0

m

0 N 0

0 _0

I

A_ 0

:

oooo

•

%0 _D

>

o O0 A_ I

CO u'% 0

•

,-4

I

C

c_

_M

! I C 0

:

t_

.r,I

o_

•

:

:

_J

:

'i'2"

c

0J

1..1

0 U

3 m

o rD

0 0

Section

B I

25 September Page 41 B

I.I.i0

1961

Jo-Bolts

The ultimate steel and aluminum Tables B i.I.i0.i

and yield allowable shear strengths for flush-head Jo-Bolts in clad aluminum-alloy sheet are given in and B 1.1.10.2.

Section 25 B

I.i.i0

Jo-Bolts

(Cont'd)

September

Page

[--I !

O0 _ _ __0__

,,.,-4

0 __

O0

O0

0

O0 _0

42

0 _

0

,-a 0 _0 I 0

I

m

[--I ! -.1"

O0000

O00

O00

0 0 _J

0

! u'3

"_ r_

4-J o_

2

00 _0__0_

O0000000

00 _ _

O0

O0 __

0 p_

o

_J
4.1 m

I "4" ,--I

O0 _0 _

_0 O_

O0 _

O000 _ _

_ _

_ _

0 CO

•,.4

&.l

"_

._ 0 0_0000000000 _ 00_0_

[-4 ! ue_

c_

U

0

_

O_

_

0 r_

0 0 CM

O_ _0__0_ _

[-.I I

O000000 _0

O00 _

_

_

_

_

0

O0

_000_

_

0

,-.-I

_0,,.,,0t,,,,,,

_J

[--I

&J

,=

0 •_ _ _0

00_ __0_ O0

_0

O000000000000

O000_

_

_ __ _

_

B

i 1961 _J

Section 25

F B

Page

Jo-Bolts

i.i.i0

B

i

September

1961

43

_o 00000000000

r..3 oq

o i-_

o_o

00000000000 _0_0_0_

,.el ! ! Ou'_ _.3

o t.q

n:l

i

0

o9 000000000000

0"0

___0_

c,h .i.J 0_

0-,1" 0 o'3 cxl

o n:l

000000000000 ___0_00 _ __

i

,-..I _0

_ >

.._0 o9

cxl

N

O_ 0 ,--.I

_9 "_

0-

w 0 .;.-I

0

000000_00000 _0____ __0__

,,.4 •,-I •IJ

¢_ ._

i:_. 0

0 0

.,-4

B

0000_0000000 ___0000

0

04

cM 0°..,.0,.°,°, ,x:

.°,°_°°,,°°°°° .°0°.,°,°°.°°° ,°.°°.°°°°°°°. .,,°°°,_°,°° ,°°,°.._°.°°.

..............

::

,....4 0 t_

._,.,,,,°,,,,,,, .,,.,.,..,.,., _,°,,.,°,,,,,,. _o,,,,,,,,,,,,. 0,.°,....,.,°. _..°°,.°,...,. _,....,°....°°, •_ O0 ___0__ _0000000___ °,,..°.°.,,..° _00000

u] 0 ,-.4 _

_

000

_

000_

_ _

O0

O0

"0 0

O0000

t_

Section 25 BI.I.IO

Jo

Bolts

Page

(Cont'd)

E.-, 00000

0

_0

!

I

00000

0

o u_ !

<

0

-,4

E.-, I

00 ___0_

O0000

O000

@q 0 0

I.I ODeq = 0 OJ ¢q

r-4 Ill

,=

_._

_z E-_ I _c_

r._

U

4J 00000000 _0___ _ _ _

_ _

O_

_

_

0 4_

•,4

o

0 0 .,4 I

<

,-4

Ol •,4 •_ _

_ o U

0

< !

m

u

00000000_ _ O_ __0__

__

0

°°°,°°,,.,°o ,°.o°°°.,,o, ,,°.,o°,,,., ,...o°.°,o,o °,°o°°,,.,°° ,oo°l,°l,,,** ,...-4 ._,,°,,.°°°°°° °°.,°°.,o°o° _,,,°,°.,,o,° _o,,.,°,,,.°° _,o°,°°,,o°°, _,,°,,',°,,°°o _.°,°°°o,°°°°

Ill ,--4

•M _ 00_ _ __ _0000000__ _000000 111

OJ

r._

_ O00 _0_

_0 _

O00000

O0 _

B

September 44

1 1961

Section 25

B i.I.i0

Jo-Bolts

(Cont'

Page

d)

B

September 45

0000000000 L)

O p..

rj

0 c',4

_J O _D ¢q

4"

4

O

¢j 00_0000000 ..D O c_

2

t

g

OOcq CO o4

4-1 m

0 m

_ 00000_0000

_0 _

r_ 0

•_

O

0 0 0_

,-.4 0 .,,.4

O I <

_ ,.,_

0 L)

0 ,--4

0000000000

,°°_._°°.°°. °.,°°...°.°, °.,.°0,..°.° .°°°°°°0o.o, .o°_°.0°°°,°0° °,°°°0.°._..° _o°,.°,°.°0°, ._°°°°,.°,°,°° °.. ......... _.°q.0.°,°,,° _°°,..°°..,°° _°°,°,.o°...°

[--i

•

,-4

o ._00_000_000 ___0__ _ 0000000__

.,-.4

o

......... °,°.

>

IJ

0

1 1961

Section

B

i

25 September Page 46 B

1.2.0

the

Welded

Joints

Whenever possible, joints to be welded welds will be loaded in shear.

B 1.2.1

metal

1961

Fusion

Weldin_

- Arc

and

should

be

so designed

that

Gas

In the design of welded joints, the and the adjacent parent metal must

strength of both the weld be considered. The allowable

strength for the adjacent parent metal is given in section B 1.2.2 and the allowable strength for the weld metal is given in section B 1.2.3. The weld-metal section will be analyzed on the basis of its loading, allowables, dimensions, and geometry. B

1.2.2

near

Effect

on Adjacent

Parent

For joints welded after heat the weld are given in Tables For

materials

heat

treated

Metal

Due

treatment, B 1.2.2.1

after

to Fusion

Weldin_

the allowable and B 1.2.2.2.

welding,

the

stresses

allowable

stresses

in the parent metal near a welded joint may equal the allowable stress for the material in the heat-treated condition as given in tables of design mechanical properties of the specific alloys.

Table Near

B

1.2.2.1

Fusion

Allowable

Welds

(Section Type

Tapered All

joints

of

of

4130,

thickness

Ultimate 4140,

1/4

inch

joint

30 ° or

Tensile

4340,

Stresses

or 8630

or

Ultimate stress,

less

b

or

heat

plate

treatment inserts

Table B 1.2.2.2 Fusion Welds

Type

Tapered

90

joints

of

or

considered

Allowable in 4130,

of

normalized 0°

b Gussets

after

taper

joint

30 ° or

weld.

with

center

line.

Bending Modulus of Rupture Near 4140, 4340, or 8630 Steels a

Bending modulus of rupture, ksi

less b

Fb, figure Ftu = 90

others

a Welded

tensile ksi

8O

after

b Gussets

Steels a

less)

others

a Welded

All

in

B 1.2.2-1 ksi

for

0.9 of the values of F b from figure B 1.2.2-1 for Ftu - 90 ksi after or

heat

plate

treatment inserts

or

normalized

considered

after

0 ° taper

weld.

with

center

line.

Section B i 25 September 1961 Page 47 B 1.2.2

Effect

on Adiacent

Parent

Metal

Due

to Fusion

Welding

(Cont'd)

450

400

350 o_

300

_" Ftu , 260 kgl

250

,,ok.,

150

_

_'_ Flw , IN

_

kjl

100

50 ......... J 0 I0

Fig.

B

1.2.3

Weld-Metal

Allowable Design of the

20

B 1.2.2-1

Allowable

weld-metal

allowable respective

Ptu'

stresses minimum

30 D/t

40

90 ksl

50

60

Bending Modulus of Rupture for Round Alloy-Steel Tubing.

Strength

strengths

are

shown

in Table

for the weld metal are based tensile ultimate test values.

B on

1.2.3.1. 85 percent

Section 25 Page B

1.2.3

Weld-Metal

•I._

_

u'_

Allowable

u'_

_

Strength

oO

(Cont'd)

0

o'1

u_

0

oh 0 _D I

O

,-4 0

U ,.--4

0 0

I

_

0 ¢_1 I

!

I

O

0 _'_

'_ 0

0

_ _o

_

_ _

0

_ co

_ _

_ _

0

O I

,---4

",_

_ O_

_

_OO

OO O_

0

I

i

I

I

I

c/_

I

I

I

I

I

I_

I_

l-.I

•l-I

i.-II---I

0

I c,,I I

I

I

I-.I

¢)

O0

C_

._..I

._,

0

>

>

co

o0 co

u'_O0 t_l _

oO

_= O

[-_ 0

0

_

Z

O_

o0

o0

O_

CY

0 c_

= cO

= O

O

,.--4 0 ,.-i ,..-I

0 ,--I ,.-.I

0 ,-.I ,-.I

,<

<

,<

O_

0 _

O0 -_'1" <1"

-_

"<1" '<1"

B

September 48

I 1961

Section

B

1

25 September Page 49 B

1.2.4

Welded

Cluster

In a welded allowable stress stress by tube that B

1.2.5

structure shall be

where seven or more members converge, the determined by dividing the normal allowable

a material factor of is continuous through Flash

1.5, unless the a joint should

Table

B

1.2.5.1

Stress

for

stresses are given

Allowable Flash

Welds

Normalized tubing - not heat treated (including normalizing) after welding. Heat-treated tubing welded after heat treatment. treated

and bending in Tables B

Ultimate

Tensile

in

Tubing

Steel

allowable 1.2.5.1 and

Allowable ultimate tensile stress of welds

Tubing

heat

joint is reinforced. A be assumed as two members.

Weldin_

The tensile ultimate allowable modulus of rupture for flash welds B 1.2.5.2.

Tubing

1961

(including

1.0

Ftu

(based

normalized 1.0

Ftu

(based

normalized

normal-

izing) after welding. Ftu of unwelded material in heat-treated condition: < i00 ksi I00 to 150 ksi >150 ksi

0.9 0.6 0.8

Ftu Ftu Ftu

+ 30

on Ftu

of

tubing) on Ftu tubing)

of

Section B i 25 September 1961 Page 50 B 1.2.5

Flash

Welding

(Cont'd)

Table B 1.2.5.2 Rupture for

Allowable Bending Modulus of Flash Welds in Steel Tubing

Tubing

Allowable bending modulus of rupture of welds (Fb from Fig. B 1.2.2-1 using values of Ftu listed)

Normalized tubing-not heat treated (including normalizing) after welding. Heat-treated treatment.

tubing

welded

after

1.0

heat

Tubing heat treated (including normalizing) after welding. Ftu of unwelded material in heat-treated condition: < i00 ksi I00

B

1.2.6

to

150

> 150

ksi

Spot

Weldin_

ksi

Ftu

for

normalized

1.0 Ftu for tubing

normalized

tubing

0.9 0.6

Ftu Ftu

0.8

Ftu

+

30

Design shear strength allowables for spot welds in various alloys are given in Tables B 1.2.6.1, B 1.2.6.2, and B 1.2.6.3; the thickness ratio of the thickest sheet to the thinnest outer sheet in the combination should not exceed 4:1. Table B 1.2.6.4 gives the minimum able edge distance for spot welds, these values may be reduced structural applications, or for applications not depended upon develop full tabulated weld strength. Combinations of aluminum suitable for spot welding are given in Table B 1.2.6.5.

allowfor nonto alloys

Section

B

1

25 September Page 51 B

1.2.6

Spot

Table

B

Weldin_

1.2.6.1

1961

(Cont'd_ Spot-Weld Uncoated

Maximum Design Shear Strengths Steels a and Nickel Alloys

for

f

Nominal

thickness

thinner

sheet,

of

Material

in.

strength,

150

0.006

.................

0.008 0.010

................. .................

0.012

.................

0.014 0.016 0.018 0.020

................. ................. ................. .................

0.025 0.030 0.032 0.040 0.042

................. ................. ................. ................. .................

0.050 0.056 0.060 0.063 0.071 O. 080 O. 090 0.095 0.i00 0.112

................. ................. ................. ................. ................. ................. ................. ................. ................. .................

0.125

.................

ultimate

ksi and above

ib

70 ksi to 150 ksi

70 120 165 220 270

57 85 127

320 390 425 58O 750 835 1,168 1,275

235 270 310 425 565 623 85O 920

1,700 2,039 2,265 2,479 3,012 3,540 4,100 4,336 4,575 5,088 5,665

1,205 1,358 1,558 1,685 2,024 2,405 2,810 3,012 3,200 3,633 4,052

155 198

tensile

Below

70 ksi

°...........

70 92 120 142 170 198 225 320 403 453 650 712 955 1,166 1,310 1,405 1,656 1,960 2,290 2,476 2,645 3,026 3,440

aRefers to plain carbon steels containing not more than 0.20 percent carbon and to austenitic steels. The reduction in strength of spotwelds due to the cumulative effects of time-temperature-stress factors is not greater than the reduction in strength of the parent metal.

Section

B 1

25 September Page 52

B 1.2.6

Spot Table

Nominal of

thinner

Welding B

v

(Cont'd)

1.2.6.2

Spot-Weld Standards

thickness sheet,

1961

Maximum Design Shear Strength for Bare and Clad Aluminum Alloys a

Material

ultimate

tensile

strength,

ib

in. Above ksi

0.012 0.016

................... ...................

0.020 0.025 0.032 0.040 0.050 0.063 0.071 0.080 0.090 0.i00 0.112 0.125 0.160

.................... ................... ................... ................... ................... .......... , ........ ................... ................... ................... ................... ................... ................... ...................

60 86 112 148 208 276 374 539 662 824 1,002 1,192 1,426 1,698 2,490

56

28 to

ksi 56 52 78 106 140 188 248 344 489 578 680 798 933

1,064 1,300

ksi

20 ksi to 27.5 ksi

19.5 ksJ and belo_

24 56

16 40

80 116 168 240 321 442 515 609 695 750 796 840

62 88 132 180 234 314 358 417 478 536 584 629

aspot welding of aluminum-alloy combinations conforming to QQ-A-277, QQ-A-355, and QQ-A-255 may be accomplished. The reduction in strength of spotwelds due to cumulative effects of tlme-temperature-stress factors is not greater than the reduction in strength of the parent metal.

Section B I 25 September Page 53

BI.2.6

Spot Table

Welding B 1.2.6.3

Nominal thinner 0.020 .022 .025 .028 .032 .036 .040 •045 .050 .056 .063 .071 .080 .090 .i00 .112 .125

(Cont'd) Spot-Weld Maximum Design Shear Strength Standards for Magnesium Alloys a Welding Specification MIL-W-6858

thickness sheet_

of in.

Design

shear

alloys

AZ31B

i Ib

72 84 i00 120 140 164 188 220 248 284 324 376 428 496

................... ................... ................... ................... ................... ................... ................... ................... ................... ................... ................... ................... ................... ................... ................... ................... ...................

aMagnesium combination.

strength

572 648 720

and

HK31A

may

be

spot

welded

in any

1961

Section

B 1

25 September Page 54

BI.2.6

Spot

Weldin_

Table

B

Nominal

(Cont'd)

1.2.6.4

thickness

thinner

sheet_

Minimum Edge Joints ab

Distances

for

in.

Edge

distance_

................... .................... ................... ................... ................... ...................

3/16 3/16 7/32 1/4 1/4 9/32

0.045 0.050 0.063 0.071 0.080 0.090 0.i00 0.125 0.160

................... ................... ................... ................... ................... ................... ................... ................... ...................

5/16

gages

next thinner gage bFor edge distances reductions in the

Fig.

B 1.2.6-1

Spot-Welded

of

0.016 0.020 0.025 0.032 0.036 0.040

alntermediate

1961

will

in.

5/16

3/8 3/8 13/32 7/16

7/16 9/16

5/S conform

to

the

requirement

shown. less than those specified spot-weld allowable loads

Edge

E_

Distances

for

for

the

above, appropriate shall be made.

Spot-Welded

Joints.

Section 25 BI.2.6

Spot

Welding

(Cont'd)

Page

55

(_IO_ o=_) vv

_v

vv

_-"_=

_,"_

_ _

_ _

_

_' _'

19z=v-bb

-'4

"o r-q CO

(_IOE

P_IO)

0

(_Og

P_ID)

_-v-bb _

9_-V-_)_)

0

O_

0 .rq 4_

q(gLOt

.'4 ,a

o c_)

P_ID)

tgC-V-bb

(OOII)

I9_-v-bb

(ooTt)

_-v-bb

o (_gO_

P_ID)

_9E-V-O0

< I

(_o0_)

6_-v-bb

(_0_

o_)

_-v-bb

_

(_0_

_)

_-V-bb

_' _

.'4

_

<

(I909)

LZ¢-V-bb

.'4

(_o_)

_[-v-bb

<

(_o_)

_[-v-bb

_'

=

CO

o .'4 0000 __

•_

_-_

0 _

000000 __

.'4 .'4

<

u_ •"4

0

CO u_

,°•°°....o°.°._ ,.°°°°°,°.°.,°_ °o°°°°°.°°°°.°_ • ° ............

,-q

CO

0

°°•°°°°°°°|

....

°•oo.°,°.._

..............

•"4

•"4

|

....

°,.°,,°o.°,.°°_ •.°°°,_°°..°°°_

_

_

_

_

_

_

_

_

_

_

&&&&&&&&&&&&&&&

_

_

_

_

B

September

_

1 1961

Section

B

1

25 September Page 56 B

1.2.7

Reduction Weldin_

in

In applications doublers are attached

Tensile

Strength

of

Parent

of spot welding where ribs, to sheet, either at splices

on the sheet panels, the sheet shall be determined

Metal

to Spot

intercostals, or at other

allowable ultimate strength by multiplying the ultimate

strength ("_' values where available) by factor shown on Figures B 1.2.7-1 through of the basic sheet efficiency in tension

Due

1961

or points

of the spot-welded tensile sheet

the appropriate efficiency B 1.2.7-4. The minimum values should not be considered

applicable to cases of seam welds. Allowable ultimate tensile strengths for spot-welded sheet gages of less than 0.012 inch for steel and 0.020 inch for aluminum shall be established on the basis of tests.

Section B 1 25 September 1961 Page 57 BI.2.7

Reduction in Tensile Weldin_ (Cont'd)

Strength

of

Parent

Metal

Due

to

Spot

020 in. = U = • u

90

__ _

ww _ _

_._

_

80

032 '0.040 '0.050

in. in. in.

063 080

in. in.

O9O in. "_ _= =

125 in. sheet

70

u

gage

6o 0

0.5

Spot

Spacing

Fig.

1.0

1.5

(Center

2.0

2.5

to Center),

3.0 Inches

B 1.2.7-1 Efficiency of the Parent in Tension for Spot-Welded 301-1/2 Corrosion-Reslstant Steel

Metal H

I00

/ _4a

u c O U

-

'°I///, ,o

in.

o,o,. 050 in. 063

!1171

,kl II1 .r,I

032

in.

o,o ,,.

-/

1090

in.

.125

in.

gage m U

_ [.-(

70_/ 60

sheet

50 0

0.5

Spot

Spacing

Fig.

1.0

1.5

(Center

2.0

2.5

to Center),

3.0 inches

B 1.2.7-2 Efficiency of the Parent in Tension for Spot-Welded 301-H Corrosion-Resistant Steel

Metal

Section

B

1

25 September Page 58

B 1.2.7

Reduction in Tensile Weldin_ (Cont' d)

Strength

of

Parent

Metal

Due

to

Spot

I00 0. 012 0. 020

.4

O. 032

___

9O .4 t&4

J

J= =

_0.

7O

U .4

0. 050 0. O. 040 063

in. in. in.

0. 090 O. 080

in. in.

125 in. sheet gage

6O

_O

0

0.5

Spot Fig.

1.0

Spacing

1.5

2.0

(Center

B 1.2.7-3

2.5

to Center),

Efficiency

of

3.0 inches

the

Parent

Metal in Tension for Spot-Welded 301-A, 347-A, and 301-1/4 H Corrosion-Resistant

Steel

I00

_4J O ;:

90

[O.02O

in.

_ID.040

in.

fILL -I_0.032

.4

in.

O. i00 --0.090 O. 080 0.071 O. 063 O. 050

80

O

?0

\ O _4 m

in.

80 ___

&J

•J

in. in.

6O

5O 0

0.5

Spot Fig.

Spacing

I

1.5 (Center

in: , in. in. in;' in. in.

Sheet

gage

l

l

2

2.5

to Center),

3 inches

B 1.2.7-4 Efficiency of the Parent Metal in Tension for Spot-Welded Aluminum Alloys

1961

Section 25

B

I

September

1961

f_

Page

B

1.3.0

Brazing

Insofar Brazing to

as

is

suitable

metal

is

furnace

allowable tures

the be

the

brazing

or

brazing

base

process

and

applicable

in

brazing

shall

the

to

steel.

by

heating

nonferrous metal.

strength

of

design.

employed is

the

filler The

filler

the

parent

accordance

Where calculated

subjected

Material

Heat-treated

a

base

structural is

in

using

only

produced

capillarity.

which be

is

the

upon the

metal

by

of by

process

silver

the

F that

joint

considered

brazing of

800 °

the

is

coalescence

below

through

of

strength

of

above

shall

brazing

wherein

point

distributed

metal

copper

herein,

a weld

a melting

effect

base

as

temperatures

having

The

discussed

defined

metal

or

59

to

with

Allowable material

Mechanical

tempera-

following:

Strength

properties

normalized

(including normalized) used in "as-brazed"

the

the

of

material

condition Heat-treated

material

(including reheat-treated or

B

after

1.3.1 The

B

of

900 ° are

stress

for

design

for

design

shall

be

shall

be

15

ksi,

for

all

treatment.

shear or

Silver-brazed

3,

heat

Brazin$

allowable

exceeding Class

shear

heat

clearances

inch.

to performed

Brazing

Silver

The that

treatment

brazing

allowable

1.3.2

properties

corresponding

during

Copper

conditions

Mechanical

normalized)

F.

listed

gaps

stress between

areas

should

Acceptable in

parts

Federal

not

brazing

to

be

be

subjected

alloys,

Specification

brazed

with

15

ksi,

provided

do

not

to

temperatures

the

exception

QQ-S-561d.

exceed

0.010

of

Section

B

1

25 September Page 60 Reference: (I)

MIL-HDBK-5, Strength of Metal Aircraft Supply Support Center, Washington 25,

Elements, Armed Forces D.C., March 1959.

1961

SECTION B2 LUGS AND SHEAR PINS

TABLE

OF

CONTENTS Page

B2.0.O

Analysis

2.1.0 2.2.0 2.3.0

of

Analysis Analysis Analysis

Lugs

and

of Lugs of Lugs of Lugs

Shear

with with with

B2-iii

Pins

.....................

Axial Loading ............ Transverse Loading ....... Oblique Loading ..........

1

2 17 23

Section

B

27

1961

July

Page B

2.0.0

Analysis

The

method

applicable loads

in

loads

are

See

Fig.

to the

Lugs

and

described aluminum

axial,

treated B

of

2.0.0-1

in or

Shear

this

steel

Sections

for

section

and

is

lugs.

oblique

B

2.1.0,

description

of

B

semi-empirical The

Transverse

B

2.0.0-1

and

load

load

load

is

considers Each

B

directions.

Oblique

and

analysis

directions. 2.2.0,

Axial

Fig.

i

Pins

alloy

transverse in

2

2.3.0

of

these

respectively.

B 2.1.0

Analysis

of Lugs

with

Axial

.

Tension across be considered.

the

Shear tear-out and are covered curves.

or bearing. by a single

3.

Shear

4.

Bending of the pin. based on the modulus

5.

Excessive

.

of

the

net

pin.

yielding

section.

This

Stress

1961

fail in any of by the methods

concentration

These two are closely calculation based on

is analyzed

The ultimate of rupture. of

B

27 July Page 2 Loadin_

A lug-pin combination under tension load can following ways, each of which must be investigated presented in this section: i.

Section

bushing

in

the

usual

strength

of

the

the

must

related empirical

manner. pin

is

(if used).

Yielding of the lug is considered to be excessive at a permanent set equal to .02 times pin diameter. This condition must always be checked as it is frequently reached at a lower load than would be anticipated from the Ftu

ratio of the yield for the material.

stress,

Fty

to the

ultimate

stress,

Notes: a.

Hoop tension at tip as the shear-bearing failure.

Do

The lug should be checked for side loads (due alignment, etc.) by conventional beam formulas B 2.1.0-1).

Analysis i.

procedure

Compute e/D,

.

Ultimate

(See W/D,

to Fig.

D/t;

load

for

obtain

of lug is not a critical condition, condition precludes a hoop tension

ultimate

B 2.1.0-1 Abr

= Dt,

for At =

shear-bearing

axial

to mis(Fig.

load.

nomenclature) (W-D)t failure:

Note: In addition to the limitations provided by curves "A" and "B" of Fig. B 2.1.0-3, Kbr greater than 2.00 shall not be used for lugs made from .5 inch thick or thicker aluminum alloy plate, bar or hand forged billet.

2

Section 27

B

July

1961

F Page B2.1.0

Analysis

of

Lugs

with

Axial

Loading

3

(Cont'd)

Bushing

W 2

P

.10P

(Minimum to

Fig.

(a)

Enter

(b)

The

Fig.

B

ultimate

B

=

2.1.0-3

Kbr

load for

assumed

misalignment)

2.1.0-I

load

P'bru

side

account

with

for Ftux

e/D

shear

and

D/t

bearing

Abr

to

obtain

failure,

Kbr

P'bru

........................

is (I)

where

Ftu x

,

Ultimate

load

(a)

Enter Fig. material

(b)

The

=

for

B

ultimate

P'tu

Ultimate tensile strength in transverse direction.

tension

failure:

2.1.0-4

with

load

=

Kt

Ftu

for

At

W/D

tension

to

obtain

failure

of

lug

K t

P'tu

...........................

material

for

proper

is

(2)

2

B2.1.O

Analysis

of

Lugs

with

Axial

Loading

Section

B2

July Page

1964

9, 4

(Cont'd)

where Ftu= 4.

Load

for

(a)

Enter

(b)

The

ultimate

yielding Fig.

yield P'y

of B

tensile

the

= Kbry

of

lug

material

lug

2.1.0-5

load,

strength

with

e_

is

Abr

Fly

e/D

to

obtain

Kbry.

..........................

(3)

where

Fty= 5.

Load

for

Tensile

yielding P_ry

of

= 1.85

yield

the

stress

bushing

Fty

Abrb

of

the

in bearing

lug

material.

(if

used):

.......................

(4)

where

Fty= Abrb 6.

Pin

bending

Compressive = Dpt

(Fig.

yield

stress

of

bushing

material

B 2.1.0-1)

stress.

P/2

l'-ITtt p/2 I t2

+(_

P/2

t 3 (a)

P/2-_--

J

P/2

I

P/2

p/2 -_--1 7 (t4/2)

(b) Fig.

B 2.1.0-2

Section

B

27

1961

July

Page B

2.1.0

Analysis (a)

of

Lugs

Obtain (See

r

with

moment Fig.

=

arm

B

the

"b"

as

(Cont'd)

follows:

compute

- -7

for

t2

smaller

(P'u)min

Loading

2.1.0-2a)

_-

Take

Axial

of

and

compute

B

2.1.0-6

the

inner

lug

.....................

P'bru

and

P'tu

for

(P'u)min/AbrFtux

the

(5) inner

Fig.

obtain for

the

near from

b

the

shear

=

and

of

the

plane.

"g" may

is

be

applies

the

Calculate

distributed

to

the

lugs

that

inner

maximum

pin

bending

stress

r"

to

compensates

pin the

and bearing

moment

load

arm

"b"

......................

between

Note

x

which

Calculate

gap

zero.

only

"7"

4

as

,,

(Pu)min/AbrFtu

factor,

+g+7

Where

(b)

reduction

"peaking"

the

with

lug

.

t

Enter

5

the

as

in

(6)

Fig.

peaking

B

2.1.0-2a

reduction

factor

lugs.

bending

moment

M,

from

the

from

"M",

assuming

in

bending

equation

(c)

Calculate an

(d)

My/l

Obtain of

ultimate

Section

B

inadequate take

(e)

strength 4.5.2.

pin

of

strength

analysis

as

Consider

a portion

indicated of

have

any

and

should

the

the

strength, excess

for

the

pin

of

the

lugs

should

it

lug

may

be

strength

by

by

use

show possible to

continuing

to

show the

be

_=haded

area

value chosen equal

to of

to

be

trial

Factors

as

B 2.1.0-2b.

to

and of

inactive

considered

sufficient by

be

Fig.

carry

error

Safety

The

active the

to for

may load

give the

lugs

pin.

Recalculate all lug Factors loads reduced in the ratio actual

(g)

pin

analysis

any

thickness

desired

approximately

(f)

the

the

follows:

by

portion

of

If

bending

advantage

adequate

and

resulting

distribution.

of Safety, with ultimate of active thickness to

thickness.

Recalculate of

Safety

as

follows:

pin using

bending _l reduced

moment, value

M of

=

P "b"

(b/2), which

and is

Factor

obtained

2

Section B 2 27 July 1961 Page 6 B 2.1.0

Analysis

of

Lugs

with

Axial

Loading

(Cont'd) v

Compute

for

the

inner

r =

Take

2

the

based

the

(P_)min/Abr 6 with

This

Factor

+7

reduced

value

excessive

structure.

In

of

=

2t4

and

"r

"

the

(Pu)min

D.

Enter

to

obtain

Then

the

inner

lug,

and

compute

Fig. the

B

should

of

load

not on

be

the

arm

stresses

cases

the

pin

uniformly

in must

across

if

the

lugs

the be

is

(9)

used

outer

2.1.0-

reduc-

moment

.....................

"b"

load

for as

Abr

bending

such the

Safety,

the

2.1.0-2b

P'tu

peaking.

of

introduce

distribute

and

_ ___4 _

eccentricity

Compute (a)

g

for

resulting

to

7.

"

B

thickness,

Ftux

"7

+

P'bru

where

(P_)min/Abr

t3 = -_--

b

of active

Ftux,

factor

Fig.

2t 4

smaller

upon

tion

lug,

adjacent

strong

the

entire

enough lug.

F.S.

following

Factors

of

Safety:

Lug p' bru Ultimate

F.S.

in

shear-bearing P

......

(lO)

p!

tu Ultimate

Yield

F.S.

F.S.

in tension p,

=

-

p

.............

--Y-P

.........................

in

=

(ii) (12)

Pin

(b)

F Ultimate

F.S.

shear

............... (13)

_U

fs

Ultimate

(c)

F.S.

Bushing

(if

in

F b = -_b

bending

............... (14)

used) p,

(15)

i)rx Yield

An

analysis

failure

of

F.S.

for the

in

bearing

yielding bushing

p

of is

not

the

pin

and

required.

ultimate

bearing

Section 27 Page B2.1.O

Analysis

of

Luzs

with

AxiJl

Loading

4-J

cq

O

oO

See

notc,_; on

I_)1 ]ow[,_;,

i_ :,,c,.

(Cont'd)

B

July 7

2

1961

B

2.1.0

Analysis

of Lugs

with

Curve A is a cutoff to be used billet when the long transverse direction C in the sketch.

Axial

Loading

in the plane

B 2

27 July Page 8

1961

(Cont'd_

for all aluminum grain direction

alloy hand forged has the general

Curve B is a cutoff to be used for all aluminum alloy plate, hand forged billet when the short transverse grain direction general direction C contains the parting direction C.

Section

sketch, and for die forging in a direction approximately

bar has

and the

when the lug normal to the

B 2.1.0

Analysis

of Lugs

with

Axial

Loading

Section

B 2

27 July Page 9

1961

(Cont'd)

o9

°

0 1.0

1.5

2.0

2.5

3.0

3.5

w/D Fig.

Legend

on

following

B 2.1.0-4

pages.

4.0

4.5

5.0

Section 27 Page

B

2.1.0

Analysis

of

Lugs

Legend - Figure F in sketch L

=

longitudinal

T

=

long

N

=

short

Curve

B

with

Axial

2.1.0-4-

Loading

L,

T,

N,

indicate

grain

(normal)

1

4130, 4140, 2014-T6 and

4340 and 7075-T6

7075-T6

bar

and

2014-T6

hand

2014-T6

and

7075-T6

2014-T6

and

7075-T6

7075-T6

extrusion

7075-T6

hand

forged

billet

_

36

2014-T6

hand

forged

billet

>

144

2014-T6

hand

forged

billet

_

36

Curve

8630 steel plate _ 0.5

extrusion

forged

die

in

(L,T)

sq.

in.

(L)

billet

_

144

forgings

(L)

(L)

2

2014-T6

and

17-4

PH

17-7

PH-THD

Curve

>

0.5

in.,

_

1

in.

(T,N)

7075-T6

die

forgings

sq.in.

(L)

sq.in. sq.in.

(L) (T)

(T)

3

2024-T6

plate

2024-T4

and

Curve

plate

(L,T) 2024-T42

extrusion

(L,T,N)

4

2024-T4

plate

2024-T3

plate

2014-T6

and

7075-T6

2024-T4

bar

(L,T)

7075-T6

hand

forged

billet

>

36

sq.in.

(L)

7075-T6

hand

forged

billet

_

16

sq.in.

(T)

alloy

casting

Curve

195T6,

(L,T) (L,T) plate

>

I

in.(L,T)

5

356T6

aluminum

July i0

(Cont'd)

transverse transverse

B

220T4,

and

7075-T6

hand

forged

billet

>

16

sq.in.

(T)

2014-T6

hand

forged

billet

>

36

sq.in.

(T)

in

direction

2

1961

Section B 2 27 July 1961 Page ii B 2.1.0

Analysis

Curve

alloy Note:

parting Curve

18-8 Curve

18-8 hard,

Lugs

with

Axial

Loading

(Cont'd)

6

Aluminum (N).

of

plate, for

plane.

die

bar,

hand

forgings,

7075-T6

bar

forged N

billet,

direction

and

exists

die only

forging at

the

(T)

7

stainless

steel,

annealed

steel,

full

8

stainless interpolate

between

hard, Curves

Note: 7 and

for 8.

1/4,

1/2

and

3/4

Section

B 2

27 July

1961

Page B2.1.O

Analysis

of

Lugs

with

Axial

Loading

12

(Cont'd)

2.0

1.5

1.0

Kbry

.5 D

0

11 0

I! 1.0

2.0

e/D FiB.

B 2.1.0-5

3.0

4.0

Section

B

27

1961

July

Page B

2.1.0

Analysis

of

Lugs

with

Axial

Loading

13

(Cont'd) 0

_4

I

i , °

IIIN

0

0 p_

i

I

I

I

IIIIIIl_ ::

(n

-if3

IllI1 0

-o-


IIlll ,--_I c,4 I

0 0

OOo_ iiiii\i\oi\ °'-° Ill/IX\t

S .4

.,-I v

oo iii_o,\ \ -4

oo /_x\ U

'

_

_

---Lfj_-c_-_ -=\ 0 ,-.4

0

C_

cO

_--

,_9

u'_

_

o'_

c_I

,-4

0

2

B2.1.O

Analysis Special I.

of Lugs

with

Axial

Loading

Section

B 2

27 July Page 14

1961

(Cont'd)

Applications

Irregular lug section entire thickness.

- bearing

load

distributed

over

For lugs of irregular section having bearing stress distributed over the entire thickness, an analysis is made based on an equivalent lug with rectangular sections having an area equal to the original section.

1

B

12sec

T

(Sec. B)

(a)

(b) Fig. Dashed

2.

Critical

B 2.1.0-7

lines

bearing

show

equivalent

lug

stress

NASA Design Manual Section 3.0.0 lists the values of the ultimate and yield bearing stress of materials for e/D values of 2.0 and 1.5, these are valid for values of D/t to 5.5. The ultimate and yield bearing stress conditions outside of the above range the

following

(a)

Ultimate obtain Fbr u

for may

geometrical be determined

in

manner: bearing Kbr

from

= Kbr

Ftux

stress: Fig.

For

B 2.1.0-3

the

particular

D/t

and

then

where Ftu x = Ultimate strength direction.

of

lug material

in transverse

e/D,

Section 27

July

Page B2.1.O

Analysis (b)

of

Yield

Lugs

with

bearing

Axial

Loading

stress:

Kbry

from

Fig.

B

Fbry

=

Kbry

=

Tensile yield direction.

With

2.1.0-5.

B

2

1961

15

(Cont'd) the

particular

e/D

obtain

Then

Ftyx

where

Fty x

3.

Eccentrically

located

If the hole the ultimate P'tu

and

ing

the

factor

by

el factor

+

P'y

e2

as in Fig. lug loads

for

+

Actual

the

Multiple

Lug-pin

B 2.1.0-8 (e% are determined

equivalent

in

transverse

lug

less than e2) , by obtaining

shown

and

multiply-

2D

Equivalent

lug

shear

are

The

load

the

total

Fig. Table

B

2.1.0-8

B

having

analyzed

carried

2.1.0.1.

the

geometry

according

by

applied

2.1.0-9 B

lug

connections

combinations

2.1.0-9

(a)

material

2D

Fig.

B

lug

= 2e 2 +

4.

of

hole

is located and yield

P'bru,

stress

each

load and

the

to

lug "P" value

is

among of

shown

the

determined the "C"

in

following

is

criteria:

by

lugs

Fig.

as

obtained

distributing shown from

on

B 2.1.0

Analysis

of

Lugs

with

Loadin_

B

1961

(Cont'd)

-Two outer lugs of equal thickness not less than C t' (See Table B 2.1.0.1)

lugs of thickness

These equal

Axial

Section 27 July Page 16

m

P

_---mm_ P2

P

_

P2

CPI..9,.---V/I/I/I/

These lugs of thickness t" Fig.

B 2.1.0-9

(b)

The maximum B 2.1.0.1.

shear

(c)

The

bending

maximum

formula,

M

load

PI b = -=7

on

the

pin

is given

moment

in

the

pin

where Table

Total

number

lugs both

including sides

OO

"b"

is given

B

2..I.0.1

Pin

Shear

in Table

is given

in Table

of

.35

.50 P1

.28

t' + t" 2

.40

.53 P1

.33

t' + t" 2

.54

P1

.37

t' + t" 2

.44

.54 P1

.39

t' + t" 2

.50

.50

.50

t' + t" 2

.43

ii

equal

P1

by

the

B 2.1.0.1.

2

v

Section

B

27

1961

July

Page

B

2.2.0

Analysis

Shape In

is

Lugs

with

Transverse

Loading

Parameter

order

transverse This

of

17

to

determine

loading,

the

the

accomplished

shape

by

use

ultimate of

of

a

the

and lug

shape

yield

must

loads

be

parameter

for

taken

into

given

by

lugs

with

account.

A Shape

parameter

=av Abr

where

Abr

is

the

bearing

area

Aav

is

the

weighted

=

Dt

average

area

given

by

6 Aav

=

AI,

A2,

Fig.

_+_--_2_

B

A 3

+

and

A4

are

_--_3_

areas

of

+

the

_--A4)'--I

lug

sections

indicated

in

2.2.0-1.

A 3 least of any radial section

_w

_ radius

all

(a)

(b) Fig.

(i)

Obtain

(la)

the

areas

AI,

A2,

Fig.

B

AI,

and

A4

2.2.0-ia

except

that

A I and line.

A4

in should

A2,

are

B

2.2.0-1

A3,

and

measured

A4

as

on

the

(perpendicular a necked be

lug,

measured

to as

follows:

planes the

indicated

axial

shown

perpendicular

in

center

Fig. to

B

in line),

2.2.0-Ib,

the

local

2

B

2.2.0

Analysis (Ib)

of

Lugs

A 3 is hole.

(ic)

the

Thought

with

Transverse

least

area

should

on

always

any

Loading radial

be given

Section

B2

July Page

1964

9, 18

(Cont'd) section

to assure

around

that

the

the

areas

AI, A2, A3, and A 4 adequately reflect the strength of the lug. For lugs of unusual shape (e.g. with sudden changes of cross section), an equivalent lug should be sketched as shown in Fig. B 2.2.0-2 and used in the analysis.

Equivalent

lug

Equivalent lug

'] 50

Fig.

(2) Obtain

the

weighted

B 2.2.0-2

average 6

Aav

(3)

=

(3/AI)

Compute

Abr

(4) Ultimate

Obtain

(b)

P'tru

(6)

load

(a)

Obtain

(b)

P'y

Check

(I/A2)

= Dt

load

(a)

(5) Yield

+

and

P'tru

+

(I/A 3) +

Aav/ for

Ktr u from

Fig.

Abr

Ftux

P'y

the

lug:

from

Fig.

of

Ktry

= Ktry

bushing

Abr

yield

Abr

lug

= Ktru

(I/A4)

failure: B 2.2.0-4

B 2.2.0-4

Fty x and

pin

shear

as

outlined

previously.

450

Section

B

27

1961

July

Page B

2.2.0 (7)

Analysis

of

Investigate

Lugs pin

modifications: [e - (D/2)] distance at

with bending Take

/t _

Transverse as (P'u)min

use for = 90 ° .

the

Fig.

B

Loading

for

axial

[e

-

2.2.0-3

(Cont'd)

load

= P' tru" _/2)]

19

with In

the term

following equation the

edge

r

=

2

B2.2.0

Analysis

of

Lugs

with

Transverse

Loadin_

Ktry-All & steel

1.4

B 2

27 July Page 20

1961

(Cont'd)

125,000 I J

1.6

Section

HT

[_50,000

alum.

HT

alloys_

•

1.2

_'_-

_

8o,

1.0 Ktr u &

x

Ktry 0.8

0.6

0.4

0.2

I

,

I

! _,_I" I

I

0.2

0.4

I

I

i

i

i

A-Approxlmate strength.

0

I

0.6

If

Ktr u

0.8 Aav/Abr

Fig. Legend

on

following

B 2.2.0-4

pages

!

cantilever

is below 1.0

this

1.2

curve 1.4

O00h_

Section 27 Page B

2.2.0

Legend

Analysis

- Fig.

B

Curve

i:

4130,

4140,

Curve

2:

2024-T4 Curve

Lugs

with

Transverse

Loading

(Cont'd)

2.2.0-4

4340,

and

and

2024-T3

8630

plate

steels,

_0.5

heat

treatment

in.

3:

220-T4

aluminum

Curve

17-7

of

alloy

casting

4 :

PH

Curve

(THD)

5 :

2014-T6 Curve

195-T6 Curve

7075-T6

plate_0.5

and

2024-T4

plate

and

356-T6

and

extrusion

2014-T6

hand

2014-T6

and

Curve

in.,

2024-T4

aluminum

alloy

casting

8:

7075-T6

2024-T6 2024-T4

>0.5

7:

2014-T6

Curve

in.

6:

2024-T3 Curve

and

7075-T6

forged 7075-T6

plate>0.5

billets die

36

in.,_

i

sq.

in.

forgings

9:

plate and 2024-T42

extrusion

i0:

2014-T6

and

7075-T6

hand

7075-T6

plate_

1

forged

billet

_16

in. sq.

in.

in.

bar

as

noted.

B

July 21

2

1961

Section

B

27

1961

July

Page

B 2.2.0

Analysis

Legend

Fig. Curve

of

B

Lugs

2.2.0-4

with

Transverse

Loading

2

22

(Cont'd)

Cont'd

ii:

7075-T6

hand

forged

billet

2014-T6

hand

forged

billet>36

All curves Note: The closely

test

by

the and

In

should

case

than

that

the

portion

load

that

may

which of can

data. best

properties no

sq.

in.

sq.

in.

are for Ktr u except the one noted curve for 125,000 HT steel in Fig.

with

obtained

>16

available possibly

the could

the be

Curves

lug

carried

be

ultimate be

for

all

means

of

very

by

very approximately by curve be below curve (A), separate is warranted.

other

the

by

Ktry 2.2.0-4

conservative load

cantilever

load

cantilever (A) in Fig. calculation

(Fig. beam B

agrees

materials

correcting

transverse

carried

under

as B

to be beam

B

have

for some

taken

places. as

action

2.2.0-3).

action

is

been

material

less of

The indicated

2.2.0-4, should as a cantilever

Ktr u beam

\

Section

B

27

1961

July

Page B

2.3.0

Analysis

of

Interaction

In resolve

by

subscripts

the

and

of

used

into "tr"

utilize

the

equation

applied

for

Oblique

subject

and

interaction

ratios be

lugs loading

"a"

separately The

with

23

Loadin$

Relation

analyzing

to

Lugs

2

to

both

to

oblique

axial

and

loading

respectively), results

Ra 1'6

critical

ultimate

and

means

Rtr Io6

of

=

loads

in

yield

loads

i,

the

is

the an

(denoted

two

cases

interaction

where

Ra

indicated for

convenient

components

analyze

by

+

it

transverse

both

and

equation. Rtr

are

directions, aluminum

is

and

to

steel

alloys. where,

for

Ra

ultimate

=

(Axial

component

P'bru Rtr

and

for

Ra

=

Analysis

(i)

applied

from

Eq.

component

(P'tru

analysis

from

load)

I and of

divided

Eq.

by

applied

load)

procedure

(smaller

of

2.)

for

divided

_

=

90

by

deg.)

load:

(Axial

=

of

P'tu

(Transverse

yield

=

and

component

eq. Rtr

loads

of

applied

load)

divided

by

(P'y

from

3.)

(Transverse

component

(P'try

Analysis

from

of

applied

load)

Procedure

for

divided

_

=

90

by

deg.)

Procedure

Resolve

the

applied

and

obtain

the

the

interaction

load

lug

into

ultimate

axial and

and

yield

transverse

Factor

components

of

Safety

from

equation:

F.So

I Ra 1"6

(2) (3)

Check

pin

shear

Investigate modified

+

and

pin as

Rtr 1.6

j0.625

bushing

bending

yield

using

as

the

in

Section

procedure

B

for

2.1.0.

axial

load

follows:

P Take

(P'u)mi

n

= _Ral'6

+

Rtrl.6_

'x

In

the

equation

term the edge the direction

r

0"625 /

=

distance of load

[e

-

(D/2)]

at the value on the lug.

/t of

use "_

for "

the

[e-(D/2)]

corresponding

to

B

2.3.0

Analysis

of

Lugs

with

Oblique

Loading

Section

B

27 July Page 24

1961

(Cont'd)

Reference Melcone, M. A. and F. M. Hoblit, Developments in the Analysis of Method for Determining the Strength of Lugs Loaded Obliquely or Transversely, Product Engineering, June, 1953.

2

SECTION B3 SPRINGS

TABLE

OF

CONTENTS Page

B3.0.O

Springs

.............................................

i

i f

3.1.O Helical Springs ................................. 3.1.1 Helical Compression Springs ................. 3.1.2 Helical Extension Springs ................... 3.1.3 Helical Springs with Torsional Loading ...... 3.1.4 Analysis of Helical Springs by Use of 3.1.5

Nomograph ................................. Maximum Design Stress for Various Spring Materials .................................

3.1.6 Dynamic or Suddenly Applied Spring Loading... 3.1.7 Working Stress for Springs .................. 3.2.0 Curved Springs .................................. 3.3.0

Belleville

Springs

or

Washers

B3-iii

...................

i i 7 i0 12 15 19 24 25 29

_f

Section

B

3

f

19

May

Page

B

3.0.0

SPRINGS

B

3.1.0

Helical

Springs

B

3.1.1

Helical

Compression

Most offer

compression

resistance

The

longitudinal

the

spring

are

desired,

springs and

loads

Where

springs

may

are

have

to

of

of

the

produces

pitch

springs

length

of

which

the

shearing

load-deflection

varying

number

helical

reduce

springs

particular

with

any

open-coil,

acting

deflection

wire.

1

Springs

springs

to

1961

spring.

stresses

in

characteristics

diameters

may

configurations,

be

including

used.

These

cone,

barrel,

hourglass.

Round-Wire

Springs

The

relation

helical

between

springs

formed 8PD = __

f

the from

applied

round

load

wire

and

the

shearing

stress

for

is

.......................................

(1)

_d 3

s where

fs

= shearing

stress

(not

in

corrected

P

= axial

load

D

= mean

diameter

minus

in

pounds

for

per

sq.

inch.

curvature)

pounds of

wire

the

spring

diameter

or

coil

(Outside

inside

diameter

diameter plus

wire

diameter) d

=

diameter

Equation stress but, due

(i)

varies

the

mine

the

Fig.

B

torsion

is

the

wire

in

based

on

the

directly

actually, to

of

the

with

stress

is

curvature.

maximum

and

This

direct f

: max

The

shearing

3.1.1-i.

f

: k s

assumption

distance

greater stress

stress

correction

shear. k

the

inches.

The 8PD _d 3

on

that from

the

inside

correction for

static

factor

gives

equation .,..°................,--.,."

for

the

the

of

factor loads

of

the (k)

is

the the

magnitude

center

cross used

found

effect

maximum

of

the

section to

deter-

in of

thc_

wire;

both

stress

is (2)

Section B 3 December 3, Page

B 3.1.1

Helical

Compression

Springs

2

(Cont'd)

where

k = kc

kc

=

Cl=

+

4C 1 -

0.615 Cl i

Stress concentration of direct shear

factor

plus

Stress

factor

due

concentration

Ratio of mean diameter diameter of the bar or

,:I

of helix wire

the

effect

to curvature

to

the

2.1 ! .

.1.9 o

AJ U

I

1.

0

1

X

0

\

0

=

1.5

_

1.

0 °_

1.1

U

=

\

\ _

1.3

r

,

0 I

o'1

r_

•

lo

0

I

2

3

4

5 SprJ:n_ Fig.

6

7

index, B

8 D C l = _[

3.1.l--!

9

i0

Ii

12

1969

Section 19

4

Page B

3.1.1

Helical

Stress

Compression

correction

for

Corrections in

elastic

This from

for

must

is

Section

be

made

of

spring

made

various

to

to

3

(Cont'd)

account

for

materials

the

materials

3.0.0

3

1961

temperature.

properties

correction

Values

Springs

B

May

allowable

at

the

Design

for

the

relation

in

elevated

stress

various

of

changes

at

strength

and

temperatures.

of

the

temperatures

spring

may

be

material. obtained

Manual.

Deflection

The when

formula

using

round

wire

8NPD

in

between

helical

deflection

springs

and

load

is

3 . ........................................

_)-

(3)

Gd 4 whe r e = total N

=

G

= modulus

number

The by

deflection of

rigidity

deflection

combining

may

Eqs.

affect

deflection

in

(I).

Eq.

coils

of

(I) to

The

also

and an

be

in

Stress

appreciable

expression

Nf

given

(3). for

terms

degree, the

of

the

concentration and

no

deflection

shearing usually

stress does

not

is

needed

adjustment

is

_D 2

_

s

......................................... (4)

Gd

Sprin$

rate

The to

spring

deflect

the

previous

rate

(K)

spring

a

equations,

the

is

defined

unit

as

length.

spring

the By

rate

may

amount

proper be

of

force

required

substitution

shown

to

of

the

be

d4G K

.......................................... (5)

=-

8ND 3

Bucklin$

A buckle

of

Compression

compression under

ever,

the

sion

spring

spring

relatively

problem operates

Springs

of

which low

buckling inside

is

loads is a

long in

of

cylinder

compared

the little or

same

to manner

consequence over

a

rod.

its as

diameter a if

will

column. the

How-

compres-

Section 19 May Page 4 B 3.1.1

Helical

As

the

Compression

critical

Springs

buckling

load

B 3 1961

(Cont'd) of a colum

is

dependent

upon

the

end fixity at the supports, so is the critical buckling load of a spring dependent upon the fixity of the ends. In general, a compression spring with ends squared, ground, and compressed between two parallel surfaces can be considered a fixed-end spring. The following formula

gives Pc

the

critical

= JKL

buckling

load.

............................................

(6)

where Deflection J L K

one

= factor from = free length = spring rate

Fig. B 3.1.1-2 of spring (See Eq. 5)

Fig. B 3.1.1-2 (curve i) end on a flat surface and

Sprinss

of

Free

Length

is for squared the other on a

buckling for a squared and ground pressed against parallel plates. with which the user must contend.

Helical

=

Rectangular

and ground springs with ball. Curve 2 indicates

spring both This is the

ends most

of which are comcommon condition

Wire.

When rectangular wire is used for helical springs, the value of the shearing stress can be found by use of the equations for rectangular shafts. A stress concentration factor is applied in the usual way to compensate for the effect of curvature and direct shear. For the springs in Fig. B 3.1.1-3 (a) and (b) the stresses at points A 1 and A 2 are as follows:

f

kPR = -s 51bc2

for

point

A1

........................

(7)

for

point

A2

........................

<,8)

kPR f

values

= -s 5 bc 2 2

for51 The

stress

and52

for

various

concentration

Fig. B 3.1.1-3(a) and concentration factors value for rectangular

b/c

factor

ratios

are

should

be

found applied

to point A 2 in Fig. B 3.1.1-3 of Fig. B 3.1.1-1 may be used wire.

in Table for

B 3.1.1-1.

point

A 1 in

(b). The stress as an approximate

Section 19 May Page

B 3.1.1

Helical

Compression

Springs

1

B 3 1961

5

(Cont'd)

2

.7 Springs right of buckle.

above

and

the

curves

to

will

the

I0

II

.6

.5 _I_ II

\\

\

.2

\

.1

0

,1, 0

3

4

5 Free Mean

6 Length Diameter

Fig.

7 _ L D

B 3.1.1-2

8

9

12

Section B 3 19 May 1961 Page 6 P

AI

P

(a) Sprin

8

index,

C 1 =- C Fig.

b/c

_2

(b)

2R Spring

2R C I =-_-

index,

B 3.1.I-3

1.00

1.20

1.50

1.75

2.00

2.50

3.00

4.00

5.00

6.00

8.00

I0.00

oo

.208

.219

.231

.239

.246

.258

.267

.282

.291

.299

.307

.312

.333

.208

.235

.269

.291

.309

.336

.355

.378

.392

.402

.414i.421

.1406

.166

.196

.214

.229

.249

.263

.281

.291

.299

.307

Table

The equation deflection (5) is

for

the

2_pR3N

relation

between

from

Table

B 3.1.1-1

b and c are as shown in Fig. B 3.1.1-3 R is the mean radius of the spring N is the number of coils G P

is is

the the

the

load

(P)

and

the

.......................................... (9)

where: obtained

.333

B 3.1.I-i

_Gbc 3

is

.312

modulus of axial load

rigidity

Section B 3 19 May 1961 Page 7 B 3.1.2

Helical

Extension

Springs

Helical extension springs differ from helical compression springs only in that they are usually closely coiled helices 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 initial tensile load does not affect the spring rate. See Figure B 3.1.2-1 for the loaddeflection

relationship

of

a

preloaded

helical

extension

spring.

Spring with initial Load jfJ / Initial

i_____Spring

/

initial

without tension

tension J

. Deflection

Fig.

Stresses

and

B 3.1.2-1

Deflection

In helical extension springs, the shape of the hook or end turns for applying the load must be designed so that the stress concentration effects caused by the presence of sharp bends are decreased as much as possible. This problem is covered in the next article. If the extension spring is designed with initial tension, formulas (I) through (9) from Section B 3.1.1 are valid, but must be applied with some understanding of the nature of the forces involved.

Section B 3 19 May 1961 Page 8 B

3.1.2

Helical

Stress

Extension

concentration

Springs

in hooks

on

(toni'd) extension

springs.

p

P

t

d

A r3

r4 r2

(a)

(b) Fig.

a.

Bending

Stress

Fig. (P).

load

B 3.1.2-2(a) The bending

fb where the

k

is

32PR =--k _d 3

the

ratio

simplified

fb

4P +-_d 2

correction

bending A is:

moment

(PR)

due

factor

obtained

from

to

the

(zo)

................................

= radius

of center

equation

is:

= _d 32P___RR . /rl 3 L_ 3 )

r 3 = inside

always higher

illustrates the stress at point

Fig.

B 3.1.2-3,

using

2rl/d. rI

A

B 3.1.2-2

radius

line

of maximum

curvature.

.................................

(ii)

of bend.

The maximum bending stress be on the safe side and, than the true stress.

obtained by under normal

this simplified form will conditions, only slightly

Section B 3 19 May 1961 Page 9 B 3.1.2 b.

Helical

Torsional At

Springs

(Cont'd)

Stress

point

portion maximum

Extension

A',

Fig.

B

3.1.2-2(b),

where

the

bend

joins

the

helical

of the spring, the stress condition is primarily torsion. torsional shear stress due to the moment (PR) is 16PR

f

C1

=

- I

_

4C I

f

(12)

4

2r 2 d

simplified

This

4CI

........................... s = _d 3

A

/

The

form

similar

to

the

one

for

bending

is

................................

=

s

_d 3

will

also

(13)

_4 give

safe

results.

IIIIII IIIIII 4

3

\\

k

i_-

k_k 2 I

0 0

1.5

2.0

2.5

3.0

R_=D c d

"-

Fig.

_D h

B 3.1.2-3

3.5

4.0

4.5

Section B 3 19 May 1961 Page I0 B 3.1.3 helix.

Helical

Springs

A helical spring Such loading,

torsional loading acts as a bending Fig. B 3.1.3-I(b).

with

Torsional

Loading

can be loaded by as shown in Fig.

a B

of a shaft. The torque moment on each section The stress is then

torque about the axis 3.1.3-I(a), is similar about of the

of the to the

the axis of the helix wire as shown in

Mc fb = kn T-

where

the

stress

..............................

concentration 3CI 2

kl

=

- CI

3CI(C I

- 0.8 - i)

3CI 2 + C I k2

C1

=

3CI(C I

2R =--h

- 0.8 -I)

factor,

Kn,

is

' .........

given

(14)

as

inner edge

Rectangular cross section wire

outer edge

; "h" is the depth to the axis.

of

section

perpendicular

2 4C 1 k3

=

- CI

4CI(C I

-i

inner

- I)

"

edge

Round cross

outer

wire

2 4C 1 k4

=

+ C1

4CI(C I +

-

1

i)

section

edge

2R

Cl = T

Angular

deformation

The deformation of the wire in the spring is the straight bar of the same length "S'_ The total angular between tangents drawn at the ends of the bar is:

same as for deformation

MS E1 Angle

8

in

some

........................................ (15) cases

may

amount

to

a number

of

revolutions.

a 8

Section

B 3

19 May 1961 Page i I

B3.1.3

Helical

Springs

with

Torsional

Loadin$

(Cont'd)

7

I

M

h

\

/

x k

\

/

f 4--

Fig.

J

B 3.1.3-1

or

d

Section B 3 19 May 1961 Page 12 B 3.1.4 Fig.

Analysis

of

l_elic_Jl

S[_rings

by

Use

of

Nomograph

The procedure for using the nomographs B 3.1.4-2) for helical springs are as

(Fig. B follows:

I.

Set

the

appropriate

wire

diam.

on

the

"d"

scale.

2.

Set

the

appropriate

mean

diam.

on

the

"D"

scale.

3.

Connectthe factor:

two

points

and

read

a.

For tension and compression on Fig. B 3.1.4-1.

b.

For

torsion

springs

read

.

Set the correction appropriate "y" or and Fig. B 3.1.4-2.

5.

Set the "calculated" on the "f" scale.

6.

Connect the two point_, from corrected stress on t le (f')

the

curvature

springs

the

"k"

factor, obtained "k" scale to the

(Eq.

I or

Eq.

3.1.4-1

correction

read

scale

and

the

on

"y"

Fig.

in step 3, on right of Fig.

14 with

steps 4 scale.

and

5,

kn

= I)

and

scale

3.1.4-2. the B 3.1.4-1

stress

read

the

Section 19

May

Page

B

3.1.4

Analysis

of

FIBER

Helical

Springs

STRESS

CORRECTION

Helical Find

Correction From

Spring

Extension

and

Factor

by

FOR

of

Nomograph

Left

3

1961 13

(Cont'd)

CURVATURE

Compression Using

Index

Use

B

Springs

Correction

Factor

Found

on

Half of Chart--Determine True Fiber Stress

f'

300,000

-I

0 OZ157'

d 0.031

i

O_

0 4.1 U

,ZO .

OO_

,19

-3

"_

2OODOO]

I_I00..4 02 0

. "17

04)

00!

._e

c ",',

007

-_5

_

009

-13

]

| 031 |

r O.

"t2

_

05_ 0

064 4

_-L, _ °_".I :_

123

I:: 6

> I-., :3 r.D

.,-1

121,, L.]9 r._

J-

01)

0 _D

1.18 116-,I

6O/)O 13

t i55" '9

6

1145-. _J

L125. ,I0

°'[

]

7

,_

5o_oo

a

4 0,000.

4_000 r._

12, 8

I.II-,

II CO

0..

1,,.= _U (3.,

°r,4

II0.

,12

109-

_13

'9 .10

!

O -,,-I r/)

7

0

C 0 -=,-I

r.o

3opoo

0 E'_

1085._.14 1.084

15

108'

16

I:: -.4

C ,,-.4

u) m O_

2_000

20po0

(,_ (t,,t 4,.I rJ3 lo,.) .,-4

_000

"_

Fig.

B

3.1.4-1

Section

B 3

19 May 1961 Page 14

B 3.1.4

Analysis

of

Helical

FIBER

Springs

STRESS

CORRECTION Torsion

Find

Correction

by Use

Factor

FOR

of

Nomograph

(Cont'd)

CURVATURE

Springs Using Correction Factor Found Left Half of Chart--Determlne True Fiber Stress

on

d

zoo_ooo.

0021

'_-_ D 0.0

"l

OO3 t

•

16£0' 0

0000_00_

_U _

0,I

=

g

1.4ei/ 1.410,

o

L367,

_-J

1332,

k

2OOpOOr._ Of" t61_ c,..) ¢J

.

°

_4Z. S tiN" IJ3@ t25,

_1

3":

05" 06-

S-.

0.7£

s7-

00_OO120-

701)00-

4.-)

I I1_1, #

04-

i 115 iiioi lOT.

t_ :;> I.

, 8 IN:

7_oo-

o,

0 ¢..)

6opoo.

_oooIK)-

_J I

rj I0_"

_000

-

5QOoo.-

o in =

4o/:X_-

4O0OO--

liosO-ll 0

09-:.

m-

1077-

II

=

30_O00-

L072. 12 20-

3oooo

m (I)

I.O?o. i3 1066. 14

zo.ooo-

1064' i5

2-

3d

I,_7,

I#

I055"

Ill _9

Fig.

zopoo-

B 3.1.4-2

.__

Section

B

19 May

1961

Page

B3.1.5

Maximum

Design

Stress

for

Phosphor Maximum

Various

Bronze Design

Sprin_

Materials

Spring Wire Stresses

120

I00 P-4

| m m U

80

&J

-,-4

60

4O

2O

0 • Ol

.02

•04

.I0 Wire

_

Fig.

Diameter

B 3.1.5-i

.20 - Inches

•40

15

3

Section B 3 19 May 1961 Page 16 B3.1.5

Maximum

Design

Stresses

for

Various

/ /

/ !

/ IS'_ - sse=]S

Fig.

u_3saG

B

mnm3x_]

3.1.5-2

Spring

Materials

(Cont'd)

Section 19

May

Page

B

3.1.5

Maximum

Design

Stresses

for

Various

S

Materials

I II

IS)i -

-

sso=3S Fig.

u_ls_ B

3.1.5-3

mnmlx_i

B 1961 17

(Co_

I

I I I1"

0

3

Section

B 3

19 May 1961 Page 18 B 3.1.5

Maximum

Design

Stresses

15DI - ss_z_S

Fig.

for

Various

uSTeo(]

Spring

Materials

v

(Cont'd)

mnmTxe14

B 3.1.5-4

_-

Section 19

B

3.1.6

Dynamic

A

freely

delivers

or falling

a dynamic

forces

may

Suddenly

be

or

Applied

weight,

or

impact

load,

analyzed

on

the

Spring

moving

basis

that

force.

of

May

3

1961

Page

19

a

structure

Loading

body,

or

B

strikes

Problems

the

following

and

no

involving

such

idealizing

assumptions: 1. takes

Materials

place

at

inelastic

2.

The

Then,

its

may

the

on

be

of

the

(a)

(b)

a

system or

of

assumed

completely

the

resisting

system,

For

very

slowly

P K --

For

dissipation

the

supports

directly

of owing

For

the

of

conservation

instant

the

of

a moving into

following

to

2P(S

G

the

energy,if

body

the

is

internal

formulas

will

stopped, strain

apply:

loads

° ........

°

° °

° ...........

(16)

° ° °

applied (17)

loads

_2

local

force.

2P K (c)

energy to

proportional

applied

transformed

° ......

suddenly

is

principle at

applied

o .....

loads

at

statically

the that

is

b

or

materials.

of

basis

energy

of

impact

dynamically

further

kinetic

energy

of

deflection

of

elastically,

point

deformation

magnitude

it

behave

the

=

dropped

+ K

_)

from

a

given

height

.................................

(18)

where: =

Total

deflection

K

= Spring

P

= Load

S

= Height

The

rate on

spring load

following

is

dropped.

problems

will

illustrate

which

compresses

such

conditions

and

their

of

load,

solutions:

Problem

I.

Given determine weight.

a the

spring maximum

load

and

one

deflection

inch

for

resulting

each from

pound a

4

lb.

Section

B 3

19 May 1961 Page 20

B 3.1.6 (a)

Dynamic Case The

weight P K

since Case

Applied

Loading

(Cont'd)

is

laid

4 i = 4 K

_

i

gently

in.

from

maximum

on Eq.

load

the

spring.

16

= 4

lb.

dropped

on

2:

The

weight

is

suddenly

5 = K2__[P ffi2(4)i = 8 maximum (c)

Spring

i:

5 =

(b)

or Suddenly

Case

load

= 8

in.

from

Eq.

the

spring

from

zero

height.

17

lb.

3:

The weight inches. 52

is dropped

= 2P(S + 5) K

=

2(4) 1

on

the

(12 +

spring

5)

from

from

Eq.

a height

of

12

18

or 5

2

- 85 =

8 +

maximum

- 96

= 0

_82

+

4(96)

2 load

14.6

=

14.6

in.

lb.

From the maximum load produced, Eq. 2 section B 3.1.1 may be used to calculate the stress produced. This should be within the limits indicated on Fig. B 3.1.5-1, B 3.1.5-2, B 3.1.5-3 and B 3.1.5-4. Problem

2.

For many uses it is necessary to know the return speed of a spring or the speed with which it will return a given weight. A typical example of this problem could be stated as follows: a spring made of 5/16 in. by 3/16 in. rectangular steel contains 4 3/8 total coils, 2 3/8 active coils, on a mean diameter of 1 5/16 in. The spring compresses 5/32 in. for 200 lb. load. If the spring is compressed and then instantaneously released, how fast will it be moving at its original free length position of 1 21/32 in.? The solution is as follows:

Section 19

May

Page

B

3.1.6

Dynamic

Weight

or

per

Suddenly

turn

=

Applied

_(i =

.0685

(2

weight

To

3/8)

of

this

active

active

.0542

weight,

if

we

lb.

coils

(1/3)

add

the

each

=

.0542

Ib

(one-third

cu.

in.

coil

of

the

the

moving

involved.)

dead

coil

at

the

(.0542

+

.0685)

end

plus

any.

The

potential

energy

of

a

distance

moved,

or

1/2(200)(5/32)

kinetic

per

in

equivalent

the

lb.

steel

material

1961

21

(Cont'd)

of

The

times

Loading

5/16)(3/16)(5/16)(.283) .0685

spring

lb.

Spring

B

total

energy

equals

Mass

= Weight g

weight

1/2

spring

is

Mv 2 wherein

is

equal

to

=

(M)

is

1/2

15.6 the

0.1227

lb.

the

in. mass

total

lb. and

load

The (v)

is

the

velocity.

or

32.16

where

g

is

the

gravitational

acceleration,

ft/sec./sec. .1227v

Therefore

15.6

2

=

or

v

=

314

in/sec

2(32.16)(12)

Often such

springs

instances

entire

energy.

typical

problem

Problem

3.

A a

30

spring

lb. that

KINETIC

are

springs In

a

of

this

weight has

used

to

absorb

must

be

designed

few

cases type

has

a

a

spring

energy so

partial

absorption

I = _ Mv 2 =

velocity rate

of

of

i0

4

ft.

ib/in,

30(4)(4) 2(32.16)

=

7.46

ft

or

7.46(12)

=

89.52 i

Spring

Spring

energy =

rate energy

in.lb.

load

times

deflection

= per

inch

= _ Kb 2 2

impact. they is

In

will

most

absorb

tolerated.

the A

follows.

per

second.

be

compressed?

ENERGY

K.E.

Load

of that

times =

deflection

89.52

in.

lb.

lb.

How

far

will

3

Section

B 3

19 May Page

B 3.1.6

Dynamic

!

or

Suddenly

Applied

Spring

Loading

1961

22

(Cont'd)

I08 2 = 89.52

2 62 = deflection,

6

17.90 = 4.23

in.

If velocity

springs are used and acceleration

Problem

4.

to propel applies.

a mass,

a parallel

attack

Let it be required to find the spring load that will l-lb. ball 15 ft. vertically upward in 1/2 second. It is that the spring can be compressed a distance of i ft. In

order

a certain

to

travel

initial

wherein:

in

This

1/2

second

can

be

the

found

Force

as

load

must

a

have

follows:

2

ft.

= ]--+

per

sec 2

2

= 38.04

ft.

per

sec.

2

Spring

l-lb.

propel assumed

gt

h = height g = 32.16 t = time

v

ft.

velocity.

h =--qt

V

15

using

= v2 _.(__._.. 04) 2 2-_ = 2 "T_l)

acceleration equals

mass

723(1) F = 32.16

times

= 22.5

acceleration

= 723

Spring velocity free height.

at

ft/sec/sec

so

Ib. avg.

The average spring pressure is 1/2 the total load. Hence the spring will compress i ft. with 2(22.5) or 45-Ib. of load. Often, it is desired to know how high the weight would be propelled. This can be determined by equating the work performed by the spring to the work of the falling weight; thus work equals force times distance.

would

In

the

spring

we

have

45 2

In

the

weight

we

have

l-lb.(h)

Hence l(h) be thrown.

= 45 2

(I)

(i)

= 22.5

ft.

ft.,

the

height

to which

the

weight

Section B 3 19 May 1961 Page 23 B 3.1.6

Dynamic

If we

we must distance

or Suddenly

were to apply _2 = 2P_S + 8) K

remember traveled

Therefore, Substituting

(S) by h

Applied

the to

previous the

g

in

i = 2(1)h 45 h

= 22.5

ft.

Loading

(Cont'd)

formula

springs,

is the height the weight is

= S +

Spring

this

the load (S + _). case

is

dropped.

The

total

Section B 3 19 May 1961 Page 24 B 3.1.7 If

Working the

Stress

loading

on

for the

Springs spring

allowance must be made in the tion. A method of determining particular spring is dependent physical

properties

of

the

is

continuously

fluctuating,

due

design for fatigue and stress concentrathe allowable or working stress for a on the application as well as the

material.

Section

B

3

19 May 1961 Page 25 B

3.2.0

Curved

Springs

The analytical springs is

expression

f = +P+A - +AR M----(I

in which B 4.3.1.

the

quantities

p

-_-_vTN +

determining

l__y__Z R+y

have

Displacement of curved Castigliano's theorem.

for

the

stresses

for

curved

..........................

same

springs

meaning

defined

is determined

_-

_N _d_+

_M

_N N _P ds +/ k_

/V

GA

_V _P

ds

_M _ ds

hy

in

section

use

of

M EAYoR

+

(19)

_M _p

ds

..............

(20) P

in which N

= normal

force

E = Modulus of Elasticity G = Modulus of Rigidity A = Cross-sectional area R = Radius to centroid ds

= Incremental

length

Yo = ZR Z+I

= - AI y =

_

"--Y--dA R+y

is measured

from

the

centroid

These expressions for stresses and displacements are quite cumbersome; therefore, correction factors are used to simplify the analysis. The correction factors (K) used to determine the stresses are given in section B 4.3.1. The expression for the stress is

f = K Mc T See

Table

B 4.3.1-1

Deflection given in Table combination of

(21)

............................................ for

values

of

correction

factor

formulas for some basic types of B 3.2.0-1. Complex spring shapes two or more basic types.

K.

curved may be

springs analyzed

are hy

Section 19 May Page B3.2.0

Curved

Sprln_s

{Cont'd) Table

Spring

26

B 3.2.0-I

types

Deflection

A B = I_R_3 3EI where

(m + _)3

(x = _ for

finding

K

finding

K

finding

K

B

8 -

C

p

3El

where

_

-

3El

where

_

+

= _2 for

8

+

= _2 for

P P D 3

where

*

ttl =

u

R

_

= _2

for

finding

K

B 3 1961

Section

B 3

19 May 1961 Page 27

B 3.2.0

Curved

Springs

(Cont'd)

0°

i

i

0.2 r,,i,--

0.1

0

_1

i

I

I

I!li

0. i

]ililllllllJllllllllnllHl[

0.2 0.15

0.3

fill

llllllil

III:IIIH

0.5 0.7 .9 0,4 0.6 .8 1.0 IJ

Ratio

Fi_.

B

R

3.2.0-2

1.5

2.0

3

4

5

6

8 7

I0 9

Section B 3 19 May 1961 Page 28 B 3.2.0

Curved

For

Springs

close

flat

(Cont'd)

approximations,

the

round

_

h R

<0.6

Figure B 3.2.0-3 spring at point A

d R

<0.6

is is

a typical calculated

u2 =

0.6

_

2.5"

The

curved spring. as follows:

Characteristics

h _<

should

U1

"_

deflection

u2

4

_i i'_i/_i

= _

RI

= R2

=

z R 2 _//_/

i"

_2 = 2 uI =

A

i"

UI

P Fig.

Solution

of

_B

h _I

be met:

wire wire diameter radius of curvature

Spring

conditions

springs spring thickness radius of curvature

the

following

B 3.2.0-3

-

The solution involves two basic types (type B and A of Table B 3.2.0-1). Type A solution is used for that portion of the spring denoted by subscript (2), and type B solution is used for that portion of the spring denoted by subscript (I). Correction

factor,

__Ul= _i = I rI I '

_

=

_A

at

= 2KIPRI3< 3El

3F.I 28.4P E1

(from

180 °

i _i

Deflection

K

Fig.

B 3.2.0-2)

KI =

.80

' =

point

u2 .... r2

2.5

_2

90o,

2.5 i

90o

A

m + _i _-

3

i + _

+

K2PR23 3E_

3El

< m + _2

2.5 +

3

=

K2

=

.86

Sc:c-t.o.-_ 15 x,-Pas_ B 3.3.0

3e!leville

Sprln_s

or

_ 3 ' "q7_

29

Washers

Belleville type springs are used where space requirements necessitate high stresses and short range of motion. A c_pleuc derivation of data that is presented in this section will be found in "Transactions of Amer. Soc. of M_ch. Engineers", May 1936, Volum_ 58, No.

4,

by Almen

and

Laszlo.

2a

=O.D.

!_' 2b i

i'D'=

I Fig.

B

3.3.0-I

in

inches

Symbols P = Load in pounds 5 = Deflection in inches t = Thickness of material

f

h = Free height minus thickness in inches a = One-half outside diameter in inches E = Young's f -- Stress

modulus at inside

k

= ratio

of

v

= Poisson's

O.D. _ I.D. ratio

circumference a b

M, C I and C 2 are constants which can be taken from the Fig. B 3.3.0-4, or calculated from the formulas given. The

formulas

are:

6

M=

(k-l) 2

logek

(22)

k2

]

6

o

CI "

_

chart,

logek

logek

, o e o,

o o,

• i o • oe

Io

(23)

• • I * • e,

- 1 j i

3 po,o,eoooo,eo

C2 - _

/

logek

Jot,,,,

_"

_"

(24)

Section 15 March, Page B 3.3.0

Belleville

The

Sprin_s

deflection-load

P The

•

orWashers

stress

h

formula

is

as

--_

constants

h

-

6

is

t + t

follows:

(,.

] ...............

: Before using these formulas to calculate a sample problem, there some facts which should be considered. In the stress formula it

are is

I973

30

(Cont'd)

formula..usin_._._he_.e

(l "- ',Z )M_ZbZ

B .3

possible

large. changed maximum

fiber

for

the

term

(h - 5/2)

When this occurs, the term to read Cl(h - 5/2) - C2t. stress is tensile.

For a spring life stress of 200,000

this might be slightly because the stress is

of less p.s.i,

to

become

negative

if

inside the brackets Such an occurrence

(5)

is

should be means that

the

than one-half million stress cycles, a can be substituted for f, even though

beyond the elastic limit of the steel. This calculated at the point of greatest intensity,

is

which is on an extremely small part of the disc. Iu_edlately surrounding this area is a much lo_er-stressed portion which so supports the hlgher-loaded corner that very little setting results at atmospheric temperatures. For higher operating temperatures and longer spring llfe lower stresses must be employed.

.040

Fig in.

1.41

It the

B 3.3.0-2 displays thick washer for is noted (from load-deflectlon

the various

Fig. B _urve

load-deflection h/t ratios.

characteristics

3.3.0-2) that for ratios is somewhat similar to

of that

of

(h/t) under of other

conventional springs. As this ratio approaches the value of 1.41, spring rate approaches zero (practically horizontal load-deflectlon curve) at the flat position. When the (h/t) ratio is 2.83 or over there lower (h/t)

I$ a portion of the curve load. This is illustrated ratios of 2.83 and 3.50.

certain point, will to return it to its The deflectlon illustrated

washers may characteristics in

Fig.

B 3.3.0-3.

the

where further deflection produces a in the curves for the washers having Such a spring, when deflected to a

snap through center original position. sometimes

a

be desired.

stacked The

and

require

so as accepted

to

a negative

obtain methods

the are

loadlny

load-

Section B 3 19 May 1961 Page 31 B 3.3.0

Belleville

Sprin_s

or Washers

(Cont'd)

q

O O _O

Fig.

B 3.3.0-2

Section

B 3

19 May 1961 Page 32

B 3.3.0

Belleville

Series

Springs

or Washers

(Cont'd)

Parallel Fig.

Parallel

- Series

B 3.3.0-3

As the number of washers used increases, so does the friction in the stacks. This is not uniform and could result in spring units which are very erratic in their load-carrying capacities. Belleville springs, as a class, are one of the most difficult to hold to small load-llmlt tolerances.

v

Section 15 April Page B3.3.0

Belleville

Springs

or Washer

B 3 1970

33

(Cont'd)

/

2.3

/ 2.2 C2/" f

2.1

/ 2.0

1.0

/ /

1.9

0.9

#'

/ C1

/

1.8

and M

1.7

j #.

v

i

1.6

/

/

f

/

C2

0.8

/

/

/

0.7

/

M

/" 0.6

/

/ / 1.5

/

! 14

! / /

1._

cl

i

/

/ i

/

/ /

/ /

F '

1.2 l I.I

I

/

/ /// /

/

/I

lO ]// 1.2

1.0

l

[ 1.6

0 2.0

2.4

2.83.2

3.6

4.04.4

4.8

O.D.

Ratio

of

, k I.D.

Fi_.B

3,3.0-4

Belleville

Spring

Constants:

M,

C Iaud

C2

Section B 3 15 April 1970 Page 34 B 3.3.0 Example

Belleville

Sprlnss

or Washers

(Cont'd)

Problem

Given: O.D. I.D.

= 2" = 1.25"

Load

to

deflect

.02"

= 675

lb.

Required:

Fig.

Determine

required

thickness,

t and

dimension

B 3.3.0-5 h.

Solution: k

= O.D. I.D.

-

2.00 1.25

= 1.6

The constants M, C 1 and C 2 may be taken from the Fig. B 3.3.0-4 or may be calculated as follows:

M

6

=

(k-l) 2 = k2

logek

The

0

CI " _

logek

C2

logek

_

6

--

Deflection-Stress

in

(1.6

3.14(0.47)

[l--_gek - i

curves

I =

- 1.0) 2 = 0.57 (1.6)2

0 El01.0 ii 1123

3.14(0.47)

= 3.14(0.47)

0.47

2

Formula

(1-v2)M,2 may

be

h

written

"

in

fM'2(i"v2)

the

CIE_

+

form

_

C2

2

CI

t

r_

Section 15

Page B

3.3.0

Belleville

Assume

Sprinss

that

the

f

or

washer

=

200,000

E

=

30,000,000

v

=

.3

=

0.02

=

0.57

=

1.123

C 2

=

I. 220

a

=

1.00

Washers

shown

B

April

3 1970

35

(Cont'd)

in

Fig.

B

3.3.0-5

is

steel

psi

max

M

C

psi

in.

i

try

t =

.04

and

200,000 h

(half

solve

for

outside

dia.,

dimension

t h 2 t .57jkl_.32j _i _t

x +

inches)

h

.02

1.220

2

1.123

(04)

__

°

(1.123)(.02)(30)106

This formula

value to

for

obtain

(h)

the

is

then

substituted

.120

_-

in

the

t3

]

in.

deflection-load

load.

e

(I_E2)Ma21

30(106

_h

- _>_h

- _ _

t +

) (. 02) 2 [('121-'01)('121-'02)('04)+('04)3]

(I-.32

Since stock

)(.57)

this

load

thickness

Then,

solving h

=

This expected beyond

is that

the

too .05

again

665

as

low,

amount

the

calculation

is

repeated

using

for

h,

the

result

a

is

this

value

of

value load

of

h

(P),

the

need

be

into new

the

formula

value

of

used (P)

is

lb.

close

this

lb.

in.

previous =

600

in.

substituting

the P

of

.ii0

Therefore, calculate

is

(t)

=

(i)

or used

as a

calculation similarly

in

the

calculated

calculation.

carried. spring

It will

is be

not deflected

to

Section

B 3

19 May 1961 Page 36

REFERENCES Manuals I.

,

Chrysler Missile Dtd. I November Convair Vol. I.

(Fort

Operation, 1957.

Worth),

Design

Structures

Practices,

Manual,

Sec.

Sec.

108,

10.4.0,

Handbook I.

Associated Bristol,

Spring

Corporation,

Mechanical

Spring

Design,

Conn.

Periodicals I.

Text

Klaus, Di_est

Thomas Issue,

and Joachim Mid-September

Palm, Product 1960.

En_ineerlng,

Design

Books

,

Seely, Fred, B. and Smith, Materials, Second Edition, New York, 1957.

.

Spotts, M. F., Prentlce-Hall,

Design Inc.,

J. 0., Advanced Mechanics John Wiley & Sons, Inc.,

of Machine Elements, Englewood Cliffs, N.

of

Second Edition, J., 1955.

-h

-4

-L

SECTION BZl. BEAMS

TABLE

OF CONTENTS Page

B4.0.0 4.1.0 4.1.1 4.1.2 4.1.3 4.1.4 4.2.0

Beams

.....

Simple Beams Shear, Moment

. .........

......0...

............................. and Deflection ..................

Stress Analysis ............................ Variable C ross-Section ....................... Symmetrical Continuous

Beams Beams

of Two Different ..........................

Materials

4.2.1

Castigliano'

4.2.2 4.2.3

Unit Load or Dummy Load Method ................ The Two-Moment Equation ..................... The Three-Moment Equation ................... Moment Distribution Method ....................

4.2.4 4.2.5 4.3.0 4.3.1 4.4.0 4.4.1 4.4.2

.........

Curved

s Theorem

Beams

Correction Formula

.......

and Axial Load

B4-iii

25 26

32 34 37 38

.............................

Bending Moment

1 24

29 30

Beam

Bending-Crippling Failure of Formed Beams ........ Bending Moment Only ........................ Combined

1

29

.......................

Factors for Use with the Straight .................................

1

..........

38 43 43 46

Section 21 Page B 4.0.0

BEAMS

B

4.1.0

Simple

Beams

B

4.1.1

Shear_

Moment_

The

general

deflection terms

are

of

and

equations given

in

deflection

relating B

load,

shear,

4.1.1.1.

bending

bending

These

A

Slope

@ =

dy/dx

M

=

EI

d2y/dx

2

M

Shear

V

= EI

d3y/dx

3

V

= dM/dx

Load

W =

d4y/dx

4

W

=

= y

A

EI

=

F J

dv/dx

= d2M/dx

x

is

positive

to

y

is

positive

upward

M

is

positive

when

d)

W

are

V

is

at

in

B 4.1.1.1

right,

lw

•

÷x

-X

the

__

compressed

i I

top. the

direction

of

y.

positive

tends

the

the

positive

negative

2

Y

a) b) c)

is

to

in

_-dx EI

Convention

fibers

and given

dx

7f_-dxEI

0 =

Table

e)

are

M

Deflection

Sign

moment,

equations

Y

Moment

1

moments.

Title

Bending

4 196]

Deflection

Table

and

B

April

when

move

vertical

forces•

The

limiting

assumptions

a)

The

b)

Plane

c)

Shear

d)

The

material

follows

cross

sections

deflections deflections

the

upward

part

under

of the

the

are:

are are

Hooke's

Law.

remain

plane.

negligible.

small.

beam

action

to of

the

the

left

resultant

of

the of

section the

X

Section

B 4

21 April Page B4.1.1

S hear_

Moment

and

Deflection

=

El dxdx

/M

+fKV_

(K) is the ratio of the to the average shearing equation:

IAb

J

maximum stress.

shearing stress on the cross section The value of (K) is given by the

b'ydy

0

(I) is the moment of inertia of the cross-section with respect to the centrodial axis and (a), (b), (b'), and (y) are the dimensions shown in Fig. B 4.1.1-1. (A)

is

the

area

of

the

shear may deflection

dx

a K

2

(Cont'd)

The deflection of short, deep beams due to vertical need to be considered. The differential equation of the curve including the effects of shearing deformation is:

Y

1961

cross-section

I Fig.

B 4.1.1-1

Section

B4.1.1

Shear_

Moment

and

Deflection

9,

Page

3

(Cont'd)

r_

O •_

4-1 F)

5" t_ II

_

l----b _q

F-.4

0 _J

V_ O

N

!

ol

II

4-

' > I

C

_

m

_

C',l

©

II

I cq

N

'

_

I

eq

cq

© +

c"3

II

0

Z

I

> od -;-I

O

II

"_

cq

oq

cch I

o

(2)

4-1

I

I

II

II

II

_

I

_-4

'_

I o_ II •_

i

I

|

II

II

II

O

O

<

_,

O

;4 o

cn_

m

t_

tl? o

xi\\o

_ \'3_'

7

0

0

_

>_

I

__ i ......

L t

t

B4

July

1964

Section

B 4

21 April Page 4 B 4.1.1

Shear_

Moment

and

,_

Deflection

(Cont'd)

> ,--, 0 _J r_7

o _._

o °,.-4 •l=J

co

*_

0

"= '_

,_i'

c_ r--1

_

+ ,z3

!

.._

°

_J

= o O

Ii +

_le_

m _

I

t

0

I

to

co

"

,-4 •_

4J

X __s

M

N

I

I

o M

...1*

U

v

11

I

cq ¢'4

t'M

c,_

=

!

-I-

3-, a q)

.=

,

co

,

+

cO

+_

c_

_

-I-

+

S

c_ ,--1

II

P_ _.... eq

o

0

!

,-I v

1_1

c-q ,_

J

t__

_,

+

+

/;

,

+

_ -,_

,,-,4

:_I,-_

:_

_

°" '<

fu t_ [.-i

_

._1

oo _

m

o= _

,

,

+

4.J Ii

il

II

li

II

I_

I_l

o

o

o

m

_

.J

_

_

--

e_,

_

>_

I_l

"

>_

>_

0 o

c_

c_ t_

_l

--

I_

II

o

:>_

If c_

1961

Section 21

B

April

Page B4.1.1

Shear_

Moment

and

Deflection

4 1961

5

(Cont'd)

r----i

Le_

o o

r --]

_._ ,,,.]

I

II

I

O .I-J

°_

o

I

cxj

>

_

<

co

t_

c_ r_

Xl

< +J

_

t_ ....

I m

t_

o

m

+

+ r_

+ m

E .I,J

_

N M _ o

I

_

I

N

Jr-

o

E

I

,

:>_

o"J

II

N ttl

•4-+

>

m

I

_ _._ ,-_1 co

_

N c_

o .l.J >

II ::m:

(lJ

v

4-1

o ,.c:

¢

Oq

-r.-I

::E:

I

°

X

u

II

<_

v _,

t_ ,...4 +

co

L,"5 _

_ V

+

_

t

_

_ + j

t

+_co _I ° +I ° _+

cc_

X

N

+

CO

II

(2) r---i + N

_

N

b--t

_-t

II

II

I

I

I

II

II

II

:>

¢N

__J

+1_ +1_

+1°++1+" ,

o

U

+

.;.-4

u

II

II

II

II

II

0 IJ

0 _+J

0

0

,-r,i

+_ 41

_.+t..,+l.._

o,0 "E3 o

i_ "_ U4 O

_ _

_

m

>-

+

_

U

u"5

+ II

E)

0 N

::m:

....

o

co

,.Q o

1'-.-.

L_

I_

++

:>

N o ,.-4 +

.1..1

.IJ 111

X M

o

I

N

_ ¢M + +

E

u_

co _ I

_

I

_1

I

_-1 v

r.D

o .-+1 co

_

!

I!

o

cN

,-+ I co

co

co

,'-N i

O

o u

i

co

Section

B4.1.1

Shear,

Moment

and

Deflection

B4

July

9,

Page

6

1964


o

_3

o -_

,--'4I'x:_

_°

_'_1

fl

4J o _2

,.-4

o

_0

_ _i _

m N

m + N

.._, ._

._

,-.4

_

I

°r-.I

,.-I

N

°g

•,4 O

II

_

,

,

II

II

+ II

I

,'¢::j

e _

._

-I_/-.. 'I

'

'I

-_,,,

Section 21 Page B

4.1.1

Shear_

Moment

and

Deflection

(Cont'd)

0

,<

0 I G

%

¢I

-%.

.1_1

+

0

+

O CO v ¢o ,-q

o_ 0

.,-4 0 r_

,2 C'4

'0

.-,

_"

_1_

¢_1

II

0

r_ 0A

°r4 4-1 _

, ×

0

×

+

r_

_

r.4

_O

_

.r4 .!..J

,-.a 0

,.O O o

.,_

H

0

_

"

II

II

II

0

0

0

0

4..1 II

II

rn

t_

I

.r'4

0 ,_

--

l 0

_

m

I

i

i _l d

B 4

April 7

1961

Section 21 Page B

4.1.1

Shear_

Moment

and

Deflection

8

(Cont'd)

o

°_

u ,,iJ

A

I

_

+

/g

,

i

II II >

.,-4

::E:

II

¢q + +

o .iJ

N

v

cq

o = o u

It

4a

v

II

I

×

+

v

_

r._

II

+

o

g

,_"

(D

II

>

¢q t

N

o_

o i

2

v

.I.J

0

u

o

+

o

II

>

o

<

o r¢3 e,J

_i_

>

_

,_, _

_'] ,t,J

e

..1 _

='_

+

+ v u

II [--i

+

I

II

II

I

II

II

m

I

I

o

II

II

fl o ._

o 4.1

+

o

0

x

0

x

Io _

I+

I o1+__0

o

_Ira ---_[

t--

_

O

[ >.

_4

+

. .i_

B

April

4 1961

Section

B4.1.1

Shear_

Moment

and

Deflection

_l _

+ o

N

:x

+

_

N

1964

o

c,4 __'_

u

[--I

c_

F---I

¢'J

O cq

+

+

II

II '4.4

II

× c,_

0

II cD

9

× ,--

_

v

'

Page

"_,

+

c-

:xI

9,

(Cont'd)

O _ I

B4

July

+

+

+

+

._ 0

c,4 _

_ O

.o,31

o

i

i

I

i

i

u_

o c.)

II

E

II O O

II

_

N

v

_

O

_ ,._ @,,1

_l_

'_l

II

U

;>

_

°

II

m

_

°

v

v

_

!

ul c._ +

II

_

_i_

_

v

.t_

k- --_v-'--_.__

C

_

_1_

_

I

I • ,-.-I

.el u3

4 (D r--4

_1oo II

o:

•,_, _l_

m

II II

I1

II

II

II

o

0

0

0

0

I

II

_ II

4_1

X X

¢B o

oO

--Ic_ i

--Ic_

O

_

>-

Section 21 Page

B4.1.1

Shear

a Moment

and

Deflection

(Cont'd)

O. 0 O0 "0

.@ O •rt

I_

+ ¢q

U

4.4

H

N

= 0 _J

4_ !

I_Im

M II

m ,-4

+

m

+

II

m

O 1.1 W 0

II

£%1.

_

_

_4 ! r-I

+

*

,

_

M

X

.

N

m

._

I

Ni

°r.l °M

• _

o"1 [._.1 H

>

_°

m

r_ II

II

_-i I

o_

O

II •_

II

II

I = "0 m 0

_J

o

=

'O

i, = 0 O

,-4

I

-"

4

April i0

B 1961

Section 21 Page B

4.1.1

Shear

a Moment

and

Deflection

B

April

4 1961

II

(Cont'd)

O oO

f_ I--I .IJ e_ ! O .M

.d-i r,-_

_ 1
_l,-e ii

Q)

II II I

0

f--I

_

e_ O

_1_ _ .-.-4

,

.'4

M

rO

I

O

r

°_

E

.,-4

I

_

_D

+

cg I

I----I O

!

M

• O _:1

E

-

%1

II

0

+ I>

>

,_ M

+

|

i

_ _"

.<

¢la °_

y.'

u? _, f-4

_q

r._

c'j

t'-xl

c-,l

!

o 11

4J

/

i,-,] I

_

I

II

0

r_

_

0 .I..,I

0 .i.i

0 ,,l,.I

,

,.-1 _

J oh .i

.Lt

'_._.__//

"_

<

cl _-I v-_

_,

°_ I ¢',1

II--t

II

,.o

> o

•,_ :_'1_

(.3

4-1

0 .I-I

u

II

v

o

o

.i.I

_

II o

,.__._j _ II

v

v

_ X

7 L

0 0

_J

II

_:,.o

nt_ ,-4

v

II (I)

Section

B4.1.1

Shear_

Moment

and

Deflection

B4

July

9,

Page

12

1964

(Cont'd)

0 ,-4 C13

_1_

0 •_

0

+

_

I I

r-I

II 4-I

m

0 IJ

°

_I II

O v

v

I

{'q

m

<

I ,-4

0 r_

_

+

._

_

'_

_-_

U

"

+

_

II

,-,

I

._

0 -\

> 0 4_ U m II

0

•_ 0

_

_

+ ,.al_

> <

I

%J

.4

o_ 1.1

o_

II

+

_

+ ___,

+

_

N

_

I

"-_--"

_ U

,1o

0

_

,

_I__I_ _

<

II

II

0 •_

0 .IJ

%1_

'

+

[--i •_ t_

0 II

_ II

I

II

II I II

0

CO

X

'

.J

cb

Section

B4.1.1

Shear_

Moment

and

Deflection

(Cont'd)

0 0

'_0

<

0 4J

-rJ C'-I

II

_J 0

E G}

0

E

_

+

I

o_

E

c._

_

_

4-1 I

0

&l

1_

o r...) _ m

v _-_

t_ _I_I_-I

•-

i

II

i

N

m o

Q? 4J

I •

_2 +

¢1 1-4 (1_

< -_

v

II _III

II

[--i

_

I--4

I _

,,.-x

o _

Ill

0 .i..1

0

0

c_ .r-I

O

O

Q_ 0

I II

X

[--4

II

e'_

B4

July

9,

Page

13

1964

Section

B 4

21 April Page B4.1.1

Shear

a Moment

and

Deflection

1961

14

(Cont'd)

_z 0

•

.,4

_l,_

__

_1_

°

o

_ %,

II

_1

II

_ m

aJ

[ 0

iJ

I

,_

4.1

_

V

O ,,_

II

,._

I

e4e4

_

i--_ _

o _

¢q

II

m

_I

_

_

II

I

•

.,4

_

_

+

+

e,i

N

N

II °_

v-

_ _

II

I

,_

m!

rn!

N

N

!

o

14

_9

-r4

•_

.,4

.Ill

i_

_

I_

_,

_"

_"

_"

_1_

_

_

u"_ .,..i

/_

II ,l_

c_

II

II

I II

_ O u

U

_,l 0,1 ,-i ..1-

0

O

II

II

II

rO

(O

0

0

Ill l.l

.= =

'°t

= .,4

i

I O _u

L

_ m

'_

I%:

_II

H

11-4

II

Section

B4.1.1

Shear,

Moment

and

Deflection

B4

July

9,

Page

15

1964

(Cont'd)

O '-0 I

O3

¢,.)

c_

'i o !

,

N 0

Y.l¢_

[--I I

II

X -_ C'4

II

_

-:_

_

,,

c_

,

I

°

"

"_

.IJ

o u

r...)

II

_

°_

_

N

O

+

t_

r.,.O

N _

0£ _.m I o0., •_

II

_

,._ C-,I

>_

II II

o

_)

×

_

"o

+

I O0

_J ,._

j

II

II

X c,'-)

I

_

•-_

I ._

+

4J

<

'

[

II_

+

_ _ ,-4

_

°_

v

v

_I_

r._

c_

IJ

II

II

II

o

o

II

I ,,-'-,

_o _1

•_

O0

u _d

_._

_

_

>

; II

o ,L)

I

_"_

"

O0

v

CD

II

II I_

+

o

II

C_

m

c_ ° r...I

"o o E o

_

_

-_

L--

- L

--

_4

>_ u

e,4

Section 2] April Page B 4.1.1

Shear_

Moment

and

Deflection

(Cont'd)

o

%

¢q

o

,_1_

!

ot,,l

4.-I U

I

4-1 II

r_

s-% P

v o_

0

_ o

o_

I 4-)

_,

I

-,-_-_

>,

r. II

0

> _

_

'_I

N

,-.4

-_

_

,

o_.,_

t_ _ ,o

._

+ II

I

,,.1-

II

_'j

,

_1

i ,o o

°_

II

I

o [..i

_o o

'

_i_

II II

< v

'1:1

80

4..I 0 r_ v

o_

IJ _l

16

B 4 1961

Section

B4.1.1

Shear,

Moment

and

Deflection

xI

O0

q-_ o (1,1 C}:

0

I'

(Cont'd)

B4

July

9,

Page

17

1964

Section

B4.1.1

Shear_

Moment

and

Deflection

(Cont'd#

+

o r-d O0

I

_Io

,

_I _

o3

II

X =o

4.1

._

.rd

II oJ

II

t',,I

_

_I_ II

0J

=la

II °_ ¢"N °

+ ¢,_

0J

o ¢,J I

,4-

°_

I

_

v

0

o

II

II

I I

,

od

o_

I _J

-.'t _

II

_4 o o

0

o I

o

II

I

NJ_

II

_0 ¢'4

¢,)I

It

_!'_ _

"

I

v

II

_

II

_

=

)

xl

B4

July

9,

Page

18

1964

Section 21

April

Page

B4.1.1

Shear_

Moment

and

Deflection

B

4 1961

19

(Cont'd)

II

II

_

r6

_J 0 P-_

I It

4.J

< +

_-,

oo

N

II O

_Ic4 cq ._

I

II

_ cq

O •,--t

cq

_ cq

_1

r,_

c; fl

UI

,._

E

I

I

U

II

>

>

r_

,-4

•r4 o3 o3 O

._ o3 o3 O

t_

_3

V

A

0

g

°_ _1

O

I,

c-4

4J U°

E: I

t

:>_

v O

L)

_

l::cl

o3 •,-.-" ,-4

O_

4J

II

•_ o

E

E

_J

::E:

O 4J

<

--,

'

<

+

N

v

U _J

cq

0

DO

O ._J

II II

_J

+

_J

+

,._cN

,-_

4J

_J

"_

_J oo

::>

II

_-_

:_1oo

_

L_

II

+ _d ¢o

,--_ II

II

!

!

II

II

I

I

II II

E_ 0

--

P_

II

O 4J

_J

,

_I

4 II

0

0

_J

4J

v _

×I o

i

°i

O0

"El 0

¢)

J,,,,I 0

:),.

_l.

/ __L. <

o_ ¢q

I

I

m

j

Section

B4.1.1

Shear_

Moment

and

Deflection

B4

July

9,

Page

20

1964

(Cont'd)

P0 o'] "0

>-, 0 -e4 _a ¢J

_1_

OJ

×, I

4-J

_a 0

A

m

_.I

_I_ rJ

I

_

.5!

O

_

_

•

I

_°_ _

×

m

,-1

.,-4

_

m

°_

_I_

_1_

II ,._

,-_

_

_I_ II

II

_

II F

_4

_¢ c'j

.c:

__.i _

._ Cxl

_

-_

I1

II

o .._

.,4 l-J

,.-4

r_

×

I_-_

_i_ _

II

II

o'_

C-,I

I--I c.m

_,1

oO

m b_

I_I

I

II

i Ii 0

r:_

r...)

4-_ O rd

0 4J

0

cJ

<

II

_i;_

_x,

_,

II _Cl

n_

N

"-"

>:

Z

-\

i_

_C

_

_

m

-

m

Section 21

B

April

Page B4.1.1

Shear

t Moment

and

Deflection

rj_

4-

f

",--" _

JI

i

II

"0 >

1961

21

(Cont'd)

<

o

4

I v

O

o_

_

_

A r6

+ <

o _) ,.c:

@ II

I

'

o u v

o4

_

oO o

c'N

II O

II

I

i

v

>

o_

I _

t

II

_._

;_ %1.

+

_1

0

_

eq

N

_

t

m

?1

,c

+

,-q

,-4

O

.,_Y_I_

U

+

_

I

V

+

+

_

u

I

>_

I

I

_

O

I-4

N

o cq

.;-4

N-d

-J

M

_

I

o

M

N

+

I

I

N I

,_I_ _<

_

_.

<

_

v

._

4_

4J

I

4-I

,.-1

r-4

II

II

o ._

o 4..i

m .-4

II

II

II

II

m

-,-41 _-)I

I

II

II

II

o ,_

_

rF

r_

'/

-el

o

o

o

_

m

' [..-4 _

c'e3

0 4J

0 ,_I

Section

B 4.1.1

Shear_

Moment

and

Deflection

B4

July

9,

Page

22

1964

(Cont'd_

O

¢q u_

II

gI ¢N

°

_I_

O I

O 4J

,-_ u_

II I

_

.4" v r_

:_ I ,-_

m

..

+

!

I

II

II

_=

II

o

_-J

4-;

O 4J

_J m

4J

.P1

° I

o

_1_

+

+ II I"

,,--4

•

_

_

0"_

_J

,_

¢q X _-_

_

I

I_1

_

II

_

I

II

m

°_

I

I

M

_

_

I II

e_

m

i}!

_, II

II

M

II

:=

>,

o

o

II

v

o=

o

II '

_

_

v

Section

B4.1.1

Shear_

_D

Moment

and

eq co X i

0

_q

+ -

_

o

_

_:

_-_

cq

_

V

•_

.,-_

i

i.--4

_ _

I_

01

U

_

II

.

II

+

_

oO

_J 0

0

O.

.,_

II

o

0

A

o

2

c,,I

I

m c_

×

Deflection

(Cont'd)

B4

July

9,

Page

23

1964

Section

B 4

21 April Page

B4.1.2

Stress The

1961

24

Analysis

maximum

bending

stress

is:

(1)

The

limitations

a) b) c) d)

The loads on the beam must be static loads. The value of f is the result of external forces only. The beam acts as a unit with bending as the dominant action. The initial curvature of the member must be relatively small. (Radius of curvature at least ten times the dept_)

e) f)

Plane cross The material

If the reduced

shear

are:

sections follows

calculated stress does modulus must be used.

The maximum is:

shearing

• • • ,,

• • • ..

where (K) is the ratio section to the average often expressed as: fs =

Q

VO It

=

plane. law.

exceed

stress

in

the

proportional

a beam

in

limit,

combined

bending

a suitable

and

v

V -_

fs=K

Where

remain Hooke's

JydA Area

• • ° • ..

• • .....

......

o o

° ° ° ° o

o.

o ° o . .

, o o ° o o o ° (2)

of the maximum shearing stress on the cross shearing stress. The maximum shearing stress

..............................................

(First moment between the fiber.)

about neutral

(3)

the neutral axis of the area axis and the extreme outer

is

Section 21 Page B

4.1.3

Variable

The beams a

following

with

by

ponents some the

cantilever

beam

a

web

of of

the

the

flanges

loads

external and

V

= Vf

Vf

=

P

+

Vw

Vw

(tan

and

tapering

vertical

figures

cross

the

force

of

resists

V.

a

P

Letting

Vf

resisted

bending.

tan

by

Fig.

5

I,

equal the

+

tan

52

+

=

P

the

webs,

shows

flange

The and

analyzing

areas

vertical tan

force

_

2,

comresist

resisted

by

then:

..................................................

(i)

51,

(2)

+

tan

B 4.1.3-1,

hi

of

B 4.1.3-1

concentrated

5

2)

................................... hi

From

25

method

Figure

two no

flanges,

force

present

sections.

consisting which

in

the

4 1961

Section

formulas

uniformly

tapered

joined

Cross

B

April

h2

c

tan

h = --. c

From

51

h2

=--, c

this

tan

Vf

=

52

h P -c'

c

and

, and

since

P

tan

51

b = V _,

then

Vf The

=

load

h o and

Vw

b V _ in

h,

we

(3)

.....................................................

the

web

is

a V E,

so

by

writing

a,

b,

and

c,

in

terms

of

have

ho = V E(h

....................................................

(4)

- ho)

vf = v ---f--- .............................................. (5)

P TANcK

P TAN

Fig.

B 4.1.3-I

_2

I

Section 21 April Page 26

B 4.1.4 The

Symmetrical analysis

Beams of

of Two

symmetrical

Different beams

the elastic range may be analyzed by an equivalent beam of one material. then applies.

of

B 4 1961

Materials two

or more

transforming the The usual elastic

materials section flexure

within into formula

The transformation is accomplished by changing the dimension perpendicular to the axis of symmetry of the various materials in the ratio of their elastic moduli. Examples to illustrate the method for various conditions follows.

Example i. Consider a beam is shown in Fig. B 4.1.4-Ia. in terms mum

stress

in member

made of two materials Assuming n = Ea/Es,

of material

(S)

(Fig.

in member

(S)

is

(A)

is

f(a)

have been transformed the same results for

(o)

max

B 4.1.4-ib)

then

f(s)

-Mh 2 = n---_

It

max

whose cross section the transformation

is then b I = nb. The maxiMh I = -_- and the maximum stress

is noted

that

in terms of member (A) (Fig. the maximum fiber stresses.

(b)

Fig.

the

section

B 4.1.4-ic)

could giving

(c)

B 4.1.4-i

Example 2. Reinforced-Concrete Beams. It is the established practice in calculating bending stresses in reinforced-concrete beams to assume that concrete can withstand only compressive stress. The steel or other reinforcing member then is transformed into an equivalent area as shown in Fig. B 4.1.4-2b. The distribution of internal forces for a beam (Fig. B 4.1.4-2a) over any cross section ab is shown in Fig. B 4.1.4-2c. v

Section 21 Page

B

4.1.4

Symmetrical

Beams

of

Two

Different

Materials

B

April

4 1961

27

(Cont'd)

-,-- b -_

d

d

J

i

(o)

satisfy

the

the

external

b

equation

of

moment.

The

(c)

B

4.1.4-2

equilibrium, mathematical

the

internal

statement

must

equal

=

nAs

(d-kd)

..............

arm

steel

area

arm

which

=

where

nA--'-_Sb _i

n

=

The

+

nAs2bd '

E

steel

E

concrete

stress

M fc

(kd)

in

the

i_

concrete

................................

fc

and

the

stress

in

(2)

the

steel

...............................................

fst

is

(3)

I

fst

..........................................

= nM(d-kd) I

where I

moment

is:

Transformed

area

kd

Rt

(i)

kd (-_)

(kd)

Concrete

From

-

.

(b)

Fig.

To

N.A,

= Moment

of

inertia

of

the

transformed

section.

(4)

Section 21

Page B 4.1.4

Symmetrical

Alternate I, a procedure force developed the

compression

the

beam.

force

Beams

of Two

Different

Materials

Moveover,

R c = _(kd_

of

the

if b

fc max,

is

beam the

(average

must

be

width

of

stress

located the times

kd _-

(Cont'd) of computing The resultant manner on

below

beam,

this

area).

1961

28

solution: After kd is determined, instead evident from Fig. B 4.1.4-2c may be used. by the stresses acting in a "hydrostatic" side

B 4

April

the

top

of

resultant

The

resultant

tensile force R t acts at the center of the steel and is equal to Asfst, where A s is the cross-sectlonal area of the steel. Then if jd is the distance resisted jd The

between R c and R_, and since by a couple equa_ to Rcjd or i - _ kd

= d

stress

in

the

R c = Rt, RtJd.

the

applied

................................................ steel

and

concrete

moment

M

is

(5)

is

M fs

= As

f

,= .

jd

...................................................

(6)

2M c

b(kd)

(jd)

........................................... (7)

N

Section

B 4

21 April Page

1961

29

i

B 4.2.0

Continuous

Beams

B 4.2.1

Casti_liano's

Theorem

Castigliano's Theorem involving continuous beams The theorem can be written

is useful with only as

in the one or

solution of problems two redundant supports.

L 8Q

- _U _Q

- Eli /

M_ _M

ds

_V _

ds

................................

(i)

................................

(2)

................................

(3)

O

_Q

L /_ JV

BUs I = _-_- = GA

o

L _U 0a

=_

M _

- E1 a

o

ds a

where

8Q

is

Q may Qa

is

M

may

a

the be the

deflection a real slope

at

or at

the

be

a real

or

energy

is

the

strain

M

is

the

bending

V

is

the

vertical

A

is

the

cross-sectional

E

is

the modulus the

moment

moment

of

Ma

of

in

the

direction

of

the

direction

Q.

of Ma.

to

beam. all

forces area

elasticity. inertia.

in

moment.

the

due

Q

load.

moment

shearing

of

load

fictitious

U

I is

the

fictitious

of

loads. due

the

to beam.

all

loads.

Section 21 Page

B

4.2.2

Unit

The

Load

unit

deflection

or

load

at

or

elastic

members

by

this

applied

to

elastic

Dummy

Load

dummy

load

or

method

is

beams

method

given

4 1961

30

Method

inelastic

is

B

April

may

be

members. in

section

written

in

used

to

determine

Deflection B

4.5.0.

integral

of The

form

inelastic

theorem

as

as

L

(i) O

L

(2) E-i-dx ..........................................

e = O

Where at

(8)

the

is

unit

the

deflection

moment.

The

section

caused

by

section

of

beam

whose

the

deflection

deflection. of

section

where

may

expression

be

the

to

be

bending

beam the

- _M

that a

a

by

as

a it

is

the

is

dummy

bending is

in

the

couple

is

and

the of

actually

is

the

moment bending

unity

acting of

moment

it

unity

at

the

is

noted

is

evident

at

any

the

point

desired at

applied

It

dimensionless.

any

moment

bending of

rotation

at

direction

desired. moment,

(8)

bending

load

(m')

slope

the

(m)

and

a

load

is

dummy

found

in

unit

loads.

moment

change of

the (M)

by

caused

thought m'

actual

caused

is The

section

(m')

the

at Moment

any at

that from

the although

the

Section 21

Page B 4.2.2

Unit

lllustrative

Load

or

Dummy

Problem:

Load

Find

point A (Fig. B 4.2.2-ia) two concentrated loads.

R=P

the

of

the

Method

(Cont'd)

elastic

vertical

simply

deflection

supported

beam

B 4

April

1961

31

of

the

subjected

to

R=P

(o)

(b)

M I---M : P L/4 M=PXI -_M=PX2

/_

--,--X

I

m

m= X2

/

.-4-

I-.--x,

X2-. _

(d)

(el

Fig. Solution:

The

x2--

actual

loading

is

B 4.2.2-1 shown

in Fig.

B 4.2.2-Ia

and

the

dummy loading is shown on Fig. B 4.2.2-Ib. loading is shown on Fig. B 4.2.2-Ic and the

The moment for the actual corresponding moment dia-

gram

B 4.2.2-Id.

for

the

The the

dummy

loading

deflection

left

and

by use

x 2 starts

,,

at

of the

shown

E1

O

dx

=

,

El

right

\ _

O

j L

4

4-_

dx2

Fig. (i)

noting

is

L

>

dxl

4

+J O

3L 4

+

on

equation

L 4

L

=

is

= 48E----I

E_

that

Xl

starts

at

Section 21 April Page 32

B 4.2.3

moment

The

Two-Moment

sponding

section and are known. to Fig.

M 2 = M I + Vld

V 2 = VI +

F

the loads applied The expressions

B 4.2.3-1

+ Fx

1961

Equation

The two-moment equation may be used to determine at one section of the beam when the shear and

another sections

B 4

to for

the the

the bending bending moment

beam between the moment and shear

at

two corre-

is

......................................

.............................................

(i)

(2)

LOADS

Fig.

B 4.2.3-1

The two-moment equation is particularly useful in determining the curve of bending moments and shears in the case of a cantilever beam subjected to distributed loads, such as shown in Fig. B 4.2.3-2. The calculations may be done in tabular form as in Table B 4.2.3-i.

,@

'' "I Fig.

B 4.2.3-2 \

Section

B 4

21 April Page 33

B 4.2.3

The

Two-Moment

Station

W

(inches from end)

Load

Equation

(Cont'd)

_-

* Bending Shear V= (WI+W 2 ) _f_x) -_--

Momen_

M=(2WI+W2)

2h

_6_VI(Ax)

ib/in i0

0 (10+14)10/2

=

0

120

567 (2.10+14)102/6

i0

1961

14

=

120

(14+18)10/2=

160

567 (2.14+18)102/6 120"]0 =

= 767 1200

20

18

280 (18+22)10/2=

30

22

26

30

increment of the relation:

M =

(_V)

(Dist.

+ Vl (_×)

(2.22+26)102/6 480.10 =

=

280

bending

from

moment

centroid

1167 4800 12_268

(2.26+30)102/6 720.10 =

1367 7200

i000 Table

*The from

240

2534 976 2800 6301

720

(26+30)10/2= 50

=

48O

(22+26)i0/2= 40

200

(2.18+22)I0Z/6 280.10 =

20_835

B 4.2.3-1

between

of

stations

trapezoid

to

may

be

inboard

calculated

station)

Section 21

Page

B 4.2.4

The

The involving

three-moment continuous

equation

is:

MaLl

+

Three-Moment

2MbLI

II

KI +

Where (L2),

2MbL2

II

K2

6E + L_

+

(KI) and (K2) respectively.

" Yb)

are

34

+

of problems supports.

The

McL2

12

(Ya

1961

Equation

equation is useful in the solution beams with relatively few redundant

+

B 4

April

12

6E (Yc L-_

functions

- Yb)

of

......................

loading

on

span

(i)

(LI)

and

span

Lir L21 t__

T

is

.......

Fig. B 4.2.4-1

system the

One equation must of simultaneous

intermediate

be written equations

for each intermediate is then solved for the

support. moments

The at

supports.

Values at (K I) and in Table B 4.2.4-1.

(K2)

for

various

types

of

loading

are

tabulated

Section B 4 21 April 1961 Pa_e 35 B 4.2.4

The

Table Type

of

Three-Moment

B 4.2.4-1

Equation

K 1 and

Loading

(Cont'd)

K 2 Values Left

Bay

for

Various

-K 1

Load Right

Conditions Bay

- K2

i.

+_2L3

+WiLl 3

412

411

r_

L

ID-

°

+WlC I

r

(3L_-CI 2)

+w2c 2

(3L228I 2

8I 1

'-T

c22)

°

wlEbl2(2Ll2-bl2)-al2(2Ll KI

b

--a--4

411L 1

i iT

I

2 - a12)]

=+

w2

[b22 (2L22-b22)-a2

2 (2L 2 2 - a 2 2) ]

K2 = + 412L 2

.

+2WiLl

3

1511

+7w2L23 60

12

I -'----------

|

.

+7wiLl3 60I 1

+2w2L23 1512

I __

.

+3L_w I i

+3L22w

|

I__

_r

L

-I

811

8I 2

2

Section 21

B 4

April

Page

1961

36

v-

B

4.2.4 Table

Type

of

The

Three-Moment

4.2.4-1

K I and

Loading

Equation

(Cont'd)

K 2 Values Left

for

Bay

-

Various

Load

K1

Conditions

Right

Bay

-

K2

o

+5WiLl3

+5w2L22 3212

3211

-FL _-

_1 -t

t

.

+wla

I _L12_a12

)

+w2b2

IILI I_

L

_L22_b22_ 12L 2

-_

o

M I

/

,

3al 2

+

M2 12

(3b22 \L_2

10.

E

1

M

x I dx 1

_o

M2

x2

the

banding

=

12L2 O jw

L 6 K2 qr

=

P

12L2

dx2

Ir Where

M

is

moment.

L2)

Section 21

April

Page B 4.2.5 The

Moment

Distribution

Method

moment

distribution

method

problems involving continuous discussion and application of (Frames).

is

suggested

beams with the method

for

the

B 4 1961

37

solution

of

many redundant supports. The is given in section B 5.0.0

Section B 4 21 April 1961 Page 38 B 4.3.0

Curved

B 4.3.1

Correction

Beams Factors

for

Use

in Straight-Beam

When a curved beam is bent in the plane plane sections remain plane, but the strains proportional to the distance from the neutral are not at equal length. at the extreme fiber of a

If_denotes curved beam

Formula

of initial curvature, of the fibers are not axis because the fibers

a correction is given by

factor,

the

stress

Mc

f =K-I

in which M__M_ [i K = AR

+

c Z(R

] +

c)

Mc I

where M A

is is

R c

is the radius of curvature to the centroidal is the distance from the centroidal axis to outer fiber

I is

the the

the

bending moment cross sectional

moment

I /_.Z_ Z = - -_R+y

Values

of

K for

of

area axis the extreme

inertia

dA

different

sections

are

given

in

Table

B 4.3.1.1.

Section B 4 21 April 1961 Page 39 Table B 4.3.1-1 Values of K for Different

Sections and Different R

Section

C

Inside

Radii of Curvature.

Factor K Fiber Outside

.

1.2 1.4 1.6 1.8 2.0 3.0 4.0 6.0 8.0 i0.0

R

K

the

same

and ellipse pendent of

for

3.41 2.40 1.96 1.75 1.62 1.33 1.23 I. 14 1.10 I. 08

O. 54 0.60 0.65 0.68 0.71 0.79 0.84 0.89 0.91 0.93

2.89

0.57

2.13 1.79 i .63 1.52 1.30 1.20 1.12 1.09 1.07

0.63 0.67 0.70 0.73

3.01 2.18 i .87 1.69 1.58 1.33

0.54 0.60 0.65 0.68 0.71 0.80 O.84 O.88 0.91 0.93

circle

and indedimensions.

0

e

i

1.2 1.4 1.6 1.8 2.0 3.0 4.0 6.0 8.0 i0.0

0.81 0.85 0.90 0.92 0.94

K independent of section dimensions .

7b

I

2b

]

1.2 1.4 1.6 1.8 2.0 3.0 4.0 6.0 8.0 I0.0

1.23 1.13 I.i0 I. 08

Fiber

Section B 4 21 April 1961 Page 40 Table B 4.3.1-1 (Cont'd) Values of K for Different

Sections and Different R

Section

C

1.2 1.4 1.6

o

3b-----_ -m-

C

-_

T I

b

±

2b I

-o-------

R

o

/-r b

Factor K Fiber Outside

3.09 2.25 1.91 1.73 1.61 1.37 1.26 1.17

0.56 0.62 0.66 0.70 0.73 0.81 0.86 0.91

1.13 i.ii

0.94 0.95

1.2 1.4 1.6 1.8 2.0 3.0 4.0 6.0 8.0 i0.0

3.14 2.29 1.93 I. 74 i .61

0.52 0.54 0.62 O. 65 0.68

i. 34 i. 24 1.15 1.12 i.i0

0.76 0.82 0.87 0.91 0.93

1.2 1.4 1.6

3.26 2.39

O. 44 0.50

1.99 1.78 1.66 1.37 1.27 1.16 1.12 1.09

0.54 0.57 0.60 0.70 0.75 0.82 0.86 0.88

1.8 2.0 3.0 4.0 6.0 8.0 i0.0

Fiber

_

Do

I

Inside

Radii of Curvature

I

1.8 2.0 3.0 4.0 6.0 8.0 i0.0

i

Table Values

of

K _or

Different

Section o

"-

A

=

1.625b

"--q"i

Sections

B 4.3.1-1 and

Section

B4

July Page

1964

9, 41

(Cont'd)

Different

Radii

of

Curvature

R c

Inside

1.2 1.4

3.65 2.50

0.53 0.59

1.6 1.8 2.0 2.5 3.0

2.08 1.85 1.69 1.49 1.38

0.63 0.66 0.69 O. 74 0 .78

4.0 6.0 8.0 I0.0

1.27 i. 19 I. 14 1.12

0 .83 0 .90 0 .93 0 .96

3.63

0.58

1.6 1.8 2.0 3.0 4.0 6.0 8.0 I0.0

2.54 2.14 1.89 1.73 i .41 1.29 I. 18 1.13 i. I0

0.63 0.67 0.70 0.72 0.79 0.83 0.88 0.91 0.92

1.2 1.4 1.6 1.8 2.0 3.0

3.55 2.48 2.07 1.83 1.69 1.38

0.67 0.72 0.76 0.78 0.80

4.0 6.0 8.0 I0.0

1.26 1.15 i. I0 1.08

Factor Fiber

K Outside

1.05b 2

I = O.18b 4 C = 0.70b

1.2 1.4

o

3

I

I

t

4t

;LJ "------ R _

0.86 0.89 0.92 0.94 0.95

Fiber

Section B4 July 9, 1964 Page 42 Table B 4.3.1-1 (Cont'd) Values of K for Different

Sections and Different R q

Section

c

i0.

_t_- 4t --1_ p, F-qr T-] 3t

3t

± _]

L_I

I

Ii.

_

2d

----_

i

I

Inside

Radii of Curvature

Factor K Fiber Outside

1.2

2.52

0.67

1.4 1.6 1.8 2 .0 3 .0 4 .0 6 .0 8 .0 I0 .0

1.90 1.63 1.50 i .41 1.23

0.71 0.75 0.77

I. 16 i.i0 1.07 i .05

0.89 0.92 0.94 0.95

i .2 i .4 1 .6 i .8 2 .0 3 .0 4 .0 6 .0

3.28

0.58

2.31 1.89 1.70 1.57 1.31 1.21 I. 13 i. I0 1.07

0.64 0.68 0.71 0.73 0.81 0.85 0.90 0.92 0.93

2.63 1.97

0.68 0.73

1.66 1.51 1.43 1.23 1.15 1.09 1.07

0.76 0.78

8 .0 i0 .0

0.79 0.86

I

m.

12.

p'-_t 4taJ

F

!

4t

_C i ±

_-

1 .2 i .4 1 .6 i .8 2 .0 3 .0 4 .0 6 .0 8 .0 i0 .0

1.06

0.80 0.86 0.89 0.92 0.94 0.95

Fiber

Section B 4 February 15, Page43 B 4.4.0

Bending-Crippling This

or

built-up

failure.

section

This

method

does

The the type

is

to be

use

that

analysis

of applied

Combined

Examples

used

moment bending

are

when

given

plastic

margin

failure

of

bending

safety

in another

is divided

as

applicable

to formed

in the bending-crippling

mode

curves

are

of

not

B4.5.

a positive

procedure

Beams

of analysis

critical

Section

loading

of Formed

methods

are

not preclude

Bending B 4.4.2

which

otherwise

It is noted analysis

contains

sections

available,

Failure

1976

derived

from

this

mode.

into two

sections

according

to

follows:

only. moment

and

to illustrate

axial

load.

the procedure

for each

type

of

loading.

B 4.4. I

Analysis

Procedure

for

Bending

Moment

Only

/Compression

Area

b

"-
Figure

I.

Bending-Crippling

of Formed

Neutral

Shapes

Axis

(n.a.)

Section

B 4

February Page 44 B4.4,1

Analysis

Procedure

Locate

the

(b)

section Divide

assuming a linear stress the compression area into

(c) (d)

axis

Bending

(a)

pages

neutral

for (line

Moment

of zero

fiber

Only

stress)

distribution. elements

(Cont

1976

sd)

of the

according

15,

cross-

to

Section

C 1,

1 1-16.

Calculate Calculate

FCCn, m according to Figure the allowable bending-crippling

moments

about

doubling

the

the

neutral

axis

for

C 1.3.1-13 moment the

for each element. by summing

compression

area

and

result.

= 2 [ Z Fcc k

n b n t n _n + X FCCm __j k .... Y

For flange members

bm

tm

_'m/2]

(1)

........... J

For web members

where: distance element. This

equation

channel (e) The

is

is

from

applicable

shown

margin

of

neutral

for

in Figure safety

is

axis

all

to the

shapes

resultant

although

force

only

of each

a formed

1. given

by:

m

(ult)

M

M.S.

(FS)ult

(z)

.............................

M

where: M

= applied

bending

= allowable (FS)ult Note: analyzed

if the

in the

at the

bending-crippling

= ultimate section

moment

factor

moment

from

in question. Eq.

(1).

of safety.

is unsymmetrical,

conventional

cross-section

manner.

the

tension

flange

should

be

Section

B

4

February Page 45 B4.4. l

Analysis

Example

Procedure

for

Bending

the margin

of safety

Moment

Only

1976

15,

(Cont'd)

1

Determine section

shown

safety

below

if the

bending

in bending-crippling

moment

is 4000

for the

in. -ib and

cross-

a factor

of

of I. 4 is desired. t = .04 Given:

Mat'l

= 6061-T6

Mech. 3.00

'

no

_',

Sht.

Prop.

a.

_

Ftu

= 42 ksi

F

= 35 ksi cy E

_

Bare

= 9.9

x 103

ksi

1.75--b Channels together.

Typ F{gure

2.

Back-to-Back

Formed

are

intermittently

riveted

Channels

Analysis

(a)

The neutral tion.

(centroidal)

axis

for this case

r

is determined

= . 12 + .02

by

inspec-

= . 14 in.

m

-1.59_

bI

= 1.59

+ . 535(.

b2

= 1.34

+.535(.14)

14) = 1.665

in.

1.48 1

4C_

1. 34

In,

a.

= 1.415in.

Section

B 4

February Page 46 84.4.1

Analysis

Example

1 (Cont'

(c)

for

35 ff = 9.9 x 103 ¥

_

Bending

Moment

Only

(Cont'

- 2...475,

.040

Fcc

1 =

1.415 .040

Ref.

M" = Z[ Mflang e + Mweb]

= 2 [(9.62)(1.665)(.04)(1.48) = 3. 36 in-kips

Margin

275(35)

=

- Z.103,

(for each

Eq.

9.6Z

ksi

Fcc z = .76 (35) =

(e)

d)

1.665

35 9.9 x 103

(d)

1976

d)

/

bl J Fcy E

Procedure

15,

Z6.6

ksi

(1)

+ (26.6)(1.415)(.04)(1.46)/3]

channel)

See page 278for and Yn

FCCn,

bn, in,

o£ safety

M (ult) M.S.

3360

= (FS)ul t M/2

- 1 = 1.4(2000)

I = +0. Z0

where:

M/Z

= 2000

in-lb (per channel) Ref.

page

278

(FS)ul t =1.4

F B4.4.2

lo

Analysim Load

Procedure

Calculate B 4.6.1.

steps If the

sider the maximum

section applied

for

Combined

(a) through (d) according neutral axis falls outside to be fiber

stressed stress.

as

Bendin_

Moment

to the procedure the cross-section,

a column

and

compare

and

Axial

of Section conwith

the

Section

B4

February Page 4 7

B4.4.2

Analysis Procedure Load (Cont' d)

Calculate neutral

the axis

section

Zcn.

3.

a.

Compute

modulus

assuming In.

-

of the

a mirror

Bending

Moment

and

compression

area

about

Axial

the

image.

a.

Cc

the

for Combined

15,

(see

Example

equivalent

Z)

allowable

.........................

(3)

stress

u

M F-

........................................... Zc

4.

The

n

.

margin

(4)

a.

of safety

is given

by

F (ult) MS

1

-

..............................

(5)

(FS)ul t fc

where: p f =-c A

Note:

If the

analyzed

Example

Mc ±

beam

normal

in the

applied

compressive

stress)

load

(P) is tensile,

conventional

the tension

flange

should

be

failure

for

manner.

2

Determine the

(maximum Ic. g.

column

the shown

margin

of safety

in Figure

3.

for

bending-crippling

1976

Section February Page 48

B4.4.2

Analysis Procedure Load (Cont i d)

Example

Z (Cont' d)

for

Combined

Bending

Moment

and

B4 15, 1976

Axial

Given: Mat'l

Mech.

= 6061-T6

Bare

Sht

Prop.

Ftu = 4Z ksi F

P

= 5000

Ib

M

= 6400

in-lb

..(FSJul t = 1.4

= 35 ksi cy

E Angles

= 9.9 are

x 10 3 ksi intermittently

riveted

together.

_._oo-. I _[Typ .o_o Typ/ I I"1 ._6Ol• 16o'.'_ c.f[.

axis

T

7c.

.7

3. 000

Figure

3.

Back-to-Back

- r

Typ-III-

'-_-

_L n.

g.

1 Formed

Z. 760 _

___

Angle

a

a.

Section February Page 49

B4.4.2

Analysis Load

Example

I

Procedure

(Cont'

2 (Cont'

for

Combined

Bending

Moment

and

B4 15,

Axial

d)

d)

°

i_

1.26

-_

_

ELE

A

Q

Ay

Ay 2

.i008

2.96

.298

.883

®

.0250

2.89

.072

.209

®

.2208

1.38

.305

.420

• 140

.675

1.512

• 140

TOT.

.3466

Z

.08

I

y

_

o

a,

2.7h Yc. g.

,

] _A__y

Yc.g.

-

EA

: LAy

: .338 P 0 =_-+

.675 -

-

I. 947

in.

. 3466

Z + EI o - y2 c.g.

(for each

2] A

= 1.512

+.

140

- (1.947)

angle)

MYn. a. Ic.g -

(a) P Yn.a.

-

A

Ic.g. M

5000 (2)(.338) =-2(.3466)

6400

:

"'762

in.

2

(.3466)

1976

Section

f-

B4

February Page 50

84.4.2

Example

Analysis Procedure Load (Cont' d)

for

Combined

Bendin_

Moment

and

I5,

1976

Axial

2 (Cont' d)

(b) _---I. Z6 _

= .16

rm

+ .04

= 1. Z6 +. = 1.575 _

_5

= .20 535(.

in.

ZO) = 1. 367. in.

+.535(.20)

= 1.682

in.

b2

(c) bl t

E

35 =

_'- =

9.9 x 103

I9.9x

I. 367 --= .08

1.016,

- I. 250, 103

.08

Fcc I = .56(35)=19.60

Fcc

z = 1.1 (35) = 38.5

(d)

= 2[ Mflang e + Mweb]

Ref.

= 2[ 19.60(I. 367)(. 08)(1.775)

= 13.60

in-kip

(each angle)

Eq.

(l)

+ 38.5(1. 682)(. 08)(I. 735)/3]

ksi

ksi

Section

B4

February Page 51

B4.4.2

Analysis Procedure -Load (Cont' d)

Example 2.

for Combined

Bending

Moment

and

15,

Axial

2 (Cont'd)

Section

modulus

about

n. a. Compression /Area

ELE

A

G ® ®

y

Ay

Ay 2

Io

•1008

1.775

• 179

.318

-

•0250

1.702

• 042

• 072

-

.1260

.788

• O99

• 078

. 0Z6

.468

. 026

1•775

TOT.

_'

1. 702 • 788

_

II

,, II il II ii i

' _--'-

Mirror

t._o ______2,'

Image In.a.

= 2[ _Ay

2 + _Io]

= Z[.468

+ .026]

Zc n.a.

3.

_ 2(.988) 1.815

Equivalent

=

= .988

1 089 "

allowable

_ 2 (M) Zc n,a•

_

in 4 (for each

in 3

stress

2{13. 60) - 25.0 i. 089

ksi

angle)

1.815 __.__[_

1976

Section February Page 52

Analysis Procedure Load (Cont _ d)

84.4.2

Example

for

Combined

Bending

Moment

Z (Cont' d)

Applied

compressive

P

stress

Mc

fc=A+i c.g.

5 Z(. 3466)

4.

The

margin

+

6.4(1.053) Z(. 338)

of safety

= 17.2

ksi

is

I

F (ult) MS

(FS)ult

where

25.0

-1=

=

-1= 1.4(17.2.)

fc

:

(FS)ult = 1.4

Ref.

page

281

+0.04

and

B4 15, 1976

Axial

Section

B4

February Page 53

15,

1976

REFERENCE

B4.

0.0

Beams

4. I. 0

Perry,

D.

Co.,

E. York,

Roark,

P.,

Seely,

Fred

York,

B.

D.

Van

Aero

Fred

Materials, New York,

Wilber,

B.

B.

First

Raymond

Edition,

J.O.,

Edition,

Wiley

of

Inc.,

Strain,

New

Third

York,

Mechanics

& Sons,

Materials,

1954.

of

Inc.,

and

Arslan,

Publishers,

and

Smith,

J.,

C.

Book

&

Sons,

for Stress

J.

O.,

John

of Stress

Mechanics

Book

Company,

and

Advanced

of

Inc.,

Structural

Company,

Inc.,

Wiley

1955.

1960.

H. , Elementary

McGraw-Hill

Formulas

York,

Fundamentals

Advanced

Wiley

Third

New

Angeles,

J. O.,

Norris,

Edition,

A.,

John

I,

Inc.,

Los

Edition,

and

Part

Company,

Seely, Fred B. and Smith, Materials, Second Edition,

Roark,

and

Inc.,

Advanced

John

Strength

McGraw-Hill

York,

for Stress Company,

Smith,

Nostrand

Second 1957.

John

Analyses,

Roark,

Formulas

Edition,

Albert

Analyses,

4.4.0

Book

1957.

Deyarmond,

New

McGraw-Hill

of Materials,Prentice-Hall

Book

and

S.,

Edition,

Seely,

J.,

Second

Timoshenko,

4.3.0

Structures,

Mechanics

McGraw-Hill

Materials,

4. Z. 0

Aircraft

1954.

Raymond

Edition,

New

PhD.

1950.

Popov, New

J.,

Strain,

New

York,

Mechanics

& Sons,

Inc.,

Third 1954. of

Inc.,

1957.

'Raymond

J.,

McGraw-Hill

Formulas Book

for Stress Company,

and

Inc.,

Strain,

New

York,

Third 1954.

SECTION B4.5 PLASTIC BENDING

TABLE

OF CONTENTS

(Continued) Page

4.5.0 4.5.1 4.5.1.1 4.5.1.2

Plastic

Bending

Analysis Procedure When Tension and Compression Stress-Strain Curves Coincide ................

5

Simple Bending about Symmetrical Sections

5

Simple Bending Unsymmetrical Symmetry Bending

4.5.1.3

1

..........................

a Principal Axis--......................

about a Principal Axis--Sections with an Axis of

Perpendicular to the Axis of ...............................

5

Complex Bending---Symmetrical Sections; also Unsymmetrical Sections with One Axis of Symmetry

8

..............................

4.5.1.4

Complex

4.5.1.5

No Axis of Symmetry ...................... Shear Flow for Simple Bending about a Principle

Bending---Unsymmetrical

Axis---Symmetrical 4.5.1.6

4.5.2 4.5.2.1 4.5.2.2

Bending Section

Perpendicular

about a Principle with an Axis of

to the

Axis

of Bending

.....

13

Shear Flow for Complex Bending--Any Cross Section ...........................

17

Analysis Procedure When Tension and Compression Stress-Strain Curves Differ Significantly .........

18

Simple Bending about a Principle Axis--Symmetrical Sections ......................

18

Simple

Bending

about

a Principle

Unsymmetrical Sections Symmetry Perpendicular Bending 4.5.2.3

8 10

Sections .................

Shear Flow for Simple Axis---Unsymmetrical Symmetry

4.5.1.7

Sections with

Axis---

with an Axis of to the Axis of 22

...............................

Complex Bending---Symmetrical Sections; also Unsymmetrical Sections with One Axis of 22

Symmetry .................... Complex Bending---Unsymmetrical No Axis Shear

of Symmetry

Flow

for

Axis---Symmetrical

"S;ctions

w'i;h" 22

......................

Simple

Bending Sections

B4

5-iii

about .................

a Principal 23

TABLE

OF CONTENTS

(Concluded) Page

4.5.2.6

Shear

Flow

for

Simple

Axis---Unsymmetrical Symmetry 4.5.2.7 4.5.3 4.5.4 4.5.4.1 4.5.4.2 4.5.4.3 4.5.5

Bending

about a Principal

Sections

Perpendicular

to the

with Axis

an Axis

of

of Bending

Shear Flow for Complex Bending--Any C ross Section ........................... The Effect of Transverse Stresses on Plastic

23

Bending

23

...............................

Example Problems ........................ Illustrationof Section B4.5.2.1 ............... Illustrationof Section B4.5.2.2 ...............

24 24

Illustrationof Section B4.5.2.6

28

Index for Bending Modulus of Rupture Symmetrical Sections ......................

Curves

for 32 32

Stainless

4.5.5.2 4.5.5.3

Low Carbon and Alloy Steels-Minimum Properties... H eat Resistant Alloys-Minimum Properties .......

4.5.5.4

Titanium-Minimum

4.5.5.5

Aluminum-Minimum

Properties

Magnesium-Minimum Index for Plastic

Properties Bending Curves

Steel-Minimum

Properties

Properties

............

Corrosion Resistant Titanium-Minimum

4.5.6.5

Aluminum-Minimum

4.5.6.6

Magnesium-Minimum Properties Elastic-Plastic Energy Theory General ...............................

.............. for Bending

4.5.7.2

Discussion

................

4.5.7.3 4.5.7.4

Assumptions and Conditions ..., .............. D eflnitions .............................

4.5.7.5

Deflection of StaticallyDeterminate

4.5.7.6

Example

Problem

34 34 35

.............. ..............

4.5.6.3 4.5.6.4

Properties Steels-Minimum

of Safety

5-iv

. . .

...............

Beams

.........................

B4

36 36

............ Properties...

Metals-Minimum Properties Properties ................

of Margin

33 33

...............

Stainless Steel-Minimum Low Carbon and Alloy

4.5.7 4.5.7.1

33

................

4.5.6.1 4.5.6.2

Properties

26

...............

4.5.5.1

4.5.5.6 4.5.6

23

.....

37 37 37 38

........

214 214 214 215 215

........

217 219

Section B4.5 February 15, 1976 Page 201

B4.5.5.6

.Magnesium-Minimum

Properties

4O Fb (ks{)

.5

1.0

k_

Fig.

B4.5.5.6-5

Minimum Symmetrical Forgings

Bending Sections (Longitudinal}

2.0

Zqc

Modulus ZK60A

of

Rupture Magnesium

for Alloy

.

Section B4,5 February 15, 1976 Page 202

B4.5.6.6

Magnesium-Minimum

Properties

o

_.o

°a

%. ,6 .4 #

°_,_

Section

A4

1 February

r

Page

Table

Inch r

A4-21.

Inch

Decimal

Fraction

Decimal

and

Metric

Equivalents

Millimeter

of Fractions

of an

Inch

Centimeter

(mm)

(cm)

1970

35

(Cont'd)

Meter (m)

0.796

875

51/64

20. 240

37

2. 024

037

0.020

240

37

0.812

5

13/16

20.637

31

2.063

731

0.020

637

31

0.828

125

53/64

21.034

Ii

2. ]03

411

0.021

034

i]

0.843

75

27/32

21.431

05

2. 143

105

0.021

431

05

0.859

375

55/64

21.827

85

2. 182

785

0.021

827

85

7/8

22. 224

79

2.222

479

0.022

224

79

0.875

0.890

625

57/64

22.621

59

2.262

159

0.022

621

59

0.906

25

29/32

23.018

53

2.301

853

0.023

018

53

0.921

875

59/64

23.415

33

2.341

533

0.023

415

33

0.937

5

15/16

23.812

28

2.381

228

0.023

812

28

0.953

125

61/64

24. 209

07

2. 420

907

0. 024

209

07

0.968

75

31/32

24. 606

02

2.460

602

0.024

606

02

0.984

375

63/64

25.002

81

2.500

281

0.025

002

81

I

25.4

0.025

4

1.0

2.54

Section

B4.5

February Page l

B4.5.0

Plastic

Analysis

of

15,

1976

Beams

Introduction The conventional maximum fiber stress range, the assumption the stress corresponds mater ia I.

flexure formula f=Mc/I is correct only if the is within the proportional limit. In the plastic that plane sections remain plane is valid while with the stress-strain relationship of the

This section provides a method of approximating the true stress which depends upon the shape and the material properties. It is noted that deflection requirements are investigated when using this method since large deflections are possible while showing adequate structural strength. The

method

is subjected The Simple

Complex

following

bending:

A

of

in this

fluctuating

glossary

This condition moment vector the

rectangular symmetrical

section

is not

applicable

if the

member

loads.

is given

This condition moment vector

bending:

Development

other

outlined

to high

for

convenience:

occurs when the is parallel to a

resultant principal

applied axis.

occurs when the resultant applied is not parallel to a principal axis.

Theory cross section will cross section would

be used in this development; yield the same results.

any

Section B4.5 February 15, i 976 Page 2

B 4.5.0

Plastic

Analysis

of Beams

(Cont,

dA

f

d)

max

/¢2

--[

'

" -7

f

e max

(a)

Section

,

(b)

fl

true Stres'

max

Strain

(c)

Stress 0

el Strain

eZ

(d) Stress-Strain (Tension Figure

the used

Since neutral

elastic

range,

a trapezoidal

stress

approximate yield stress,

the true stress distribution, a buckling stress, an ultimate

stress

above

level

the

proportional

Curve

and Compression)

B4. 5. 0-I

the bending moment of the true stress axis is greater than that of a linear

in the

£max

distribution about Mc/I distribution as distribution

is

fmax may be defined stress or any other

used as

to a

limit.

Let

fo be a fictitious stress at zero strain in the trapezoidal stress distribution as shown in Figure B4.5.0-Z. The value of fo may be determined by integrating graphically the moment of (not the area of) moment

the true stress of the trapezoidal

distribution stress

and equating distribution.

this

to the

bending

Sect ion B4.5 February 15, 1976 Page 3

B 4.5.0

Plastic Analysis

of Beams

(Cont'd)

f max

f

f

max

max

f o

Stress

f

f

max

max

6

Strain

(a) True

Stress

(b) Trapezoidal

Stress

Figure Mba

=

(c) Stress

Strain Curve

B4. 5. 0-Z

Allowable bending moment of the true stress distribution for a particular cross-section at a prescribed maximum stress level.

Mb

c Mc

a

Fb

= Fictitious allowable I stress or the bending modulus of rupture for a particular cross-section at a prescribed maximum stress level.

I

I Mb t = "_-fmax

+ (ZQ- I_)c fo =

The bending moment of a trapezoidal stress distribution that is equivalent Mb a

Where

for symmetrical

sections,

c

I

=

(

y2 dA

(moment

y dA

(static

of inertia)

c

Q

._

C

"

max

0

Distance

from

eentroidal

axis

to

moment the

extreme

of cross-section) fiber

to

Section February Page 4

B 4.5.0

Plastic

Analysis

of Beams

Mb

Therefore,

from

Fb = fmax

B4.5 15,

(Cont,d}

c

t I

Fb =

we

obtain: (4.5.0-I)

+ (k-1)fo

Where, k

--

2Qc i

(4.5.0-2)

Two types of figures are presented for plastic bending analysis. One type presents Bending Modulus of Rupture Curves for Symmetrical Sections at yield and ultimate (Reference Section B4.5.4). The other which

type are

presents necessary

Plastic Bending for the following:

1)

Limiting

stress

z)

Tension

and

significantly 3) Figure

B4.5.0-3

Flanges

k = 1.0

] Thin

k'= 1.27

yield

{Reference

or

stress

Section

B4.5.5)

ultimate.

strain

curves

which

are

different.

shows

Generally, aircraft

Tube

than

compression

Unsymmetrical

sections. for typical

I

other

Curves

cross-section. values

of

validity sections

Hourglass

k for

various

Rectangle

Hexagon

I

, SHAPE

symmetrical

cross-

of the approach has been shown only which are geometric thin sections.

FACTOR

Figure

B4.5.0-3

]Solid

Roun_

Diamond

1976

Section B4.5 February 15, Page 5

B4.

5. I

Analysis Strain

B4.

Procedure Curves

When

Tension

and

Compression

1976

Stress-

Coincide.

5. i. 1 Simple Bending Sections.

about

The

procedure

is as follows:

io

Determine

k by

Equation

a Principal

(4. 5. 0-2)

Axis---Symmetrical

or by

the use

of Figure

B4. 5. 0

-3.

For

2,

yield

or

Rupture determine

.

For

,

on

stress

the

other

(See

k curve

Step

3 and

B4.

use

section

fb,

the

the bending

and

of safety

yield

the Bending

B4.

Modulus

5. 5 for index)

or

ultimate,

use

5. 6 for index).

(or k=l)

to read

in determining the margin

than

section

stress-strain

appropriate

F b from

stress,

k - see

of

to

F b.

Curves

stress

limiting

(F b vs.

a limiting

Bending

the

ultimate

Curves

curve

F b for

calculated stress

for pure

and

the

directly

may

combined

as

Plastic

the limiting up

to

strain.

stress,

for

bending

move

same

Mc/l

ratio

the

Locate

be

used

stresses

follows:

fb

Rb-

(4.5.1.1-I) Fb 1 -

M.S.

Where:

B4.5.

1.2

S.F.

Simple

Axis

(4. 5. i. I-2)

with

about an

Axis

(yield

a Principal of

or ult) safety

factor

Axis---Unsymmetrical

Symmetry

Perpendicular

to

the

of Bending

procedure

of its principal

1

is the appropriate

Bending

Sections

This

Rb(S.F. )

axes.

also

applies

to any

section

with

bending

about

one

Section B4.5 February 15, 1976 Page6

B4.5.1.2

Simple

Bendin

Sections Axis The

.

procedure Divide

the

,

as

Axis---Unsymmetrical

of Symmetry

Perpendicular

to the

(Con t'd)

into

(not

B4.5.

a Principal

Axis

follows:

section axis

Figure

an

of Bending. is

principal

G about

with

two

the

parts,

axis

of

(1)

and

(Z),

symmetry)

on

either

similar

to

side that

of

shown

the in

I. Z-l.

Calculate: QlCl k1 -

ii

(4.5. I. Z-I)

where I1 is the moment of inertia axis of the entire cross-section.

3.

section

made

may be sections

used where shown.

Calculate

up of part part

of part (I) only about the principal This would be identical to k of a

(I) and

its mirror

image.

(i) and

its mirror

image

Figure form

B4.5.0-3

one

of the

similarly:

Qzcz k2 = 1

Assuming use

iZ

part

the

(4.5. I, Z-Z)

(I) is critical

P_astic

Bending

in yield

Curves

and

(or crippling, locate

strain ( or k=l) curve. Move directly and read for the same strain. Fb 1 5.

Read

the

6.

Calculate:

E27.

8.

strain

to the

ultimate,

stress

on

E Z on

priate

k curve Mba

Mb a =

the

appropriate

etc. ) stressk curve

el.

ElcZ c1

Locate

Calculate

up

this

(4.5.1.2-3)

the

strain

to read

scale

and

move

directly

up

to the

appro-

Fb2

by

Fblll 5_

+

Fb212 E_

(4.5 .l. 2-4)

Section February Pa 9e B

4.5.1.2

Simple bending u_n_s_ym_letrical symmetry

M_

mustbe

7

axis-axis of

to the

axis

of

(Cont,d)

used

an_dath--emargin

a principal with an

perpendicular

bending.

1

about sections

B4.5 15,

in determining

of safety

for

the

pure

moment

bending

ratio

for bending

and

as follows:

M Rb

_

(4.

5.

1.2-5)

Mb a 1 M.S. Where:

=

S.F.

ab

-

(s. F. )

is the

1

(4.5.1.Z-6)

appropriate I

(yield

or

ult) safety

factor

!

(i) rincipal

Axis

(z) Figure B4.5.1.3

Complex

This

B4.

not

condition

parallei

with

occurs to

5. i. 2-i

Bending--Symmetrical

cal Sections

is

B4.

One

when

a principal

Axis

the

axis

as

Sections;

Also

resuItant shown

applied

moment

Figures

B4.

in

5. i. 3-2.

vector 5.

1. 3-1

,y I

M

i

;/ X

X

__ x

-

"C.G.

Y

Y Symmetrical

Section

Unsymmetrical with

Figure

B4.

5. i. 3-I

one Figure

Section

axis B4.

of 5.1.

or M

I/

Y/ X

Unsymmetri-

of Symmetry.

symmetry 3-2

1976

Section B4.5 February 15, 1976 Page 8 B 4.5.

1.3

Complex

Bending--

Unsymmetrical The

procedure

may

be

is

very

as

follows

conservative

1..

Obtain principal

2.

Follow

Rb=

Rbx

4.

For

pure M.

Rbx

is

components

also of Symmetry.

always

conservative

and

shapes):

of M with

in Section

respect

B4.5.1.1

to the

or

B4.5.1.2

Rby (4.5.1.3-1)

+ Rby bending, S.

1

l

Rb (S.F. ) Where: S.F. B4.5.1.4

Axis

cross-sectional

outlined and

Sections; One

procedure

some

the

procedure

to determine 3.

My,

with

(this for

M x and axes. the

Symmetrical Sections

is

the

Complex

appropriate

(4.5.1.3-z)

(yield

or

Bending--Unsymmetrical

ult)

safety

factor.

Sections

with

No

Axis

of Symmetry This not

condition

parallel

x

occurs

to a principal

when axis

the as

resultant shown

y,

y

\

I

applied

in Figure

moment

vector

is

x' (Principal

axis)

B4.5.1.4-1.

M

-

-

x(Ref.

X

y' (Principal ! y (Ref. axis) Figure

B4.5.1.4-1

axis)

axis)

Section

B4.5

February Pa ge 9 B

4.5.1.4

Complex Axis

The

Unsymmetrical

Sections

with

no

of Symmetry.

procedure

i.

Bending--

15,

is as

follows:

Determine

the principal

axes

by

the

equation

2I tan

where

2 0 -

x and

Ix

xy Iy - Ix

y are

,

4.

and

dA

(Moment

of

Iy =

x

dA

(Moment

of inertia)

(Product

of

inertia)

of

M

dA

Obtain M x, and principal axes.

Follow

the

Rbx

, and

The

stress

For

axes

y

the

My,,

procedure Rby

= Rbx

pure

components

outlined

in

inertia)

Section

B4.

with

respect

to

5.

1. 2 to

determine

for

complex

, + Rby

complex

bending

(4.

5. 1.4-2)

(4.

5. 1.4-3)

is

,

bending,

the

margin

of

safety

is

I M.

S.

ab(S.

F. )

where: S.F.

is

the

,

ratio

Rb

So

centroidal

=

Ixy=/Xy o

(4.5. I.4-i)

the

appropriate

(yield

or

ult)

safety

factor.

1976

Section B4.5 February 15, 1976 Page I0

B4.5.

I. 5

Shear

Flow

for

Symmetrical

only

Simple

Bending

About

a Principal

When plastic bending for cross-sections

occurs, withk=l.O.

unconservative. The derivation follows for an ultimate strength

the

classical formula For k greater than

of a correction assessment:

factor

__

f

/ ///J

Jr

SQ/I 1.0, ;3 SQ/I

__l

ma-_l

•

J/i

P.A.

-t

,%q

Cross -Section

Stress

Figure Let

P = load

P=

From

on

the

cross-section

geometry

"

fot c

-,y Let

A= area

y and

c

dy

+ ff

of cross-section

/

Jy

t dy

B4.5.

i. 5-1

" fo) y-c

_c A=

5.1.5-1

between

of Figure

f = fo + (fmax

•".P=

B4.

Distribution

tfdy

jy

Axis--

Sections

then,

(fmax

" fo)

between

tcy

y and

dy

c

°"

is correct SQ/Iis is

as

Section B4.5 February 15, Page 1 1

B 4.5.1.5

Shear

Flow

Axis--

P=

fo A

+ (fmax

P=

fmax

I--_)4

From

for

Simple

Bendin_

Symmetrical

about

Section.

- fo) A__

and

a Principal

(Cont'd)

since

Q=Ay,

c

Equation

fo (A

(4.

(a)

Q)c

5. 0-i) Mc

Fb

Since,

= fmax

c

dM

I

dx

by

+ fo (k - I) _

df

df

max

0

dx

dx

definition,

c -_--(S)-

then,

I

(k - 1 )

S = dM/dx

max df dx

[ 1 +

and

df

q = dP/dx,

o

(k

- 1 ]

and

from

(a),

(b)

max

df

df in ax

q

=

df

Q

dx

o +

c

m ax

df

dx max

Let

X

= df

0

/dfmax

and

_

=

1 +x l+x

(c/F(k-l)

l)

then,

df max

q

But

and

=

since

from

substituting

(b):

(d)

(c)

dx

into

df max (c)

we

/dx-

ScI

/

[ 1 +k

have

(Ac/Q q

Since

Q

=

= A-y-', SQ

q-

_-y-

[1

+k

(k

-

- 1)] 1)]

(k

-

1)]

(d)

1976

Sect ion B4.5 February 15, 1976 Page 12

B 4.5. I. 5

Shear

Flow

Axis-The

method

for Simple

Symmetrical outlined below

shows

the margin of safety for simple of a symmetrical section. The

procedure

1.

Determine

Mc/I

Determine B4.5.0-3.

k by

o

1

Refer

is

to the

as

Equation

applicable

how

to correct

SQ/I

about

and calculate

a principal

axis

(4.5.0-2)

Plastic

or

by

Bending

Move c.

across

the

use

Curves to the

of Figure

and

locate

appropriate

k curve

By use of the stress.-strain (or k = 1) curve and the fo curve, determine the rate of change of fo _vith respect to f at the true strain E, which would be expressed as df O

)t =

To determine determine B4.

area

5.1.

axis

-

dr/d,

(4.5.1.5-1)

the shear the distance outside

5-1.

Determine

/d_ O

df

of the

1

a Principal

follows:

d'f

o

about

(Cont wd)

plastic bending

Mc/I on the F b scale. to read the true strain, o

Bendin_

Section.

This the

shear

flow from

of the is

point

defined flow

at any point the principal in question as

at

on a cross-section, axis to the centroid as

Shown

from

the

in Figure

_.

distance

"a"

principal

by SQ a

qa

= _

I

(4.5.1.5-2)

where,

=

Qa

=

l+X (c/7- l) l+X (k-l)

£

y dA

(4.5.1.

5-3)

14.5.1.

5-4)

Sect ion B4.5 February Page 13

B

4.5.1.5.

Shear

Flow

Principal

.

Calculate

for

f

Bending

about

Symmetrical

= (r s)

Sm&x

f

Simple

Axis--

a=

0

a

Section.

(Cont'd)

and

= (q/t)

a=

15,

the

stress

ratio

is

0

S l_fl aN

R

(4.5.1.5-5)

s

,

For

F s

the

margin

of

safety

with

pure

bending

and

shear

use

1 M.S.

,

For

=

the

2-

-1

+ Rs

margin

of

safety

with

(4.

axial

load,

bending

5. 1. 5-6)

and

shear

use

1 M.S.

=

1

(4.

5.1.

5-7)

(S. F. )v_R b + Ra )2 + R2s where,

B4.

5. 1.6

Ra-

fa Fa

S.F.

is the

Shear

appropriate

Flow

for

Simple

Unsymmetrical dicular

B4.

A procedure 5.1. 6-1 is

•

Divide the

similar follows:

the

in

Axis to

section

principal

shown

the

axis Figure

with

of

that

into (not B4.

or

Bending

Section

to

as

(yield

ult) safety

About an

a

Axis

factor,

Principal

of

Axis--

Symmetry

Perpen-

Bending shown

two the

in

Section

parts, axis

(1) of

B4.

and

(2),

symmetry)

5.

1. 5 for

on

Figure

either

side

similar

to

of

that

5. 1. 2-1.

Calculate:

k1 where principal identical image. mirror

QlCl I1

(4.5.

I 1 is the moment of axis of the entire to

k

Figure image

of

a section B4.

form,

5. one

inertia of cross-section. made

0-3

may of

the

up be

of used

sections

part

(1) only This

part

(1) and

where shown.

part

about would its

1.2-1)

the be mirror

(1) and

its

1976

Section B4.5 February 15, 1976 Page 14 B 4.5.1.6

Shear

Flow

Axis--

for

Bendin_

tl_syrnmetrical

Symrnetry B

Simple

Section

Perpendicular

Calculate

about with

to the

a Principal an

Axis

Axis

of

of Bending.

similarly: Q2c2

k2 -

.

5.

to the

MCl I 6.

Mc 1 -I

Calculate

Refer

(4.5.1.2-2)

i2

on

Find

I

applicable

the

(use

Mc 2 and

Plastic

F b scale.

k 1) curve

Move to read

• E 2, similarly,

Bending across the

true

Curves

and

to the

appropriate

strain

_ 1'

locate k

_1c2 or by

_ 2-

(4.5. i. 2-3) cI

.

By

use

of the

stress-strain

determine the rate true strainq which

xl = e

=

Calculate

(or

of change would be

k = 1) curve

of fo with expressed

a,_d the

respect as

fo curve,

to f for

7l

the

(4.5.1.6-i)

similarly: (4.5.1.6-2)

.

To determine determine of the

area

Figure 1 0.

B4.

Determine neutral

qa

axis

=

the shear the distance above

or

flow from

below

5.1.

6-1.

This

the

shear

flow

at any point the principal the

point

is defined in part

on a cross-section, axis to the centroid

in question as

(1) at

_a

or

distance

as

shown

_b" "a"

from

by

_a t -T-SQa)

in

(4.5.1.6-3)

the

Sect ion February Page 15

B 4.5.1.6

Shear

Flow

Axis--

for

Simple

Bending

about

Section

with

Unsymmetrical

Symmetry

Perpendicular

to the

a Principal an

Axis

Axis

of

of Bending.

where, i

/3a

=

+k

I c(_ a

1 +X1

(kl

i )

- 1)

(4.5.1.6-4)

c 1 (4. Qa

11.

=

_a

Determine neutral

the axis

qb

=

5.1.

6-5)

ydA

shear

flow

in part

(2) at distance

"b"

from

the

by

_3b

I

(4.

5.1.

6-6)

(4.

5.1.

6-7)

where,

i3b =

Qb

12.

For

l

=

the

parts

+%2

_b

(4. 5.1. 6-8)

the

cross-section be

at

the

side

developed

of

so

each

and

that in

the

the

amount

of

the

neutral

axis.

entirely on

principal and

made

axis would result side was used in

determining

ments follows:

I)

ydA

could

neutral which

buted

2

1

-

flow

of

each

(k2

shear

analysis

)

-

1 +_Z

side

Equations

from is

axis, use

the

shear

external 5.1.

and

S1Q a qa

=

/3a

I1

at of involve

distributed

(4.

the

is shear

corresponding 5.1.

6-6)

S 1 =

M1

+ M2

distrimo-

become

M 1 where

the

to

moment

then the

rigorous

regardless This would

bending

shear, to

6-3)

A

both

calculations

shear the

proportional (4.

value

transverse

q using

larger. flow

the same calculations.

If

calculate

as

Fb 1 11 S,

M 1 = (See

C-1 B4.

5. i. 2)

B4.5 15,

1976

Section B4.5 February 15, 1976 Page 16

B 4.5.1.6

Shear Axis--

Flow for Simple Unsymmetrical

S]rmmetry

Bendin_ Section

Per_.endicular

about a Principal with an Axis of

to the

Axis

SZQ b qb

=

_b

of Bending.

Iz

M Z

IZ

where

S Z=

M1

+ MZ

M2=

S,

Fb 2 C2

(See

f

i

c1

B4.5.1.2)

t

m

a

_i

f

_ Principal

axis

I

b

Yb

cz -i

tz Figure 13.

Calculate

f

S a

ratio

s:

B4.5.1.6-I

qa = -,t1

and

f

= sb

qb -t2

to obtain the shear

f S

1%

a

(4.5.1.6-9)

.=.

F s

S a

fs b F

Rs b -

14.

For

the

(4.5.1.6-10:) S

margin

of safety

with

1 M.S.

bending

- 1

=

(s.F. )_+

pure

and

shear

(4.5. i. 6-II)

RZs

where: S.F.

is the

appropriate

(yield

use

or

ult)

safety

factor,

Sect ion B4.5 February 15, Page 17 B

4.5.1.6

Shear Axis--

Flow for Simple Unsymmetrical

Symmetry (Cont ' d) 15.

For

Bendin G about a Principal Section with an Axis of

Perpendicular

the

margin

M.S.

=

of safety

to the Axis

with

axial

of Bending.

load,

bending

and

shear

use

I -1 (S.

F.

)_(R

5.1. 6-12)

R 2

Ra )2

b

(4.

where: f a

R a - Fa S.F.

B4.

The

5.

1. 7

Shear

procedure

l

•

is as

S x,

The

or

ult) safety

factor.

Bending--Any

Cross-Section

axes

by

or,

inspection

if necessary,

(4.5.1.4-I),

and axes.

Sy,, the components The principal axes

in

the

Figure

same

principal axes at a prescribed

4.

Complex

the principal

indicated

Follow

for

(yield

follows:

Equation

Obtain principal as

.

Flow

Determine by

,

is the appropriate

shear

obtain point.

stress

x'

to the and y'

B4.5.1.4-1o

procedure to

of S with respect are denoted by

of Section the

ratios

shear

about

B4.5.1.6 stresses

at this point

fsx

, and

both fSy,

are

f S X

_

Rsx'

fs

So

For

w

(4.5.

F s

Rsy

' _

the

margin

:.7-l)

,

F sY

(4. of

safety

with

complex

bending

and

5.1.7-2) shear

1 M.S.

= S.F

-1 b

+

Rs

(4.1.

5.7-3)

use

1976

Sect ion B4,5 February 15_ 1976 Page 18

B 4.5.1.7.

Shear Flow for Complex Section. {Cont'd)

Bending--

Any

Cross-

where,

Rs S,

o

=_R

is

F.

the

Sx' 2

+ Rsy,2

appropriate

For the margin shear use

(4.1,

(yield

of safety

with

or

ult.

axial

) safety load,

5.7-4)

factor.

complex

bending

and

l M.S.

=

-1

S.F. B4.5.2

Analysis Strain

This

may

in the longitudinal is higher than the B4.5.2.

1

Simple Sections

5.7-5)

R b 4- Ra )2 4- R s

Procedure Curves Differ occur

(4.1.

When Tension Significantly

in materials

such

grain direction compression Bending

as

where curve•

about

and

the the

Compression

AISI

301

tensidn

a Principal

Stress-

strainless

steels

stress-strain

curve

Axis--Symmetrical

f

f

tC

N.A. P.

_]

A.

N.A.

•

f

Section

oC

A

C

(a)

A.

t

f C

Figure

C

(c)

(b) Strain

B4.5.2.

o

True

I-I

Stress

(d)Trapezoidal Stress

Section February Page

B

4.5.2.1

Simple

Bending

Sections.

about

a Principal

Axis--

19

Symmetrical

(Cont'd)

F tu

Tension

f

6U

(e) Stress-Strain Figure

The B4.

procedure

for

5. 2. i-I is as

I

,

Curves

B4. 5. 2. I-i (cont.)

a symmetrical

section

such

as that

in Figure

follows:

Determine

k by

Equation

(4. 5. 0-2)

or by

the

use

of Figure

B4. 5. 0-3.

,

.

For

yield

of

Rupture

ing

the

or neutral

been

taken

Note

Step

For use on (or cept

ultimate Curves axis into

the

fol.

away in

use

F b.

from

account

Plastic

stress

curve Note

(or

Bending

appropriate

k=l)

stress,

determine the

the

the

The

Bending

correction

principal

axis

development

of

the

Modulus for

by

shift-

A has

curves.

8.

a limiting the

limiting to

strain) Curves.

Locate

or

compression)

(tension and Step

call 9.

other

this

fl

at

_ 1 with

than

yield the

or

limiting

ultimate, stress

stress-strain its

trapezoidal

inter-

B4.5 15,

1976

Section February Page

B 4.5.

Z. I

Simple

Bendin_

Sections.

Locate

,

about

a Principal

Axis--

the neutral

difficult

axis

axis

in Figure

sections.

See

which

toward

to determine,

provided

Symmetrical

would

be

the tension but

B4.5.

Example

trial and

Z. 1-2 for B4.5.4.

some

side. error

I for

7.

Find Fbl and Fbz for E I and k I and k z which may or may

Calculate I 1 and I Z, the moments with respect to the neutral axis

Calculate

Mb

Mba

11

are crossin

that k I and

correct

k2

values

of

of inertia of the elements of the entire cross-section.

Fb Iz Z

Cl

margin

formulae

procedure

Note

using the be equal.

A from be

symmetrical

typical

+

c2

(4.5.2.

cases as in Step 2, F b may be used stress, in determining the stress

For Mc/I the

_

type

may

by

a

Fbl

8.

_ Z' not

distance This

several

determining A, e Z' fz' fo2' kl' and k 2. are with respect to the neutral axis.

.

20

(Cont'd)

the principal

,

B4.5 15,

of safety

for

pure

bending

with fb' ratio for

I-I)

the calculated bending and

as follows:

I M.S.

=

-I (S.F.)

(4.5.2.

I-3)

Rb

where:

S.F.

e

For

is

cases

the

as

in

appropriate

Step

3,

(yield

M_. gJa

moment

ratio

bending

as

for

bending

and

mustbe -

or

ult)

used

safety

factor.

in determining

the

_

the

margin

of

safety

for

pure

follows: M (4.5. Z. 1-4)

Rb

=

Mba I

M.S.

=

(S.F.)Rb

-1

(4.5.2.

I-5)

1976

Section B4.5 February 15, 1976 Page 21

B 4. 5.2. ] Simple Bending about a Principal Sections• (Cont,d)

Axis--

o

_

_

¢,a

u

n 0

Symmetrical

_1 II

0

_1 r..)

<_I"_

;_ r_

o

_

'

o

_

._

+

I

Z

,

£

cO

_

-.D D.-

o0 (3". ..D

÷


+ u

u _i _

v

_ ed

_

[..-i

?1

_2:d)

-..D

rd

_

_

°_


c;

_1_

_

_

cq

o.1 -..D

I

•

(y.

_

E

;>

, ._ =--.

I_';

i

,,

+

_ _ i

_) O ,._t

v

o

_

_ ::::;

_

_i

_ _I_

m mM

_ M N


+

X

O

I

o_

Z

0 O3

I,L

oI , O

_

0

No

A

<_

r_ o i

..D

rq •

•

I

_

4,_

0 _

C_

m

,

_

_

+

0

_
u

+ .r-i

,

,-4

_o

..D _

u_4

0

¢_

_

t_ ÷

_ll

! 0

_m

_

E

"_

+

+

t'-

O

_d

Section B4.5 February 15, 1976 Page 22

B4.5.2.2

Simple Bending about Sections with an Axis Axis

The .

procedure Locate

in

require

the

fmaxc, bending higher trial Refer Z.

the

which

would

the tension B4.5.4.1 for

derivation

of an

the

identical

exception

be

some

Complex cal

Refer

expression

relating

the

F b on Plastic

Sections

with B4.5.1.3

2 which

is

expressed

Follow

the

and/or

B4.5.2.2

applicable

Complex

One and as

from

A,

to that

neutral

axis

instead

Axis follow

side

Sections;

lot,

loads due to the likely to contain be obtained by fo curves.

of Section

the compression Bending Curves.

the

to that This will

fmaxt,

(or k = 1) and procedure.

Bending--Symmetrical

to Section

Z. 4

distanceA

side by a method similar symmetrical sections.

procedure

of using

Remember that the compression

B4.S.Z.S

B4.5.

axis toward Example

follow

axis. from

Z.

axis

and f_vc by. use. of. the equilibrium of axial stress dlstrlbut_on. This expression is powers of Aand the solution for _ can best

Now,

Step

follows:

and error using the stress-strain to Example B4. 5.4.2 for typical

with

for

as

neutral

principal outlined

Axis--Unsymmetrical Perpendicular to the

of Bending is

the

a Principal of Symmetry

B4.5.1.2 of the is

principal

obtained

also

Unsymmetri-

of Symmetry the

identical

procedure

except

follows:

procedure to determine

outlined Rbx

and

Bending--Unsymmetrical

in Section

B4.5.2.1

Rby. Sections

with

No

Axis

Symmetry

for

Refer to Section Step 3 which is 3.

Follow mine

the Rbx

B4.5.1.4 expressed procedure and

an(t follow as follows: outlined

the

identical,

in Section

B4.5.

procedure

except

Z. 2 to deter-

Rby

i

of

Section

B4.5

February Page 23

B4.

5. 2. 5

Shear

Flow

for Simple

Symmetrical

Refer

If shear

B4.

flow

5. I. 5 and

is being

Plastic

The

compression

5. Z. 6

Flow

pendicular

the following

flow

tension

for Simple

similarly

Refer

Axis

bined

Shear

the

Flow

the

The

The

elastic

loading

Therefore, have to be affected.

follow

the Equation

similarly

using

about

a Principal

Axis--

an Axis

of Symmetry

Per-

the

identical

The

for

procedure

same

do

with

plastic

correspond

under modified

compression

follow

portions

ratio.

combined if the

side

Bending

use

the

for

Equations

is considered Curves.

Cross-Section

the identical

of Section the

of Transverse

shape

of X

Bending--Any

procedure

not

and

side,

procedure

except

as follows"

and

of Poisson's

tension

evaluation

Plastic

Complex

B4. 5. I. 7 and

Effects

on the in the

compression

is expressed

5. 3

the magnitude

use

of 71 for

is considered

with

Curves

(4. 5.1. 6-2).

to Section 3 which

the effects

side,

Curves.

determined

principala prescribedaxes point, and obtain

B4.

with

of Bending

If4. 5.1. 6 and

Bending

using

Follow

.

procedure

evaluation

Bending

Sections

is being

Plastic

(4. 5.1. 6-i) or

Step

Axis--

exceptions:

If shear

for

side

Bending

to the

to Section

5.2. 7

identical

the tension

in the

compression

Unsymmetrical

B4.

the

on

Curves

Plastic

Shear

Refer

follow

determined

Bending

(4. 5.1. 5-3).

B4.

a Principal

exception:

tension

the

about

Sections

to Section

the following

Bending

15,

of the

shear

on

Plastic

bending curve

the plastic

bending curve

com-

due

depend

of a given

stress-strain

fSy,__at

for

curves

curves

both

Bending

curves

of the uniaxial

plastic

loading

fsx, and

of stress-strain

stesss-strain

uniaxial

5. 2. 6 about

stresses

Stresses

to those The

B4.

curves

to on

material. may

is significantly

1976

Section B4.5 February 15, Page 24

B 4.5.3

The

When

Effects

of Transverse

modification

procedure

is

as

plastic

Modify

.

Determine

the

uniaxial and

E.

bending

stress-strain

construct

Refer

ground; the constructed

Plastic

curves

curve

the

Bending.

is

to Section

the

necessary,

B4.5.0

plastic for

Cross-Section

the

A3.7.0.

bending

theoretical

plotted and (4.5.0-1).

of AISI

Elongation) Stainless Steel Bending in the Longitudinal the

Shift

in Neutral

When the Limiting Is Not Used Since the

Section

curves, back-

then

k curves

Problems

A Diamond

Find

by

modified

fo curve must first be by the use of Equation

Example

Reference

on

follows:

,

F b vs

of the

Stresses

Minimum

Hard

(2%

Sheet Is Subjected to Pure Direction at Room Temperature. Axis

Stress Is Fty > Fqy. Plastic

301 Extra

(A) from (a) )

Ftu

Bending

the

and

Curves

(b)

Principa Fcy.

! Axis (Fty

on page

133

is

and

134. _ide /_tenslon

(a)

Ftu

= ZOO ksi

ct

-= . 020

lot

= 146

From Figure Awhen k = 2 (for

in.

ksi

B4.5. Z. 1-2 a diamond).

ft + Zfot-

PlA,

/in.

use

1

(I - A 2)

the

equation

[fc +fo

c

for

determining

(Z + 6A-

3AZ)]

By trial and error, equality is reached within 0.8% is assumed equal to 0.187c as shown below: From

Equation

(4.5. i.2-3) Et

E ic2 c2

-

c =

c1

or

0. 020 x I.187c .813c

_c -

cc ct

= 0. 0292

in. /in.

when

! 976

Section February Page 25 B 4.5.4.1

l_.xample

Problem

From

the f

B

4. 5.4. 1 (Cont

stress-strain

= 149

ksi

= 107

ksi

(k=l)

B4.5 15,

'd)

curve

page

135

[ 149

+ 107(2

at

c

c

C

f O C

Therefore,

+ 6 x

0.187

1 - 0. 1872

492-_,488

(b)

f

= 97

ksi

cy c

= 0.00575

c

fOc

A

= 27

in./in.

ksi

From Figure B4. 5.2.1-2, when k = 2 (for a diamond).

use

the

equation

for

determining

1 -

ft + 2fo t

By A

trial and

is assumed

From

(I -A

error,

equal

2)

[fc + f°c

equality

to 0. 024c

Equation

-

ct

=

From

the ft

shown

for=

within

]

0. 60%

below:

_ c ct or

=

c t

0. 00575 x 0.976c i. 024c stress-strain

16.5

- 3A 2)

(4. 5. I. 2-3)

Cl

= iZl

+ 6A

is reached

as

c I c2 c 2

(2

ksi

ksi

=

Cc

=

(k=l)

0. 00548 curve

in. /in.

page

132 at

ct

when

1976

Sect ion B4.,5 February 15, 1976 Page 26

B 4.5.4.1

Example

Problem

B 4.5.4.1.

(Cont'

d)

Therefore, 121

+ 2 x 16.5_

1 1.0.-0242

I97

+ 27

(2 + 6xO.

024

154_154.93 B4.5.4.2

Calculate the the Principal

Ultimate Axis for

Allowable Bendin_ Moment About the Built-Up Tee Section Shown.

The Flan@e Is 60 ksi.

Is in Compression The Material Is

Steel Sheet Longitudinal

at

Since strain Section

the

Room Grain

longitudinal

AISI

and the Crippling Stress 301 1 Hard Stainless

Temperature Direction. tension

with

and

compression

curves are significantly different, B_. 5.2.2 will be followed.

For rather

ultimate than

Fcc

design,

crippling

the

is the

= 60 ksi

the ing wide.

principal the load (See

c

stress-

procedure

limiting

of

stress

/----tension

o.o_

;

-_

fOc -----

and

in the

Ftu.

.c=O. A trial

Bending

1_

_._.L

1

T

error

equation

axis toward on the tension Section

"2

for

shifting

the

neutral

axis

the tension side is determined by side to the load on the compression

from equat-

B4.5.2.2):

=

Cc

'Fcc

+

cc

Section February Page 27

B

4.5.4.2

Example

Problem

Equality 0. 065:

B

is reached

c

4. 5.4. 2 (Cont'

within

= 0.345

+ A

0.70%

= 0.410

84.5 15,

1976

d)

whenAis

assumed

equal

to

in.

c

c

Using

= 1 - c

t

(4.

Equation E

ct

= 0. 590

c

in.

5.1. 2-3)

C

c t c

-

0. 0046 x 0. 590 0.410

-

0. 00662

=

in. /in.

c

From

page

ft

112 at

= I13

fot

Substitution equation

results

Section

e t

ksi

38

ksi

of

these

in

equality

properties 3 2 tc t It-

3

Qt

=

Z t c t

Qt kt

-

(which

Ic -

values

into

within

about

0.

ct

above

2 x 0.05

0.05

x

_Z 0.

x 0_

x

3

= 0.

0.

01740

3

B4.

5. 0-3 t,

+ 2 x 0.450t

+ 2x0. 450x0.

05x0.

in.

= 1

Figure

in.

590

0. 006845

with

error

= 0. 006845

590

0. 01740

and

axis:

3

It

checks Ztc3 c

trial

70%.

the neutral

-

=

the

(c c \

_-_2

-_-)

5

for 2

a

rectangle)

2x0. 05x0. =

= 0. 008967

3

in. 4

4103

4

Section B4.5 February 15, 1976 Page 28

B 4.5.4.2

Example

Problem

B 4.5.4.2

Z Qc = tc c +2

x 0.450t

(Cont'd)

(Cc _ -_-) t, = 0.05 x--0.4102+

2x0.450

%

x 0.05 x 0. 385 = 0. 02572

Qc

c c

k

0. 02572

I

C

3

in.

x 0. 410

=1.17

0. 008967

C

From

page li2 using _t and k t fb = 132 ksi t

Let's check

this value

by numerically

evaluating

Equation

(4.5.0-I).

Fb t = ft + (k - I) fOr = 113 + (i.5-I)38

From

page

Fbc

114 using

= 64.5

Calculate

_

and k

C

allowable

It -

the

Problem

+

Flow

B4.5.4.2

at

Shear larger

flow

section of the

at

the

properties

2.94

=

the

Neutral

two

from shear

axis should each

flows

The procedure of Section B4.5. I. 6 will be used.

side

should

B4.5.2.6

of the

Shear,

Moment

be

of the be

in. - kips

Axis

If the Transverse

neutral

(4.5.1.2-4)

0. 590

Equal to 5 kips. and the Bending Ultimate (2.94 in-kips. )

the

Equation

13Zx0. 006845

cc

64.5 x 0.008967 0. 410

Shear

by

Ic c

ct

+ Find

moment

Fb

t

B4.5.4.3

c

ksi

the ultimate

Mbult

= 132 ksi (check)

Example (s), Is

Is Equal

determined neutral

(answer)

to the

by using axis.

(conservatively) and consequently

The used. Section

Section February Page B4.5.4.3

Shear

Example

Problem

Flow

B4.5.4.3

From

Tension

The required B4. 5.4. 2 are:

section

(Cont 'd_

properties

in.

Ic = 0. 008967

in.

already

known

from

4 (tension

side

only)

4 (compression

side

only)

4 I

= I t + I c = O. 015812

c t=

0.590

Qt=

0. 01740

kt=

i. 5 (rectangle)

From

the

in.

in.

t = 0. 00662

in.

in. /in.

stress-strain

/_\[d-v:-] =7 ksi/. \co/ 1

(k= i} curve

001

(df d-_-_ I = 8 ksi/. 001 1

Using

Equation

(4. 5. I. 6-i)

x i =_d-Y-71 = _Ta_ 8 7

XIUsing

Equation

_a = i+_ _ Ya-

-1.14 (4.5.1.6-4)

l+X

1

-

1 (kl - i)

ct 2

29

Properties

= 0. 006845

It

B4.5 15,

- 0.295

in.

Jl

on

page

112 at _t

example

1976

Section B4.5 February 15, 1976 Page 30 -._J

B 4.5.4.3

Example

Problem

B 4.5.4.3

(_ 295 590

1 + 1.14 I + 1.14

_a = Shear

flow

at

the

/3

[Note

From

that

the is

= 7.48

5-2)

36%

kips/in. SQ --

conventional

l

unconserva_ve] J

Compression

required are:

section

I = O. 015812 c

(4.5.1.

a

]formula L he re.

The B4.5.4.2

= 1.36

Equation

5 x O. 01740 O. 015812

qa = I. 36

Flow

(1.5 - I)

I

qa-

Shear

d)

1)

NA from

SQ

a

(Cont'

= 0.410

Properties

properties

in.

already

known

from

example

4 (Reference

Page

82 )

in.

C

Q

= O. 02572

in.

C

k

=1.17 C

E

= 0. 0046

in.

/in.

C

From

the

stress-strain

= 4.

dr)

6

(k = 1) curve

ksi/. 001

(df ==/I= 5.o ksi/0oi 2

on page

114 at

E C

Sect/on B4.5 February 15, 1976 Page 3 1 B4.5.4.3

Example

Using

Problem

Equation

X2

(4. 5.1.

=\dr

(4.5.1.

Yb

=

6-7)

-Yb

-

=

qb =

-

b

7

2 [0.05

c c

(0.410

O. 0Z572 O. 086

at

the

NA SQ

/d b

= 0. 086

+ 0.450)I

from

= 1.18

i'2quat*on

(4. 5.1.

6-6).

b

I

i.18

5 x O. 02572 O. 015812

= 9. 6 kips/in.

Note

that

the

--]SQ

Iformula

conventional

is 18% unconservative

here.

The ._

larger

in.

_ O. 299 in.

t + t. 09 @410299 t) • 1 + 1. 09(1. 17 - 1)

/gb = flow

Q

b

A

Ac =

qb

z

1 +x z (k z - t) Q

Shear

dr/de

- 1.09

Equation

/Jb

3 (Cont'dl

6-2)

/2

5.0 4.6

)t2Using

B 4. 5.4.

shear

flow

q = 9. 6 kips/in.

should

be

used

which

is

Section B4.5 February 15, 1976 Page 32

B4. 5.5

Index for Bending Sections

These

curves

for symmetrical ferent tension

provide

for of work

all fibers Modulus

in tension of Rupture

yield

sections only. compression

and

corrections the case

It is

Modulus

shifting hardened

and

of Rupture

Curves

ultimate

modulus

For materials stress-strain

of the neutral stainless

axis steels

of rupture

with curves,

that

are already in longitudinal

MIL-HDBK-5

or

the

other

Where

material

etc.,

only

Therefore, used in the

Modulus

for materialproperties given curve, the

modulus

the

B4.

Stainless

or

vary

two

(provided 5.5.1

allowables one

% elongations Steels

are

with

thickness,

of Rupture slightly values

practically

- Minimum

included. bending

sources

values section,

be

correspond the modu-

cross-sectional Curves

are

higher or lower may be rafioed the

In with

transverse

official

used for allowable material properties. Where these directly to the values called out on the graphs of this lus of rupture values are applicable as shown.

area,

values

significantly difthe necessary

(as in pressurized cylinders), Curves are applicable.

recommended

for Symmetrical

presented. than those up or down

same).

Properties Page

AISI

1/4

Hard

Sheet

*(RT)

...........

39

AISI 301

ltZ

Hard

Sheet

(RT)

...........

40

AISI 301

3/4

Hard

Sheet

(RT)

...........

41

AISI

3/4

Hard

Special

(RT)

...........

42

(RT)

...........

43

(RT)

..........

/#4

(RT)

...........

45

301

AISI 301

Full Hard

AISI

Extra

301

Hard

Sheet

Sheet Sheet

AISI 321

Annealed

AM_

355

Sheet,

355

Special Sheet, Forging, (RT) ............................

_ "_ AM

*(RT)

301

- Room

Temperature.

Sheet Forging,

Bar

and

Tubing Bar

and

(RT)

......

46

Tubing 47

Section February Page 33 B4.5.5.1

Stainless

Steels

- Minimum

B4.5 15,

Properties Page

B4.

17-4

PH

Bar

and

17-7

PH

Ftu

= 180

ksi

(P.T) ...............

49

17-7

PH

Ftu

= 210

ksi

(RT)

...............

50

15-7

Mo

(THI050)

Sheet

...............

51

PH

15-7

Mo

(P.H 950)

Sheet

...............

52

19-9

DL

(AMS

5526)

&

19-9

DX

(AMS

5538) .....

53

19-9

DL

(AMS

5527)

&

19-9

DX

(AMS

5539) .....

54

Carbon

and

Alloy

5. 5.2

Low

Steels

::-" - Minimum

Properties

AISI

1023-1025

AISI

Alloy

Steel,

Normalized,

Ftu

= 90

ksi

(P.T) ....

99

AISI

Alloy

Steel,

Normalized,

Ftu

= 95 ksi

(P.T) ....

100

AISI

Alloy

Steel,

Heat

Treated,

Ftu

= 125

ksi

AISI

Alloy

Steel,

Heat

Treated,

Ftu

= 150

ksi(RT)...

I02

AISI

Alloy

Steel,

Heat

Treated,

Ftu

= 180

ksi(RT)...

I03

AISI

Alloy

Steel,

Heat

Treated,

Ftu

= 200

ksi(P.T)...

3

Heat

Stainless A-286

Monel

Resistant

Steels

Inconel-X 5. 5.4

Ti-8

Mn

Ti-6

A1

Ti-4

Mn

Steels

Properties

(RT)

and

32

S.e¢ _Page

I19

...............

Hardened Rolled

I04

120

............

Annealed

121

(P.T) ......

- Minimum

Properties 129 130

........................... (RT)

- 4 A1

include

(RT)

lO]

(P.T) .......................

(P.T) - 4V

- Minimum

(P.T)...

..........................

Titanium Annealed

Alloys

Treated

- Cold

(P.T)

Pure

98

.......................

Sheet--Age Sheet

.......................

(RT)

Alloy-Heat

K-Monel

*Alloy

48

(P.T) ..............

PH

B4.5.5.

B4.

Forging

........................

131

(P.T) .......................

AISI

4130,

4140,

4340,

132

8630,

8735,8740,

and

9840.

1 976

Section B4.5 February 15, 1976 Page 34

Page B4.

5.5.5

Aluminum

- Minimum

2014-T6

Extrusions

2014-T6

Forgings

(RT)

2014-T3

Sheet

Plate,

(RT)

and

Thickness 2024-T3

&

T4

<

Sheet

....................

...................... Heat

.250

and

Treated

in.

Plate,

140

(RT) .............

Heat

14l

Treated,

Thickness

.250_<

2024-T3

Clad

Sheet

and

Plate

(RT)

.............

]43

2024-T4

Clad

Sheet

and

Plate

(RT)

.............

]h4

2024-T6

Clad

Sheet

- Heat

Treated

2024-T81

Clad

Sheet

- Heat

Treated,

Worked

and

. 50 in.

Aged

(RT)

(RT)

and

...........

Aged

Sheet

7075-

T6

Bare

7075-

T6

Clad

7075-

T6

Extrusions

7075-

T6

Die

7075-

T6

Hand

7079-

T6

Die

Forgings

- Transverse

7079-

T6

Die

Forgings

- Longitudinal

7079-

T6

Hand

Forgings

- Short

Transverse

7079-

T6

Hand

Forgings

- Long

Trans'_erse

- T6

Hand

Forgings

- Longitudinal

5. 5. 6

Sheet Sheet

Treated

AZ

61A

Forgings

HK

31A-0

Sheet

HK

31A-H24

ZK

60A

Extrusions

ZK

60A

Forgings

Sheet

Aged

(RT)

146 .......

147

(RT)

..............

148

and

Plate

(RT)

..............

149

(RT) ..................... (RT)

150

...................

(RT)

Magnesium-Minimum Extrusions

and

145

Plate

Forgings

61A

....

and

Forgings

AZ

(RT)

................

T6

- Heat

142

Cold

6061 -

7079 B4.

Properties

151

.................. (RT) (RT)

(RT)

152

.........

153

.........

154

(RT) ..... (RT)

.....

.......

155 156 157

Properties (RT) (RT)

(RT)

....................

......................

........................

(RT)

197 198

....................... (RT):

(RT)

.................... ......................

201

Section

B4.5

February Page 35

B4.

5. 6

Index

for

These curves stress-strain

the axis.

These

sections mate

are

when

materials

with

stress-strain When for It

be

is

used

the

the

Where etc.,

those or

that

to

one

in

down

(provided

Stress

the

directions

bending

modulus

for

direction

(in

Separate in

the

curves longitudinal

curve,

with

slightly

higher

direction, therefore

compression.

the is

presented

these

on

graphs

in

the

pure the

tension

AISI for

301 AISI

of

are

of

higher

sources

this

shown.

cross-sectional presented.

or

lower

may

be

ratioed

the

same).

practically

headings

are

values as

values

by

the

the

Curves

caused than

for

official

Where

higher

are

compres-

both Bending

thickness,

modulus

in

ulti-

presented.

applicable

slightly

stress

compression

grain are

out to

out are

% elongations

only

are other

Bending

the

or

and for

Plastic

or

called

Plastic

called

apply fact,

only

properties

the

Curves

longitudinal

the

curve

vary

two

given

Bending

the

or

material

used

slight,

values

allowables

only for

wdues

yield

tension Curves

properties.

rupture

than

sections.

MIL-HDBK-5

the of

material

Therefore,

is

material

modulus

is other

Bending

difference

allowable

for unsymmetrical

different

Plastic

stress-strain

directly

section,

stress

significantly

recommended for

correspond

area,

the

lower

useful

unsymmetrical

curves,

presented.

of rupture values relative to on either side of the neutral

particularly

or

1976

Curves

the allowable

symmetrical

For

Curves

Bending

represer_t modulus curve for a section

curves

and

for

sion

up

Plastic

15,

the

than

Plastic

bending. transverse modulus)

stainless stainless

The grain than steels. steels

in

Section B4.5 February 15, 1976 Page 36

B4.5.6.

1

Stainless

Steels-Minimum

Properties Page

AISI 301

1/4

Hard

Sheet

':-'(RT)

..........

55-5A_

AISI 301

1/2

Hard

Sheet

(RT)

..........

59 -64

AISI 301

3/4

Hard

Sheet

(RT)

..........

65-70

AISI 301

3/4

Hard

AISI 301

Full

AISI 301

Extra

AISI 321

Annealed

AM

355

Sheet, (RT)

AM

355

Special Forging,

17-4 PH

Hard

Sheet

Hard

(RT)

Sheet Sheet

(RT)

(RT)

75-78 79-82

...........

83 -84

.............

Bar

and

Tubing

Heat Treated Sheet Bar and Tubing {RT) Treated Bar .......................

71-74

.....

............

(RT)

Forging, .......................

Heat (RT)

Sheet

and

t

......

Forging 87

17-7

PH

Ftu

= 180

ksi

(RT)

..............

88-89

17-7

PH

Ftu

= 210

ksi

(RT)

..............

90

PH 19-9

15-7

Mo

,

DX

B4.5.6.2

.

.

.

.

19-9DL

,

.

,

.

.

°

.

,

.

.

.

,

•

and

Alloy

Steels*-':=-Minimum

Ftu

AISI

':" (RT) Alloy

Alloy

- Room Steels

.

.

91-93

,

Properties I05-I06

AISI

Alloy

.

(RT) ...................... Ftu = 90 ksi

AISI

.

94-97

AISI Alloy Steel, Normalized, (RT) ....................... Alloy

.

...................

Carbon

Low

AISI 1023-1025

**

Special

Steel, Normalized, (RT) ....................... Steel, Heat Treated, (RT) ....................... Steel, Heat (RT) ......

Treated,

I07-108 = 95

ksi 109-110

Ftu

= 125

ksi 111-I13

Ftu = 150 : ................

ksi 114

Temperature. include

AISI

4130,

4140,

4340,

8630,

8735,

8740

and

9840.

Section February Page

B4.5.6.2

Low

Carbon

and

Alloy

Steels

- Minimum

Properties

37

(Cont' Page

AISI

Alloy

AISI

Alloy

Steel, (RT)

Heat Treated, .........................

Steel,

Heat

(RT) B4.

5.6.

3

Steels

A-286

Alloy

K-Monel

Inconel-X B4. 5. 6.4

B4.

Treated,

- Age

(RT)

Ti-8

Mn

Ti-6

AI-4V

Ti-4

Mn-4

5. 6. 5

(RT)

Hardened Roiled

(RT)

and

(RT)

(RT) AI

(RT)

Sheet

&

135-136

T4

138

(RT) (RT)

and

Sheet

160-161

...................

Plate, <

158

.................

Heat

. 250 and

in.

Treated, (RT)

Plate,

Heat

Treated,

.250

Clad

Sheet

and

Plate

(RT)

...........

Clad

Sheet

and

Plate

(RT)

...........

Clad

Sheet

- Heat

Aged

(RT)

Clad

Sheet,

Sheet Bare

and - Heat Sheet

to . 50 in.

Treated

162-163

...........

Thickness

Worked

7075-T6

133-134

Properties

Thickness

-T6

127-128

Properties

........................

2024-T3

2024-T81

.......

..........................

Forgings

2024-T6

(RT)

............................

2014-T6

T4

126

.............

Annealed

36 123

............... (RT)

__

........................

Extrusions

2024-

Properties

...........................

2014-T6

2024-T3

ksi

S.e.e.P.age

Aluminum-Minimum

2024-T3

= 200

Metals=Minimum

Titanium-Minimum Annealed

Ftu

I17-I18

Treated

- Cold

Pure

6061

115-116

Resistant

- Heat

Sheet

ksi

......................

Sheet

Monel

= 180

.........................

Corrosion

Stainless

Ftu

(RT) ........

166-167 168-169

and

170-171

..................... Heat

Treated,

Aged

(RT)

Treated and

Plate

165

Cold .............

and (RT)

Aged ...........

173 (RT)

....

174-175

176-177

B4.5 15,

d)

1976

Section

B4.5

February Page 38

B4.5.6.5

Aluminum

- Minimum

Properties

(Cont'

]5,

d) Page

7075-

T6

Clad

7075-

T6

Extrusions

7075-

T6

Die

7075-

T6

Hand

7079-

T6

Die

Forgings

(Transverse)

7079-

T6

Die

Forgings

(Longitudinal)

7079-

T6

Hand

Forgings

(Short

7079-

T6

Hand

Forgings

(long

- T6

Hand

Forgings

(Longitudinal)

7079 B4.

5. 6. 6

Sheet

and

Plate

(RT)

Forgings

180-181

. . . ..............

(RT)

, Magnesium-Minimum

178-I 79

.............

...................

(RT)

Forgings

(RT)

183__

................ (RT)

]85

........

(RT)

Transverse)

....... (RT)...

Transverse)(RT)

....

(RT) .......

]86 188-]89 19l 192-193 ]95

Properties

AZ

61A

Extrusions

AZ

61A

Fo rgings

HK

31A-0

Sheet

(RT)

......................

HK

31A-HZ4

Sheet

(RT)

. . . ...................

ZK

60A

Extrusions

ZK

60A

Forgings

(RT) (RT)

........

•..........

....................

202 204-205 206-207

(RT) ................... (RT)

....................

2]2-2]3

1976

Section February Page 39 B4.5.5.1

Stainless

Steels-Minimum

B4.5 15,

Properties

300

;

75,000_p_;i 27 x 10 _ p 3i

Elongation

= 25%

T -!

_+ U ltimate

200

(Transversc)i

100

1.0

k= Fig.

B4.5.5.1-1

2.0

1.5 2Qc I

Minimum Benmng Modulus Symmetrical Sections 1/4 Steel Sheet

of Rupture Hard AISI

Curves for 301 Stainless

1976

Sect|on B4,5 February 15, Page 40 B4.5.5.

I

Stainless

Steels-Minimum

Properties

soo_....

200_

Fb

(ksi]

lO0

5O 1.0

Fig.

B4.5.5.1-2

1.5

Z.O

Minimum Bending Modulus o£ Rupture Curves for Symmetrical Sections l/Z Hard AI$1 301 Stainless Steel Sheet

1976

B4.5.5.1

Stainle

s s Steels-

Ftu

= 175,

Fty E

= 135,000 psi = 26x 106 psi

Elongation

300

Minimum

Sect ion

B4.5

February Page 41

15,

Properties

000 psi

= 12%

Fb . (ksi)

,2OO

I00

k

Fig.

B4.5.5.1-3

_.0

1.5

1.0

=

2Qc I

Minimum Bending Modulus Symmetrical Sections 3/4 Steel Sheet

of Rupture Hard AISI

Curves for 301 Stainless

1976

Section B4.5 February 15, 1976 Page /42

B4.5.5.1

Stainless

Steels-Minimum

Properties

300

Fb

(ksl)

ZOO

I00

I.O

1.5

k=--

Fig.

B4.5.5.1-4

Minimum

Bending

Symmetrical 3/4 Hard

AISI

z.o

ZQc I Modulus

Sections 301 Stainless

of

Rupture

Curves

for

Special Steel

Sheet

Section February Pa ge 43 B4.5.5.1

Stainle

s s St e els-Minimum

Ftu

= 185,000

Pr.ope

rtie

B4.5 15,

1976

s

psii

Fty = 140, 000 psil E = 26 x 106 psi Elongation = 9_o

_44

3OO

"ltimate

Transverse). ;ii

Fb

:ii

(ksl)

Z00 i

"

"

!ft

ttt

LH I00

1.5

1.0

k Fig.

B4.5.5.1-5

Minimum Symmetrical Steel Sheet

Bending Sections

Modulus Full

Z.O

ZQc of "Rupture Hard

AISI

Curves 301

for

Stainless \

Section February Page 44 B4.5.5.1

Stainles

s Ste els-Minimum

Properties

F E 300

Fb (ksl) 200

100

1.0

Fig.

B4.5.5.1-6

1.5

Minimum Symmetrical Steel Sheet

Bending Sections

2.0

Modulus Extra

of Rupture Hard

AISI

Curves 301

for Stainless

B4.5 15,

1976

Section

B4.5

February Pa ge 45

15,

1976

t

Stainle s s Ste els- Minimum

_B4.5.5.1

Prope

rtie s

t_ _t

120 _r

*;lh';l'ii_ i ....... _ ;IIHT_IIII

11

'" • 100

H_!

ftI:_ttI_If I_1;,;11i_1

tlt::!II_!: _!tf_tt|:tt

tl

!!I12:"-8O

':If:::I!"

i!II_ilIii_

iit Fb (kai)

:_!!:11!t:t

,If, I iitt_!t!i!!

60

_!;¢!;tt_: J_

it:,t ftl_:

4O

ii:;ii_i:_i

_t:!!:! t_:

.'tt_ ZO

_!!!!i!!!il

!::

i

:'/!_!i!!:i!J :!!tl;li_Id ittpllrl+_4

l,tf,t,_,J

2"t,

0

t_tHISH;1 1.0

1.5

ZQc T Fig.

B4.5.5.1-7

Minimum Bending Modulus of Rupture Curves for Symmetrical Sections Annealed AISI 321 Stainless Steel Sheet

2;.0

-A

Section B4.5 February 15, Page 46

Graph to be furnished

when

available

1976

Section February Page 47

Graph

to

be

furnished

when

available

B4.5 15,

1976

Section B4.5 February 15, 1976 Page 48

B4.5.5.

l

Stainless

Steels-Minimum

Properties

'.:..:-.-l.i-..'-.-i.

•...... , :...... i..J. ............ .... . ....": ....... I !.-! ...... I...:......I _.-----t---_-_+-_--_ ......, :,..1 ... _...i..+, _ *-__--_----r .... : ..... -F-_-.-_ L....... t----i_oo_ _,_,,e-_,,r_ :...t..;-I'-I :': i : i _ !..-:.--.i-' i""i • 1 : : ...... "" " :" I ":'.1

I , ,

3Z0

""

"

'

" "

m

"

)-_-r-"-¢--_-t :-+-Jr-:-'-:--'_".-+-_ l

'

'

:

I--.;-.-_..:-..I-

,.....

'

I":

'

.....

1....... !--.!..

'

'

'

:

'

:

I

•

"

i '!

_.:..h-r--.;--.j-.q.-i__.-.-.'.__._.:i..:._L...i_.j_

Z80

Z40

I

.... | ....

.;

:

""

_'"

".

..........

":"1"

";

.....

:"

"

_-: '-I-+-_-: -l--_._"-t- , ' I

/

/

'

!.....:.-,:._!.:_.l !-.' '--.. !_ " . :j t_,_ _,_-..,..: ,: I:

I

:

:

•

:"

:

,

"k ........ :" . "-'-:'-i.--'_-::-l:---_..---I----. :. .1 1__ ..i. '[Ultimate :[ _ | : | --

_.'

'

"

'

I

•,:,-. !.:...: I _._._f-_t_l ' : " ! ;: ). _I _-t--.,-.-, ..... +---_-+--t......_--t--:-;-t--.---i--.---I.: i I :1 l--..i...-i._;---!..-: L.!.-.i...I...._...-i--. :..i ': i.....i..-_..--,:-=-.-. ..I-. ;1 i...,--.;-..-t.-:._:.-i---,.-..i _._"

• '" "'-";'

•: _._/

"

"-''-:'fl"-:-i-'":

: I.---

'---

__.__.--

:

i ,i;

::

I i: !: ! _.i_i_tflgiH_g_fl_i_g_i_i_gmfigi_i_flff_m___

i.:.:L.)...:..::L.q.._dlllfillllllHlilfllggglggiHlfllgr_MgfilBHlgmLW .";_'_.-! "_! -i i ;: _.;,,_tl!lillmlilllHMIgglgglgmll,,HH.fl.flllgg_lP.!!":." F. !-_ _"_d"!"'T T-K:] )i.,. I..,/,ddlIMIllMIHIIIHIHJHIlgiluuRw," "I !--[..--I' i " i :.:_ ) : ! )./_;/i._!iiil!iillilllililii!lllllllit_."._;.__.L.i_..i-. _ .-_i _ ]i_._:7!i:7_T ::T.-i I- .... I: ":... I1::1 ' _ '1.1..: I : 1.L' • : ' :,_: " , • I -:.L.]

200

160

¢ 1_.._4__,._..'" • ",. ':

I" ":....

,::

""'l" ....

"" ;', ....... :" I

,:

l I"..................... ....

I,. . _

-T "_.:

"

"-""';

I .:---Jr i" I "-::.....F" .

........ :

--1":''"[

"1' ' "_"

............

I

"

l"

r . ".... "--' , . -_--L--J .

" :,

'

I"l

I

;

'.

'

"

i:

i' . ._---J : I

t-:--: .....-,........... -'....... '................ I--'---_ ........ _,,................ -, ...............' ,

• , "_,_. !" :. .':

i ..:

tI::[ .":"_'.

IZO

:

-F.-'--.--'.--'.':-'/ • '" " '

...... •

1------!.'-" • :

, .'-T-l": • I

I I

•

:

,"

1

.1

,'--,::-_:::-1--:-:1--iI .. .-i.......i .... i....... .-,.-:.-'

" " !..:. !-i..i ,: L : _ L:' : _ L , .; __-..,_o.o,,o,,,_ . , , : / : /.: I : / • : • . : _',:.-16s, ooopsi , .... r::!:i,' LI __-tr--:.-r-i.--I;-.-!--.v -r-_:' ;--_7.,,,1o_ ,,_,).--.--z,.--,• , :' " : - I-----I • ..I,I

i :,, t, _ I, :: i

_--..r'-.-. -', ....... ..;....)i .... ;..-.:..._-:.: .....

i

i....

; "d-'---'-I"-'-_'-t-. t......... :, .......

......

, , '

• ',

"

: =

-,' I|' :

!

O'_c

"-T,"_..."

I:Y .

i '

. 'L, :

_

i..--: -- i""- ""I......... !.-..'.:-1 ....... L..'-.J

I.Z

:--1 _

j.._.._i.;;

.

,

•I

1.4

1.6

- Bars

and

, I

•

"-""

•

F.._

1.8

of Rupture Curves for PH Stainless Steel -

Forgings

I .

_ .

•

i

I

.::

i......... l-!i:.-!..i

ZQe I

Minimum Bending Modulus Symmetrical Sections ]7-4 Treated

,

• I ................

"

.--_

I

[.. ,

.

"--I

:

"

i

, I

......

•

I :

.

. '|

"i .....

L

j

.... •

...........

•

_ J

L

........

;

Heat

..... t-,zorl@,atl.on

_''"

I . L..;_./ -I,--"+-_I_ . - .) ....ii. :...I..:.L. i .... _.... _-l-.-n

k=-B4. 5. 5.1-10

I :";

*'

i----:..._ ........v-,..._........ i-.--..:..-...-..i ............ [ , h-r-----.-r---.---! I

_.-4--."--L_-. "1 • l •

• •

_

I

_::

UI_"

1.0

,•

.t.,I.

:

1 !:,., • .................

...... i-

J

•

i'

t •'-

"";"--'i"'_'_

_ ,...-r-. !.,.,

" !1 ",: .... "1

I

I

_

L

•-_--

_

,: I_,.

r ;rfl

....

4O

: ,

" ......

.

"

::-,"t--'-;I'

;

.....

',;

.....

...............

'-'..'_', • I ...............

"_"'_-'-T--

_

"--I". ....

_

"_'.'-""

8O

•

:-!"-:",; _:1_t:I; I.....i :I ....... i-: r-T::_ ........ ! ,-"i"'---j .... +-:--+-----!'.-----H .... "+'-'---F'--.'--F ..... i-_ , "" _ :

Z. 0

Section

B4.5

February Page 49 B4.5.5.1

Stainle

s s Steels-Minimum

Prope

rtie

15,

1976

s

300 ..

_ ::-:!::':'_:: ,:_:::": _:i:,:_i ' i:i:_.... ,: :::: I_.. :_':"_::l :_i-_-..:--i.-:'._ :1.!:r.i-.:-I .._ :::.i.::I'_""I: ]':'l-:.i

.:

:

.;:

..

:';:

.

...:!_

- ,,'_.." l :-:_- '--_

1Room

• :

".

.

•

;

.

T :

Temperature-T

• .:.._:

•

..

,.

; '

.

.

:

: '

: I.: ;:_.1".-TT..'-.[:li::!.. I'... ]' .I.: :.. '/.

;

._.i_i../_...:.;"

-_L. -.I. :.::.

..!"-:4!"'[';"i .....

.i::i_:-_:.i-:--.:.i,_: .:-..-:.....: I:::_:4-!:-:.':_-I::!:--i-I]-':. :#.':i::-_:::!i:-_:i:::;-i-1-._:: -:-.i::- :-::!::. :-.!--_

_0

:.:::.t--::ti:'#:.4:-:.ii::.-I :.i::.l_.?::.:;l_:;.i_-:il?_i:; ::I.:-i',:.-:1:!_41 _:1__!.l:._i: :... :l..-_i.._.:..i...L:.:i.}:. !_:i:.l..i::]..I

1. ! I i_,:i-P:.l

!i:ii:i.i i i l._.:.l._!:i

1:].

i...

i,[!:':

]. :I

._i:q:::__.,1:: i:.__-_ .q..-T,., T!-iT _.ltu_,_mat_ __!_[:'::,,'::!i_TiF' 1

,40

•

:'L":!5" ":'" "''iT" "!..... "7F!"v!'Y-'-I"q[F .... !'r"i'V':':'"":'":':" .......:,. _. .::. .-_.._:_.._..-... • • .: . .. ...... .. ;F:L"?:"7"'!: :_-712i.._ .- .:":"1":...... :_+,• :...-.. • v""1• '. -_:. .:. i-.. :,:• :...-_._

--"T--'t::I:I.:" •:: !i

:: :_

! ii :

.l::.._

Fb (ksi) 1Z0

:

::

_ :i

I

"

-_--

,

.'1: :I:".: : /:'"

" .'.|':I

•

++-i--

I

:

I

+-_÷

_:-:-

.:..::........ :: .,... -r

,,-['i:--T_,:_

[

:'

_:+--

:.. " .

:::-7..' Y!Iz.-i::?77 _i!Z":-:F!iF" iii_ !iiii_i_ _! Z.

So

:,.

: :

-

!_ ! •

i; :'!;

,:

'

• :

: '

-_:

_-7 i57.:-,_ •

' "i,

.

':

!

...__

...._-;:': .....::__:: .......... _' _i!._.""i. S:!Z :::T=[[III._:. ]:__Z.!_!Zi:T:-.i:: ':Zi.:].2:i._ .::.]:;.:].i:._.]::Z:.,.i:...i:.:. "

'

": :: : ":'":::: :::: ":

:.v".: 1_';:"::

':::

" "'!:

I:::.!'.:. I:: '.:: '::;:!_:_

,r

-_,-,

"^6

":'i

• • _ '.

" i; :.: .

i_ii_h.....jii.qi_iit.!!il._]:l:ii'l:;]ii_7,?_..-..._li_];ii_iti:!;i._!_:i;!_il ]_, = _v x tu ps_ ::TZ-T:V_T_" :_t:._:iifi:t:ii:_:-l:]_-,-i_q.-.-t-:i: .P.i:F.-÷ti..iit-.=..-t.-.,:-ti_:i:ii_ - _.longation = 6% :ti:+[-]_:!" :iiiT_]:jlZ[il.]

ill:

iii!:}[];

:

]L::-:I::]]!LI:

::Ii]ZF :!!!KZZ|F

I :_

•

I!_i

•iiil!::::/l:..4iVzi!ii_i::?ii:ii.li_i _i_ ]_i:J_iiiiii_it_!:.ih=..l;!::il_!i:_hiqii ili_!]if!t:r_:!_i_i_.._;l..;_ .!_[:";-F_.]_:--_ 40

i:::_':_'iT:';;::i _:]!;::_',::ifi:i':t!ii:i:i_Yiii]:-iti:::_i!!:tii]i}iF:.,'ii!liii:i'iii:::i!_:.]i_:! : i-,!:b:.l :.:; i: I::i::! I:::1.: :::::::::::::::::::::::::::::::i]ti::l ._]ii-!: ...::::.,:::....',...ui:.,:.,:. :..Iq:: _...,_ _.;::..:;::._.: :!::__ ...._..,.._ .....:. :::![ii:':!iii!'_!:l _:iiii[:: ! iili!i ! !:]:_-'!;! :q: ii t I ;_!!-;1::.!'::i'l:::!ii_.t:t!':i!i;:!

o

:7-'

1.0

l.Z

1.4

:i_._: ii ' _!_;,._i:i:,iii! '": ..... :1

1.6 k--'_

Fig.

:i.-,-t..

,..:..... :,.::. :::,::.: :,,.:., ..... ...... . : i:.ii:il :;ii'!.:i_di]i:::.'_iiii;i]ii!iTi}:i: ::;[[ii]i'i:.!i-i['-!..i.i:i_

B4.5.5.

I-II

1.8

2Qc Ir

Minimum Bending Modulus of Rupture Curves for Symmetrical Sections 17-7 PH Stainless Steel

2.0

Section

B4.5

February Page 50 B4.5.5.1

Stainless

Steels-Minimum

15,

1976

Properties

!iilt!_i!i 360

;:'!i !i: ::': :':',

-

";:'"-!:':

:.l:_:!:

'.:h,:':::i:.

':

:;_!

::"!::!

.';:,

"!:..

:!;;!_!i

:_:,! .7,.

..!:

':_:i_ti:ti!'._ '._ :::•::_' _"i1:F'_iEH_'_i :!.H:]IiL:i:_::=_ _._il:i:.}.,...i._==-iiiH,l:: "r:itiii! ii:i =,_-_...._..:_:_:_......... i_i!!i!i!;..,._. "'": _i)_ ii!]t!i!i!'-__ "!ii:!ii

_!'==H':_

iiTii_i!i

• El!E'il!':Eiil!:!i::!::i!':':;:iiii:i,:;!:!;i.:i :" 1;_ . [..; i,.. :: r",|." |. _ :::: ::.[: ,', ; 1:'; .[.. -!..

I'":'.._"T,..'_',.... !: , J;;: I:':

;. ":',

::::'::":::':;:"

, ."1"*,"?"[-., ..'"_1"". .. I";1', . ]:'," I;'" I .....

, .-'_-r*-..,':Tl:... i'"|i .........

.'-','*"_._

._...,,':';_;:_!1_:' ":i i:i.;ii:lii:!:i:_iiH_ i:;]}Ultimate :_::t. :ii!::i :.::i]i!!i

it_.q!i!;iiiilttti _fit!li; _ii;1;_. I.,..;1_,:, ;i_!l;i.:i:: [_i_ ii:!li::[ _.

ii!:'ilii-" ":;;[:'.",

:;1:_::

: :11:1%

: ::./l"

.11: ; ::;:

' i.t::."

:.:.h:

: :."Ji4f.,

:: :.:..

_!iiiiiii!!ii!i,!!ilti_ii!i!i: i_!i,ilil.,iH..:_.r_:_l_!: ;:,; ;;I; I';; i": ..... I:: ......... -. |.;l ." ;;., .-'1.-': .:,...li" ,_'_:_-'_-

280

...... i: ....... ,'.'.!FHli:.It:,.':t:I,:

'" ....... :t:: ;_:: "l.: iii!t!::! '_'" :'

•":|':!:1::1;1!'1'11"'

i';'l

"'

•.: "::':.'.:',::l}.::[ •:""l':" :'Y."Ei:J""k:.:E_. _.i:.Li:_:i;iL_!: :.::iiU::! : _':;':,i !?:::i::._!:ii:ii::.:i

I',

I;-;;

_: i: -_!:j.

"" :_:r, z: _'"t_!

:;I

.I--,..;. :•

_-:4_"

-";_.,:::!]:

!!l,:

::;r

.....

. ':'

.

,.ll -.,iiili:;i !. ._i:i!ii!i

;.11;.:.i.i' :' • ,.

,

: ," _'.:

!

..,

• "'1"" " " "'!'' "" ' :::';::_|:':" :_.,.:.._ .::_].;.: .-.['_.. _._ :.,.. i_ield :!_...i_.J_:.:i:ii::;i •"l::l i: ! !::]:'!i.::: ": l': ' "l.: ":_": ""i'::

'::i_

!:7_i:i. :'._i'!'.

• .[

.............

I

;:

I

I

;

.

:

;;

-

;;

.:1

, ';

L.:J.;._'

"

l J.I

.'i"

'. :,':"

. ...!...

::.;!I:.-!]:i:'*F.-:i:

....

i:.,-,..:';

,-;

• l" " '.: "' .

_11,/ -% .. :,ii:ii!i' :.._, .,; :,.,..,,:H_ii i i.l:,.ii

!]ii .... _oo ii]iii!i_,;iii'.ll__.

:: ";_-'

•L".: :_ F:_ i:.:f':!i

;;..;I

,

i I '::

_:._.:F!l:/a , .. "i:;

'

i:'_'

:

:

•

,'::

".':|*.":

"'"I" r'-; ':_!_""_'1'.-'-:,':':

:ii;

;

I

'_.'.!

;;-.

"_:'

_ '_ :, !:.i'_ii_li:,:_',:;:l,., _..q_:'::_,::_.I:_: -:.:':':i "L:I!i:"

ii.;

d]!|':ll'l_l."i

!!:

.,"_t."TT':,i;,;' :;i .;i- . .... ....... i .... i. h ..... J .,I :::::::::::::H:H::::::|::::|::::::::, i::!i.!:i,i!:i!_i::l, ":'::

i:l:i

.l::::l;|.:.;:i

"

":

iJ_i:..

i:"

;"

:.'.l;':'....'.1;:'1

::';

';"

......

:H:t!

::

[i !::.l;

'::

i I;I;.

•

;;I

;.

.;'.1

7F."_:i':i_il!!i! : ":1,::',,_ .!, ,: tii!li!:. ..,:.., .-a,_:.._l?i ,.:.h .. ;.:,, :] ;l ;'-:, :.:r:_., .,:J.:l: • !;_"; ::1.i

;

I

;;;-

I;

i

:

-;

•

|-.

i

,

i

.

;

;

.

;

J

,

-

.

.::!.!

.

_-:._i..... :"_:', :::i :!;_._ -,:.:l_,: ,_1:./:I,, , _ __l,_:;,!_.r :: ': I : I:::_.;! , ,:,: .., .... r- .,_.-..-._,:":',:'-'.,'-"r'-:r'",

:.::...::i:: _.,.._-..:., - :i, ., .::. .:', ... _:...,,,, i_.:4!i:.. !ii__ ,..: i:i::-i i: i.'/.i: ............ ,.1'!.! i:':., I i._'il ;];:i ':;!"i ";i;.: '1::i d-

,

t ....

,.':{:

:.

':;

I;

t.

1.

!::_i:i:.::i';!{!!Ftu,.. •,-.I-.i- :_F.,..

._.

.t ....

' :. '.|

:-'T':.

.', ..".1,

= ZlO, = 190,

.

.I

O00 psi 0OO psi

.,/

":iJ

"::

°t'::

,'_."

I'! ''::H:i • I......

;I.

k.:-'tT:'"':"" i:'" :t?:":1," i'::' :_!i:'-_

_" i:,"r[:'_] ':l:l::i

':' ':' o

;i

r

....

"!t:.ii":

i

,., . _:r,-;S = Z9 x ,.o ps_ -1- ....... -.':-:. ::.:r.,-,.-: i_ _H':i:_:;i_ ..... ::_ ::i:. :.i_l.[V._o,,gatio_, = 6% . .. :_!i::_i;!H .._i._÷i],.,_ 4:,i_H-_ :.,. :_:;::,::.:i:r.;:_,.i!-_ !:l:.i!:i:_ii.-,.:, :il!! ............... ,.,,. , .... ;::.:,.. ;

4,0

","r i

! i.}i;lii!:

........ ,............

80

....

• ' ' : ....

• "!':"!

":"":

_!i It!.

:'";

ii. L ["ilL!

"!

';'"

: .....

;!

: ....

!! !_ i: !'i'l

:'l::';

.!:Iri

!

:L,.L"...!L:."...:;.:. ;t...:;,:.[ .:.h.:.; =: I_. ;. , . ; .,. ......

:'?T|':F.

i!.,.,ii:,.,. i:i:l_iii i ::i_: _,,. ::'['i::':'"i]q: I '_7"IL': ....."":" :.,':;

1.0

l.Z

1,4

1.6

1.8

2.0

_c

_ -= -'r Fig.

B4,

5.5.

l-IZ

Minimum Symmetrical

Bending Modulus Sections 17-7

of Rupture PH Stainless

Curves Steel

for

Section

B4.5

February Page 51

B4.5.5.1

Stainle

s s Steels

-Minimum

Properties

350

300

F b

(ksl) 250

2OO

150 1.0

1.5

2.0

k = ZOc I Fig.

B4.5.5.1-13

Minimum Bending Modulus Symmetrical Sections PH Stainless Steel 0. 185 Inches.

Sheet

of Rupture Curves 15-7 Mo (TH 1050).

& Strip

Thickness

0.0Z0

for to

15,

1976

Section

B4.5

February Page 52 B4.5.5.1

Stainless

Steels-Minimum

15, 1976

Properties

400

300

Vb (ksl)

200

I00 1.0

Fig.

B4.5.5.1-14

1.5

2.0

Minimum Bending Modulus of Rupture Curves for Symmetrical Sections I=H _15-7 Mo (RH 950) Stainless Steel Sheet & Strip Thickness 0. 020 to 0. 187 Inches

Section

B4.5

February Page 53 B4.5.5.1

Stainless

Steels-Minimum

15,

Properties

++, tlt

!!i :_

160

rtHtVt h'1111 It

140 ÷#*

tt

ttt

_t H_tVf H_HH:

}ii

:

HIHIU

Jill?! it:

!t: !iT

lZ0

_t [T_![![[ _: [71T:![T

!iti!tf_ i!i

;_

_tl

t-I

"f!ttltr

I00

_!!!!!i!

F b (ksi)

"!+*t*t

80 t:!

!t

:!Fi_!i !ii_Fi _*-+-,-+-e.,_

:!!!i!!!. !;ri!TY

60

'+lt!lN

:i!!!,h_

tlt

_t**.

-_IUttt: 4O t

:_

I i" J_t

,

t;:',1+,: tl

i-tl_t-

!i1t',t HIH;.t

ttttti_

Z0 1.0

1.5

k=--

Fig.

B4.5.5.1-15

2.0

ZQc I

Minimum Bending Modulus of Rupture Curves Symmetrical Sections 19-9 DL (AMS 5526) & DX (AMS 5538) Stainless Steel

for 19-9

1976

Section

B4.5

February Page 54 B4.5.5.1

Stainle

s s Ste els-

Minimum

Prope

rtie

15,

s

240

200

160

120

rb (k,i)

8O

4O

0 1.0

Fig.

B4.5.5.1-16

Minimum

Bending

Modulus

of Rupture

Symmetrical Sections 19-9DL (AMS DX (AMS 5539) Stainless Steel

Curves 5527) &

for 19-9

1976

Sect ion B4.5 February 15, 1976 Pa ge 55 B4.5.6.1

Stainless

Steels-Minimum

Properties

k=2.0

140 k = 1..7

k=l.5

k=

1.25

lod k=l.O

8O Fb

(k.i) 6O

40

20

t

0 0.002

0.004

0.006

0.008

0.010

(inches/inch)

Fig.

B4.5.6.1-1

Minimum Plastic Bending Curves Stainless Steel Sheet-for Tension Compression and Stress Relieved Tension or Compression

1/4 Hard AISI or Transverse Material-for

301

Sect

ion

February Page 56

B4.5.6.1

Stainless

O

f,-

II

II

Steels-Minimum

Properties

u_ N

_

II

II

O

II

Q

o N

B4,5 15,

1976

Section February Page

B4.5.6.1

Stainless

Steels-Minir_am

Properties

120

100

8O F b

(ksl) 6O

4O

2O

(inches/inch) Fig.

B4.5.6.1-3

Minimum

Plastic

Bending

Stainless

Steel Sheet

Curves

1/4 Hard

- for Longitudinal

AISI

301

Compression

57

B4.5 15,

1976

Section B4.5 February 15, 1976 Page 58

B4.5.6. l Stainless Steels-Minimum

0

_,

II

u_

II

N

II

Properties

O

II

II

O

.r_

N

0

_

0

I

4

0 _D

0 _

0 N

0 0

0

0

0

0

0

Section February Pa 9e 59

B4.5.6.

I

Stainless

Steels-Mini].num

84.5 15,

P±'o_erties

240

ZOO k=2.0

k=l.7 160

k=l.5 k=

F b

1.25

(kst) k=l.0 120

8O

4O

O. 002

O. 004

O, 006

O. 008

O. 010

(inches/inch)

Fig.

B4.5.6.1-5

Minimum

Plastic

Bending

Stainiess Steel Sheet-for Compression and Stress Tension or Transverse

Curves

1/2

HardAISI

Tension or Transverse Relieve_Material-for Compression

301

1976

Section February Page

B4.5.6.

O

II

1

Stainless

Steels-Minimum

r--

u'_

II

II

N

II

B4.5 15,

1976

60

Properties

O

II a_

'.O O

"_o N g-4

c; _ O p=4

A

U _

w

0,=1

._ ¢II

_

O

._ ,_

! .0 _

O

I

O

4 O O

0

0

0

0

Section February Page 61

B4.5.6.1

Stainle

B4.5 15,

1976

s s _St:f: ,i.:_i? _ i:,,! -_'2)_! [ [.t"_._[c_:i_F_!:_____!_2..

120

k=2.0

k=l.7 100 k=1.5

k=1.25 8O

(kF_)

k=l.0

6O

4O

2O

O. 002

0. 004

0. 006 (inches/inch)

Fig.

B4.5.6.1-7

Minimum

Plastic

Bending

Stainless

Steel Sheet-for

0. 008

0.010

I

Curves

I/2 Hard

Longitudinal

AISI

301

Compression \

Section B4.5 February | 5, Page 62

B4.5.6.1

Stainless

II

Steels-Minimum

II

II

Properties

II

II 00 N "_

0

O r_

o4

_U

O

r_

0

.,_

O N

d

0

_

,,D

d

< ,.¢

N

d

! 00 O

4 m & O

0

0

N

N

0

0

v

0

0

0

0

] 976

Section

B4.5

February Page 63 B4.5.6.1

Stainless

Steels-Minimum

15,

Properties

240 T:!

EH Room

iii

Temperature

+÷+ ttt

200

= 2.'0

=1.7 :t:

160

Ftu = 150, 000 psi _cy

Fb

(kst)

=1.5

= Z7.0 105,000 x106psi psi =

Elongation

:fl

= 1.25

= 18%

1i

=1.0

120 rt

80

F 4O

0.002

0.004

0.006

0.008

0.010

(inches/inch)

Fig.

B4.5.6.1-9

Minimum 1/2 Hard

Plastic Bending AISI 301 Stainless

Longitudinal

Compression

Curves Steel

Stress Relieved Sheet - for

1976

Section B4.5 February 15, Page 64

B4.5.6.

l

Stainle

s s Steels

-Minimum

Prope

rtie

s

0

II

II

II

II

II

! .I.a

m

0

u_ @ 0 er_ 0 A

U

0

0

e.i

U

14

_o ¢) i-q !

ow.i

o ao N

0 _p N

0 o N

0 _D ,-_

0 m

_'_ ,11

0

Q

0

1976

Section

B4.5

February Page B4.5.6.1

Stainless

Steels-Minimum

15,

1976

65

Properties

240

k=Z.O 2oo k=l.7 k=l.5 k= 16o

k=l.O

12o :T

Stress-Strain

Curve

80

40

fo

H-IHH-t-H_t_: o 0.002

0.004

0. 006

0. 008

0.010

(inches/inch) Fig.

B4.5.6.1-11

1.25

Minimum Plastic Bending 301 Stainless Steel Sheet verse for

Compression Tension

or

and Transverse

Curves 3/4 - for Tension Stress

Relieved

Compression

Hard AI_I or TransMaterial

-

Sect

ion

February Page 66 B4.5.

6.1

Stainless

Steels-Minimum

B4.5 15,

1976

Properties

360 k=2.0

320

k=l.7 280 k=l.5 Z40 F b (ksi)

1,25

k= 200

v

160

120

8O

4O

(inches/inch) Fig.

B4.5.6.1-12

Minimum Plastic Bending 301 Stainless Steel Sheet verse

Compression

for Tension

and

or Transverse

'u Curves 314 Hard AISI - for Tension or TransStress

Relieved

Compression

Material

-

Section B4.5 February 15, Page67 B4.5.6.1

Stainless

Steels-Minimum

Properties k=2.0

140

k=l.7

k=l.5

120

k=

l. Z5

100

k=l.O

8O F b (ksi) 6O

4O

2O

0.002

0.004

0.006

0.008

0.010

c (inches/inch) Fig.

B4.5.6.1-13

Minimum 301 Stainless pression

Plastic Steel

Bending Sheet

Curves for

3/4

Longitudinal

Hard

AISI Com-

1976

Section B4.5 February 15, 1976 Page 68

B4.5.6.1

Stainless

Steels-Minimum

Properties

_0

! ....... r ....... , "..... ....... . -• s 'I ..... "1 r'_ •

'

!

, •

I

,

-;

I .... ;-.+ • I.._r. '" - "'r ......

:

..t'':"';i _ .......

u,

ii _

. ,

:

i

I

I

•

....... I

.... , .... _...... .....

]

'

"

!

I

'

•

l..............

!

[ • "" "i." .... I _ ........ ..I-.......... " • : """r...... ,

4........ i

.....

,

I

i .....

I.....

•

0

I.......

_--_.---II • _ "-'-"i

•

"_l

._

:1 , Ii ...... . t ":-'-_.'---:---!.

_ _

-'-"F-- ...... • I

...........

.i

t ......

i ..... .

I

I

- _: -' .... •

,

........

....... " _ .......

j

"_ i

• l,

I _...L...I i :

.....

C:_

i

, ' i _ !'-':"i'-! .......... 1...... " I ' . • c_ ." ....... i .... ,...... i"! " ......_.........1....... "'"'-"i'"':-'i m

!i!*

.... : ........... ........... _4.--1 '"

:"'_

:.

t

: ......

..... i'":.

-i-.- _ • ................... i.!.I

_0

'oc_ " _ of" ' " I....... I I

_

"'_

I

....... i-,.__--i---;--: .......

, • "l

.........

I, ..:..,.: t:_!

.........

'

•

I "

I

'

!..... _'---t--'--r--.----. : ....

'

[.!

- _[_.

I:

:..:

.

I

I

....-1..

.....

.....

I

:..I..._

!i :

i

_ •

i _! " ;"1 " i _ .-_.

7

i

I....,4....1_,_ , i s

I,

:' i

_ " _

..,..

_4 4

Section February Page

B4.5.6.1

Stainless

Steels-Minimum

B_°5 15,

69

Properties

240

200

k=2.0

=

L7

=1.5 160

= 1.25 =1.0

Fb

1Z0

(ksi)

8O

40

0. 002

0. 004

0:.006

0. 008

0.010

(inche's/inch)

Fig.

B4.5.6.1-15

Minimum 3/4 Hard tudinal

Plastic Bending AISI 301 Stainless Compre

s sion

Curves Steel

Stress Sheet

Relieved for Longi-

1976

Section

B4.5

February

15,

Page 70

B4.5.6.1

Stainless

Steels-Minimum

Properties

k=2.0

k=l.?

k=l.5

i

k=

k=l.O

O. 02

O. 04

O. 06

O. 08

O. 10

&. 1Z

e (inches/inch)

Fig.

B4.5.6.1-16

Minimum Plastic Bending 3/4"Hard AISI 301 Stainless tudinal Compression

Curves Steel

Stress Sheet

Relieved for Longi-

O. 15

1.25

1976

Sect ion B4.5 February Page

B4.5.6.1

Stainless

Steels-Minimum

15,

71

Properties

240

200

k:2.

O

k=l.7 k:l.5 160

k:

Fb

1.25

I_: 1.0

(kst) 120

8O

40

0.002

0.004

0.006

0.008

0.010

(inches/inch)

Fig.

B4.5.6.1-17

Minimum Steel pression

Plastic Bending Curves Special 3/4 Hard AISI Sheet for Tension or Transverse

301

Stainless Com-

1976

Section

B4.5

February

15,

Page 72 B4.5.6.1

Stainless

I

320

"

:

Steels-Minimum

"

'

"

:

'

Properties

" "I "

'"

:

I ......

' ....

_+-_-.!-:_:_ .........!........".......i"_. + k=2.0 •..... !...I..,.l...i..i .'. +' •...........t...: i _ .i ..-. _.... ,..-..L...-;.._.LY...:.. :'.-. :

_

:.+;

,

'

-" "-i"" I .......

280

..... i

...,_.,

•

]r[

, ......,

: _:+

,

t"- .... ""

"

i

._7-_

k=1.7

i-::.1171171. _, ........ .

--+;..+i-,.+--i k=l.5

Z40

!-iT-ZOO

I.

-:...!.::i -. I. ! !-i; i-i.i ......ii-i : i;i ....---.I-:-I.-i ......... _....... I--:-.-,---4k=

•

:

•----

'

1.Z5

;

t............... !........i....... I.-..-.,L ---I

;. •

,_ ! _

I

._ ,..i. • i. :.l : i.. I..-.,.-.t...i.--i

_....i........ ::._t_:.s_-st ,._

_

.:.

l._..__.t_.__. +

_i_ :,.-v_,:_

k=l.O I

160 :" : ; !

+

...,.

i,;'; i

i

•.

I '

8O

!"

|

|

• •

"

I

•.-I........ ;..... 1...... ,........ 1........ ]....... • _ )_: • : _t, _ l _<_ .... ,; ))(, 'i .: .,. l .....

] I;"

....

ix"

"':"'

-J2o, ooo L,.+i

i

i.>"

"" :

•

I

--:...--4--...+ ............

..... •

I

: !

I" l'"

..L.J ....:...... -..-.......... :........ :

•

i

•

•........ / ..... .....1.--.-..] • • i i: ".--I l" "'" "', ....... '"' :'"'1

:._.J : L-_L_i

--,i.....i'":'---'"----'

"

"

i.......

: " J .....

I ........

! ........

I

,"

'

le.......

_,-, ;..,

| ........

I Roonq "['ernl_eratt, !::..... I.....; ' •

"

"

I

.... ;........i..................... i-!..L :. i,.._..i .......

'

re '

I.- .... •...L.._...J

;

-"

-..

' '"

-" ...u.__..,..

-" _+

I • _• ; i _ ":........ l"""l':!"l'"; "''J :

'

"

"

"

"

- ....... ,........i.-..-.l---.'.-l.-.... .-I--..+-. !. :. I ,....i .......i........t....:.t....-.:.i

-_-+.-I-.-+..:_.-.:.--I---:--+.--....! ........ l-.-l---+-.-,-..:._ • ''

'

"

'

::|

_ 4-" ,., ' _1,+ .... ::" " _,= 2¢" ()x-- LO :>si I[-L I' ! , '/ "" " /;"/: : '! P.lohga+,.,n IZ..o I ..... "" "--'--T .... --'r--._

i :"' :. .: ÷ -" --"--' •

4O

i....... !

; i i "" "i

J..... i....... "..'. ...............

i.

I

.

+......

..i.

• .., 120

1

"

"

+

_

::;: 0.02

:

"

i i:

O. 04

O. 06

!

i.

:."

i i O. 08

O. I0

O. IZ ILl

((Inches/Inch) Fig.

B4.5.6.1-18

Minimum Steel pression

Sheet

Plastic for

Bending

Special Tension

Curves.

3/4 Hard AISI or Transverse

301

Stainless Com-

1976

Sect ion B4.5 February 15, Page

B4.5.6.1

Stainless Steels-Minimum

Properties

150

k=2. "'E .... "-- i"

140

""

"R._.'_m

O

i

I---

k=l.7

:.... __.:LL_. .... i I

.... :" ............ Y" "!.... ........ k=l.5

I t

IZO

)

:

•"

f k=

.I I

I.Z5

:..... _ ,._o : : "

I " i'

"

! "

i I00

k=l.O

...)...... I :PP"

i

I......... I......... i..........

Fb

(kst)

'

{ '" St:.q:._s-Strair

C,,r'.'e

: •

....,....i:....'i.......t.... i....-i .......!....:..... I

8O

.......i i.]-i!i •,_

6O

: ! i--

:

:

4O

zO

{ (inches/inch) Fig.

1976

73

B4.5.6,

1-19

Minimum Steel

Plastic Bending Special 3/4 Sheet for Longitudinal

Curves Hard AISI 301 Compression

Stainless

..............

i: i _ .....

i......

Section B4.5 February 15, Page74

B4.5.6.

I

Stainless

Steels-Minimum

1976

Properties u_

I"4

_

II

II

,-=I

,..=l

II

II

•

IF

: ' i il :ii:l:i_l ,....:.. , .. !i _ :--::t.:--! . !."'F,:" "_,':::_. • , ...... ...,, ........ i'-]i_]_ LJ._i. [:::: : : ;:"::::!i" li_ ,,_

IF!

:::

ii!!! Iiill fir-4." .........

fin

i!!":! _i.z.2 _ -i ,_ ......

_

1:

;'.

m

:!

L:.i i: q:ii :_: .!i ......... _::::--i_f:::i h"1:Fl

" ":::;:: :::".'_T:" '-!17] i:l :i _i:i! :_:! : ;::::

E'_.... :..,:,;.. _H £'"

?1'"

..I. : : .... _ :

i-..":

.....i-i[ ....!ff:ri] i_:_.-:-'_i:

-,.. ::L.

":,___.

i! L:' i'il

i :!i:l

..... ..

_..._.

',', ." :I;.'.. .:;;.:. ,_ :_

i!!:" .. ::i:! ".... . :,..... .........

..

.r,_ ....

....

_ 14

;.

°:T"

:'" .': :T:-i':

0

".--1.

i:i:_Z_i:i_i_::4

::! :"!T! :::½"':-........ ' : :;:;': ..-:-._-'._ ":" :" ! 1 ' ..":t':,_; ! : .....

....

•.......... : i: '"':;

...... 111

::[i

' -.":

i

I

m !

;:"

,..2. :.] !.:_..::[::]..........

i • 1... i •

,|

:.1:;;.:,1 " ::'_I .....;;; .I

.

0

'.---1 ..... :'T_°'''_

:"

v •

:.

..

: ::

.:.

:!

...:. l l

:.

0 i

.: .. ;; _..:

::'.: .... .- - ,,.-

I-_.-t.

...

•":

I! ,:

[:._!:

; ._ I:::

-

.,

li iT_ q!:i-

i:

i .... .. .'. -:-:.: ?, .

g !.

_'-

:

-

l;

.

,

t . _•...

:

f[iF: ._, : _!h.! .

4

"--"_'-

• ':

.1

:, ',i:"

_

:i!.

::!.

.

i:!i!::i!T!i : .....

:...

...:.

:-

::_:: ,:_-:_: '_::i: _ ii.F _z'" .... .

,.

...t

L!

'

__:: : !_i::_

:l.::

: ......

__ _:

:

w

O

: :::."

"-T.'..... ":_::"

T:':" : - • -','-I-....

:

..i.lL

. .::..

'

!..:L:=

•

• ,.

.:,..

...._..:.:..,; .... ..i:::".i ;:-:1:-;-;: o

,.it. !:1

.

'::....._.... : i : ..,_ ..

!:

! i:i !'_T..... _'-

F_ :il., ; ..; :._._ ;..

:_:, ....

:i:

_!:.:.i

._!I! ;:i..

. .!-, _!i_l:!:

.._ .-.p.,

-.. _

*

!

o

......

:1 7i !

:!1.:.:

!!h

;

i

.iz-.:;L:_

: !ifi:ii: ': • .!:: y:!

i_,:

--'_-

:.'.ihL : _!:i iHF

I,

";: ! .P'E_

;" ,..;.

;T O

_

7:: :i i

• ,i _T_l:i_::

i,:: o;:.

.!

_

,_°_

L__:.__+_

.... ,

U

, -:]IF.

.

. ..:

ii ....

u o

,...

i ; :-_ .......... _".:

':',,

"--'I"-_'

O

;,;,I : I

c_

d 4

_

Section

B4.5

February Page B4.5.6.

l

Stainless

Steels-Minimum

1976

15,

75

Properties

Z40

HIH

7_f

!_!ii tilii _!!!t

EH_

T!

'! H.,+!

Nil! 7i!

200

_]tti

k=Z.

!ii{

:}i

k=l.7 I

k=l.5 k=

i

ili ++*

iii

HI , :!1

1 i_

:tl

it '

,

!t

tts

,+

IZ0

f: Fb

tt

!i

(ksi) Stress

Stra1!

,,!iI W!

II:

Curve

80

t,,

Li !y

4O

0.00Z

0.004

0.006

,_ t

21!

0.008

O. 010

(inches/inch)

B4.5.6.1-Zl

I.Z5

k=l.0

160

Fig.

0

Minimum Plastic Bending Curves Full Hard AISI 301 Stainless Steel Sheet - for Tension or Transverse Compression and Stress for Tension or Compression

Relieved

Material

-

Section B4.5 February 15, 1976 Page 76 B4.5.6.1

Stainless

Steel-Minimum

Properties

3O0

20O

(ksi)

lOO

O. O1

O. 02

O. 05

O. 06

O. 07

O. 08

O. 09 u

: (inches/inch)

Fig.

B4.5.6.1-ZZ

Minimum

Plastic

Stainless

Steel

Compression Tension

and or

Bending

Sheet-for Stress

Compression

Curves

Full

Tension

or

Relieved

Material

Hard

AISI

Transverse - for

301

Section February Page 77 B4.5.6.1

Stainles

s Steels-Minimum

B4.5 15,

Properties

200

160

k=2.0 k=l.7 k=

1.5

k=

1.25

120

k=l.0

Fb

(ksl) 80

40

0 O. OOZ

0.004

0.006

0.008

(inches/inch)

Fig.

B4.5.6.

I-Z3

Minimum Plastic Bending 301 Stainless Steel Sheet pression

Curves Full for Longitudinal

Hard

AISI Com-

1976

B4.5.6.

!

Stainless

SteeJs-Minimum

Sect ion B4.5 February 15, 1976 Page 78

Propertiee

,/3

O_

"

tl

•

II

,

II

{I

II

0

0

uo_

i

:" ;:',

I

_4 4

0

0

0

0

0

N

,"q

.-_

,--4

v

0 00

0 _"

0

Section

B4.5 15,

1976

_:

"

February Page 79

B4.5.6.1

320

Stainless

Steels-Minimum

!ii_:_iii!_i!ii!i.ii!!l:;d:::'i:il#_.t: ;.i_i._: ::i:. i _ !!i i!ii!!:iiii]iZii::iii[iiiilF_

280

Properties

Room

:I::_1._l_::i._: _ i_:i i.liiFTi_i_Tl::_

Temi)eratu'r'e

"ii

i!

ii!

i::ii i :;iiii'.[

_-_i'i[i!!:Iii

,-...:.: ..'.-:,.- _iE:!_: '--:' .'::.::; :..: ; .'..1: .:.1 _ ". ::. _:;,: : ::::i: -: ;:::F. ':-;. ::' ::.:_ll .:::: 1 :'): . ............. _:"-.... 11t=:':"_::" ii:i:.!..... I:.....:::. ................. _,'._:-v::"::":::,::v !i- :'li!: i'i'_."

...... I . .... " I:tt ................... :::t..................................... • • ::'.:.; ! : ............ , : :::,'-=,.:: - k: : ............................................ _ I............... I......... :....... '

.

II:!_

'

":. 1":::'_

"'

'

'

'

":' I:::" "."h":

'

'

"'....:'.:

'

'"

..............

:::'. "":

: : ": "::':

• :

.'.1::'.

........ i-L-'!

:'

_.:'-_-_I;;::

".

:::

:"

t"

". ::: ' :::. :"

""

7":I:_"

1":

l"

: " ..'

"! .....

,"I.'

; l_l:

; ...........

'

-"

.

II"

k

"_

:11

_:_. o

":_::

:

m

:

[

::J::::

"::

"

"

:

m

Z

:

"

::

:

,

'

I)"

:

'l

_

,

1

[

_ml_

"

_

"

lull

_ .........

m

I

m

'

:

1

;

m

nm

:

1

:

m

m

'

:

:

l

:

m

In

m

m

,

:

:

1

I

I

::::::::::::::::::::: i_i,F.:i-_-_f_-h _-_::_--_.;_a,,,,i,,_--:--I;_

:

::m_::

_F ....... my:t'.']:4"': ''_" [email protected] "|'I'_'L:::)E'" ....... ;!:--: I. . _!.11 .... __

"

m:

.ram

..

Y

_

:

m

"::i:':i .r. ...... I"6_'I'%::;. "ii} ;_ i[ ] ' t )::'! ......._-;':i:" h,. n "'7I... .................... , .

' '

:

a"

'

= Z%

.

g

'

m

"

Elon,atxon ' '

psi

:::::::

....

"I

(ksx) !_ti!!i!:i;)l_!! _::,_!i !!l!:;:l; :!. :!i ":_:F_::'!_.,_Z.:_ Wi!];ii:.:::;::,.!F!:::: i' ' .... ......... ::I.!!': i_b;-- "_:' ;.-" .... .ii:1-22

_u

I ,-_: - : .

'---It.l" •

_ "T':%_.'_ ....... '""il:i_ ii.:!.i..,7.:ji:y:

"_ .....

,7"!!_-......

:

m

mmmll'

m

m

:

:

'

:

: :..i: .:).i .i ......... ::::.;

"

:

'

........

77"_ _

120

t -_ _tress-_tra_ ' ......

"

.....

t

::::1:':

_'!!i

'.' '

5_.Z _l_, _._:.

'

160

_'"" "--_'"':'"" _';-_

iiiihi!_;_!::-i::_i.:......

.,

.....

8O

T';i._ i i

ii:U:!!

_i

:"_t-:' i_'-

,

! :

.

. .

.

.

: ;,:

......

:

: .

;

•

....

.........

i!iiii!: .................""i', ......-,4O "'!??I"_

:Fii::_

_)ir. ::_. ,:. ili: .i F 0.00_

0.004

ii/TT 0.006

! 0.008

_":_ 0.010

0.012

0.014

(inches/inch) Fig.

B4.5.6.1-25

Minimum Plastic Bending Curves Stainless Steel Sheet -for Tension Compression and Stress Relieved Tension or Compression

tu Extra Hard AISI or Transverse Material - for

.

'i)lllTI!!ml :TI: _)i":i :]itl)l!)]'::'_):iIT::]: )FT)I;)I:_i!:ZTT:I.-I:F:I_

= 160,000

[

!;TI_: .:-:_i. _T!:_i-Ii_IT 7:T!T..-II_ ......

i::)" Ft.

:

ii::_ _'tu :zoo,ooopsi -'i::.

200

'

J_

'1

240

I...........

'

5

..I ......

"

Z

'

301

Section B4.5 February 15, 1976 Page 80

B4.5.6.1

Stainless

Steels-Minimurn

Properties

0

_ _._ _

0 !

0 0

4

o

Sect ion B4,5 15,

1976

Hard AISI Compression

301

February Page 8 1

B4.5.6.

l

Stainless

Steels-Minimum

Properties

160 Fb (ksi) IZO

0.004

O. OOZ

0.006

0.008

0.010

' {Inches/inch)

Fig.

B_.

5.6.

1-27

Minimum Stainless

Plastic Bending Steel Sheet for

Curves Longitudinal

Extra

Section

B4.5

February Page B4.5.6.1

Stainle s s Steel s - Minimum

0 0

_--

II

II

u_

II

Prope

uq e,]

CD

II

II

1976

15,

82

rtles

o 0

0 .'_ U_

0

m 0

_'_ A

I2

w

.

_

N !

,6

i

4

0

_q

0

Section February Page 83 B4.5.6.1

StMnless

Steels-Minimum

Properties

60

5O

4O k=

F' b . (ksi)

1.Z5i

3O

2O

10

0 O. 001

0. 002

0.004

0. 003

0.005

(inches/inch)

Fig.

B4.5.6.1-30

Minimum Stainless

Plastic Bending Steel Sheet

Curves

Annealed

AISI

3£1

B4.5 15,

1976

Section

B4.5

February Page 84

B4.5.6.

I

Stainless

Steels-Minimum

15)

Properties

140

120

•--i ................... :.-. L..--.._ t

v

:

i

i'

I

I ;

k=l.7

' i i i "'': i ............ ' n

• -.

100

|.

.

..:

.;

'

i

Fb

....

(ksi)

i

•

- ..

i...... i :

i

k=

%,

•

_

;

k=l.O

I

"

•

S-r(:_

•

-Strain

o

Curve

I

-i' 'i " "i:": L " ' " , j. i ..... .......ii ......._............ 4......._---." -1 ....... ..-' ...... ' .. i ! : • i. ! : i_..#_

I

I

I

• ....

,

•

i

i

"

:

•

+,:k..... ,....

: "_-

' _. i:_b.4-"_-!-!_i '.'_-" ' -.................

: i -I_

:

:,.

r. "i "TT!"!"

:7"_;

I

•

"

""

-±-: .... : i......i.......... ;-,

.. _"q.--I

! 2

T

t "

i

I

.

I: .....I '_'--: "1 ": i-:

; ....

I ....

I ""'-.%.1_1 _"

_

."

-._1

" ..... LL.........

:

-L

!

"

'

, : ....... , .......r-.-l-;.--_

: ) ! ........ ! _:_ _,: i:,_ !......... t-"'.'--'t"-""-['-:"'-1

"....... ;'"'_.........

....

f " .! _. !..... l...... i ..-...:.......1

......I---I-_...1........ f+--I-!--F.....j.----4....... -......t......... !--L4--:...4....-4--.-i_.i..1 'il_-i:--I"-÷-!-.:.-._.--t ..... :i.--:..-'-. ..._.'_.:")." . ! . I • ; ' :, _ _ : )-"-----T-""._.'"""

i

i.. .......

i. i . : ...I I , ' • r ...... i........ i...... _'-_"!

.......... •"i " j: .... ]: i: i . _ -, ...... ---.-: ...... -. ...... ..'..... :....... :...... i O. 10

O. 20

! .....

i. :

_t" !_T" _-"t-+_

B4.5.6.

1-31

Minimum Stainless

Plastic Steel

Bending Sheet

i : ........... ....

i • i i.._..... I :.L..l..:.'..i.......l ! • !. !_ i ... J O. 30

0.40

t (inches/inch) Fig.

I

i

-

1.25

.

I

:, I

.

!

i

"'"

20

'

i.....i- .....)--:-.-!

--..i--.'.-.i-:-_ ....... 4---'--i

8O

4O

k=l.5

!

i

60

.

...... _........ i

Eu Curves

Annealed

AISI

321

1976

Section February Page 85

Graph

to

be

furnished

when

available

B4.5 15,

1976

Sect

Graph

to

be

furnished

when

available

ion

B4.5

February Page 86

15,

1976

Section

B4.5

February Page

B4.5.6.

I

Stainle s s Steel

s-Minimum

Prope

15,

87

rife s

0

--!

!,

'! .....

! '

:

:

I

• i

Lr! ......r i: .... i.... ,_ Z[ffiTii3 77_[ili ii 77i _ ........ t........ + ......i ,_ ....

_ .......

--!

"_

i ...!-._.

i'"

'

-4-- .....

_i. :_--!.---_--.; :-e'| .°_

"';"

_°__,,:.! ...... I l..::.lii.;.I Z..:: gN .....

_

..:.-

|.-..,

t[_.

....

•,Ii

•

_

"

...... _

"

_

"

i

"" •

!...... i

o

.......

0

.,.<

' ............

.......

_, ......

:"

";

:_fi'"'i

.......

i

_7 ._ :,_ _ ...... i...... i ....

I 0 N 0

ff

.el

.,D

J

_a _a

N

_i..._....,.:...,_il ....i;-. !:T .......!T !........ g -_:--i ,_-_....... i-i ...... d--!!....... ;- ._i.... :: :":" i".'%:"'! ..... r- :" .... h ..... ! i'

l, :' :[ _ '; _:!_:;;Tr 7,T:II _i t_ _i. :i

:1 • ' "" ;ill: !7,71; '

i

00 0 0

!

-_"'"'"r • ;

'

":

; ::.

;.

: " ,,,.,._L_I •

,i

i:

-T77T_':-_'..-T

::;i|_:,!

.: ! 7'

.

::

i

_ -:?

1-1 _::i

0 0

M 4 d_ .,-.I

0

0

0

0

0

0

eq

N

N

N

_

"4

0

0

0

1976

Section

B4.5

February B4.5.6.

1

Stainless

Steels-Minlmam

_i

.J

,-:

If

II

II

Propreties

Page

15,

88

.J If

I!

0

c_

m m

e_

0 !

I....

0 A

o

l_O

u

o.._

o O

° _= oo

I

N 0

4

-F,I

0 ,_ "f'_l

0 0 ¢",1

0 _

0 N

0

0

0

1976

Sect

ion

February Page

B4.5.

6. l

Stainless

Steels-Minimum

B4.5 ]5,

89

Properties

k=Z.O

320

k=l.7 280

240 k=l.

ZOO Fb

k:l.O

(ksi) 160

120

8O

4O

Z5

1976

Section February Page 90

B4.5.6.

1

Stalnle

s s Steel

s- Minimum

Propertle

s

k=2.0 360

k=l.7 320

k=l.5 280

k= 240

k=l.O 200

Fb (k.t) 160

120

80 :

I

•

I

4O

Fig.

B4.5.6.1-40

Minimum Stdel

Plastic

Bending

Curves

17-7

PH

Stainless

1.25

B4.5 15,

1976

Section February Page 91

B4.5.6.

1

Stainless

Steels-Minimum

Properties

k=2.0

350 k=l.7

k=l.5

300

"I

k = 1.25 250

Fb (ksi)

k=l.O

2OO

150

100

5O

O. 004

O. 008

O.012

O. 016

' (inches/inch) Fig.

B4.5.6.1-43

Minimum (RI-I 950) 0. 020 to

Plastic Bending Stainless Steel 0. 187 Inches

O.02 u

Curves for Sheet & Strip

PH 15-7 Thickness

Mo

B4.5 15,

1976

Section February

Page 92

Graph

to be

furnished

when

available

B4.5 15,

1976

Section B4.5 February 15, 1976 Page93

Graph to be. furnished when available

Section February Page 94

B4.5.6.1

Stainless

Steels-Minimum

Properties

100 k=2.0

k=l.7 8O k=l.5 k=

60

1.25

k=l.0

F b

(ksl) 4O

2O

O. OOZ

0. 004

0.006

0.008

0.010

, (inches/inch)

Fig.

B4.5.6.1-46

Minimum 5526) &

Plastic 19-9DX

Bending Curves for (AMS 5538) Stainless

19-gDL Steel

(AMS

B4.5 15,

1976

Sect

ion

February Page B4.5.6.

I

Stainless

$teels-Minimam

0

C" • II

II

15,

1976

95

Properties

L_ _ •

Lr_ •

II

0 •

II

II

il!!!!]i o "

...... F_

<

•:_ I ,i ,

-

.t :

' •


:

Ifi ..

A

:

": •

""

: ............

......

i_

.o..

: I • .... ! .......

_

o--

.....

' •

_ .........

'_ 0

t'..i

.A _4 4 i

0 -_D

0

0 _,_ A

0

m

. c_

iii..i,i.... -.......

0 0

B4.5

0

Section

B4.5

February Page 96

B4.5.6.

I

Stainless

Steels-Minirnurn

Properties

ZOO

k=2.

O

160 k=l.7 k=l.5 k= k=l.O

4O

0.004

0.006

0.008

(inches/inch)

Fig.

B4.5.6.

I =48

Minimum 55Z7) &

Plastic Bending Curves 19-9DL (AMS 19-9DX (AMS 5539} Stainless Steel

1.25

15,

1976

Section

B4.5

February Pa 9e _4.5.6.1

Stainless

'

Steels-Minimum

I:i-I

:

I.

l'e:X:l)e

:.,.tu

:.e

i.. i .... I,b._c,rn

97

Properties

:........ r---........ : I.:..:.. '

15,

I

i.

] I .....

I i'

: :

k=2.0

i

t

I-

'i

;

.....

....!i • ........

k=l

•

7

"! .... i

200

I.

"" , •.....-.'...........Ii....i.-

I

....

160

k:_

1.5

k=

1.25

o,

Fb (ksi)

!

i...... t.........IW....

120 I

i...

I I

k=l.O

i

1

I

I I

8O

O. 01

O. 02

O. 03

O. 04

O. 054

(inches/inch)

Fig.

B4.5,6.

1-49

Minimum 5527)

Plastic & 19-9DX

Bending (AMS

Cu

Curves 5539)

Stainless

19

9DL Steel

(AMS

1976

Section February Page 98

B4.5.

5.2

Low

Carbon

and Alloy

Steels-Minimum

B4.5 15,

1976

Properties

IZO

Room

Temu_

100

Ultimate

8O

Fb (ksl}

60

40

ZO

1.0

Z.O

Z_c I

Fig.

B4.5.5.2-1

Minimum Symmetrical

Bending

Modulus

Sections_Carbon

of

Rupture Steel

Curves AISI

for

1023-1025

v"

Section

B4.5

February Page B4.5.5.2

Low

Carbon

and

Alloy

Steels-Minimum

15,

1976

99

Properties

[._i. :i_i i[ I_i:-;Tti!:ITt:T ih_ :_7.:]i .-i_:_-iT:i:iliii_:i: i_ii:iii

i"--f-.:.-_:t;::!....;...._o_i_m ':,,_,_.r_,_ __,-H-t_-i ....."i!-.:'"_ ...... :-t 16oI_:[i_ ifi:ll:....!/"l::l!:,:!-i--:r--_!-mi:-:.t---I-:--,--::_ .....:-_+d: ....I

,_.:_-_:i:-F ::t:"d:i_--i-_..i.-i.l::.-l.._.l_i.:d:. _:..,?._,.:,51_ _'d: 'i •

4

f ....

-_

: !. : • ' • L" .... "...

: i"

.., ................ ' _....... ! ...... :

i .............. : "

"

[''I

:.

:

................. L

:""I"": ....... I ._'

...... ,,

.. ....

:!

]:

.... '

ii : :

, !.

1-!: ' :

)_:._.._!_i.. _i_.Z:i__:Z__i..iI..ii!i..: '.... _.0,i:_ .........._t._.,- __i.._._t

I ......... :" "

•

t....

.

i

i ....... :.......... ! [ '

.._/;:I2____..L_ ....... _::: ....:!"_ :i ....i

.....

"-"":"

"

!.......

_'-

':!

_ "':i_7-

_:--: --_--

i--.:..I.-_..:._!:i _...... : _it -.:t: t :.1.. .!-1:.._-.,__!..-i ......... i...._.' ..,...t.-..: ........... /: `_....:._.__..:_::.:_:._:_:_:._.._:_._*_._ ........ , _h (ksi)

::

....... |-!-,......."i .:'-:. ....!-" _.-_-i. _-....i".-':..! F

::::_ _--_-h_: 60

: _ [!}!iiiii!]

............

. . , , "']""t" _ .. :: ;'":':';, ...... ....... "_" "'i':.!'_.... ""'i. i,_"":: '. :-..... _-'. :........ I...... J _-i..;'"'i........... ! ,.--4 .

_----_-_-_-.:1_..!..-1-_.-i __-.i]:...i-_-.I....:.:./..::.:,:...i....:.:

.!

o "Z.J."2.Z:jI]_il...... d-_i..... i:"12]',:: __ddt:i "17:_k::-I! '_jt:.!Z :'j:] d ,

i

11 ...

i

I

o,,:: :IZI, : .......

.

1.0

.

:...

' ::!

i

_-_: ......_.

::.:::

"

: _

-'--t..

l.Z

B4:5.5.

Z-Z

.

I__:, ......:_:_:-, ....... -................. ...:-.... Td ......... t ......_.-_-:_-:: _,_lr [I I......... -.'.

i

"'..'I:"

.;:::.1.

1"_';':1

1.4

"1::

: : .:

•

1.6

:

•

:

'. '

1.8

Minimum

Bending

Modulus

Symmetrical Sections > 0. 188 Thick

AISI

of Rupture Alloy

Steel,

Curves

'

.

g.O,

k = _ZQc I Fig.

-']_

for

Normalized,

.':.

"

Section

B4.5

February Page

B4.5.

180

Low

5.2

Carbon

and

...... "r- .... _ "'"" ";....... : : : i " , : ........ I "" i "''; • : : " ' '

'"[ •

_-..... --............ ..... _ ....... ! _.. ' , ..... .---..---_"

160

,._ i.: ' J'"':"'IRoom

Alloy

" i ' i . • "'" ! • • •

_

i : ' • i " .4-":" ...... -I- .....

_ ' 'rempera(ure

Steels-Minimum

i _

•

1976

lO0

Properties

......... I ..

, •

15,

L ..... '

I _

• '

I _ _ • ...... "1. ............ ;

_ : .. • •

,

..... ' ...! ..._,.r.._. l' • . --.._....; .... --4

: '

,_

.... i..... i

" Uitimatc_;.-_+----..J '! T ; _.

"!

i !

_ ..._.:...'.... t..._, i.... L, jq__;...!.:._!I • i : : . . _ ! ! • _ --i

; _A

I

.. .... ÷-----t-:'-r-_"-. ..... ....... r..... 1....... F-_.-i-"----;-9 '.1. • _ _ I ..::.._..:..i ...I.. ,.._'-._P-:i 140 I"i'i":._':.']"' I..... i " '_ ._.L.i_..'. ........ [-_ |._._-..1 :_' • , _ .. ., : i :" •...... I :

.................. '-....... r":'"'l

,

_

JC_...__i .u-,-:....... -i--_

:" .-_".

!

. ..

l ....

:

I

--

,

_

"

"

._"_i

_

'

: . j,,,r.__

!

• i:

!'_ ..... ,:,. : ....... i. .... , _-!,___I . • .

'i."-_-."I

.

[ !

:

-_-.- i..... .

I

........

;..........__ ..... ._; .........

,_o_--._-_._ ....... --__

, , .-i .... -.'.-_... i i

: i

!...

;......t. ....... I....'. _

.-_I_..L]

_-_-I_'-=

I00

..... •

F

;

"'" "

: .'. I

•

i' .... ;I

:: .... :-:;-F:;;;:;;: :7; i ".... ........ -
: • "_.., : [ .

_-"I-"'-...... • : .

i

-........ |....... '......

•

'

.

'

:

..

, I

,

,

[

:

i ..:... ........ -. ..... I....

.: ..... '...... •-._. _.-.:_---*-.--.__,;-!;;_-! ,_'*, .<-L.LL.,__ -___.._.._ '.-L.:" """ ----. : _ '_ ....... 1--'--'. .... --I .... I ' "

--4 ...... ;...... "-

...... i

.... _

80_t-=;_ _!.._; i-;:;<-;_:<::i!..... !:i-:r-_-.! _I " _. ! : . I" " "'_I

! : I _.......___ • :

:

! • I . I

I : .i I -----t- ..... ! l

,..!._.i: i:..!- -..I. _ i:-I"!i:

.... I

• " ?...... :'--': ........ i • i

!:

'

__ ._l ..... , .... :

"

•

•

i

J.'.._---L ..... .4 . t . : [ • l

! ..... I.... i:

:, :.i

_].

- 75

"'._-- .......

_/

_0,.... _._,.-+--_---_-_--_-_--; --_-_t-_-:-,i;=S_--_. ..... _--:-..;:_;;i_ {.-.:-4-..i..".: .i.:...... :':--I ........ I-:"I :! _'tu: ,_,' , I.... : '!:::I : ......

i

I

• ......

'

--,- ......

i . i ..i :'':''l'"

.

'i:: r

I

".

s.......

_.......

i :'

l

_........

''

_L. 9F

"l|

L---I

......

"_'|

:' ': ..........

"_"

"

.!: • _ , .J . i_.L

_0 "-r-r_i-:"T.

F*-

1

! : • ' :.... L:'

"

0",n '

'

' _' "

._si :

-r--_

:

"

'

!

'

]

"

:" '

"

"

Z9 xli:¢,_s_'. :- I....... !......... [

"

_'_i

i...... F_.,,,._,'i,_ _z_..', 4.... t---+---,

.- _.--T- .... r...... l

." ' i" " _

: I . ._t

•

_

,

• . .,

. . i...... I...: .[.......... i

..._...!-,._-..I. .I..'..!'--:!_!...... ;...... l .... !...... F- .... l" : i _ i : _ _ i :_-_Li=-i.--!.._ ..... 4--=--,-----_:--4 ..... --i----e .'T-'N, ":-'t-'_} ' ! '_.-"i-i,,"-.:

,o

' .....

•

i : ]......•

! : ......

..

i .

l

:

!

I

•

:

....................

• ._____...;._L---t.__a.._l__-I-_..-,-----b-.-_........----_-.-r • . • • • . .. i-_'.'-[:. : • J . ::1 1 :. ..:.....'.-.:.--.r--.i.---,-._--.-i.--._-: i:l _I-_--I : i :-::......... ........ ,-.-,.-I ...... I- r--L- ..... ,_ :'--4"-':'_ ,,:: , • . , : • ' • : : ! ": I " -L ..... •......

•;-..._....+--_. ....... !.....G.---_--.-.T---.-----.

o i !.....I i: 1.0

B4.5.5.

Z-3

Minimum Symmetrical <

!:

t

I- 1

• :

" "l:l ....... J _...... _'i _ ....! ' ;-.._-;;;-

1.2

1.4 k=

Fig.

: _ ". i .....: ...I....... i..J.-: ,, , | : •.... , ,

0. 188

1.6

Zac

_

I

Bending Modulus Sections AISI Thick

1.8

of Rupture Alloy Steel,

Curves for Normalized,

2.0

Section 84.5 February 15, 1976 Page 101 B4.5.5.

Z

Low

Carbon

and Alloy

Steels-Minimum

Properties

Fb (kii)

4O

0

Fig.

B4.5.5.2-4

Minimum Symmetrical Treated

Bending Sections

Modulus AISI

of Alloy

Rupture Steel,

Curves Heat

for

Section B4.5 February 15, Page 102

B4.5.5.

2

Low

Carbon

i .: i

and

i

• _;

Steels-Minimum

: I

I .

!

Alloy

......

,

_

•

• " T" ! '!.... " II ..... . . F:--4 ...........

:;.

.

!I ............. ] 4

!" . .i ............ ,.......

: .._ /

:

,..i. ' :

:. i .... I !.

i......... "."_-I .......!...... -..:1

_........T........... 7-[ .......

.

i :---;

.

........... ,--4.:

....I',' ": "'."I" "'! .....!

; .I. ._ I .... I-':.--.I . I . /

.....I""q- ......i....... --r"-i

......

I /

!

'

Fb

160

120

I

.

!

I

I

"

'

:

- .: ..... i : I

..... ;:--.-t ...... ....... " ! : :'-! ..... : .... i-": ..... 4--....-

i.: ..--_ I .... I } ...... I ..../.. ,..... I..i ......... _..... I........ ; " ; • , • I I : . : I L ..... +-+."i ...... |•....... 1-----.---[.-----_ ........ I' .......+....._ ............. _......... 200

i

' .........4. - -':........ | ;, ..........

'---[....... 1-..... -r:i ....... ;--r:-t "1 ' / ...... :i '".

I.

" 7_

........

_ :-1 ........

_" .... . !

: _-:!Room Temperature .... Jr.... I....__...' "/ 240

properties

...... ":

.

• ",.-...... I, ........................ "I_ •

280

[

I i:

1976

:

:

:.... | i ' --L.__I ...... ,! "

I

'

'

I.......... Jl......,, . . ,,................ ;, ..... -T', . ...... . .{....... ..4.+--" " t_ .....1""'."....... I," ": ...... ,....... . ..... ;............ .I . '.,1 • ---T"7"".'""."-I--'"1 ....... :"Jl-":'"'l" ._ ....... I. --. -I • _ : ; : ' ' • .! • ' •+ I + . : ,.. ,.:+ _: .... :.... I ......I: .... iY,ehl"l-._ --:"I ..... l ...... I' ' ' ' ' ' ,,,,,;, .... _----="-' ........ j..._.., ."_ii_IIIIIIIIIIIIIIIIIII_IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII_ .-. :: _: :" i _..J'_iaIIIIIIIIII_IIIIIIIIIIIIIIIIIIIIImIII_IIIIIIIIIIIIIIIIIiIIi._ '"!i i :: ........ ::i ........ ' FTi • I ': _ •, , ...... "!.. ...... -"i- ---t"':'"T---.':'"---!". "";--

,_•• i • I i •i• r: , •'•1 ........ I " ! •_••_ I ......... ••1 :•••!•_ ':••:-I•: ......... :........ ".....' 'I-= +"-....... I - !,:--q'......... i, ...... [ _.... _'-I:. I+....._-::;' .....--:-* ......... I

,

•

.I

•

I

'':

:

+"

..........................

_

:

:

............. .

,:'"

....

: i .... +;...... , . • +:+", ""............. !-+r! ......... i......... !---1, • __.;.._1! '"-; !. ..... • .r, :...,..__+_-...I......... i !. i ...... r,...._ + i ......... t:.lq, ,._ _:-._.: _..... 8O

• I

i

" i

• / ......!-!:-I

..... !.... I...... !"

:..... 4- ...._.....-F--i---4..... _---..

......4.... '

•" -; ........ I......_ "-: ........ i--!-"-t 4O

.... •

I • , ! '

..................

i

..I. :

, ..I. I l

_ ........ : ...........

'

I = "

i

'

• _ ' " Jr-.....__Z.

..........._,,.- 13z,::,oop.,,,----!--

: i ...,

.-g ....

i." ..... !.-.:....., ...... :..i-.-

i r

...i..:

-4--- ............

i.''

/ 1--

"I

z+;
"0_'

"

E'°nRat"_n-"

'

-'

"8"_:°

"......... !: ' " ?-!....... i;......... ::.:---i--....-...... '!-........ =!"";--++!i: ............. .......:-1........iT: !......... _.............. t....... I-----.i--..,.;+ ..... :

"

:

•'. :+ ......... .... _...................... 1.0

1.2

'

t- .....':

:

•

I

......... I '1 ! L...... :..... I....... I __L 1.4

'

_

1.6

1.8

Z-5

I%4inimum Bending Modulus of Rupture Curves Symmetrical Sections AISI Alloy Steel, Heat Treated

•.: z.i0

k = __2qc l Fig. B4.5.5.

|

...... + .... I " i ........ ........ ,...... J. __.: ........ ! ............

for

Sect

ion

February Page

B4.

5.5.2

Low

Carbon

and

Alloy

Steels-Minimum

Properties

360

320

280

240 F b

(ksi) 200

160

120

80

40

Fig.

B4.5.5.2-6

Minimum Symmetrical Treated

Bending Sections

Modulus AISI

of

Rupture

Alloy

Steel,

Curves Heat

for

103

B4.5 15,

1976

Section

B4.5

February Page 104

B4.5.5.

Z

Low

Carbon

and

Alloy

steels-Minimum

15,

PropertieB

4OO '

i

32O

'

Temperature

_'

Iit_!! ill!i] _t!tiii iltli! fltt_

!._It_)t_ttfltttt_,....l'[t!ttttlllllltttl!tltI!t!iii!!l!ttl!t!f!_llf _

!;,!II!!

" :-i_:"-: ......... :...........;..... _.......__:;

_-k-! -_---.___, --." ' , 160

•...... _--_" i":.... h: _

.r :-i ....:....I....7...... l ....... i....... 1.........i .........i .........I....... u.....: I--...i--..i

........

" ..............

; ....

_h_ttltllltlittlllttlilk_.t_ 8O

i

tttttttttltIt ft14 ,ttlllllltlItlttt t111tllt IIIitf!ttttt_ ttiftt_ttltl t[t tlttttlltllttttttltltttlt!_tf t_t_ttttttt ftttl[tlttflttltttl_ttltfI_iliJllftillt ittttfttttttl [tt_i Ittttttltttt_tttttltltt!IiI1_it!1 #i_i_t_ ti11 t_tt!)iltiitltt111Itt l_ti!lttt_t

_i _i

24O

Room

t' .....

' '_ F.

- zoo, ooo psi

' -_

_tttt lttttlltltlilliIl_li!llfll t_, _**:,,6,ooo _, _,,,_ It_tt!tt)ttttt]ttt!tlftt!ll1II!t! /il!lllttlt!_. =_.o_1o6 psi

0 1.0

Fig,

..

B4.5.5.2-7

1.5

Minimum Symmetrical Treated

Bending Sections

2.0

Modulus AISI

of Rupture Alloy

Steel,

Curves Heat

for

1976

Sect ion

B4.5

February Page 105

B4. S.6.

Z

Low

Carbon

and

Alloy

_te_ls-Mtnii-durti

15,

i3ro_erttl_i

k=Z.O

70

k=l.70

6O

k=1.5

k=

5O

1.25

k=l.O 4O

Fb (ksi) 3O

20

I0

0 0. 002

0. 004

0. 006

0. 008

0. 010

c (inches/inch) Fig.

B4.

5.6.2-1

Minimum Plastic AISI 1023-1025

Bending

Curves

Carbon

Steel

i 976

Section

B4.5

February Page I06

B4.5.6.

lO0.

Z

Low

Carbon

and

Allo

7 Steels-Minimum

_

1976

15,

Properties

k=2.

44+PP

it

O

_Room

Temperature

_,

H

I IHIII IIIII!iN il!HI!III!I!!III I !!!!! k=l.7

k

1.5

f_ t_M_.,TitlIIIP_;_',..'_ iLtjlLt-ttttl ,, iiiiiiiiiiJi _u!!_ l_ ]_;;,tlllll!W_';;iftttlltttti ttlltft_Jt Ii !];:ttJlftftt tittttt !!!! k=I.25 II__t-_flt!!!It_iii_itIltt!I!Jl _!Iiil_l_IIIi!!!li IIIiill _'..m,_,,

60

k=l.0

II _ .-._"'_"'"_'"""'i'"". . •.... i. ....... : , i__-' ..... !l'..;,Iillli,'.'.:i_.nn.nIli,.ll,-Iiqllll..,,_""t'_ ......... I , _.",,". ......

i

't .,,iuilI. I--

"

I

=.

•

i

..........

......... F

I

I

--'"

I

I .......

!1;_ "'

I

"

t

lit

"

I

...............

....

,I'AII

I

40.

t

I _

""

I, -'IUl.._lllllUllllllllllllll,lllll

: ;dl'IjliilliilliililJllllW"" :,,-.:...pp,,!,,,, :

: I II " : _'-:mtm ,,_m,,,_tlt-_tm-_m,tlttliiiimiiiirt_ tI !_ _tfiltitltm _-tttJ_' _ttttJitH

i_II ::::::: _,. =_,ooo_=_tt_lftftt- : IiilIiIJiJlJlilIiI ";;";:

ZO

_tv

Ili,Ilt

= ao, vuu.psz

'_ittlill

llTltlltltttll?!lttl

O. 04

O. 08

O. 12

O. 16

1

'

}

'

t

I

_

:: _

.+._:+_

,Hii.i4litlt,tlitHilltiiilfllttHlltltll]i

I

'_lfltttl!liltlttltf iiiiiit I!I: II t.i,itil_tlhtitJ '_ tttiJI tJ ....liiihi l,Itfttli tl,,,it-ht,t.'ul _i 11t,it +_ti ttt 'ilI _ tit,I!IIt_tttttfllt-tI1tt!ttt ;;;;i I i tfi tHt 'li tflitltl _ ,, tttilllltlifttfttttf. •lttff tt,l,ld,_ddII,,t,,t,t_,_d_fittt '_ :::: ,,tili_,tlnltmtui,_iltlJ, tItH7 _-+r'.. + .......

iiiiJiii

71-ii _iiiiii =-= _-<_ x_o. ,>._i__tii !iiii'_+_!"!!!!' El°ntati°n = _Z_/o

:r

O. ZO

(inches/inch)

Fig.

B4.5.6.2-2

Minimum 1023-1025

Plastic

Bending

eu

Curves

Carbon

Steel

AISI

Sect ion B4.5 February 15, Page I07

B4.5.6.2

Low

Carbon

and Alloy

Steels-Minlmum

Properties

140

k_2.0

!iiilililhiill 120

k=l.7

k=l.5 !

100

k=

8O k=l.0 Fb (ksl) 60

4O

_0

.....

!1 .......

O. 002

O. 004

O. 006

O. 008

¢ (inches/inch)

Fig.

B4.5.6.2-3

Minimum Plastic Bending Curves Normalized, > 0. 188 In. Thick

AISI

Alloy

Steel,

1.25

1976

Section B4.5 February 15, Page 108 B4.5.6.2

Low

,-

Carbon

and Alloy Steels-Minlmum

.... • •

i...... i"'_

[

.

'

.

.

're:rperaturei

.

:

I

:

i

r..... -i-_ .....;............. _ .....I..... ....... _...... 7........ t-_ . ......I.:.--i .... :-_............... _ " ,"_! p....-.i..-.----...--....i.-:..-.,-...-.-.. ......._.._---!

....... I _:_:,_J ......._1 :.i_.,_

150

'

!

!

•

•

•

•

"

:

-

"

!

i_'r-.

•

I

! " !

i

I

:

!

, "

."

'

•

i',".... "_--,',"

.

I¢,. --.-

,.; .

_

,, .. "

I

i

[

-

'i

i

i

..

_

!

.....

" "

• "

J '.

I I

"

[

I

I

i

! i

!

........

i

.

/. ,......i-,-

• •

.: !.. !. .i.. '

:"

r_-i......... :' " ! " ! ........ i......i.........

_-T,

i

I /_ "" l"

_,,_;

.......

i

.,,__ i

•

•

" ....... _ !i........... : lk=,., , ,-

I

i--r" ,-r-:' _-_.; :. :.....:._ •

I..............

.

•

Ro:_m

•

Properties

"......................................... ;

,

1976

k-,.zs

:

'i ;_,iilIIMIUlIIIIe"_ ................. :'_ n'. , -....... i....... ............ - .....'...... -t....... ;........ !.............. ;........

100

;

i

i l

Fb (ksl)

............. I

:.

,

I

I

i

!

I

:

.

'.: I:!

! i

".'"'" '

:

; .........

:"l

"

,

i

........ :

I

"

• S.r.;..,._.Str,"liil

.

,

•

I

" Cur-L:

'ii,,,!,,,,,;-""i""7i .... •

;

i

'.

_

:

.

;

'

:

....

I

.......

•

,

I

:

:----t........ .--._-t---"

.[__-::

......

._....;

:

:

I

"

'

:

-

i

.- ,

.i,.

'

""

"

.--

.:

-'

....

,

•

'"

,

. ,l.r

,

........... :-i:-!........... i.,:_ !'!---......

.....................................

!..

i

i.---_.

•

.,, '

_-!

,

•......

: ...........

i

!

>'

'......

i...... ' I • ......... '

- . ....

: ................

! .......

"

• • i. - I t, p, -_

:: i,_::.; :;-_i....::: ........, ........ I_7: ir_,

it__ t__ _ol_I-i-_

:

..b

•

i;......

":" i......

i...." _

Ii

I........ '" ,. I .... "'i ........

i

i

i I

.......... it

;

.

i

:

.

i

:

.

"-..... :,...... .4--;...:,- ............ I.•

i

i

:

.

i

•

J

it:,, ..,,. :.,, !, : I-' ......... ...., ......... :._, ....... :., :, ...,, !i ' I,,. ,, !.............. .....I'_ i !! '! ....,........ _..... I '.............. :: :. i I,.......... i ..... ,............... ,.....................

I

_.. i

n-"--I , ' i; .:._J :..-.........1.--...-I.......-i........t........................ .I..... i-'t!......... : ,...-....l... ! ', • II--.i.0

! ........

li...-... ."

•

!...... ! .....

, .... ....... '

O. OZ

O. 04

I.

t

i...

i

:

.'

I

I

: ... ............ :

.

O. 06

,.......

O. 08

O. i0

B4.5.6.2-4

Minimum Normalized,

Plastic Bending Curves AISI Thickness > 0. 188 In.

.....!

..: ..I ......." "_;

,.................. O. IZ

i........ i

O. 14

E (inches/inch) Fig.

-

,

'u Alloy

Steel,

,

O. 16

Section B4.5 February 15, Page 109

B4.5.6.

_

Low

Carbon

and

Alloy

Steels-Minimum

Properties

k=2.

140

Room

O

Temperature

k=1.7

IZO k=

1.5

k=

l. Z5

I00

F b (ksl) 80

k=l.O

60

40

20

O. 002

0.004 (inches/inch)

Fig.

B4.5.6.2-5

Minimum Normalized,

Plastic

Bending

Thickness_<

Curves 0. 188

AISI In.

Alloy

Steel,

1976

Sect

ion

B4.5

February Page ] I 0 B4.5.6.2

Low

Carbon

and

Alloy

Steels-Minimum

]5,

Properties

0

<

u_ _Vl

0

e_

_z I

0 _0

0 ,_

0

1976

Section B4.5 February 15, Page Ill

B4.5.6.2

Low

Carbon

and Alloy

Steels-Minimum

Properties

24O

k=2.0

200

k=

1.70

16o k=l.5

k=1.25

120 k=

Fb (ksl)

8O

4O

O. OOZ

O. 004 I

Fig.

B4,

5.6.2-7

Minimum Plastic ' Heat Treated

O. 006 (Inches/inch)

Bending

Curves

AISI

Alloy

Steel,

1.0.

1976

Section February Page 112

B4.5.6.2

Low

Carbon

and

Alloy

Steels-Minimum

•

°

II

II

B4.5 15,

Properties

o

0

;,:1

o

"i_ !!i! I IM

}iii

,d _4 4

iIi: ;!I!

m

!!!! i

I 0

0 0 r,,l

,0 ,,.0

0 ¢,,.I

0 (13

0 ',_

0

1976

Sect ion B4.5 February 15, Page I 13

Graph

to be

furnished

when

available

1976

Sect ion

B4.5

February Page ll4

B4.5.6.2

320

Low

Carbon , J

and,, Alloy

Steels-Minlmum

:I._i T_:T: ..... -ff:-I-:¢

_:_'...._'

" , d_i :_ i _:_ :' :::'_:.:: _ --_--'_.0 '1 I" I _:

_

"i_.i:_! -.:I.: .... i.r__

I

280 ::-!-i:!:':'!!'"/'e_"'l

..........

240

!",',

-]

_

t

?....

" ........

:!lit!it

; il I

200

i]i!::ji

i

_

!i i!!i

Fb (ksi) 160

i _'

S "c_ain

Curve

_.._ k

l_--4

b IZO

8O

It_

I_ t

___Fty 1:! :i.!_

'.... :iii;: ! !_: "I]:::

. : ,..I.:'_ !:l::

7;iir]i!iT_!--_<.,

4O

:i7!

I::l:::i!

:':;!::'

::,

; :!

! ;:

! :

! 1:

,'ii iiiT--::i:i,,: 7![ i:] ...... ......... _:::v::

i

....

IF t IE

= lJ2, O00psiBZ, 000psi = 2-9 x 10 6 psi

;Elongation

:= 18.

5%,

::t-i-:

::

:iiii::

ITT1777T ;:: i:T;II71)"

O. 02

O. 04

O. 06

O. 08

O. 10

O. 12

O. 14 E

(inches/inch)

Fig.

B4.5.6.2-10

1976

Properties

ill•i'_; RooJ_T_ _p_r<,,r _ ; :.... i "-I_ d_ .... _

15,

Minimum Plastic Heat Treated

Bending

Curves

AISI Alloy

Steel,

U

1 25

Section

B4.5

February Page I I5

B4.5.6.2.

Low

...-:_..:-: ,

Carbon

and

.........

Alloy Steels-Minlmum

i ..................

15,

1976

Properties

!

.... _ ....

.......

•

:

.

•

k_Z.

280

240

i:":" 5-:_--- _'_-- .... _ ........ JF" i_i: :_ :i:i :i : "i ':':[" t. ! : : ! zy ": ', "'::-:': :i-÷!!i ..... -":.... :"'!:"': ....... : ...... :":............ ::._.::. :i_i:iT:. " ::-" . :,. ::: ,_::1_.... i

z_x

::.:.: •

I: ........... '" "..

"

•

".

•

•

'

;

:

'

_

i '_ i_--_ _

IP

D

::. '

i"

: .

.

•

k=

1.70

k=

1.50

.:: i

--.......

:_..

,;

• : k=

ZOO

:::i::.:,l -F- T_-.:-lT.r:q-':

'!

l:!

1.25

:., l l:- • k=l.O

160

:_:_!:i:::[ l::!

• ! -i_.:[ -_-'_

_i'i

!!:I-÷:

Fb

_-I

t,

!

....

(kst) [::::iif!i [

,--'_,J

!" ........

120

_!i_ii:!

J Stress-Strain

.:_:_ !::1:

.

:

Curve

..........

./_

:

l!i!!ii!tli!_.ii_.:_":'_?'""_':_'":-..., :__:,. _i_if_!;ii.:_!! _'::: ....:..,._ ..... ,:,......... _...... 8O

l:i::::...i!::.:-_/.++_--"--t-. F'_!':_t! 40

_........

, ......

_:-H----t-/, .......... _

i

.iiFtu

_

:

JFtv:

180,000

163,000

psiPSiI'"':-i:, ''F" i

......... :" +_-"'= 7" I'----T-" --T" .. :. " I:,_:,_A..._pl-r. :-'"-.7...___. =,__,o6 ,,._,i .....

_:. :i.i:_-,..-:-:[_.-..._,-, _..H-..t---t-_.,oo,_,,o_ =,_o,o_.-._--i i • i': :i

: _ ..-::K_:::_:t:: 0. 00Z

......... z ......

:...._[_ :

.......__....":: : "::'

0. 004

0. 006

...."......:"/

0. 008

"...... _..:.J :::: ,, " ""L_ ............. l"i_J 0. 010

0. 012

0. 014

(inches/inch)

Fig.

B4.5.6.2-II

Minimum Plastic Heat Treated

Bending

Curves

AISI Alloy

Steel,

Section B4.5 February 15, Page 116 B4.

5.6.

Z

Low

Carbon

and

Alloy

Steels-Minimum

Properties

k=Z.O

360 _0 tIII

k=l.7

280

i:= k=l.5

Z40 k=

1.25

200

_b

k=l,0

(k.{)

160

120

8O

4O

O. 08

O. I0

(Inches/tnch)

Fig.

B4.5.6.

Z-1Z

Minimum Plastic Heat Treated

Bending

Curves

AISI

Alloy

Steel,

1976

Section February Page l 17

B4.5.6.

2

Low

Carbon

and

Alloy

i.i!"....'.:'::;:.. I I.T7_-:]--F-7-FI Room Temperature

.........

3ZO

Steels-Minimum

Propertie_

"'''I

280

Ftu

= ZOO, 000

psi

_'-

-'""...,., .... .... Fty = 176, 000 psi

.=

7_- _ _-,_9.0 x 10Gpsi"

240

:i ....

Elongation

= 13.

5%

_

".i;....

200

120

l

1

1

I

}

1

1

:: i

i':i.i'!

"!

.... ..... i:i:i--:'-1

....

]

...............

l?F!l_

:.!....i_::..:...i._ ..... • i_ _

160

:

:

(kF_)

•,..

!"

" !" F

.......

80

:i:i!:_iI :....i- i i::r .....

:

i/'li!

.:sl

[

-

.L _..:_:._

40

....-__'- -i-:-;!-?ii_ 7:17i:_ _ii., :i!i.... :;_ TIT T'!"i :i:11 ....

0

0. 002

0. 004

._r.-

0.

O. 008

O. 0l{)

O. 013

((inches/inch)

Fig.

B4.5.6.

Z-13

Minimum Plastic Heat Treated

Bending

Curves

AISI

Alloy

Steel,

B4.5 15,

1976

Section February Page B4.5.6.2

Low

CRrhon

and Alloy Steels-Minlmum

2_

Propertlen

118

B4.5 15,

1976

_-"

Section B4.5 February 15, Page 119

B4.5.5.

3

Heat

Resistant

Alloys-Minimum

Properties

_00

160

IZO

Fb

(ksi) 8O

4O

0 1.0

1.5

k

Fig.

B4.5.5.3-I

=

2.0

ZQc I

Minimum Bending Modulus of Rupture Curves for Symmetrical Sections A-286 Alloy, Heat Treated

1976

Sect ion B4.5 February 15, Page 120

B4.5.5.3

Heat

Z40 :ttit_.._

Resistant

Alloys-M[nlmum

:_[_: ,_[_::;i_R°°r nl;::-i ......

_;-_-_'zoo _*+_ L_{_i

" _

'!

It

........

_

-AIJ-14_ -trTT_ _$_

I

,

Properties

i.iii,_:i_i_i_=_iii._Temperature_. .... ; _-_ i

.......... i

i i

1976

"

" ! ...... _",'-....f-/_ . ! ! .':,_,_'I?L! •" ,

: •,

•....

ttii

i

'160

....._:i!l_ '_ " r_l_llil' ll!I

"_'_ II Fb (ksi)

..... liLI2_ll,

, i

li i Hliiili!iilIillliIIillllll IllllfIIll

i

" ".,

....

"

'. •

lZO .......... :.... •

l!li l

"

; ,

: .................

"

'. ....

',

.

_:

i

•

" " : __.'-

: :..:

.

'_

i:!: !!

..

_

..

i

,

•

i

::; '.,

: '

i_i!

•

I.

•

I

•

Alloy

Bending Sheet

.I

-

•

!

6

k=_

Symmetrical

I

'_

1.5

Minimum

:

_................;.........i ....... ;

.... :

'

-.I........... ! I

1.0

Fig. B4.5.5.3-2

i

_..... _.......

i

8o_-_ ' _

•

_

! ................ '

I I

•

Sections

Z. 0

ZQc I

Modulus Age

of

Rupture

Hardened,

Curves K-Monel

for

Section B4.5 February 15, Page 121

B4.5.5.3

Heat

Re ststant

Alloys

-Minimum

Prope

rtie

s

1ZO

100

Ultimate:

8O

6O

4O

ZO

2.0

1.5

1.0

2Qc I Fig.

B4.5.5.3-3

Minimum Symmetrical Annealed

Bending Modulus of Rupture Sections Monel Alloy-Cold Sheet

Curves Rolled,

for

! 976

Sect ion B4.5 February Page

Graph

to

be

furnished

when

available

122

15p

1976

Section B4.5 February 15, 1976 Page 123

B4, 5.6.3

Corrosion

Resistant

Metals-Minlmum

Properties

200

160 F b

(ksl)

120

(inches/inch)

Fig.

B4.5.6.3-1

Minimum Plastic Heat Treated

Bending

Curves

A-Z86

Alloy,

Section B4.5 February 15, 1976 Page 124

Graph to be furnished when available

Section B4.5 February 15, 1976 Page 125

Graph to be furnished _en available

Section B4.5 February 15, 197 Page 126

_4.5.6.

3

Corrosion

Resistant

Metals-Minimum

Properties

240 k=2.0

200

k:l.7 I ..::

....

i ......

'

k:l,5 I .....

I....:.--

160

k:

l. Z5

k=l.O IZO

8O

4O

O. 01

o. 02

0. 03

0. 04

0. 05

0. 06

Eu

c (inches/inch)

Fig.

B4.5.6.3-4

Minimum K-Monel

Plastic Alloy

Bending Sheet

0. 07

Curves

for

Age

Hardened

Sect ion B4.5 February Page

5.6.

B4.

3

Corrosion

Resistant

Metals-Minimua'n

127

Properties

60 k=Z.O Room

Tempe

rature

k=l.7

50

k=l.5

.....

, .......

j

..

_".........

k=

1.25

(kFs_) k=l.O 30

20

._...... |

I

!

Ftu

= 70,

Fty

= 28,

E

= 26

Elongation

I0

000

psi

000

psi

x 106

psi

= 35%

i!iiiii_ii

i

0 0. 002

0. 004

0. 006

0. 008

E (inches/inch)

Fig.

B4.5.6.3-5

Minimum Plastic Bending Curves Cold Rolled, Annealed Sheet

Monel

Alloy

15,

1976

Section February Page 128

_4.5.6.3 Corrosion

Q

0

0

Resistant

Metals-Minimum

Properties

B4.5 15,

1976

Sect ion B4.5 February 15, Page 129 B4.5.5.4

Titanium-Minimum

Propertle

s

130 t't_

tt_ _4 +H

120

ttt _4

_tt H

t_ trt

Itt

H_ H, tt-!

LH

iL

110

ii

F b

(k,i)

H .H

.+.

Ji .o

./i

100

.ii U II

..+ 1 I

90 ¸

1 i

:

1 l

........i.... .! ....... _......I......... _........ ! !il]t!t_ _! !,ii!ttfl_[!!!]_], |

,

"

•i

|l

. ]J,tt!j,

,

I

,..;.

_.. i _ii_tt_h_!Ii!_Jlt

,

_ th,_¸ ,,_,,_IlJi_ _-,_ii¸

8O

l;tliiti

, iiittIll IIitittI!r.itt! i!!!_,Ii!

t t_

i!! tli _t

f

..,

.

i_;

_

!_!ii!t i_l'!i li II I tt!i!i!! ;;' fiJ 7O 1.0

1.5

k-

Fig.

B4.5.5.4-1

Minimum Symmetrical Titanium

2.0

ZQc I

Bending Modulus of Rupture Sections Commercially

Curves for Pure Annealed

1976

Section February Page 130

B4.5.5.4

Tita_[um-Minimum

Prope

rties

Z30

ZlO

190

170

150

130

II0

k-

Fig.

B4.5.5.4-Z

2.0

1.5

1.0

Minimum Symmetrical

Bending

2Qc

Modulus

Sections

of

Ti-8Mn

Rupture Titanium

Curves Alloy

for

B4.5 15,

1976

Section February Page B4.5.5.4

Titanium-Minimum

Propertie

s

240

220

200

Fb

(ksl) 180

160

140

fli!llttlttftttllli_lltli!f!tttt 'il 120 1.0

1.5

2Qc k

Fig.

B4.5.5.4-3

Minimum Symmetrical

=

I

Bending Modulus of Rupture Curves for Sections Ti-6A1-4V Titanium Alloy

Z°O

131

B4.5 15,

1976

Section February Page

B4° 5.5.4

Titanium-Minimum

Properties

Room

Temperature

Z50 Ftu

= 140,001

Fty

= 130,000

E

= 15.5X

psi I06.psi

= I 2a/o

Elongation

220 Ultimate Fb (ksi)

t,--.

'

•

I

"

: ''

I

",'

t_._..... I.

,

;

i

.

:

,

'

:

I"

l

.

.... _......... _, _oi/ _1_=_...._"

...................

1.0

""

..;......... L......

L..... --.......................

B4.5.5.4-4

I

'

:

,

•

I

;

"

I

:

I

•

"'

,

•

I

!

":, :

•........ ,.. :...'.2 .............

i!

"_ - .........

!

'

• ........

i

..... _...... __.. i ! ,_ •

i .....

,

..

_J .............

1.5 k=_

Fig.

|

"': ...... [........ I'-'" ....I..........!...........

Minimum Symmetrical

2.0 20c I

Bending Modulus of Rupture Curves Sections Ti-4Mn-4A1 Titanium

for Alloy

132

B4.5 15,

1976

Sect

ion

B4.5

February Page B4.5.6.4

TitanlumoMinim_/m

15,

133

Properties

160

k=2.0 Fty

• • "1

140 :

:

!

,i!'

i

!

":',.

_k=l.7 i

]

.... !: ...... _

.......

..... !. 2...i

i

k = 1.5

_...__.i ;

:

:k

!

} :

= 1.25

lZO i.---i

• : '" ": ' "ik = 1.0

,!

100

"'--T"--

I-i-.

Fb

.,: •

(ksi)

,.

..F..i.

Stress

.d .....

,

Strain ......

i

.

......

...i

i.

8O

60

4O

ZO

( (inches/'inch) • Fig,

B4.5.6,4-I

Minimum

Plastic

Titanium

Alloy

Bending

Curves

for

,.,

Curve

Ti-8Mn

1976

Section

B4.5

February 15, Page 136,

B4.5.6.

4

Titanium-Minimum

Properties

i

i

=I.7 I

" " I

'I

,

i

k-

1.5

I

I

_' l

"[

- --. --F---i ..... I---

.....

1Z0

-_--

S_ress-Strain

:

, "

!|

I-"

"

Curve ....

FI"-T "'-'''''_

I

[

. ..................

[

1

I.

........

I

7-5

I O'

I

:

I

'

i.

:

k

i-

•

k

I "

"1' .l

"

'

"

'

l

II

(k.i)

....

Ila I.m,."--""l

l

..1 .......

:..j,

i

_!

[. •

" :" I

•

.

:

160 Fb

! .... iI

i

II

....:

Curve

fo

8O

.I .........

........... 4 ..................... r........ b........... • t _ i i " "'_

0.005

O. 01

0. 015

O. 02

O. 030

1

O. 035

¢ (inches/inch)

Fig. B4.5.6.4-Z

Minimum Alloy

Plastic Bending

Curves

.....

, _

.........

1

I1

1

1

1

1

1

I

O. 025

"--I-'"'"

I:}

I li

' _

..........

I

,

....

I .....

i

i

_ ..

....

,..... ll ......... i!: " ; ' : }

_

4O

J

:,. ....

2

I

_

i

f

_

'

I

, • • ......

}

b

" i

....

"i

O. 04 'u

Ti-8Mn

Titanium

1976

Sect ion B4.5 February 15, Page 135

B4.5.6.4

T_tardurn-Mtnirnum

[H:[

b.'_._l' _

F I,_L_L_.L_

'

•

___ .... _.

Propertiee

II

,_

[

:_

i:

II

[]_ :; :_

_

*

i

,:_: I:::: :_i:_

**

*["

'I"

[ *_

I:

,.

_.............. ............. :. ........ i....... ........ i....... :. ........ i.....

> l

l

o

i [:

::._f. _._-_, 1

llti

_i

ii

l

!ll.

I

_

"

"

ll

"

>

!

_,

i

.

_, :

_,

!_,

/

I

i_

i

•

i

........

1 !

....

_

....

l_

l

_

"1_

"ll'_--!"

l

:

.....

_

_

.......

_

......

"V--;

"

....

_

_4--_

"--41_

'

L)

I

;"

•

tel

::

C_ I_

:_

i ...... _':_

_

_

:

"

.

•

:

J

: "

,

l

._.l."= l

_:

_,

_

_"

_

_,

:

"'_

--t

•

.

_

I

•

I

I

:_'':

I l

: _ :

.....

•

" "l

_1 ,

•

..

_

....

•

:

_

"

I

\ I _

:

"

.

I

....

_

•

i

. :

_

_----_---

:---i

.....

•

r

-

l

7

--

_

"I

:: :

i

•

_

:

_+,_1, ° _ _ ,, Lh_,_, i_,,_,_,_i_,_,_, ....

.......

•

%.

_-._--..i...

•

_--'_

•

_i_ .

,._

•

_

_i

•

"_,

:

! _

.

i_'_

_

I

•

,

"_ I_=_

........

....

l

I

_

:,'-,.

........

--

l

.

e_P

_

I

_

I

.......

....

"

"

"

"

:

l

--"

"

F

.

"

•

--

•

:"" !l/ ....

,_.

l

;

I

.....

i"

'

•

:

"...............

: ;.

•

:

.

, '

.

;

: :. " .:

,,---,_

....

'. 'l

_

'

_

.

.

_.

I--_l

il

m "'!

i :

_i_l,_,_l

•

"

,i

I

0

_Q

&

'". ::I: ..... i !_71.....-ii I L.........: ....i I

I

I

I

.......

....t.... r ............... "...................

I"

:,....... _; _ ........ !..... !....ill i

....

I ......

o d

N

o oO --_

0

0

_

_

0 N _-_

o O

1976

Section

B4.5

February Page 136 B4. 5.6.4

Titanium-Minlmum

15,

1976

Properties

.240

200

160 Fb (kei) J

120

8O

4O

O. Ol

O. 02

O. 03

O. 04

O. 05

t (inches/inch) Fig.

B4.5.6.4-4

Minimum Titanium

Plastic Alloy

O. 06 _u

Bending

Curves

Ti-6A1-4V

Section February Page

Graph

to be

furnished

when

avialable

137

B4.5 15,

1976

Section February Page

B4.5.6.4

Titanlum-Minimum

138

Proportioo

280 _=2.0

Z40 k=l.7

k=l.5 200

k= 160

k=l.0

120

8O

4O

Fty

= 130, 15.5 000 x106psipsi

0 O. Ol

O. 02

O. 03

O. 04

I (inches/inch)

Fig.

B4.5.6.4-6

Minimum Titanium

Plastic Alloy

Bending

Curves

for Ti-4MnI_4AI

1.25

B4.5 15,

1976

Section February Page 139

Graph

to

be

furnished

when

available

B4.5 15,

1976

Section B4.5 February 15, 1976 Page 140 _4.5.5.5

Alumlnum-Minlmum

Propertle

•

120

II0

150

90.

(kst) 80

70

60

50. 1.0

1.5

2.0

k = __2Qc I Fig.

B4.5.5.5-2

Minimum Symmetrical Forgings,

Bending Modulus of Rupture Curves for Sections 2014-T6 Aluminum Alloy (Transverse) Thickness <4 In.

Sect ion B4.5 February Page 141

B4.5,

5.5

Ahlmintlm-Minimum

Elongation

: Iggo

•

:

1976

Propertiill

I tID_ttNttt!tttt

15,

_1{

_

i

.....

,

,........ :......... ......,..._....,,,_...,.,....:,.. .... .

Fb

(kst)

i i

/,

I

80 ......... I ..... l i ............ _.'"'IT._ ........,.,.;. •,,,..i ...... . i ffflTfttttltfftlttt,t,l,_ ttttt+t h lf-llfhlTitpltttittfliTi_IflIti!+71ti_ ir rt +t_

_tt

ftIHHIitt_t titlt:1i tltItlittll!iltltltttltt!llf_411}i I i!" + i_tttlT: Ii-"i'i'_:"i_'+'+'"tt+ti'_i_t+_iiti<,

!i 7t

i_tltii!

6ol+i+l!lll,,+ll+ll,ll ++_ l+,,Isl. tlilttllt+t_ll+ttti+t+ lftlll:++tll+!li_tll:rlill tll+tllt+_+++tll +_ + .l- ' I+I +iI_.¥._, llil_+: <, _!l_iil.i + ! I+' :' _ i , _ _++i,-+ I+I+H++I+II++ll,++II+ +l++i_+_ +I-i++ +++++_+++ +,.

40

_+i

I:

:,ii+

+ ++_++iH+++ ++,++++i++++l++++i+ +!i +!+ +!++i+ ittlE_i

H t[t'HriltlltH_+l t! t+i ttl+l+tif_f+tf

1.0

_llli_llt+!tl-+ittI_ !t/ftltt..fii-I +Hl+f

I+ it ++f-i+Til[f: Htlt-ir

1.5

2.0

ZQc I

Fig.

B4.5.5.5-3

Minimum

Bending

Modulus

of Rupture

Curves'for

Symmetrical Sections Z0Z4-T3 Alloy Sheet Heat Treated. Thickness _ 0.250 In.

& Plate-

Section B4.5 February 15, 1976 Page 142

B4.5.5.5

Aluminum-lvtlnimu_n

Properties

140

IZ0

100

Fb (ksi) 8O

6O

40, 1.0

1.5 km'-'-

Fig.

B4.5.5.5-4

2.0

2Go I

Minimum Bending Modulus 0f Rupture Curves Symmetrical Sections 2024-T3 _ T4 Aluminum Alloy Sheet & Plate - Heat Treated. Thickness 0.50 In.

for

Section February Page 143 B4.5.5.5

Aluminum-Minimum

Properttes

110

I00

9O Fb

(kit)

8O

70

6O

5O

40 1.0 k=_ Fig.

B4.5.5.5-5

Minimum

Bending

1.5 2Qc

2.0

I Modulus

of

Rupture

Curves

• Symmetrical Sections ZOZ4-T3 Aluminum Alloy Clad Sheet & Plate - 1teat Treated. Thickness 0. 010 to 0.06Z In.

for

B4.5 15,

1976

Section B4.5 February 15, Page 144 t

B4.5.5.5

Aluminum-Minimum

Properties

120

1o0

8O

"!

[

|

.

[

.

: :

. i

:

:

.

•

. . •

i

:

: ....

|..-

60" ,

. .

[ •..1... :...1..

:1

i; :.: _.: : I:_: ; :__:___ : Ii...... ............i i::::i !

4_

1.,.5

k = ZQ....._c I Fig.

B4.5.5,

5-6

Minimum

Bending

Symmetrical C_ad Sheet 0.25

to

0.50

Sections & Plate In.

Modulus

of

Rupture

Curves

Z0E4-T4 Aluminum Alloy - Heat Treated. Thickness

for

1976

Section February Page 145 B4.5.5.5

Aluminum-Minlmum

Prope

rtle s

II0

I00

9O Fb

(ksl)

8O

70

6O

5O

1.0

1.5

2.0

ZQc

k =T Fig.

B4.5.5.5-7

Minimum

Bending

Modulus

of Rupture

Curves

Symmetrical Sections ZOZ4-T6 Aluminum Alloy Clad Sheet - Heat Treated & Aged. Thickness < 0. 064 In.

for

B4.5 15,

1976

Sect ion B4.5 February 15, 1976 Page ]46 Alumlnum-Minlmum

Prol_rtle

m

IZO

110

100

9O

80.

7O

80

50 1.0

k=-Fig.

B4.5,

5, 5-8

2.0

1.5

Minimum Symmetrical Clad Sheet Thickness

Bending

ZOc l-

Modulus

of Rupture

Curves

Sections 2DZ4-T81 A1uminim - Heat Treated, Cold Worked < 0. 064 In.

for

Alloy & Aged

Sect ion February Page 147

B4.5.5.5

Aluminum-

Minimum

Prope

rtie

s

90

80

70

60 Fb (ksi)

5O

4O

30 1.0

1.5

k

Fig.

B4.

5.5.5-9

Minimum

Bending

Symmetrical Sheet - Heat

Sections Treated

=

2.0

2Qc I

Modulus 6061-T6 & Aged.

of

Rupture Aluminum Thickness

Curves

for

Alloy >_0.020

In.

B4.5 15,

1976

Sect ion B4.5 February Page

B4. 5.5.5

Aluminum-Minimum

148

Properties

150

140

130

IZO

110

i!-_ Room

............ Tem, perature-4

.....

.......

°

4

' : i I: l i °.....t

9O •!........

:""

8O,

.....

i

7o

Fig.

B4.

5. 5. 5-10

Minimum Bending Modulus of Rupture Curves for Symmetrical Sections 7075-T6 Aluminum Alloy Bare Sheet 8_ Plate. Thickness <. 039 In.

:'"]

15,

1976

Section B4.5 February 15, 1976 Page 149 B4.5.5.

S

Aluminum-Mtnimu_n

Propertie=

130

illIl!!II

!!!!!!i!! t,ttff,tt

,°,_,*°,

ttlttttlt

t_IHHII t"t_it'

!IN! N!I}II !!!!!!!!_ !!!!!!t!_

!!!!!!!!! !!!!!!!!i Fb

tttlrHlt

(kat)

Room :]iiii_ii

_!!!!!!!! Ill!!!!! ;ii_iiiii

:_!!!!!!!

[![!![!!

1,0

Z.O

1,.5

k

=

2Qc

1" Fig. B4.5.5.5-t1

Minimum

Bending

Symmetrical Clad

Sheet

Sections &

Plate.

Modulus

of

Rupture

7075-T6

Aluminum

Thickness

_< .039

Curves Alloy In.

for

Section February Page 150

B4.5.

5.5

Aluminum-Minimum

Properties

140

120

I00

Fb (ksi) 80

6O 1.0

1.5

k=--

Fig.

B4.5.5.5-12

Minimnrn Symmetrical Extrusions.

Bending

2.0

2Qc ! Modulus

of Rupture

Sections 7{)75-T6 Thickness < 0.25

Aluminum in.

Curves Alloy

for

B4.5 15,

1976

Section B4.5 February 15, 1976 Page 151

B4.5.5.5

Aluminum-Minilnum

Properties

;Y t_

140 i:::

i_ll

_:It i;

, ii

_F, ',,_._ Ultimate

_

I ':j,,_r'': _'.,

if::

120 ....

Fb (ksi)

',

i00

:,, ,It,

i

_

_:'tit'.

t::t

._ t_t;

I::I

_;t:I:_iI_

i! :,i ii }iil: l :i!

:::

_

:;HII:

lll_

80

:i;!

t1:itt_r;'ltt_'ltt I I i; ......

i]i: i!t{I',It ;r_

6O

:i][

k

B4.5.5.5-13

: :!1 _ ':1,

It:{

f!ilt]/i!ti_Jt

ltt_

Temperature " ; ii _ ;!i

t : I

'{

ttlt

:_ t:l

2.0

1.5

1.0

Fig.

Room

!: J_tl'.!fli_t'_

Minimum

Bending

for Symmetrical Alloy

Die

ZQc =-I

Modulus Sections

Forgings.

of Rupture 7075-T6

Thickness

Curves

Aluminum < 3 in.

Section B4.5 February 15, 1976 Page 152

]_4.

5.5.5

Aluminum-Minimum

Properties I

i

i

Fb (ksi)

100

8O

6O '1° 0

Fig.

B4.5.5.5-14

1.5

Minimum

Bending

Symmetrical Hand Forgings

Sections Area

2.0

Modulus _

7075-T6 |6 In.

of

Rupture Aluminum

Curves Alloy

for

Sect ion February Page 153 B4.5.5.5

Aluminum-Minimum

Prope

rtie s

140

120

I00

8O

Fb (ksi)

6O

4O

2O

0 1.0

1.5

k

Fig.

B4.5.5.5-15

=--

2.0 2Oc I

Minimum Bending Modulus of Rupture Curves for Symmetrical Sections 7079-T6 Aluminum Alloy Die

Forgings

(Transverse).

Thickness

<6.0

In.

B4.5 15,

1976

Section B4°5 February 15, 1976 Page

B4.5.5.5

Aluminum-Minimum

PrOperties

160

140

120

(ksi)

I00

8O

60 1.0

1.5 k---

Fig.

B4.5.5.5=16

Minimum

Z. 0

ZQc I

Bending

Modulus

Symmetrical

Sections

Die

(Longitudinal)

Forgings

of Rupture

7079-T6

Aluminum Thickness

Curves

for

Alloy < 6.0

in.

154

Section B4.5 February 15, Page ] 55

B4.5.5.

5

Aluminum-Minimum

Properties

1.20

110

100

F b

(ksl) 90

8O

7O

6O

Z_c I

Fig.

B4.5.5.5-17

Minimum

Bending

Symmetrical Hand <6.0

Forgings In.

Modulus

Sections (Short

bf Rupture

7079-T6 Transverse)

Curves

Aluminum Thickness

Alloy

for

i 976

Section B4.5 February 15, Page 1 56

B4.5.5.5

Aluminum-Minimum

Properties

tu 130

ty = 58, psi psi 10.3000 x106

II0

m

(ks_) 9O

7O

)erature!

5O 1.0

1.5

k=--

Fig.

B4.5.5.5-18

Minimum Symmetrical Hand

Z. 0

ZQc I

Bending Modulus o£ Rupture Curves for Sections 7079-T6 Aluminum Alloy

Forgings-(Long

Transverse)

Thickness

<_ 6 in.

1976

Section B4.5 February 15, 1976 Page 157 B4.5.5.5

Aluminum

- Minimum

Properties

140

120

100

8O F b (ksi)

6O

4O

2O

0 1.0

1.5

k--_

Fig.

B4.5.5.5-19

Minimum

Bending

2.0

2Oc I

Modulus

of Rupture

Curves

for

Symmetrical Sections 7079-T6 Aluminum Alloy Hand Forgings - (Longitudinal}. Thickness < 6 In.

Section February Page

B4.5.6.5

Aluminum-Minimum

B4.5 15,

158

Properties

100

k=2.0

8O

k=l.7 k=l.5 k=

1.25

60 k=l.0

4O

2O

fi , ifllHi _" I I IilU WIHHHII _ | 0

O. OO_

O. 004

O. 006

O. 008

Oo 010

(inches/inch)

Fig.

B4.5.6.5-I

Minimum Aluminum

Plastic Bending Curves 2014-T6 Alloy Extrusions. Thickness _< . 499 in.

1976

Sect ion B4.5 February 15, 1976 Page 159

Graph te be furni,'_hed _11_en available

Section B4.5 February Page 160

B4.5.6.5

Aluminum-Minimum

Properties

100

8O

6O

ty c

= 52, 000 psi 10.5 x 106 psi

4O

_ ;!.hhi,WIHl lltlgmlglll n n nlq"

2O

Fig.

B4.5.6.5-3

Minimum Aluminum

Plastic Alloy

Bending Die

Forgings.

Curves

2014-T6 Thickness

< 4 in.

15,

1976

Section February Page 161

B4.5.6.5

•

120

Aluminum-Minimum

Properties

: ...:l:i.:J::::::'.i:::ii.i:: .::I.: .: .

.:

:

i!i_], Room

:

" I .!!:.I :!..,

.

"

-,

: ..T.: :. :: • r-_-_Tr-[--..--

:

.......

...::':i":l::!:i:i_l::::;i_il:::i!i_:_l:::.: ....... .,..... :.i.i ::::!:i::ii_! :;i!li:ii• '_ .....: :::::. 1,......

Temperature,

,..::.:...... ]:::,.::.:

.,

.,: .... ::.. ..... ... : -.: :.....

k=

Z.O

-.-__i-:_,:___: :._i.!_ :_-:::.:I:::_': .-, ..,_::.:_,_,-_,_:-_,_'_:._ :: ::_ ii_:-i ii _..:,...::,.. ._ .,::., .... _............. .i_.: ...._.: 100

.i.I :_:,:_.l:ii;i:: .::_:ii_:::iiii!i:::_-::_i:.i_i ii!i!!_!li::::!ili_.i _ l i:i:::_r:ii:! : i • :-_ k :_:: _ ]-:iil:.!!.,,'lllfllllllilfllllflllllllllilliimw',',,.".',"--"-t_!:_:l_i_ :_i:: __!::_:'_]

_.

k=l.5

8O

-/_,: ::, :::::t:i:: i:_-_l:_ ::___---

..

".;

e

--

] : :'_

_ _

'_

'"

!"l]:

""

"

--'.I .....

_....

'

.'.

.

- '.

'

.'.

.

.

k=

I.Z5

Fb

tt:-__iiii!

(ksl)

•

.

[:

i!..i_7:i_:$;:_!_;ii:_i!St_e_s-_':ra_n Cu_i

:..;

....

I .....

_,__.i.___

_: ....

6O

4O

ZO

:I ::!:: _ _: ! j]'..i !i :_i

-=10.5X 106 [ong&t[on = 10%

!ii. i!:,i: P si ;;i!_ ::!!i:::ill

r!!_::i i:!:[:

'::i:i":lill

•

0. 01

0.02

Ec

_:_:_: ::1 !.

0. 03

(tnche

Fig.

B4.5.6.

5-4

;;: ::::::1::::!:: i!': i: o!i --_ _i-i-!i-ii:r-.li.:,:::i:_ !li--::_i: ilii: :-I:-ii-I :i_!-: :'t __.. :_:.t":: ' I, .it.i: Ii::.:: I : _:1.: , _:1 :i I.:[.! : ".."Ti': ;: _':: ;i;: i: I [ .i t 'i! !:

Minimum

Plastic

Aluminum

Alloy

0. 04

0. 06

s linch)

Bending Die

0. 05

Curves

Forgings.

0. 07

e

2014-T6 Thickness

_< 4 in.

k=l.0

B4.5 15,

1976

Sect ion B14.5 February Page 162

B4.5.6.5

Alumirlum-

Minimum

15,

Properties

8O

k=2.

k=l.7 7O

k=l.5

6O k=1.25

5O k=l.O

4O

3O

2O

10

0. 002

0, 004

O. 006

O. 008

(inchesinch) Fig.

B4.5.6.5-5

Minimum Aluminum Thickness

Plastic Bending Curves 2024-T3 Alloy Sheet & Plate - Heat Treated. /_ 0. 250 Inches

O. 010

O

1976

Section February Page

B4.5.6.5

Aluminum-Minimum

B4.5 15,

163

Properties

un o

t,-

II

II

u'_

II

N

O

II

II

I ¢,4

o_ ¢fl

4)

I

N o

uu_m _

.!

M

4

0

A

v

_

1976

Section B4.5 February 15, Page 164

Graph

to be

furnished

when

available

1976

Section B4.5 February 15, Page 165

B4.5.6.5

Aluminum-Minimum

Properties

120

:

:

:

I

i

I

'i'" i

"1 "'""

:

-:.: -i...:...: ...... ..... ' •..... : i :

[

:

•

]

,]

I

.]

io0 : .

,

....._i,,;Ifilllilll!l!llllllniUn"_"':I

I

t

• ,,:

:

8O

'

:

60

','

'

n

,m

k=l.5

e ........

J

i

I!

]

........ I.... i..... _......

i

•

,

i •

: ......

:

: •

:

": ,

I

k=

m

i. i

"

I ,,

I 1 : .... • .... "............... -" ;

i :

:

I

: :.

I k=l.0

I

:

.... i..... t"

:

:

k=l.7

i

"lllll_--"

:

I

• :,

:

I :

_

,

:

,

:

4O

f .... ....

i

:

"_

• --

• ,.......I..... ii :

:

I

, Fb (ksl)

k=2.0

i

,,:!:i,-.....i........!

f!:

..... i .... i l

;

I I ....

:

• [ .......

i

L..........

i .............

:

i "

"

,[

.

;

>

(',, :I'',:'

'':

I . . .; l

.....

:

'' i

,-

'

"'i"

:

I

" i .......

J

]'t', '

_"','"'";

l.'t..

;'_.

C(;':

!:"_:

:

:,',_

:

Z0

.................

,f .

.4.. .......

:

."

in

."

t ..................

I

i.... • 0. 02

•

: .......

•

n

•

B4.5.6.5-8

"

i .......

I

:"

I]

.. ......

t ......

:

:

I

:

!

::

ill"

t ........

} ...................

i

.

I

.".....

.||

'

,....... I.....i........ :.... I ....l......i........ :......... 0. 04

0. 06 E

Fig.

]"

Minimum

0. 08

0. i0

0. 12

(inches/inch)

Plastic

Aluminum

Alloy-Heat

Thickness

<

0.50

Bending

Eu

Curves

Treated-Sheet Inches

for

2024-T3 _

Plate.

&

T4

1.Z5

1976

Section B4.5 February 15, Page 166

]B4. 5.6. 5

Aluminum-Minimum

Properties

7O k=Z.O d_h

=1.7

6o {ii_i_H _ H_

IHil

_:': k=l.5

_0 _

40

k=1.25

k=l.O

!iI_

!Hi

ii!i _o i!ii _!ii"_i_!_ '_ '_'_ zo if!!: ,,!!ii : [!

4 Ir

!i! _ ir_i: _

!

!t:

ii{!i _4_;',

I,_.i!

o

:_i!ii_ __!i_ O. 002

0.004

O. 006

0.008

0.010

{inches/inch)

Fig.

B4.5.6.5-9

Minimum Aluminum Thickness

Plastic Bending Curves 2024-T3 Alloy Clad Sheet g_ Plate - Heat 0. 010 to 0.06Z in.

Treated.

1976

Section B4.5 February 15, Page 167

B4.

5.6.

5

Aluminum-Minimum

0

Properties

0

I|

II

o,1 0

I_ _ ,..-t

L_

*._

.

U

•_ •

d

4

0

.Q.,_

. _

_

1976

Section

B4.5

February Page 168

B4.

7O

5. 6. 5

Ahminum-Minirnurn

15,

Properties

k=2.0

k=l.7 60 k=l.5

5O

k=

Fb

(ksi) k=l.O 4O

30

2O

10

O. 010

Fig.

1.25

1976

__=

Sect

ion

February Page

B4.5.6.

5

Aluminum-Minimum

Pr6perties

120

k=2.0

100

k=l.7

k=l.5

8O k=

1.25

=i.0

6O

4O

2O

:

O. OZ

O. 04

•

O. 06

.

,"

:..

O. 08

O. I0

E (Inches/inch)

Fig.

B4. 5.6. 5-1Z

Minimum

Plastic

Aluminum

Alloy

Treated

Thickness

Bending Clad

Curves

Sheet

0.25

and

2024-T4 Plate

to 0. 50 in.

- Heat

169

B4.5 15,

1976

Section

B4,5

February Page

B4.5.6.5

Aluminum-Minimum

170

Properties

k-2.0

80 • "i :::'": !

:

:

I

"-!:'"

_

.....

|

• " ; " " .,l'tY.

70

=I.7

I':::i

"

k=l.5 60

50

k=l.O

% 4O

!._::]:._::i: ..... ii;/; ._ .... ""

"

i'i. ''':

30

...... i

,

;

I ...... '

.:..

•

"

i"" .....

J

, '

i/I

" ". ......

........: '.

::.......""

!

• ......

•

/

,

; ..... ' _:

"_

"'

........

:

::"

.i

• •

"|!'"!.i.":':':'_

i :...:.:.. "

i

:"

.

•

• ..i

:.":i: _,, i [/[[i ,/' / :,, i,-_._', i,-:-_,_ . : ...i/... • ':' i ' • /___-'/i::!_::!::2]] _-0 !:................. /p.......... ;, -....... , ........... .----.----; /

:

:

r

i

/

I_'

_

i:: :/_ ,: ,:/ ,of..... .....,/i: ....,,: i" =,_; ': ........

:

l

........

:i/-_ i I: :,l :'!

]!/"

0

___

!:

; "

••

". , . / *

.

:; l/...

: .......

;I

........

O. 002

.

i ::

. _i--__

'

.

/",7,::.'.,'.--

'!..... i'-i........... r:-_l :i:::_:i_i_::i_i_' .... ........ ':_ ...... !'

'" : .....

'

_ .:

I':'"..

..........

"":"

L- :....L

:_"-T!

?.......

/__]'

O. 004

,'

;

O. 006

: :"

"'. ....

;

"'_': .....

' "::

I.:

."

.:.:

:

I':':'"",':-:::.-:-:

O. 008

O. 010

• (inches/inch)

Fig.

B4.5.6.5-13

Minimum

Plastic

Alloy Clad Sheet < 0. 054 Inches

Bending - Heat

Curves Treated

2024-T6 & Aged.

Aluminum Thickness

15,

1976

Section February Page

B4. 5.6. 5

Aluminum-Minimum

Properties

120 V*+++4+

iRoom

Temperature

k=Z.O

ii0

i00

"U!!_ k=l.7

9O

_titt

k=l.5 8O

k=

1.25

7O F b

(ksi)

k=l.0

6O

50

40

30

20

I0 0

(tnches/_nch)

Fig.

B4. 5.6.5-14

Minimum Aluminum Aged

Plastic Bending Curves Alloy Clad Sheet-Heat

Thickness

< 0. 064 in.

'U

Z024-T6 Treated and

171

B4.5 15,

1976

Section February Page

Graph

to be

furnished

when

available

172

B4.5 15,

1976

Section B4.5 February 15, Page 173

B4.5.6.5

Aluminum-Minimum

Properties

120 k=2.0

k=l.7

I00

k=l.5

80 k= Fb

(kst) k=l.0 60

40

Ftu _ty

= 62, 000 psi = 54, 000 psi 9. 5 x 106 psi

Elongation

ZO

= 5%

e (inches/inch)

Fig.

B4.

5.6.5-16

Minimum Aluminum Worked

Plastic Alloy and

Aged

eu

Bending Curves Clad Sheet-Heat Thickness

2024-T81 Treat, Cold

< 0. 064

in.

1.28

1976

Sect ion

B4.5

February 15, Page 17.4

B4.5.6.5

Alurninum-Minimum

Properties

k=2.0

60 k=l. 7

k= 1.5

50

k= 1.25 40

N Stress-Strain

k=l.O

Curve

30

fo Curve

10

0

Room Temperature

0.002

0.004

0.006

0.008

0.010

a (inches/inch)

Fig.

B4.5.6.5-17

Minimum Plastic Alloy Sheet Heat

Bending Treated

Curves & Aged.

6061-T6 Aluminum Thickness >_ 0.0Z0

in.

1976

Section

B4.5

February Page

B4.5.6.

/'.,'

i

5

;

Aluminum-Minimum

i

•

•

.;

i:.':, i....L.i..:...i, :

,

.

•

I '

......

I

i...! .......i...

..... "......... [ .............. ....... i.. :...I ....

'... i .J.... :

.

I • ..;...

T "

t'

I......;

| ....

:

I .

.

:

i

:

I

....

i I I

.

i

, :;,

k=

r"

•

:

1.7

I {

.....

......... .i ....... !.......

"--I- .... r ...... I

-.

:

!

'| ............. ....

i

!

.! i

! •

;''" k=2.0

I

]. :

i

i

i......

.,

:k2. i l...

.... i ........ !.......

!

....

[.-.-...i...:: .......... :...',

L-If ::!;::

175

Properties

i......... ,.:..L._ .......i....... -_i--i--.-l: ......... 8o

15,

!

I

k=

1.5

k=

1.25

k=

1.0

60 •.! ......

I.........

l

i t

i ....

.....ii_ i

5O

: i 1....... -_!-_"!

'

.

Fb (ksi)

!.i ....

il

../i'i_, i: St res s-Strain

i ......... t

,

Cu¥_'_----_*......'

! ""i"

I

40 .....

"-30] ....

,

i.i

i.

i

_!i_;

:

" •

-'

'''

_'

: :, :

'

: |:

f

!

•

"I , ...L ' ....

I

';............. ! :

!

I

;' .....

:.

Curve

:

:

:

:.......... ;

l .............. , I

: " t-_............. i......i-..-i..._

: ............ !

i.-.q .... I ' I

,

": ............. t

i!!'_l.!i i!!i

i'" ":"

. " ........

.! .i.........1..' : t......... • IIt_ ....

•i :: L_ :. , I_-F-

"!1......

i lfo

.

_, ........ .' , ....... ., ' J • ....... t,....... .......... ...... ................. .......... 1...i......... 1 i

.... l i',..1..,,, /i

|.

i

..........! ......

_::

'! i! E

=

9

•

9 xl0 6

E:],on.gation

P

'

/., !I ;:iii'

•

I

,t

:] .. I : i _:_ :ii

:i

,

.i. :i:l : .

!: _i! i!i

si l'::;;:i _! ::!i!::i::,:il :!::ii ! i:ilii ]!'. I [!i 177|ii i!ii _: !I

= [40/0

"'t' • "i ...... I ' 't ......1'"I J...

!, ;ii :

;

,,

.' ii! i::

:i i:i!i:! i ]

i

I

i ..I.......

!

lo I..-.t!--:---t.-_-!...!..+u-_--.:-..-+....l...+ ......... +-,-.!-:-!....,: .....:.1-:...... !----:-.-.1--.-I.-!-+ ........ l.--._-i........ 1

.tl.-.i

.. _.._..,.... _............ I.. .... , ....,..., ....' _.i...'._.'

0. 01

0. 0Z

0.03

0. 04

0. 05

0. 06

. (Inches/Inch) Fig.

B4. 5.6. 5-18

Minimum Aluminum Thickness

Plastic

Bending

Alloy Sheet > 0. 020 in.

0. 07

0. 08 ' u

Curves

- Heat

6061-T6

Treated

_ Aged

1976

Sect ion B4.5 February 15, Page ! 76

B4.5.6.5

Aluminum-Minimum

Properties

100

k=2.0 k=l.7 k=l.5

8O k=

l. Z5

k=l.0 6O

Fb. (kst) 4O

2O

0 0

0.002

0. 004

0.006

O. 008

O. 010

(inches/inch) Fig. B4.5.6.5-19

Minimum Plastic Aluminum Alloy <_. 039 in.

Bending Curves 7075-T6 Bare Sheet and Plate. Thickness

1976

Section February Page

B4. 5. 6. 5

Aluminum-Minimum

B4.5 15,

177

Properties

!i,t;

F-UT-7 ....

k=2.0

_,.

140

I

........... ,

120

71

i'ii'i;

i

'

?i

,I -i.....

±_. ,

.......

I111

k=l.7

i;

.,..,

k=l,5

Ii,;;

100

• •

7

:t!

Fb (ksi)

_:.

ii_ii ri.t "

.....

'.i!.!2

1.25

k=

1.0

Ii:

till!!il_i_i! _ii_Curve _iil_ii_ 'stress-Strain

8O

I ii ii!ii !

lil

_

_-:' .2. i ...........

._t_

;i:illlI I [.2-

! i!:i

"'71:7::1:::rf o Curve

.l,,,t '11

6O

!:?:!

' !i1 I till

:..ill t

4F+

Tr"

pr.

......

::2' 2,ii!

:::

.....

;i

ii__i_ :ill ii! iii':iiili:: _tu = 7_, 000 psi +i_iiii2i..i!2i _tr = 65,000 psi siii!iiiiiiitii_L.

4O

Elongation

= 7%

li!ii/!

*:!li]i / ;;,

2O

_r

I i1!I1! Itt_lil,i

_ l!f Ill

i

i!,ii!1, ,t,,ti!,,

ill i!lttil_i

llt,ll=lt 'i_:!llll

0. 03 (inche

0. 04

0. 05

s/inch)

If U

Fig.

k=

B4. 5.6.5-20

Minimum Aluminum < . 039 in.

Plastic Alloy

Bending Bare

Sheet

Curves & Plate

7075-T6 Thickness

1976

Section B4.5 February 15, 1976 Page 178

B4.5.6.5

Aluminum-Minimum

Properties

I00

k=2.0

k=l.7

8O

k=l.5 k=I.25 k--l.0

6O Fb

(kst) 4O

2O

0

0.002

0. 004

0. 006

Oo 008

O. 010

, (inches/inch)

Fig.

B4.

5. 6. 5-21

Minimum Aluminum 0. 039

Plastic Alloy in.

Bending Curves 7075-T6 Clad Sheet & Plate. Thickness

Sect ion B4.5 February Page 179

B4.5.6.5

Aluminum-Minimum

15,

Properties

140 k=2.0

120

....

l! i ::1!

÷t+÷

L2:i

L_

.H

k=l.

7

k=l.

5

!TF ,,; iti: :2"' 1 !U, ,_ :1[? i00 :t"

i

[:

7i_i f¢

80 _L_

k.J

k=

TY/:, .... _ttt

t_

::; :I[2:t H÷t

Fb

(ksi)

!_

H !rT!! t_ L tt*f I 1LLL L;

k=l.

HH

E;Xl 'r

_f

:i ....

r2

?tltl

:fi:!l_:LLiL

20: _}I +4 _4

;rn

_

_+_

1!!! 0

_?t

44_"tt'}-

0.01 E_

(inches/inch) Fig.

1.25

L_

B4.5.6.5-ZZ

Minimum Ailoy

Plastic Clad

Sheet

Bending & Plate

Curves Thickness

7075-T6 < 0.39

Aluminum in.

0

1976

Sect ion B4,5 February 15, Page 180

B4.5.6.5

Aluminum-Minimum

Properties

k=2.0 I00

k=l.

7

k-l.

5

k-l.

25

k=l.

0

8O

60 Fb

(kst) 4O

2O fo Curve

.illlllllll O. 002

0. 004 ,

Fig.

B4. 5. 6. 5-23

0.006

O. 008

(inches/inch)

Minimum Plastic Bending Curves 7075-T6 Aluminum Alloy Extrusions. Thickness <0.25 in.

0.010

1976

Section

B4.5

February Page B4.5.6.5

Aluminum-Minimum

Properties

Fb (ksi)

0.05 (

(inches/inch) Fig.

B4.5.6.5-Z4

Minimum Plastic Aluminum Alloy < 0. Z5 in.

Bending Extrusions.

U

Curves

7075-T6 Thickness

181

15,

1976

Section B4.5 February 15, Page 182

Graph

to be

furnished

when

available

1976 _J

Section B4.5 February 15, Page 183

B4.5.6.5

Aluminum-Minimum

Properties

k=2.0

k:l.7

k=1.5

I00 k=

k=l.O

Fb (ksi)

!

Fig.

B4. 5. 6. 5-26

Minimum Aluminum
(inches/inch)

Plastic Alloy

eu

Bending Die

Curves

Forgings.

7075-T6 Thickness

I.Z5

1976

Sect ion B4.5 February 15, 1976 Page 184

Graph to be furnished

T_hen available

Sect ion B4.5 February Page 185

B4.5.6.5

O

Aluminum-Minimum

I_"

II

II

_

1976

Properties

N

II

15,

O

II

II

N

I

o,-i

i

I

_

vI

U m

(M I

,d ,4 4

O

0

0

0

0

0

0

0

Section B4.5 Februrary |5, Page

B4.5.6.5

1976

186

Properties

Aluminum-Minimum

tO0 k=2.0

k=l.7 80

k=l.5

k= k=l.O 6O

Fb

;tre

(ksl) 40

2o

ti J,.t IIII .. ,,.I !!

I.I ....I..R..,lll..dl .... il! , ; .M .I I, m, .... ,....m, NUlil "II

-" I |III

"'"

II" 'I

|l"il'"

O. 002

O. 004

O. 006

O. 008

. {inchos/inch}

Fig.

B4.

5. 6. 5-29

Minimum Aluminum Thickness

Plastic Alloy <__6.0

Bending Die in.

Curves

Forgings.

7079=T6 {Transverse)

O. 010

1.25

Section February Page

Graph

to be

furnished

_en

available

187

B4.5 15,

1976

Section February Page 188

B4.5.6.5

Aluminum-Minimum

B4.5 15,

1976

Properties

W Room

Temperature k= 2.0

100

k=

1.7

k= 1.5 8O k= I. 25

k= 6O

Stre

s s- Strain

4O

!if!f! 2O

iiuqi. !!qqqlJ,::i. :i immnlUU,Hmnul .,||I,,IIU,I.I. ,lU .11.|11111 |111.11111 |11111111111111 O. 002

O. 004

O. 006

, (inches/inch)

Fig.

B4.

5. 6. 5-31

Minimum Aluminum Thickness

Plastic Bending Curves 7079-T6 Alloy Die Forgings (Longitudinal) _< 6.0 in.

1.0

Sect ion B4.5 February 15, Page

B4.5.6.5

Aluminum-Minimum

;,-

,

'_.

li]" _......

;:

tii;t

;;::tltl;|::/l

;z};

.it

140 •

,1

....... ;2

t ' ....ttt. Ill:

:ii ,1....

_ ,._.

::;it:ill

;_i

_

_

t .....

;i;

."

.;

'J

_r_r

]

,.

;1'

+_.

i

;.

tJ

;

:_.I

,+.,*e',

_w_.k = 2. 0 t:l; t,t, *,¢t'

I : ,! 1

...... ;:i!

!ill

t............. :

i

....

]ii

I

[i;

i

_'t

_ ............

;

-!

:i

,1I, .: _.t I,; t i.,.' i ,!+

-_..... t

Properties

4

if

_i_

4

I

,

'11

.:

;J

;

ii: iii.

i ......

:!i: .i? k = I. 7

120 ,

i_

J .........

t_.,t]ii_f

:':i

t

....

i!tlt,t'.)._fl

t_!t_i_?tl;;llll;;

i

'Ll,_ff'_. _._l_il

.... :.... i:.lii...:..i.:,,,_-,..

.i"_'"

,_"

i

.....

lti: Itt;

i!tii! ! _ $t ! ':i!1t},._]1,;!; ,t_t1;_i I,:, _,!i

.!:

.... ![

.......I!jiL

_1

'_"

........ .

,

_ ....

'.

':It

:?i; ii;:

k = 1. 5

::

._1,_ .....

_

i

I00

!'1[

;i; ;1 :.<:.,

I'_ !,,,::.; '''II i_i

I

,._....

, _ ,:_11! _. ,: i, _: .:!_![

I ....

i,

_

+_

2z_ Lal

. _

! y',:!il. _ ,,l_j ,:i!l,,.,., iiil

t

,, i:11

'_i

..........

k =I. 25

',;It

_

1t; :[i;;

8O

l;'_-

ii_;_[

i

;

]l

,i

_

i

_

_

l}Jl;;!ili![:t

_j_

:i

i

_,

_$;:

[;lIl/_llllill_[il[;:

I_Stress-strain

_l'i

"

li;]

Curve

:t

[

I_

.......

I;i:

!lli

Fb :;'

(ksi)

,l_i !_; r..............

fJ:'lt!_:lt!ti/t1!!/ i1;, , I[

60 !

ii

'l

;

;;

:

i

;

[ _,/,

:flt,!_!l ;;j I:

i

[

:, : [ 'ill

i

J

#'1

,_i;

t

,.

!i,

....;_[

t

J]

':'1'_ ; ]ii

;;]|

;:].._,_,._-:d___:_:.,!_!!!l_lt;t

: ': ;::II:" J

I

[I;

_ ;;I[ f.....

:I ;i:

];:

:

T

:t :

[

]

!

i

I

r:

]

_ 't

::1 l

[

t

t

,l ,,1+

il_:

_N

,

I; 1,,

_urve -

O

It!i]44'l_'_l

i

'

,4,,

+_k=

tF,: i

!:::t

::

_ S

::i

:

rr :

:

;

:Ii! i,

':! t':

4O ;

J

.....

]

;:::iil£4tu =

74,000

.....

,,

Ill !t ,

!|!1!_1 ,

_

....

2O

J]

psi

::!I!

Elongat,on _ "too

i]!l,_iil:::T_._i iil_ t!ttt!:_l_!!t'!r!!

t

,tl

_

,

,,t

:l!tl

!,d,,

Itr

ii:

i

Ii,

,,,, f .... :1,,I,_,,,_1,I .112i::i I

!t .... t.........

I.... t

I _

I ....... ;_1!!

_[l: ;],]]I,H':[],I

0 0.01

],', Ill! lli',l !l:ff!t!t]i!

0.02

lJJJll]f':Ifi:i 0.04

0.03 (inches/inch)

Fig.

189

B4.5.6.5-3Z

Minimum Aluminum Thickness

Plastic Alloy < 6.0

Bending Die in.

Forgings

,]_,t [ii!

ll!;1

0. 05

(

Curves

7079-T6

(Longitudinal)

U

1.0

1976

Sect

ion

February Page 190

Graph

to be

furnished

when

available

B4.5 15,

1976

Sect ion B4.5 February 15, 1976 Page 191

B4.5.6.5

Aluminum-Minimum

Properties

: ;

......

i I

i

,.

) I

"

I .... i n

0 _0

0 rJ) v

_I_ "_ •_

o

ovm

.....

!

!

""

I

II

•

I

l

:

,.

•

:

:

i | :

-,-

:

:

I

:

1...:., I

•

) : 1 •

:. i •

•

I

: •

: ! U_

......... ___. _!.... _ ..... I

I

:

4

i:.......

_.-" _- ...... i ..... :.... i,.- -_ .... -.... '. n

_.

i

__.._........ ........ i.......L

!......L........ i ...... _ : L ...... ,_'g

Sect ion B4.5 February 15, Page

B4.5.6.5

Aluminum-Minimum

1976

192

Properties

100 kin2.0

k=l.7 8O k=l.5

k= k=l.O F b

(kei) 40

20 fo

Curve

0 0. 002

0. 004

0. 006

0. 008

0. 010

(inches/inch)

Fi s.

B4. 5. 6. 5-35

Minimum Aluminum Thickness

Plastic

Bending

Alloy Hand < 6 in.

Curves

Forgings

7079-T6 (Long

Transverse)

1.25

Section B4.5 February 15, Page 193

B4.5.6.5

Aluminum-Minimum

Properties

k-2.0

k=l.7

• k=

1.5

k=

I. Z5

k-

1.0

8O Fb (ksi)

.._..;.._.L.....

6O

:

4O

2O

0.01

0.02

0.03

0.04

0.05

(incLes/inch) _U ¸

Fig.

B4.5.6.5-36

Minimum Aluminum Thickness

Plastic

Bending

Alloy Hand < 6 in.

Curves

Forgings

7079-T6 (Long

Transverse)

1976

Section B4.5 February 15, 1976 Page 194

Graph to be furnished when available

Section

B4.5

February Page

B4.5.6.5

Aluminum-Minimum

Properties

k=2.0

40

O. Ol

0.02

0.03

0.04

0.05

(inches/inch)

Fig.

B4.5.6.5-38

Minimum

Plastic

Aluminum

Alloy

Hand

Thickness

< 6.0

in.

0.06

0. 07 {u

Bending

Curves

Forgings

7079-T6 (Longitudinal)

195

15,

1976

Sect ion B4.5 February 15, Page 196

Graph

to

be

furnished

when

available

1976

Section B4.5 February 15, Page 197

B4.5.5.6

Magnesium-Minin_urn

$;;tl

iiil !!iii::i:i:!iti_:

Properties

_!! !t!:::I!;!ili_:_i

;_

:_!i

ii

:IF i!ij ii_!|ti!!!![:li[ii|iiii"_i_-iJl_oom

.,._:it i]]

Temperature

i! f!_

60

i ......

[t t:

*;4

iiii t_.

*t

"

._ ...........

t,,

._1,

!,

t_f

!i

4 , t

50

:

;: i! i!i

Ult_mate

ilt: ,:i: :i!

_; iii !

r!_t

_ .... _I;_._,_ ,,r ....

! i! I

[!i,'t!il; }iii _ i 'Wl:',ii :: _i!i i.ll _I_'I '_',_I:'_' ; i':'l"i _ r' "lI ;"_

! I

;i !_ _t !t

_[l:! _._;

40 ..... I......

_ .........

_[,I,.!

....

h.,

:!ii

;:St_

Fb (kst)

:::! ....................

1

', il

t

t!22!'i;t,,t

iiii l _! ![ il

;:7, :IL:

iI

30 .......

. :!i!i

i[::i

..,..

,t., k.,

itli: !i,,

t:;:

....

Itl:

!!

i: :

,

:: t :::: :_ :i

.:::

it::iili*

::tl

t:!l

[i :ii ii

ii?itliii............. .... ; ...... ii:_

it

--_TT *_'YtM,.

.......

tl;l

:t}

t_ ;i I t

it,

.:.

:ii! !::T

;i::

iI:_ !t}!liill i!i!tiiil

.t,r

10

,,-,-d

l,

',t!:

20

11!{ 'i, _1 !

ii,

!.i!i::_

fill

Fry ::: E

ii!:;][!i

,t

: 22, 000 psi = 6. 3 x 106 psi = 6_o

Elongation

!iil

,

_,q+Si-_!!!:!: !iil ;ill i,

_1Tr

1.=l;!i:

::! [._ :: i

..... _4=_! :;'l!:Ir

!ii"

11_i '::'

ii[iil:

''' [["

I] ,

i

i

!:[_,][',J_];II;:.:T :',ii:I "ill[ !iri[:jl!l'l'llliit

I_I

I_.H

.ii.., [:llil

1.5

1.0

2.0

ZQc k

=

I

Fig.

B4.5.5.6-1

Minimum

Bending

Modulus

Symmetrical Sections AZ61A Forgings (Longitudinal)

of Rupture Magnesium

Curves Alloy

for

1976

Sect ion February Page 198

B4.5.5.6

Magnesium-Minimum

Properties

;:

$77i

!!!!i!_ !If iii:

_i; !11 ;;!t ,.. !r

if:

lii

1i i_=,; ....

Itli:tt

i!:i! I !1ti t_t!

H!t

]!I_ :,ill ;_

Iiii

!!H

*l+

,++,

!!1 ![!': ',hl ;:;i

',i!

ttfi

;ii

Iic]

tit

Ill',

t_:

Itll

if i[t Z.O

Fi x .

B4.5.5.6-Z

Minimum Symmetrical Alloy Sheet.

Bending

Modulus

Sections 0.016 a

of Rupture

HK31A-O Thickness

for

Magnesium <- 0.250

in.

B4.5 15,

1976

Section February Page

Graph

to be

furnished

when

available

199

B4.5 15,

1976

Section B4.5 February 15, 1976 Page200

Graph to,be furnished when available

Section B4.5 February 15, 1976 Page203

Graph to be furnished when available

Section B4.5 February 15, Page 204

B4.5.6.6

Magnesium-Minimum

Properties

5O

4O

k=l.7 k=l.5 30

k= k=l.O

2O

10

Fig.

B4.5.6.6-3

Minimum Magnesium

Plastic Alloy

Bending Forgings

Curves

AZ61A

(Longitudinal)

1.25

1976

Sect

ion

February Page

B4.5.6.6

Magnesium-Minimum

Properties

:1. • : : ,. ..... .:: ...... , :!.... J .rI.............. . :_'......... :::i ".:'"'t'-'r-. '.'"_ ................... :---" ............. [ ............... ÷..... :" 6O

]

1: !

.

.

,

.I

/

]

Room

: . : ! , :, :[...... , ::I 1 ......... _........ ' ......... _....... _i

'Fe:inperature

!: _ ,:1 [::_ ' '::iFtu

= '8'

!....iLrtv : : i:..,,.:E.

---zz, 000 psi 1 L i I 6 "/'-_-7 .................. =_>.3xtO i>__. i...... ] '

()001'si

I

: !: : _ I _ ..... l : ; " "I ....r!--! .........i-t; .... .---,,_............ i: ::! i ' 'i: ...... 1i .... I J '

'

J L..4 _ -r--.

i

.!..l.

,

I

.

• ".I

: F_.;_l t...' -T-I _:-I • "

=2. oo

:, .:, '.l"j£,

" _.::.ik=

1.7

5o

7!!7-

:

40 -, Fb

(ksi) 3O

2o

10

Fig.

B4.5.6.6-4

Minimum Magnesium

205

Plastic Alloy

Bending Forgings

Curves

AZ61A

(Longitudinal)

k=l.O

B4.5 15,

1976

Section B4.5 February 15, 1976 Page 206

B4.5.6.6

Masnesium-Minimurn

Properties

0

o

v,

A

U

T !

Sect ion B4.5 February 15, Page 207

B4.5.6.6

Magnesium-Minimum

....

Properties

i /...

0

t_

_o AI

h

U

t.J

_J

• "4

_o

,4D I

_4 0

0

0

0 0

0

0

°1-1

v

0

1976

Section February Page

B4.5 15,

1976

208 _J

Graph

to be

furnished

when

available

Section B4.5 February 15, 1976 Page209

Graph to be furnished when available

Section B4.5 February 15, Page

Graph

to be

furnished

when

available

210

1976

Section B4.5 February 15, 1976 Page211

Graph to be furnished when available

Section B4.5 February 15, 1976 Page212

B4.5.6.6

Properties

Magnesium-Minimum

0 0 °

._--pI

4.1 0pI

0

:..-.2!-.:._i- -I-::-i--- " '_:i----".........i.. :...: ..L_i..__.:.i ....... I_.'.

;

!. :....

_-.----T. --_...._

••22

_ _

_"

°

"_'

:

:'.

:

'.'i

i

i:i: i

;

,-....... : _L.: ; ....-:-..

_

lid

• '" :' ..:

_

" :''_ " I •

• . .... :

"

•

; :

.... "

f "':_.: .... i :_." ::_:F

: ...... -_'_ : i 1," 1,"'_ u -c' " ' ' " .... : ::: ....." ....:.: ..;.L.L.'..: '-" _ _ ....... _,_ ...i .......... : ....... ..... . .:"_'- _--' _ ". : _ :.": !- : :

'= ....

""--"

........... li:

• :!

":

_

i

.................. [ "

.

.

t",'

.,.

: '

,_ .........

I

I..... :.-

1

!

I : ,

!

_1_-

o

]

r '

;'

:'_-r

.....

' : ":::

:

'" ...... :"1" "'" : ............ . :. l- .! -- .:'---: ..l " . .:. ..... I. i ........... .-'.

•

.

•

..

.'%.

_ ....--.-:--'"-4 ......•......;..... :'_ ............. "'""_"

._::. _:, ........ . .._o _ • o

.'

•

,. i....:.:._ .L.._..:.. : .. '.._-."

. .:

;

; .......... :

I

"._ .................................. .....

_'':

•

:

q_

o

" "" , ", •

"

_ _

•

• "

" " : ....... " ............................. _..... :.: [ " :" I; ........

i

r

. '

:

.." ..... :: : : 1...

•

" !

...... ;

i

.. i

'

:

._'

4

:......_............. ..... '.

"

'

[. _ . I ...................: :" ................ i • '

"..-.: .............. :

"

......

I

•

, :

:

" ' ,_..." :

,.

, •

: ....................

:

.. •

:............. ...... :...... I.. .. '..-. ., .b.. :- "":. ..... : ...F.:.. ":!:..H....:.. i !':. !...'.! I

.

"....:.. ! ....... i.............. -: ...............

,.'..._...._.....'..,."

:

:

, - :

• :

1 •

i

.... -.'..

• :

" • .'

_4

"1 I ..... : "

:

t ................ : ' :

,

...:

"

•

.,_,

•

:1 ...

:1..... .-...... :............. !:'--:.!......... ! .].:.... • "'_.'_'..-'_E_"" .... :... : ".:....'.. :...;... :.-_i.:. : ..!-.:-}._l_ .-,-I ' ":, : • • '.... • : • ' : ";'." -' ""_-'1

.

i

e4

°

. ":

,

.:

,'"

"1

:_

:

,--I

o

: ,

I

::

, :

-

1

•

o

Section B4.5 February 15, Page 213

B4.5.6.6

Magnesium-Minimum

Properties

Fb (kai)

4O

2O

Fig.

B4.5.6.6-1Z

Minimum Magnesium

Plastic Alloy

Bending Forgings

Curves ZK60A (Longitudinal)

1976

Sect ion B4.5 February Page 214

B4.5.7

B4.

Elastic-Plastic 5.7. I

the

Elastic-Plastic

Elastic

This

section

which

may

bending of

will be

in

or

also

as

Energy

In the

Z

as

e.g.,

be

be

of

the

the

on

deflections

margin

of

safety

M.S.

margin

the

maximum

may

the be

permissible

used

safety

margin of

loads,

part

a

material

in

the

a

the is

some

is

plastic

beyond

propordepen-

cases

Therefore

when involved,

Theory

usage.

structure

analysis

are

Elastic

(no

In

been

common

of

levels

the

this deflec-

an should

be

are

then

the

most

at

yield

as

loads

and/or

of

safety

for

or a

excessive critical

ultimate, stress

Permissible

Load permissible

deflec-

structural

design

may

condition,

the

a permissible by

a trial

and

(4.

calculated

load Equation

5.7.Z-I)

load error

level process.

corresponding (4. for

5.7.

occur. the

Load)

deflection.

obtaining

deflection

-1

(Applied

etc.;

element

deflections

Load

is

used.

levels,

becomes

Permissible

in

of well

Factor)

all

Safety

basis

(Safety

to

no

strains

the

=

where

of

as

if

Energy

Margin

a positive

shown

plastic

Elastic-Plastic

considered

although

may

the

and

but

of

more.

or

have

and

involved.

or

used could

deflection

strain

any

structures

those

effects

stress

elastic

100%

factor

calculating must

as

with

simplicity

only to

an

plastic

much

Discussion

In tions

of

as

such

of

to

limit

designed

amount be

accurate

Deflections

determined.

theories

its

proportional

error

a limiting

analysis

If

the

may is

are the

bending

bending to

Plastic

required.

indeterminate

elastic

due

stresses

both.

be

extension

a material.

bending

readily

similar

plastic

chosen

structures

limit, on

include

an

of

or

to be

statically

Other

theories beyond

strain).

5.7.

to

can

procedures

to

6 will

due

range

plastic

was

Elastic

error

structures

by

well

due 5.

as

range

range,

B4.

plastic

defined

plastic

plastic

Section

Theory.

Theory

stressed

tion

Bending

is

the

energy

or

in

the

solved

Energy

tional

only

elastic

completely

be

extended

B4.

the

determinate

Partially

into

consider

in

stresses

dent

for

Theory

Theory

found

statically

Elastic

Energy

Energy

curves

fiber can

Theory

General

The of

Energy

1976

]5,

5-11)

a maximum

Section February Page Assumptions

Energy load

energy

Plane

sections

4.

The

5. 7.4

plane;

work

is equal

due

to a virtual

to the internal

that deflection. i.e.,

the

strain

is linearly

dis-

cross-section.

effects

deformations

materially structure

B4.

any

deflection

during

remain

ratio

the external

a real

developed

across

Poisson's

i.e.,

through

strain

215

Conditions

is conserved; moving

tributed

.

and

B4.5 15,

are

are

negligible.

of a magnitude

affect the geometric to one another.

so

small

relations

as

to not

of various

parts

of a

Definitions

dA

cross-sectional

c

distance

area

from

of an

the neutral

infinitesimal axis

to the

volume,

extreme

dV.

fibers

of a

cross-section. real

deformation

of an

infinitesimal

volume,

dV,

in the

x-direction. A

real vertical deflection of a virtual load. total (elastic

Cb

dV,

c

-

in the

extreme

plus

of a beam

plastic)

strain

at the point

of application

of an infinitesimal

volume,

x-direction.

fiber

strain

of a cross-section.

bmax

F v m

W

virtual

normal

virtual virtual

bending load.

force

acting

moment

dA.

in a beam

1

internal strain energy equal to a summation virtual forces times their real deflections.

fb v

virtual

unit load.

virtual

bending

stress

on

dA

due

load

to the application

external work deflection.

Q

to a virtual

due

e

W.

equal

on

moving

to m.

through

of a

a real

of internal

1976

Sect ion B4.5 February 15, 1976 Page2_6

B4.5.?. 5

Deflection

Consider

of Statically

the infinitesimal

Determinate

volume

dV

Beams

of Figure

B4.

5.7. 5-1(a)

and

(b) fb

myI

-

(4.5_v 7_5-I)

V

Fv

= stress

>< area

= fb

dA v

= -I

6

Since

dA

i4. 5.7. 5-2)

= c b dx

plane

(4. 5.7. 5-3)

sections

Cb

= Cbma

5

= c

remain

x

plane,

c

(4.

5.7.5-4)

and

bma

By

x

Y---dx c

(4. 5.7.5-5)

definition,

W

= Q A

(4.5. 7.5-6)

=Y

(4.5.7.5-7)

e

Wi Since

energy

is

F v 6 conserved,

W e

= Wi ,

(4.5.

7. 5-8)

QA

= Z Fv 6

(4.5. 7.5-9)

or

Substituting (4.5.7.5-9)

Equations since

Q is

(4.5.7.5-2) equal

to

unity,

and

(4.5.7.5-5)

into

Equation

Section February Page

B4.

5.7.5

Deflection

of

Statically

fAiL

(mY_.T._ dA)((

by

definition,

I

,',A

cb

can

dA

d)

(4.

dx,

C

5.1.

5-10)

=

L

dA,

mob max

:_ #

dx

(4.5.7.5-Ii)

c

(4. 5. 8. 5-ii) be

217

Yc dx)

max

I

.So Equation

(Gont'

1976

mob

y

y2

/e.

Beams

maxb

2

-/ So But,

Determinate

B4.5 15,

determined

can

now

from

be

solved

a plastic

graphically

bending

curve

to findA. for

the applicable

max

I

material rupture) bending the

as shown in Figure B4.5._.5-i(c). Enter the F b (modulus of scale with Mc/I and move horizontally across to the plastic curve for the specific cross-section; this intersection locates

corresponding

¢b

•

varylng

.

cross-sectlon,

on

max

the

c may

be

c (strain)

scale.

For

beams

with

a variable. _,

L

"-

_X

(a)

Beam .I:/

(d)

Real

Load

(e)

Real

Moment

Diagram

Cross-Section dz

Fv

I I I

(b)

DifferentiaI

Volume,

dV

Figure

B4.5.7.5-I

Diagram

Section B4.5 February 15, 1976 Page 218

B4.5.7.5

Deflection

of Statically

Determinate

Beams

q

(Cont'

I lb. !

(f)

Virtual Load (Real Deflection

Diagram Shown)

M_.X I

%

l,

• (in./in.

(c)

Plastic Beam

)

Bending Diagram Cross-Section

Cb

for

(g)

Figure

B4.5.

Virtual

_. 5-1

Moment

(,Cont ' d)

Diagram

d)

Section February Page 219 B4.5.-';7.6

to

a

Example

problem

A rectangular concentrated

beam, vertical

deflection

of

the

Material: Beam Load:

beam

simply load at

its

2024-T3 dimensions: P= 400

supported its center.

at

the Find

ends, the

is subjected vertical

center.

Aluminum 1/Z

at

Alloy

in.

x

Plate

1 in.

x

20

in.

lb P A

I

Procedure:

i ,

Apply the

a virtual virtua!

2.

Construct

3.

Calculate

Enter

5.

to

at

the

beam

center

and

construct

m.

moment

real

-

diagram,

bending

M.

stress.,

F b,

for

each

value

of

x by:

k=l.

5 for

a

Mc T

= the

plastic

bending

curves

cross-section)

determine

obtain'a

Q,

diagram,

real

the

Multiply

load,

moment

rectangular and

unit

the

Fb 4.

2o

the

c b

value value

of of

with for

max

m m

on each

each

by

Cbmax

the

value

page

Z18

value of

(at of

corresponding for

each

F b

from

Step

value

of

Ebmax

x.

vaiue

of

x.

3

B4.5 15,

1976

Section February Page

Example

problem

Tabulate

,

the

results

Construct a plot under the curve. (See

8.

Figure

B4.

f_ L

of the previous

of m _ bmax This area

steps

vs. x and represents

(see

Table

determine _'g

5.8. 6-I)

J0

the

m

Cbmax

B4.

area dx.

mob max

= J J

dx

Ref

C

eq

(4.5.7.5-11)

0

c

=

0. 250

in.

L f

m

dx

c bmax

by graphical integration

= 0.165

0

• •

•

By

0. 165 /%

_

an

which

0. 661

analysis,

is 7.7%

stresses

considerable beams

greater etc.

was

in error.

existed

this example.

/or

in.

0. 250

elastic

only The

more

lengths;

found

Partially over

elastic in error

in which

the

beams

to

be

for

0.6104

plastic

middle

analysis

the plastic

e.g.,

220

d)

A.

Calculate

A

(Gont'

B4.5 15,

in.

fiber

8 inches would

higher

in

be

loadings

stresses of constant

exist

and over

moment,

)

5.8.6-i).

! 976

Section B4.5 February 15, Page221 B4. 5.7.6 P = 400

Example

0

(Cont'

M

Mc/I,

d)

lb.

x, in.

problem

m in.

-lb.

in.

-lb.

0

ksi

0

0

c bmax in.

/in.

, x

0

m 10-

3

in.

e bmax -lb.

, x lO

0

I

.5

200

4.8

.4570

.2285

2

1.0

400

9.6

.9140

.9140

3

1.5

600

14.4

1.3710

2.0565

4

2.0

800

19.2

1.8280

3.6560

5

2.5

i000

24.0

2. Z850

5.7125

6

3.0

1200

28.8

2.7420

8.2260

7

3.5

1400

33.6

3.14

I0.9900

8

4.0

1600

38.4

3.80

15.2000

9

4.5

1800

43.2

4.64

20.8800

i0

5.0

2000

48.0

5.83

29.1500

iI

4.

1800

43.2

4 .64

20.8800

12

4.0

1600

38.4

3 .80

15.2000

13

3.5

1400

33.6

3 .14

I0.9900

14

3.0

1200

28.8

2 .7420

8.2260

15

2.5

i000

24.0

2 .2850

5.7125

16

2.0

8OO

19.2

1 .8280

3.656

17

1.5

6OO

14.4

1 .3710

2.0565

18

1.0

400

9.6

.9140

.9140

19

.5

2OO

4.8

.4570

.2285

2O

0

5

0

0

Table

B4.

0

5. 7. 6-i

0

-3

1976

Section

B4.5

February Page B4.5.7.6

Example

problem

(Cont'd)

28

k _AREA

24

15, 222

UNDER

CURVE

EQUALS

_

L "--" m.

I I

J" me b dx 0 max

_i

I I I i

2O E

v

O

x X m

16

/ / I

\ \

A

12

/

\ \

I

/ / J

\

/

/

¢

\ \

/ f

f 10 I

x (inches) Figure

B4.5.7.6-1

16

1976

SECTION B4. 6 BEAMS UNDERAXIAL LOAD

TABLE

OF

CONTENTS

Pa ge B4.6.0

Beams

Under

Axial

Load

.................

l

4.6. l

Introduction

4.6.2

Notation

........................

3

4.6.3

In-Plane

Response

5

4,6,3.1

Elastic

4,6.3,2

Inelastic

4,6.4

Biaxial

Elastic

4.6.4.2

Inelastic

Combined

Elastic

4.6.5.2

Inelastic

Recommended

4.6.6.1

In-Plane

Practices Response

Inelastic

Biaxial

Elastic

4.6.6.2.2

Inelastic

4.6.6.3

Combined

Elastic

4.6.6.3.2

Inelastic

9 9

Response

10 .........

..................

10 12

.................

13

..................

14

.................

16

................

Analysis

16

...............

Response

Analysis

21

.............

22

................

Analysis

Bending

4.6.6.3.1

References

Twisting

Analysis

In-Plane

4.6.6.2.1

...............

.................

Analysis

4.6.6.1.2 4.6.6.2

and

Analysis

Elastic

5

..................

Analysis

4.6.6.1,I

5

.................

Response

Analysis

4.6.5.1

4.6.6

..................

Analysis

Bending

l

...................

Analysis

In-Plane

4.6.4,1

4.6.5

.....................

and

23

............... Twisting

Analysis

Response

................

Analysis

...............

...........................

B4.6-iii

24 .......

24 25 27 42

LIST

OF ILLUSTRATIONS

Pa ge Tables

B4.6,1

Beam Column Elastic

Tables

B4.6.2

B4.6.3

Analysis

Analysis

-

Tensile

Beam Column Formulas Elastic

Tables

Formulas

-

Axial

Loads

.......

Axial

Loads

29

Compressive

Reference Listing for Other Beam Column Loading Conditions ................

B4.6-iv

....

32

40

Section

B4.6

February Page l B4.6.0

15,

BEAM-COLUMNS

L B4_.6. I Introduction Beam-columns

are

eously

to axial

bending

loads,

couples

from (or

end

applied

moments

at both

columns.

of

are

many

For

superimposed

since

characteristics

overall

instability

If

the

elements

is relatively The analysis on

short,

analyals simple

shear-web

beams

ever,

considered It

overall

has

are

the

are may

beam,

eccentricity

the

are

to be

of

of

subjected arise

the

simultan-

from

surface

transverse

shear

axial

are

buckling

load

As

loads,

or

at

end

In

one

the

section

shown

that

on

scope shear

below: Response

2).

may

of

the this

be

deform-

strength

the

and

lateral

must

and

distor-

the

where

instability)

is distinctly

beam-columns I and

cases

crippling

beam-

cannot

affect

Both

thin,

of

curvature,

all

factors.

a consequence,

beyond

effects

conditions

beams

analysis

column

relatively or

the

Initial

(torsional

the

(a) In-Plane

and

important

shear-web

(References

with

considered.

section

local

beam

restraint

f

indicated

on

associated

instability

are

been

response

and

beams.

in

bending

interdependent.

and

need

of

of

from

individual

effects,

lateral

point

problems

they

ational

lacking,

any

The

which

the member.

example,

inelastic

is

at

members

loads.

resulting

ends)

There

tion,

and

structural

be

support

considered.

beam-column

occur. different

effects section;

of

from

axial

they

the

loads

are,

how-

beams. can These

have are

one

shown

of

three

in Figure

types i as

of

1976

Section B4.6 February 15, i 976 Page 2

Z I

(a)

(b)

IN-PLANE

BIAXIAL

RESPONSE

IN-PLANE

RESPONSE

i \ 1 (¢)

COMBINED

FIGURE

BENDING

I

AND

BEAM-COLUMN

TWISTING

RESPONSE

RESPONSES

1

v

Section

B4.6

February

15,

Pa ge 3

f-. !

The upon

five

(b)

Biaxial

In-Plane

(c)

Combined

particular

Bending response

conditions.

/f-

(a)

Cross

(b)

Span

Response

These

and for

Twisting a given

Response member

is

dependent

mainly

are:

Section

Shape

Length

(c) Amount

of

(d) Restraint

Intermediate

Lateral

Conditions

(Torsional)

Support

at Boundaries

(e) Loading The in

analysis the

following In

with

combined

inelastic

bending

of

beam-columns are

based

the

member

These

on

are

developed

and

strengths

well

responses

design

will

procedures for

procedures

twisting

be

discussed

response

both

for are

for

the

beam-columns

elastic

blaxial

and

In-plane

limited,

in-

response

especially

in

the

range. powerful

referred

a column are

techniques

employed

to as "interaction

experimental

as

these

and

However,

the more is

for

analysis

ranges.

stress

One

the

response

stress

employed

paragraphs.

general,

in-plane

elastic and

procedures

data, and

then

the

and

they

strength

expressed

in

in

the

equations;" require

as terms

that

a beam of

be

stress

analysis

these the

equations

strength

determined ratios

of

of

separately. (applied

F load/strength) presses

the

be

in

used

and effects

the

incorporated of

following

the

in

combined

discussion.

the

interaction

loading.

equation

Interaction

which equations

exwill

1976

Sect ion B4.6 February Page 4 B4.6.2

15,

Notation

Area

A

of

Cross

Torsion

r

Section

Warping

d

Eccentricity

E

Modulus

of

Tangent

Modulus

Stress

(ibs/in.

Et f

Nominal

fcb

Constant

(in. 6)

(in.) Elasticity of

(Ibs/in.

2)

Stress

at Lateral

2)

G

Elastic

h

Distance

I

Moment

of

Inertia

(in. 4)

Moment

of

Inertia

about

I

2)

Elasticity

Fiber

(Ibs/in. Shear

(Ibs/in.

2)

Extreme

Buckling

(in. 2)

Modulus

between

(Ibs/in. 2)

Centers

of Flanges

Minor

Axis

(in.)

(in. 4)

Y K

UnlformTorsion

J

El p

L

Length

L'

Effective

M

Bending

x,M

(see

Constant

Tables

I

of Member

Principal

Axes

and

3)

(in.)

Length Moment

2,

(In. 4)

(in.) (in.-Ib) Bending

Moments

(in.-ib)

Y

Mac tual

Bending Moment Loads (in.-Ib)

Me,

Bending Yield

Moment Point

due

to Both

Bending

at the Elastic (in.-Ib)

and

Stress

Axial

Limit

(Ky)e (Mx)u,(Sylu

Bending Moment (in.-Ib)

at

the

Inelastic

Stress

Limit

or

1976

Section B4.6 February 15, 1976 Page 5 MI_ M2

Applied Loaded

Moments with

at Each

Unequal

End

End

of

Beam-Column

Moments

(in.-Ib)

Meq

Equivalent Moment for Beam-Column with Unequal End Moments (in.-Ib)

MoS,

Margin

P

Applied

P

of

Safety

Axial

Load

(Ibs)

Critical Column Buckling Load Stress Range (Euler Load);

e

Pe

=

Loaded

in

the

Elastic

(_2Elmin)/(L')2

(Px)e ' (Py)e

Critical Column Buckling Elastic Stress Range for Axis (ibs)

Loads in the Each Principal

P

Critical Column Buckling Stress Range (Ibs)

Load

u

U

L/j

W

Transverse

Load

(Ib)

w

Transverse

Unit

Load

Y

Deflection

(in.)

Z x, 8

Zy

(see

Section

Tables

Moduli

i, 2,

about

and

(Ib

in

the Inelastic

3)

per

Principal

linear

in.)

Axes

(in. 3)

Slope of Beam (radians) to Horizontal, when Upward to the Right

Positive

Section

B4.6 15,

February Pa ge 6 B4.6.3

In-Plane

Response

In-plane

response,

which,

when

plane,

respond

in Figure

subjected

la.

sional)

support

Beam-columns sively these

for are

B4.6.3.1

the bined the

yield

occurs

or when

torsionally

does

correct

in

always

present,

fections

are

discussion 84.6.3.2

respond

the

of

creases shown

above

manner

stress

analysis

range

acting

in

one

same

plane,

as

shown

the

when

adequate stiff

have and

loads

been

the

lateral

sections

(torare

investigated

inelastic

used.

exten-

stress

range;

below.

or

occurs

stress

consider

the

case

contribute

considered, this

of

the

concept

strength when

in

strength

and

the

the most of

beam-columns

computed highly

of

a pure the

This

buckling;

as

response.

Thus, remains

is given

by Massonnet

strength

of

it

in

is not

imperfections these

valid.

(Ref.

reaches

definition,

if

on

the com-

fiber

such,

Initial

above

of

stressed

column.

definition

is based

value

the material.

danger

to

of

are

imper-

Further

2).

Analysis bending

elastic

in Figure

the

yield

limiting

the

for

failure

bending

maximum

hypothetical

elastic

this

separately

Inelastic The

in

in

to beam-columns

Analysis

not

the

response provided,

that

point

a sense,

as

usually

elastic

and

bending

This

Elastic

axial

and

twisting

which

assumption

axial

refers

without

discussed

The

to combined

herein,

substantially

is

both

as mentioned

limit

yield 2.

strength.

moment, By

a beam As

yielding

comparison

the

actually applied

penetrates

of Figures

2b

is

higher

bending into and

moment

the

2d,

than

cross

it can

the

insection be

seen

1976

Section B4.6 February 15, 1976 Pa ge 7

Z

3 Q. a .J

>-

f

i,i Z m f

Z o. a .J ,,i >-

Z 0

IZ

,,,

Q .J ,,i

"r

>-

_D Z w rr oO uJ

> 3 v

<

row w rr

,,i

e_ uJ o-

X

U-

.J rr

w Z

Z

Ii1oo A

that

a greater

distribution. strength shown

are

strength

i_ obtained

Analysis

procedures

known

to vary, (a)

as

and

inelastic

to depend

The Mmax/P the

case

of

increases,

which

utilize

three

- The

pure

considering

analyses;

on

Ratio

by

factors

reserve

buckling

toward

the

this

this extra

(Ref.

and

that

is

-

For

B4.6

February Page 8

15,

inelastic extra,

stress

or

strength

-

reserve,

has

been

2).

strength

(M = 0)

value

the

Sect ion

is

almost

zero

tends,

as Mmax/P

associated

with

for

pure

bending. (b)

The

Shape

of

strength

the

is

smaller

rectangular (c)

The is

Nature

Figure

Several for

the

criterion (a)

for

the

higher

a

an

Metal

for

(The

flat

failure

used

of

3.

strength

Section

example,

I-section

than

the it

is

reserve for

a

section.

much

have

Cross

has

as have

beam-colunms one

of

the

Stress

Criterion

maximum

stress

level

(b) Maximum

Strain

Criterion

maximum

strain

level

collapse, material.

Load load

A

than

in

in

Criterion which

the it

is

diagram does

the

been

used

(Refs.

Maximum

(c) Ultimate

example,

stress-strain

portion,

been

For

material

criteria of

-

3,

for

for

diagram in 4,

reserve

5,

material

B in

material for

the

strength

A does

material

B).

inelastic

and

6).

occurs

at

not

analysis Basically,

the

following: - Failure the

inelastic

- Failure the

- Failure

utilizes

the

prescribed

some

prescribed

range.

occurs

inelastic

some

at

range.

occurs

at

inelastic

some

ultimate,

behavior

of

the

or

1976

Section B4.6 February 15, Pa ge

F

¸ n

w

UJ 0O I m-

LU

I-Z .J

DC I-CO I.I.I

nr"

° \

I.l.I rr £9 I I.I_

uJ

9

1976

Section

B4.6

February Page I0

B4.6.4

Biaxial This

In-Plane

section

stiff

beam-columns

cause

either

about

both

in

all

Case

which

of

one

without of

the

discussed

the

information that

as

short

in

the

if

they

On

the

other

in

a direction

hand,

the

the

forces

bending

respond will

direction Elastic the

plane

of

were long

slender

to

the

cause about

or

be

the

the

forces

from

members of

no

loss

of

principal than

type

of

2.

bending

applied

the member

less

beam-columns

for Case

of

both

only

or

that

bending

are

free

response

can

to

deflect

occur

as

Direction

large

plane

forces

paragraphs.

restrained

strength

then

Strong

with

bending

axis

torsionally

conditions:

following

members

applied

This

is available

normal

essentially

may

in

the

of

The

two

in

methods

principal

twisting.

following

analysis to

ib).

Bending

strength

In

(Fig.

(b) Primary

essentially

B4.6.4.1

the major

Bending

predicted

in one

axes

subjected

Biaxial

I is

strength

are

strength

(a) Primary

Little

bers

the

about

principal

directions,

a result

Response

considers

bending

15,

However, forces

and

deflecting with

small

the bending loaded

strength. axes

it

will

develop in

respond

bending

weak

and

Intermediate

simultaneously,

strength

for

the

limiting

stress

direction.

forces

responding

buckle

develop

concentrically,

the

been

as much

the

forces

has

that length and

is, mem-

the

exclusively

other.

Analysis

elastic

range,

when

failure

criterion

is

1976

Section February Page l l

r

used,

the

determination

principal flexural

axes

for

knowledge

of

of

In-plane

but

it has

more

can

Thus,

the

of

interaction

few

of

strength Hence,

extended

to

i)

that

cases

of

response

about

equation

be

one

for

the

axis.

in-plane

to

as

test

(Ref.

for

due

independently,

although

can

found

response

stresses

response.

response been

the

prediction

in-plane

conservative

cases

of

be made

actions.

procedures f

B4.6

bending

there

data for

is no

are

such

about

the

two

coupling

of

available, cases

15,

the

practical

are

based

on

the

case

elastic

solutions

include

biaxial

in-plane

response;

limiting

stress

criterion

is even

about

both

Austin

(Ref.

response

to

for

a

axes

than

i) extended

include

it

is

one

biaxlal

for

form

of

the

in-plane

response. B4.6.4.2

Inelastic

Austin based by

on

it can

has

response

appears

theory

methods

extended

and

Structural

members

of

twisting

(Figure

Ic).

the

axes one

form

This

of

thin-walled forces, phenomenon

precise of

for

the

this the

open may is

beam-columns From

interaction for

the

inelastic

fail

a practical

equation

for

range.

In general,

in-plane

in-plane

studies

which

inelastic

phenomenon.

biaxial

Respons

theoretical

twisting.

response

inelastic

Twistln_

no

without

to predict

the

bending

been

strength

in-plane

used

Bendin_

and

have

is available

Combined

axial

of

principal

to predict

combined

there

to biaxial

information that

that

extended

B4.6.5

to

both

Austin

test

be

K_rman

about

viewpoint,

Little

states

the von

bending

in-plane

(I)

Analysis

response,

response.

_ cross

respond commonly

section, by

when

combined

called

subjected

bending

torsional-

and

1976

Section

B4.6

February Page

flexural

buckling

a result

of

such which

as

I,

tion

of

4).

The

low

channel

follows

to dlrectly

force

the

or

it

or

loading

if

Is

by

torsion-bending. of

members

It

be

noted

should the

open

couples,

does

not

defined

the member

is

as

such

pass the

as

point

to bend

twisting

under (a) Axial both (b) Axial

open any

of

the

through

without

section

the

following

pTincipal compression

in

action

cross

the

section

discussion

are

not

subjected

when

the llne

shear

center

which

is

the

of

ac-

(Figure shear

twisting.

CENTER

4. SHEAR CENTER

cross

compression

open

members

SHEAR

FIGURE

of

that

arise

through

twisting

with

section

CENTROIO

Columns

The

stiffness

that

torslonal

center

pass

angle.

Is assumed

transverse

must

torslonal

applled

shear

buckling

15,

12

and

wlll

respond

three

by combined

bending

and

conditions:

moments

acting

to

cause

bending

about

moments

acting

to cause

bending

only

axes; and

1976

Sect ion B4.6 February 15, 1976 Page 13 about the major principal

axis when the momentof inertia

about the major axis is muchgreater than about the minor axis; for example, an l-section

memberwith axial compression

and end momentsacting to cause bending in the plane of the web; and (c) Axial compression and momentsacting to cause bending in a plane parallel

to the plane of either principal

axis, when the

principal

axis does not contain the shear center as well as the

centroid,

as may occur for a channel, tee, or an angle section.

Little

information

is availn_le on the strength and behavior of mem-

bers subjected to the first

and third conditions (Ref. i).

of study has been devoted to the second case, primarily (Refs. 4, 6, 7, 8, 9, 10,and ii).

The greatest amount

for 1-section members

The second case is discussed in the follow-

ing paragraphs. B4.6.5.1

Elastic Analysis

As mentioned above, most of the elastic

analyses for beam-columns

subjected to combinedbending and twisting have been limited I-section

members. It has been found that the behavior is similar

the lateral only.

to

buckling action of an 1-beam subjected to transverse forces

However, exact formulas for critical

buckling are complex (Refs. 4, 6, and Ii), have been used. These interaction closely with available data. by Hill,

to uniform

loads for torsional-flexural therefore,

interaction

equations

equations have been shown to agree

An interaction

equation has been proposed

Hartman, and Clark (Ref. i0) for aluminumbeam-columns; this equation

has been verified

by Massonnet (Ref. 2) for steel beamcolumns.

In

a

equation

theoretical used

combinations buckling were

of

by

free

ed with

axial

rotate

respect

tapered

Hill,

Hartman,

in

the

(Ref.

and

compared

The

outcome

the of

test

the

studies

of "solid"

tapered

and

interaction

curves

degree

of

taper

B4.6.5.2

Inelastic

Again, devoted

in

are

Massonet into

the

plastic

with

results

of

also

flanges.

on

the

elastic

tests

suggests

on

members

have

obtained

tapered

been

are

approximated

by

Butler

beam-columns,

curves. curves

can

_Ref.

independent

the

be

analytical

by Gatewood

essentially

of

However,

Also,

made

restrain-

response

steel

Salvadori's

the

ends

elastically

load.

beam-columns.

for

whose

interaction

that

as uniform

of

studies

members. used

inelastic has

the

axial

interaction

elastic

members

of

and

the

produce

were

done

15,

predictions

but

to Salvadori's

closely

the majority

equation

the

which

B4.6

February Page 14

of

results

of

range

have

14), the

Salvadorl.

Analysis

to 1-sectlon

interaction use

and

well

safe

that

web,

planes

performed

comparison as

gave

found

considered

the

been

12)

bending

bending

results

to

of

the

has

have

applied

the

tapered

in

combined

13)

Clark

and

plane

work

under

(Ref.

Salvadori

to rotation

members

and

compression

analytical

Anderson

Salvadorl

torsion-bending.

to

Little

and

by

study,

Section

Hill

in

range

elastic

by

extended

domain. their

using the

The

tests

of

in

the

and

steel

Clark

analysis the

elastic

results

inelastic

of

tangent

(Ref. can

9) have

also

modulus

interaction this

1-section

shown

that

extended

for

concept.

equation

extension columns

be

been

were and

for

steel

compared

found

to be

the

1976

Section B4.6 February 15, 1976 Page 15

f

in excellent agreement for oblique eccentric loading, momentat one end only, and equal and opposite end moments(Ref. I). a thorough study of the inelastic applied in the interaction B4.6.6

Recommended

Detailed ensure

comply

and

with

reliability. given be

given

In

prime

the

of

not

be

made

liminary

of

the

particular

member

as

these

no

the

the

impair

from

the

or

the

distribution.

of

problem

in

section the

feasible,

to determine

of for

etc.

the

as

to

design

Thus,

safest cross

and

the

of

in general,

section

of

at

any

member

that

not

so

or

be

nearby changes

is

choice

of

and

axial

the and

of

a shape,

choice bending

as

trials

along

in

section

such

section.

station

shall

loads;

successive

economical

e.g.,

in collapse

undesirable

direct

design

those

designed

shall

combined

selection

most

so

a beam-column

the

the

designs

design

loads

produce

variables, the

the

of

to withstand

except

the of

the

number

I-section,

selection

so large

be

resulting

limit

function

follow

weight

shall

of

and

to

requirements,

shall

minimum

application

performed

structure;

analysis

member

be

vehicle

instabilities

resulting

loading

of

shall or

or

methods,

and

from

to

cross

the member

the methods

occur

analysis

structure

components,

Because

rectangular,

integrity

utilizing

immediate

usually

and

Deformations

a suitable

loads.

stress

shall

large

The

comprehensive

consideration

(a) There

(b)

equation.

In general,

herein.

buckling of beamswhich may be

Practices

efficiency

shall

lateral

Galambos(Ref. 15) presents

The

is round,

must pre-

the member

Section B4.6 February 15, 1976 Page 16 may, in many cases,

be

based

on

the

Zx

where

the

interaction

Naturally, In

this

the

be

provided.

then be

degree

an

of

lateral

For

Also,

_y

be

and

improved

shape,

restraint

or

little

stiff

section

such

type

response

of

by

end

if

the

axial

particular

example,

when

formula

for

Stress

'

transverse

must

optimum

a torsionally

used.

the

selection

choosing

to

of

elementary

or

analysis.

attention

should

which

lateral

a box

is neglected.

a refined

restraint

no

as

loads

or

is not

will

restraint tubular known,

be

given

or

can

is

provided,

section all

should

three

con-

ditions

should

In-plane

(b)

Biaxlal

(c)

Combined

be If

analysis the

(a)

response In-plane bending

analyzed

the

response

to

analyst

procedures,

and

twisting

response

determine

the

critical

or designer

has

the

the

following

response.

choice

factors

of

should

elastic be

or

inelastic

considered

in making

choice. (a) Member

function

(b) Material (c) Deflection inelastic (d)

limitations

- Deflections

may

be

excessive

in

range

Thermal

conditions

- Little

thermal

effects

the

in

information

inelastic

range

is available

on

the

Sect ion B4.6 February 15, 1976 Page 17 (e) Dynamic conditions - Little elastic

effects

information is available

for in-

due to dynamic loadings

(f) Reliability (g) Analysis procedures available

for the method of analysis and

types of loadings considered. The recommendedpractices and procedures for the analysis of beamcolumns are discussed in the paragraphs which follow. design for local buckling and crippling methods presented in Section CI.

The analysis and

should be in accordance with the

In complying with the references and

recommendationscited above, the designer should keep abreast of current structural

research and development. With this approach, optimummethods

and designs should be achieved. B4.6.6.1

In-Plane Response

B4.6.6.1.I

Elastic Analysis Tensile

Axial

combined

flexure

commonly

encountered

tabulated

for

some

methods

of

able

several

in

and

for

compressive of

the

more

conditions

the

cases.

In Tables

these

for

axial commonly are

cases.

other

references

case

strength

axial

Compressive tained

- The

tensile

of

analysis

Loads

17,

Loads

of beam-columns loads.

Results

encountered

discussed

has

below.

been

B4.6.1

18,

19

- Many

are

given

conditions.

and

for

many

results

are

solutions

and

conditions

are

avail-

20).

exact

subjected

B4.6.3

adequate

loading

subjected

investigated

and

In general,

beam-column

(16,

Axial

loads

of beam-columns

solutions to combined

have

been

flexure

in

Tables

2 and

A

number

of

and

3 for

these

ob-

and

some other

to

Sect ion B4.6 February 15, 1976 Page 18 Beamcolumns with intermediate supports have been investigated. Probably the three momentequations (Refs. II, powerful method of analysis for this case.

18, 20,and 21) are the most

A variety of conditions is

covered in this technique; for example: (a) Any type of transverse loads can be included (b) Span lengths may vary (c)

Niles

inertia

may

of

rigid

(e) The

effects

of

intermediate

(f) The

effects

of uniformly

Newell

above.

(Ref.

consists

demonstrates

the

analysis

good for

compressive For

source

this

shown

problem. See

cases

of

to give

method

(Ref.

cross

section

26) and

for

He

Table

results

particularly

can

be

extended

loads

be

form.

II p. (Ref.

many

used

of

for

23) 21).

the

cases

this

Saunders

matrix"

for

an

(Refs.

technique

elastic

22,

to

axial

of

a savings

include

in

of

numerical time

and

to beam-columns many

(Ref.

the

tensile

illustrations members,

foundation

Het_nyi

solution

both

applicable to

(Ref.

for

25).

and

stepped

with

on

24,end

considers

and

llne

on multl-supports.

17,

B.4.6.3

can

"transfer

formulation

tapered

good

is

(6,

results

to matrix

the

straight

axial

which

beam-columns

references

loads. the

of

to span

supports

tabulated

adapted

for

span

in a

spring

beam-columns

solutions

in several

not

method

application

from

distributed

present

a solution

vary

supports

analytical

nonuniform

and

particularly equations

of the

of

Methods found

18)

Another

23)

been

of

effects

problem

be

moment

(d) The

and

listed

The

commonly

can

17)

is a

differential and

some

axial of

these

methods labor. of

cases.

have Newmark's

variable

encountered

Section B4.6 February 15, Page

loading a

conditions.

sequence

finite of

of

successive

difference

Other found

in

lytical

methods been

18,

19,

The elastic

which

can

be,

Ii,

can

20,

26,

and

16,

be

be

used

for

(Ref. the

by

27)

define

analysis

conditions. in

Tables

B.4.6.2

and

26,

28,

29,

and

adapted

to

the

solution

of

available

in

are

several

B.4.6.3 30).

can

The

problems

references

be

anawhich

(4,

ii,

29).

of exact

solutions

interaction

equations.

is_used

in many

is

basis

the

can

integration

Baron

19,

beam-columns

equation

a numerical

18_

of

and

is

Salvadori,

which

covered

investigated

analysis

using

loading

not (4,

abundance

for

line

conditions

method

methods

many

references

not

16,

with

the

approximations.

numerical

beam-columns

have

Basically,

19

and

which

respond

However,

instances. for

methods

several

solution

in-plane

the

The

of

reduces

interaction

following

interaction

for

the

the

need

equation

simple

straight

equations,

M P -Pe Since

Mactual

verse

loads,

However,

for

is it

the can

be

(b) Must

have

(c) May

be

maximum it

has

been

shown

difficult

simply

(2)

all

both for

of

the

the

axial

and

complicated following

trans-

conditions. requirements:

supported

subjected moment

from

to obtain

satisfy

uniform

(Refs.

I.

resulting

which

(a) Must

=

Me

moment be

members

actual

+

cross to any

at

or

I, 2,

section combination

near I0,

the

and

center

31)

that

of bending of

the

a good

forces

producing

span approximation

of

the

1976

Section

B4.6

February Page

actual

bending

moment

20

by

is given

M x MactuaI

Here

(Px)e

applied due

to

dition Eq.

is

the

moment the of

(2)

Euler

and

axial

due

the

the

(3)

P/(Px)e

critical

maximum

interacting

load

moment,

in

the

not

plane

considering

with

the

deflections.

of

the

axial

to interaction

of

load

For with

the the

moment

the

con-

deflection,

becomes

p -Pe and

1 -

elastic

M x is

load

moment

-

Mx +

= 1 (Mx) e

corresponding

{ 1 -

margin

M.S.

of

P/(Px)

safety

(4)

e is

=

given

by

M

"

P

1 For

eccentrically

at both

ends,

loaded

Eq.

(2)

__P Pe The

margin

Eq.

(5). The

moment, in Figure

of

safety

interaction Meq, 5.

for

members,

takes

the

for

this

equation

a beam-column

{1

P/(Px)

d is

following

+ (Mx) e

where

-

the

may

can

be

be

(5) equal

form

= 1.

determined

written

subjected

- 1

eccentricity,

Pd - P/(Px)e[

case

]

e }

in

analogously

terms

to unequal

(6)

of

end

to

an equivalent moments

as

shown

15,

1976

Section B4.6 February 15, 1976 Page 21

F

M 2

M I

FIGURE 5.

Equation

(2)

UNEQUAL

END MOMENTS

becomes

M __P P e

where

a good

+

eq (Mx) e

approximation

Me q

for Meq

= 0.6

+

=

{ I - P/(Px)e

MI

(7)

as given

by Austin

(I)

for

> M2 > MI

- 0.5

M2 -M1

0.4

1

}

1.0

is

(8)

and M eq MI Accurate been

cases

restrained

both

is conservative maximum analysis

for

interaction

developed

example,

= 0.4

moment without

for

with

each

to use

Eq.

the member regard

to

which

with

end

respect

M2 -- _ MI

> -

formulas

beam-columns

where

in

-0.5

can

be

and the

are

than

free,

hinged,

and

Mx =

of

and

simple

general

(Mx)max,

fixed,or

axial

where

by

an

load.

have

supports;

translation.

is determined

effects

(9)

simple

other

to rotation (4) with

-i.0.

not

for

elastically However,

it

(Mx)ma x is

the

ordinary In

structural

utilizing

this

Sect

method, Euler

the

effective

length

Inelastic

Analysis

concept

should

formula

for

be

employed

ion

B4.6

February Page 22

15,

in determining

the

load.

B4.6.6.1.2 A

practical

interaction

metal

beam-columns

which

respond

i, 2,

7,

under

in-plane

10,and

31)

combined

and

in

predicting

compressive

the

inelastic

the

axial region,

strength

loads has

and

of

bending,

been

shown

(Refs.

to be

p P--_ +

where

Pu

range. ii), the

the

The

strength

value

Johnson's section

bending the

is

Pu

the

can

(Mx)u,

methods

presented

for

the

zone

As

Plastic

in

tangent 31),

withstand.

In

equation

the

the

inelastic

modulus

method

or by methods

the moment

(Mx) u is

Cl0)

due

to

ultimate

determining above,

one

presented

transverse

moment the

of

(Refs.

which

ultimate

the

three

4.4.3.2, Distribution

Design Elastic

used.

trapezoidal

The 5,

29,

M x _s

and

= i.

a column

the

(Refs.

interaction

Stress

as

by

before,

load,

(c) Double

(Refs.

found

in paragraph

(a) Trapezoidal

be

he

parabolas

axial

{ 1 - PICPx) e }

the member

inelastically

moment,

can

can

columns.

without

(b)

of

modified on

section

of

(Mx) u

29)

design.

It

alloys.

In general,

Modull

is widely

is especially the

stress used

adaptable trapezoidal

in

distribution the

aerospace

to metals stress

which

method

developed

by

Coz-

industry

in

structural

behave

like

aluminum

distribution

method

will

be

4 in

1976

Section

B4.6

February Page

preferable; ods

16

may

however, prove

and

32)

been

Obviously,

ject

in

to

t_le

(Fig.

I0), be

limitations,

substituted

required

to

equ_tJon

for

other

In

_mploying

In

general,

except

no

methods

a

M x

for

prespecified

in

limits

boundary

and

the

for

(Meq)

the

deflection

other

method

meth-

(Refs.

3,

a

of

the

consequence,

supports,

it

uniform

with

by

sub-

8

and

investigation of

and

end-moments

equations

Additional

is

sections,

unequal

defined

one

this

9 is

interaction

conditions.

method,

of

the

extension

applicability

loading

e_se

calculating

i0.

of

inelastic

the

as

design

an

As

simple

equation

the

is

4).

of

steels.

beam-columns

moment

for

dttermine

for

one

plastic

10

(Eq.

i.e.,

where

for

equation

section

equivalent

the

developed

However,

the

cases

example,

primarily

previous

same

end-moments.

For

certain

interaction

the

equal

may

are

superior.

has

presented

there

15,

23

plnstic

resulting

excessive

deflections

design,

there

deflections.

limitation

the

inelastic

procedures

for

combined

may

are Thus,

method

few

or

if

there

may

not

occur.

is

be

adequate. Inelastic bending

have

methods

can

B4.6.6.2

analysis

not

been

probably

Biaxial

be

directions

res_.Iting

c_nsist

from

of

may

which

have

either

compressive

However,

extended

In-Plane

Beam-columns

all

developed.

to

previous

axial

elastic

loads

and

analytical

inelastic

behavior.

stiff

free

Response are

torsionally

a biaxial of

cover

tensile

two

axial

in-plane

loading

load

and

and

response

conditions.

to

condition These

two

deflect

(Fig. conditions

in

i)

1976

Section

B4.6

February Page

(a)

Primary

Bending

(b) Biaxial For always

case

direction. weak or

This

direction,

stiffness

response

and,

it will

have by

if necessary

that

Direction;

the

strong

only

or

checking

For

(Ref.

the

I) and

direction,

in-plane the

providing

direction. Austin

be

seen

that

practical

the

response buckling

adequate

in

member the

value

strong

for

intermediate

condition

where

Massonnet

(Ref.

the

supports

a biaxial 2)

is

both

inplane

provide

an

only

interest.

blaxlal

in-plane

Therefore,

it

response

is

for

discussed

in

case the

2

is

follow-

paragraphs.

B4.6.6.2.1

Elastic

"The principal

axes

can

actions

In-plane

response

clude

bending For

of

be made

in

the

problems

about

about

both

interaction

equation account

--

Pe

the

(Paragraph principal are

in-plane

blaxlal

due as

range"

principal

for

stresses

independently

which

both

the

elastic

beam-columns

moments

into

Analysis

determination

flexural

take

In

accomplished

occur,

can

primary

Ing

bendlng

Strong

discussion.

It

ing

thal

is

for

does

excellent

of

so

the

Bending.

i, primary

designed

in

15,

24

(Ref.

by

Austin

response,

loading.

Thus

the

is no

coupling

I).

Thus,

solutions

can

simple

subjected

about

there

4.4.6.1.1)

axes

axes,

to bending

to

extended

superposltlon

axial has

be

Equation

stated

4,

Equation

can

of

the

for

elastic

to inof

compression

also

two

results.

and that

bend-

the

be modified

to

4 becomes

+

=

(Mx) e

{ I - P/(Px) e }

+

(My) e

{ I - P/(Py)e

}

I.

(II)

1976

Section B4.6 February 15, 1976 Page25 Equation II is subject to the samerestrictions the _uivalent utilized

as Equation 4.

momentconcept as discussed in Paragraph 4.4.6.1.1

Inelastic

Analysis

The recommendedprocedure for inelastic of the interaction plane response.

--

+

problems are

flange,

restrained

torsionally

various

a high weak

angle

is

possibilities

apply

in

the

and

torsional

wherein

restrictions

} discussed

Response

be

by

sections applied

adequate

are

only

bracing

applicable

to beam-columns or

to beam-

open

as well

section

such

as

during

bend

twist

may

be

involved

axis

and

centroidal

as

are

the widethe

response.

summarized

follows: (a)

If

the

shear

incident, twisting failure

center

the member and

may

bending,

increa_in_

to

rigidity. of

twist

- P/(Py_

here.

should

twisting

to

same

preceding

beam-column apt

the

Twisting

response,

possess

or

also

and

against

(M_)u{l

equation,

summarized

in-plane

tee,

will

Bending

methods

which A

interaction

4.4.6.1.2

of

columns

The

this

Combined The

=i. (12)

(Mx) u { I - P/(Px) e }

Paragraph

in-

Equation I0 then is

+

Using

4.4.6.3

analysis is an extension

equation from in-plane response to include blaxial

Pu

which

can be

for unequal end moments.

B4.6.6.2.2

in

Also,

respond

with

for very

the

by

axis

are

a combination

tendency

thin-walled,

toward

not of

twist

torsionally

co-

as

Sect ion B4.6 February 15, 1976 Page26 weak, short column sections. (b) If the shear center axis and the centroldal incident,

axis are co-

as in the case of the I- or Z- shapes, buckling

by pure twist may occur without bending. The recommendedformulae given herein for both elastic tic analysis are in the form of interaction

and inelas-

equations and are applicable

only to cases with bending in the strong direction.

These equations have

a simple form, are convenient to use, are accurate, and have a wide scope of application. and llcontaln

If a theoretical analytical

solution

investigations

manycommonsections and loadings.

is ueslred, References 4, 6,

of torsional-flexural

response for

However, most of the work has been

done for axial compression and momentsacting to cause only bending about the major principal

axis when the momentof inertia

is muchgreater than about the minor axis (e.g., proposed

by Austin

include

primary

experimental B4.6.6.3.1 For subjected action

or

that

bending

about

is

p __ Pe

both

equations

axes.

available

l-sectlon). can

However,

to verify

be

there

It has

extended are

few

been

to data,

this.

Analysis

elastic

to primary

equation

interaction

analytical,

Elastic the

the

about the major axis

analysis bending

in

of doubly

symmetric

the

of

plane

the web

1-section the

members

following

inter-

recommended:

M +

x fcb

Zx {I

. - P/(Px) e}

'

= 1.

(13)

Section B4.6 February 15, 1976 Page 27 The value of fcb is the nominal extreme fiber stress at lateral

buckling

for a membersubjected to a uniform momentcausing bending in the plane of the web. The value of fcb is given by

J

2El yh fcb

= 2ZxL2

A

complete

study

Reference

ticularly

be

the maximum

interaction

moment

M x.

true

if

The

uniform

when

an

moment

all

shapes

and

appropriate

support

buckling

be

moments

is

=

(14)

given

for

+

that

symmetric

that

the

expressions

by

Clark

and

applied

with

sign

Hill

2)

+

1-shaped

discretion,

in

span,

is

parend

above

can

substituted

of

the

for

equivalent

is

.

(15)

equation

sections,

but

lengths,

adopted. as

cited

and

0.4 MIM 2

effective

the

the maximum

equation

interaction

are

of This

and

determination

(M2)2 the

center

calculated

(Ref.

fcb

the

conservative.

is

the

proper

for

near

opposite

moment

(Ml)2

states

or

interaction

Massonnet

_013

to doubly

of

the

uniform

by

at

excessively are

expression

also

be

is not

However,

as given

provided

should

may

equivalent

Massonnet

sions

end

recommended

limited

moment

for M x.

Meq

arily

lateral

equation

is used

used

this

_ 2E r

8.

When the

of

KGL

I+--

is not can

used

equivalent

However,

little

be

necess-

work

these

has

been

for

moments,

extendone

to

this. Relatively

of nonuniform twisting.

or

Butler,

few

studies

tapered

have

beam

Anderson

been

columns

(Ref.

13),

conducted

which and

fail

on by

Gatewood

the

elastic

combined (Ref.

14)

stability

bending have

and investigated

Section B4.6 February 15, 1976 Page 28 tapered members, and equatlon=my ing

be

conditions

their

used. will

results

However,

be necessary

indicate

that

additional

the

tests

to establish

previous

under

this

interaction

a variety

possibility

of

load-

conclusive-

ly. 84.6.6.3.2 It the

inelastic

has

been

Interaction

adequate used.

for For

Analysis

predicted equation

inelastic

this

case,

by

various

recommended analysis

Equation

sources for

provided

(Refs.

elastic that

analysis

the

fcb: and

the

associated

2Z_L2

interaction

_'

1+

equation

_2E

elastic

lateral

31)

also

modulus

that

be

is

•

2 r

(16)

=I.

Pu

(Ref.

will

and

Is

+

Galambos

tangent

I0

14 becomes

KSL _2Etlyh

I, 2,

fcb

15)

buckling

Zx

should value

{I

(17)

- P/(Px)e}

be to be

consulted used

In

for the

further interaction

study

of

the

equation.

in-

Sect

ion

February Page

15,

29

0 C_ ,! X

0

;I I X

_C C &J

X t_

.3 _C U

0

i

C

II X

0

I

C I I C &J

N m

O0 C

.C C

0 M

_

C 0 &J

I

!

I

I

II

II

tl

II

_-_

C 0 U

X

X

X

0

I::

0 0 O4 C_ D Ul

I

C

!

I _ ] i..._

C "0 0

0 14 C

._.

11o

ii

i

B4.6

[ I

I

---'1r

1976

Section February Page 30

B4.6 15,

1976

_vj'

uJ z Qo o <.,)

N

u o cL el-

1

I

i__

_

-i --_ _L

o P-O

0

,J

Section February Page

I¢

r----i

cll Izl

II ._

,,i oQ

x

cn o

+

_1_

U

_I _

i,i

_l r_ _.._-

_leq

I

0 0

I

_

e

.,,,.4 I@I

(N

V

II

41

II

I

U II

x

+

II

X

x

J

+

_

_

X

Ig

l _

c'M

__ _\L.

,,-3

j

_2

3 1

B4.6 15,

1976

Section B4.6 February 15, Page 32

0 II X 1.1 I--I

I

t_ II

x + 0 tO

II

u

_J c_

j.-_,,'

0 l-J

0

I

II

,...1 + ,-I 0

_"'I I _"_

I

+ (11

_0

0 O

I o_

I

oM

_

_

""_

I

_J

Q o_

!

I

u_ 0 O

,-4: I

I

II

II

0

L___I

1:

"7

II

N

II

N

rQ I-' X

\\

\\\%\\\\

/ I

L &

1976

Section

B4.6

February Page

o_

0 .,.4 4-) U 0 q-I "0

II

,i

N O m

II

N II

o

II

J 4-1 4_ O N

!

II

o_ I

0

_I _ _

O I

_--\

"N

!

m

I

_ =1_

+

0 0

O U O

N _ _I

_

,--I

c'_l

0 U I

oq ! c_

_ "_ _1_ if)

v

,

:_1_

I1 Q

! II

I!

I

I

II

I|

N r_ rO

F-

X

m o

o 0

X

33

15,

1976

Section February Page

34

_J C_ 0 I

I

I

o °M I

_0

,M

_l._.r,i °_

ffl

V

i

II

II

°_,,_

•", J II

u

o II

II

o o

_

II

_ o

_

V .t--_ (N

o

0 .u

t_ II

0

A

0 4.1

II

II o

_1

_4

u

.,ii

o ._I _J

I

II 0 •_4

II 4..i

II I

0 4J

II

o _

o °r,l

m

IT' _J

II 4_

ul

0 4_

c

_

m _J _J

c r_

_J

,.Q

¢t

-,-4

I o

0 4.J 0

°'l

.lii u

i

,-4 kl4

ii

II

CD

C_

B4.6 15,

1976

Sect

ion

February Page

I

II

O

.1..I

_1 "'-

×I_

,-_lm

• _-I

O

I

m

,A ,--i

._

+

°_

+

o

.,-

"O

E I .r-I

x I E

II

II II

(1)

°_ O E

O II N

II

N

II

II

4-1 0 4-1 O

•,4 _ _

O 4..1

0

.,--I 0

m I

II

0

¢'+'

O CD

_ %

°H

N _

I

O .,-I r_

_1_

o

_

0

, •r-

O

m _I_N

_

m % I

_ °,4

.,.4 IJ

4-1

_

_3

4-1

o

-_ .1_1 0

_j

,

_lm I

.I-I

II

II

N

CD

-r.t

O

>-

Ela" I

35

B4.6 15,

1976

Section

B4.6

February Page

15,

1976

36 I

NI_ O _4 co

m O U

II

+

N 4J

U

co

I

'_m

_1_

r----m

II

"J-

I

_1_

_i

_'_

m q

o o

_le, I

0 I u

CO

..

I

_

'

o

o L)

N

o

l_lt_l

0

,

,

CO

I

4.,I L____I II

o m La_ I C_

co io ill ¢:I

iJ

II

_l

II

.IJ

o ,_1

m

,-I

,a

"i ,a

I

I

,_le_ _ II

il II

II

II N

H

4J i

p.-

.o x

0 •-,il(li _

0

>-

I/ H

,,

Section

B4.6

February Page

.0

!

!i

I

0

b q

o3

+

l i

_l_

I

I_I_ _I_

-i o

i

I

,-q

0 L_

II

bZ

_I_]

. _I-'_I _: _

' 0

i_l_

I'_

L_l°_

0

_J °_._ I H -r

12

1

N

l °r'-_

i

.,e 1

_l _

0

II

4-1

II _0 I ¢q

"c ('1

U

H

! .IJ

II

X II O

c_

;2 x F--

i

: I i, _1 i,r:

t:.J ._ >-

!'. _

_

°

i

,--4

_1

__

37

15,

1976

Section February Page

B4.6 15,

38

0 0 !

0

0 II t-I

+ 4-I 0 0

_0 0 0

0

4J

I _N

+

I

!

"PI V

_I_

II

I

_1"_ 4J

+

4J v

O 0

g

@

I_1_

I

o

v-4 _)

m

I=

_

_1"

,._

II II

0 I=

_

N

I:1

_:_

0

I !

oI

.=

!°

I

_

-,.4

_

o

I_ -_1

II

._I

° _ _I

II

L) I

M

,d 4

-r-1

_ I

._-I

tJ

II

,

m

0

I_

°

_1

xl'_-_ @ ,_.-)

(d

Q)

II

-PI

:_

-_I

L__._.J

:_I_

I

I

!

!

II

II

II

II

_-I ._"1

I

..Q (II f--

)

I °-

(1_

0,--_ _-i -el

i!/'

0

u_ 0

¢)

_ 0

r

0 -_I

m.= rj

._ ._I -_I

rj

(_ -_I .u

1976

Section

B4.6

February Page

J 0

_J

=0 4-J 0 _J

N

M F--1

,._

o 0

0

O

ed

0

+

m I

_I_ 0 P_ I0

_

.r-_

I

I

_I_

o

_

,

' 0

a_

_'_

•r-4

_I_

I

O

L__

aJ o c_J Q _0

_

t__l

:3: _Im

_I_

I

II

II

_ 0

II II

II

N

N

_

c_

+

"

M

cd

.Q

O _D

_-_ 0

o c_

0 _

._

_

(_ _-_ 4_

u_"m 0

_J m m

4.1 _ O

1.-1 _

0

r._

,,D

12u

_

N

! fl

_

o 0

_

r_

N • .,.-I t_-4

• O N

N

39

15,

1976

Sect ion B4.6 February Page 40

15,

,-i

m 0 o"_ O0 0 &J o O 0

","4 o

..4

I--

E

--.._ '1

1976

Section

B4.6

February Page

g i-4

o w-f

O_

0 0 0

Z o

U

_

t_

m

IJ

0

_

0

I U

_

X $ ,.-_eO

L

-I

_L

i

C'I" _ -

I

% %

-i

i

E

< 0

.t (_

I

-]

c!

('-_

41

15,

1976

Section February Page 42

• =

0

o

_J

0

•,.4 N I

-

L

-

I

m

!

g.\

z o

,-1

eL.

,

0

B4.6 15,

1976

Section February Page

43

_FE_N_S:

I.

Austin, W. J.: Strength and ASCE, April 1961, page 1802.

2.

Massonnet, C.: Stability Considerations in the Design Columns, Proceedings, ASCE, ST7, September 1959.

3.

Beedle, L. S.: Inc., New York,

4.

Blelch, F.: Bucklin_ Co., Inc., New York,

5.

Cozzone, Sciences,

6.

Seeley, F. and Smith, Wiley and Sons, Inc.,

7.

Clark, J. W.: Eccentrically ASCE, Vol. 120, 1955, page

8.

Clark, J. W., and Hill, A. N.,: Lateral Buckling in___, ASCE, Vol. 86, No. ST7, July, 1960.

9.

Hill, H. N. and Clark, J. W.: Lateral Buckling of Eccentrically Loaded I- and H-Section Columns, Proceedings, First National Congress of Applied Mechanics, ASME, 1952.

J.: May

Plastic 1958.

Bending 1943.

Design

Design

of

Strength 1952. in

the

J.: New

of Metal

Steel

of Metal

Plastic

Frame_,

John

Structures,

Range,

Advanced Mechanics York, 1957. Loaded 1116.

Beam-Columns,

Aluminum

Hill, Alloy

Ii.

Timoshenko, S. and Gere, J.M.: Theory of Elastic edition, McGraw-Hill Book Company, Inc., New York,

12.

Salvadori, M. Transactions,

G.: ASCE

13.

Butler, D. J. Beam-Columns,

and Anderson, G. Welding Research

Wiley

Materials,

and

Sons,

of Eccentrically

Book _

of

John

Transactions,

Beams,

H. N., Hartmann, E. C., and Clark, J. W.: Design Beam-Columns, Transactions, ASCE, Vol. 121, 1956.

Lateral Buckling Vol. 121, 1956.

Steel

Aeronautical

Columns,

I0.

of

McGraw-Hill

Journal

of

Proceedings,

Proceed-

of Aluminum

Stability, 1961. Loaded

Second

1-Columns,

B.: The Elastic Buckling of Tapered Journal Supplement, January 1963.

B4.6 15,

1976

Section B4.6 February 15, 1976 Page44 14.

Gatewood, B. E.: Buckling Loads for Beamsof Variable Cross Section Under CombinedLoads, Journal of the Aeronautical Sciences, Vol. 22, 1955.

15.

Ga]_.mbos,T.

16. 17. 18. 19. 20.

21.

22. 23. 24. 25.

97. 28.

Ine]_etlc

L_.ter_1

Bresler, Boris and Lin, T. Y.: Wiley and Sons, Inc., New York,

_ckling_

Design 1960.

of

Niles, A. S. and Newell, J. S.: Airplane Sons, Inc., Volume II, third edition, New Roark, R. Hill Book

J.: Formulas for Stress Company, Inc., New York,

Timoshenko, S.: Strength Problems, 3rd ed., D. Van 1956.

of

Steel

Hetenyi, M.: Beams on Elastic Foundation, Press, Ann Arbor, Michigan, 1955.

Be_.m_:

Proceedings,

Structures,

the University

Structures, York, 1949.

and Strain, 1954.

Third

John

of

John

Michigan

Wiley

Edition,

of Materials Part II, Advanced Nostrand Co., Inc., Princeton,

Laird, W. and Bryson, A.: An Analysis of Continuous with Uniformly Distributed Axial Load, Proc. of 3rd Congress of Applied Mechanics, ASME, N.Y., 1958.

and

McGraw-

Theory and New Jersey,

Beam-Columns U. S. National

Saunders, H.: Beam-Column of Nonuniform Sections by MatrlxMethods, Journal of the Aerospace Sciences, September 1961. Saunders, Supports,

H: Matrix Analysis AIAA Journal, April

of a Nonuniform 1963.

Beam-Column

Lee, S., Wang, T. and Kao, J.: Continuous Beam-Columns Foundation, ProceedinKs , ASCE EM2, April 1961. Sundara, Foundation,

26.

V.:

Ne_mrk, Moments,

K.

and Anantharamu, ProceedlnFLs

, ASCE,

S.:

Finite

EM6,

December

N. M.: Numerical Procedure for and Buckling Loads, Transactions,

Salvadorl, M. Prentlce-Hall,

G. and Inc.,

Woodberry, 1959.

F.

R.

H.:

Beam-Columns

of

on

on

Multi-

Elastic

Elastic

1963.

Computing Deflections, ASCE, Vol. 108, 1943.

Baron, M. L.: Numerical New Jersey, 1962. Analysis

on

Beam-Columns,

Methods

Design

in EnHineering,

News,

April

Section B4.6 February 15, 1976 Page 45 29. 30. 31. 32.

Bruhn, E. Tri-State

F.: Analysls and Desig n of Flight Veh_le Offset Co., Cincinnati, Ohio, 1965.

Griffel, W.: Engineering,

Method for Solving September 14, 1964.

Beam-Column

Guide to Desisn Criteria Research Council, Engrg,

for Metal Compression Foundation, 1960.

Plastic Design in Steel, Inc., New York, 1959.

American

Institute

Structures,

Problems,

Product

Members,

of

Steel

Column

Construction,

v

SECTION B4.7 LATERAL BUCKLING OF BEAMS

--.._j

Section

TABLE

B4.7

OF CONTENTS Page

B4.7

...._

Lateral

Buckling

4.7.1

Introduction

4.7.1.1

General

4.7.2

Symmetrical

4.7.2.1

I-Beams

of Beams

1

......................

1

................................ Cross

Section Sections

.........................

1

.........................

5

..................................

Ie

Iklre

Bending

II.

Cantilever

III.

Simply

IV.

Simply

5

..........................

Be,am,

Load

at End

...............

Supported

Beam,

Load

at Middle

Supported

Beam,

Uniform

4.7.2.2

Rectm_mlar

4.7.3

Unsymmetrical

4.7.4

Special

Conditions

4'.7.4.1

Oblique

Loads

4.7.4.2

Nonuniform

Cross

4.7.4.3

Special

Conditions

4.7.4.4

Inelastic

4.7.5

RE FERENCES

Beams

End

5

...........................

I - Sections

.....................

............................ ..............................

Buckling

Section

...................... ........................

............................ ..............................

B4.7-iii

Load

6 ........ .........

7 8 10 13 14 14 14 14 15 16

Section B4.7 15 April, 1971 Page 1 B4.7

LATERAL BUCKLING OF BEAMS

4.7. i

INTRODUCTION

A beam of general cross section which is bent in the plane of greatest flexural rigidity may buckle in the plane perpendicular to the plane of greatest flexural rigidity at a certain critical value of the load. Concern for lateral buckling is more significant in the design of beams without lateral support when the flexural rigidity of the beam in the plane of bending is large in comparison with the lateral bending rigidity. Consider the beamwith two planes of symmetry shownin Figure 4.7- i. This beam is assumedto be subjected to arbitrary loads acting perpendicular to the xz plane. By assuming that a small lateral deflection occurs under the action of these loads, the critical value of load can be obtained from the differential equationsof equilibrium for the deflected beam (Ref. l). Beams with various cross sections andparticular cases of loading and boundary conditions will be considered in this section. 4.7.1. i

General

The expressed

f

Cross

Section

general expression for the elastic by the following equation (Ref. 3).

cr

S (KL) c

.y

2

2g + C3k+

buckling

(C2g+

C3k) 2+

strength

of beams

__ w I y

1 + GJ(KL

can be

2

(1) where: f

=

critical

E

=

modulus

of elasticity,

I

=

modulus

of inertia

=

distance twisting,

between in.

or

Y

L

stress

for

lateral

buckling

lb/in. of beam

Doints

2 cross

of support

section against

about lateral

the

y axis, bending

in. and

Section 15 April, Page

B4.7 1971

2

o P

p m

2 !

0

Z

m

Y -U

|

!

!

FIGURE

4.7-i

LATERAL

BUCKLING

-

Section

B4.7

15 April, P,_e C

torsion warping

constant,

1971

3

in. 6

W /

g

distance

___

from

shear

load (positive when other%vise),

shear

G

J

S

to point of application shear

center

of transverse

and negative

in.

modulus

torsion

__

center

load is below

of elasticity, Ib/in.2

constant,

in.4

=

section modulus

for stress

k

=

c

e

=

distance

from

centroid

and compressi(m

in compression

flange,

in. 3

C

CI, C2, C3, K=

+

i m 21 x

rj (x 2 _ y2) dA, A

constants

In the equation pass through

the shear

beam in such a mmmcr directions as the beam

depend

coefficients

mainly

of Ci, C2, C3,

depend

to centroid, flange,

mainly

_)sitivc if bct_vcen

in.

on conditions

of loading

and support

(Tsl)Ie 4.7- I).

above, center

it is assumed

that the lines of action of the loads

and the ccntroid,

and that the loads attach to the

that their lines of action remain parallel to their initial deflects. It is also assumed that the shear center lies

on a principal axis through

The

shear center

which

for the beam

in.

the centroid.

Ci, C2, C3, and K

on the conditions

of loading

and K given in Table

4.7-I

are derived

in Reference

and support

for the beam.

have

been obtained

from

3.

They The

values

Reference

3.

Section

B4.7

15 April, Page Table

4.7-

I.

Values

of Coefficients Strength

in Formula

for Elastic Buckling

of Beams

VALUE

OF COEFFICIENTS C

M

M

SIMPLE SUPPORT

1_--4

K

C1

1.0

1.O

1.0

05

1.0

1.0

1.0

1.31

0.5

1.30

I0

1.77

0. S

1.78

1.0

2.33

0.5

229

1.0

2.56

0.S

2.23

1.0

1.13

0.45

0.5

0.97

0.29

1.0

1.30

1.SS

0.5

0.86

0.82

1.0

1,35

0.55

O.S

1.07

0.42

1.0

1.70

1.42

0.5

1.04

0,84

1.0

1.04

0.42

FIXED SIMPLE

l

SUPPORT

FIXED M SIMPLE SUPPORT

+

t FIXED SIMPLE SUPPORT FIXED SIMPLE SUPPORT FIXED w SIMP.LE SUP PO.R.T

t

+ FIXED w SIMPLE SUPPORT FIXED P SIMPLE

+

SUPPORT

+ FIXED

C2

t---+:--! =4

SIMPLE SUPPORT FIXED SIMPLE

+

SUPPORT

I FIXE D CANTILEVER

BEAMS

WARPING RESTRAINED !1 AT _,UPPORTED

WARPING

END

1.0

1.28

RESTRAINED 10

AT SUPPORTED

END

2.05

I

3

6.S

P

12

4

RESTRAINT

LOADING

CASE NO.

1971

0 64

2.5

Section

B4.7

15 April, 1971 Page 5 4.7.2

SYM ME TRIC AL SE CTIONS For

a point to zero.

sections

that

(channels, zee The expression

f

_

are

about

2g+

z

(C2g)

horizontal

w

+ _I

a

1 +

axis

GJ (KL) ; EC

y

Values

of C t, C 2, and K can be obtained

4.7.2.

i

or about

from

2

(2)

w

Table

4.7-1.

I-Beams

Given conditions be used.

below

are

solutions

for I-beams.

I.

value

the

sections, etc.), the quantity k in equation (1) is equal for elastic bucMing strength can then be written

y _KL_ 2

or

symmetrical

Pure

For

cases

particular not considered

cases

of load

below,

and boundary

equation

(2)

should

Bending

If an I-beam moment M

of the

for

is subjected is

to couples

M at the o

ends,

the

critical

o

(M)o

This

cr

-_

expression

Lrr

EIy GJ

can be represented

1 q

_--_L---_/

in the

(3)

form

EI GJ (M)

=

o

K 1

y

L

cr

where

K1 = Values

of

K 1

_ J

are

w

I + Ec GJLT_2 given

in "Fable 4.7-2.

(4)

Section

B4.7 v"

15 April, Page

Table 4.7-2. L2GJ -EC

Values

0

0. t

of the

Factor

K 1 for

i

2

10.36

7.66

I-Beams

4

in Pure

6

8

5.85

5.11

4.70

32

36

40

3.59

3.55

3.51

1971

6

Bending

10

12

W

K1

L2Gj EC

oo

31.4

16

20

24

28

3.73

3.66

4.43

4.24

100

oo

W

Kl

3.83

4.00

II.

of the

Cantilever

Beam,

If a cantilever end cross section,

P

=

cr

K 2

Load

3.29

at End

beam is subjected to a force applied the critical value of the load P is

at the

centroid

L2

(5)

where 4.0t3 K 2

For

values

=

of

L2GJ greater

EC W

4.7-3.

For

values

of

values

of K 2.

0.1,

values

of

K2

are

given

in Table

L2Gj less

EC W

for

than

than

0.1,

see

Reference

1, page

258,

Section

B4.7

15 April, Page Table

4.7-3.

L2Gj

Values

of the

Factor

K 2 for

Cantilever

Beams

1971

7

<)[ I-Section

0.1

EC W

K2

44.3

L2GJ

15.7

10

EC

12.2

10.7

9.76

S 69

12

14

16

24

7.20

6.96

6.73

6. 19

8.03

4O

,)r)

W

7.5_

K2

III.

Simply

Supported

supported

If a simply the

eentroid

P

Values

er

of the

middle

=

,/EIyGJ L2

K3

of K 3 obtained Table

.

L()ad

I-beam

(.'ross

at is

secth)n,

the

Reference

I, page

Middle subjected

to

critical

264,

Values of K 3 for Simply Concentrated Load at

L,,md

L l (;J

a l,m(I

value

,ff the

are

given

Supported Middle

w

At

(t 4

4

_

11;

21

2t2

4_

Iplmr _lallgt.

_il

5

2il

('{'ntl'OllJ

W_;

|

31

117

-tq_

f;4

_11

1

lti

9

15

4

15

II

14

!1

[4

!)

27,

_,

21

h

211

{

1!1

_,

I_

o

:1_

2

FILl 3

27

1

25

1

_'_

I, 2 (1.1

L,_ad Apph¢_l !'q

P applied load

I-;(" w

!l(i

]fi(t

2.1'_

121_

41)0

|'lql.r Fi:m_u'

13

ql

15

It

15

('lqll[rl,ld

1_4 3

1_.

I

17

l'langc

22

21

7

21

I

I

9

I

in Table

I-Beams

I-2("

AptAt_,d

I. langr

5.64

at

Pis

(6)

from

4.7-4(a)

Be.am,

5.£7

15

3

t5

l

15

ti

15._

17

5

17

I

t7

2

17

2

Lql

u

193

19

i1

I_

7

4.7-4(a) With

Section

B4.7

15 April, Page If lateral

support

in Table

4.7-4(b).

Table

is provided

4.7-4(b).

L2Gj EC

Values

0.4

4

8

466

154

114

at the

of the

middle

Factor

of the

beam,

for

Lateral

K3

values

of

Support

16

32

96

128

86.4

69.2

54.5

52.4

1971

8 given

K 3 are

at Middle

2OO

4OO

w

K3 If lateral

support

in Table

4.7-4(c).

Table

L2Gj EC

is provided

4.7-4(c).

Values

at both

of the

ends

Factor

0.4

4

8

16

24

268

88.8

65.5

50.2

43.6

of the beam,

K 3

for

values

Lateral

32

49.8 of

K 3 are

Support

64

47.4 given

at Ends

128

200

30.7

29.4

320

w

K2

IV.

Simply Supported Beam, If a simply

critical

value

of this

supported

load can

40.2

Uniform

I-beam

be expressed

34. i

28.4

Load

is subjected in the

to a uniform

load

q,

the

form

EIyGJ (ql)cr

=

K4

L2

Values of IQ obtained from Reference

(7)

l, page 267, are given in Table 4.7-5(a).

Section 15

Page Table

4.7-5(a).

Values

of K 4 for

Simply

Supported

I-Beams

with

B4.7

April,

1971

9

Uniform

Load

F"

Load Applied At

L 2 GJ/EC w 0.4

4

8

16

24

32

48

Upper Flange

92.9

36.3

30.4

27.5

26.6

26.

Centroid

143.0

53.0

42.6

36.3

33.8

32.6

31.5

223.

77.4

59.6

48.0

43.6

40.5

37.8

Lowe

1

25.9

r

Flange Load

L 2 GJ/EC w

Applied At

64

80

128

200

280

3(;0

400

Upper Flange

25.9

25.8

26.0

26.4

26.5

26.6

26.7

Centroid

30.5

30.1

29.4

29.0

28.8

28.6

28.6

36.4

35.

33.3

32.

1

31.3

3 1.0

30.7

K 4 is

given

Lower Flange If the

beam

has

Table

lateral

1

support

4.7-5(b)

.

at

Values

Load

the

middle,

of K 4 with

Support

Table

4.7-5(b).

at Middle

L 2 GJ/EC

Applied At

Lateral

by

W

0.4

4

8

16

64

96

Flange

587

194

145

112

91.5

73.9

71.6

69.0

Centroid

673

221

164

126

101.

79.5

76.4

72.8

774

251

185

142

112

85.7

81.7

76.9

128

200

Upper

Lower Flange

Section

B4.7

15 April, Page If the Table

beam has 4.7-5(c).

lateral

Table

L2Gj EC

support

4.7-5(c).

at both

Values

0.4

4

8

488

16i

i19

ends

of the

of K4 with

beam,

Lateral

K 4 is given

Support

1971

i0 by

at Ends

i6

32

96

128

200

400

91.3

73.0

58.0

55.8

53.5

5i.2

w

K4

4.7.2.2

Rectan_lar For

rigidity

C

w

f

Beams

a beam

of rectangular

can be taken

= cr

y (KL) 2

S

section

as zero;

therefore,

2g +

of width

(C2g) 2 +

e

b and height

equation

(2)

GJ(KL) _ EI

h,

the warping

becomes

2 (8) y

If the load is applied at the centroid, g = 0; therefore,

f

cr

=

C 1 _2_"_yGJ S

(9)

KL c

3

By taking

G=--_-

f

cr

=

1.86

E,

K

J=0.31hb

C1

Eb 2 Lh

3, I=

hb 3, and S = bh-_2 c 6

(iO)

or

Eb 2 f cr

= Kf

Lh

where

Kf

=

i. 86 C 1 K

(ii)

Section 15

April,

Page

Values

of

cases.

Kf

For

Table

4.7-1

of the

given

are cases for

given

in

not

available

values

Equation

Figure

4.7-2 in "[',able

of C 1 and

8 must

be

and

K for

used

for

4.7-6

use

loads

Table

4.7-6

and

Figure

in equation not

for

applied

B4.7 1971

11

several

load

4.7-2,

refer

to

10. at the

eentroid

for

any

cases.

3.0

_'-L

1 Kf

2.0

t"

1.0 0

L _

ill

m

111111 0.10

1

0.20

I 11

0.30

11 0.40

0.50

LATERAL

STABILITY

c/L

FIGURE

4.7-2

CONSTANTS OF

FO[_ DEEP

DETERMINING

RECTANGULAR

'FILE BEAMS

Section

B4.7

15 April, Pa_e

Table 4.7-6.

Constants

for

Determining

Rectangular

Side

View

the

Lateral

Beams

Top

View

L/2 L/2

6

7

8

9

I0 11

12 1

17

Stability

12 of

1971

Section

B4.7

15 April,

_v

page 4.7.3

UNSYMMETRICAL

1971

13

I-SECTIONS

f_

about

For I-beams the horizontal

approximate

f cr

equation

=

symmetrical about axis and subjected for the

elastic

(KL) 2 I e + J _S_ EI c

the vertical to uniform

buckling

e2 +

C Iw y

stress

axis, bending

but unsymmetrical moment, the following

should

( 1+ GJ(KL)2_ ,r2ECw

be used

j1/

(Ref.

3).

(8)

Section

B4.7

15 April, Page 4.7.4

SPECIAL

4.7.4.

i

Loads

case

of a beam

subjected

to a uniform

not lie in one of the principal planes of the cross References 4 and 5. Reference 5 shows that the takes

the

replaced the x axis

4.7.4.2

14

CONDITIONS

Obliclue

The

1971

form

of equation

by the

expressicn

is the

axis

Nonuniform

lwith

C 1= C 3=

I:Ix/Ix,y

1.0,

bending section equation C 2= 0.

moment

to the

Cross

Section

plane

does

is discussed in for the critical The

in which

normal

that

y- and x- denote of bending.

quantityI

moment

Y

principal

is

axes

and

A concise solution for the lateral buckling strength of a tapered rectangular beam, subjected to constant bending moment and simply supported at the ends, is presented in Reference 6. Tapered cantilever I-beams have been investigated experimentally in Reference 7.

4.7.4.3

I-beams

Special End Conditions Solutions have under a load

acting perpendicular with various degrees plane.

Each

fixity.

In all cases,

rotation

that

type

about

to the principal plane having of restraint against rotation of restraint

beam

was

the beams

a longitudinal

Frequently, extends over

cantilever

been obtained (Ref. 8) for the buckling (either uniform or a concentrated load

a cantilever two or more may

considered

were

axis

considered

perpendicular

beam supports.

not be fixed

against

maximum bending of the beam about

to vary

between

to be fixed to the

plane

constant,

Cw,

to be zero

in the

rigidity either

zero

lateral

bending

ends

of the

cross

of the

buckling

formula.

and

and complete

at the

is simply the overhanging end In this case, the supported

but some restraint is supplied by continuity at the support. conservative estimate of the buckling strength can be made warping

strength of at the center)

beam

against section.

of a beam end of the flanges

In such cases, by considering

a the

Section 15 Page

If a beam for any one

is continuous

beyond

span are generally

one or both supports,

between

the cases

B4.7

April,

1971

15

the end conditions

of complete

fixity and simple

F support

covered

References

4.7.4.4

in Table

9 and

I.

The

effect of continuity has been

It is explained

in Reference

II that it is possible

buckling

tangent

E t, corresponding

modulus,

elastic modulus, show

buckling i2 and

E,

stress 13).

when

Tests

below

in the inelastic range to the maximum

in the elastic buckling gives

the bending

of aluminum

i. 0 to -i. 0, resulted

percent method.

stress

that this method

with the ratio of the moment from

in

Inelastic Buckling

to the theoretical

beams

discussed

10.

to 39 percent

formula.

a close

moment

alloy beams

subjected

at one end to the moment in experimcnt.al above

the values

in the beam

for the

Tests

on aluminum

alloy

to the experimental

along the length to unequal

(Ref.

end moments,

at the other end varying

critical stresses computed

limit

stress

approximation is constant

to obtain a lower by substituting the

varying

by the tangent

from

8

moduhls

Section

B4.7

15 April, Page

1971

16

REFERENCES

D

3.

Timoshenko,

S. P., and Gere,

McGraw-Hill

Book

Structural Design Manual, Clark, of the

J. W., American

Division, .

.

Company,

and Hill, Society

Go.diet,

J.

Flexural-Torsional

Bulletin 1942.

No.

McCalley,

Cornell

.

Krefeld,

e

.

W.

I0.

J.,

Butler,

Beams.

The

ASCE,

Salvadori, ASCE, Nylander,

of Bars

Engineering

of Open

Section.

Experiment

Vol.

D.

Welding

M. Vol.

G:

t20, H:

81,

Torsion,

J.,

and

Anderson,

Journal

S.,

Station,

Vol. 121,

G.

Research

and Tung, I-Beams.

B:

Welding

Supplement,

T. P: Separate

Cantilever

March,

1959.

Lateral Buckling of No. 673, Proceedings

1955.

Lateral i955,

Transactions of the ASCE,

Buckling p.

of I-Beams.

Transactions

of the

1165.

Bending

and

Lateral

Buckling

of I-Beams.

Bulletin No. 22, Division of Building Statics and Structural Royal Institute of Technology, Stockholm, Sweden, 1956. II.

Proceedings Structural

Lateral Buckling of a Tapered Narrow Rectangular of Applied Mechanics, September, 1959, p. 457.

Austin, W. J., Yegian, Elastically End-Restrained of the

Buckling

University

Clark.

Lee, L. H. N: On the Beam. ASME Journal

Wedge

1961.

R. C., Jr: Discussion of Paper by H. N. Hill, E. C.

Hartmann, and J. W. 1956, p. i5.

.

York,

H. N: Lateral Buckling of Beams. of Civil Engineers, Journal of the

1960.

28,

Theory of Elastic Stability.

Northrop Aircraft, Inc.

July, N:

J. M:

Inc., New

Bleich, F: Buckling Strength of Metal Company, Inc., 1952, pp. 55 and 165.

Structures.

Engineering,

McGraw-Hill

Book

Section

B4.7

15 April, Page RE FERENCES

(Concluded)

1971

17

:

f

i2.

Dumont,

C.,

Aluminum t940. 13.

Clark,

Alloy

J.

Subjected

W.,

Hill,

1957.

H.

I-Beams

Mechanics

N:

Lateral

Subjected

and Jombock,

to Unequal

Engineering July,

and

End

J.

Division,

to Pure

R:

Moments.

Stability

Lateral Paper

Proceedings

of Equal-Flanged

Bending.

Buckling No.

1291, of the

T.

N.

770,

NACA,

of I-Beams Journal ASCE,

of the Vol.

83,

SECTION B4. 8 SHEAR BEAMS

Section B4.8 15 October 1969

TABLE

OF CONTENTS Page

Shear

Beams

....................................

Plane-Stiffened 4.8.1.1

Shear-Resistant

Stability

of Web

Beams

Panel

Transverse

Stiffeners-Flexural

II.

Transverse

Stiffeners-Effect

L,,cation

Stiffeners-Flexural

IV.

Transverse

and Central

Longitudinally Compression Combined

4.8.1.2

Flange

4.8.1.3

Rivet I. II.

III.

Stresses Design

5

.........

Thickness

Torsional

Longitudinal

and

Plates

20 21 21

..........................

22

..............................

Design

Approach

4.8.1.5

Stress

Analysis

4.8.1.6

Other

22

............................. Procedure

of Web

Tension

Field

Beams

4.8.2.1

General

Limitations

4.8.2.2

Analysis

of Web

4.8,2.3

Analysis

of Stiffeners

Design

t0

20

.............................

Web-to-Flange

8

16

...............................

Stiffeners-to-Flange

........

. . .

in Longitudinal

...........................

W eb-to-Stiffener

Rigidity

Stiffeners

..............................

Design

Types

Only

7 and

Stiffened Web ...............................

4.8.1.4

Plane

Rigidity

..............................

Transverse

VI.

3 4

of Stiffener

Ill.

V.

..................

.........................

I.

RIg('!

4.8.2

1

22

......................

23

......................

23

.......................... and Symbols

25 ..................

............................. .........................

B4.8-iii

26 28 38

SectionB4.8 i5 October 1969

TABLE

OF CONTENTS

(Concluded) Page

4.8.2.4

Analysis

of Flange

4.8.2.5

Analysis

of Rivets

4.8.2.6

Analysis

of End of Beam

4.8.2.7

Beam

Design

........................... ............................ .......................

...............................

B4,:8-iv

46 47 48 50

t

Section B4.8 15 October 1969

f--

DEFINITIONS t t U

b b C

d C

thickness

of web

thickness

of attached

spacing

of intermediate

clear

web

clear

depth

effective

o_

plate

plate

;

OF SYMBOLS

stiffener

distance

of web :_,:t

I(:g

stiffeners,

or width

between

of unstiffcned

web

plate

stiffeners

plate

ratio;

e

= b/d

c

for

unstiffened

w.eb plates,

or web

plates

reinforced

by single-sided

stiffeners; = bc/d c for web

plates

reinforced

D

flexural

rigidity

of unit width

E

Youngts

modulus

#

Poisson's

I

moment

yL

limiting value of T

K

critical

shear-stress

limiting

value

by double-sided

of plate

= Et3/12(

stiffeners 1-u 2)

ratio of inertia

of stiffener

about

base

of stiffener

(next

to web)

coefficient

S

KL T}S'_B

Af

of K

S

plasticity

coefficients

elasticity _=1

for

area

of tension

stress

which above

or compression

account the elastic

flange

B4.8-v

for

the

limit;

reduction within

of modulus the

elastic

of

range,

Section

B4.8

15 October

DEFINITIONS p-D

C

rivet

OF SYMBOLS

i969

(Continued)

r

factor P

r

D

rivet

diameter

r

fb

applied

bending

f

applied

web

critical

(or

S

F

stress

shear

stress

initial)

buckling

stress

in shear

S cr

M

applied

P

rivet

q

applied

web shear

S,V

applied

transverse

A

parameter

h

height

BT

=

EIT,

"/T

=

BT/Db,

bending

moment

spacing

used

shear for

of beam flexural

flow

type

between rigidity

nondimensional

on beam of shear

beam

centroids

selection

of flanges

of transverse

stiffeners

flexural

rigidity

parameter

for

transverse

stiffeners torsional

CT

rigidity

BL

=

EIL,

TL

-

BL Dd ' nondimensional c torsional

CL FL

flexural

=

of transverse

rigidity

rigidity

stifleners

of longitudinal

stiffeners of longitudinal

stiffeners

,_arameter stiffeners

C L/Ddc

B4.8-vi

for

longitudinal

stiifeners

Section

B4.8

15 October

1969

f

DEFINITIONS

FT

C T/D

G

modulus

KL H

limiting

distance critical

F B

SYMBOLS

(Concluded)

de of rigidity value

of K

for

web

reinforced

by

vertical

stiffeners

S

central nl

OF

horizontal from (or

stiffener edge

initial)

of plate buckling

to longitudinal stress

stiffener

in bending

eY

I

stiffener moment

of inertia for longitudinal

O

B4.8-vii

stiffener

and

a

Section B4.8 15 October 1969 Page

4.8.0

SttEAR The

riveted

1

BEAMS

analysis

and design

or welded

to web

of a metal

members

beam

are

composed

of flange

eommon

problems

effieient

type

members

in aerospace

struetural

design. Shear

be_'uns

denote

the

moment

resistanee

the

extreme

fibers,

connecting The

the

generally of the

based

web

may

type

depend

economy

the

If,

however, shear

as a tension-fiehl

on many

of weight,

bc'anl

most

factors. tt.

Wagner

of weight:

4V h

wh ere V

= shear

load,

h

= depth

of web,

lb

and in.,

of beam.

which

In shear

are

is provided

beams

eoneentrated

by the

near

thin web

caps. beams

response

within

causing

flanges,

resistance

of shear

the web

o1" shear

the economy

shear

desit,m

upon

is known

by the

and compression

and

of load

beam

the

beam1.

application

The

and

is inhibited

shear-resistant

the

is provided

tension

analysis

a particularly

as structural

to the

design

ultimate

applied

components

shear

load,

the web

is allowed

to be resisted

in part

the

loads. beam

to buckle

are If bueMing

is known :ffter

by tension-field

as a

some action,

beam. suitabl_'

for

One of the

most

[ 1 ] offers

a particul:_r" common

the following

dcsiRn

factors criterion,

apl_lication is based based

on upon

SectionB4.8 15 October 1969 Page 2 with

the

when

recommendation

A > 11 the

shear-resistant

choice

between

of web

to be used.

The data

the

criterion

designs

for

Shear-resistant Paragraphs

4.8.1

have

shear-resistant beams and

4.8.2

is best.

other

should

techniques

A < 7 the

web

two; factors

above

and design

weight

that when

than

tension-field When

weight

not be adhered become

web

7 < A < 11,

will

then

to rigidly,

available

that

is best,

and

there

is little

determine

however,

have

resulted

the

because in reduced

beams.

and tension-field respectively.

beams

will

be discussed

type

in

new

Section B4.8 15 October 1969 Page

3

/f-'_.

4.8.1

PLANE-STIFFENED As previously

not buckle The

stated,

under

analysis

the

of this

beam

of the tension

stiffeners

are

thickness, weight the

of the

thickness

strength The

shear

plate

requirements)

weight

The

1.

3.

increase

it does beam.

is,

with

the

[lange,

and

than

from

rather

axial,

the a

the

usually

(just

stability be less thickness

between

and

with

solution

beams

web

by increasing

be economical

as possible

tr:,nsverse

the transverse of the

be increased

economical

in the

of the

of the

Elements

compression

always

of shear-resistant

bending,

Elements under

will

buckling

shear, 2.

A more

stiffeners

criteria

Local

That

standpoint

not always

as small

by an adequate design

can

"rod increasing

of such

introduced

will

used.

of the

that

to as a shear-resistant

the

a stability

web

a desigm

material

is so designed

one of stability.

the web,

from

web

standpoint.

of the

but such

is referred

flanges,

BEAMS

whose

is primarily

stress

stability

RESISTANT beam

loads

all designed

allowable

The

a shear

applied

exception

material

SHEAR

the

transverse stiffeners

respect

is obtained

thick

enough

the

to the by keeping

to fulfill

by introducing than

its

stiffeners.

additional

weight

of the plate. may

be stated

stiffeners

under

stresses must

not

as follows:

must

combined not occur.

buckle

locally

stresses.

flange

must

not buckle

locally

under

the longitudinal

stresses. If the

criteria

above

are

not met,

the

procedures

of analysis

that

follow

are

applicable. Design following

analysis paragraphs.

techniques

for

shear

resistant

beams

are

given

in the

not

SectionB4.8 t5 October i969 Page4 4.8.1.1

Stability

The

of Web

critical

Panel

buckling

stress,

F

S

of a web

'

panel

of height

d

C'

width

b,

cr

and

thickness

F

t,

s

is given

by

K 7r2E 12 ( l-p2 )

for

_b

dc

(1. a)

S

or

F s

K 7r2E

2

Us

12 (l-pz)

,

where

K

S

is a function

by the stilfencrs Fil4urc increases

(1. b)

c

of the

aspect

ratio,

d /b,

and

e

the

edge

with

how the

decreasing

value

values

of the critical of the

aspect

shear

stress

ratio,

b/d

edge are

In vicw stiffeners, made

so effective

of the

also

and

included

indicates

effects

the

of obtaining

relationship

buckling the

figure

in increasing

of theoretical the

the

This

importance

a number

to determine

stiffeners, have

conditions.

stress

and

buckling the

correct

desi6m

of the

the

size

stiffened torsional

for

of rectangular

rigidity,

webs.

of intermediate

and spacing plate.

a number

why vertical

investigations

web

Ks,

C

clearly

stress

experimental

between

of stiffener

quite

cocl ficient,

(d _b), C

stiffeners

offered

and flanges.

1 shows

of different

restraint

have

been

of intermediate These

stiffener

investigations thickness

Section

B4.8

15 October Page

and

rivet

location,

stiffness. given

The

central

longitudinal

procedure

for

the

stiffness,

desi_ma

and

and

5

single-sided

analysis

of these

1969

or effects

double-sided will

be

below.

16

14

12

\

CRITICAL SHEAR

STRESS

COEFFICIENT K

10 $

\

PLATE ON

CLAMPED

ALL

FOUR

EDGES

\ -"v---PLATE

SIMPLY

SUPPORTED ALL 4

i.0

FOUR I

1.5

EDGES I

2.0

1.

K

VERSUS

ASPECT

#%

3.O

2.5

ASPECT

FIGURE

''f ON

RATIO

RATIO

FOIl

3.5

Go

10/d c

DIFFERENT

EDGE

RESTRAINTS

S

I.

Transverse

Stiffeners

Thboretieal

between and these

Ks

and

boundary

the

has

been

points

be

Rigidity

performed

noted

2 is a typical that

of discontinuity

Only

[ 2-4

parameter

Figxwe

It will These

Flcxural

nondimensional

conditions.

parameters.

K /3/ curves. S

work

--

points

] to ascertain EI 3_ (=D--b) plot

for

showing

of discontinuity denote

where

the

relationship

various the

aspect

ratios

relationship occur

changes

on in the

of the buckle

Sectioa

B4.8

i5 October Page

pattern

occur.

It can

there

is no appreciable

value

of _,

since

also

be seen

increase

higher

values

30I

that

when

in Ks.

This

would

result

a certain value

V

COEFFICIENT 10

25

50

75 ¥ =

2. THE FRALICH

AT THE

design.

PANEL SIMPLY SUPPORTED ON ALL FOUR EDGES-

I/

CRITICAL SHEAR STRESS

FIGURE AND

_'L or limiting

...... VALUE OF.

ORTHOTROF

Ks

of _ is called

in an inefficient

....

25

15

of _, is reached,

7 I

2O

value

1969

6

100

125

175

EI/(Db)

THEORETICAL K/_ RELATIONSHIPS DERIVED BY STEIN FOR INFINITELY LONG PLATES SIMPLY SUPPORTED

EDGES

AND REINFORCED

BY STIFFENERS

0.5d

APART. C

For

design

purposes,

it should

be noted

thickness

is equal

7L

=

27.75

5, L

=

21.5

that

the following these

relationships

to or greater

(d e)-2

relationships

_ 7.5

((_e) -2 _ 7.5

than

the

are

for

valid

thickness

for double-sided single-sided

should only

for

of the web

stiffeners, stiffeners,

be used.

However,

stiffeners

whose

plate.

(2. a) (2. b)

Section

B4.8

15 October Page

K L

=

7.0 + 5.6

=

b /d

(OZe)-2 for both single-

1969

7

and double-sided

stiffeners,

(3)

where

a e

c

a

=

b/d

stiffeners

parameters

Stiffeners

--

have been

Effect

(b->d)

of Stiffener

than the web-plate

investigations in Figure

of single-sided

[ 5].

to evaluate

3.

It was

The

their

shown

Thiclmess

and

out to investigate

carried

that affect the behavior

legs thinner

c

c

Investigations

shown

(b _-d ) and

c

Tr:msverse

in these

stiffeners

c

for single-sided

e

II.

for double-sided c

that

the

on

stiffeners having

Ks,

primary

Location

fully the various

parameters

effect

Rivet

attached

tu/t and c/t were

where

influence

studied

t,

t u,

and

c are

on

K

was

the

s

as value

c and that the tu/t variation had little effect on K

S

.

Figure

parameter

4 shows (t

/t)

the

(t/c)

variation

1/2.

From

of the

K

S

with

figure

the it

U' ttj !

'"

I

can be seen that for the stiffener to provide

l-[

't

wdue

of K S equal to KL,

it is necessary

(t /t) (t/c) '/2 -> 0.27

a

that

.

(4)

U

FIGUR

E 3. tu ,

VALUES AND

OF

t,

This

equation

can

be

used

to determine

the

e position

of

the

rivet

for

fully

effective

stiffeners

Section B4.8 15 October i969 Page

for

various

value

of K

values

of tu/t.

should

be reduced

S

For

values

of (tu/t)

as shown

(t/c)

in Figure

1/2 less

8

than

0.27

the

4.

25

. ..I I / KLEC UAT,O. (3>

// 2O

GROWTH

LINEAR

K -VALUE

15

MARKED

A

/T I

10 20

0

40

60

80

100

120

Y

FIGURE

4.

VARIATION

0.4

0.2

0

OF K

WITH

THE

0.6

0.g

(tu/t)(t/c)

STIFFENER

THICKNESS

S

AND RIVET III.

Transverse

Stiffeners

Theoretical

results

between ratio

K

s

and

the flexural

of torsional

longitudinal disposed

the

midplane

FIGURE

and Torsional

been

[4,

obtained

to flexural

It was

PARAMETER

-- Flexural

rigidity

rigidity

edges. about

have

LOCATION

assumed

that

of the web

5. VARIOUS

6-8 ] that

of the stiffeners rigidity the

for

Rigidity

for simply

provide various

were

as shown

in Figure

SHAPES

values

supported

stiffeners

plate

relationships

or clamped

symmetrically

OF STIFFENERS

of the

5.

r

Section

B4.8

15 October Page

Figures

6 and

7 give

K

/31 relationships

for

b = d and

1969

9

b - d/2

with

the

S

longitudinal for

edges

b = d ,and

of CT/B

simply

b = d/2

T plotted

is

supported.

with for

the

Figures

longitudinal

a closed

circular

8 and

edges

clamped.

tube.

This

K

0.6

value

lo

AND

.r/..

s

/3/ relationships

The is

cT/B =

u

Ks

9 give

maximum

value

0.769.

t

0.769

__

_-I_ _ LONGITUDINAL EDGES SIMPLY SUPPORTED

9

8

-_

_

#

c

dc

THEORETICAL BUCKLING

P/M MODES

1/0

7

/

3/1 2/I

6

/

4/I

24

FIGURE

From buckling place

these

figures

resistance, of the

a thin-walled

6.

K S VERSUS

YT

it is

that

Ks,

open-section circular

are

shown

obtainable

stiffeners tube

for

IIELATIONStIIPS

very by

significant using

stiffeners

used. (CT/B

b :-d

increases

closed-section

so frequently the

FOR

T

For = 0. 769)

C

in the stiffeners

ex,'unple, the

in by

gain

using

in K L

Section B4. S 15 October Page

35--

o._ _-_

30

0.2-0.4 -._ O.OS-

1969

i0

° 0.6 0.1--_

' 0,769 _ KI

LONGITUDINAL

20

SIMPLY

EDGES

SUPPORTED

,.Eo,, UC L.O MO0,/0 S

,o

2/I S

...........

3/i 4/1

24O

FIGURE

is 25 percent longitudinal gains

are

and edges

Transverse

may

make

VERSUS

60 percent

5,T RELATIONSHIPS

supported.

and 43 percent design

FOR

b = dc/2

for a = 1 and c_ = 2, respectively,

are simply

weight

closed-section

The

Ks

13 percent

if a minimum

IV.

7.

For

when

the case of clamped

the

edges,

for a = 1 and a = 2, respectively.

is desired,

consideration

should

the

Thus,

be given to

stiffeners. and Central

use of deep

beams

it desirable

Longitudinal with webs

to employ

Stiffeners

having

a high depth-to-thickness

both vertical and horizontal

ratio,

stiffening.

When

St, ction

B4.8

15 October Page

19(;9

11

f

yi

14

A,'f'

] 12

/

#r

10

"_

=

....

--.,,.-'r-

.....

_

Do, K L =

.....

.

LONGITUDINAL.

_

14JJ2

_

EDGES

t

_

CLAMPEO

'

TH:O:,TIC,L 2

t_

,UCIk_I/:GMO)E$ 5/'1 ....

I

3 I 41

.... --.--

1/0

......

9 0

2

4

6

8

10

12

14

16

18

20

22

24

26

Y=

1 "T

BT

=

0.769. 0.40._ 0.20_

.or--?0_\" /

I

/o'os.\'

!

),r = \

_\ \

T.... r _--f_--------

x

iN"

KL=

41,55

m,

KL. =

29.17

l

-

,_+.+_-.7:'lX--_--7

c_,

I

y.r =

LI

Ks

LONG_TUDtNkL

20

i,

EDGESCLkMPEO

,

i THEORETICAL

BUCK

P 'M

k ING

MOOE|

P/M

2

t

4'1

3

! ....

$

----I ....

10 0

20

40

60

80

100

120

140

160

I_

2@0

2_

¥'r

FIGUIIE

9.

K,

"/T RELA'I?IONSIIIPS;

ASPECT

I{ATIO

o_

2.0

2_

2JO

Section B4.8 15 October 1969 Page

a web

is subjected

stiffener can

to shear,

is at middepth.

result

in more

stiffeners

are

the This

economical

employed.

For

buckling

stress

achieve

a given

as little

as 50 percent

most

effective

position

combination

of vertical

designs

are

than

example, with

the weight

when

when

horizontal

vertical

stiffening

only

of stiffening

and

required

a single

and horizontal

possible

horizontal

of the weight

for

12

vertical

required

to

stiffening

only vertical

can

stiffeners

be are

used. If neither rigidity then

and

of the the

the value

transverse

vertical of 7LH

or central

stiffeners

have

necessary

_LH

=

11.25

KLH

=

29 +4.5(b/d

longitudinal a rigidity

to produce

stiffeners

equal

the limiting

has

torsional

to or greater

than

value

is given

of KLH

EILv

(b/de)2

, by

(5)

and

Additional used

weight

in either

in Reference weight

the

respectively,

4 and

can

.

(6)

bc achieved

gains

in the

of up to 25 and

given

value

For

of K L (which for

stiffeners

parameter

strong

direction.

60 percent

closed-section

have

if torsionally

or longitudinal

shown

by using

Investigators

)-2

transverse

6 have

of the web)

only.

savings

e

_ equal in the

studies

stif[cners example,

are studies

is proportional to one and

transverse

to the

two, direction

on the following

in References

9: 1.

Transverse

and

2.

Transverse

stiffeners

stiffener

longitudinal

possessing

stiffeners

of closed only flexural

of closed

tubular rigidity.

cross

tubular section;

cross

section.

longitudinal

Section

I34.8

15 October Page

3.

Transverse

stiffeners

stiffeners Figure

being

using

used

in conjunction

c_ _

1,

using

it

of closed

10 enables

from

one

torsionally

is evident

little

strong

in buckling

resistance

stiffener.

However,

an

as

a central spaced

stiffeners,

but

(that

benefits

longitudinal

that

as

the

result stiffener

is

stiffeners.

buckling

When

resistance

a considerable

a torsionally

is,

that

longitudinal

in the

by using

c_ increases

the

transverse

is obtained

be obtained

rigidity,

of the

when

increase

13

section.

assessment

of equally

transverse will

flexural

cross

stiffeners

a system

that

only

tubular

to make

strong with

torsionally

possessing

1969

strong

transverse

by

increase longitudinal

stiffeners

are

6O

55 FLANGES

ANO

LONGITUDINAL

STIPFEIqER

PROVIDE

SUPPORT.

TRANSVERSE

STIFFENERS

SIMPLE

ALL

EDGES

CLAkIPEO_IaC_'/CLAMPE

O

_ SUPPORT

•

5O

45

Ks

40

'

_

\

35

Y'. \

"-.

I ,CA.CEoR"'o,._ I

_

L °"or Tu °l"* "__ __.___2:':"0. STIPFENER

PROVIDES

I

3O , ALL

I!/OGE$

_

P :uOp/DER

TCLFA:::

GE D

S AN "_

25

5;.; _ iUPPORTED

I51MPLY

SUPPORTED

LO_IGITUDINAL

STIPFENER

20

2

1 CX:

FIGURE

10.

K

VERSUS S

ASPECT

RATIO

0

dc/b

FO[_

VAIIIOUS

EDGE

RESTRAINTS

Section

B4.8

15 Oct()ber Page

more

closely

strong

spaced),

stiffeners

resistance

the

relationships

above. b = d and other

11,

different

values

and

s

and

positions

of the

strong

the

ratio

is

readily

= soo lOO._.-----_ 30 ,,-v

(_ = 1.7,

obtained

from

of torsionally buckling

stiffeners

and

a

to that

obtained

stiffeners that

is

is

for

relationships strong

use

i4

the

equal

seen

torsionally be

when

propertics

typi(':_[

can

the

transverse

stilfener

13 give of the

aspect

and be

from

transverse

rigidity

it will

K

12,

until,

stiffener

Thus,

between

Figures

significant;

torsional

longitudinal

rigidity.

in K resulting

torsionally

without

strong

flexural

more

with

stiffener

torsionally

increase

becomes

obtained

longitudinal

only

the

1969

with

a

possessing necessary the

to know

cases

between

enumerated

K_ and

YT

_tiifener.

Culves

for

Reference

4.

points

The

]

for

°

i

3o 20 10

2C

K

I

b--d

3I;c"

f_

10 / /

P/M

2/I ........

3,"2

THEORETICAL

5/I 4/il

20

40

80 y.r =

FIGURE TORSION

11.

Ks'

YT

RELATIONSHIPS;

(C L :_ 0.769 FLEXURAL

BL);

RIGIDITY

EID

100

140

b

LONGITUDINAL

TRANS\rERSE

BUCKLING MODES 120

STIFFENER

STIFFENERS

(C T -: 0) ; ASPECT

RATIO

WITH

WITH

a

_ l. 0

ONLY

Section B4.8 15October 1969 [>age

30

°U

15

= 500 100

A

2o -'--d

c

I-------,H

fi

I/ -E rdc c,f2 ¸¸2 3/1 2/'1 } iS/'1 120

4:1

0

1

0

20

40

60

80

100

THEORETICAL P,'M MODES i 1,10

BUCKL_NC

'y.r= EI'Db

FIGURE

12.

Ks'

SUPPORTED: (C T

%1" I1ELATIONSIIIPS;

TIIANSVEI1SI'2

_ 0.7(19

BT):

STII"FENERS

IAI)NGI'FUI)tNAL

RIGIDITY

(C

LONGITUI)INAI_

L

WITII

STIFFENERS

_-tl);

ASPECT

EDGES

SIMI'LY

TOIISIONAL

RIGIDITY

WIT II ()NLY

FLEXURAL

RATI()

_'

1.It

/'7..---1 ff 3O :if'////

20_///

K

YL-5

1/

1

:

o b=d

/

dc/2

ilt

4/I / 5/1 /

m_:.km_ i_li

dc/2

o

2o

40

60 Y.r =

FIGURE EDGES;

13. ALL

K S ' YT RELATIONSHIPS; STIFFENERS

WITI!

80

100

120

SIMPLY

SUPPOI_TED

TORSIONAL

RIGIDITY

I,ONGITUDINAL 0. 769

(C T

C L=

0.769

BL);_

I 140

El 'Db

ASPECT

RATIO

ff

1.0

B

T'

Section

B4.8

i5 October Page

marked give

A on the

95 percent

point

for

increase

of the limiting

in the

of Ks would

value

beneficial

In deep

from (1)

The

the

neutral

region stiffener

from

it is often

in locations

where

high.

stiffener axis

(Fig.

at a distance

at the 14a)

from

the

itself

does

fb

STIFFENER

_

_: \

Thus,

the

to stiffen

longitudinal of the

longitudinal (2)

the

of the

fb

%

stiffener center

stiffener plate

compressive

POSITIONS

cutoff a small

in TT would

Compression

compressive

line

by longitudinal resulting

be considered

of the web,

located (Fig.

plate

stresses will

that

fb

.

14b).

In case

STIFFENER

fb

t_,

\

(b)

LONGITUDINAL

is,

1, the

dc

OF

here:

in the compressive

stresses.

(o)

14.

only

increase

the web

dcl 2

FIGURE

would

as a good

chosen,

extra

economical

edge

not carry

of _/T are

in Longitudinal

and

of TT which

is suggested

Plates

the

values

standpoint.

Two positions

located

13 are

This

values

occur.

Web

and

of K . s

a weight

Stiffened

are

12,

If greater

beams,

bending

11,

value

design.

Longitudinally

stiffeners

of Figures

an efficient

not be very V.

curves

1969

16

STIFFENERS

at

Section B4.8 15 October 1969 Page Adding

this

assembly

cost;

structural of beam A.

longitudinal therefore,

depth,

that

span,

Stiffener

stiffener

such

arrangement

For

stiffener

and

results

in another

construction

will

save

external

structural

is not widely

structural

17

weight

used

part

and more

although

under

certain

practical

value

it is a conditions

loading.

at the Centerline

a stiffener

moment

at the

of inertia

eenterline,

the

largest

of the

is

I =

O. 92t3d

_'B

(7) c

With

this

value

of I O , the

critical

bending

stress

is

FB cr

rib

For

<)2

12(1_g2)

values

of F B

for

(_) for

stiffener

c_ >- 2/3

moment

of inertia

less

than

and

Neutral

Io,

see

er

Reference B.

10. Stiffeners The

the eenterline the

inelastic

in improving

Located

increase

in buckling

of the web range. the

Between

amounts

Stiffeners

stability

of web

Compression strength

that

to only at the plates

Edge can

50 percent

centerline in ease

are

be obtained of the

by a stiffener

unstfffened

therefore

of pure

Axis

bending

not very stresses.

plate

at in

effective

v

SectionB4.8 15()ctobcr 1969 Pagei8 For

a stiffener

is given

spaced

at a distance

m -- d /4,

the critical

e'

buckling

stress

by

FB cr_?B

if

1017r2E 12(1-hz)

(_c)

(_ -> 0.4),

2

(9)

3: -> Yo

Where

y

EI

=

T/BDd c EI =

O

To

Dd

=

(12.6

+506)

,

a = b/d

ot2

-

3.4

c_3

(o_

for

a stiffener

1.6)

and

C

6

A

=

rldct

Comparison of the region

plate shows

Limited longitudinal

c

of the

with

the

that

numerical

of the

plate.

This

value

of the

buckling

K = 129 and

is larger

from

the

results

the

stiffener

compression

results

obtained obtained

reinforcement results located

than edge.

for

a plate

stiffened

in the latter have

been

at a distance

information strength

above

is plotted of the

in the case

plate

case

obtained m = dc/5 in Figures stiffener

of a stiffener

at the in the

is much

for

plates

from 15 and system located

centerline

compression more

effective.

reinforced

the

by a

compression

16.

edge

The

largest

corresponds

to

at the

distance

dc/4

Section B4.8 15October i969 Page i9

K

129

m

120

_m=

_TJ

°/..7 8O

ii//y r

4ole

tb

'

_,,_

--_----

_',l

m

t dc

! El

I 10

FIGURE

15.

Dd C

20

K VERSUS

y FOIl

30

VARIOUS

4O

VALUES

OF c_, 5 = 0

K

120 12_ _---;I_ .....

AT/ =5

"-i

4011-

gj

0

FIGURE

16.

] dc

i

El

I

Dcl c

I 10

K VERSUS

20

30

y FOIl VAIIIOUS

40

VALUES

5O

OF c_, 6 = 0.10

SectionB4.8 15 October i969 Page20 VI.

Combined Stresses The above mentionedbending, shear, and possibly axial andtransverse

stresses that act upon the web shouldbe interacted by the following equations. R2+R _1 s cL R

2+R

(10)

<1

s

(11)

cT

R S 2 +RB2 RB1.75

(12)

_-<1

+Rc

(13)

< 1 L

where

fS R

=

s

fe

Fs

RB

'

=

FB

cr

and

the

c

-

FC er

cr

subscript,

transverse.

R

'

L,

The

indicates

critical

longitudinal

values

of F C

and

the

should

subscript,

T,

be obtained

from

indicates Section

C2.1.1.

cr

If the

interaction

must

be performed.

4.8.1.2

Flange

The The

beam

ultimate

material, connections, of the

equations

cap.

above

are

not satisfied,

an iteration

of the design

Design flanges

allowable

are

designed

stress

reduced

by the

the

efficiency

for

attachment factor

for the

tensile

tension efficiency

is the

ratio

and

compressive

flange

is equal

to Ftu

For

riveted

factor. of the

norma[

net

area

to the

forces.

of the or bolted gross

area

Section

B4.8

15 October Page

Compression on

the

compression

shear

at

the

techniques

should

flange

web-flange are

4.8.1.3 I.

flanges

be

connection,

available

Rivet

can

be

in Section

designed

for

a combination and

column

transverse

2 1

stability.

of normal

The

force,

forces.

1969

loads

longitudinal

Specific

analysis

B4.4.0.

Design

Web-to-Stiffener Although

attachment

no exact of the

information

stiffeners

Table Web Thickness

to the

B4.8-I.

is available web,

on

the

data

Rivets:

strength

in Table

B4.8-I

required are

of the recommended.

Web-to-Stiffener Rivet

Rivet (in.)

the

Size

Spacing

(in.)

0. 025

AD

3

1.00

0. 032

AD

4

1.25

0. 040

AD

4

1.10

0. 051

AD4

1. O0

0. O64

AD4

0.90

0. 072

AD

5

1.10

0. 081

AD

5

1.00

0. 091

AD

5

0.90

0. 102

DD

6

1.10

0. 125

DD

6

1. 00

0. 156

DD

6

0.90

0. 188

DD

8

1.00

Section

B4.8

15 October Page

II.

1969

22

Stiffeners-to-Flange No information

stiffeners than

to the

that

same

used

size

III.

is available flange.

in the

be used

on the

strength

It is recommended

attachment

whenever

of the

required

that

one

stiffeners

of the

rivet

attachment

the next

to the web

size

of the larger

or two rivets

the

possible.

Web-to-Flange The

rivet

size

(bearing

or

spacing,

gives

undue

shear)

stress

4.8.1.4

value

q,

proper

margin the

b,

of a stiffener

lie designed the

applied

of safety. rivet

factor,

so that web

the rivet

shear

flow

allowable

times

For

a good

design

and

Cr,

should

not be less

the

rivet

to avoid than

0.6.

Approach of stiffened

of t and

should

by q x p,

concentration,

design

Assuming

spacing

divided

the

Design

The

and

h,

and

shear-resistant

E are

known,

compute

f

required

to develop

s

= q/t.

The

beams

is a trial-and-error

the first

step

problem

is to find

an initial

is to assume

buckling

method.

a reasonable

the moment

stress,

F

of inertia , in the web

s cr

greater

the

than

desired

1

by the desired

8

Fs

' and

Fs

cr as a function

of F

margin

/_s

of safety.

if required.

The

procedure

Ks is then

found

is to choose

from

equation

(1)

cr S

/7

and d/t

the

required

or h/t.

Then,

depending

on the

type

of stiffening

cr

arrangement 4.8.1.1.

Particular

an efficient given

used,

design.

to longitudinal

attention For

I of the should

be given

a minimum-weight

stiffeners

and/or

stiffener

is obtained to stiffener

design,

torsionally

from properties

consideration strong

stiffeners

Paragraph that should

provide be

as discussed

Section B4.8 15 ()ctober 1969 Page 23

in Paragraph and

rivet

4.8. I.I. Attention should also be given to stiffener thickness

location

as discussed

in Paragraph

4.8.1.1-II.

4.8. I.5 Stress Analysis Procedure The stress analysis procedure for the web of a stiffenedshear resistant beam all

is straightforward

tmown,

and

the first

4.8.1.1,

depending Then

etc.

F

to apply.

step

is to obtain

upon

the

can

S

easy

K

aspect

be obtained

Since

from

s

ratio,

q, b, h, E, t, fs'

the appropriate

torsional

from

equation

C2.0.

Then

curve

rigidity,

(1).

and I are in Paragraph

stiffener

If necessary,

thickness, values

of

cr

77 c,'m be obtained

from

S

Section

the margin

of safety

for

the web

is

F S

M.S.

er

-

-1.

f

(14)

S

4.8.1.6

Other

The

web

trade-off Three

light

as,

previously

lightness. be kept

types

expense

or lighter

than,

however,

is introduced

into

The

nonbuckling

a web with

Actually, separate

a stiffener

the beam.

webs

The

must web

the

of parts low,

is one

are

the

of weight

frequently

web

in most

stiffeners.

There

be provided

wherever

types

used cases

in is as

is a general a significant

are:

Web

with

formed

vertical

beads

at a minimum

II.

Web

with

round

lightening

holes

having

spacing

costs

problem

I.

various

number

expense.

of stiffeners.

in that

a large

manufacturing

to a minimum.

of shear-resistant, the

require

To keep

manufacturing

to save

limitation, load

must

versus

Design

discussed

to achieve of parts

design

of Web

designs

(stiffeners) number

Types

45 degree

spacing formed

flanges

at

SectionB4.8 15October i969 Page24 III.

Web with vertical

round formed

The webs

with holes,

hydraulic

and electrical

Procedures Reference

11.

for

lightening beads

between

II and III, lines

the design

holes

also

having

of these

beaded

flanges

and

holes.

provide

that are

formed

built-in

sometimes beams

access

space

for the many

required. should

be obtained

from

Section B4. 15 (_tober Page

4.8.2

PLANE If web

buckling by

buckling

occurs

is resisted

shear-resistant

defined

TENSION-FIELD

as

an incomplete

some

by pure

action

25

BEAMS :ffter

in part

of the tension

application

tension-field

web

(Fig.

field,

T L

17).

or

¢ff load, action This

partial

tile of the

action

tension

of

shear

(a) NONBUCKLED

(b) PURE

17.

STATE

("SHEAR-RESISTANT")

DIAGONAL-TENSION

OF

STRESS

and

the

web

field.

WEB

WEB

IN A BEAM

load

web,

tl-]l

h

FIGURE

S 1969

WEB

beyond in part is

Section B4.8 i5 October 1969 Page

The In this pure

theory theory,

after

tension-field

in practice,

partial

action not

Peterson,

the

The

limited

to beams

alloys

here.

Levin

from

other

4.8.2.1 The reasonable normal

[ 12 ],

developed

verification

of these

beams.

an extension

the

analysis

Extension

of the

the designer

of Kuhn's

work

analysis

exercise

in the

analysis

for

indicates

of beams

restricted

the

must

analysis

range

was

by

nonexistent

results

the

As a consequence,

alloys. and

within

[ 1] .

is resisted

is essentially

experimental

beams for

load

a semiempirical

with

for the

by Wagner

shear

behavior

Correlation

alloy

published

the total

be discussed

is not documented,

in attempting

buckling,

is conservative

aluminum

was

Such

experimental

7075S-T

beams

of the web.

beams.

analysis

tested.

web

and

tension-field

that

tension-field

initial

and will

Kuhn,

and

of pure

26

Kuhn

to 2024S-T analysis

to include considerable of beams

is

other caution

fabricated

alloys. General

Limitations

methods

of analysis

assurance design

and

Symbols

and

design

of conservative

practices

and

given

herein

strength

proportions

are

believed

predictions,

are

used.

The

to furnish

provided most

that

important

points

are: I.

The

uprights

tu/t> II.

III.

should

not be "too

thin";

keep

0.6.

The

upright

0.2

< b/h

The

method

with

webs

When flanges

h/t

spacing

should

be in the

range

< 1.0. of analysis in the < 115,

must

range the

be taken

presented 60 < h/t portal into

here

is applicable

only

to beams

< 1500.

frames

effect

and

account

by using

the

effect

Reference

of unsymmetrical 13.

Section 134.8 15October 1969 Page27 SYMBOLS

(in addition area

Af A

to those

of tension

given

in the front

or compression

matter)

flange

actual area of upright (stiffener) U

A

effective

area

of upright

ue

C1,C2,C3

stress

F

column

concentration

factors L_

yield

stress

(the

column

stress

co

allowable

F

column

at m P

-_

O)

stress

C

F

ultimate

allowable

compressive

stress

for natural

ultimate

allowable

compressive

stress

for

ultimate

allowable

web

average

moment

crippling

max

F

forced

crippling

0

F

shear

stress

S

If I

of inertia

required

moment

moment

of inertia

of beam

of inertia

flanges

of upright

about

its base

S

I u

Msb

secondary

P

applied

P

bending shear

of upright

about

moment

its

in the

base

flange

load

upright end load U

distance fcent ff

U

centroid

to web

compressive

stress

at centroidal

compressive

stress

in flange

component f

of upright

average

of the lengthwise

diagonal

axis because

of upright of the

distributed

tension

compressive

stress

in upright

vertical

Section

B4.8

15 October Page

fsb f

secondary

bending

stress

component

of the

diagonal

maximum

U

compressive

in flange

because

of the

1969

28

distributed

vertical

tension stress

in upright

max

k

diagonal

b

spacing

h

effective

tension of uprights depth

of tension qr

k

of beam

centroid

of compression

flange

shear

angle

of diagonal

load,

web-to-flange

and

web

splices

tension

of gyration of upright with respect to its centroidal to web (no portion of web to be included)

critical

to centroid

flange

rivet

radius parallel

P

factor

shear

stress

axis

coefficient

SS

restraint

R d, R h

angle

apD T

coefficients of diagonal

tension

transverse

load

N

applied

Q.

static

moment

h

height

of stiffener

U

L

effective

e

stiffener

load-per-inch

o)

4.8.2.2 The

Analysis web

shear

of cross

for

pure

tension

field

beam

section

length

acting

normal

to end bay

stiffener

of Web flow

can

be closely

approximated

by

V

q: T

(Is)

Section

B4.8

15 October Page

From

this,

f

it follows

=

S

The

that

the web

shear

stress

1969

29

is

q/t

(16)

critical

buckling

stress

of the web

is given

by

F

c =k ss

12(1-p

z)

b

Hh + 1/2(Rd (17. a)

and

F

scr k

ss 12(1-p

z)

- Rd)

d + 1/2(Rh

b > d c. (17. b)

The restraint

value

of kss

coefficients,

is obtained are

from

given

Figure

in Figure

18. 19.

The

Figure

values

R h and

20 provides

Rd,

F

for S er

case

_7 ¢ 1.

calculated

When from

disregarding

the

disregarded

and

R h is very

the

equation

small, above

the value may

presence

of stiffeners.

the

value

latter

is used.

be less In this

of the than case,

critical the

shear

value the

stress

computed

stiffeners

the

are

the

SectionB4.8 15October 1969 Page30 10

/ /

/ /

i

J

f r!

k$$

i d

f$]

¢

r± m._r------_

_mm._--------

r b, dc:

o

0.2

CLEAR

PANEL

0.6

0.4

qlr-----_

b

I

vI

DIMENSIONS

1.0

0.8

dc 'b

FIGURE

18.

k

VERSUS

d /b

SS

The

loading

ratio,

fs/Fs

, is used

e

to determine

the

tension

field

factor,

k.

cr

It may

be calculated

k-:

tanh(0.5

by

logl0 fs/Fscr)

f

> F S

, S cr

(18)

Secti_,n

B4.8

15 ()ctol)er l)age

1!)6_)

31

Rk, R d 0.8

/ °, // /i

0

/

t -

/

0.5

WEB

FLANGE

t

-

STIFFENER

THICKNESS

Rh -

RESTRAINT

COEFF.

ALONG

STIFFENER

Rd -

RESTRAINT

COEFF.

ALONG

FLANGE

1.0

19.

EDGE

RESTRAINT

THICKNESS

1.S tu T '

FIGURE

THICKNESS

t! -

2.0

3.5

3.0

t

COEFFICIENTS

FOR

\VEB

BUCKLING

STRESS

SectionB4.8 15 October t969 Page

32

4O

3O 2024-T3

F$cf

20

lO

I

l lO

20

30

40

F scr"

FIGURE

20.

VALUES

OF

GO

60

q

F

WHEN

77 ¢ 1

S cr

or

it may

read

from

Figure

21.

For

values

of f

-

F

, the

S er

unbuckled

state.

web

is

in the

Section B4.8 15 October Page

1969

33

f

1

N

© ;I

<

b

.I

< ©

R

m

2 u

9

M

bl

2 41

N © r..) <

qr_

2: r_ i

Z

f-

r_

o o • -.

¢D

0

q' _iO.L_)¥:l

0

NOISNIJ.

"IVNOOVI(]

0

Section

B4.8

i5 October Page

The shows

angle

of the diagonal

the variation

of tan

tension

is then

o_ as a function

obtained

of k and

from

tb/A

34

Figure

.

i969

22,

For

double

For

single

which stiffeners,

ue

A

is equal

ue

to the cross-sectional

area

of the

stiffeners.

stiffeners,

A U

A

=

ue

,,. 1 + _e/p) 2

It is recommended to a maximum

that

(19)

the

diagonal

tension

factor

at ultimate

load

be limited

value,

/

k

max

to avoid

excessive

fatigue

average

maximum

f

where

wrinkling

and

,

(20)

permanent

set

at limit

load,

thereby

inviting

failure.

The the

= O. 78 - (t - O. 012) 1/2

S

web

web

= f max

S

stress.

factor

The

web

allowable

+kC2)

web

from

F

be appreciably stress

is given

smaller

than

by

,

(21)

obtained

c_ differing

arising

stress,

may

coefficients for

factor

fs'

maximum

empirical

accounting

stress-concentration

stress,

The

(1 +k2C1)(1

C 1 ,-rod C 2 are

correction

shearing

from the flange

, is given

from

Figure

45 degrees.

23.

C 1 is a

C 2 is the

flexibility.

in Figure

24 as a function

of k

Sail and

cePDT,

the

angle

that

the

buckles

would

assume

if the web

could

reach

the

Section

B4.8

15 October Page

1969

35

I --

z 0

c)

Z ill

•-'-"

i...I

.<

Z

Z

o es_

<_

_ O-

< Z 0 q_ <

0

Z <

=7J iI 7,/j / i I/1/,ss /

, jj,i

J,.

J

/ /

o_.

0

•

•

O

II

uol

0

C'I-

;.q

r_

.<

Section

B4.8

15 October Page

1969

36

0.12

a- ANGLE

OF DIAGONAL

TENSION

0.08

cl

0.04

0 0.6

0.7

0.8 tan

1.0

0.9

O[

1.2

c3 0.8 cob = 0.7 b

"_ 1/4

t .(I c

c 2, c3

+ It) h e .

Ic = MOMENT OF INERTIA OF COMPRESSION FLANGE

0.d

It = MOMENT OF INERTIA TENSION FLANGE

OF

J 0

1

2

3

4

cob

FIGURE

23.

IN WEB

EMPIRICAL AND

FOR

COEFFICIENTS SECONDARY

FOI_ BENDING

MAXIMUM MOMENT

SHEAR

STRESS

IN FLANGES

Section 15

B4.8

October

Page

1969

37

S

]0

t 25 *p,W 45 48 35 30 25 20

I0 O.2

O.4

0.6

0J

1.8

k (e) _)24-T3

ALUMINUM

ALLOY.

DASHED LINE IS ALLOWABLE

)5

|tv| t

• 62 ksi

YIELD

STREW

I I

r

-

3O

I

e_h. DEG 4$ 48 35

J

30 2S 2O 2O

150

(b)

FIGURE

0.2

ALCLADT075-T6

24.

BASIC

0.4

ALUMINUM

ALLOWABLE

0.6

0.i

ALLOY.

ttult

VALUES

1.0

72ksi

OF

f s

max

Section

B4.8

15 October Page

state

of pure

diagonal

tension

without

The

rupturing.

values

1969

38

of F

have

been

Sall established

by tests

connections,

they

I.

Bolts,

II.

and may are

just

between

Bolts,

snug,

washers

flange bolt heads

to be tight;

IV.

Rivets,

assumed

to be loosened

webs

flange,

stresses are

given

are

bolt heads,

use

basic

different

or web

plates

allowables.

directly

dimensions

Figure

25 to obtain

increase

basic

in service;

valid

not exceeded.

of unusual

use

For

"

on sheet;

reduce

basic

10 percent. assumed

or rivets

under

bearing

Rivets,

allowable

allowable.

angles;

III.

sheet For

"basic

as follows:

heavy

sandwiched

allowables

The

applied snug,

just

be called

They

use

if the

allowable

are

not valid

arranged

allowables

10 percent.

basic

allowables.

bearing for

stresses

countersunk

unsymmetrically

with

on the rivets.

respect

to the

F Sal I

4.8.2.3

Analysis

Stiffener not carried

of Stiffeners

loads

result

by the web.

from The

the web

stiffener

diagonal load

tension

is given

and

the

transverse

load

by

S

P

=

ktbf

u

tanoL + Nb s

I

1-

where

N is positive

for

a transverse

by the

stiffener

the

effective

with

the

and

stiffener

ma_

be assumed

CI"

fsF (td tb+ Au)

compressive web.

The

'

load.

effective

to be given

(22)

1

width

This

load

is resisted

of the web

working

by

b e

b

-

0.5(l-k)

.

(23)

f

Sectiol_ 15 Page

.......

B4.8

October

1969

39

j

20

_

,! I0

2024

BASED ON F_ ] 0.4

[ 0.2

0

: 62ksi T 0.6

0.8

1.0

4O

3O

F'eli,

kli

20

I0

7075 0

0

FIGURE

6ASEDON

0.2

25.

0.4

ALLOWABLE 7075

Ftu

AT

WED HOOM

STRESSES

TEMPERATURE

FOR

2024

AND

Section

B4.8

15 October Page

The

average

stiffener

stress

1969

40

is then

P tl

f

=

u

A

+ 0.5(l-k)

(24)

tb

ue

The

maximum

compressive

stress

of the

beam.

The

ratio

stress,

fu

/fu'

is obtained

in the

stiffener

of the maximum from

occurs

stiffener

Figure

stress

near

the neutral

to the

average

axis stiffener

26.

max Stiffeners by true

may

elastic

stiffener

stress

excessive

bowing stress

II.

The

average not

ratio

hu/2O.

effective

only

must

U

stress

exceed

column

the

stress,

allowable

length

compression properties.

the following

must

the column

the column stress

for double

Column double

and stiffener

not exceed over

crippling.

in (symmetrical) loaded

of the web

column f

must

or by local

is an eccentrically

and

The

action

is possible

is a function

I.

The

by column

instability

A single failing

fail

cross for

stiffeners

yield

failure

stiffeners.

member

whose

To guard

be adhered

against

to:

stress.

section, a column

feent with

= fu Aue/Au' the

slenderness

is

h L

U

= e

L

'41

= h O

+ ka(3-2

b < 1.5h b/hu)

b-> U

(25)

1.5h

(26)

Section

B4.8

15 October Page

1969

41

//j

p/r -

//I,

Lq Z _q

f GO

'1I

Z

_q E_ O

_q L_ ,< _q > < 0 E_

/

;.q E_ D_

V

/

/

i

© ©

J W £Xl

/

A

,

M II

I I"

9

o

Section B4.8 15 October 1969 Page

To

avoid

less

column

than

of the

failure

the allowable

stiffener Forced

the

attached

web

shear The

stress

material, crippling

stiffeners,

taken

with

stiffener

from

the average the

column

the slenderness

of stiffeners

leg of the

must

curve

ratio

f,

should

for

solid

sections

allowable

forced

crippling

/__\t

1/s

\

In this

by being

forced

mode

of failure,

to adapt

itself

to the

stress

is given

by the

empirical

equation

L/

C is a constant

as follows:

Double Stiffener

Single Stiffener_ 2024-T

(Bare)

C =26.0

21.0

7075-T

(Bare)

C =32.5

26.0

Nomographs

for

proportional equation

be

Le/p.

be considered.

is deformed

stress,

wrinkles.

O

where

of double

42

F

limit,

o

exceeds the material

are given in Figure 27. If F O

a plasticity

factor,

77, equal

to Esec/Ec

is used

in the

above.

Torsional

stability

of single

stiffeners

is provided

by meeting

the

following

criteria:

(fs- Fs

)hI/3e t = 0"23E cr

(J-_bh2)

,

(28)

Secth_n

B4. S

15 ()ctob(,r Page

19(;!)

43

it'--" Y)

6O

1.0

10

0.9

9

.tO

8 0.8

S

7 4O 0.7

6

0.6

--

$

-_:

4

-

3

30

0.5

0.4

-

20

.-

1.5

t u

k

F o. ks,

Fo,

--

kS,

2

0.3 DOUBLE UPRIGHTS

SINGLE U PRIGH

10 -_

TS --

-

8

1.5

l0

0.2

9 7

-:: I.o ---

0.9

8

6 -

--.

7

0.8 0.7

--

-

6

5

0.1 3

Iol

FIGURE

27.

2024

0.6

-

0.5

-

0.4

---

T3 ALUMINUM

NOMOGRAM STRESS

-

ALLOY

FOR ALLOWABLE

(FORCED

CRIPPLING)

UPRIGHT

t

SectionB4.8 15October 1969 Page44

10

7r

9

0,9 6fJ

08

07

If

0.6

0.5 _0

04 f:

k*.l O'

I'

o

w .

20 0.3

DOUBLE UPRIGHTS

t

SINGLE UPRIGHTS

2

O2 1.0

0

10

0.9

9

0.11

8

0.7

7

0.6

6

0.5

1

0.4

qb_

7075-T6

ALUMINUM

FIGURE

27.

ALLOY

(Concluded)

Section B4.8 15 October 1969 Page

45

where

(f S -F

=

)het

S

total

web

shear

load

above

buckling

which

can

be carried

sheet

stiffeners.

cr

before J1

=

the

effective

1/3

(developed

the

polar

stiffener

moment

width}

cripples

of inertia

t 3.

This

of the

applies

stiffener

for formed

U

To prevent

:_ forced

extern_

compressive

diagonal

tension,

evaluate

this

crippling load

+

and

F

U

allowable

such

as the

upright load

resists

resulting

following

must

an from

be used

to

= 1

,

(29)

the

maximum

upright

compressive

stress

and the

ax

forced and

f

co

respectively,

crippling and

F

resulting

An effective

column

the

to the compressive

equation

__2

are

O In

The

when

ce/

f

allowable

of failure

effect:

Fo

alone,

in addition

an interaction

m_______,_

where

type

area

crippling effect failure

ce

stress, are

the

from of web

stress,

of the

external

of the

upright.

respectively, actual

external plus

Fmax, load

and

for

allowable

compressive

upright

may

may

be used

should

To prevent

also

diagonal

stress,

stress

alone.

acting

in computing

f ee"

The

Fee.

be investigated

cohmm

acting

compressive

be used

for

tension

failure

under

with

respect

combined

to

SectionB4.8 i5 October Page

loading

f

the

u

+f

following


ce

criteria

should

1969

46

be fulfilled:

(30)

co

and f

+f

cent

4.8.2.4

Analysis

The

flange

stresses: the

<-F

ce

(1)

of Flange stress

result

bending

of the

stresses,

component

stresses

diagonal

is the

primary

flange-parallel

bending

(31)

c

(2)

of the web

because

of the

bending

stresses

superposition axial

diagonal

stiffener-parallel

of three compression

tension, component

individual because

and (3)

of

secondary

of the web

tension.

The

primary

_

c

fprim

s

1-

If

are

by

If

cr

--_s

given

(32) ( 1- _- )

where I

= moment

of inertia

of section

and If

= moment

The diagonal

total

of inertia

axial

tension

and

load

of section

because

applied

axial

(web

neglected).

of the flange-parallel load

component

of the web

is S

Paxial

= khtf s cotc_

+ Pa

1-

I cr S

(33)

Section B4.8 15October 1969 Page47 P

is positive

for compressive

axial

load.

The

axial

flange

stress

is then

a

Paxial faxial

=

A

+A c

where

A

c

The

_md A

t

(34)

+0.5(l-k)th t

are

the area

secondary

bending

of the stress

compression is given

and

tension

flange.

by

(35) sec

see

where

f Pb u

M

sec

= C3

(over

12

s tiffene

r }

and Pb U

M

sec

= C3

C 3 is an empirical The

allowable

of Section

C 1.0.

F

material,

tu

of the

4.8.2.5

Analysis

(midway

24

stress

concentration

stress

for

The

where

modified

stiffeners)

factor,

compression tension

given flange

stress

for

by the attachment

in Figure can

be found

a tension

efficiency

23.

flange

by the methods is given

by

factor.

of Rivets The

V = h-V (I +0.414k)

h' = beam

the

allowable

Web-to-Flange:

q

between

depth

flange-web

shear

flow

at the line

of attachment

,

between

is

(36)

attachment

line

of flange

web.

SectionB4.8 15 October 1969 Page48 Web-to-Stiffener: develop

The

sufficient

a unit

until

=

b The

i

t

failure

b

strength

occurs.

The

for double

to make

shear

the

strength

stiffeners,

must

two stiffeners

should

act as

be

Q (37)

e

= outstanding

S

shear

rivets,

cy L S

where

longitudinal

column 2F

q

stiffener-web

stiffener-web

--- 0.15t

F

= 0.22t

Ftu

tu

stiffener

flange

connectors

must

width. carry

a tension

component

as follows:

(double

stiffener)

(38)

(single

stiffener)

(39)

and N' The

interaction

of shear

and

Stiffener-to-Flange: the

empirical

P which into

U

gives the

4.8.2.6

axial case.

in the connectors

stiffener-to-flange

is given

connectors

in Reference are

11.

designed

with

relationship

U

A

(.i0)

tie

the load

in the

The

stiffener.

connection

must

transfer

this

load

cap. Analysis

The beam.

=f

The

tension

of End of Beam

previous The

vertical

compression Since

discussion

the

has

stiffeners loads,

diagonal

been

in these

as presented. tension

effect

concerned areas The results

with are outer,

the

subject,

"interior" primarily,

or "end

in an inward

Oays of a only

to

bay, " is a special pull

on the

end

SectionB4.8 15 October 1969 Page49 stiffener, it produces bending in it, as well as the usual compressive axial load. Obviously, the end stiffener must be considerably heavier than the others, or at least from

supported

edge

component

members

w

to reduce

of the running-load-per-inch

that

is given

the

stresses

resulting

edge

parallel

in such

formulas

moment

edge

it must

There

I.

II.

to the

member

three

(stiffeners).

to w,

loads.

the

greater

k,

additional

and

additional

a combination

for

thereby

end reduce

(This members

reduce parts.

the

of the

member.

thereby

the weight

is inefficient

the thickness

Provide

of dealing

or strengthen

(This

edge

ways

to keep

"beef-up"

Increase

and

axis

subjected

being

or to reduce

III.

neutral

in general,

Simply

in the

(stringers)

and

(42)

the object

its

axis

The

longer

the unsupported

will

be the

bending

carry.

are,

to bending,

to the neutral

_

normal

of the

Actually,

bending

(41)

members

members

length

by the

produces

-- kq tml o'

_) = kq cot

for

members

bending. The

for

by additional

edge

member

so it can

long unsupported bay panel

either

the running

(stiffeners)

methods

edge

member

subjected

load

moment

because

might

be best.

of w.

of

it nonbuckling

large the

all

)

producing

for

to support

carry

lengths. to make

inefficient

) of these

the

down.

is usually

its bending

with

bending panels.

edge

member

(This

requires

)

SectionB4.8 15October 1969 Page50 4.8.2.7

Beam Design

This

paragraph

of tension

field

for

strength.

b.

The

spacing,

permanent than

which

conform

The

b

dashed

spacing

aspect

line

can

be seen

b/h

= 0.2

from b/h the

Stiffener

Area

structural These required

right

"oil

the curve

for

= 1.0 are

sheet

thickness,

is the

approximate

when

b/h

the on b

the possibility

design

at full

strength.

b'

of k

t,

the

and

is loaded , the

max

shear given

max

to

m:uximum wrinkling spacings

flow

availables

in Paragraph

the absolute

m_iX'

flow,

boundary

Stiffener

by using

value

sheet

shear

of excessive

4.8.2.2.

maximum

stiffener

"

in the construction has

only

0. 050 she(-t,

plotted.

The

of Fig_ure

a small where

effect

on the

additional

relationship

for

28 is that curves,

curves other

the :is

for

sheet

gages

is

same. Estimation:

index.

This

area

preliminary

allowable

is a limitation

canning.

made

as a function

are

line

establishes

plotted

curves

beams

on the

Varying

left

if necessary,

limitation

= 0.5.

the ultimate

on the

to minimize

assumptions

approximately

ratio

dashed

to prevent

b/h

and

efficient

of the

tension-field

be used,

at the

of the

line

the web works

can

max

28 gives

dashed

central

to the

ratio

to facilitate

as a function

and

when

in order

One

area

The

in order set

greater

Figure

sheet

shear-resistant

stiffener and

Flow:

Alclad

spacing,

between full

Shear

7075S-T6

stiffener

methods

beams.

Allowable q,

presents

index

to be used

of stiffener

for

Figure of b/h

29 presents and x/q/h,

is a measure

stiffener

the square

of the loading

only as a me:ins preliminary

the

of roughly

desig_

and

root

are:_

to web

,)f ttm

intensity

on the

approximating preliminary

the weight

beam.

Section

B4.8

15 October Page

Ioooo 9000 1

i969

5i

aMAX TO PREVENT EXCESSIVE I WRINKLING AND PERMANENT SET| WHEN WEB WORKS AT FULL I ',TR ENGTH

J

7000 t 6OO0

._00

4OOO

3OO0 'I

f

2,$00

20GO

I,$00 quit lb. tn.

I000

BOO 70_ 6OO

I 30O

,

2. WEB SI/_PLY SUPPORTED 3. THE STRESS CONCENTRATION FACTOR DUE TO FLANGE

--m

Q

FLEXIBILITY.

(2 = 0.2

I i

.

f

_0 Io

12

h b.

NOTE: FIGURE

FOR 28.

BARE

7075S-T6,

ULTIMATE

MULTIPLY

ALLOWABLE

16

la

20

1 22

i 24

26

28

,_.

BY 1.07

SHEAR

FLOW

ALCLAD

7075S-T6

SHEET

Section B4.8 15 October 1969 Page

52

Q

i

i C/3

I

LC_ L'--

D

N

c_c_ _Z

Z_

! _-

! zr_

r_m Z_

M b-

i

o

C)

°1v-

Section B4.8 15 October 1969 Page

estimation. area

might

cannot f

If the be larger

always

Curves

for

found

stiffener

that

2.

The

desi_m

This

fixes

The

h,

3.

flow,

q_-_,

the

is desirable

a zero

7075S-T

material,

that

the

root

b,

of the

for

and

square

similar

required

margin

of safety

single-angle

stiffeners.

to Figure

of 7075S-T

is often

29, can

web-stiffener

be

can

be

excessive

secondary

bending

0.5

to around

0.8

are

The

required

web

thickness,

b are

known.

This

the maximum

4.

Estimate

the

required

5.

Compute

the approximate

figure

value

equal

spacing, Au/bt

with

cross-sectional

might

b the

beam

of depth,

from

system.

induce b/h

tension-field

be used

not

web-stiffener

In general,

for

also

known.

inspection

to the

spacing

can be obtained can

case,

of the

in the flange.

t,

allowable

b,

design

used

usually

by considerations

stiffener

commonly

are

index.

is the

spacing,

wide

h,

structural

If such

weight

that

of beam,

determined

designer.

minimum

it is possible

depth of the

a stiffener

However,

against

for

design

q,

spacing,

29 shows

q and

29 since

the

manner:

the control

Figure

are

preliminary

shear

stiffener

under

curves

sections,

12.

Method:

The

to standard

in Figure

or 2024S-T

at in the following

1.

is limited

given

The

stiffeners

in Reference

arrived

than

be obtained.

double

Design

design

53

ratios

from

beams. Figure

to check

28 since the

stiffener

max"

aid of Figure area

of stiffener

29. as follows:

Section B4.8 15 October 1969 Page

e

Choose or unless

a stiffener there

is an efficient resistance of failure.

with are

other

design. against

the proper

forced

design

Also,

area.

considerations,

a stocky,

crippling,

Unless

the beam

is very

a single-angle

equal-legged which

54

is usually

angle

gives

the

dominant

deep,

stiffener greater mode

Section

B4.8

15 October Page

1969

55

REFERENCES 1.

Wagner,

H.,

"Flat

Sheet

Parts

I-III,

NACA

TMS

Cook,

I. T. , and Rockey,

Metal

Girders

604-606,

with

Very

Thin

Metal

Web,

"

1931.

f

2.

Supported

Infinitely

Aeronautical 3.

Cook, with

T.,

Mixed

Rockey,

and

K.

Plates Vol.

Rockey,

Boundary

November 4.

Long

Quarterly,

I.

K. C. , "Shear Reinforced XIII,

K.

C.,

pp.

349-356.

C.,

and

Cook,

"Shear

1962,

p.

Buckling

Aeronautical

I. T.,

of Clamped

by Transverse

February

Conditions,"

1963,

Buckling

"Shear

and Simply

Stiffeners," 41.

of Rectangular

Quarterly,

Buckling

Plates

Vol.

XIV,

of Orthogonally

Stiffened

/-

Infinitely

5.

Long

Flexural

Rigidities,"

Rockey,

K. C.,

the

Rockey,

Rockey,

Infinitely

Transverse

Long f

1964,

Plates

Aeronautical

Quarterly,

upon

and

Cook,

-- Influence Quarterly,

Buckling

Reinforced

Rivet Webs,"

and

p. 75.

Postition

upon

Aeronautical

T.,

XV,

I. T.,

of Torsional February

XIII,

"Influence

August

of the

Buckling

May "Shear

1964,

p.

p.

Torsional

p.

212.

Rigidity Plates,"

198. of Clamped

of Transverse 92.

and Simply Transverse

1962,

of Stiffened

Buckling

Rigidity 1965,

of Clamped

by Closed-Section

Vol.

the Shear

Vol.

1969,

and

on Shear

"Shear

Quarterly, I.

Thickness

Torsional

97.

Plates

and Cook,

Having

February

Stiffeners

I. T.,

Long

Quarterly, K. C.,

p.

Cook,

Stiffeners

Aeronautical Rockey,

C.,

-- Stiffeners

of Stiffener

Aeronautical K.

Plates

of Single-Sided

K. C. , and

Stiffeners,"

8.

"Influence

February

Supported

7.

Supported Aeronautical

Effectiveness

Quarterly, 6.

Simply

Infinitely Stiffeners,"

of

Section B4.8 15 October 1969 Page

o

Rockey,

K. C.,

Supported

10.

12.

T.,

Plates

"Shear

Buckling

by Transverse

Aeronautical

of Clamped

Stiffeners

Quarterly,

Vol.

and

XIII,

and Simply a Central

May,

1962,

95-114.

Bleich,

F.,

Bruhn,

E.

Buckling New

F.,

Analysis

Company,

Kuhn,

P.,

York,

" Parts L.

Ratios

of Web

of Metal

and Design

J.

P.,

"Strength Depth

of Flight

Ohio, and

I and II, NACA

R.,

Structures,

McGraw-Hill

Book

1952.

Cincinnati,

Peterson,

Levin,

May

Strength

Inc.,

Offset

Tension, 13.

I.

Long

Stiffener,"

Company, 11.

Cook,

Infinitely

Longitudinal pp.

and

56

to Web

Structures,

Tri-state

1965. Levin,

TN266

Analysis

Vehicle

L.

R.,

1 and

TN2662,

of Stiffened

Thickness

"A Summary

Thick

of Diagonal

1952. Beam

of Approximately

Webs 60,"

with

NACA

TN2930,

1953.

BIBLIOGRA

PHY :

Rockey,

K.

Subjected Stein,

April

"The

to Shear," M.,

Simply

C.,

and

Supported

Design

of Intermediate

Aeronautical

Fralich, Plate

R. W., with

Vertical

Quarterly, "Critical

Transverse

Vol. Shear

VII, Stress

Stiffeners,"

1949.

Structural

Design

Manual,

Northrop

Aircraft,

Stiffeners

Inc.

on Web

November,

1956.

of Infinitely

Long,

NACA

TNi_51,

Plates

SECTION B5 FRAMES

TABLE

OF

CONTENTS

Page B5.0.0

Frames

5.1.0

Analysis The 5.1.1

of

Statically

Method

of

Discussion

5.1.2

Sample

5.1.3

Application

5.1.4

Particular

5.2.1

Sample

the

Problem

Arches Analysis

of

Indeterminate

Frames

Moment-Distribution of

Problems

5.2.0

I

................................................

Method

By

...............

of

Moment-Distribution

................................ of

Moment-Distribution

to

9

Advanced

.................................... Solution

of

Bents

and

ii

Semicircular

...................................... Arbitrary Problem

Ring

by

Tabular

Method

................................

B5-iii

i 2

II ......

27 39

v

Section 12 Page B

5.0.0

B5

September

1961

1

FRAMES

This

section

indeterminate circular

deals

with

structures.

rings

are

specialized The

given

in

methods

procedures

detail

in

of

of

analyzing

analyzing

Section

B

rigid

5,1.0

and

B

statically bents

and

5.2.0

respectively.

A given

sample in

problem

each

to

of ion,

Statically

Moment-distribution indeterminate does

\,

not

structures

involve

a

series

of

precision

the

methods

and

procedures

is

section,

B 5.1.0 Analysis Moment -d is tr ibut

of

illustrate

of

the

required

is

a

convenient

to

a

problem

solution

converging by

Indeterminate

of

cycles the

Frames

method in

simultaneous that

problem.

may

of

statics.

by

reducing

Method

of

statically

Moment-distribution

equations, be

the

terminated

but at

consists the

degree

Section

B5

12 September Page 2

B

5.1.1

Discussion

of

the

Method

1961

of Moment-distribution.

The method of moment-distribution requires a knowledge of momentarea theorems and slope-deflection equations. The five basic factors involved in the method of moment-distribution are: fixed-end moments, stiffness factors, distribution factors, distributed moments, and carry-over

moments.

Only structures comprised of prismatic translation are considered in this article. to be elastic.

members with All members

The fixed-end moments are obtained through "end-moment" equation derived in the development method. (See Fig. B 5.1.1-1)

the of

use the

no joint are assumed

of a general slope-deflection

MBA: --f--

where

MAB MBA E I L 0

is the is the is the is the

moment moment modulus moment

acting on the "A" acting on the "B" of elasticity. of inertia.

end end

of member of member

AB. AB.

is the length of member AB. is the rotation of the tangent to the elastic curve at the end of a member and is positive for clockwise rotation. is the rotation of the chord joining the ends of the elastic curve referred to the original direction of the member and is positive for clockwise rotation.

(_o) A

is the static moment about a vertical axis through "A" area under the M o portion of the bending-moment diagram on Fig. B 5.1.1-1.

(_O)B

is

the

static

moment

about

an

axis

through

"B",

of the shown

Section B5 12 September 1961 Page3 BS.I.I

Discussion

of the

Method

Any

of

Moment-distribution

,(Cont'd_

Loading

MBA

_A

El

B

MAB

i

AA

Fig.

B 5.1. I-I

[

Section B5 12 September Page 4

B

5.1.1 If

Discussion 9A,

eB

of

and

the

_AB

Method

are

all

of

equal

member are completely fixed against member is called a fixed-end beam. are therefore equal to the so-called fixed

end

moments

FEMAB

= _2

FEMBA

_

as

I

FEM

(_°)

and

Moment-distribution

CA, _B

2 (_°)B

2[

2(_°) A

(_o) B

zero,

then

(C0nt'd) both

rotation or translation The last terms of Eqs. "fixed end moments".

setting

A

to

and _AB

1

1

equal

ends

of

....

,°°°,°°

the

and the (I) and (2) Denoting to zero.

...................

°°"

1961

(3)

.........

L Fixed

end

moments

for

various

type

of

loading

Equations (3) and (4) are summarized calling the near end of a member "N" and

then

KNF

= stiffness

factor

the

fundamental

slope

MNF

= 2EKNF

N +

(20

for

member

are

by the

NF

one far

equation

9 F - 3_r NF)

+ FEMNF

in Table

B 5.1.1-1

general equation by end "F". Also let

INF = -LNF

deflection

given

. .............

(5)

...............

(6)

is

The conditions to be met at a joint are: (i) that the angle of rotation be the same for the ends of all members that are rigidly connected at that joint and (2) that the algebraic sum of all moments is zero. The method of moment-distribution renders to zero by iteration any unbalance in moment at a joint to satisfy the latter condition. To distribute the unbalanced moment mentioned factor DF is required. This factor represents the the unbalanced moment which is reacted by a member DFbm

is given

above, a distribution relative portion of and for any bar bm,

by

%m DFbm-

where

the The

. .......................................... E K b summation

is meant

distributed

moment

Mbm Equation

= -DFbmM

(8) may

be

to

include

in any

bar

all bm

members

is

meeting

as

at

joint

then

....................................... interpreted

(7)

(8)

follows:

"The distributed moment developed at the as joint 'b' is unlocked and allowed to rotate moment 'M' is equal to the distribution factor moment 'M' with the sign reversed."

'b' end of member 'bm' under an unbalanced DFbm times the unbalan,

b.

-r B 5.1.1

Discussion

of

Table Notation:

the

Method

B 5.1.1-1

Moments

2.

L

t i _

L

l_ 2

PL

PL

M^ - -F..

aB =

_

for

unit load (Ib per (in.-Ib) positive

P

^

Ib/In.

MA

L wL 2 " I-_--

MA

(Cont'd) Beams

linear in.). when clockwise.

a

3_ I

.I

b L pa2b

"7

MB w

-

e2

lb/in.

It flll_, a

"V

1964

9, 5

P

o

w

ttttttltttt

July Page

Pab 2

F"

o

ltf_!

B5

Moment-dlstribution

Fixed-end

P - load (ib); w = M - bending moment

I.

of

Section

,m|

L

•

MB

wL 2 = " i-_--

I_

11wL 2 192

MA" 6.

L 2 = _ 5wL 2 _

MB __

w

Ib/In.

A+_..,'mmilt, iiiih,..

A

_

B

B-

2 MB

. . wa_._._ (4aL.3a2) 12L 2

Q

_B a ! wL 2 " T

MA

"

MB

|

2

wL 2 " 3-"0-

MA

.

wa (10L2_10aL+3a 60--U

MB

wa 3 - - -(5L-3a) 60L 2

2)

10.

B

A a

-i-

r MA

b L

"-_-

(

- I),

__ -

M B-

- -_- (3

-I',

MA

wL 2 " 3"-_

MB

"

wL 2 " 3"-_

Section B5 July 9, 1964 Page6 B 5.1.1

Discussion Table

of

the

B 5.1.1-1

Method

of Moment-distribution

Fixed-end

Moments

for

12. Ii.A _.--

a _P

a

P

tMA = Pa(1

L

MB = - MA

L

L

13.

L

MA

e

=

-

/

15PL 48

MB

14. !._-_ ___

IllIIII" llt11111: B A_'.._ ,,__.. I I.__. _ __._

, ,_B

/--

-,

- _)

(Cont'd)

A_

B

L

Beams

(Cont'd)

w

Iblin

__

= - MA a _.

.... flIItifIIIl A_

i_ _'

L

e MA

i5.

2 (3L-2a) = wa6---L

_

_

" -

MB

w (L3_a2L+4a3) MA = 12--"L

= - MA

__..t_

16.

MB

=-M A

------ L J

A

B

,_--

A;, ;=

L

,, L¸

v _ i

I

wL 2 MA = 3T

MA

30

MB-

3 _v_ ZOLZ

17.

..w

MB "

3wL 2 160

(5L-4a)

elliptic

load

18.

:IItIItItttitt7ttl_'_,,

A_

MA

=

wL 2 13.52

MB

wL 2 = - 15.8"----_

L.

L

MA = l___ L2

x(L_x)2f(x)d

x

o L

MB

_ -i [ L 2 ,)

x2(L_x)f

(x)dx

O

_J

Sect ion B5 12 September Page 7 B5.1.1

Discussion

of

the

Method

of

_,j--'-

Fig.

The @m = _bm one-half

Mbm

"carry-over"

moment

Moment-distribution

r_b _

b

1961

(Cont'd)

Mbm

B 5.1.i-2

Eq.

(6)

considering

= o (see Fig. B 5.1.1-2). The carry-over moment of its corresponding distributed moment and has

is the

equal to same sign.

= 4EKbm@ b and Mmb

is obtained

=

by

applying

2EKbm0 b

Henc e, Mmb

I = _Mbm

..........................................

(9)

f

The sign convention adopted for this work deviates from the usual convention used in elemen_ beam analysis as found in Sec. B 4.1.1 of this manual. The positive sense for moments has been adopted from the convention used in slope-deflection equations; namely, moments acting clockwise on the ends of a member are positive. (See Fig. B 5.1.1-3)

M

Fig.

B 5.1.1-3

Positive

The following procedure distribution analysis: i.

Compute as

shown

the in

stiffness example

sense

is established

factor problem

for one.

for

bending for

each

the

member

moments process

and

of moment-

record

Section

B5

12 September Page 8

Discussion

of

the

Method

of Moment-distribution

(Cont'd)

Compute the distribution factor for each member and record as shown in example problem one. o

o

o

.

Compute the with proper

J

Draw a horizontal sum of all moments be zero. Record

the

span one.

and

joint

record

line below the balancing at any joint above the

carry-over moments moments

moment

at

the

moment. horizontal

opposite

ends

The algebraic line must

of

the

member.

have the same sign as the corresponding and are one-half their magnitude.

Move to a new joint and repeat the process for the balance and carry-over of moments for as many cycles as desired to meet the accuracy required by 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. Obtain

the

algebraic total of be zero. g

moments for each loaded shown in example problem

at each

Balance the moments at a Joint by multiplying the unbalanced moment by the distributor factor, changing sign, and recording the balancing moment below the fixed-end moment. The unbalanced moment is the algebraic sum of the fixed-end moments of a joint.

Carry-over balancing .

fixed-end signs as

1961

Reactions,

final

moment

at

the

end

of

each

sum of all moments tabulated at the final moments for all members

vertical

may be found final moments

through (step

shear, statics 8).

and by

bending

member

moments

utilizing

as

the

this point. The at any joint must

the

of

the

above

member

mentioned

Section B5 12 September 1961 Page9 B 5.1.2 PROBLEM moment

Sample

Problem.

#I. Compute diagrams for

the this

end moments frame.

and

draw

the

shear

and

bending-

'D_Z-T 1=2 I P1 = w

5 kip

i0

kip

= 2 klp/ft i_

15-,-

_A 1=2 _ B 1=3.5 _ 25'-_35

AB

BA

× I04

.08

.I

.44

.56

-104 +38

+18 +48

+2

-4 +2

+2

+2

-62

+62

+19

-4

+I

--- +124 _

BC

Fig.

I=5

C

_

40'

_!

I

I DF

25' P2 =

DC

× 0

CB

CE

CD .08

.i

.129

.33

.41

-25 -8

+50 -i0

0 -7

+24 -8

-I0

-6

26 -5O

-5

-5

-3

-6

EC

-17

B 5.1.2-1

+30

-13

-60

Section

B5

12 September Page i0

B 5.1.2

Sample

Problem

(Cont'd)

5 kip

i0 kip

2 kip/ft I

i

II ----20'---_

--'--25'----"-

+27.48

35

40'

I,,,

kip

+3.42

+4.25

kip

kip

-1.58

'-5.75 kip

kip -22.52

kip - SHEAR

DIAGRAM

-

-6 klp-ft [+189

kip-ft +13 kip-ft

i

|

+7

kip-ft

\

+55

kip-ft -60

-62 -124

kip-ft

kip-ft - FINAL

Fig.

MDMERT

DIAGRAM

B 5.1.2-2

-

kip-ft

1961

Section

B5

15 February Page II B 5.1.3

Application

of Moment-dlstribution

to Advanced

1970

Problems.

f

In article B 5.1.1, "Discussion of moment-distribution", the basic principles of moment-distribution were founded. Moment-distribution may be applied to complex structures involving joint translation, settlement of supports, non-prismatic members, symmetrical and unsymmetrical bents and other involved structures. For information on the technique of solving such structures see the references listed in Section B 5.0.0. Methods for accelerating the convergence and other short-cuts may also be obtained in these references.

B 5.1.4

Particular

Solution

of

Bents

and

Semicircular

Arches

In the tables that follow, formulas for computing reactions are given for several load cases. In all cases constraining moments, reaction, and applied loads are positive when acting as shown and

I2h K

_

IL 1

for

cases

i through

for

cases

19

18

IIS 2 K

_

m

12S 1

through

28

Section B5 ,,12 September 1961 Page 12 B 5.1.4

Particular Table

i • VERT. LOAD

Solution

B 5.1.4.1 Two

CONCENTRATED

of Bents

and

Semicircular

Reactions and Constraining Legged Rectangular Bents VA

= Qb L

Arches

(Cont'd_

Moments

VE = Q

- VA

a =b=

L 2

in

30ab H =

,1 !g c

!

2Lh(2K

D

F

FOR

h

'9_-"--

+

SPECIAL

3) CASE:

L "-"_

i

V A = V E =Q 2

I !

_-I I

l

Ii-_ !

H

4

=

3QL 8h(2K +

H

I 3) L

VA 2.

="

'

VE

VERT. LOAD

CONCENTRATED

VA

Q

H

=QbL

=

I I +

3qab 2Lh(K !

@(b - a) VE L2(6K + i) -I

= Q

_ VA

2)

V I

'_--'-

L

_

__Qab |

l

L

+

2)

1 2(K +

2)

MA

I i

[ 2(K

(b,- a) 2L(6K + I)

h qab L

i

ME

E

A

VA

=

FOR

SPECIAL

VA

= V E =R

v=q

Q

B

a_

13_

CASE:

2

FOR

J

_

SPECIAL

V . 9-bL

L

E

.__ HE

V

V

=

HE

HA

= Q

= HA = Q 2

CASE:

.L, qL

8(K +

2)

" HE

QaibK(a + h) 2h h2(2K + 3) +

= j I

' hI

HE

D

I/

L

l

L a = b = -

MA = ME

CONCE_RATED

C'

2L(6K + 1)_

2

VE

3. HORIZ.

+ (b -

] I

b = O,

a =h -i

Section

B5

12 September Page 13

/--

B 5.1.4

Particular Table

HORIZ. LOAD

of

Bents

and

B 5.1.4-1 Two

4.

Solution

CONCENTRATED

Reactions and Leg Ied Rectangular

Semicircular

Arches

HE

= Qab h 2h 2 L b

MA

= Qa 2h L

ME

Qa [ 2h [

FOR

SPECIAL

(Cont'd)

Constraining Moments Bents (Cont'd)

_ 3(_a2K V = Lh(6K + i)

HA

h +

b + K h(K +

b(hh(K+ +b +2) bE)

in

= Q

a)

(b 2)

1961

HE

I

+ h

f /

E;

V

5. VERT. UNIFORM RUNNING LOAD

3qhK = e(6K +

-b(h + b + bE) h(K + 2) CASE:

b

]

+ h (6K

= O.

a

+

i)

= h

i)

HA

= HE

= 2_

_-I

MA

= ME

: Qh(3K 2(6K

VA

:

+ +

I) I)

"I

_

_

wcd

= wc w

r

BC_'_'_1

d4" ]Xl _

H =

)

+ x2 + _\

12dL-12d2-c

3wc = 24Lh(2K

+

+

3

2

3)

ii

where: I-,---L h

X1

I1

:

wc 24L

24-L

+ 4c

- 2_d 2

L

I_ F

X2

=__[ wc 24L

d 3 " bT 24_-"bc2

+

3_+

2c2

- 48d2

+ L

FOR

SPECIAL

CASE:

a = O,

c = b = L,

d

=2

VA

wL

VF

VA d=L

a 2

b 2

VF :T wL 2

H= 4h(2K

+ 3)

Ip

u

24di

Section B5 12 September 1961 Page 14 B 5.1.4

Particular Table

6.

Solution

Bents

and

B 5.1.4-1 Reactions and Two Legged Rectangular

VA

VERT. UNIFORM RUNNING LOAD w

of

_

wcd = _

+

V F = wc

h

H

X 2 are

X 2)

+

2)

MA

X1 + X2 = 2(K

FOR

7. VERT. TRIANGULAR RUNNING LOAD

given

in case

5

+

XI -

2) +

SPECIAL

2(6K

CASE:

a

X1 + X2 = 2(K + 2)

X1

2 wL = 4h(K + 2)

2(6K

i)

X2 + I) = O,

L d = 2

c = b = L,

wL 2 MA

= MF

=

12(K

+

2)

n_

,r

2-iwc

we

+

_--VA'2-_

2c

_-

3 [x3+x4 3wc+ 3) [ de H ffi _ 2K + _'_= 4LN(2K

c182

d2 ]

WHERE:

i

h

X3

=

we " _

c + _-

51c 3 + _ +

c2b 6L

. d2

I1

I

X4

t d

+

wcd

VA= VF=

.,-

X2

wL ffiV F = _-

VA

H

in

- VA

= 2h(K

MF

_Cont'd)

XI " X2 L(6K + I)

3(X 1 + _--- b ----_I _------L -.-_

Arches

Constraining Moments Bents (Cont'd)

X 1 and

d_-r

Semicircular

_L

a 3

ffi

wc 2L

FOR

SPECIAL

V

wL -6

s

2b 3

d3 c2 L + _ + CASE:

c2b 6L

2d 2 + L

a=o,

c=b=L,

d

=_

.==c_ wL 2

H

51c 3 810L

= 8h(2K

+

3)

dL

Section B5 12 September Page 15 B

5.1.4

Particular Table

Solution

of

Bents

and

B 5.1.4-1 Reactions and Two Legged Rectangular

8. VERT. TRIANGULAR RUNNING LOAD

Semicircular

Constraining Moments Bents (Cont'd)

X3 VA

=wcd 2L

+

L(6K

+

X 3 and X 4 are given in case

i)

- VA

H =

X 3 + X4 = 2(K

+

X3 + MF

_

SPECIAL

= 6--

]

IO(6K

I +

20(6K

a 3

VF

V F = _--

2b 3

H

UNIFORM LOAD

8h(K

+

wL2

I)

+

I) L 3

I

c=b=L,

d =

i 1 +

i)

+

I)

iI

2

i

6K+

c27

H4

2L

----

HF

=

w
CASE:

1

= w(a

Ii j

h

V

HA

=

8hB(2K c=o,

b=o,

--

2L

= wh

V HF =

- HF

4E

I +

2(2K

- c)

KIw_a2-c2)(2h2-a2.c2)

wh 2

V

2)

2)

I K+ 5

v =w%_"

FOR

__[

+

wL 2

=

MF=_

D _---iL

+

a=o,

wL 2 [ --+6K+ 5 2 MA--Z-=12v K+

9. HORIZ. RUNNING

2h(K

- X4

2(6K

CASE:

+ X4)

MF

VA

d =L

X3

2) 4

7

- X4

2(6K

X4

wL I

VA MA_

X3

2)

= 2(K +

FOR

in

3(X3

V F = _-

(Cont'd)

- X4

wc

MA

Arches

1961

K 1 +

3)

+

3)

a=d=h

- HF I

Section B5 12 September 1961 Page 16 B5.1.4

Particular Table

i0. HORIZ. RUNNING

_---L

Solution

of

Bents

and

B 5.1.4-1 Reactions and Two Legged Rectangular

UNIFORM LOAD

w(a 2 V=

HA

-----

Arches

Constraining Moments Bents (Cont'd)

- c2> 2L

= w(a

Semicircular

- c)

MA "_-

(Cont'd) in

MF -L'-

- HF

E

D

• 2

2,

X=

X_(K

- I)

%-12 nF

-

4h

- 2-h _ _

_

WHERE:

HA

JM A V

_

-_MFHF

X5

= l_h2

[d3(4h

- 3d)

- b3(4h

-3b) 1

X6

= w--w--- [a3(4h 12h 2

- 3a)

- c3(4h

-3 c)l

V (3K + MA

I)[ w_a2 -2 c2)

=

2(6K

2

+

i)

K +------26K +

(3K +

l)[w(a2

MF =

2_6K

2" +

i

+ X5

c2)

- X51

I)

6K + FOR

_PECIAL

CASE:

- X5 J

c=o,

f b=o,

a=d=h:

wh2K V

= L(6K

+

i)

HF

wh(2K + 37 = 8(K + 2)

MA

= 2-4--

6K +

HA

= wh

- HF

wh 2 F 18K MF

- 24

1 + +------2 K

|

+ 5

6K +

i

w 213°K+7 ii H

K+2

Section B5 12 September 1961 Page 17 B 5.1.4

Particular Table

II. HORIZ.

Solution

of

Bents

and

B 5.1.4-1 Reactions and Two Legged Rectangular

Semicircular

Arches

Constraining Moments Bents (Cont'd)

(Cont'd)

in

TRIANGULAR

RUNNING

LOAD

V = _w

(a 2 + ac

VL

--I

a

HF

= _

X7

=

HA

= w(a2-

c)

_ HF

KX7 +

(2K +3)h

w 120h2(d-b)

SPECIAL

WHERE:

13(4d5+b5)

20h2(2d3+b

h FOR

- 2c 2)

3)

CASE:

_ 15h(3d4+b

4)

+

_ 15bd2(2h_d)2 b=c=o,

a=d=h:

wh V HA

_

_ V

HF

= 6--L-wh2

HA

wh

7K

2 =--

_ HF

_ )

V

=

12. HORIZ. TRIANGULAR RUNNING LOAD

V

1 + 10(mE + 3)

w " _

(2a +

c)(a

- c) KX

HA

D --L

= w(a 2

c)

_ HF

HF

_ 2-_ VL + h(2K I0 + 3)

WHERE: I

f=-

I I b

i

W

Xlo

=

-30h2c

(a2-c 2) +

+

15c(a4-c

4)

h

_ V

FOR

SPECIAL

V

wh 2 -3L

=

CASE:

HF 1 '

HA

wh =,_-

HF=_

20h2(a3-c

120h2(a.c)

- HF

whI4K 2K+3+

5 ]

b=c=o,

a=d=h:

- 12(a5-c 5)

3)

Section B5 12 September 1961 Page 18 B5.1.4

Particular Table

13.

Solution

of

Bents

and

B 5.1.4-1 Reactions and Two Legged Rectangular

HORIZ. TRIANG_ RUbbING LOAD

V

"

wl_a2

_

Semicircular

Arches

Constraining Moments Bents (Cont'd)

ac

2c 2)

-

6L

(Cont'd) in

MA L

MF L

X8 " _

. x9(K-l) _

w(a - c) HA

=

u "F

=

2

" HF

--- L

°I,II

w( a2 + "

ac - 2c 2) i mh'

WHERE: X_

= ._.._w

_

[15(h+b)(d4.b4

60h2(d-b)

i.Jc/

|

) _ 12(d5_b

5)

3

-20bh(d3-b3)I

HA ,../MA V

_._M F F

Xo

V

= ----_m 60hZ(d-b) b2h_b 3)

[lOd2h2(2d.3b) L _ d4(30h+15b)

+ +

lObh(4d3+

12d 5 +

3b 5 ]

J

Ew ac 0a2+ 2c2 x8 1 MA

"_"

+

2(6K

x9 _-

[K +I

(3K +

2 + 6K

i)[

w(a2

MF

I)

3K + i]+X s

+ ac6

2(6K

xsl _'- K+ 1 2

+

+

2c2)].

X8

I)

6K+3K i 1 ii=....i.

FOR

SPECIAL

CASE:

blclo

, a=d=h

wh2K V = 4L(6K

+

wh I)

HF

wh(3K + 4) = 40(K + 2)

MA

wh2 " _O-

I

27K 2(6K

+ +

7 I) +

HA

=-_

HF

wh 2 [ '2(6K+I) 27K+7 =6-'_

3K + K +

7 1 2

- HF

K+2

Section B5 12 September 1961 Page 19 B 5.1.4

Particular Table

14. HORIZ. RUNNING

Solution

of

Bents

and

B 5.1.4-1 Reactions and Two Legged Rectangular

TRIANGULAR LOAD

V

=w(2a

=

Arches

Constraining Moments Bents (Cont'd)

+ c)(a 6L

w(a

HA

Semicircular

- c_

MA L

(Cont'd) in

MF L

c)

. HF

2

E H

/

F

=

w(2a2_ac.c 12h

2)

_ XII 2h

+ XI2(K 2h(K

+

I) 2)

where: r

x H "_-'-'_}B

"_---_

_A

"_ V

5 -20hdb3-12b

60h2(d_b)

4 ] (d+h)

I F

A

5hd4-3d

= ----w--w I

II

b.a_MF

X

= w-----E-60h2(a_c)

12

15 (h+c) (a4-c 4 )

12(a5-c 5)

L

V

- 20ch(a3-c3)]

x,:l MA

=

2(6K

x:2 +--2--

MF

=

I K +1

+

2

I)

6K 3K + I ]+X:I

I 3K+I](w(2a2-'ac-c2) 6 2(6K + i)

K+2 FOR

+

- XII ] _ X22 2

6K+l

SPECIAL

CASE:

P

3Kwh 2 V = 4L(6K

!HF

=

+

I)

wh(7K + II} 40(K + 2)

HA

" w__h_h . 2

MA

wh 2 = _-_

V97K+22 L 6K+I

3 " + K-_

f-

MF

wh2 " _-

I 21K 6K ++ 6i

K +1 2']

Jl

Section B5 12 September 1961 Page 20 B 5.1.4

Particular Table

15.

MOMENT SPAN

B

Solution

of

Bents

and

5.1.4-1 Reactions and Two Legged Rectangular

ON HORIZ.

Semicircular

Arches

Constraining Moments Bents (Cont'd)

(Cont'd) in

M V

_.

w

L

H

= 3(b

- L/2)M

nh(2K + 3) FOR

SPECIAL

CASE:

a=o,

b=L

M I

h

V=-L

I1

_-_M I

l MOMENT SPAN

V

= 6(ab

+ 3)'

V ON

HORIZ. + L2K)M

L3(6K

H

+

i)

= 3(b - a)M 2Lh(K + 2)

MA =

M

-6ab(K+2) b

ME

= VL

FOR

V

i

i

t

V 16.

H

3M = 2h(2K

V

V

- L

[a(7K+3)

- b(5K-l)l

J

2L 2 (K+2) (6K+I) - M

SPECIAL

- MA CASE:

a=o,

b=L

CM

_._

L
I {

H -

3M 2h(K

i + 2)

(sK- I)M IdA " 2(K

+

2)(6K

+

I)

Section B5 12 September 1961 Page 21 B5.1.4

Particular Table

17.

MOMENT C " I'

'6

Solution

of

and

B 5.1.4-1 Reactions and Two Legged Rectangular

ON

SIDE

SPAN V=--

D

Semicircular

, b

h Ii_ _----- L -P

H____

A

Arches

Constraining Moments Bents (Cont'd)

M L

H

=

3 [K(2ab+a 2h3(2K

_b FOR

f

Bents

SPECIAL

CASE:

a-o,

(Cont'd) in

2) + h_M + 3)

b--h

M V

=

--

f-_M

L

k

_L

H

E

3M 2h(2K +

=

lv 1'8.

MOMENT ON SIDE

3) q)

SPAN 6bKM

V=

3bM

y--L

+

I)

2h 3 (K +

-M MA

[ 4a 2+2ab+b2+K

!A __._,-

MA_-J

2)

(26a2-5b2)

+ 6aK2(2a-b)]

12 ME

r

+ b ]

= 2h2(K+2)(6K+I)

b

b

[2a(K+l)

H

hL(6K

_ ! BM:

q)

I

=VL-M-M

FOR

A

SPECIAL

CASE:

a=o ; b=h

_ E _,_

H

6KM

V =

L(6K

I

+

i)

+

2)

V 3M

H=

2h(K

= MA

2(K

M(5Ki) + 2)(6K +

I)

Section

B5

15 February Page 22 B

5.1.4

Particular Table

19.

VERT. LOAD

Solution

B 5.1.4-2

of

Bents

Reactions Triangular

and

Semicircular

and Constraining Bents (C,:'_'d)

Arches Moments

1970

_Cont'd) in

CONCENTRATED VA

= Q

VD

=L

- VD

Re

_

Re = h

20.

VERT. LOAD

rb

d _a+c)

Li

+ 2a2(K

+

] i)

J

CONCENTRATED VA

=Q

- VD

] D

L

,

L

4,

M

= A

-

2a 2

6La_h(K+l)

^qcd 6a_(K+l)

J

[L

r(a+d)(BK+4) L

qc2d MD 21.

HORIZ. LOAD

2a2(K

+

I)

CONCENTRATED C

!H ,A=Q

-H D

d _h+c) ] 2h2(K + i)

+

2(2L+b)(a+c)

- 2(a+c)] J

+ 3ac

Section B5 12 September 1961 Page 23 B5.1.4

Particular

Table

22. HORIZ. LOAD

Solution

B 5.1.4-2

of

Bents

Reactions Triangular

CONCENTRATED

and

Semicircular

and Constraining Bents (Cont'd) r-

Arches

Moments

(Cont'd)

in

]

v 11- 1 = L

2h 2

HAQ Ho

J;

=

-

C i HD

= Qc Lh'

4 va_ q_- b --_r

HA

e

_

V

VERTICAL RUNNING

d 6h2(K+I)

(h+d)(-b[3K+4] [ + 2(2L+b)(h+e)

- 2L) +

_MD I_ HD

V

23.

b +

UNIFORM LOAD

MA

qcd + , I) [ (h+d)(3K+4) = 6h2(K

MD

=

VA

= wa

' B

_cd 6h2(K +

. i)

(h +

-

2c +

2(h+c)

I

d)

I - a_2L]

wa 2

Vc = 2-i1_ ii_j

_S

2

I2

H = wa2 8-_- 14b l- + 1 +-----f i ]

24.

VERTICAL RUNNING

uNIFORM LOAD

V A = wa

LB

12

H

! I 3a I - 8-_

wa 24Lh(K +

__

i)

S2 MA

=

M

=

wa2{3K + 2) 24(K + I) 2

VA

___

VC._ L_D-

H

"dMc

c

wa 24(K

+

i)

:

b(10

VC

+

9K)

--

+

3wa 2 8L

2L +

a

]

I] 3acl

1

Section B5 12 September 1961 Page24 B 5.1.4

Particular

Table

B

25. HORIZ. RUNNING

Solution

5.1.4-2

of

Bents

Reactions Triangular

UNIFORM LOAD

and

Semicircular

and Constraining Bents (Cont'd)

Arches

Moments

(Cont'd)

in

wh 2 V

:

2L

W

I1

12 _A = wh wh

_c "_HA

_I_

.

vf 26.

HORIZ. RUNNING

4b i Hc

_-+_ K+I

_

L----_v UNIFORM LOAD

3wh 2 8L

_A

" wh

-M C wh

k_

_

8L(K

+

I)

[ b(3K

+

_A "

24(K

+

I)

V

27.

APPLIED AT APEX

MOMENT

M L

.I.b1

!s,iJ vr

L

_'_h'L

L!

K+I

a]

wh 2

wh2(SK+ 2) V

4) +

MC

= 24(K

+

I)

Section B5 12 September 1961 Page 25 B 5.1.4

Particular Table

28.

APPLIED AT APEX

Solution

of

Bents

and

Semicircular

B 5.1.4-3

Reactions

and

Triangular

Bents and or Arches

Semicircular (Cont'd)

Arches

Constraining

Moments

(Cont'd) in

Frames

MOMENT V

=

H

=

3M(a - bK) 2he(K + I)

MA

KM = 2(K +

I)

MC

=

M 2(K +

I)

/

_-

3M 2L

b b_

S.

c

jMc L

V

29. _k

-T V

S INUSOIDAL NORMAL PRESSURE

V

-

C_R 4 CR 4

I b(Ib/in.) M0

- CR24

(Positive f

*V

b=c

sin@

Vt

section

[(n-28)cos

moment ahead.)

acts

8 - _ +

3 sin

clockwise

on

Q]

Section B5 12 September 1961 Page26 B5.1.4

Particular Table

30.

Solution

of Bents

and

Semicircular

Arches

B 5.1.4 Reactions and Constraining Moments Semicircular Frames or Arches (Cont'd)

(Cont'd)

in

S INUSOIDAL NORMAL PRESSURE

V

-

C_R 4

b(Zb/in.) 13_8 2 _2- 32 ]

H=_L

M_

CR2 4

[_3i0_.]_ 8 2

sin0

31974CR

=

.05478CR

2

F

/ b=C

=

M 0 = CR 2 |.81974

rv

sin

0

cosO _ - o) } - .s4018 + -7-(7 (Positive section

31 • UN IFORM PRESSURE

moment ahead.)

acts

clockwise

on

NORMAL

I M m

0 at all points since pin points permit a uniform hoop tension. T, where :

T m V = in. ) H_O It

bR

B5.2.0

Analysis

The originally

may

procedure define

is

given

is

with by

in

a

ends

the

Tabular

1968

Page

27

Method

corrected

be

built

version

in,

procedure

tabular

to

by

B5

9,

of

work

taken

6.

frames

analyzed

sequence

Ring

analysis

Reference

or

be

the

Arbitrary

following from

Rings pinned

of

Section July

form

elastically

outlined

(Table

in

B5.2.0-1)

restrained,

or

this

section.

The

with

added

notes

to

illustrate

the

pro-

followed.

f-

A cedure.

This

loads. sample

sample

The

problem

sample effects

is

problem of

given

in

See.

considers

shear

flow

B5.2.1

the

are

to

energy

due

presented

as

to

a

direct

and

supplement

shear

to

the

problem.

Procedure Procedure

(i)

(2)

to

Obtain

Initial

Set

out

the

neutral

and

O.

(In

a

Divide

the

twenty the

complete

a

not

T

into

a

usually

Elastic

Data

ring

between

the

O

of of

should

of

segments

equal

length

sufficient the lie

end

points

T

coincide.)

_umber

give

points

segment

and

and

necessarily

central

complete

the

ring

will

and

of

axis

but

segments

Joints

rings

axis

neutral

conveniently

Geometric

which As.

Ten

accuracy.

segments. each

are

In

side

of

to

Mark

symmetrical

the

axis

of

symmetry. (3)

Calculate

the

values

As/E1,

As/EA,

As/GA'

at

the

segment

centers.

(4)

For a which

built-in defines

Figures

(5)

B5.2.0-I

For

a

Use

the

and

-4,

For

a

_'

at

and

pinned-end X

and

built-in

directly. sidered.

ring, calculate a the elastic center

the

axes

ring

segment In

from equation the axes X'

and

(I) below Y' (See

0

is

-2.)

ring, Y

and b C and

no as

obtain centers.

syLi_etrica]

translation

they

are

the

co-ordinates

For rings

of

defined

a only

pinned half

axes in

from

necessary.

Figures

B5.2.0-3

x',

and

ring the

y', obtaiL_

ring

need

x,

the y, be

angles and con-

Section

B5

July 9,1968 Page 28

B 5.2.0

Analysis

of Arbitrary

Angles Measured in Direction of Arrow

Ring

by

Tabular

Method

(Cont!d)

Positive 5

fo

4

--

9

3

/

'

Neutral_

Fig.

B5.2.0-I

General

Ring

with

Built-ln

y

Ends

Note :

i

Neutral

7

6

9

'

Fig.

B5.2.0-2

Symmetrical

X

j

2 0

4

T

Built-ln

Ring

e=O

Section

B5

July 9,1968 Page 29

B 5.2.0

Analysis

of Arbitrary

Ring

by Tabular

6

Method

(Cont'd)

5

4

with

Pinned

O

Fig.

B 5.2.0-3

General

Ring

End

Y 7 Neutral

6

Axis

9

4

_S

3

"%\1 O Fig.

B 5.2.0-4

71/' T

Symmetrical

Pinned-End

Ring

Section

(6)

Note

that

The

_,

angle

8,

must

and

_'

are

originate

counterclockwise

to

measured

at

the

the

end

For

the

slope

of

the

tangent

the

slope

at

the

midspan

of

As/GA'

are

:

As/EA

and

most

cases

compared

the

to

with

the

II.

Limits

of

30

in

from

neutral

1968

Figure

O,

and

B5.2.0-I. be

measured

axis

of

the

element,

element.

not

usually shear

bending

9,

Page

T.

the

and

July

shown

coming to

the

thrust

General

I.

end

going

use

NOTE

as

B5

required,

energies

since

are

in

negligible

energy.

Notes

Application

The method deflections

may be applied to any are linear functions

elementary

formula

connecting

ring or curved beam of the loads and in

curvature

in which the which the

and

bending

moment

holds.

as

redundant

reactions

can

calculate

them

Calculations

(1)

The

advantage

due

to

without

(2)

The

The

referenced

General

In

but in

cos

(5)

Mc

one

is

tabular be

form

used

set

out

the

supports In

is

using

and in

for

finite

segments.

exceptional

steps

arbitrary

T

all

work

IV

an

and

other

essentially

numerical notes

equations.

cases

an

applied.

6.)

form

is

directly

and

of

additional

columns,

Accuracy.

Although

is

0

built-in

"give"

cases the

the

same.

possible

when

ring,

size A

of

the

considerable

the

ring

is

VI.)

the

calculation

e.g.,

for

x

may and

y

will

not

four

or

exceed five

results

three-figure

figures shall

in be

the correct

the

data

accuracy remaining to

three

in it

columns is

be one

conveniently requires

1-6

necessary

columns figures.

in

but

elastically

x'

etc.

Numerical

final

the

certain

with @,

also

III,

the

(See

practice

done

in

may

when

Note

reduced

and

simultaneous

be

table

Y, Thus,

shown

may

applicable

symmetrical

(4)

are

method

reduction

X,

solve

integration

also

(See is

to

calculations

analytical

is

choosing

orthogonality.

having

Graphical

(3)

of

their

order

and to

24-26 retain

that

the

table

Section B5 July 9, 1968 Page 31

III.

#-

Arbitrary

Built-In

Ring

Equations

required

and

(Fig.

the

B

5.2.0-i)

stage

at

which

they

are

used

are

as

follows:

Z F

(i)

x o As

E1

a .

YO

&s

b=

E1

(2)

After

column

12,

A_Ks _ 2Ex'y' tan

20

(3)

(4)

M

x'

, N

.0

0

tree

at

After

As >; sin

2_h'

_,(

=

y,(x

x =

E1

'rz

-

.,)

As _--

sin

0;

y'

cos

0 +

y'

, and

S

are

y

>;

cos

=

y'

calculated

2, r_,

cos

for

0

tile

As

As GA'

- x'

sin

ring

built

0

O.

colunu_

33,

,fAs

-EMoY X

=

_

+ _3N o

cos

_' _ As

_

LSo

sin

_" As _,

-

Z'Y_ As

.-, _, As

As

0.

in

at

T and

Section July Page

B

5.2.0

Analysis

of

Arbitrary

Ring

by

As EM

Tabular

Method

As

ox _-__ _I +

ZN

sin

o

B5 9,1968 32

(Cont'd)

As

_-

es o cos _-GA'

y

As _

Zx2

ZM M

=

+

_

sin2

_

As E-A + _

c°se

As G--A'

_

As -E1

o

-

c

As It. E-"_

(5)

After

column

M

: M c-

N

=

N

Xy

39,

+

Yx

+ M °

+

Xcos

_ +

Y

sin

+

X sin

_

Ycos

0

S

"

S

-

o

(6)

Effect the

of

Elastic

supports

T

coefficients, moment

Supports. and

kTm

required

force

to

distance,

direction

the

elastically support

For

the be

to

in

the

purposes

of

represented

centrated

at

T

and

the

and

the

two when

i

As

1

E1

_m'

EA

kTn'

As

I

As

1

two

defined

by

force is

is no

are

three is

is

$_

move At

displacements

, there

kTm k la

distance.

a

of

radian,

to

at

sets

Thus

required

unit

i.e., @_

koS.

one

force

the

the through

T

in

a

each assumed

applied rotation

to and

the no

_$ the

calculations

by

two

O,

of

additional which

the

respectively: As

kon,

the

direction

through

direction to

a

characteristics

by

kom,

in

is

_

rotation

normal

may

kTS

elastic

T through

T

orthogonal, T

movement

rotate

and

The represented

krs;

move

normal

support

are

, kTn, to

required

unit

O

As GA'

As

1 kTS

I

'

the

support

segments "elastic

flexibilities

considered weights"

conare

Section B5 July 9,1968 Page33

"r

B

5.2.0

Analysis

of

Arbitrary !

r--

Note

that

Symmetrical

(I)

and

_0

(2)

is

the

as

X,

Y1and

Any

For half

the

X

(4)

20,

and

may

the

external

28,

30,

are

loading.

half

Thus, 36, N,

V.

that

i.e.,

elements

(I)

(2)

required

an

the

the

supports

of

this

tional

segment

and

anti-

and

12,

O,

18,

columns

in

M c and

in

N

and

III

need 0 .

S

20,

X are

of

determinate equal and about

C

external

22,

27,

29,

31,

expressions

omitted.

considered

For

in

Y

is

course,

only

elastic by in

to

For

anti-

and

S

be

taken

symmetover

characteristics introducing

III

table,

loading

antisymmetrical

applies

the

symmetrical

antisymmetrical.

are

(4).

the

the

and

omitted.

(4). be

described

B

III

21, in

columns

are

statically XC and

19,

required,

still

(Fig.

Y

load

symmetrical

as

table, required,

in

are,

have

the

involving

formulas

being hence be

not

included

support

section

Ring

are

external

and

M

at

terms

ring

symmetrical

method

S

in

39

be

formula

half

When

Thus and

involving

the

7-12

determinate, and should

the

terms

ring.

X

and

of

the

the

an

the addi-

(6).

5.2.0-3)

23,

27,

28,

30,

33,

35,

38,

and

39

are

required.

is

the

(3)

entered

not

hence

table,

half

4,

moment

not

summations

Y

are

symmetrical

the

the

rical.

not

4-12

loading X and M c are load in the direction

should

loading

Columns

a

38,

N,

are

are

Pinned

M,

37

only

The

35,

from

the

symmetrical

Arbitrary

columns

as

in

from

N

of

are

directly.

and

S

Note

and

values

5.2.0-2)

loading.

for

they

determined

M

the

coordinates

Y is statically the direction YC

determined

to

M,

_Cont'd>

be

and

and

33,

For antisymmetrical being half the total

for

B

analyzed

loading load in

respectively;

(6)

be

in

Mc

34,

necessarily

calculated

23,

opposite

(5)

(Fig.

expressions

32,

not

symmetry

are

Iymmetrical the total

18,

Method

loading.

included

in

of

_

loading

need

Ring

line

symmetrical

(3)

Tabular

The values above of the table.

Built-ln

CY

by

!

_T

_' at T and O. in rows T and 0

IV.

Rin_

is

statically external

determined

determinate

and

should

hence

loadillg.

from

the

formula

in

Ill

(4).

be

included

in

Section B5 July 9, 1968 Page 34 B5.2.0 (4)

Analysis

The

M

(5)

of

Arbitrary

formulas

• Mo

Effect

for

- Xy; of

N

Symmetrical

(i)

Any

Elastic

Pinned

loading

symmetrical

the

total

For

considered

(4)

(Fig.

be

the

When

_

; S In

=

So

this

noting

(Cont'd)

*

X

case

that

sin the

the

_. procedure

terms

of

kTm , kom

B5.2.0-4)

analyzed

loading, in in

the

as

a

the

the

solved

an

additional

IV

(6)

and

this

is

in

have section

(5).

the

purely

segment V

table,

loading load

supports

of

the

symmetrical

load

direction

procedure

total is

method

_Method

S are:

cos

Ring

the

problem

X

Tabular

Supports.

antisymmetrical

half

and

applied,

load

Otherwise,

(3)

by

and

an

anti-

loading.

For be

N,

be

may

symmetrical

(2)

M,

- No +

III (6) should do not exist.

Vl.

Ring

by

Y

Y,

i.e., the

X

is

should

O

is

only

elements same

as

in

X.

obviously half

the

half

the

7-12

statically

direction

and

ring

general

in

O.

case.

determinate, Thus,

need

support

this

being case

the

statics.

symmetrical still

at and

elastic

applies. bu

For

introduced

characteristics

the

the

loading

at

symmetrical O,

as

described

in

Section B5 July 9, 1968 Page 35 Notation

Mo,

No,

A A' C E G I

: cross-sectional area = effective area of cross section for shear stiffness : elastic center : centroid of elastic weights As/E1 = Young's modulus = shear modulus

k

: elastic

Mc

- moment

SO

- bending

= moment

of

inertia

constants at

elastic

the

at

T,

reactions

(e.g.,

in

a

determined).

to

• total

outer

any

T

s

terminus, area

s

- length

_' xp

y

of

ordinates

X,

Y

direct

and

may Y

be is

in

ring

any

of

statically

always

positive is

for

positive

com-

for

com-

acting outwards (Subscripts do

load,

convention e,d

as

point

right-hand

end

referred built-in

axes

and for

of

on not

shear

Mo,

load

No,

at

SO

ring

point

of

where

- angle

between

CX'

- angle

defining

built-in

CY

in

ring

or

cross-sectional

OX

passes

or

in

of and

built-in

ring,

elastic

ring; axes

through

axes

or,

OX,

OY

coof

T

CX of

to

axis

built-ln

orthogonal

and

orthogonal

geometric

of

neutral

CX(_

directions

ring,

neutral

to

slope

respect

arbitrary

to

CX,

ring,

= reactions

of

to CX' referred

referred

with

to ring

slope

with respect = co-ordinates

ring,

load

segment

orthogonal

8

N o

111,6)

"O")

def_nlng

pinned

is

note

ring

co-ordinates OX', OY' of - angle

shear

positive when of a section

left-hand or

of

is side

(see

loading

ring

M o

O

ring

which

fibers.

Sign

or

Mc,

and

and

pinned

moment,

section.

= origin,

Yo"

point

bending

O

xo,

load,

statically

refer S

T fixed

external

to

determined

So

in

due

and

in

section

direct

Y,

pression and the right-hand

N,

supports

X,

pression

M,

cross

of

center

moment,

supported

of

CX,

=

axis _'

CY

for

built-ln

at

directions

elastic

OX,

OY

center for

for

pinned

ring x',y'

,

coordinates axes

CX'

referred and

CY'

of

to

geometric

built-in

ring

and

elastic

orthogonal

Section

[e__] gV sVJ

!

o

L_

' _Z uT

B5

July

9,1968

Page

36

Section

/f

"0

X v

!

0

i

B5

July

9,

Page

37

1968

o= v

I

0

Section

B5

July Page

1968

9, 38

Section

,aO

¢q

Z

÷ '.0

cO I

p_ on 4t_ ¢,q

oO ¢,3 ._.

tr3 -a-on ¢.4 "44,, --..1"

o

p_ ,.-.4 O_ on u o oO on

! ,--4

o o4

p-. on

¢0

soo

X

_u._s

H_

X

0 Z_

< E-4

X

co u_ oO

X >*

X

>4 on

bO

-,-4 0 I

B5

July

9,

Page

39

1968

Section

B5.2.1

Sample The ring

segments

for

segment

are

section

of

at

results

each

to

be

analyzed

illustrated this

as

The

is

in

shown

plotted

problem

Given

section for

Figure

calculations

are

been

this

in on

data,

sample

5.2.1-1.

values

shown

necessary

in

Figure The

average

ring

segment have

in

problem.

used the

The S

9,

Page

40

1968

Problem

frame

built-ln

B5

July

properties the

is

ring

is

at

entire

the

symmetrical

divided

the

into

center

segment.

A

of

typical

determine form

graphs,

which

statement,

and

the

on

pages

are

shown

magnitudes

of

42

46.

through

on

conditions

pages

47

M,

= :

M

= I000

R

= 50

o

I00 I00

Ib Ib in.-Ib in.

follow.

Ra E

= 47 I0

in. 6 x I0

C

=

A

= 11 x

l

A'

= 5/6

A

3.85

x

psi

106

(See

load

S

the for

the

bending given

moment conditions

M,

direct

load

by

of

use

N, Sec.

psi

Ref.

and

N_and

through

7)

Problem:

Determine

cross

The

:

P Q

24

each

5.2.1-2.

to

tabular

problem

The

shear

B5.2.0.

49.

B5.2.

i

Sample

Problem

Section B5 July 9,1968 Page 41

(Cont'd)

>

> T

/ 0

o

O_

|

Imaginary Bracket

Figure

B5.2.1-1

Frame _-"

Section Figure

B5.2.

1-2

Typical

for 1.0

Sample

Problem

in.

A-A Frame

Cross

Section

Rigid

Section July Page

B5.2.

i

Sample

Problem

(Cont'd)

I

2

3

AS

AS

4

5

6

9, 42

B5 1968

7

8

.=

Segment or

AS

r_ X

EA

EI

¢q

_q

i

!

GA'

0 o

Points m this

Operations T

10-6

10_-6

10-e

1 2

.11608 .05156

.24631 .18591

.76774

3

•11608

.24631

.76774

4

.11608

.24631

.76774

5

.05156

.18591

.57946

6

.I1608

.24631

7

.11608

.24631

.76774 .76774

8

.05156

.18591

9

.11608

10

•11608

II

.05156

p"

24631

[.24631 .18591

row

described

in terms

of column

numbers

.57946

.57946 .76774 .76774 .57946

i

12

.11608

.24631

13

.11608

.76774 .76774

14

.05156

.24631 18591

15

.11608

_24631

16

.I1608

.76774

17

.05156

1.24631 i.18591

18

.11608

.24631

.76774

19

.11608

.24631

.76774

20

.05156

.18591

.57946

21

.11608

• 24631

.76774

22

.11608

.24631

.76774

23

.05156

.18591

24

.11608

•24631

.57946 .76774

.57946 .76774

.57946

0 E

2.26979

Sum

of Columns

Columns for

4 through

a symmetrical

12 not built-in

required ring;

Section July Page B5.2.1

9

Sample

Problem

i0

iI

9, 43

B5 1968

(Cont'd)

12

13

14

15

16

17

_9

_9 0 0

W !

!

Y ! g_

O o

4x 5x I

(42 _5 z )

(2-3) x7

×i

_J

CD .,-4

0

i

(2-3) ×8

47.

See 070

Note IV(i) 277.5 6. 197

.99144

.i3051

43.

15_5

17.

875

292.5

.92387

.38267

37.

666

28.

902

28. 902

37. 666

307.5 322.5

.79334 - 60875

.60875 .79334

17. 875

43.

155

337.5

-. 38268

.92387

6. 197

47.

O7O

47.

O7(}

• 13051 • 13051

•99144

- 6. 197

352.5 7.5

-17. 875

4:].

155

22.5

.38267

.92387

-28. 9(12

37.(;(i(i 28.

D02

.60875 .79334

.79334

-37.

37.5 52.5

17. 875

67.5

.92387

.38267

6. 197

-43.

G(i(; 155 07O 070

m

o

o2

%

-43.

155

-37.

(;66

.60875

82.5

.99[44

197

97.5

.99144

875

I12.5

.92387

-•13051 -.38267

-6. -17.

.99144

.13051

-28.902

127.5

• 79334

-.60875

-28. 902

-37.

666

142.5

.60875

-.79334

-17. 875

-43.

155

157.5

•38267

-.92387

- 6. 197

-47.

070

172.5

-.99144

G. 197

-17.

070

187.5

.13051 .13051

--13.155

17. 875 2_. 902

-37.

202.5

•38268

-•92387

(;6(;

2[7.5

902

232.5

•60875 .79334

-.79334 -.60875

•92387

-.38267

37. (;(it;

-2_.

43.

-17.875

247

I_ 6.1!)7

262.5

155

47. 070

J

-.99144

5

•991.44

.13051

Section B5 July 9, 1968 Page 44

B5. 2. I

Sample

Problem

21

2O

19

18

(Cont'd)

% 14 I

172x2

16zx2

23

%

0 o

u)

t3Zxl

22

24

25

26

M o

No

So

o o

162×3

172×3

See

Note

10 -6

10 -6

10 ±_

10 -6

.24212

.00420 .02722

.01308 .08485

11.

.15868

.75466 .49459

.15503

.09128

164.69

•09128

.15503

•48322 .28451

.28451 .48322

22.

7

16.47

96.02

.02722

.15868

•00420

.24212

.49459 .75466

7

257.19

_,08485 •01308

-1.

4.46

-63.0

10-_

10-6

4.46

257.19 96.02 164.69 96,96

16.47 96.96

4.46

257.19

•00420

,24212

013o8

16.47

96.02

.02722

.15868

.08485

96.96 164.69

164.69 96.96

.09128 .15503

.15503 •09128

.28451

96,02

t6.47

.15868

.02722

.49459 .75466

257.19

4.46

.24212

.00420

257,19 96.02

4.46 16.47

.24212 .15868

.00420 _02722

164.69 96.96 16.47

96.96 164.69 96.02

.15503

.09128

.09128

•15503

2.356

.962

1.661

1.441

•

-1.996 -8.366

-5.799

-211.2

-26•356

-13.065

-500.6

-35.790

-23.116

-43.784 -49.024

-35,664 -50.058

.48322 .28451

-880.3 -1457.8

.08485

-2259.2

-50.361

-65,318

-3169.8

-46.935

-80,198

-2527.4 -2244_8

47.798 29.656

18.909 27,523

-1920.2

12.470

31.835

-1535.4

-2.385

32,319

.75466 .49459

708485

.48322 •28451

.28451 .48322 1.49459

29.736 25,028

.75466 .49459

-532•4

-25. 028

19.205

.15868

.01308 .08485

-337.5

-24.849

13.224

.15503

.28451

.48322

-171.

4

-21,647

7.890

.28451

-81.

4

-16.406

3,772

• 24212

•02722 • 09128

,24212

.08485 .01308

96.96

.15503

.09128

.48322

96.02

16.47

•15868

.02722

.49459

.08485

-41.4

-10,278

257.19

4.46

.24212

.00420

.75466

.01308

-11.2

-4.448

2543.

16

*Calculations

2543.16

2.7141 on

pages

50

-1,251

-16.810

-21.484

•00420

164.69

• 957

-814.6

257.19

164.69

9

16. 2

-1166.6

4.46 96.96

.246

.75466

.02722 .00420

.15868

257.19

96.02

• 320

7

-13.896

4.46

16.47

.48322

.75466 .49459

II1(3)

2.7141

8.45964

through

53

8.45964_'.__

1. i5t 0.000

Section

B5.2.1

Sample

27

Problem

(Cont'd)

28

29

31

30

U3

<

U3

32

< U_

B5

July

9,1968

Page

45

33

34

35

L0

,3 Lt_

t-4

03 _.1

O

O

O

03

O O

-,q 09

O

03

O

z

0 U

-r4

O3

See 24x I

10-6

-1.35

24×13×i

111(5)

10-6

10 -6

-63.7

-8.4

-.06000

.00791

-.24357

.03206

32.403

-1193.318

3.6

-.40466

.16761

-.51500;

.2[332

93.465

_-i094.065

.24905

-.87768

.67347

151.123

-954.908

-.44727

.58289

196.948

-732.723

.27741

-.66972

225.649

-4,53.

.58105

-4.4140

246.120

-157.106

8.7

i0-6

70.7

54.2 '''t-

. 32457

76.2

99.4

.29928

-.39003

-.09

-I

-3.7

.59519

-1.4369

.6

-24.52

152.0

-25.81

461.4

-102.19

2953.4

-344. t .54037 -4.1050 -li54.2 F-.84724 -6.4362 -1113.9 -2.5462 -6.1472 -3849.0-6.5650-8.5568

_169.22

6374.0

-4890.0

-116.48

5026.7

-2082.1-8.

-367.97

17320.4

-2280.2

-293.39

13810.1

1818.2

-115.74

4994.7

-222.90

8395.8

-178.23

5151.2

6713.2

!- .3576

-60.14 -94.56

1075.1 586.0

2595.6

-.9886

4450.9

-61.80

-382.9

-45.3 )

-17.40 -19.89 -355.8

-2.13

-92.1

-1.30

-61.3

64567.7

Yxt3

10-6

10-6

[ 67

-9.9447

246.120

157 .106

-5.1259

-12.375

225.649

453

-16.668

-21.722

196.948

732.723

-7.3507

-30.489

-23.395

151.123

954.908

-3.5828

-34.968

-14.484

93.465

1094.065

-11.462

-1.5088

-61.044

-8.036

32.403

1193.318

11.672

-1.5365

14.393

-1.895

-32.403

1193.318

2068.9

5.094

-2.1098

14.734

-6.103

-93.465

1094.065

6442.3

2. 437

1.8698

19.390

-14.878

-151.123

954.908

-9.5797 6498

-1.

3091

.46605

15.10.5

-[9.685

-196.948

732.72

-.6906

2.3867 5.2464

6.594 2.508

-15.919 -19.051

-255.649 -246.120

453.1 157.1

2908.8

.8046

6.1119

-1.924

750.9

!1.7679

749.3

-9.45

Xxt4

Note

10-6

i .88

-1887.15

26x17x3

25x 16x2

2.64

-7.31

26x16x3:

24x 14xl

.2O

f

125xt7x2

>*

0

O

Z

x

273.0 38.1 8. I

13248.01

3. 2458 13.2059

.167

-14.618

-246.120

-157.1

-2.932

-'7.079

-225.649

-453.

4.2300

-3.688

-4.806

-196.948

-732.723

2.4599

-2.267

-]

-151.123

-954.9

4.2679

.763

67 06 06 l 67

1.7654

.7312

-.6162

-.25523

-93.465

-1094.0

i. 0862

.143

0.0000

0.00000

-32.403

-1193.32

-9.9626

-18.563

-89.533

-199.59

08 7

Section July 9, Page

B5.2.1

Sample

Problem

36

X×

16

38

O o

.=

X

>. Note

Xx

17

-.682

.682 2.001 3.183 4. 148 4.831 5.184

.682 2.001 3.183

5.184 4.831 4.148

-5.184 -4.831 -4.148

4.148 4. 831 5.184 _.184 4.831 4.148

46

(Cont'd)

37

See

B5 1968

HI

39

O o

(5) Y×

16

Yx17

41

M

N

25+

-406.0 -352.3 -258.4

26.063

-9.702 -15.433 -20.113 -23.422 -25.135

-3.309 -9.702 -15.433

-25.135 -23.422 -20.113

3. !83

-20.113

2.001 .682 -.682 -2.001 -3.183

-23.422 -25.135 -25.135 -23.422

-15.433 -9.702 -3.309 3.309

177.4 -427.2 -1177.5 -470.3

9.702 15.433 20.113 23.422

-225.8 17.2 225.6 343.6

25.135 25.135 23.422 20.113 15.433 9.702 3. 309

420.0 388.0 266.3 124.2 -53.8 -210.6

-15.433 -9.702 -3. 309 3. 309 9.702 15.433 20.113 23.422

25.!35

42

M c + '24 -34 +35

25.135 23.422 20.113 15.433 9.702 3.309

-20.i13

-3. 309

40

37

+ 38

365.2

27.779 24.957 17.585 6.167 -8.317

531.2 558.4 486.9

-24.481 -40.661 -55.069

-34o.?

-65.954 -71.782 -71.388 2'i.981 4.233 -10.826 -21.966 -28.429 -29.997 -26.903 -19.978 -10.362 .524 11. 143 26.005

26 + 36 - 39

-1.555

5.83_ 12.726 17.887 20.170 18.654 12.752 2.307 -12.368 -30.477 -5O.785 -71.705 20.784 22.652 20.550 15.389 8.315 .575 -6.612 -12.199 -15.406 -15.809 -13. 382

July Page

9, 47

1908

P

:

:

•

"

:i ......

i

i |

•

: ..... '''

i........

i........

...'.

. •

i": ....

|

':

.......

, .....

•,i....i ........ .._..! ;_-;i[ • .'

:

.....

"..

1

,

;

S "1

i

""

[ :.:fill

'

:

i

:.'>.iii

:!'"

_:"_ .... i .................

,1..

i

....

i

•

,

•

:

:

I

i

r

l

i.,,:-,;-:i,; :t

.'I :

...

l

,,

,

,:'. ':'..

",

!..,:1.

....

:," I

:

i

] ...._

;

_

!

,i.....

"', ....

.'I:

;

":'l

" "".

l

• ,

'.

".

,

Figure

B5.2.

I-3

(See

Bending

Moment

Fig.

B5.2.1-1)! Diagram

of Sample

i" ':: ' :"" I ....

,

::

Segment

"•

Problem

..] ..,.....,.........::::::::::::::::::: "."

.:..'::" :': ":;: : :

I I

!

: ":

:'.':: ::1.:::: ,,.

....

, ;',,::':::I , ......

Section B5 July Page

Figure

B5.2.

t-4

Shear

Diagram

of Sample

Problem

9, 48

1968

v

'l: :: ; I;'L .:::".'.:'::".

V.':'::':I ::"::': : ".

:L• - .... i'i'i]lt5 .

2• I

"

l':..::.'"! " ::':: " ":1:':"• '"' ." '

S;irnldC ..

i)roi,l('rn

":': " "s'.::--": _:'" •

ICont'd) . ....

.......

!: :

'

.:: :::.::i..iLF:r

:

I . :" . :

:.....:1', .... ::i: "} :l . . .............

.-_-.*'" ."%'.'"---.*:-.'-."'--.:

_/...:hl..'i-_7-:..-.,T-.-::.!7.:i_;!:.:iiT_!:.i:.:.L:: l::i:::i i::. : _i: :.:.I ..:': :! :!:.. i i ::_:.i _ i..;;. .F.:: i... i.: .. i I'--o-:

o-'_.

.......

:.."

"-'I'--":--.:','-'T"T""

! "

."

"i.':'i

'"

: ....

'.':'"I:

'

: "i'"

.........

_,.--:::--"I'."

i.

........

! .....

": :'.J.

:'d':

.:'.L

'

......

::,_":

..'

L ": "

.....

':

I:"::':" • .: }

...............

•

"'

}:... . •

:

: ......

.q........

" :

"

i"

|_.:::::'":"

"':

. ""._1

i i

: .......

:

.

. : .

.!. !.. .I:_ !: : .. :;.... _l:_.i.4

.

":i

..

'i

:.

: .....

"':"

:

:"

•

"

"':

!

'"

I-IL...L-;:.I i :_::i:...:I_.:. !...L:.L.::!....:.:.; ....:::::::::::::::::::::::::-:.:;..].. i:.:.::....i:.-l:i..i:::.ii.-!:.:.;l:i:.:.:..! ....:...:.. i:.. i

'

.....

l:i :...;!i i.:ii_l:::;...!...:..;..: : .. ! ....._.::.;..,:....._..:.. .:::..,::.;.:.. ....!..;.:..!::..;.:, .: ....... .... .... •

|.'"

:

.

••

• .'.l•

' I:::• ..:..I.... .: ..... .... : ".:'.I" :!'_'..:: .... !--I;I,'i • i

Jt

•

i

" ". • I ::.:" '. ---: ::;-- i :::. _

" : ": ...........

"

I......

"" :

"

: .... ," :

I

I

::.;'"

I'

",".

• ...;

.........

i

,

.

,

.

'_!

.

: "l'

• :: • ,. i .......

,.

I

,

.

."

I_ --

.......................................

I::..,_1...L • ''.

'

•

. . .

•

":

,

. .

,

•

I_1

.

|'

" ....

;

' : '; " : -.i._ i.-.

"1" :

:'

'""'" .,...1;;..{

.....

.

":..

;

:

•

l

•

,

", ...... • "I";'. : .... I .... : • ;!.!i _- '- ;

: :.,

' ................

I_

'

"

"'.

t

"

.

i:!

,i ",;

'':"

":

':

;'.:

.....

""

|

.:

:

"

.

.'

_

"".,l'

.......

:"

[

I

l_.l--"'! i. .-".:: , !_".-_-! " I " : ::.... : :_ _!

"

'-'!" I I

'

..":: : I,

I .......

":

;'...:

::

.

:

.....

•

,':.

,

_

•

i

"

•

:

""

":

:

.......

:__

.

:

• ;

....

: .....

. '

'':

:....

'

:""

;

"

"

_:: :.,I...; !:............ :......... ,_. ....... !.:.'..:: -=....,-t..... I:..,-',..:, ......... _...-: ......_ i_..::i..._ ..... '::: Pi" I • ' ;" ; ": I_L : t" • "1 ' ! " I ; "t :'" " F ": 'i

:" ...... •:

_i!i_.] ;; ! .... i.-._:.: ....... !\ ....I......::1::......:II...:!

i.

":;"

.-_"

: "

::....

I.::i"

'.-,,':T, ..... • ,: ......

","

" "'

"!

.

"

:

_

•

"

!

:

......:..... ITS:::i: Ii:: :,

:"

:1" :!" 'i::':i"

.....

",

i

I

I];

"

l ,

;

.

"

' I

I

_

:

•

........

I ........

:l "........

..:.:...,

:%'::':"

iL'._2:.I-..:...

I

::::::;

"I

. " _ ..

''

i{:::!._;i .. •":-E::._-ta":

..

i : .:!

'. " i:

"

........ :;.... :.-.:

':::

_i_ii_!:::l i: :: " :.:j !:: " ::.

'::

::

.

.... -=...;:..-

'

•

::" :" 1"" .... i :::":::':::

::'-'-_:T-_:'..: .... :_,,|,,,.. .....

:

:

!

....

I

•

.......

'.

'

!

:;::'"::r

.i

"

"

.:

!

"

"

"

,

• ;

: i

•

•

•

Fig_ure B5.2.

"

''

"

:" 'T"

;

I

,

:"

'

'

:

.., '!1"1, ...,,.

• I

/;

'

"

]

•

-..---..-

:';i: .,

I!!

_'-

:

I '" I,d ....... '

:

III

'

;: "t.,,, "

"

'

'

'.

'.

-.;.';: ..........

-'_

:i:Y _i:!il

: " i .:;[;:-

'

i'i;:: ......

!........ :!'-;H'i'i,"

,...: ..:.; • ..:_::::'.::;;:;;

_ii,!__ .

"

"|:

":

I ..... "'", .... ",............ "|

I

."

""

Load

•

I},) I

"

"

:;.:':'.,,i : • ':....

_..,.:::L.;;;_...;_I: |' Ig

: "

:

;

I!i : "!!i_i;:. :l: !;-:;1::. :-.!....:..,!...."-'l_;:;!;-i.::!.. • :;::_i:_i_!i

(S(_('

' •

"" ';'

I

:.1_;: +_ I

Direct

!

I..... _

1"

..."1

_

.

'

................ ,:....... . ...... i : i::: i

r__

" ,.r.'

::-...... ::: ", ,";....""1........ ,....... : t'rh

•

";:

.. • -_,•

i

r

:

k-i ---! .... ;'t--.=1.,-;;

"

" ",.4._."

_' ""

"

.... ! ..... _£.:....i:_;....._.........

_'

:

'

...... ,_,;...._, .,L._ ....,ll....,., t..l.. 1,.,.....i,.,....

... , _ ,...:L.,:: _ .....

I-5

' •

, !

! ; : •

"":.,-.':"";

I

, "" ':'1 ......

;:;i

".... _..',_=-...=_ ...... ,....... :,.......

;-

. ::.,..: .;:.:

i"

'I

.'::

.:.I

..1',]

L ..........

I

.-:1 ...... :;.:

i

':: ::;:,! " • ,I-- : ,,i " .... I : ..'. :.._( gll

__.-,.l:;'''-::"'

•

:::.i.... I: :ItL

.,: ! .....

-'"

:

:'1 ....

"

I ; : • .I .':'" I .:: : i " •'"":r .... I ..... • i ......... ' _ •l_ _1'" ...... • _ :':":" ........ "::

.:--:-'.-:-.: .......

:

: li

'

,

,,1

i. i !: ::!

............

• : ..::1 .. ,

•

...... ;..... :'-": ....... ..I.......... ;.

........ _ ............. :::;':: ..... • • : •,":---_;=._.-

.......

"

....... .... i: ii: ',

•

,':_;i: i : : :_'. ';" :_ "" ' i:.:' r • .."--:...! ......... 1I........ ...... i'-::' ...... "1_ .................|,,..I...:--:: _ ; • ..... ''

I

:

:

,

..... !' I

"' '

,

I

_

."

:."l:

_

:

• ........... i ::\

I

"

';

!k:

.

'

..

:I...

..........

T -"; . I': .'i''_"'i"

. ""

.I.

:

._

-'

:

. ._ ..i... i...:..:;_

"

" :

""

I :

.......

'

....

:

:•

•

. i ' i .:ii. : i

I ....... :::i..'...:

:

|

i

: •

.i ....

. :'

%

:

:

| ................

'"

•

:" ; " "'" J_ ;""' •: i'" .- "" .... i......... ":'

..... :..l

I .'. ......

I..,

,', :

•

: ,. :..: •

"

;

•

I :':.: ...... I "-_.i .!

........

I_: _, _-,,I._ ."ri.;.;.......I

:

•

..li:._. _ .:.: :+:;: ; ;:1

1

.... ""

:

:_i:"':i.:. _

:

"i

":"

............ . ' -:.

_ ":..

i:..

.._

"

..

:l

" " "

,..

:

.......

,_i;''

,

i_:::.i.I... ::.i:: .: :.i .:_:_i;...r i i, i,::l ::: i:!_::l i

•

I ....:

....

.......

:. -I •l Y :i; I:i !:,,::; .....

:"

...

....

: " ..' " ! "".J i:: ;:::i:-,

li::.:::...;-... I. ....__. : : _::.i: :_. ! :: lli:Ti :1 ii :i i i:i__ i:: .....

• "

2

'

'

:

"'"":"':I .'".."

'

Diagram

.....

"

•...... I :";":'I'.-.':'::';':";"E; , I,

I.....

I

!-'_'-

_ ,-_"

of Sample

:

i

"|

:

;

" _ _

:'"

i :i

I

:

I

'":' ;................ I ......... _.... :.::'::::':" :::::':::" " '::':'::"

.: .l::".:J'W:::: ,, ,,,;., ; '-":,-"

1- ! _'

: h .,,,,_t_'..;.. I .... .....

'1

I,

I:'" :':

:""'1 • - _

• ": "'!"

I:.'.; .... I:":":: -_"''-":

-!'!

Problem

.....

'1"""."" .. ........

:1

:'; ..... :" :'.:::::::1

"'" :1

'" '::!!

!

"''_

"

.......

Section B5 July 9, 1968 Page 50 B5.2.1

Sample

Problem

(Cont'd)

The values for Mo, No, and S o are calculated and shown in the tables on pages 52 and 53. The necessary formulas for computing the shear flows are shown below. In these equations, counterclockwise shear flow is considered positive.

Figure

B5.2.

t-6

Symbols Mo,

and

Sign

N o and

Conventions

for

Computation

of

So

P

-M

qPot

= lrR---o sin

_

qM_

qQc_

= ---Q'-sin 7rR °

(c_ + 90 ° ) - 21rR _

o

qa

-

2_R:

= qPoL + qQo_

+ qMc_

In order to perform the calculations for Mo, No, and So, the shear force acting on each segment must be known. This shear force is obtained by multiplying the average shear flow acting on the segment by the length over which it acts.

Section

B5.2.1

This the

Sample

Problem

average

shear

typical

segment

l

July

9,

Pa_e

51

1968

(Concluded)

flow

in

qavg. = _ {qa

B5

is

Figure

÷

determined

by

B5.2.1-7,

tt

using

this

Simpson's

average

Rule.

shear

flow

F_

r

is:

tit)

4qa + q_

where: t

cl_

= Shear

flow at left

end of the segment.

tt

qc_ = Shear

flow at center

of the segment.

q_!

= Shear

flow

at right

end of the segment. f!

q_

Figure B5. 2. I-7

The and

53

number No, and omitted.

formulas

used

for

are

listed

and

its

range

are

applicable

M o

under is

Shear Flow Acting on a Typical Segment

computing

the

cables.

the The

indicated

where

for

values

all

values index necessary. of

n,

in (n)

the

tables

on

refers

to

The

formulas

therefore,

the

the

pages

52

column for

index

So, was

Section

B5

July 9, 1968 Page 52

B6.2.

i

Sample

Problem

(_Cont'dJ_

e_

+

+

t!

+

h

+

W

fl e_

H

I$ el

e_

II

>

0

II e_

II

II

II

fl

II

Section B5 July 9, 1968 Page 53 B5.2. I

Sample

Problem

(Cont'd)

+

0 i@4

_

_

_

u_

_

_D

_D

_

¢q

Q0

x-_

OO

Cq

OO

_

c_D ¢q'O

r _- P-

¢q_

_--

o

I r"

0

Z

O

_

I

......................................... Ill

I[I

t._

d

lllliJi

+ O I

IIIIII I

II O

.............

............

%,_°_=_ II

e-,

il

O

I

.= iilllll

I II O

4_'_'_'_'4'_'_'d'_'C'_'_'_'d'_4 i'_ _ _ _ ,I

_, _

_

_

_

z

"'_'_'

,

II

c_ °,

.,

°.

°,

°.

°,

°.

0.

..

°

.I

°1

°i

.i

.i

.i

°1

01

°1

ol

°1

.i

°

" _ " ®" '<>-=_ _,i_ '° " _ " '7

+ I

I

I

I

I

I

O c_

>

!

_

q

O

Vl I

--;I

II

II

II

Section

B5.0.O

-

B5

July

9,

Page

54

1968

Book

Company,

FRAMES

References Timoshenko,

1

New

,

.

H.

Wiley

Wilbur,

&

J.

No

E.,

Co.,

D.

J.,

York,

Argyris,

Structures,

H.

Inc.,

McGraw-Hill

Theory

H.,

Co.,

of

York,

Structural

York,

C.

Book

New

L.,

New

Norris,

L.

Perry,

of

Bowman,

and

Grinter,

and

•

B.

McGraw-Hill

New

.

and

Sons,

Edition,

Macmillan

o

Theory

Inc.,

1945.

Sutherland, John

,

S.,

York,

Theory,

Fourth

Edition,

1954.

Elementary Inc.,

Modern

New

Steel

Structural York,

Analysis,

First

1948.

Structures,

Vol.

II,

The

1949.

Aircraft

Structures,

McGraw-HiLl

Book

Co.,

W.,

Structural

Inc.,

1950.

J.

Data,

H.,

Fourth

Dunne,

P.

Edition,

C.,

Tye,

The

New

Era

et

al.,

Publishing

Co.,

Principles

Ltd.,

London,

date.

Roarke, McGraw-Hill

R.

J., Book

Formulas Co.,

for Inc.,

Stress New

and

York,

Strain,

p

1120,

Third

Edition,

1954.

r

SECTION B6 RINGS

TABLE

OF

CONTENTS Page

B6.0.O

Rings

...............................................

6.1.0 Rigid Rings 6.1.1 In-Plane 6.1.2

..................................... Load Cases .........................

(Index to In-Plane Cases) ................... Out of Plane Load Cases ..................... (Index to Out of Plane Load Cases) ...........

6.2.0 Analysis of Frame-Reinforced 6.2.1 Calculations by Use of

..._.

B6-iii

Cylindrical Shells.. Tables ...............

I

2 7 8 57 58 59 70

.,.r..z

B 6.0.0

and

with

loading,

respect the free

p

Only load the

section

--._..

B be

are

presented

shells.

Section

to the resisting body

is

given on of shear

B

"flexible"

should

6

July

9,

1964

Page

l

of a ring

to facilitate B

6.1.0

medium,

supported

by

the analysis

deals

with

i.e.,

for an

rings

of rings that are

out-of-plane

a thin shell

is as

follows:

".

bending

cases effects

Section by

tables

ring-supported

rigid

B

Rin_s

Charts and

Section

6.2.0 rings.

6.1.0 gained

considered

in

the

pages 9 through 54. and normal forces.

deals The

for any on both

with choice

given

circular between

problem

methods.

deflection Refer

curves to

page

cylindrical the use

for

the

56.1

to

shells

supported

of this section

is left to the analyst.

in-plane include

over

Experience

Section

B 6

15 September Page 2 B

on

6.1.0

Rigid

Rings

In general, a ring loaded

four basic loadings in the plane of the

Loading Loading

by by

3. 4.

Loading Loading

by a single moment by a distributed load.

for

a single a single

are required ring. They

I. 2.

Special cases B 6.1.2.

1961

out-of-plane

to define are:

all

loads

radial force tangential force

loadlngs

are

considered

in

Section

The procedure for calculating the Bending Moments of the basic in-plane loading is briefly reviewed in the following discussion. Many other loading conditions may be analyzed by using these basic cases by applying the principal of superposition. The general a cross-section) dm ds

S =

The

general

N+dN

equation is

for

dm rd_

equation

S+dS

I

the

transverse

force

(shearing

force

................................

for

the

axial

force

is derived

(i)

as

follows:

X

(N+dN) 7P_Rd_ N_N

(a)

(b) Fig.

B 6.1.0-1

de 2

on

B6.1.0

Rigi d Rings

(Cont'd)

7_Fx = 0 = -S +

(S + dS)

= dS

Neglecting

N

The

the

dS = - d-_

second

+ Nd_

second

- P_R

term

(P_R)

The procedure i11ustrated by the

+

+ dN

order

term,

B

July

9,

1964

Page

3

P.Rd_

QdN

_

_

..........................................

occurs

in the

to obtain following

6 .

d_ + P_ Rd_ + N -_--

(N + dN)

d___+ 2

Section

case

of

a distributed

an expression for specific case:

the

(2)

load.

bending

moment

is

Load

q

q

P

2

(h)

(a) Fig.

B 6.1.0-2 Because

is is

The shear flow distribution is obtained the static moment of half the ring, S is the moment of inertia of the section. ¢ =

S _R 3

2

f _ O

Rd95 R

cos _

=

S

of

symmetry

Y = 0

from q = SQ/I, where Q the shearing force, and I

sin _5 _R

=

P sin _R

Section

B 6

15 September Page 4 B 6.1.0

Rigid

Rin_s

(Cont'd)

dMq

= q_

R [1 - cos ( _-e)]

Rde

= P-_R

Mqfl =

R [1-cos

sine

TPR

_ PR ¢_

(I

R2de

(C-e)] [1-cos

(¢-9)]

sin0

[1-cos

(¢-0)]d@

_ cos

¢

sin ¢)2

o P

T Fig.

P = -_ R

Mp¢

sin

¢

B 6.1.0-3

Z =+i R_Y

- +I R

sln¢

0

Mx

- -R cos_

1961

My

=-R

sin_

Fig. B 6.1.0-4

M z = +I

Section 15 Page B

6.1.0

Rigid

The

Rings

B

6

September

1961

5

(Cont'd)

displacement

"i"

due

to

load

system

"k"

=

= f

5ik

MiMkd E1

s

_R 3 E1

5xx

=

/

(-R

cos

_

)2

Rd_

2

o Deflections E1

5yy

=

/

(-R

sin

¢)2

Rd¢

R3

to

2

shown

o

E1

5zz

--

unit

Fig.

f

(+1) 2 Rd_

due loads

in B

6.1.0-4

= _R

o Displacements

E1

due

5xo

to

applied

=

(Mp

forces

+ Mq)

and

Mxds

reactions

Because

of

symmetry

MpMx

=

0

3PR 3 / o

E1

5zo

=

<

I _ cos__

(Mp

/_

o/

_

+ Mq)

PR_

_ -2sin

P

_

PR

(-R

c°s_

) Rd_

Mzd

_2 sin_

+

i _

cos_ ._

psinp_ (+I)2 _

Rd_

-

o

_ xo

3P

X; 4_ xx Redundant 5 Z

obtained

by

equating

deflections. zo

3PR

-

= 2_ gz

By

superposition

M

= Mp

_

+

dM R_

-

dS d-_

XM x

+

3PR 2 _

S"

=

+

PR 2 sin_

+

N

Mq

=

ZM z

<

_I

(+i)

=

- cos_

_

( _-¢

_

_sin_ 2 n

) sin

¢

_

PR

+<

- _3P_

-

- sin_ 2

[ (_"

+

_)

sin_

(_-_)

+

cos

3

cos 2

_

R cos

i 2

[

=

+

]

2_

P 2--_-

]

2nP

3P"2

Section

B 6

15 September Page 6 B 6.1.0 Sign

Rigid

Rings

(Cont'd)

Convention

Moments

which

Transverse

produce

forces

tension

on

act

upward

which

the

inner

to

the

fibers left

of

are

positive.

the

cut

are

positive. Axial

forces

which

produce

tension

in the

frame

+S

+M

Fig.

B 6.1.0-5

Positive

Sign

Convention

are

positive.

1961

Section

B 6

15 September Page 7

B6.1.1

In-Plane

Coefficients

Load to

and axial force along some of the frequently

1961

Cases

obtain

slope,

deflection,

with equations occuring load

for these cases.

bending values

moment, are

given

shear, for

Section

B 6

15 September Page 8

B 6.1.1

In-Plane

Load

Cases

1961

(Cont'd)

Index .

I0.

o

i

p

IP

13.

_.d

M'

18.

20. P_ =Pmax cos (2_

P_ =Pmax c o s

ax

21. P_=PmaxC

23.

os

25.

p _a

P_ =Pmax c o s

x

P_ =Pmax

co_

P_= Pmax (a +b cos_ +c cos2_)

Section

B

6

15 September Page 9

B6.1.1

In-Plane

Load

Cases

k

r

1961

(Cont'd_

_f

f

'-

il . I N

/ f

/

/ /

\

./

I

\

"o cu

i

'b o

\

\ }

>-...

/ C jf J

/r i, r

I

jl

"o

f / /

/, 0

q

o

o•

o.

9 9

I_

_-

9

?. o,.

Lf'__

Section 15 Page B

6.1.1

In-Plane

Load

Cases

B

6

September

1961

i0

(Cont'd)

/

/

/

J

j/ r

I

......

/

J

/

•

!

g¢_

r

5

r

/

/

.o

_,.

----+p--

0

8

/

/

/ J

\

O

0

/ _b

\\

e4

/ •

Q

F

+

\

i IN ".\

\

8 N

b

\ % \

0 Q

0

\

\

. J

" ,/ /j/I %%

/X

"

"/

+

....

o °

! _

0

0 __

__

_,]

S /

//

/

/

d

/ / i \

/

\

t

v

t

......

N.

(D

__

O

0

t "_

I

/

O

i

- ..........

0

eo

O4

O4

(D

o.

I

_'_

___

o. I

v

............. ..... _

0

o/

/ io

_--

f

/ O E

....

/ \kk// /-

!< Z

/ /

/

o.

O

f

I'

I'

I"

o

q

-

-

C_

I

I

I

I

I

i_

,t °0

Section

B

February Page

B 6.1.1

In-Plane

Load

Cases

l l

'd )

(Cont

,f

4\

n

yl J J

f b_

_

I,

/

LI f

J P f

h

f

I

r Z

f

J

f

f

(

f

Xx

.. _<

f

-i

6

15, 1976

Section

B 6 1961

15 September Page 12 B6.1.1

In-Plane

Load

Cases

(Cont 'd

f

i

j

o 0

I 0

J

f \

j

J

I' f

\

/

J

f

o

I

/

i fl

t

0

i

o

i" 0

o J _;oz

¢q

_

0

oz

o _0 oJ

\

L

%

o-

Y-. J

\.o

-.

)o j \

i

11

° \

\

7

-... J

k

i

) J J

j-

A

_J z

-j

_J

cO

1

/ O/t

(

J f/

\

i i

/ /r

•

I

i

t

f

,i f

_

v

_"

o

_

N

oJ

o

-

o

_'

o

o

I"

_

co

o.I

(0

o

q

-

-

I

I

I

I

o I

Section

B

6

15 September Page 13 B 6.1.1

In-Plane

Load

Cases

(Cont'd) J i'

b ,/

/

b

h

I_..

/ I

a. k

b J i

i

f

\

r

\

i

/

f

f

J

i

i

J s v

i

i

\

f

i Fs

4

w

\ \

\ \

k )

_>

l

\

jf J

J

J

J i

f

J

/

J f

_J

/ f

-_.

q

O.

O.

q

Q

o

o.

0

0

0

I"

I'

I"

0 I

1961

Section 15

In-Plane

Load

Cases

f jr

o

/

.,_"

/

/

//

i

n

ii

1961

14

(Cont'd)

Q.

--_

6

September

Page B6.1.1

B

/,

/

!--

o

7-....

,o

,I

ON

,/

Ii

o

• _

5:oz

i

b

i

0

'N,_" •

\

b

",_,,,

_

k'M 0

/

j,-

/

/

i"

/

,-

i

,

t

,

i

vv

..,, "

/

_

•

-0

"

<' 0

i o ° 0 o

\ _

.,_ _

b

_

, _.

o

f

\ "\

-

._

j'/

"\

'

/

b

....

-_

/_ _oz_

/ t

( i;

v

/

/

/

\_

o

," ----7-

/

I//

__ z

/"

,i I

i

_. -.

o

7

e_ I'

_

I"

_. I

to. i

Section 15 Page B6.1.1

In-Plane

Load

Cases

B

September 15

(Cont'd) I

I

\

i

f ....

J

\

O

O

J f

,F

oJ O

O O ro

\

O

o

a.

O

O oJ

J

@

O

J

a.

jf

\

J ro

oJ

)

J J

/

f

J

/

f

O ¢q

J

I! J

O

J

f

0J

O GO

J J

\

o

J

O

O

/

O

x

(

O

O oJ

\

\ O

"\\

8 O

S o r

jJ

\

_f J

7 J

_/

f

f

o O

/ --%.

<)=

0

--. I

0

o.

O

o. I

04 I'

rO l"

CO

Od

o I

I

6 1961

Section B 6 15 September 1961 Page 16 B 6.1.1

In-Plane

Load

Cases

(Cont'd)

V _J J

J

f

0

0 od I¢)

_J

/

J

f J

/

I

l¢)

o,I --.

/

_o

,

E,

oJ

v II

=E

o.

o-

0 X

Z

t

L

oJ

iJ

||

0

\

Z L

od

\

0

0 oO

\ j

O.

_p

i\

0

J

_f I.

0

J

f

I J

i

/

r

S J

J

f

/

/

f

/

0

0 0

/

oD

f

@

0 _P

f

_r

_Z

0

I 0 0 V.

N

0 o,I

"

I

c_ I

to !

I"

I

I"

I

I

0 I"

T v

Section 15

6.1.1

In-Plane

Load

Cases

_-'-

6 1961

September

Page B

B

17

(Cont'd)

,

-_._

to O

L

f-

O

O

J

J

I ,f

a,.

_'_1_.

f

I

I

f

J

j

)

o

O _

f 0

O

0

O 0

O N 1

) L.._.._

o

_

_....-

O

n

f

0

_

O

o

O

O

J f J

) o

lJ

f

f-

_

_

s

J

8

_ Q

O (10

t

_ i

) f I J i

0J

r

I

f

l_

ff N

Q

--

Q

O

O --

_ O

O

Q

o

?. O

9.

O m

9.

Section

B

July

1964

9,

Page In-Plane

B6.1.1

Load

Cases

J

/

(.I

o

_;v

2"

v

v i|

I|

i

i

/

z

18

(Cont'd)

/ W

6

i

/ f

/ ,I

(!

II

f

f

,

fA

0.

\

I'\,"" /

- _

O4

..2

l

/

J

]

/

,i

1/

/

/

t

\

t

/

f

f

_p.-,

/ I

% /"

,/

I

ir

i

i

f

J

/

i

I

I

/

(

I •

\

/

f

i/,,

_

>'..._

/ \

/

/" /

i

i

I

/

/ /

/

f

i

/

i J

(

f

t

/

\

\ •

ix

\

A

__ t

/

¢t,

,¢

-

o.

I"

I

I

o

o

-.

I

I

I

¢0

-

O.

0

I

I

I

-vf

Section July

B

9,

Page B

6.1.1

In-Plane

Load

Cases

6

1964

19

(Cont'd) I ]

I

I.--

b, ---

- +----

_

I

_....,. i i

I

-

---

I i

[-

I

ro

[

b

I

,

-1- -K

?:>

b '- .,,,

i

--4 -I _

_I

.

_

÷___

_

oJ 1

J ....

-_!__

I

-1 ....

'

+

i

--*-

, t--

--I

\

I

_

,

/

1

\,

I

_'[_. /

,

_

/

i ....

__ ____,

....

1 ____+

....

1

-

44

I

1

I

l

r

I I

\ C

Q I

I

0

0

0

0

0

0

--

o.

o.

o.

_.

o.

I

I

i

o. I

!

Section

B 6

15 September Page 20 B6.1.1

In-Plane

Load

Cases

(Cont'd)

\.

i

f" O

_L O

o i'M rO 0

Z

_--

........

"7 O

Ivv I

II

\

II

I"

%.

iOZ

I

-

t

li

i GO ¢J

'\/I

O

O

I

/i

"_

%

\\ k,

N o

_=

................

_

o

\

(Xl i'M

\

I _-..

/ p

_

,,s

J .,._

_

_ _

o o

_t.., 7 .............

7

/

0

0

/

_o

1\/" I/'_, X" / /

F

0

o

\

t

_

L ........

/

/

0

/

Oa

o

-

_/ o.

_-" 7-"

.......

/ _. --.

/

_ --

_. 0

N. _

•

o

N. 0 I"

_,I"

0 I"

_.

q

_

0

'

I'

--.. I

I"

T

I"

1961

Section 15 Page B

6.1.1

In-Plane

Load

Cases

B

6

September

1961

21

(Cont'd)

i r I

k_

O

0

-\

0

/ 0

0 0 0

11

O

"+"

/

/

_N

11/

/

fl

/

( \

_'o O4

q

LO

N

cO

_1"

o

o

o o•

o o•

•

0

_

_

oa

_

0

o 9°

o ot'

-oI"

-9"

q=

Section

B 6

15 September Page 22 In-Plane

B 6.1.1

Load

Cases

1961

(Cont'd) f

./

I I

t J

/

,,j

/ /

/

\ \ \ \ \

\

N

/

\ \

/

e"

/

,

Q

/

\/ /

t I /

fl"

/ o. z

/

/ /

II

/

/

/

!

f

Z

% v

!

! \

I

/

II

% o

(3

\

\

IE n v II

=E

"o

(M

0

Z--

Section

B

]5 September Page 23 B

6.1.1 In-Plane

Load

C,'_ses ____

6 1961

Section B 6 15 September 1961 Page 24 B 6.1.1

In-Plane

Load

Cases

_J

(Cont'd)

,, ,, ,

\,

IF. o;z

•

/

\

/" \

"_

.

_

"

pJ

/"

/°o

// _-

oo

/

I

f /

/

\

>-

a

/

J

. ,-

N

1

%

J

-

)oo

%

\

.i

\

/

_ o_

\ Z

I'

v_9

_.

Q

o

o

i •

o"

i

Section 15 Page B

6.1.1

In-Plane

Load

Cases

B

September

6 1961

25

(Cont'd)

°J,I

J

jr I

) a.

11

J I

m-

f

'° _I

_

f_

J i

)

a. J f f f

J

_I

I

f

fJ

/ \

I

"o

J

f

L

J J I"-J

I

I

f _v

11

_.

, I

_ I"

' I'

.

Section 15 Page

B 6.1.1

In-Plane Load Cases •

(Cont

ro

oJ

I

q

o

0

X II

0 II

_j_

_

¢_

r0

q

q

q

/ /

\ O_

1961

26

'd )

0

\

B 6

September

0

0

Z

0

__

II

\

_'_OZ

0

0

0 "-L

0

/

// 0

0

_I I I vP D

0

0

0 l

o.

0

0 0

\ \

\

\

I°

0

0

/ "\

I

_D 0

0

f

J

/J

/

J/

0

0

./

J 0

0

I

/

T

J

¢.f /

/-

J

N.

o

f

/

°0

,./

/'o

N

"

I I"

0 0

I

l

--.

0 _

_

/

_zj

v

I

0

9

_ 0 l"

oo 0 I"

I"

_ -I

I

_D -I

I'

0 /

]_

SecEkun

15

Suptembec

Page B

6.1.I

In-Plane

Load

Cases

27

(Cont'd) o

0 \_

If

k

(.D to)

_0

0

0

0 to

<1

n

!_

_

-"

_

"_

.....

0

0 0 0

3.

"_A/"J

_'--r',_

.........

\\

_

0

:

--

00 0

0 cxl I1.

/

._t

-----

Y

0

0

0

0

t

cJ 0

0

0 _'_

,

.........

0

_

0

_1 _ _

__. .........

kO_

.......

jf J

_--''"

0

f

0 I

/

0

f

/,I

X

0

/

0

$ \

J

jl


0

o

0 (0 0 Q

o. _ 0 8 o •

o

9 8

_. 0 o I"

I

I

D

]-_JU I

Section 15 Page B 6.1.1

In-Plane

Load

Cases

B

September 28

_Cont'd)

b

o

0 0 0

0 CO oJ 0

0 OJ 0

0

0

0 oJ OJ 0

0 0 0

0 co 0

0

0

0

0

0 oJ 0

0 0

0

0

0

0 0

0

2_ o

_

_

_

o

_

6 1961

Section

B 6

15 September Page 29 B 6.1.1

In-Plane

Load

Cases

(Cont'd)

11 J Itl

J

Q 0 O,J r'O

F I

/i" 0

f -

/I

0,,]

I

0 0

\ k

OJ

1.1(_

J N O,J f

/

I I.

0 N

f

r

J

I

L r \.

J I J f

,q"

) J

0,,]

\ jJ

_J f

J

J

j

'

0

I

I

f

I

/

f

J

/

(

J

\

..p--

_

-''

% N

"\ o.

Q

¢D 0

_. O

¢M O

o.

q

=

_

_

0

0

0

O

o I"

0

o I

?.

1961

Section 15 Page B

6.1.1

In-Plane

•

-

Load

O

C@ses

o

B

6

September

1961

30

(Cont'd)

O

O

0

v

0

0

0 0 0

0 (1) 0

0 cO 0

0

0

0 oJ Z

"

q

Section

B 6

15 September Page 31 B6.1.1

In-Plane

Load

Cases

1961

(Cont'd) 1

1

_o J ft_

/

! f I

°o

J I'-f J f

jo

I

...i o

I r_ I

0,I

J _k

_

J

.......

J

"8

I

a.r.

,Y

I

jJ

_f II

.s

f

0o

jl

II

W

J J _p

\

.ff

.# h

f

f

oo ,-...,..

J

I

r

I f I

J

I i

J

'r

I I

J J

J f

,i

Q_

I

J

J 0

° 0

I N

,o

9

o

Section 15 Page B

In-Plane

6.1.1

Load

Cases

B

6

September

1961

32

(Cont'd)

ro

QZ

j

_..-

m

m-

\

_I

f

_'-"

\ mr

I!

N

N

OZ

_Z

_L

r

/

...--

/

/

Y \

f-

/

-

I

11

I

/

/

J

11 J J f

J J I

J

O,4

o_:1

i

v

N I _

_1"

_

l"

I"

I"

!

_.

0 '

,-r_

0 !

I"

uu

Section 15

B

6.1.1

In-Plane

on

U


_

Load

Cases

6

September

Page

_r

B

1961

33

(Cont'd)

°c> 0

o

o

•

"

o

o "

o

o

o

I"

I"

I"

o r

Section

B 6

15 September Page 34 B 6.1.1

In-Plane

Load

Cases

_Cont'd)

I,

\

\

\ \ l

\

\

\

\

j

\

\

\

\

l

,

\

\

I

\

r'

\

/

\ x i

Z2_ I! _

Ig V

• I! _

L

/ 24 //

!

J

/,vJ

J

!

,

/

1

/

:/

/ / /

-i

l

(

/

Ii

i

J

/ / [o/

,"\

/

\

"4

\

\

N.

\

1

\\

/

l"

i

\

/ 0

"7 I"

I"

I

I'

1961

Section 15 Page B

6.1.1

_n-Plane

Load

Cases

B

6

September 35

(Cont'd_ f I

;I

f ff

O

t_

f

/

f

/

f

f \

,,i

/

b GO ,OJ

O0

||

\

\ %

J

\

J

L

f I

Jl

f J

:b GO

/

J I I 14

I f

f \

/

f

J f

/

/ _o

.J J

¢q

f

J

<

\

\

J J j-

\

I -%.

0

_,-_i_ 11 1 PO "_. OJ I

0

q

q

o

o

_

l"

I"

_. _ I"

_ _.

u'_

i

I"

_ I"

o!

1961

Section 15 Page

B 6.1.1

In-Plane

Load

Cases

B

6

September

1961

36

(Cont'd)

\ \

\

\ \ \

\

\ \ \ \ \ \ "_ IQ= • o _ _ 0 II

II

0 \ \ \

II

\

\

\ \

0

\ \,

0

\

\

\

I

0 \

\ \ ]

\

Q

\ \ \

\

\ \

\ N

0 I

|"

I"

$

I'

Section 15 Page B

6.1.1

In-Plane

Load

Cases

I

B

6 1961

September 37

(Cont'd)

\

i ro

,_;I_ _;lW !

0J rO

k

b

@

b_

j

o I jl I I

I

f

f f \

f

..ow

f"

@

O

I

0J @

8

\ \

b

L

Q

O

t f !

/ f_

@

O

@

O i

o

& --

\ @

O Lo

f

\

j

@

O f

.-_

[ f

J

L

I

°

J

/

@

O N

i

f"

! Q

o.

q

5

O

,

?

o

o

o

I

I

--

{q

o I

!

N

Section 15 Page B

6.1.1

In-Plane

Load

Cases

(Cont

9 J

/

-%

J

=

/

\

/

II

,x,

J J

/

J J

J

0

/

0 0

\

0

/

J

j-

N

#

J i

/

j

I

f

__D

/

CM J

/

jl/

/

J

/ b.i--

fd

/

J

f

-e.

/

0

0

J

/;

_0

1961

38

I

)

\

n

6

'd )

\

II

B

September

/

!

0

/ (

/ /

_b

/

\,,

/ 0

0

,\

f

\ \

\

f

/

0

0

X i / /

\

\ \

!

I

\ 0

0

\ \

0

0

\

z

\

\

\

%

\ 0 Z

I

.. N

0

0

/,

\

0

0

0

0

N r

N I

I

I IL

i

r

I

Section 15 Page B

6.1.1

In-Plane

Load

Cases

B

6 1961

September 39

(Cont'd)

\ \\

J

J Od rO

J

f N

o

I

_y

O0 OJ

> l'=Z

,
J

____

o OJ

-----_7

jl

F

f

f

i D

J

o

f_

<

/ CO

J

\ J

I

>

Od

J

o

11FJ_

\ /

I _F I --I

J f

I"--

f j_

<-

fl

...I--

,4_L o

o ¢x;

\

v

o

o

o

N

o

,o.

r.j

o 0'

i'

¥

i" P

0

,q-

f,O

¢_

--

0

0

0

0

.

.

.

0 |.

9

0 I"

Section 15

Page B

6.1.1

In-Plane

Load

Cases

B 6

September

1961

40

(Cont'd_

/ /

_ II

z II

.

\ .i/

°_

II

I

/ / /

\

f

/

f

N

/

J

J f

"_

/

/

_1N

o_

/

/ /

/

/

/

/

/

/

/

/

N

/ /

/

! /

%

/

v"

/

\ f

\

I J

\

\ \

\

/

/

\ o_

\ \ \ \

Z

\

\

\

_.

N.

o

O_ I'

O

!

I"

m I

Section 15 Page B

6.1.1

In-Plane

Load

Cases

(Cont

B

September 41

'd

I

\

i [

o

0 ro

\

)

0

0 oJ ro 0

J

0

/

f

f

0 co cxl

f

J

\

0 0 rid

h

J

J

J F

7

(

J

LO oJ 0

J w

eJ

f 0

0 oJ oJ

h,

0

0 0

/ II

II

<_

..i r-f f f J

i

f

r

0

8

J J

f-

J J

(

0

s

\

J 0

0

/ \

0

\,

\ 0

0 00 0

_J

0 10

f

J

I j f

)

f'
J

0

0

f 0

0 oJ

ro

oJ

q

q

¢J

GO

_

oo

5.. 0 0 •

o

o

9

9

0 0

0 0

O.

,I"

I"

I

6 1961

Section

B 6

15 September Page 42 In-Plane

Load

Cases

v"

(Cont'dJ

0

0 0

0

0 o 1,0 G

o oJ 0

o o,I 0

o @

o 0,I Q

o o II,

o

CO 0

o i

¢I) ,0

o 0

o oJ i

1961

0

o o 0

o 0

o 0

o ,,¢ 0

o

Section 15 Page B

6.1.1

In-Plane

Load

Cases

(Cont'd)

B

September 43

6 1961

Section 15 Page B6.1.1

In-Plane

Load

Cases

B

6

September

1961

44

(Cont'd)

0

0 I¢) 0 N (.)I

0

0

_b 0

8 N

0

0

_b _b 0 0

0

0

0

0

0

_b cJ Z

N.

0 I

I 0

0

I tD

0

o. I

0 I

!

I

Section 15

In-Plane

Load

Cases

6

September

Page

B6.1.1

B

1961

45

(Cont'd)

_r

X

X

X

0

-I _ UI

o 0

×

.

n

.

rO

.

0 c_

_:

0

i

....

_.-

0 0 rO

GO c_

._ _._;

...

o

0

j _leJ J J

oJ

/

i

0

0 ed c_

I f

/

0

/

8 C_l

/

0

0 GO

CO X

0

0

\ c_

_,

o 0 IZ) 0 °

o 0

0

OJ

7

Q

N

0

eu I"

_

_. I

m

I"

o

.-F

Section 15 Page B6.1.1

In-Plane

Load

Cases

(Cont

B

September 46

'd) f

I f J

J_

_k.,, H

x

,.,<_

\ 0

\ II

II

0

\

)

N

/ J f

J J

J

o. f

L/

(n 0

/

r

/

J

t

/

0

0

o

./ f

X

J f

\

dl

-

/

0

0

X /

\\

v

,%

\ 0

0

\ )

e 0

I

°

J jjF

j_

f

_

_J

0

o

J

f

0

J

J

ql) J

/ (D O v

(4) O

_

o

q

OJ o

q

o

e4

9.

_ I

_

o I'

_ I"

_O

9

_

6 1961

Section 15 Page

B 6.1.1

In-Plane

Load

Cases

B

6

September

1961

47

(Cont'd)

0

0

aD 0 I" I'

I"

!

I"

Section 15

6.1.1

In-Plane

II

Load

Cases

6

September

Page B

B

1961

48 v

(Cont'd_

II

I)

/i

,,,

/

o

I_I_

I

_.._ _

/

"'-

I"

-

o

....

o

o

o

o,

o

I"

o

l'

o

7

Section 15

September

Page B

6.1.1

In-Plane

Load

Cases

B

49

(Cont'd)

k

_/j

,\

0

f./

0

<,

ro 0

0

-

°

(

ro

X

(._1

0 ro 0

I_.

0

&

J1____ jJ /

J

N

:EO ,_, E

/"

J

_<<

o

/ v

Z

0

/ s

/ k

o

J

N ---

f

J

•_

_s

--

0

.J

0 _0 --

0

0 _0

* I _-'

0

0

(/

0

\

0 oJ

\

° 0

(2I _ . 0

/I

_o

J j-

Z

)

j f

J

/

J J J

o-

/

J '

r

XZ., o4

--

f

O

I

I"

1901

Section 15 Page

B 6.1.1

In-Plane

Load

Cases

B

6

September

1961

50

(Cont'd) f I J

\

0

s _

0

I J

0

0 oJ iv')

F"

¢

0

0 0 Iv)

oJ 0

0

rq 0

0

o

0 0 ¢J 0

0 co

0

0

0

\

)

8 0

0 co

/ 0

0 tO J 0

I

0

@

0

_0

o

_•

to

o

o.

,N

_.

O

o--,.

N

Iv)

o.

9

_; ,.

LO 0 I"

Section 15 Page B

6.1.1

In-Plane

Load

o. 13. 13. :E (_ Z "4 _ "I " I' "

Cases

B

51

(Cont'd)

/

\

/

'\

/I

t

I L/

J

.

"9"

ro o 0 OJ

_°x

o" 0

13..

(_

"

¢d_

_

/

6

September

N

\

! f

/ f_

/

/

/ /

/

/

/

/

f

/

/

/ /

b

/

!

I

/ / J

I

I j"

\

/

\

m

f

i

\

t

\

\

I I

\ \

®

\

\

\

\

0

0

\

\

\

0

\

_?.

F)

N

0

I'

(_1 I"

i_i"

,_. I

_'_ i'

1961

Section 15 Page B

6.1.1

In-Plane

Load

Cases

B

September 52

(Cont'd_

\

f

f

I

/ i /

(\

N o-I_

n

II

0

0 0 ro

II

\ cO 04 a.

/ 0 U

_0 04

J

/"

x J f

Dl!

o4 04

/

/

o.

/

/

0

/

0 0 04

/

0

J

\

0

f

0

0 /

B--

/

0

/

0 0

0 04

\ 'N

\ \

0 0

0

/ "X J

0

0

f

fJ

J

0

0

lJ

f

/

/

I

04

J

/ _D

Q 0

•

I¢) 0

0 .

04 Q

04

?

{_.

o

(.0 0 I"

_0 0

0

Q

I'

oJ

0

o

0 I

i

I"

I

&

6 1961

Section 15

B

September

Page B

6.1.1

In-Plane

• o

Load

_.

Cases

53

(Cont'd)

_

_

•

o

o

I'

o

o

I"

I"

o I"

o I"

\

.

/

I_

N

o,I II

II

II

\ I _I

gl

_c •

0 C_l

I I I

!

....

\

I

\,

X

x

B

\

| I I

_EOz

j--I ......

__

/

/

n-

/ jj/ t

,q f,,¢ __

m_

f"

____

8

__

\

8

\ \ I

,..,, I

I"

9

o

_'

_

I

I"

(_. 0

,,i

I'

!

6 1961

Section 15 Page B

6.1.1

In-Plane

Load

Cases

B

6

September

1961

54

(Cont'd)

I

J

ro 0

0 N rO 0

0

\ 0

O N

J /

N

J

-)

j" O-Wa"

II

<::1

J

J II

U

(_

n.

/

D.

I

/

_

- ,r_

_v f f 0

/

0 0 N

%

\ \, 0

0

/ 0

0

\ 0

f

0

0

0 0

\

\

0

0

/

0

/ J f

"_

J


0

I

¢,,

/

j

o i 0

._

/ v

! LO

o

J

oa Q

0

o I

o,I 0

re) 0

_I" 0

I"

I'

a'

lo 0 I'

Section 15 Page

_r-

B

6.1.1

In-Plane

Load

Cases

B

6

September

1961

55

(Cont'd)

0

0

o

0 It') Q

0

Q

0 N o

0 0

0

0

0 N 0

0 0 o

0

o

0 o

0

o

0

0

0

0 I

I

0 -p It

0

0

0

Section 15

In-Plane

Load

Cases

(Cont

I"

I"

I"

I'

I"

I"

I"

I'

r

"r

¢

\1

mo

\

\

Nffl

56

\

]

-e.

\

I °

0 0

1961

'd)

Orn ZZ,,--

\

6

September

Page B6.1.1

B

_'_

f

%

_o

:J, /

_L

\

7,,

mo #

/

0

J

I

O0

m

÷

"

"" X

Inn Ii

oJ 0

!/'

m I

\

/ !

,,

Z

:;

'

/

_

0

0

_ O

j

(M

_o

r'x

n

II

_

O.

_;

_

Z

II

/

/

0

0 0

'v

..

O

X

0

I

j

II

Z

OJ

\

j/ X

/

/

0

8

/

/

0

i/ /

!

_x21-

.) :t

t

._-

!

_

I

0

I

J/

I

o

• 0

0 0

N

i!

0

0

[

_0

, ....

\/

/

j

!

/ J

, I

! /

.J f

i

I ?-,

!

i.-

!

[

Or

II

o 0 _'M

tO "-r

0 _

I_ 0

-.

_.

,0.

'_.

_o

I

t

I

I

r

0 t"

0 _ _

Lo _ I

0 oa r

0

Section July

B 9,

Page

6

1964

56.

1

f

B

6.1. I

In-plane

Deflection normal F

forces

Load

curves are

Cases

for

displayed

(Cont'd)

the three on

basic

the foI!owing

(/_) that is to be used with the curves cross-sections is tabulated below.

Cross-Section

for

Shear Area

of

AQ=

th

to shear

A

shape

deflection

and

factor

of vario,_s

Shape

Factor,

Web

i_=

Area

A Q -- b h

Entire

due

pages.

shear

h

_i_

cases

Area

Entire

--f-

load

1.00

_ = 1.20

for

b

> 0.50h

l_ = 1.00

for

b

<

0

50

h

Area =2.00

AQ=

27rr

t 13q

Entire

Area =[1+

AQ = (w) z -

"1ab3

[

t t-O--tip 2 .I

( 2a)(w-2t) N.A.

3-(bZ-- aZ) a t__ll ]

,---:--e, p

= radius

of

If the flanges nonunifo r m

gyration

with respect neutral axis

to

the

they by

may an

section have

Fig.

B6.1.1-1

are thickne

be

of s s,

replaced

"equivalent" the

and

area

the

actual

whose same as

flanges width '.hos_

section.

of

Section

B

6.

1.

I

In-Plane

Load

Cases

B6

August 15, 1972 Page 56.2

{Co,[',l)

\

N\ \

\ \ \

_

i

"/

_m

m<

(_ I

-! __/

\

/ , ,z

\

/

/

/

/ /

/ J

/

I f

f

°

J

I

!

_.J o

o

_

o

o

o

_

J

• t

0 i

i

I

i

i

o

h r

Section B

6.1,

!

In-Plane

Load

Cases

(Contld)

B

July

9,

Page

56.

6

1964 3

r

f

i

oo

J

I

r

I J f o

f

/

/r_

/

\

\

<

0

,,1 -%

\ ,?.

) /

/

o"

J

J

oo

J z

f

J f

J

J

I

f

fi

\

I i"

i"

/

i_

i'

o

_oction B

6.

I. I

In-Plane

Lt,ad

Cases

July

(Cont'd]

Pa_e

Z

o

o

o

o

_

o

o

o

o

B 9, 56.

6

1064 4

Section March I, Page 57

B 6.1.2 Sign

Out-of-plane

Load

B 6 1965

Cases

Convention

The following sign convention directions for out-of-plane loads.

is

given

to define

Moments which produce tension on the inner fibers Torque "I"' and lateral shear "V" are positive as shown B 6.1.2-1.

+T

Fig.

+V

B 6.1.2-1

the

positive

are positive. in Fig.

Section

B6

March Page

B6.1.Z

Out-of-PlaneLoad

Cases

1,

1965

58

(Cont'd)

Index .

PA

For

0 all

cross

sections

0 For

all

3"

cro_s

r MA

V

sections

..

Fo .cross

Fig.

sections

B 6.

1.Z-g

v

_

Section March B

6. 1.2

Out-of-Plane

Load

Cases

(Conttd)

Page

"

B6 1, 58.1

1965

B

6.

I.Z

Out-of-Plane

I_,_,iidCas,.s

(Contld) l'uK{_ 5x. 2

B6.l.g Out-of-Plane Load

Cases

Section

B6

(Cont'd) March Page

i,

i965

58.3

o

B

6.

1.2

Out-of-Plane

Load

Cases

(Contld)

M

r_'h

,

o

i

_t+';5

m

-

Z-

----

1,

-7. /

....................

-_

o

X

\ \, H

B

6.1.2

Out-oi-Plane

Load

Case8

(Cont°d)

Secclon March Page

B6 I+ 58.5

1965

o

o

.

_v

°

°

.

Section

B 6

15 September Page 59 P

B 6.2.0

Analysis

of Frame-Reinforced

Cylindrical

1961

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 second-order importance. account for presented.

Procedure for modifying the important certain non-uniform properties of the

parameter structureare

to

Notation 2.25

A Lr

2

E

Young's

modulus

Ef

Young's

modulus

of unloaded

Eo

Young's

modulus

of

loaded

Esk

Young's

modulus

of

skin

base

natural

logarithms

G

of

axial

force

shear

modulus

moment

of

-

in

ib

loaded -

inertia

/in 2 frames frame ~

frame

-

of

a typical

Io

moment

of

i

I/_o

-

ib/in 2

ib

ib/in 2

of an unloaded ~ in 4

inertia

~

ib/in 2

Ib/in 2

moment of inertia the loaded frame

of

-

the

unloaded

frame

frame,

loaded

frame

l_Lr_

2

-

distance

~

in 4 "_"

from

in4

in 3

n2

Kn 3 distance

from

loaded

frame

Lc to undistorted

shell

section

-

in

Section 15

6.2.0

Analysis

Notation

Lc

of

Frame-Reinforced

Cylindrical

characteristic

length

(see

Glossary)

=

characteristic

length

J_o

frame

~

in

M

bending

in

loaded

Mo

externally

applied

concentrated

moment

Po

externally

applied

radial

-

P

axial

load

per

q

shear

flow

in

r

radius

of

skin

S

transverse

shear

force

s

transverse

shear

per

To

externally

applied

t

skin

thickness

t'

effective

t e

weighted

spacing

moment

panel

ib/in

line

~

in

(Cont'd)

radial

x

axial

co-ordinate

7

"beef

up"

N

to

of

of

parameter

nearby

heavy

in.lb

~

ib

~

Ib/in

load

~

Ib

for

the

bending

the

loaded

shell

-

in.

-

in.

shell

shell

Io/2i frame

-

Lc

in.

in.

loads

material frame, in.

~

of

axial

~

shell

of

~

in ~

ib/in

frame

perimeter

displacement

displacement

i/4

in

all

the

I

%E/_-_-V G--I--

~

shell

thickness

of

displacement

for

in

t'r2 T

Ib

~

loaded

inch

panel

around

w

in

r -_-

[

in-lb

shell

tangential

distributed

tangential

the

--[--r A/_

=

~

load

~

adjacent

v

frame

shell

average

axial

Glossary)

in

skin

u

(see

inch

stiffeners)

y

1961

60

(Cont'd)

Lr

7_

Shells

6

September

Page B

B

-

(skin assumed

in

and uniformly

in

Section 15 Page /I

B

6.2.0

Analysis

of

Notations

polar

In a

the

the

a

the

co-ordinate

a

frame.

finite

attack

(Cont'd)

radians

frame

and

shell

along

length the

the

(i)

solution are

free

no

axial

(3)

The

(4)

The

away

from

in

the

frame

has

or for

shearing

of

shell

consists the

moments

of

inertia.

bending

stiffness.

the

to

plane

nonFor

the

skin,

to

but

the

sides.

skin is

has

of

attachment ignored

and

of

It It

moment

different

have

the

for

frames with

longerons

properties

or

both

twist.

possibly

and

and

one

constant.

longerons,

skin

on

Its is

of

frame,

frame

either

flexibility.

and

frame neutral axis unloaded frames.

All

some Procedures

and

infinity

bending

The

at

frame.

on

bending

its

loaded

perturbations

loaded

away

extends

of

to

of

propagated

sections.

to

eccentricity

the and

similar

the

flexibilities.

the

be

a

sides

made:

circumferential

respect to the loaded

can

loaded

following

distance

of

both

negligible

the

in-plane

out

frame

concerned,

obtain

on

practice,

applied

shell

mainly to

discontinuities

are

are

is

used

worst,

for

infinite

warp

effects

with both

inches

The

to

inertia

at

loads

loaded

is

In

are,

an

sides.

any

shell.

account

is

infinity

of

assumptions

reacted

The

to

theory"

discussed

Concentrated

both

the

section

6.2.0-1)

effects

to

following

are

(2)

"Lc"

this

B

stretching the

beam

which

(Fig.

shell

distance

properties used,

with

model

Clearly

"elementary

modifying

uniform model

of

uniform

characteristic for

~

of

structural

loaded

from

displacement

method

for

only

Shells

1961

61

Assumptions

simplified

solution

Cylindrical

6

(Cont'd)

rotational

Basic

Frame-Reinforced

B

September

shell

no are

uniform.

(5)

The

longerons

giving

an

effective

(6)

The

shell

out"

in

are

"smeared

equivalent skin),

frames, the

equivalent circumferential

for

but

direction moment

out"

constant

of bending

axial

the

circumference t',

(including

loads.

not of

over thickness,

the the

inertia loads.

loaded shell per

frame, axis,

inch,

are

giving "i",

for

"smeared an

Section B 6 15 September 1961 Page 62 B6.2.0

Analysis

Basic

The bear

Frame-Reinforced

Assumptions

slight

discussed

structural

is

in

the

Glossary

of

system at

x

=

provided

distance brating that

required stress

the

frames

of

to

I/e

for system

Shell

In

(e

to rigid

with

is

lowest

the

basic

assumptions

vehicleshells, of

of

skin

section the

certain

however

parameters

of

decay

to

in

bending.

I/e

Flexible

are

two

required

as

characteristic for

self-equilibrating

natural

panels

envelope

there

distance

order

~ base

the the

this

Lc

the

that

are

-

follows:

envelope decay

by

space

pages.

length

o,

Cont'd_

Terminology

as

to

Shells

described

practical

by, modification

following

defined

exponential

to

compensated

Characteristic lengths,

model

resemblance

difference

Cylindrical

(Cont'd)

simplified

only

the

of

logarithms)

are the

rigid lowest

of

its

value

in

of

its

shear.

order at

the

stress

x

=

o,

provided

to_ v

Loaded B

Frame

6.2.0-1

/

u / f q W

B

6.2.0-2

placements at

x=o)

the

Frames

_-.Externally

Fig.

is

self-equili-

H Fig.

value

Lr

in

Load

per

the

shell

inch

and

(Loaded

disframe

Section 15

6.2.0

Analysis

of

Evaluation Case

In listed

of

be

the

previous

area,

and

and

Cylindrical

Lr,

Lo,

and

Shells

1961

63 (Cont'd)

7

shell

where

the

axial

Parameters

uniform

cases in

stringer the

of

Frame-Reinforced

6

September

Page B

B

shell

happens

pages,

and

shell-frame

to

in

moment

circumferential

satisfy

all

particular, of

inertia

directions,

the

if

the

assumptions

the

are

skin

thickness,

uniform

following

in

both

formulas

may

used:

1/4 Lc=

Lr

__E__r V 2

=

E t' Gt

.......................

(1)

.......................

(2)

I o

..................................

7

(3)

2 iL c

Young's equal. in the

modulus

for

Coefficients tables. These

mations

when

use

modified

the

substituted

of

non-uniform

(a)

In

the

over

-

that

r

=

the of

formulas

I

V

2

Eo

and

The

be a

stiffness

averaged characteristic

over

frames

is

assumed

thru in

21. the

In

(Lc, Lr, and defor-

non-uniform

following

shells,

equations:

Esk G

properties,

shell

to

]

i,

t,

a moderate are

and

t',

degree,

vary the

appropriate:

. .....................

I/4

(4)

.........................

t' t

(5)

Io ................................

2Ef

neighborhood

all

of these parameters the required loads

definitions

r2

=

the

14

shell the

Esk te Ef i

_-

Lr

Eqs. indicated

surface

following

and

by use yield

shell

case the

stiffeners

into

parameters

Case

Lc

skin,

are obtained coefficients

(6)

i Lc

factors, of

the

a

length

length

Gt,

loaded of from

Esk,

te,

frame. shell the

and

The

extending loaded

Efi,

factors

frame

must Gt

be and

approximately in

both

averaged Eski

one-half

directions.

in

shall of

7)

Section B 6 15 September 1961 Page 64 B 6.2.0

Analysis

Case

of

(b)

When

of

Frame,Reinforced

non-uniform

shell

unloaded

frames

unequally

spaced,

computing

Efi:

Efi

=

(Efi)fw

d

Shells

(Cont'd)

(Cont'd)

_ave

the

+

Cylindrical

(Efi)af

-

i Lc fwd

E

(Efi)fwd

=

I Lc----aft

Z

(Efi)aft

unequal

moment

following

t

of

weighting

inertia

factor

or is

.........................

(WEflf)

(WEflf)

are

used

for

(7)

...................

(8)

...................

(9)

Where X

W

=

1

=

o

for

is

measured

(x

The frames

heavy,

to

rigid

be

applied,

the

applying

Eqs.

is

near

end,

the

the

for

a

the

effect

the

more

The

Since in

end

shell

of

to

free

of

in were

of

or

are

the

the

of

correction an

Lc

used

be

(7),

Eqs.

be the

in

are

ignored loaded

in frame

beyond

the

(9),

frame

and

extended

loaded

the of (8),

exaggerates

when

compared

next

section.

Efi and

as

frame.

sub-section

estimate (7),

correction

and

the

this

such

(8),

average

given

particular

the

continued

loaded

in

initial in

if

Eqs.

than

less

corrections must

of

and

single,

symmetry,

those

sides

heavier

I/4,

a

all

discontinuities

must

the

indicated are

of

over gives

frame

of

particular,

of

both

extended

loaded

If

about on

calculation

neighboring

shell

summations

Lc

be

of

case

a plane

In

frame)

to

discontinuity

(9).

which

(Efi)

the the

other

other

symmetric than

to

or

loaded

method

For

for

shell, the

method on

(9)

closest

calculation

calculate

and

applicable.

and

the

of

The

end,

is

(8)

aft

away. or

frame

frames

accurate

L c depends

order

frames

a

greater

method

(8)

and

frame.

discussed

(7),

length

Eqs.

frame,

heavy

< Lc

Lc

farther

fictitiously,

though

>

x

forward

to

those

bulkheads, to

x

loaded

neighboring

factors

for

in

the

importance

importance

as

Lc

summations

except

greater

-

is (9).

required

with

Section

B 6

15 September Page 65 B6.2.0

Analysis Corrections

to

The

form

general 7"

where

of Frame-Reinforced

7

fb, and etc.

=

7

7

of

by

factors

Modification

"Beef-Up"

parameter

the modified

"beef-up"

.fa.fb.fc.

is computed fc are

, the

for

Cylindrical

, etc.,

the

different

(Cont'd)

parameter,

7 *,

........................

methods

accounting

Shells

of for

the

preceding

effects

value

is: (i0)

section,

of nearby

1961

and

heavy

fa,

frames

of Lr/Lc

The value of Lr/L c 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, the following formula may be

7 * =

7

+

Lr

h*

Lc

/

2

Lr i +

•

where (Lr/Lc)" is the (Lr/Lc)* is the value graphs

are

2

(ii)

/

,,

value of the parameter for the shell, of the parameter closest to (Lr/Lc)",

available.

Modification The

2

for

modification

7 * =

7

finite for

1 +

frame

finite

2LcK 2

spacing

frame

spacing

1 +

2 7 K2/

is as

follows:

where z distance

from

loaded 2

K2

-

Lc

/

2

frame

to adjacent

frames

2

and for which

Section

B 6

15 September Page 66 B 6.2.0

Analysis

of Frame-Reinforced

Modification for discontinuities.

nearby

heavy

The corrections to "7 account for discontinuities form of the correction for

7 *

:

7

frames

Cylindrical and

for

Shells

other

(Cont'd_

similar

"

in a previous section are in circumferential bending these effects is:

nearby

not intended stiffness.

......................................

-f(2)

1961

to The

(13)

Fig. B 6.2.0-3 shows f(2) plotted for nearby heavy frames and for nearby rigid bulkheads. Fig. B 6.2.0-4 shows f(2) plotted for a finite length of shell terminated in various ways on one side 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 B 6.2.0-3 and B 6.2.0-4 are for Lr/L c = 0.4, but their variation with Lr/L c is negligible for conventional shell-frame structures and adequate in other applications for Lr/L c < 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 concentrated

of moment loads is

of inertia 4.0 in 4 that is subjected to supported in a uniform shell whose characteristic

length, Lc, is 200 inches and moment of inertia per unit length, i, is 0.i0 in. 3 A heavy frame having a moment of inertia 16.0 in 4 is 50 inches to one side of this frame. The loaded frame and shell loads are

required. The

parameters

7

=

7_

=

needed

4.0 2(.1)(200)

are:

= 0.i0

by

Eq.

(3)

16 = 2(.1)

_

50

Lc

Using

7_

and

0.40

200

= 0.25

200

_/L c

-'- 7"

in Fig.

= 0.75

B 6.2.0-3

(0.i0)

=

0.075

yields

by

f(2)

Eq.

= 0 75

(13)

Section B 6 15 September Page 67

F¸

B

6.2.0

Example

Analysis

of

Frame-Reinforced

by f_

Shells

(Cont'd)

i (Cont'd)

Use 7 = 0.075 instead presence of the heavy frame. Example

Cylindrical

1961

of 0.i0 in the curves frame on the stresses

to account for the in and near the loaded

2

A shell whose characteristic length, Lc, is 250 inches is supported a large number of identical frames whose moments of inertia are 2.0

in4, 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 opposite 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

2

i =

-

-

Io

Io 2iLc

7 =

-

f

Lc

.0833

24

2 = 2(.0833)(250)

48 250

= 0.048

by

Eq.

(3)

- 0. 192

For the tangential loads there is a plane of syn_netry midway between the loaded frames, while for the radial loads a plane of antisy_netry exists at the same place. From Fig. B 6.2.0-4 it is seen that for the radial load stress system, f(2) = 0.32, while for the tangential loading f(2) = 1.75. Hence, the values of 7 * to be used in

the

graphs

Eccentricity

are

0.015

between

and

skin

0.084,

line

and

respectively. neutral

axis

of

the

loaded

frame.

In the three types of perturbation just discussed, it is possible to account for the effects by modifying 7 only, since the "elementary 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 "elementary-beam-theory" solution is also affected. This particular aspect is discussed in Appendix E of reference i.

-

Section B 6 15 September 1961 Page 68 B6.2.0

Analysis

I

of

Frame-Reinforced

Cylindrical

Shells

(Cont'd_

I

7£=0.1

7s= o.5 7_= 1.o 7_ =

2.0

Two Rigid BuIkheads Sy_'_etrically Placed about the Loaded Frame

I --

0 0

0.2

Lc Fig.

B 6.2.0-3

A single frame on one side of loaded frame or two rigid bulkheads symmetrically placed about the loaded frame curves of f(2) and f(3). Lr/L c = 0.4.

Section

B 6

15 September Page 69 B6.2.0

4.0

Analysis

I

I

of Frame-Reinforced

Cylindrical

I

1

i

I

I

,,..----

_

I

Plane

End

at x

I

I

(Cont'd)

I

I

---__

Loaded

x

Free

Shells

1961

Frame

=

of

Symmetry

At

Plane x =

Built-ln

At

x

of

Anti-Symmetry

at

=

_/L c Fig.

B 6.2.0-4

Finite vs

length

_/L c for

of

various

shell

on

boundary

one

side

of

conditions

loaded at x

frame =£

f(2)

, Lr/L c - 0.4.

Section 15 Page B6.2.1

Calculations

Eqs. of

a

(14)

thru

(21)

concentrated

computed

by

is and

by

load

using

axial

in

load

at

displacements

The

given

in

moment

on

previous

this

a

section,

in

loaded

parts

the

frame

of

the

by

1961

70

which

shell-supported The

section.

points

the

following

tabulated

are

6

Tables

tabulatedcoefficients.

a

all

of

of

or

the

indicated

Use

B

September

These and

be

computed.

to

overall

method

the

solution

be

computing

the

shear

internal

are

effects

may

of

enable

shell

the

frame

flow

loads

omitted

and

in

the

coefficients:

(i)

The

"elementary-beam-theory"

which

(2)

The

(3)

is

calculated

part

from

beam

"elementary-beam-theory"

intensity

in

from

theory.

beam

The

rigid

of

part

longerons

which

translations

of

should

and

skin

shear

flow

theory.

rotation

the

axial

load

be

calculated

of

the

loaded

frame.

As

a

consequence

intensity

in

directly The

to

shear

assumed

the the

flow to

shell

that

axial

loads

possible

of

an

unsymmetric

about

derive

a

indicated

shell

a

the

correction

of

on

be

obtained

of

"p"

adjacent

to

At

the

present

time

internal

forces

bending loaded l_/i

"m",

simple and

moment frame, (see

in

load

in

for and

other an

then

Appendix

the it frames

matter

the

is

in

loaded

frame.

this

tables

frame.

the

load

added

calculation.

the

loaded

frame,

for are

be

shear It

effect,

are

In

a

flows is

but

and

not the

exact

applicable.

on

one

frame

concentrated The axial

loads "q"

on

are

may

be

obtained

by

loads into which the load and shear flow in

several

tabulated

frames in

by

Ref.

a

similar

3 NASA

TN

D402

x/L c.

Frames

a

the

axial

can

frame

a distributed

since

function

inch,

a

the

2

the effects of the load can be resolved. can

however,

by

simple

reference

to

and

theory"

given

loaded

ahout

load

effect

the

flow

tables,

bending

respect

symmetric

in

shear the

distributions

is

not

(2), from

"engineers

load with

superposition, as

axial

symmetrical

superimposing distributed

and

results

Distributed

the

(i)

calculated

be

to

The

items as

and

are

solutions

of

shell,

to

frame

is

possible

not

adjacent tabulate

internal

adjacent

of

forces

by

reference

by

to the

frame,

obtained D

loaded

due

the

of

tables frame.

frame-bending

moment

as

a

function

of

to

a

force

multiplying I).

use loaded

"m"

applied at

the

to

compute

It

is,

per

x/Lc. at frame

The the station

Section

B 6

15 September Page 71 B

6.2.1

Calculations

Effect It charts,

of

by

local

Use

of

Tables

reinforcement

is not practical the many possible

1961

(Cont'd)

of

the

loaded

to attempt to cover, reinforcing patterns

frame

by a set of tables or that can be used to

locally strengthen frames in the region of applied concentrated loads. solution is presented in Appendix A of reference 3, 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 rein-

A

forced

to produce

the

actual

inertia

variation.

Tables

F

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.

along

Coefficients for displacements with coefficients for "q" and

q

= Cqp

Po _+r

To r

Cqt

M

= Cmp

r

Por

+

Cmt

........................

(16)

M ___2_o r

........................

(17)

M _._.9__ r

........................

(18)

F

-- Cfp

Po +

Ctt

To + Cfm

V

= Cvp

Po

W

: Cw p po

Csm

Cvt

+

To

To Cwt

+ C8 El °

•

Mo

Cmm

To t

Cpm

_r3 El o

_r3 + El o

_r2 El °

+ Cvm

Cwm

+

(19)

(20)

7 r2 El o

Mo m

r

2 7 r El o

Mo

Mo

Ce

7

(15)

+

To +

_r3 El o

o

r

Cst

+

..................... (14)

__2__o

Po +

-_ El o

Ref.

CP t

S = Csp

8 =, C e p Po _

i

Tor

Mo r--_--

+ Cqm

+ p m Cpp

v, w, and 7 are tabulated in "p" as a function of x/Lc.

___ El °

.. .....

(21)

3

Section

B 6

15 September Page 72 B 6.2.1 Sign

frame

Calculations

by

use

of Tables

(Cont'd_

Convention

Loads, moments, and displacements as shown in Fig. B 6.2.1-1.

M

are

positive

in the

loaded

o To

Neutral

eo

F M S

M

axis

1961

Section B 6 15 September 1961 Page 73 f-

B 6.2.1

Calculations

by Use of Tables (Cont'd_

Frame Loads

Index of Tables

Coefficient

Lr/Lc=.200

Lr/Lc=.400

Lr/Lc=l. O00

Cmp

B 6.2.1-1

B 6.2.1-5

B 6.2.1-9

Cmt

B 6.2.1-13

B 6.2.1-17

B 6.2.1-21

Cram

B 6.2.1-25

B 6.2.1-29

B 6.2.1-33

Csp

B 6.2.1-2

B 6.2.1-6

B 6.2.1-10

Cst

B 6.2.1-14

B 6.2.1-18

B 6.2.1-22

Csm

B 6.2.1-26

B 6.2.1-30

B 6.2.1-34

Cfp

B 6.2.1-3

B 6.2.1-7

B 6.2.1-11

Cft

B 6.2.1-15

B 6.2.1-19

B 6.2.1-23

Cfm

B 6.2.1-27

B 6_2.1-31

B 6.2.1-35

Shear

Cqp

B 6.2.1-4

B 6.2.1-8

B 6.2.1-12

Flow, q At Ring

Cqt

B

6.2.1-16

B 6.2.1-20

B 6.2.1-24

C

B 6.2.1-28

B 6.2.1-32

B 6.2.1-36

Bending Moment, M

Shear, S

Axial Load, F

qm

•

i r-¸

Section 15 Page

0

_ _ _ _ 0 _ _00000000000000000000000000

_

_

_

_

_

_

_

00

B

6

September

1961

74

_

_

0

_

|IIIIIIIIIIIIIII 0 II

x o! _00000000000000000000000000 ,,.,,.°.t°,...e°...,_,..°,oQ IIIIIIIIIIIIIIII

_ _ _ _ _ _ _ _ _ _0 _ _ __0_____0000__ _000000000000000000000000000

0 c'%

_ _

_ _

_ _

0_ _

_ _

_

_ _

_ _

O_ _0_

_ _0

_ _

__ _

_

0 _

IIIIIIIIIIIIIIIII

,-4 !

0 ! c'4

C%1

____

___0_0__

0_00__ _0000000000 °o.°._....._.°°.°.....,.°°,.

___0000000_ 00000000000000000 Iiiiiiiiiiiiiiiii

_ __ _ 0 _ _ _ _ _ _ 0 _ _ _000___000000000000000 0000000000000000000000000000 ....,..o,..,°.....,.......,, iiiiiiilliiiiiiiii

,-4

II

___ _0______00__ _00000000000000000000000000 0000000000000000000000000000 ..°_°......o°..°o..°.o°,,.., IIIIIIIIIIIIIIIIIII

u% CD

__0____0____ _0_____1__0000__ _00000000000000000000000000 0000000000000000000000000000 ,,._,,,..,.......°,.°.,..... IIIIIIIIIIIIIIIIIII

____0_

on 0

0 °i

_______0__ _0______0000000_ _00000000000000000000000000 0000000000000000000000000000 IIIIIIIIIIIIIII1111

0

_ _

_

_ 0

_ _

__ _

_

_

_0_

_

_ _

Section

B 6

15 September Page 75

oo

o__o_z__o::o_:o_ o_o _0 _

0 0

__ o

_

_

_

ooo0oo

_

_

_

_

_

o_

_ oo

_ oO

_

_

o

o

_ o

IIIIIIIIIIIIII

nI o

o_

_

_

o_

_

_o_

_

_

oooo_ooog_

_

_

_

o

IIIIIIIIIIIII

f

o

O4 !

O_

_

O_ _

__ _ _oo0ooo0oooooooo00oooo

_

_

_

_ o

o

_ 0

_

o_

_

o

_

_

_

_

_

_ _

_

_

_o _

_

oo

IIIIIIIIIII

0 0 Oq

0

_° I .I II _j

_0_

_ _

__0

,-1

_

000 0000

_o_

_

0000 O0000000000000

__o_

o

00_

000

0 O0

IIIIIIIII

00_00000 _ _000o

o

O00000ogg 000000

OO

o

oo

gg_

O0 O000

O0

0o00 O0

IIIIII

o.

_

0000000000000000000000 Illll

ID _.)

o_°_°_°_______o_ _oo

04 0

0_ _ _ IIII

o

0oo

oO

oo

oo

oo

OO

o

oo oo

o_ 0 0

ooo oo

oo oooo

O0

o

_

o_

ooo o oo

00oo 00o

1961

Section

B 6

15 September Page 76 _____0___0__ 0______0____ _ __ _ ____ o.°l°.°.,°°,°°..°.°.°°._,°°. Iiiiiiiiiiiiiiiiiiiiii

0 0

0__

__

O_

0__ O00_

___0

1961

_ _

o 0 II X

o

0

!

___0_0_0__0___ ___0__0_0_ __0_0_ _0___000000000000000000 .....°..,..°.° NN_IIIIIIIIIIIIIIII III

0 oq

0 0 o_

_0_ ___00000__ .......

.....,°

__0_0___0__0__ ____0__0__0_ ___0__00000000000_0_0 ___00000000000000000000 .o.,........_,°...,....o.... NNNIIIIIIIIIIIIII III

0 II

_ _ _0_ _____0_0__0_00 ___0_0000_0000000000_0 __0000000000000000000000 _°,..°°..°.°,..°°_°°°._°,.. _NNIIIIII iii

Lrh 0

0__00_____0__0 ____0_0__000__ ___0__0_000_0000_0 __00000000000000000000000 ....°.°°,...o..°,°...°°°.°.. _111 III

0

I

_

_

_

_

_0_

__

II

I

0_

III

II

I

I

_

_

I

I

I

I

I

I

_

_

I

I

I

I

Q _0____0___0_ _ _ __00000000000000000000000 _NIII III

0

_ I

0_0 II

__ I

I

O0 I

I

Section

B 6

15 September Page 77

1961

0 o

.4

0 o

f--

o_

,-4

_0_o___oo_o__o _o_ ___

__o_ _ o

oooo

_ o

_ o

_ o

_ oo

_ o

o_ _ o

_ o

_ o

oe°le,lo.I°o.,°o.°°.oo,°°°

o

IIIIIIIIIIIII

II o

o__oo_o___o__ ___o______

f

•

°-°°°Io,°°°°,o°°

..... IIIIIIIIIIIII|

o 0_ °oei,°,eJ°o.oot°o,°,°...°° _

o I

_o 0 o

1.4 u'h o

c_ o

¢'4 0

0 _'_

Illlllllltllllll

°,°,

o

_ o

o

Section 15

B

6

September

Page

1961

78 ,,,,,......

0 0 c"q '-'_

'-I

000000000

0000000000000000 I

I

I

I

I

I

I

!

I

I

!

I

I

I

I

_________0_

O_

o.i ,.-4

_ 0_ O_ _ _0000000000000000000000000 -o.......e.°....°0.o..i°...i IIIIIIIIiiiIiiii

_

_

__

_

00_

_

_

_

0 II _0_0___0_____ ___0__0__0___

XO t_

_00000000000000000000000000 o.-*.oo.°°.........o.0..°.. IIIIIIIIIIiIiiii

_0_________ _0__0__0_____ __ O_ _ _ _00 O0000 O0 "'..-..--°........o......... IIIIIIIIIIIiiiiii

0

_ 0 _ _ 0 _ _ _____0__0__ _ _ -T _ 00 _ _ _000000000000000000000000000 _.--,--,.....°...,.,,,,...,, IIIIIIIIIiiiiiiii

0 I

.

_ O0

_ _ O000000

_

_'_

_

_

_

_

_

0

_

_0_

_

_

_

c_

_

_

_

_

_

_

_

_

_

_

_

0000000000000

_

_0000_ O0

O000000

_

_

_

_

_

_

0000000

_

_0

oOi 0 ,-4 II

___0_0_0__0____ __0___0__0__ _ _ _ 00 _ _ 0000000000000000000000000000 _.°_°°°._°.°°.°°..°.°..°.._. IIIIIIIIIIiiiiiiii

_

_0_0_0____0__0 _0_0_00_0___000__ _00_ 0000000000000000000000000000 ,,_.°.,°°°,,°.°.°,°,°°,,°,_° IIIIIIIIIIiiiiiii

t_ 0

___ ____ _00000000000000000000000000 0000000000000000000000000000 °,°_°°...°.°_°°_°°°°°.°,..°_ IIIIIIIIIIIIiiiiii

0

__0_0__0___0__ _0____ _00000000000000000000000000 0000000000000000000000000000

L) 0

IIiIIIIIIIIIiiiiii

0

_0000000000000000000

__0___

0_0__ _oJ__O00__

___000000_

_

_

_

_

_

0

_

_

_

Section 15

B

6

September

Page

1961

79

I0111111111111!

o I!

X o Lr_

0 m

I O4 _

m

m

_

_

_

_

_

o

o

o

o_

o

o

o

o

o

o

o

o

o

o

o

o

o o

o o

oo oo

_

_- o

o

o

o

o o

_ o

_ _

oo oo

IIIIIIIIII

< E_ 0 0_0

I

0 _o __o

o

ooo

oo

_o oo

ooo ooo

oo oo

_ o

IIIIIII

o __ oo_ 0 _ _ _

0

___ _oo _ oo

o o

_o __ooooooooo oooooo

0

___m_

___ ___ oo

ooo

oo

o _ _ ooo ooooo

_ ooo ooo

_o _o oo

IIIIII

L)

0__ 0_0_0___ 0_ _ _00

c_ 0

IIIII

o

_ _0

_ _0000000000000000000 O0 O0 O0000000

__ _ _

_ _

O000000000

_

__0 _

_00

o

Section 15

B

6

September

Page

1961

80

o 0 °°.°°_o°.°°.°.°°°°°°°°°°°°°° IIIIIIIIIIIIIIIIIIIIII

____ ___0_

____ __0__0__

_0_

0 0 • 0

,--_

.°°°..°°.°°o°.°.°°.°°°°

....

IIIIIIIIIIIIIIIIIIIII

II

X 0 Lr_ .°

....

°°._°°°°..°°

..........

IIIIIIIIIIIIIIIIIIII

_ 0 O_

________0_

_______00___ _ _ _ _ °°.°°°o°°°°°°°.°..°..°°.

_

L_

_

_

_J

o_

_

_

0000000000000000 ....

IIiiiiiiiiiiiiiiiii

I T--I

0 Oq

_J

____00000000000000000 • . ° ° N_IIIIIIIIIIIIIIIII Ii

_D

.

.

....

°

....

.

.

°

.

.

.

°

......

0 _

_

_

0

_

_

_

_0___ ___0000_00000000000000 • .....

0

0

_

_

0

_

_

,_

_

_

_I

_

_

_

_

_

_

°°..°

..............

____1'_____0_ _ _

_ _ °°

t_ _1

_ _

_ _

....

_ _

L_ _ °,

_ ¢J 0 cl _ 000000000000000000000 ....................

_

_

00000000000000

Iii

C_ 0

_0,_0_00_0_00000000000000 __0000000000000000000000 °°...°°.°.°0

....

.

.....

iii

________0_0_ _00________ _0___00_0_000000000000 __0000000000000000000000 °.°°.°°°.°°°.°°°.

Oq 0

iii

0

_

_

°.°

III

w_ 0

,_

_000000000000__

...........

°°°°°.

_

Section 15 Page 0 0

81

_00____0___ 0

c_

_ ,_lt°°o,°°,°t.o°llo°°°_.°0

000

000000

0000 IIIIIIIIIIII

0 0

O' II

0 u_

0 oq

00 I

B

September

0

0 0 .._'0

III

0

0

0

O0

6 1961

Section 15 Page

0 0

B

6 1961

September 82

_0_0_0__0__0___ __000000000000000000000000 .t,..o,..o..e...o........o.. iiiiiiiiiiiiiii

__0___ _0000000000000000000000000 _°°_l°°°.°._°..°.°..°.°.°°.°

0 0

___0__

IIIIIIIIIIIIIII

0 II X

_0__0_0__0_000__ 0 __0_____00__ _00000000000000000000000000 ...........0........._...... IIIIIIIIIIIIIIII

_ __L_O_ _______0__0_

0

______

_00000000000000000000000000 ..i......°°......o....°.°°.. IIIIIIIIIIIIIIII

!

0 _00000000000000000000000000 ..... ..°..°..°..°°._.......° IIIIIIIIIIIIIIII

,-1

0 0 0

o

_____0___0__ _000000000000000000000000000 °.......°.o..°...._.°..o_°.° iiiiiiiiiiiiiiii

_ _ _ _ 0 _ _ _ _ _ _ ____t_O__ _00___00000000000000 0000000000000000000000000000 .. .... .°°.°°.°.o..°...o.°o°. IIIIIIIIIIIIIIIII

0

"I

____0__0_0___ _ _ _ _ _ _ _ _00__00000000000000000 0000000000000000000000000000 °.°0.°...°..°°o°°.°.°....°._ IIIIIIIIIIIIIIIII

0

______0____

E

_00_00000000000000000000 0000000000000000000000000000

0

IIIIIIIIIIIIIIIIII

0 O.

_

_

_

0

_

_

_

0

_

_

0

_

_

_ _ 00__

_

_

L_

0

_

_

_

_

_

_

_

_

_

_

_

00

_

_

e_

_

_

_

Section 15

B

Page

oo__ •

__

___

___

_

_

ooooo

_

1961

83

__0

oo

6

September

_

_o00

_

__

_

o

_

o

IOIIIIlJiOIllll

o_

_

_

__

_0

_o

_o_o__

o '0 _

_

_oo

o

o_ oooo

oo

ooo

oo

ooo

o

ooo

lllifltillOlfl

0 I!

)4 o u% /

0 0%

o ,-4 I ,--4

0 04

< E_

°io o _

o

__o _oo

IIIIIIII

o'I 0 r

0

0_0_0__00000000000000 __00000000000000000000000

0

IIIIII f

o

_

_ __ ooooooooo

_

_oo

o o

o o

oo oooo

oo

Section

B 6

15 September Page 84

1961

0 o

o o 0,--_ II

o

0 c_

_Ot_O__O_ _____0000000000000 ....0°

___0___ .....

°..°.°°..°...°..+

IIIiiiiiiiiiiiiiiiii

.--I

0 ¢',1

! ,--I

__ _____000000000000000 °...°...°°°..°.+...+

ol

__0___0___ .....

°0°

Iiiiiiiiiiiiiiiiiii

o o o

_J

.--4

0___i__000____ _ 0 _ ......°

II

_

_

_1101111111111 ii

L_ 0

0,1 0

O__O__t___O__O ___000000000000000000000 .°°...°°.,..+,o..°°...,.°... III

o 0

t_

_

_

_1 ....

_

_

00000000000000000 ,..°°.....

.....

..

Section 15

B

September

Pa_e

0

m 0000

____ O0 O000 t°Io...,o.o,..I,e.....e.,o

_

_

0

O0

_

_

O00000000000000000 iIIiiIiII|i

o II

0 u_

0

.,-4 I ,--I

o c.4

oq

r-_ ,-1

o o 0 .o ____oooo___0oo II iiiiiiiiiiiiii o

.1 o

0

0

_

_

1961

85

_

_

6

0

Section 15 Page

0 0

_ _ O_

0 0

0 u',,

O000

_0

_ _ _

_ _ O0

_ _ _ 0000_ O000

_

_ _

_ _0 O00

0_ _ _

_ _ _ O000

O_

_ _

_ _ _

_ _ _ O0

___0_0_0____ 0___0___0__0_ ____000000__000 00000000000000000000000000 °.°..°°..°..°o.°..°.°°.°°° IIIIIIIIIIIiiii

____0____0_ __0___0___ 00___00000000000000000 00000000000000000000000000 ......°..°°...........°°.. IIIIIIIIIIiiiii

o °i

,-1 E-'

_ _ __

_

_ N__O_____ ___0__0___ 00___000000000000000000 00000000000000000000000000 0.o...°........° IIIIIIIIiiiiiii

0

0_ _

....

°.....

0 0 c_

II

0 ,-4

0000000000000000000000000000 00000000000000000000000000 .°°...°......°.......°..°. IIIIIIIiiiiiii

.1

0 _____ _ ___ O000000000000000000000000000 00000000000000000000000000 °....°...........o.°..°... IIIIIIIIIIiiii

u_ 0

0

O0

_ 00_______0_ ____000000000__000 00000000000000000000000000 0 0000000000000000000000000 • ...°...oo.. IIIIIIIIIiiiii

_0_0_0______ _ ___00000000000000000 0000000000000000000000000000 00000000000000000000000000 ..°......oo.......°..°.... IIIIIIIIIIIiii

oq

0

____ _00000__

_

....

.o........

6

86

__0_00___0_0__0_ _____0__0_ 0_____00000___00 00000000000000000000000000 i°oo°.o°.°l°°o°°°°°°o°°°°. IIIIIIIIIIiiiiii

C II

oq I

_ _ _ _ _ _ _00 __ _ 0 O00 O00 °.o.e...o°l°..o.0°...oo..° IIIIIIIIIIIIIiii

B

September

O0

O0

1961

Section 15

B

6

September

Page

1961

87

0 0

0 0 e-4

0

_0_ _ _ _ _ _ _0 __ _0_____00__ _000 00000000000000000000000 °.,,o°,.,.,..°..b,.,..,.,e.. IIIIII

_

_

_

_

_0

__0

_

IIIIII

II

0 Lr_

0

_

_

_

_

_

_

_

_

_

_0

_

_

_

_

_

_

0

_0

_

0

0

0

0

0

__0_____0000__ _000000000000000000000000000 IIIII

IIIIII

,-'4 I ,-'4 ___000_____00 0 C'.l

_O00000CDO0000000000000000000 ,4.,. ..... IIIII

..........°....... IIIIII

_q _q _q

___0_______

<

0

r04_O00_O__O00000000000000 0000000000000000000000000000 ...b....°..........°........ IIII

0 0-4

IIIIII

n

_0000000000000000 00000000000000000000 °....°..°... III

C_9

0 • III


O0 ....

(n

000 000 ..... IIIIII

Cj .......

_

_.

_

r_

-, q.

'_ _ .-_-...--....''"

0000000000000 IIIIII

_______0_0_ _0____]__000000_ _00000000000000000000000000 O0 O0 O0 0 0 ._..°°..°...°°°..°..°.°°..°° Ill

O4 0

00000 00000

..°....

0

0

0

0

0

0

0

0

0

c_

0

0

0

0

0

0

O0

iIiIII

0 0

_0

_0_0

_0

_0

_0

_0

_0,_

0

0

0

0

0

0

0

Section 15

B

6

September

Page

1961

88

0 0

0__0___0_0__0_ 0_0_0_0__0____ 0____0_____ _ _

0 0

0 _

_

_

_

0

0

0

0

0

0

0

0

O0

0

0

0

0

O0

O0

oo..°.l.°°Q.°o°°°.°o°.,°°e° IIIIIIIIIIIIII

II X

0_ _______0_ 0__ 0____00____0 ___00000000000000000000

0 u_

•

....

_0__0____ 0

...°°..

....

°.°°.,°0°..

IIIIIIIIIIIII

0 e_

0_______0__ 0__0_0__0____ 0_0___00_____0 __000000000000000000000

0

°.°.0°.,°°°.°0°°°°.°,°°.°°0 IIIIIIIIIIII

,--I !

0

O_ _____0___0_ 0__0_0____00_ 0___0000___ __0000000000000000000000

__ __00

°°°°.°°,0°°

.........

0

,,°°0°°

IIIIIIIIIII

0 0 ,-..1

0

0_0___0___0_0_ O_ _ _ O_ 0 0__00000000000_000000 __0000000000o000000000000

II L

_

0

_

_

_

_

_

_

_000

_

__ 0

°.°°.°0.°...,°.°

,--1

_

.....

.°°°°,

IIIIIIIII

u_

0_________

o.I

00_0000000000000000000000 _000000000000000000000000 •

....

0

°°

....

,°.°°

.......

°.°.

IIIIII

0___0____ 0________00 0__0000000000000000000000 _000000000000000000000000

o.i

__ 0

0°°°°.°°.°°°.°0°..°°,._°°°. IIIII

4-

0_0_0_ 0_0__00_0__ 0_000000000000000000000000 _ 0000000000000000000000000

0

• IIII

0_0

I

.°.°°

.....

_0___

__ _

,0°.°°.°°°°,,..,

0_00

Section 15

B

6 1961

September

Page

89

o II x 0

0

0

,5

0 _'_ 0 • ,-_ II

O_

_

_

_

0

,_

_

_

_

_

_

_

_

_

_

_

_

00000000000

_

_

_

_

_

0

_

_

_

_

_

IIIIIIIIIIIIIIIIII

0

_

_

_

_._

_

_

_

_

_

_

_

0

O0

_

0

0

IIIIIIIIIIIIIIIIII

0

_ _____®o_____ 0_0_ _

,.%

0_0_0_0

_0_

_0

_

_0_0

_

__

O0

00000

IIIIIIIIIIIIIIIIIII

_0_0_0

_0

_0

_

O0

0

0

0

0

0

Section

B 6

15 September Page 90 0 0 c_

0_ 00

_ O00000

_

_

_

_ O00000

_

_

_

000_ O0

_ O000000000

_

_

_

_

_

_

_

__

_

_

_

_

_0

llOllOllllilitli

0 0 .-4

00000000000000000000000000 ...,.°.o....°.....°..°...-. llllOltIIllilil

It

___0__0__0__0_ c 0____000000__000 00000000000000000000000000 o..°....0°°...°..°.°o...°. IIIIIIIIIIIIIii

0

o

I

_J

_ _ O_ _ _0_0_0_0__00__ 00____000000000_00000 00000000000000000000000000 .....°......°.°........°.. IIIIIIIIIIIiiii

_________0 _____0_ 00___000000000000000000 00000000000000000000000000 °...°..°°.°.i°.°.°..°.°... IIIIIIIIIIIIii

0 0 ,-1 II

C ,'-

U

lip C

0

_ _____0_0_00 _0_0___00____ 00__000000000000000000000 00000000000000000000000000 .....°......°..°..°°...°°. IIIIIIIIIIiiii

_0____0__0_0_0_ ____00000___0 0000000000000000000000000000 00000000000000000000000000 °.°.°.°°.°°°..°°°°....°... IIIIIIIIIiiii

_____0____ ____00000000__000 0000000000000000000000000000 00000000000000000000000000 °....°°...........°°...... IIIIIIIIIIIii

4J 0

___0__0____ ___00000000000000000 0000000000000000000000000000 00000000000000000000000000 ..°.°..°_.°°..°.°°....°... IIIIIIIIiiiii

_

____

_

_

1961

Section 15 Page 0 0

_0____0____ __0____ _0000000000000000000000000 .**..0_,,°.,.°.-°..°.,°00..,. IIIIIIi

_0__

IIIIII

_________0_ 0 0 0

,--I

_0_0_____00__ _0000000000000000000000000 ''',,*.*.*,...,.,0°°,°°,..... IIIIII

IIIIII

U X 0 u_

0

_0_________ _ 0__0__0_____ __ 0____ _000000000000000000000000000 °'°°*°'°°'°°,-°°°.°.°,°°°l°. IIIII

_0000__

IIIIII

cO !

0 c_

_0_0__0_0____0_

_000000000000000000000000000 °,,° .... IIiii

0q

0 0 .-.1"0

___0_0_

0000000000000000000000000000 • ,,.,...°. IIii

II

CD

0

r_

0q 0

_0_0_0____0__0 _0_0_00____000__ _00__0000000000000000000 0000_000000 *°°.,.0.,,.°..0 IIII

.0°..°..,,...,°,.,.. IIIIII

0

__0__

__

.........

....0..°, IIIIII

_DO000000000000000 ....

°..°.,..° IIIIII

_____0___0_0__ N_____NNO]_O00__ _000000000_0000000000000000 0000000000000000000000000000 ,*°..**,°°°°,°°°°°.°,°.._,.. III

_o_s__S____ooooo._ _ oooo oo ooo oooooooooooooooooooooooooooo ,,°°_,**°°,°.,.°°°°...**°.°. III

IIIiii

oo

oooooo IIIIII

B

September 91

6 1961

Section 15

B

6

September

Page

1961

92

I0 0

0 0 0 II

X !0 Lf_

0 0 __

_ _

_ _

_m

_ _ _ 000

_

_

_

_

_ _

0

_ _

_

_ _ O0

_ 0

0 _ _ O00

_ _

_

0

_

_

_

_ _

_ _

_

_ _0_

_ __ O00

_

_ _ O00

_

_ _ _ _ O00

0 _

_

_

_

_

O0

_ _

_ O_ __000

_ _

O_ _ _

O0

0

,,,.,°.,,,.°..°,..,,o,...., IIIIIIIIIIIII

0_________ 0____00___0__ 0 _ _ 0 _0 ___00000000000000000000 0,°°,°,°°°°°..°..°,o..,.,,. IIIIIIIIIII

0

I

0 _ O_ 0 _ _ _ _ O_O___O0_c_ __000000000000000000000 4.e°°°,.° iiiiiiiiii

0 oq

A

r_ ,-1

< [-4

_

_

_ _ __

.........

_

_

_

_ _ °.

....

_

_

°.,

0 0 0__0_____0__ • 0 ,--4

0___00____000000 __00000000_0000000000000 o°°.o.°o,.°°...°.,....,°°,. IIiiiiii

II

,-I

Lf_ 0

0_____0__0_0_ 0___ 0_0_00_00000000000000000 __00000000000000000000000 °,°,°..°..°...,°...,..,.o,, IIIIIII

0_______0__ 00_00_0______0 0__0__000000000000000000 _ 000 ......_o.°..°°.°.°o.°..°°°o iiiiii

0

Cxl 0

_

O00

0__0_____ 0_0_0______000 0_0_00000000000000000000 _ _ O0 O00000 ,,,°°°..°°.°°,°°.°°,o.°,°°° iiiii

__

O0000

____

O0000000000

O0

__ O00

O0000

.

O0000000

0

/

Section 15 Page

0

_0__0______0_ 0000000000000

O000000

O0000000

IIIIIIIIIIIIIII

0 II

0

0 o'3

0 ¢q 0 __0000___00000000000 .t..Q.o.*.b....,t....o...... IIIIIIIIIIIIIIII

0 0 -.I"0 • ,-4 __000____0000000000 .,t.,o....t..........,,o.... IIIIIIIIIIIIIIIIl

II 0

u% 0

ItIItIIIIIIIIIIII

j

0

___000__0__0__ _ _0 _ _ __ ,....,..t.....o............. IIIIIIIIIIIItIIIIl

O"

u) 0

o

_

_

0

O000000000

B

September 93

6 1961

Section

B 6

15 September Page 94

o

0 0 O_

_____ OO

o0o

0oo

oo0

_oo oooo

ooo

___

1961

_o oo

o

OO

ooo

IIIIIIIIIIIIIIII

___o__ 0 0

o

,--4

_____0o ooo

__ ooo

ooo

o

oo

_

ooo

o

oo_ oo

ooo

_

_

__oo ooo

o

o

o o

oo o

|illlllillilllll

o U

o u_

0 o 0_

________o_ _0_0 _ 000000

_

_0_

_

_

___

___000000_

_000 O00

O00000000

O0000000

IIIIIIIiIIIIIII

,-4 O4 I ,-4

o o cq

_ o

_ oo

_ 0

_ _ oo

_ o

_ oo

_ o

_o oo

ooo ooo

o oo ooo

o _ oo

_ o

_ o

_o oo

t.e.ool..m,o,oeo°ooeo,eeoo IIIIIIIIIIIIIII

_0 _a o o oo •

00 r-I ¸

____ _ _ _ 00000000

_ __ _

_

0_ _0000000 O0000

_

_ _0

__ _

O00

_ _o ____ O00000000 O00 O000000

_ O0

IIIIIIIIIIIIII

u_ I _-_ 0

o

_o_o__oo____o o__oooooooooooooooooooo 0oo0o0o0oooo0ooooooooooooo

0

...o..°.°,.e°.o°....,.,,.. IIIIIIIIIIIII

o 0

_o_ __ o oo o oo

_oo_0 __ 000o oooo

___o _oooo ooooo ooooo

0oo ooo

_ _ ooo ooo

_ _ ooo oo

o

__ _ _o ooo000 ooo

o oo

.,..e....,,...o,.,,..o.,,. IIIIIIIIIIII

_J 0

o

___ ___ ooooo ooo0o IIIIIIIIIIII

0

oo oo

o o

_ _ __o _ _o oo ooo ooo ooo oo oooooooooo

__ _ _ _ ooooo ooo

_

_ _ _ _oo ooooo o oooo

o 0

Section 15 Pa_e

O O

_0______ __000000000000000000000000 °.°°° .... Illllll

B

6

September

1961

95

_0__ °°°°o°°°oo°°°°o°o°° IIIIll

0 0 0 II X _O__O_O_ 0 U_

__O_OO

O__

__O_____ _OOOOOOOOOOOOOOOOOOO .°0°_° ............ IIIIII

O o_

_

OO__ OOOOOOO 0.°0.°.°.o iiiiii

__100_____ _OOOOOOOOOOOOOOOOOOO 0°°0. ...... IIIIII

OO__ OOOOOOO .°°

....

_

°°°°°0°°.° iiiiii

O I ,--4 ° c'q _D

_,_000000000 • ........ IIIIII

_ 0000000000000000 ................

.

.o IIIIII

O O O ,-.i

,.4 II

O ,-_ _OOOOOOOOC_OOOOOOOOC o°°00°°°°0. iii1|

.4

OO ....

°°

OOOOOOO 0°..0°. iiiiiii

....

-1

_ 0

O

_ 0

e,i 0

°o. iiii

e_ 0

0 0

0 0 ........

_ 0

_ 0

_ 0

_ 0

_ 0

_ 0

_ 0

_ 0 °0°..-

_ 0

0 0

0 0

,_ c)

_ 0 ........

C 0

0 ,_

_D 0

0 0

0 0

0 0

0 0

III

O

0 0

O

O

iiiiiii

______O___ _OO_OOOOOOOOOOOOOOOOOOOO OO O O O OO O OO

0 0

iiiiiii

_OO__OOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOO °,_°o°0oo°°°..°0°°°o.°°°°°o0 IIII

oq O

0 0 0°o°

_ OO

OOO

O

O

O

O

O

O

OO iiiiiii

O

O

O

Section

B 6

15 September Page 96

1961

0 0 __

_

_

_

_

_

0

0000

_

_0_ _ _

0

O0000

O0

_

_

_

_

O00

***,.,**,,.,..,.....,,..... II|lll|lll|llll

0 0

0_ _

_ _ ,°.l**...°.,..°°.....,°..,. IIIIIIIIIIIIII

0 II

_

O_ _

0_

_ _ O000000000

__

_

0_

_

0

0 _ _ 0___0____0__

0

0

_00_

00

_0

ooo °"

_

_

0

_

0 O0

ooo

,°**°..,..**°°o,.°.°,,.o.., llllllllllll

I

0 cq

I:Q 0 oo II

o u_ 0

0_0________ 0_0___00_____ 0__00____000000000 _ _ _ _00 °....**..°°...***.,°°,°,°., IIIIIIII

O0000

0____0_0_0___ 0 _ _ _ _ 0_ _ 00_ _ _ _ O0000000 ****.°...,°°°0..,...°..°..° IIIIIII

O000000000

_

_ _

_

_ _ _ O0

O0

0 _ _ O0 O000

_ O0 O00

_ _ O000000 O0

O00

_ O0

4-. u,_ rj

0_________ c'4 0

0 _

0

_ 0 _ _ O0000 °,°.**°,..o.,.°°,°,...**... iiiiii

0

_

_ O0

_

000 O0000

0000 O0

000000 O00000000

O0

0

Section 15

B

6

September

PaRe

1961

97

___0___0__0__ g

_____0_0__0_0_ __0000____000__ 0000000000000000000000000000 ,.o,.IoIe..,,....°...,..,.., IIII1|111111111

O0

O000 O00000 ,,,°.,,,,ao,.,...,.,..o,°..,

O0000

O000

O00

O000

IIIIIIIIIIiiiiI

o II o t_ _

0000000

O0

O00

O0

O00

O0000

O00

IIIIIIIIIIIiiii

0 o')

__

_o_

_

0

_ O0 ,°,,°,°ool°m.,,,,l,..,,°,ool

____ O0

0

O0

0

0

0

0

0

0

0

0

_

_

__

GO

O0

O0

0

0

IIIIIIIIIiiiiii

...le.,.i

,5

___00_____

C)

_

__

_ _ °llil.o.eo.......°..°.0°....

0

O

CD 0

_

_

_

_

_

O

_0

_0

0

00

O0

0

IIIIIIiiiiiiiii

!0 iO 0

E-I

o

II 0

,-1

t_ 0

_O_O__g_ __000____0000000000

__0_0__

IIIIIIiiiiiiiiii

0

_0 _0 __00____0000000000

__

_

_

_

_

_

_

IIIIIIIIIIIIIIIII

4-1

/

__ __ _0__ __00_

0,1

o

•

0

_0 00_

___ __

__ ___0__0 _ _

00_ _ _ "

"

"

"

I"

I"

I"

I'

I"

I"

I"

I"

_ I"

_00__ __ _00000000000

I"

I"

I"

I"

_0 I"

I"

_0 _

__ I"

I"

"

"

"

"

"

"

Section 15 Page

B

6

September

1961

98

0 0

0 0 ,,.°,°oo°,..,o**..°,,oa,,., o

IIIIIIIIIIIIII

ii

X_

o_G_o_o_

0 u_

®

.

_

_

___00000000000000000000 ,°°.°.°.°°,°°..°°,°._°..°°° Iiiiiiiiiiiiiii

00_0_00_0____0__ 00__0_____0_0_ 0__0_000_____000 __000000000000000000000 °°.o0o°.°.°°.0°......°.°.°_

0

IIIIIIIIIIIIiiiii

I

0_______0__ 0____,___0__ 0 __o_000____00000 __0000000000000000000000 o.°....°.°.o°......°_°..°..

0

&

IIIIIIIIIiiiiiiiii

r._

D •'_ •

0____0__0_0_0__ 0_0_0_00_____ 0__0000_000000000000000 __00000000000000000000000 °.°.°..°0.o.°°.°..°°eo.o°..

0 r-I

II

II1|111111111111111

0______0_0__ 0________00 00_o100000000000000000000000 _000000000000000000000000 °°.°....°°°°°°.°°..°.°.°°.° IIIIIIIIIIIIIIiiiiiii

u_ 0

oq 0

0__00_0__0__0_0__ 00_0_0___ 0_00_000000000000000000000 _000000000000000000000000 .°°°°o°.°°°.°°.°°.°._o_°o°° IIIIIIIIIIIIIIIiiiiiii

C

0__0__ 0____0_0__0_0_00 0_0_0000000000000000000000 _0000000000000000000000000 IIIIIIIIIIIIIIIIIIIII

0 0_0_0_0_0_0_0_0_0_0000000000

___0_00

__0__0_

I

Npction 15

Spptember

Pa_e

0 0 eq

•

.

I

0 C_

•

I

_

°

I

_

.......

I

_

_

,

I

_

I

I

tq

o,.,,

I

-T

_

I

_

I

_

I

_

....

°

I

_

.

I

_

,

I

_

.....

I

_

,

I

O

I

I

O

I

O

O

.°,.,,,°

I

O

]_ 6 1961

99

....

,

•

,

I

O

O

O

O

O

.......

O

O

O

%,..

IIIIIIIIIIIIIIIIIIII

II X 0 u_

_0

____oJ_

0

_0_

0__0_

_

OO___J_OOOOOOOOOOOOOOOO *,.,,,,°,,,,,,,,

.....

,,,,,..

TTIIIIIIII'II'IIII,

0 _

__

_000000000000000000

................

,

........

,,.

_1111111111111111 II

_0

eq

Cq

c_i

00".-1"

I'--

u'h

r'_

r'f)

_-_ _'4

_D

_

OJ

m l --I

I'--

_0

I'--

r'l

_D

I_

I"D

000

_

_0

O0

oJ

OJ

-1"

I_-

Lth

I

u_

TM-

O0 cq

ch ,-41

_

_ 0 _ .....................

_q

i

I

tD

00

o!

C.._ 000000000 ,

.....

i

_q

_:_ _-_

0 00 c_

Oh r:q

,-_

I"-. ,_ ,_D C-I

•

•

O cq O0 u7

•

•

t',,l cq

•

I',-O" _ C)

•

•

_th C

,_ (-

_ -b

C)h C) C:,

O O

C, O

O O _'D 2,

C_ O

0 ,2_

•

•

O O

O O

C) O

O O

rD O

C, <_

0 O

0 O

O O

O O

O O

O O

O O

O O

O O

.....................

II I

I

I

q_

0

O0 C4 p'B O • C_I

_

I

0

f_ e,I

•

_

•

1_-4 I

0 __

_ _ ,,O ¢'q •

CO -1" 0 ,._ {-'} (D

•

I

I

, I

• I

0 O

,'-4 _D

,

.

I

"_J 00 O O

•

.

_ O

O .

I

•

•

I

"_ (D

0 O

....

o

I

I

0 O

0 O

.

•

I

00 O .

0 O

O

o

.

I

0 O .

.

I

I

I

_

_

_

_ c,_ ol 0 _ _ 00000000000000000000000

00

_

0

_

0

_

0

_

0

_

000000

.,.,,,,,o.o,°°,.,,o,..°.,.,, _111

I

II

I

II

I

I

I

I

I

I

I

I

III

__00000000000000000000000 ,,,,,,°.°,,°,,°°.°°,.°°,o,,° _111 III

0

I

I

I

I

Section 15 Page

B

6

September

1961

i00

!0 0

0__0__0____0 ,...°...,,,.,.o....,,..°., IIIIIIIIIIIIIIIIIIIIIIIIII

_0________ 0 0 .,.,,,.,....,..,..,,..°... 0

IIIIIIIIIIIIIIIIIIIIIlII|l

II

X 0 L_

0_0__0__0____0_000 __0____0000000000 ...°.°...°..o°._....°°.°°. N_NNNIIIIIIIIIIIIIIIIII IIIIII

I

o

I

0 oq

&

0__0_____0_

_

___,__0000000000000 .°°.°....°...o.0..,.°....° IIIIII

,-1

o Oq

__0___0__0__ __0_0__0_0__

_

,-4 It

___-_00_00000000000000 ....°°...o.°.....°.......° IIIIII

L_ 0

0

0

_

0

,_r_

0

u'_

0

,r_

0

_

0

_r_

0

_,'_

0

u"_

0000000000

v

Section 15 Page

0 0 cq

0 0 r-4 0

X 0

00 ! C ¢-d ¢,q

0 0 c'q 0

< II

0

cq 0

0

0

B 6

September 101

1961

Section 15

___0___

1961

102

__ _

6

September

Page

0_0 ___

B

_

_00000000000000000 IIIJIIIIiiii|

0 n

X 0 un

0

O_ c'q I 0__ 0

_

_

0_ _ __0000000000000000000000

___

__

O_

_

__

_

_

_

_ 000

IIIIIIIIIIIiiiiiii

,-1 < 0 0

0

0______0_0__0 0_0_____0___ 0__00___000000000 _ _00000

O00000000000

O00000

IIIIIIIIIIIIiiiiiii

I!

0

0

0___0____0__0 O_ ___ 00_ _0000

_0

_

_ O0000

_ _0000000000

_

_

_

O0000

O00

_

_ O0000 O0000

_

_00 O0 O0

IIIIIIIIIIIIIiiiiiiiii

oq 0

O_ __ _0 _0 0___0_____0_00 O_ _0 _ _00000000000000 _ _00000000000000000000 IIIIIIIIIIIIIiiiiiiiii

I

_

__

__0

_0

_0 O000 O000

0

Section 15

B

6

September

Page

1961

103

0 o

0 o ,--4

o II o

f

o

o i

o 04

_D Im 0 o 4o • ,-i II o

o

u% o

__o___oo0oooooooo0o ___oooooooooo0ooooooooo0 _

_

_

I

I

I

I

I

I

I

III

_ t_ o

o_

_

_o

_

O_

_o_0_0__o_oooo0ooooooooo _ o _ _ o oo _1111 III

E 0

0 "8

_ oo

o

_

_

_

oo

o

o

_ o I

o

_

o

o I

o I

_

_

o

o I

o

_

o

o_

o

o

oo I

I

Section 15

1961

104

____0____0 0_____0__0__0 ________000 °,°°,°o°°o°°.o°..°.ooo°o..

c_

o o

6

September

Page

o o

B

IIIIIIIIIIIIIIIIIIIIIIIIIl

0__0__00____0_ __0__0___0__ 0_0____0_0__0 _______000000 °°°°.°°°°.°°.0.°°°.°°..°._

o

IIIIIIIIIIIIIIIIIIIIIIIIII

II

o u_

______ _____0____ 0________0_000 _00____000000000 °0°°°°0°°°°°.°.°°°°.°°o°°_ I__111111111111111111111 IIII

_0_0_

0

,--4 eq I

0 Cq

_4 0 0 v II

0 ,-_ 0

cJ

_ _

_ _ _ 0 °.°.°..°°0°°

_ _

_ 0

0 L_

_ _

_ _

_ _

_ _

0 _

_ _

_ ,_ _ _ 00000000000000 ....

_

_T

00

_

_

_

_

_

_

0

°°°°°o°°°0

IIIIII

0

c_ 0 0

___'_ ____ .°°.°°°°°°°.°...°°°.°°°°°° __lll IIIIII

Oq 0

_

_ __0__0 00_0_0000000000 II

I

II

I

I

I

Section 15 Page

B

6

September

1961

105

0 0 eq

0 0

,0 II X 0 L_

0

c'q I

0

0 0 .0 ,--I

___ _ ___00_0_0__00_ °°...°....0..°°..°°...°°.°.° _0__

II qJ ,-1

Ii

0

cq 0

L_ c'q 0

0

0__

_

_

I

__

I

0_0_

I

I

IIII

I

Section 15

B

6

September

Page

1961

106

o II

0 Lf_

0

0____0_00____0 0_ 0____0_____00 _ _ _

_

__ _

_0_

_

_000000

_0

_

_

_0

_

_0

_

0_

O0000000000000 iiiiiiiiiiiiiiii

I

0 o4

0_0__00_____0 0__ _ 0_0_0_0_____000 ___00000000000000000000

_0

_

_

_

_

__

_

_

_

_

_

_

iiiiiiiiiiiiiiii

,.-1

< 0 0 0

0

0___0______0 0_00________ 0 _ _ _ _

_0'0 _ _00

_00

_ O00

_ _ O000

_ _ O00

_

_ O0

_

_

O0

_

_ 000000 O000000

IIIIIIIIIIIIIIIIII II

Lr_ 0

0_____0___0_0_0 0______0___ 0 _ _0 _ __0000000000000000000000

0 0

_

_

_

_

O00

O000

O00

O000

IIIIIIIIIiiiiiiiiii

0

0____0____0_0 00_0__0_0____00 0__0___000000000000 _,_0000000

O00000

O00

IIIIIIIIIIIIIIIit111

0___0__0____0 0____0____0000 0_0_00 __00000000000000000000000 ....o0°o°.0.0°.°.°,..,...._

_

_

IIIIIIIIIiiiiiiiiiiii

_

_

_

O00000000000

O0

Section 15

B

6 1961

September

Page

107

oo

_0

__

_

_

__

_

0__

_

_0

o /pk

O ,-4

oooe0eI._.o.°o0°i°°.I..QI.° IIIIIIIIIIIIIIIIIIIII

O II

O u% /

_0_0_0____0_0__ _000__00__0_0_0__

0 e_

_

I

_ _ ........°_.o..°..°.°°..°o.°° IIIIIIIIIIIIIIIIII

_

O0

0

O0

000

0

0 ______00____ ____0000000000000000 IIIIIIIIIIIIIIIII

I:Cl

o 0 0 .0 ,-4 ,_ II (J

.I

u_ 0

_0_0_0______0 _0_ __

0

_0__0___ _

_

0

_

_

_

__

_

_ _

_ 0

0 0

_ 0

O0

o

ooo oo o 0

O0

_

_

_ 0

_ O0

O0

0

_ O_ O0

0

0

0

0

_

_

_0

_ 0 O0

0 0

0 0

0

O0

_

_

0 0

O0 O0

0

0

0

0 0

0 0 O0

_llllllO Ill

_ _0 ____0__00__0_0_ _ _0 O_ _

eq 0

_ O0

_ 0

_

0 0

_

z _N_IIIIII III

0

I

I

I

I

Section

B

6

15 September Page 108

1961

O O

O O Oi II XOu_

___O_O_OO_O___ _ _O___ O_O___O_____O _ _ __ °°°.°°°..°°°°°°°.°..°°.°°. IIIIIIIIIIIIIIIIIIIIIiiiii

___O_ _

_

_

_

_

_

_

_

_

_

_

O

O

OO

O

O oq

u_ oq !

O

<

O O O• O r-4 II _J

____ OO_O_OOO____O_O_O_OO _ __O____OOOOOOOOOO .°°°0.....°o.°.°°°0..°°°.. I_NN_NIIIIII|IIIIIII IIIII

___O_

_t_

___O___O_O__ OO___ __O___OOOOOOOOOOOO ..°°°)°°.°°.°°.°°....=°°.°

O_O_,_O__

O_O_OO

IIIIIII

O

O O

_n_

__

_ _ _ _1 _ _ _ __O_O__OOOOOOOOOOOOOO ...............°..°.......

_

__O_ _

_

_

_

_

_

_

O_ O

_

_

_

_

_ _

_

_

_

_

O

O

IIIIIII

O

OO_ _O___O__OO___ O____O__O_OO__O __ °.°_......°.°°°°°.°°°..°°° __NIIIIII IIIIII

O_O__O___

_

__OOO_OOOOOOOOOOOO I

I

I

I

I

E

t4-

O

_

O

_

O

_

O

_

O

_

O

t_

O

_

O

_

O

_

O

O

O

O

O

O

O

O

O

O

v

Section

B 6

15 September Page

1961

109

,7--

0 0

/

0 0 _=4 0 II 0 u_

0

@_1 II

IIIIIIll

_1 II

IIIIIIII

,,O I ,-i

0

0 0

0 _l:

II

_BIN II

0

/

0

II

U F

0

0

NNN

NN

I

I

I

IIII

I

Section

B 6

15 September Page ii0

1961

References

,

1

MacNeal, Richard H., and John A. Bailie, Analysis of FrameReinforced Cylindrical Shells, Part I - Basic Theory. NASA TN D-400, 1960. MacNeal, Richard H., and John A. Bailie, Analysis of FrameReinforced Cylindrical Shells, Part II - Discontinuities o_ Circumferential-Bending NASA TN D-401, 1960.

,

Stiffness

in

the

MacNeal, Richard H., and John A. Bailie, Reinforced Cylindrical Shells, Part III NASA TN D-402, 1960.

Axial

Directions.

Analysis of Frame- Applications.

I

\