NASA Astronautic Structural Manual Volume 3

i

NASA

TECHNICAL

MEMORANDUM NA SA TM X- 7 3307

ASTRONAUTIC STRUCTURES MANUAL VOLUMEIII (NASA-T

_-X-733|.

HANUA/,

90LU_E

Structures

August

7) 3

and

AS_FCNAUTIC (NASA)

Propulsion

N76-76168

ST?UCTUE_S

676

00/98

Unclas _4u02

_..,=--

I'IASA ST| FAClUiY

Laboratory

197 5

NASA

\%

marshall

'-"'

i.Pu_ B_.c. i/

Space Flight Center

Space Flight Center, Alabama

MSFC

- Form

3190

(Rev

June

1971)

TECHN_,_AL

NASA

TM

T'.TLE

X-73307

"NO

2.

GOVERNMENT

ACCESSION

REPORT

NO.

I 3.

RECIP)ENT'S

1

SUBTITLE

5.

ASTRONAUTIC

STRUCTURES

7.

AI;T

t 9 i •

ur'P

197 5

PERF0qh_ING

DPGANIZATION

8._i-'_FORMING

:

_"PFC'-'.'ING

ORGANIZATION

C.

Marshall

NAME

Marshall Space

Space

Flight

AND

ADDRESS

Flight

Center,

10.

S;-{,_JRING

Center

1.

Alabama

_,_JZ_N_,Y

NAME

UNIT

CONTRACT

OR

and

D.C.

20546

,

", '_LEMENTA:("

NO,

35812 TYPE

Space

OF

REPRR'_

&

PERIOD

COVE_ED

Memorandum

Administration I.%

i Washington,

GRANT

AI3nREc;S

AND

Aeronautics

REPD_r

NO.

Technical t National

• :

CODE

O_CANIZATIL')N

_©Pt_

13.

i

NO.

Ill

George

" "

PAGE

DATE

A____ust

MANUAL

TITLE

CATALOG

REPORT

6.

VOLUME

STANDARD

"

",,_O_,"

L,

A,SENCY

CCDE

NCTLS

.

I Prepared

by

!

This

Structures

and

(Volumes

document

; aerospace I cover most

strength structures

Propulsion

I,

analysis that encountered,

Laboratory,

II,

can

and

III)

Science

presents

and

a compilation

be carried out by hand, and that are sophisticated

used

to methods

Section

D is on thermal

on composites;

These

17.

TM

KE_

of strength

analysis;

stresses;

Section C is devoted

Section

three volumes

X-_on42,

usually

elastic

enough accurate and

available,

in

scope estimates

in to

inelastic

but

also

as

a

supersede

Volumes

WC_DS

SECURITY

Form

3292

I and II, NASA

thll

December

TM

report_

1972)

DI_T/{IGUT"

"_

X-60041

5,,:T__',.

Unclassified

SECURITY

Unelas (R..v

mechanics;

stability;

Section

F is

and Section H is on statistics.

!0.

CLASSIF.(of

to the topic of structural

and

respectively.

Unclassified MSFC-

the

E is on fatigue and fracture

Section G is on rotating machinery;

OR_GII,_AL P._,.C._ ,Z, OF POOR QUALITY

19.

for

general to give

methods

An overview of the manual is as follows: Section A is a Keneral introduction of methods and includes sections on loads, combined stresses, and interaction curves; Section 13 is

devoted

NASA

of industry-wide

that are enough

I of the actual strength expected. It provides analysis techniques P stress ranges. It serves not only as a catalog of methods not i reference source for the back_zround of the methods themselves.

i

Engineering

CLAS3IF.

(,J

_htl

:

-- Unlimited

pa{_)

• -_I.

sifted F'.)r,_ale

"_3.

OF

_;,,ES

673 by

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:,on

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22.

PRICf

NTIS ..,,,.

c; i rm.-fi,'hl,

Vir_,ini;,

221¢1

._i-

APPROVAL ASTRONAUTIC STRUCTURESMANUAL VOLUME III

The cation. Atomic

information

Review Energ:/

Classification be unclassified.

This

in this

report

of any information Commission Officer.

document

has

report,

also

been

reviewed

concerning

programs

This

has

has

been

for

Department been

in its

made

entirety,

reviewed

and

security

of Defense

by the has

approved

classifi-

MSFC

been

or Security

determined

for

to

technical

accuracy.

A.

A.

Director,

McCOOL Structures

and Propulsion

•t_

U.S.

Laboratory

GOVERNMENT

PRINTING

OFFICE

1976-641-255,P448

REGION

NO.

4

j

TABLE

OF

CONTENTS

Page

Do

THERMAL

STRESSES

1.0

INTRODUCTION

2.0

THERMOE

1

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

LASTICITY

3

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

2.0.1

Plane

Stress

Formulation

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

3

2.0.2

Plane

Strain

Formulation

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

4

2.0.3

Stress

Formulation

2.0.3.1

3.0

1

........

STRENGTIt 3.0.1

3.0.2

Sohuli,'m

(,f AiryVs

Function

.....

5

Plane

Stress

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

5

II.

Plane

Strain

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

5

Unrestrained

S()I,U'II()NS

I_eam-Therm'd

3.0.1.1

Axial

3.0.1.2

lfisl>lace)nmlts

l/cstrained

Stress

¢)nly

......

:)

.............. I_(m(ls

InteRrals

Sections

Simply

()r, ly

for

. ......

II.

Fixed-

lIl.

F ixe d- 11in ged

IV.

Deflection

Fixed

D-iii

12

l,:xamples

Supported

Beam

L_mm

Plots

l(i

Vqrying

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

Beam

l,cstr;ine(1 , • )

7

7

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

Ewlluation()f

Io

7

.......... I,oads

Bealn--Thernml

Cross

3.0.2.2

Stress

I.

()1,' MA'I'I,:I{IAI,b

3.0.2.1

4

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

Be a m ...........

........

14

........

14

.........

53

.........

56

58

TABLE

OF

CONTENTS

(Continued)

Page 3.0.2.3

Representation of Temperature Gradient by Polynomial ..........

70

I.

Example

Problem

1 ..........

74

II.

Example

Problem

2 ..........

76

3.0.3

Indeterminate

3.0.4

Curved

3.0.5

Rings

3.0.6

Trusses 3.0.6. 3.0.6.2

3.0.7

Beams

Beams

Rigid

Frames

.......

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

80

..................... i Statically Statically ......................

3.0.7.1

Circular I.

II.

III.

II.

Determinate

..........

Indeterminate

80

..........

8i 81

Plates

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

81

Gradient ............

Temperature of the Radial

Difference Coordinates

Rectangular I.

80

Temperature the Thickness

Disk

78 80

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

Plates

3.0.7.2

and

with

Central

Plates

Temperature the Thickness Temperature the Surface

D -iv

Shaft

Through 81 as a Function ..... ........

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

91 101 104

Gradient ............

Through

Variation .............

Over

t04

119

TABLE

OF

CONTENTS

(Continued)

Page 3.0.8

Shells

3.0.8.1

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

.

I,_(_tr()pic

Circular

],

Analogies

II.

Thermal

with

Radial G radie.nt

and

Thermal

Stresses

Constant

l{adial

3.0.8.3

'2 lsotropic lsotropie

Shells Shal)e

I.

Under

Sphere

4.0

THEHM()EI,ASTIC 4,0.1

lleated

d.0.1.1

4.0.2

.

133

Axisymmctrie 149

Deflections--

Gradient

.....

170 179

..........

Hevolution

of 191

............. l{:,lial

Variations

132

Gradient,

Shells of

Arbitrary

131

l)cllcctions--

and

Axial

('onical

....

Pcoblems

Gradient, ............

Axis 5 mmetrie 3.0.8.

Shells

Isothcrnml

Stresses

Linear Axial

IlI.

Cylindrical

.

Temperature 201

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

STA BII3TY

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

20"_

Colunms

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

20.'l

lk;am

Ends

Axially

Unrestrained

1.

B,,)th

Vnds

Fixed

II.

Both

Ends

Simply

III.

Cantilever

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

Ends

Axially

Supported

Restrained

Thermal

Buckling

of

4.0.2.1

Circular

l'latcs

D-v

20(;

Iqates

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

.....

20(; 206

...........

,1.0.1.2

203

........

20 209

209

TABLE

OF CONTENTS

(Concluded)

Page 4.0.2.2

Rectangular

Plates

I.

Plates

Heated Edges

II.

Plates

Edges

5.0

INELASTIC 5.0.

6.0

Thermal

1

Buckling

EFFECTS

Creep

in the

Restrained

Edges

222 in Plane--

Loaded

Post-Buckling All

4.0.3

Loaded

Unrestrained

Heated

III.

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

Plane

Deflections

of Cylinders

....

222

in Plane--

in the

Simply

Plane

.....

225

with

Supported

......

230

...........

234

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

245

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

246

5.0.1.1

Design

Curves

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

248

5.0.1.2

Stress

Relaxation

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

251

5.0.2

Viscoelasticity

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

253

5.0.3

Creep

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

253

Buckling

5.0.3.1

Column

5.0.3.2

Rectangular

5,0.3.3

Flat

THERMAL

of Idealized

Plates

Column and

Shells

General

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

6.0.2

Stresses

and

Section

....

......... of Revolution

SHOCK ..................

6 0.1

REFERENCES

H-Cross

255 255

.....

256 263 263

Deformations

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

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

264 2_ J

D-vi

SECTION D THERMAL STRESSES

rDEFINITION

OF SYMBOI_S

Definition

Symbol A

Cross-sectional

A0

Cross-sectional area of beam

Amn,

A

area; area at x = 0

Coefficients for the series by which the stresses are Pq expressed,

in.

C(mstants based (m the boundary conditions, equations (9?0

Ai, A2, A3, A4

and (96). dimensionless Constants,

psi

(Figs.

5.0-8,

5.0-9)

1

l,imiting

a

value

(lower)

of middle

surface

Maximum

value

a 0!

Constant,

° l"

al

Constant,

o F/in.

a0

Coefficients

Bran, Bpq

NOTES: 1. Bars

over-'any

2.

The

subscript

3.

The

superscripts

particular 4.

The

thermal

er

and

subscript

lcttc)rs

inside

radius

or radius

of initial

imperfection

series

by which

the stresses

arc

in.

denote

denotes I'

radius;

of cylinder

lov the

cxprcs3ed,

for

and

complementary R denotes

mi(hlle-surface_ eritic:_l

C

_alues

identify

required

deformation_.

D-vii

for

quantities

:mlutions, w_lucs

values. buckling. associated

with

respectively. to completely

suppress

the

DEFINITION

OF SYMBOLS

Definition

Symbol BI, B2, BS

Constants,

b.

Breadth

in./(in.

Specific

P

C1, C2t C3t C4

C-I'Co'Ct"'"

o F/in.,

heat

Constants

limiting

value

(upper)

representation

and • F/in.2,

of the material,

of integration,

of the temperature

respectively Btu/(Ib)

(°F)

in.

Coefficients

in polynomial

representation

Ib/in.,...,

respectively,

refer

stiffness

or shell-wall

of

to equation

U P,

In.-Ib,

Ib,

(106)

Diameter

D

Plate

bending

Constants o F, d

section;

radius

in polynomial

T1(x ) ; ° F,

5.0-i0)

of cross

outside

Constants

b 0, bt, b_

) ( hr)(Fig.

(or width)

for radius;

C

(Continued)

-1

,d

O'

d,...

in polynomial

° F/in.,

Coefficients

representation

stiffness

of the function

and ° F/in. 2, respectively in polynomial

representation

of

1/in.

respectively;

refer

V P,

!

dimensionless, equatton

,...,

(106)

E

Young t s modulus

of elasticity

Eb

Young t s modulus

of support-beams,

E

Young t s modulus

of plate,

P

bending

D-viii

psi

psi

to

in.,

T2(x ) ;

DEFINITION

OF

Definition

Symbol E

(Continued)

SYMBOLS

Secant

modulus,

psi

S

Et

Tanzcnt

e

llasc

F. E.M.

Fixed-end

FF

I" ixc(I-

fixed

FS

Fixed-

s upl)ort

G

Variation

in (lepth

G T

Modulus

of rigidity

H

V:/riation

ha width

It A , H B

Running

c(Igc

positions

A

psi

modulus,

for

natural

logarithms,

dimensionless

(2. 718)

morn(rot

ed of beana or

shear

of beam

forces

acting

and

along

the

length

modulus

along

the

normal

B ,

respectively

centroidal

moment

length to

tile

axis

(Figs.

of r(woluti()n

3.0-51

and

lb/in. I

Moment

Ib

Support-beam

I,I y

z

Area

of inertia

moments

respectively, i

Imaginary

K

Thermal

of inertia

taken

of inertia

about

the

y

and

z

in. 4

number,

_.-rTT-

diffusivity

of the

material,

ft2/hr

= k/C

p P

k

An

integer

axes,

(1, 2,3,,1,

D-ix

,5) exponent

3.0-52),

at

j"

DEFINITION

OF SYMBOLS

Symbol

Definition

k'

Thermal

L

Length

L( )

Operator

M

Moment

M A, M B

Running

conductivity

defined

edge

respectively

MT Mb

of the

moments ( Figs.

Thermal

moment

Thermal

bending

3.0-51

Mr0

Running

twisting

moment,

M r' ' M0'

Bending-moment

Mt Mx, My

resultant,

bending are

B ,

in.-lb/in.

in.-Ib/in. in.-lb/in. (Table

moments

to the

(positive

compressive),

acting

when in.-lb/in.

Moment

about

y

axis

Moment

about

z

axis

Moment

in beam

z M0

3.0-52),

and

in.-Ib/in.

Y M

A

3.0-5

and

Figs.

3.0-15

in.-lb/in.

perpendicular

respectively

M

(ft) (o F)

3.0-19)

Running

are

and

parameters

Temperature

which

at positions

parameter, moment.';,

Btu/(hr)

(103)

acting

bending

through

material,

by equation

Running

Mr, M 0 , Mx, M_

(Continued)

at x = 0

D-x

on sections x

and

associated

y

of the

plate

directions,

upper-fiber

stresses

DEHNITION

OF

SYMBOLS

Definition

Symbol Temperature

m

m k

me N

Moment

coefficients,

Surface

moment

also

Nr,

distribution

Exponent

N O , N x, N

Nr0 N r ' , N 0'

(Fig.

limit

load

per

for

length

mernbran(,

loads,

ilunning

membrane

shear

3.0-46,

in.-lb/in.

2

along

resultant,

n

Temperature

distribution

the

indices,

on

plate

load,

PT

Io r_'_

Axial

Ior_'c

P0

Column

P

Ila,lial

in the plotted

P,q

Summation

Q

Heat

From

indices,

3.0-6),

dimensionless

y-direction in Figure

temperature

load

pressure,

dimensionless

lb/in.

resultin_

psi dimensionless

input

D-xi

beam;

lb/in. (Table

coefficients,

Axial

of the

edge

_li mensionless

l)

length

dimensionless

Ib/in.

parameters

Temperature

Hoop-force

Figure

summation

Ilunning

Membrane-force

in

wtriation

unit

z-direction

3.0-53)

Nt

nk

in the plotted

of thermal

upl_er

Axial

N T

(Continued)

3.0-49,

I)E FINITION

OF SYMBOLS

Definition

Symbol

Qx

I{unning

q

Temperature

qk

Shear

transverse

shear

Radius

SS

Simply

S

Meridional

load,

distribution

coefficients,

r

lb/in.

in the x-direction

plotted

in Figure

3.0-46,

dimensionless

supported

truncated S*

(Continued)

coordinate cone

Meridional truncated

(Fig.

Temperature

T

Average

3.0-50),

coordinate cone

T

measured

downward

top of the

in.

measured

(Fig.

from

3.0-50),

upward

from

bottom

ol the

in.

I

value

Weighted

TD Tedges Tf

average

m

OF

value

difference

Temperature

at edges

Final

uniform

Inside

Average thickness

for

T,

°F

between of the

temperature

the plate

plate, which

faces,

oF

°F the

body

reaches

at

long times

temperature;

the body, T

T,

Temperature

sufficiently Ti

for

also

initial

uniform

temperature

of

°F value

for

at any

temperature single

D-xii

position,

distribution oF

across

the wall

r-

I)EFINITION

OF SYMBOLS

(Continued)

.F

Symbol T S

T xy

Definition Temperature

of the

supports,

Temperature

at any

location

T O

Outside

T 1, T2

Temperature

t er

V

*F in the plate,

functions,

Time

(hr)

Time

to the

°F

or thickness onset

of creel)

in the

circular

plate

Function

representing

buckling,

x-direction

or r-direction

temperature

Vp

Component

of deflection

without

VT

Component

of (h_flcction

ineludin4a?" tbermal

Vo

Shear

V

Displacement circular

W

x=

Deflection through

in a meridional

in yplane

therm_tl

for

('floors effects

0 in lhc v-Hirection

or O-direclion

for

plate

Displacement i

.for

variation

also

at

rotations

hr

directions;

W

°F

temperature

l)isl)lacement

u

..

in the parameter

3.0-19),

z-direction (Table

(timcnsi
D-xiii

3.0-5

and

Figs.

3.0-15

and

z-

a shell

I)I,',I,'INI'I'I(}N()F SYMBOLS(Continued)

Definition

Symbol Displacement,

W

are A W

simply

in the z-direction, supported,

Displacement

meant

wk

A

Deflection

coefficients, axis

Y

Coordinate

axis

Z

Upper also

c_

(orT0b2/t2) cr

limit

the

loads,

Coordinate

axis

Coefficient

of linear

Critical

value thermal

factor

x-y

Time

plane,

rate

strain

z-direction,

plotted

in Figure

summation

all

for in.

symbol

index

normal

thermal

(Note:

The

and

is not

3.0-45,

dimensionless

k , dimensionless;

to undeformed

plate

expansion,

in./(in.

) (° F)

parameter

(value

at which

occurs),

dimensionless

dimensionless (Fig. in planes

4.0-17), parallel

dimensionless to and including

in./in.

of change

for

D-xiv

edges

shell

psi

buckling

Knockdown

where

exponent.)

of temperature

factor,

case

deflection

an identification

measured

Knockdown

Shearing

%

for

surface

initial F

is merely

the

radial

in the

to be a generalized

Coordinate

x

in. ; also

component,

superscript

for

"Yxy'

in./(in.)(hr)

the

DEFINITION OF SYMBOLS(Continued)

Definition

Symbol 6

Maximum the

V

x-y

absolute plane,

value

for

deflection

measured

normal

in.

Del-operator Unit strain Strain

•°

intensity

defined

in equations

(1),

in./in.

1

l"ime

_o

rate

of change

for

1

Normal x

strains

acting

c. , in./(in.)(hr) 1

in the

respectively Time

(positive

rate

in./(in.

and

y

directions,

of change

when for

•

fibers

defined

Plasticity

t)y equations

reduction

factor,

coordinate

(Fig.

o( )

Vunction

defined

by equ_tions

coefficients,

X

A constant

F

Poisson

P i

,/(1-,)

pl_tted

in str:_in--su,'ess

v s r'ltio

3.0-14), (58)

in Fixture

D-XV

(76),

dimensionless

(78),

dimensionless

rad and

3.0-46,

dimensionless

re_l:_tionship

(s(,me_imes

(>f l m,o matc':i:_l,

and

dimensionless

Angular

l)en_ilv

in./in.

E , respectively, y

(56)

0

Slope

and

x

len_hen),

) (hr)

Function

Ok

x

y

written

It)/ft:

;_,m

)

to

I)I,:FINITI()N ()I,' SY M I_()I,S (Continued)

l)cfinition

Symbol

o'f 0",

Stress

induced

by restraint

Stress

intensity

defined

in equations

(1),

psi

1

( i)cr

Critical

value

for

the

stress

intensity

(r i , psi i

Axial

%,%,%,%

stress

Normal

due

artificial

acting

in the

stresses

respectively

(positive

In-plane

shear

Normal

stresses

stress

Lateral

axial

stresses

Plane

t,

O, and 0

r,

in the

(positive axial

PB

' psi

directions,

psi

psi

acting

Critical

force

in tension),

stress,

respectively

O"

to the

x

an(1

in tension), for

y

directions,

psi

buckling

of the

cylinder,

psi

stress

yz Shearing

T

stress

acting

in planes

parallel

to and

including

xy the

x-y

Stress

pl,'me,

psi

function

"meridional"; Function Paramctcrs respectively,

[Airy* also

defined

s stress

function

angular

coordinate

in equations

(76),

tabulated

in Tables

dimensionless

D-xvi

I(x,y)

] ; also

denotes

dimensionless

6.0-1,

6.0-2,

and

6.0-4,

DEFINITION

OF

SYMBOLS

Definition

Symbol

%,%

Parameters

tabulated

respectively,

in Tables

6.0-3

tabulated

in Table

6.0-1,

6.0-5,

dimensionless

Value

of

_I,2 at

r/R

= 1,

Value

of

_3

r/R

= 1 , dimensionless

Function

and

dimensionless

Parameter

)

(Concluded)

at

dcfined

in equations

D-xvii

dimensionless

(78),

dimensionless

_J

Section

D°

THERMAL

1.0

INTRODUCTION.

October Page 1

STRESSES.

Restrictions imposed on thermal expansion body or by the conditions at the boundaries

of the

the body. body are

problem

In the absence self equilibrating.

of constraints

at boundaries,

further methods

or of some

simple

combination

will yield good results. shape, the finite element

results.

The

idealized simpler

method

structure elements

connected

of finite

which (rods,

at a finite

rectangles, of the actual

or thermal The linear

tion of large depend upon

number

problem streng*h

deformations. deformation,

have led the of materials

element

analysis

etc.)

in a

not represent

[or

only at vertices

compressive

stresses

problem formulation involved in solving researchers and finite

to resort elements.

use

on _n

of smaller, plates, etc.) of triangles

the

or

configuration

resulting

from

ti_cr-

may produce instability of the strucof tile problem excludes the ques-

materials;

loads must

(e.g., beam-column analysis). the nonlinear thermoelastieity to the

approximate

associated with The phenomenon

methods

high temperature of the increase

of

is th:_t in str:_ins

is subject to constant stress and constant higl_ The general formulation remains the same :_s in

or strength of matcri:_ls, by a viscoelastic mode/. many

of mate-

Thus, for buckling, or for problems where nonlinearity ti_at is due to large deformations

when the specimen is called croci>.

thcrmoclasticity tion is expressed

is suggested

approximately

loading formulation

curved named

of strength

by a large number plates, rectangular

(e.g.,

and finite a

has a complex geometand yields satisfactory

to provide

One of the important problems deformation :md relaxation.

with time temper;tturc

of material its geometry,

metimd

if the structure is easier to use

of points

and mechanical, thermoclastic

the

However, metimd

structure,

be incorporated in the The extreme difficulties

of creep

stresses

following: rod, beam, of one of the elements

of them,

can be represented beams, triangular

or ends of rods, structure.

In a constrained real, ture.

by continuity stresses in

thermal

approximations leading to the strength are used extensively. Depending upon

structural clement is classified as one of the beam, plate, or shell. If a structure consists rials rical

or contraction induce thermal

1970

15,

Except for a few simple cases, the solution of the thermoelasticity becomes intractable (see Ref. 1). Therefore, for thermal stress

analysis, element

above,

D

but the

except theft the stress-strain relaThe linear viscoelastic model does

complexities

multiply

if the

nonlinear

SectionD October 15, 1970 Page 2 model is used. Relatively little work has beendonetowards the solution nonlinear

viscoelastic Vibrations

with those

resulting

of

theory. that result from

from

mechanical

thermal load.

shock

are

quite

small

They

are

not considered

in comparison here.

Section

D

October Page 2.0

THERMOE

strain

can

spherical

Plane For

in plane

be found

in Ref. for

Stress

below

rectangular

the

are,

stress,

cylindrical,

for

the

T(x,

y)

most

of the axy'

Exx'

eyy,

eight

(no

body

form

Exy,

u,

in a long and

v

equations. forces),

0_

xx Ox

+

xy

=

0

YY

=

0

3y

Off

_ff

_÷ Ox

Oy

Stress

-Strain

Relations,

1 =

_

(or

xx

E

-

1,a

xx

)+

aT

yy

1 Cyy=

_

(O-yy-

1 ¢xy

=

vO-xx)+

=I

2-Yxy-

Ou ax

ceT

1 2G

Strain-Displacement

E

or

part,

two-

coordinates.

following

of equilibrium

0a

displacement,

of rectangular,

given

axx , Cryy,

concept,

Equations

equilibrium,

1 in terms

distribution

quantities,

stress

3

Formulation.

a temperature

eight

for

Formulas

expressions

2.0.1

body,

equations

coordinates.

dimensional

1970

LASTICITY.

Three-dimensional and

15,

axy

'

relations,

;¢

=_ yy

Ov Oy

1 ;

Exy

=

_

Yxy

2

3x

prismatic

satisfy,

Section D October 15, 1970 Page 4 and in the case

ff

=ff

zz

¢

of plane

=ff

=0

xz

zz

2. O. 2

yz

v

-

stress,

E

(axx

+

aT

ayy)+

Plane Strain Formulation. In the case

of plane

u = u(x,

y)

v = v(x,

y)

strain

defined

by equations

w=O

replace

E,

v,

and

a

of the stress-strain

relations

of plane

stress

E

tion

by

E 1, vl, and

a 1 = a( 1 + v). remain

body.

The

Stress

equations

where

of equilibrium

E1 = _

;

"1 = i--Z"_v ; and

and strain-displacement

relations

Formulation.

The solution of three condition gives the The

equilibrium

xx+

a_

partial stress

equations

xy

ax

Oy

differential distribution,

equations satisfying the given (r , (r , and (r in the ×x xy yy

are

+ X=O

_cr

xy+ ax

and the

respectively,

unchanged.

2.0.3

boundary

al,

formulaI'

YY 0y

compatibility

V 2 (axx + cr

yy

+ Y=O

condition

is,

for

+ o_ET)+

(t+

v)

a simply

(

OX ax

+

connected

=

body

o

i

Section

D

October Page Solution

of Airy's

Plane

For the

connected

of this

Function.

regions

problem

$(x,y).

(See

o- xx -

5

Stress.

simply

solution

function

Stress

1970

15,

OyV

;

The

relations

above

tion

of these

relations

is

+

(xg

V2T

V 4 4)

-

V2(V24))

= 0

'• a xy

equilibrium

the

of the by

body using

forces, Airy's

X,

Y,

stress

Then

_

the

into

absence

considerably

AI.3.6)

=

satisfy

V 44)

simplified

Section

o yy

in the

stress

-

Ox0y

equations

compatibility

identically, equation

and

substitu-

yields

,

whe re i)24)

For stress

those El-

this

problem

=

the

function

boundary

Plane

Strain.

For

plane

strain

1 -y

V44)

2 024) +

_)x_)y

conditions

044, +

Oy-_

should

be

expressed

in terms

of the

4).

II.

above E

O-_x

problems

by substituting _

;

+

_1-

_

c_E

_(1÷

V2T::

E 1 and p).

0

the

governing c_1 for

equation E

and

_

can

be obtained

respectively,

from where

'CEb NG pA¢-EBLANK Section

D

1 April, Page 3.0

STR ENGTH The thermal

fore after

1972

7

OF MAT ERIA LS SOLUTIONS.

assumption that a plane section normal to the reference axis beloading remains normal to the deformed reference axis and plane

thermal

loading,

along

with

neglecting

the

effect

on stress

distribution

of

lateral contraction, lays the foundation of the approximate methods of strength of materials. The exact results obtained by the methods of quasistatic thermoelasticity show that the accuracy of the strength of materials solution improves with the reduction the length of the

of depth-to-span beam is smooth.

siderable

results

error

vicinity

of abrupt

changes

in the

cross

along a consections.

If the temperature is either uniform or linear along the length of the the assumption of a plane section is valid, and the strength of materials

beam, method method.

are

in the

ratio, if the variation of temperature As in the case of mechanical loads,

gives

the

same

Since

the

effect

zero;

ence

e.g.,

(r

yy

results

as those

of lateral = a

contraction

= 0

zz

given

in the

case

by the

plane

is neglected, of a beam

with

stress

thermoelastic

lateral

axial

x-axis

as the

stresses refer-

plane.

3.0.1

Unrestrained

3.0.1.1

Axial

is given

For by

Beam

-- Thermal

Loads

Only.

Stress.

an unrestrained

beam

PT (_xx =-_ET+

(Fig.

Iy

--_--

+

3.0-1)

the

longitudin;d

MT

z II yz

-I z yz

y+

where T

= T(x,y,z)

PT

=

f

aETdA

Iz

:

f

A

MT

z

MT y

J' A

(_ET

=

f

aET A

y2dA A

=

Y

z

stress

dA

I

dA

I

Y

yz

=

f

=

f

A

z 2dA

y zdA A

/ Iz

(axx)

- Iy z MTz_

Section D October 15, 1970 Page 8 y,¥

z1

\ \

Yl

J

o

r_-A

'

CENTROIDAL

/

AXIS I

X,U

I

CENTROIO

Z,W

I,I_A

Figure CASE

a.

The

y-z

3.0-1.

axes

General

are PT

(rxx

=

-viEW+

_

tan 0 I y MT

In the

new

principal axes, this coordinate

y z

xx

= -aET

+

MT

MT

- I yz M T - I yz M T

z

-7"

the

MT

I

A

= zl

Yl2 dA1

f

A

(2)

z

axes. with

A new coordinate y-z axes such that

system

in general

neutral

axis,

does and

not constitute equation

(1)

in

Yl zl

I

c_ ET (x 1 Yi zt) Yi dA1

= f

y

y

which

where

z1

= 0)

y

Zl

MT

yz

(3)

axis becomes reduces to

+

(I

beam.

z

system,

PT a

axes

not principal an angle 0

coordinate

the z system

principal

+ --T--y+--Tz

CASE b. The y-z axes are Yl, zl is chosen which makes Iy MT

unrestrained

'

(4)

Section

D

1 April, Page 3.0.1.2

1972

9

Displacements. Axial

displacement

u(x,

y, z)

with

respect

to the

u(0,

y, z)

is given

by y u(x,y,z)

=u(0,

y,z)

+

+

-g

o

I

T MTz -Iy zM IyI z - Iyz2

Yl

Y

(5)

dx

+ t Iz

The x

average

displacement

Uav(X)

of the

cross

section

at

a distance

is x

u

(x)

_tV

:

u

z)

= w(x,

v 0, 0)]

d2v

1

dx T-

I-_,

T

dx

(6)

0

Displacements w(x,y,

f-x-

(o) +

aV

P

1

and

are

Y I

w

given

oftherefereneeaxis[v(x,y,z) by the

following

v(x,

differential

0,0);

equations:

z M IT I - -I Iy z 2MTy 1 y z yz (7)

IyzyzMTzt E1 l Iz MTIIyS'z--I2

dxYd2w

If the

y-z

axes M

principal,

equations

(7)

reduce

to

T

d2v

d2

are

z

EI z

(s)

M d2 w

T

d2

l,:I

y

Y

Section

D

October

15,

Page In

yl-zl

axes,

defined

by equation

(3),

equations

(7)

1970

10

reduce

to

MT

d2v "_x

z£ EI

-

zi

(9) =0

3.0.2

Restrained Considered having y-z

tions

The at any

cross

restraints the

values

Loads

Only.

henceforth in this paragraph axes for the principal axes.

are

M , y resulting

section

constraining

=

-- Thermal

P,

against

Mz

Beam

thermal moments

M°z

+

V°z

and

M

z from the

expansion; and

shears

x

,

are

the

axial

external

force

M

y

of beam

and

forces

therefore, at the

cases

bending

and the and

M

cross

z

sec-

moments

reactions depend

to the only on

restraints.

(10) My

= MOy + VoyX

where the sign in Fig. 3.0-2.

convention

,

on moments

and

shears

and

M 0 and

V 0 are

shown

y v

v

M

M

M

M 0 V0

Figure

3.0-2.

Sign

convention

of moments

and shears.

V

Section

D

October Page The

displacements

v,w

MT

are

given

1970

15, 11

by

+ Mz Z

EI M d2w

(ii)

+M T

y

Y

dx _" -

EI Y

Solutions (10)

of equations

(ii) for the special

case described

by equation

are

x

x2 M T

_Iz (xl)dx

v(x)=- f f 0

(Yi)

0

f

2 +

c°z+ c'zx - M°z

z

X

-V°z

1 dx

_)

EIz(xl)

dx2

X2

f

0

x (x2_ fo

'EI

0

Xl

z

dx

(x,)

1 dx

2

(12) x

x 2 MT

w(xt =- f

f 0

_i y

0

-Voyf

moment

Coy

+

Cly

x - Moy

X

X 2

0

0

f/

El

y

(xO

X2

x1

f 0

bending

dx 1 dx 2 +

y (xl)

X

The

(xl)

l,:I (x,)dx 0

1 dx

2

y

and shear

force at any

cross

section

are

d2v M

Z

:

- E1

Z

- MT

-_X

' Z

d2w

M

= -EI y

y

_ -M dx _

(13) T Y

dM V

=

z

dM z

_

dx

;

=

V

y

dx

Section

D

October Page

which 3.0.2.1

notation

Each of the two equations are calculated from four Evaluation

b=boh

cross

d=dog(xl)

where

b o and

d o are

for

section

(xl)

1970

(12) has four unknowns, Co, C1, M0, V0, boundary conditions, two at each end of a beam.

of Integrals

For a general is chosen:

15, 12

Varying as shown

h(xl)

= 1+

g(xl)

= i+G(-_)

reference

width

Cross

Sections.

in Fig.

3.0-3

the

following

H (-_)

,

and

depth

at

A

= Aoh(xl)

x_

O;x l-

x L

g(xl) ....._,¢

do

I

I Z

I

Y

h(xO

g3(xl)

h3(xl) YO

g(xl)

z 0

= I

b_

Figure Letting

T(x,y,z)

3: 0-3.

the temperature

= f(x l) V(y,z)

General variation

,

cross

section.

bc represented

by

Section

D

October Page the

necessary

integrals

2 T

=

f

agTdA

f

= y

= ag

f(xl)

g(xl)

h(xl)

f

crETzdA

VdA

o

,

= al_f(xl)

g(xl)

h(xl)

f

VzdA

o

VydA

o

Ao

= f z

13

Ao

A

MT

1970

become:

A

MT

15,

crET

dA

A

= crEf(xl)

g(xl)

h(x_)

f

Y

,

Ao

M x J' 0

T

x 1 Ydx=

EI

---E-a

y

x

Ioy

0

0

The

J' 0

_h

(x1)

dx

o! _-zo

=

f VydA A 0

dx 1

o j" 0

xl xdx I_I

1 z

dxl

x 1

x f

V zdA° Ao

MT EI z z

f

f

-

gI

integrals

J

z0

0

necessary

x x l (Ix) h(xl) g°(xl)

to evaluate

. '

x1

,f

dx lg-"-i-"z

0

PT'

MT

' y

particular

cross

section

and

follows :

Let

F o= f

VdAo

,

Ao

Fly

: f

VydAo Ao

,

temperature

distribution

can

1 gI

and

z0

'f 0

MT

z be evaluated

dx 1 h(x 1) g'a(x 1)

for

a as

Section

D

October 15, Page 14

1970

and

/.

F1 z

= J V z dA 0 A0 n

Then, the

letting

V(y,

temperature

be evaluated several values cross and

variation for

of

sections Fly

Fly

when

3.0.2.2

and for

m=

for several

y- and

Simply

z-directions,

0 and

Supported

3.0-1

values

rectangular,

Beam

is a polynomial

Table

n=

of

0-

5.

gives m

3.0-3 values

Beam.

.&--- "

I.

I

= 0@x=0,

Conditions:

L d2v

Mz =

-EIz

_

Vo = Mo = 0

- MT

Table

gives

3.0-2

and

m

since deflection, moment and are similar, only the results are given (i. e., m = O).

A i,

v

can for

gives of

Flz

diamond

values and

of

F0

n.

Examples.

t

Boundary

n.

of

evaluations

elliptic,

¥

A.

Fly , and

these

and

Table

for various

representation

F0,

triangular,

shapes

In the following examples, tions along the y- and z-directions ary value problem in the y-direction I.

which

various

standard

Restrained

,

shapes.

shapes

F 0 and

z

in the

common

common

m

z) = VmnY

=O@x=O,

L

shear equaof the bound-

Section

D

October

15,

Page TABLE

3.0-I. EXPRESSIONS

FOR

F0,

Fly,

AND

FOR

Flz

1970

15

COMMON

SHAPES.

RECTANGULAR

7I •

2

t

! ,

2

_I_ I_

I_

V_T 2

4V

_N=I 2

n+l

,ITImll

(re+l) F 0

m,

(n+l)

n: 0, '2, 4, 6..

=

4V (m+l)

m or

n=l,

re=l,

:3, 5,..

3, 5..

n+2

inn (n+2)

Fly=

4V (m+2) FIz=

n+ 1

m+ 2

mo (n+l)

'

2 ;

,

n=0,

2, 4, 6

Section

D

October Page TABLE

3.0-1.

15, t6

(Continued)

TRIANGULAR

Y

"_d o

Z

=

/I: {°\"1 2

do

2

["m+l n+m+2 2Vmn

(__)

m+l

m: d°n+l

[]

_=1

0,2,4

B.+(-2)L

F 0 = 0

re=l,

3, 5

m=0,

2, 4

where (n+l) ]3.

'.

=

1

(m+2-i)

I

(_

n! I (n+i)

_)n+,

2Vmn

n+m+3 d0n+2

[ - _ +2

Lil

I

3,?

C. + (-2)

'

Cm+2]

m

where

C.

1

(m+l)

=

(m+2-i)

,_+,,, (__)n+,+, _

! '

(n+l+i)

:

0

FI2

_ (_)m+_ 2Vmn

i_l

where

Di

=

_(m+3-i)

!

n'. (n+i)

'

(, _)n+i_

Di+(

1,3,5

1970

ORIGINAL

PAGE

IS

Section

OF POOR QUALITY

October Page

TABLE

3.0-1.

D 1970

15, 17

(Continued)

ELLIPTIC

r

-r 2

2

n n'.

m--TT-\ =,/

\ _!

(_)z

(re+n-l)

n+m-l)

v,

(m+n-:l)

(in+n÷2)(nl+n)

.... ....

(7)

(:-,)

(:_)

(1)

Ill,n ;In(l

(H)(I;)(,I)

IJ,2,.l,(; nI+ll

II

F 0

m

I)]"

II

1,

:;,

5...

m

7rVmn( m+l

\

1_2)n/+l 2

[d /"-_)

\n+2

(')"

(n+l)!

(n+m)(n+m-2)...

(n÷m)_

(n+m+:_

(7)(5)(:_)(t)

.I

o,2,

u

t,:l,5

,-T (m+n+l)...

(_)((;)14)

Fly n

m+

" v,,,.( ,,,'CI+?.(__ ?'_'

¢_z" nl

1,::,S..

l

n_ (n+Ili}'.

FI

0,2,.1,(i

(n+nl)(n+,u-2)... (rn+n+:_)

(7)(5)(::)(1)

(llt_ll÷l)...

m

(x)(I;)(.I)

n

Z

o

I)

l,:t,

5

or

m

11,2,4,1;

1,::,7, O, 2,4

1,t;

Section

D

October Page TABLE

3.0-I.

15,

1970

18

(Continued)

DIAMOND

2

Vmn

mi

n! 4\

2]

\2]

m, n=O, 2, 4..

FO

m orn=l,3,5

4Vmnml(n+l)

4V

!

(m+l)

Flz

=

3, 5..

m=l,

3, 5..

!nl

mn

(n+m+3)

m=l,

!

n=0,

2, 4..

or

n=0,

2, 4

v

SectionD October 15, 1970 Page 19 TABLE 3.0-1.

(Continued)

T-SECTION

b

_t J

Z _-----

r I ¢

0

I w

I'

I

'1

b

2V mn (re+l)

c+w)

(n+l)

2V (re+l)

c n+l]

,+,)_ +'

mn

F 0

n+l-

(n+l)

m

0

1,3,5

n=0,

1, 2,

{(-_--)m+l(an+2-cn+2)+

(-_-b2)m+l[cn+2-

(c+w)n+2

2V

0,2,4,6

]

n

n

(n+2)

m

1,3,5

n

0,1,2,3,4

6

:3, 5..

0,2,4 0,2,4

m:0,

mn

0

m

m

Fly (re+l)

O, 2, 4,

:1

mB (n+2)

O, 2,4,

n

n:l,

2V (m+l)

m

2,

1, :3, 5

4

(i

Section

D

October Page TABLE

3.0-1.

15, 20

(Continued)

I-SECTION

_

[

•o

o-t

t

b

w

"[

1 4 Vm n(_-) )(m+l)

n+

(n+l)

F0 m,

0

4V

m or

(d'_

n

n

:

even

odd

n+2

mn\,-£-] (m+l) Fly

(n+2)

= n+2] 0

NOTE:

z-Section

m

can be :Ipproximated

principal axes.

The

: odd

by I-Section

results above

} or

m=:evcn n

even

with respect

are applicable

to its

to this section.

1970

Section October Page TABLE

3.0-1. HAT

(Continued)

-SECTION

y

d

L+

t

b

•

C

IT

0

F 0

m

1, 3, 5...

2V mn (n+l)(m+l)

{ [(-c)

n+l

o (-c-w)

(_)'n+l[

n+L]

n+l

I(_-

(a_k)n+l]

+ t + i)m÷

l - (_

+ 1)m+

}

o. 2,,1... 3,4. nn, 0,1,'>

m Fly

=

l 1

1,3,5...

o

2V Inn (n+2)

_ m+l)

,, m,,(_5°+, ] n+2 (.__)m+

FIz

-

1 [;

(a_k)r_21

m 0, 4. n 0,1, ,, 2, 3, ,1.

}

m

0

0,2,4...

2V mn (n+l) (m+2)

{ [(-c)

+ [an+l

n+l-

-

{-c-w)n+l]

( -c-wj

n+l]

[(_-+

f{b [3 2

t +p)

+ t)m+2

m+2

-([2 )--

- ( J_-) I m+2

+t)

]

m+2]

. .

21

D 15, 1970

Section

D

October 15, Page 22 TABLE

3.0-1.

(Continued)

CHANNE

L

1

_------

F o

=

d -------_

0.0

n=1,3,5...

2V mn (n+l)

(m+l)

{I(c_w)m÷i-cm+ll

.

(b)n+i+

_

Ccr_l

-(d-c-w)m÷ll,

_

n:0,2,4... m

Fly

=

0,1,2,3,4,5...

n0,2,4...

0.0

2V mR (n+2)

(re+l)

I_c,+w

) m+l

[(2b_) n+2

Flz

=

_cm+l

(b

1

(__)b n+2

+

[cm+l(d_c.w)m+l],

n:1,3,5... m=0, I,2,3,4,5...

t)n+2]}

n-I,3,5...

O. 0

2V mn (n+l)

(m+2)

+ { [(c+w)m+2-

r b ,+I

•

cm+2

b

]

[cm+2

-

(d-c-w)

m+2]

(_-2b) n+l

"+:

]

n

0,2,4...

m:0,1,2,3,4,5...

*

1970

Section D October Page TABLE

3.0-1.

(Continued)

RECTANGULAR

TUBE

Y

T

=

_t 2

F0

=

:

0.0

n

1,3,5...

0.{}

m

J,3,

n

0,2,4...

m

0,2,4..

n

0,2,4...

0.0

Fjy O. 0

4V m n (n+2}

FI z

•

n_2

b

m+

1

(re+l)

0.0

0.o

4 Vmn (n+l)(m+2)

m

1,:LS..

5..

Z

23

15, 1970

Section

D

October Page TABLE

3.0-1.

24

(Concluded)

CIRCULAR

TUBES ¥

Z

b

m=5

F0

:

0.0

n

1,:1,5...

0.0

m

l,:t,5

4_mo[(:_) (b J' .... ' m÷|

I'd

"m+n+z

-

b

rn+n4

2 ]

I_._

-

-

Z{

114 :l)

÷

_ (I',')

;_ )

(m+l_l'n-!)_r_l-:l)

{m÷J)(m-l_{m-:_(m-5

4_(n+7)

)

:IM4(

÷

n

(1+2,4,

m

0,

2,

0.0

n

0,2,4.

0.0

m

n*

,

]

!*)

.

4

,

Fry

-

'Vmn m+n+:'m+,

[(_-)

-\-_](b_

....

Ill ljL

n'-_z

-

_!

•

(re+l)_{

(n*¢l)(m-l)_n_-?:l

+

1,3,5

n*,i){ n,-l)

im*I)(_l_l)lm-:_}ln_-,%l

4_(n+:_)

n m

FIz

=

/

:l_4(n+

O. 0

n

fl.O

ell

1,:_,5

10)

.

]

.

_L2.4

1,

II,

:J, 5...

Z,

4

'"_,,[cor ''_ c,,_ ....'][, o,._,.,,-,,,, -

(*nlC2)

(In¢2)(rni(_ll-_)itln-4)

illQ{in-2) l_(n+7)

+

;g_4l

n m

II+Z.4.., 1,

:_, r,

n*!O

/ --J

15,

1970

Section

D

October TABLE

3.0-2.

VALUES COMMON

RECTANGU

OF

Fo

AND

FOR

Fly

1970

t5,

Page FOUR

25

SHAPES

LAR

TRIANGULAR

2 Z

I

% 2

b_o I-

._

-:

2

2

b0

m=O

Fo

Fly 0

bodo Voo 1

12 1 bd

"_

3

2

3"

m--O

2

_I -I

o oVo2

bod _ Vol

Fo

n 0

2

I bod_ Vo3

3

0

4

0

0

1

0

Fry

1 -_- bod o Voo

1

1

27O

1

3 b0do '_ o2

bod_

1

4

27"-_ b°d° v°3

4

bodo Vo3

1 270

bod_ Vo 3

2

270

2 7(243)

Vol

bod _ Vo4

bod6o

7(243) 31

V°5

6

b°d° b dTv

o(729) o o

V°4

Section D October

15, 1970

Page 26 TABLE

3.0-2.

(Concluded)

E LLIPTIC

DIA MO ND

¥ J_

Z

= Z

r

m

2

p

2

2

m=0

m--O

n

Fo

0

lr bod0 Voo

Fly

n

0

o

32

2

_

I 7r b_

3 4 5

b_ 0

o

Vo2

12-_bod_Vo3

6 I-_

V°I

o

V°4 15 _r 32(256)

bod_ V05

1

2

3

4 5

Fo

Fry

I bodo Voo

0

0 1 3 4--8 b0d0 v0l

I bod_Vo_ 4-_" 0

o 1 bod_ Vo 3 48--'6"

t__ bod_Vo4 480 0

o 1 28(i20)

b "_ _oVo5

Section

D

October Page TABLE

3.0-3

VALUES

•

OF

COMMON

*----b

F o AND

f

f

Vy dA o

d!+ L

I

!

VdAo

Fly

-

V

1 mll

1

3

5

7

0

0.207

0.121

0.093

i0.084

0.004

2

0.030

0.011

0.004

0.001

0.006

0.002

4

0.018

0.006

0.002

0.001

0.004

0.002

6

0.013

0.004

0.002

0.001

3

5

7

0

2

4

0

0.531

0.207

0.121

0.093

2

0.084

0.030

0.011

4

0.050:0.018

6

0.036

0.013

FOR

= 2.00----_

I/ 1 mn

F1 z

,I

|

V

27

SECTIONS.

!

F°-

AND

Fly

15,

6

--b

= 2.00-----_

!

i

o+t

t 1

d = 3.75

o.,f,1/

I

I

t m_

6

m_

1

1.379

3.117

0

0.784

1.379

3. 117

8. 152

0.079

0.075

0.073

2

0.079

0.075

0.073

0.076

0.050

0.047

0.044

0.042

4

0.047

0.044

0.042

0.039

0.036

0.034

0.032

0.030

6

0.034

0.032

0.030

0.028

0

2

0

0.719

0.784

2

0.084

4 6

4

1970

Section D October 15, 1970 Page 28 TABLE 3.0-3.

_'--

3.00

(Continued)

---_

|

|

3.

0.1 |

|

!

1 F o-

V 1

fVdA

o

m_

0

2

f

Fly:

mn

Vy dA o

mn

4

6

m_

1

3

5

7

0

1.043

1.085

1.731

3. 603

0

1.085

1.731

3. 603

8. 892

2

0.352

0.326

0.303

0.285

2

0.326

0.303

0.285

0.277

4

0.474

0.438

0.405

0.376

4

0.438

0.405

0.376

0.350

6

0.762

0.704

0.652

0.605

6

0.704

0.652

0.605

0.563

3

5

7

-I

3.5

------_ !

I l

3.5 0. L

m_

0

0

t

|

m_

1

2

4

6

t.750

1. 663

2. 323

4. 198

0

1.663

2. 323

4.198

9.096

2

0.898

0.791

0.705

0.639

2

0.791

0.705

0.639

0.600

4

1.641

1.445

1.279

1.139

4

1.445

1.279

11.139

1.021

6

3. 590

3. 160

2. 798

2. 492

6

3. 160

2. 798

2. 492

2. 232

Section

D 15, 1970

October Page

TABLE 3.0-3.

_----

4.50

29

(Continued)

-_

o.. Lp 1

4.624

5"11

|

|

F0 :

t

1 $ V dA o

_-mn

Fly

VydAo

3

5

7

0

2.774

6.745

21.551

82.620

4.416

2

2.072

3.014

4.416

6.584

13.245

4

6.279

9.110

13.245

19.295

47.892

6

22.705

32.942

47.892

69.767

4

0

1.715

2.774

6.745

21.551

2

1.430

2.072

3.014

4

4.336

6.279

9.110

22.705

J

1

2

15.681

1 Inn

0

6

-- V

32.942

6

---- 5.00 ----_ !

#__

I 1

4.376

1

!

I 0

2

4

0

2. 925

4.325

9. 228

2

3. 261

4.488

6. 237

6

1

3

5

533

0

4. 325

9. 228

8. 783

2

4. 488

6.237

4

16.766

23.199

32.289

45.199

74.845

103.564

144.135

201.741

25.

4

12.

188

16. 766

23.

199

32.

289

6

54.

408

74. 845

103.

564

144.

135

25.

533

7

5. 783

85. 468 12.

697

Section

D

October

15,

1970 J

Page TABLE

3.0-3.

30

(Continued)

t 0.923

_

.,---

t

0.125

0.125

0.202

__t.

.

_ 1.5

1

1

_1: V-

go= W- fVaAo mn 0

1 0.001

fVydAo

inn

2 0.046

3

m_

0.018

0

-0.009!0.003

-0.001

0

2

-0.009

0.003

-0.001

0

0

4

0.012

-0.003i0.001

0

0

4

-0.003

0.001

0

0

0

6

0.005

-0.001

0

0

6

-0.001

0

0

0

0

1

2

3

4

0.309

__k _----

m_

0

1

2

3

4

_

0.019

0.018

4

0.035

----,.

0.046

3

2

1.191

0.001

2

0.328

I

0

1

0

0

0.019

4

0.013

0.125

0.125

.. ! 1.75

-_

m_

0

0

0.406

0.001

0.102

0.051

0.064

0

0.001

0.102

2

0.056

-0.021

0.008

-0.003

0.001

2

-0.021

0.008

-0.003

0.001

0

4 0.026

-0.010

0.004

-0.001

0.001

4

-0.010

0.004

-0.001

0.001

0

6 0.014

-0.005

0.002

-0.001

6

-0.005

0.002

-0.001

0

0.051!0.064

0.058

0

0

Section

D

October Page TABLE

3.0-3.

1970

15, 31

(Continued)

1 -.---

O.788

t

0.125

0.125

0.087 |

|

2.oot---_

1

1

_ V__

F°-

fVdA o

F 1 -

f

V

mn

m_

0 0

_.359

2

D. 083

4

D.050-0.

6

0. 036

1

2

0

4

0. 001

1 0.026

2

3

4

0.011_0.008

0.005

0.011

0.008

0

020

0

0

2

-0.

012

0.020

0

0

0

007

0. 001

0

0

4

-0.

007

0.001

0

0

0

005

0. 001

0

0

6

-0.

005

0.001

0

0

0

0.001 -0.0

-0.

3

Vy dA 0

mn

0.026 12i0.

1

.,,--

0.985

t

0.156

0.156

0.109

I _'-"--

m_

2.50

m_

0

1

2

3

0

0.002

0.064

0.034

0.030

0.024

0

2

-0.038

0.008-0.001

0

0

-0.001

0

4

-0.036

_.007-0.001

0

0

-0.002

0

6

-0.040

D.008

0

0

0

1

2

3

0

0.561

0.002

2

0.203

-0.038

0.008-0.001

4

0.190

-0.036

0.007

6

0.213

-0.040

0.008

0.064

4

0.034!0.030

-0.002

4

Section D October 15, 1970 Page32 TABLE 3.0-3.

(Continued)

1 1.544

......4 _

0.156

t 0.30

0.156

....L

F°-

V1

fVdA

1

IJ

°

F 1 = _--

mn

m_

0

1

i/ Vy

dA o

mn

2

0.002!0.261

3

4

m_

0

0.195

0.284

0

0.002

-0.019

0.008

2

1

2

3

4

0.261

0.195

0.284

0.348

-0.132

0.051

-0.019

0.008

-0.002

0

0.756

2

0.352

-0.132

0.051

4

0.474

-0.179

D. 069-0.027

0.011

4

-0.179

0.069

-0.027

0.011

-0.004

6

0.762

-0.288

0.110

0.017

6

-0.288i

0. II0 -0.043

0.017

-0.007

-0.043

I

--

2.292

t

',,--.-0.188

0.188

0.52

_t_,

.. ]

0

1

2

3

4

m_

0

1.281

0.007

1.049

1.115

2. 492

0

2

1.004

-0.614

0.383

-0.233

0.156

2

4

2.406

-1.478

0.914

-0.570

0.358

6

6.875

-4.222

2.612

-1.629

1.023

m_

,

o

1

2

3

4

1.049

1.115

2.492

4.471

-0.614

0.383

-0.233

0.156

-0.081

4

-1.478

0.914

-0.570

0.3581-0.226

6

-4.222

2.612

-1.629

1.023

O. 007

-0.647

OEiGINLL

P#.C_

15

OF. POOR

QUALITY

Section

D

October

15,

Page TABLE

3.0-3.

0.125

1970

33

(Continued)

]

olYi 0._25

L--1.75J

_-1.0]

IllI] 0 0

0.656

1 0.7:}4

1

0.0

2

O. 804

3

O. 0

2

x

10 -3

0.55

0.0

3

_ 10 -I

0.165

0. [)

-0.

152

×

10 -2

×

10 -1

_

10

O. 559

[}. l)

0.659

O. 0

4

x 10 -1

-[).

[I. l}

O. 172

5

0.0

101

×

-0.

449

0.14

0.0

1

f

Vy

:,: 10 -2

-0.

10 -!

-0.

O. 0

409

0.

IL l}

If.I)

0.34

x

10 -3

0.0

0.(;27

-I 4

10 -2

0.0

135

[I. (} -0.

x

5

125

x

10 -2

_

10

O. 0

1,1 /

-2

O. 0

393

0. 0

(bX o

121n

O. 55

0

0. 734

1

O. 0

O. 0

2

-(}. 152

0.66

3

0, 0

O. 0

×

10 -3

y

10-

×

10 -1

U. 1(;5

!

,(

-2

10 -2

o.

x lO

I).559 0.0

0.0

34

1 u -3

x

l)._;IZ

O. 11

-

10

-3

0.0 -2

4 5

-0.

449

-0.

O. 627

x: 10 -1

(}. (}

O. 1,Is

0.0

135

-0.

0.0

x

10 -j

-t}.

0. (}

409

x

10 -I

0.

0.0

125

x

,0.

1(}

0.0

14

x

10 -t

-0.

0.0

(;;_K x

10 -3

0.1) 393

x

10 -z

0.13U

0.0

10 -z

x

0.0

1 .I' Vz

FI A

(b% o

Ii'(n 0 0

1

0.0

1

0.

2

0.0

3

0.

4

0.0

5

0.

2 O. 0

O. 0 G39

-0.

0.5S1

155

0.0 158

×

1() 1

0.

-0.451

O.

O. f} 455

x

101

-0.

134

3

0 141

(1.0 x 101

0.40(i

4

O. 0 x

10 -I

-0.

138

O. 0 × 11) -1

0. 0 -0.41:1

O. 561

(}. 0 <

10 -2

O. 0 •

111-1

0.

131

123

-0.

13

10 -2

0.0 -t < 10

O, 0

0.0 -(}.

5

O, :I,_|,_ 111-1

-0.397

z 10 -2

(1.0 -0.

11!1

_ 10

-1

Section

D

October Page TABLE

3.0-3.

1970

15, 34

(Continued) Y

Fo = _-L

fvdAo

mn 0

1

0

0.972

-0.901

1

0.0

0.0

2

1.

-0.534

3

0.0

0.0

4

O. 209

5

0.0

X 101

-0.

2

× 10 -z

0.603 0.0

0.0 x 101

× 101

0.161

0.517

0.289

0.0

0.0

0.0

0.352

-0.121

0.0

-0.838

0.0

Fly=

5

0.0

O. 118

0.0

4

-0.273

0.554

148

3

0.0

0.094

0.0

-0.476

0.0

0.0

fVydAo

_n_n

0 0 1 2 3 4 5

-0.

901

1 × 10 -2

0.0 -0.

534

× 101

0.0

4

5

0. 517

0. 289

0. 514

0.0

0.0

0.0

0.0

0.0

-0.

0.0 148

3

0. 161

O. 554

0.0 -0.

2

O. 603

273

0.0

0. 118

× 101

0.0

-0.

838

0.0

0.0

1

0.82

-0.565

2

0.0

0,0

3

0.2×101

4

0.0

5

0.572×101

-0.140× 0.0 -0.431×101

1 fVz mn

0.0

0.0

3

0.0

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D 15, 1970

October Page TABLE

3.0-3.

35

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October

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3.0-3.

1970

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36

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Section

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October 15, Page 38 3.0-3.

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15, 41

1970

Section

D

October Page TABLE

3.0-3.

(Continued) y

0.37!

1 ".'_----0.375

F, v'--:v o mn

0 0

0.506×

1

0, 0

2

0,973x

1 10 -t

10

-3

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0.0

0.0

0.0

0. 142

3

0.0

0.0

0.0

4

O. 256 x 10 -4

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0.339×

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0.0

0.0

0.0

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m_

0

1

3 l0

× 10 -4

10 -_'

Vfi--'- f qln

0.0

0..q73x

2

I

0.0

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0.0

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0.0

0.0

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0.0

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5

0.0

0.0

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10 -4

10 -6

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0

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0.o

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0.0

15, 42

1970

Section

D

October Page TABLE

3.0-3.

(Continued)

y

1.0

_Z

L I

0.049 ---,--

1.0

,I

1

Fo

V-- Iv"a° rll n

0 0

I).

1_(;

1

O.

()

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0.2x2

:t

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4

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2

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0,

0.2x2x I)

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5

4

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I).577×

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t1, I1

l).(l

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(i. (I

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15, 43

1970

Section

D

October Page TABLE

3.0-3.

15,

1970

44

(Continued) Y

i 1

1 1 0.095

2.0

0.095----

2.0

mn

m_

0

1

0

0.

724

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|

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0.0

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3

0.

439

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o.

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lO

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0

OF POO;:i 'e_:,'-&i ;'.... _ .... IY

Section

D

October Page TABLE

3.0-3.

Fo

(Continued)

_,

/ V d.",o Hin 3

2 0

0.

[

O, 0

177

2

O, 24

:l

0,

,I

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15, 45

1970

Section

D

October Page TABLE

3.0-3.

(Continued)

V 0.277

_Z

1 1

Fo

_

.j VdAo mn

1

0

2

0.508x

1

0.0

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10 -4

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l0 -5

10 =7

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0.0

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0.0

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1

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15, 46

1970

Section

D

October Page TABLE

3.0-3.

(Continued) ¥

T1.0

=Z

L =

1.5

t

()

1

•

[ V(IA0

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4

:l

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15, 47

1970

Section

D

October Page TABLE

3.0-3.

(Continued) y

2.75

L F 0

m_

0

0

0.114x

1

0.0

2

0.116x

3

0.0

4

0. 183

5

0.0

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0.0

0.122×

0

0

0.0

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0.0

0, I;47

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0.0

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0. 597

0.0

0. 744

0.0

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15, 48

1970

OR,_,I,o,....

PAGE

IS

Section D

OF POOR QUALITY

October Page 49

TABLE

3.0-3.

(Continued)

f

5.844

_ z

F o

1_ V

V dA

j

o

llln 0 0

O.

147

I

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0

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15, 1970

Section

D

October Page TABLE

3.0-3.

(Continued) ¥

z

-------6.0------*

_--! mn

Fo=

0 0

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I

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0.0

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0.282×

0.0

0.0

0.0

0. 293

0.0

0.0

0.0

0.918×

0.0

0.0

5 I03

0.0 0.0

× 103

0.0 0.0

103

O. 0 O. 0

l Fry

_

/

VydAo

mfl

0 0

1

0.0

O. 533

2 x 102

I

0,0

0.0

2

0. o

O. 851 x

3

0.o

0.0

4

o. 0

0.317

5

0.0

0.0

x

102

103

Flz o 0

o.0

l

O. 508

2

0.0

3

O. 262

4

o.o

5

o. 154

0.0

0.2X2x

0.0

0.0

0.0

0.293x

0.0

102

× 103

x lO 4

5

0.0

O. 173 × 104 O. 0

O. 0

O. 12:) x

O. 0

0.0

0.0

0.0

0.918 x i()3

0.0

O. :175 x 104

0.0

0.0

0.0

O. 0

_ 1 mn

_

103

4

0,0 I03

104

•j VzdAo

2

I

x

3

3

4

5

0.0

0.0

0.0

0.0

0.0

0._51 × 10 z

0.0

0.29,3×

0. o

0. o

o.0

O. o

O. 0

o.o

o.317×

o.o

O. 91_ × 10 7

0.0

0.0

0. o

o.0

0.0

0.0

0.0

0.133×

0,0

0.25×

1o 7

Io 4

0.0 10 3

10 4

0.0

0.0

15, 50

1970

Section D October Page TABLE

3.0-3.

(Concluded) ¥

/

I11n 0

i

0

O. 934

i

O. 0

2

0.159x

3

O. 0

4

0.414w

5

O. 0

1 ×

101

103

10 '_

2

O. 0

O. 166

O. 0

O. 0

0.0

0.135

O. 0

O. 0

0.0

0.244

O. 0

O. 0

Fly

3

×

10 a

×

1() 4

×

] ,

V

105

VytlA '

4

O. (1

O. 446

O. 0

O. 0

0.0

O. 225

O. 0

O. 0

0.0

O. :i32

O. 0

O. 0

5

x

104

x

105

0.0 O. 0 O. 0 0.0

106

x

O. 0 I). 0

o

Inn m

\

0

1

0

O. 0

O. lt;G

1

O. 0

O. 0

2

0.0

O. 135

3

O. 0

O. 0

4

O. 0

O. 244

5

O. 0

O. 0

2

×

|0 3

×

104

×

105

F1 z

3

O. 0

O. 441;

O. 0

(_. 0

O. 0

O. 225

O. 0

O. 0

0.0

0.:]:12

O. 0

O. 0

V1

./

Vz

,t x

104

"_" 105

,_ 10 I;

5

O. [1

O.l:_3x

0.0

I). II

O. 0

O. 465

O. 0

O. 0

O. 0

o.

O. 0

O. 0

619

1(I _:

×

10 +;

x

107

dA o

n/n 0 [I

I). 0

I

0.159×

2

O. 0

3

O. 414

4

0.0

5

O. 118

2

1

103

x

104

x

10 _

I). 0

O. (I

0.0

0.135

(}. 0

O. 0

0.0

0.244x

0.0

0.0

0.0

0.4_1

×

3

104

105

x

lU G

4

O. [)

0.0

0.11

0.225

O. 0

0.0

0.(}

0.3:]2

0.0

0.0

0.0

O. 412

5 O. 0

_

105

0.0 O. [I

×

106

×

107

O. 0 O. (I O. 0

51

15, 1970

Section

D

October Page B.

52

Results:

x v(x)

=-f0

v(x)

="_

I1=

fl fl

x2MT f0

(xl)

EIz

L

z(xl) ax,dx ÷±f L

(_-II-IIx)L

Io z

0

x 2 MT

(xl) Z

f0

dx 1 dx 2

z

2

where X

0

u

_v

dxldx

0

(x)

M (x)

2 and

Ilx=

f

f 0

g'(Xl)

1 ?1

f

:_-0_A

o_ ET dA dx =

X 2

0

_ g (xl)

dxldx

xj_ f(xl) 0

a F 0L A°

2

dx 1

=0

Z

MT

PT a XX

= -nET+

-_

+ My Y I

+

+

y

Y

M Y

may

or may If end

B

not be zero, is hinged,

Uav(X) = 0 V0 = M 0 = 0

v(x)

a

= same

depending

=

the

boundary

, ,

as above,

+

Iy force

upon

then

= -nET+

axial

i5,

P = /" c_ ETdA A

\

Y

_

condition.

1970

Section

D

1 April, Page II.

Fixed-Fixed

1972 53

Beam.

¥

B

-t A.

Boundary

Conditions:

dv V

--

:.: O,

dx

(or

X

x=

O,L,

X 2

MTz(Xl) 0

+ EIM0(xl) z + x 1 V°z I

dx 1 dx2

Z

0

I Z0

where x2

x

dx dx 2

0

I2 :

j l jx2 h(_1)dx_a(x,)_ 0

0

0

X2

I3x = f

J 0

h(_,) g_(×,)

'

o o

0

13j.1 _ g_(xl) 0

dx,2

x I dx_ -I_ j' 1 h(x,) _ (xl) 0

F 2 -

xl ) dxg3(xl) i 12 fl h(xl -13 fl h(x,) dxg_(xl) ,, 0 0

x dx h(xl)

g3(x_)

dx2

Section

D

October Page

l

II

- 12 f!

dxl h(Xl ) g_(xO

0

f(xl)

F3I2/

h(Xl)xldXtg _'xA_11 - I3

0

(Table

3.0-4

gives

M0z

c_ E F 1 F 2

Voz

o_ --_

=

Mz(X

values

for

F 3)

) = MOz + x Voz

"Jxl A°

B

is

Uav(X)

restrained

= 0

f(x 1) dXl

0

against

= -a

xx

longitudinal

motion,

ET

+

(.) (.M) My

Y I

MT

z ÷

y

v(x),

CASE

a:

EI

Mo, Vo

z

v (__)

(x)

-

are

same

c_ F 1 x 2

I

2 _ E Fl(q-1) (q+l) (q+2)

z

z

y

I

z

as above.

= constant,

Z

M° z -

then

,

MT

cr

and

g_(xl)

E F 1 F3

•

end

0

dx 1

t_(xO

,

Uav(X)= _

If the

F2

f'

dxl

gZ(xO

0

+n

15, 54

1970

L_

ORIGINI_.L PZ._E OF

POOR

{3 Section

QUALITY

D

October Page

TABLE 3• 0-4. II

;

VALUES

OF

CONSTANTS

q

_ I

II

,

r--,

{_

F2,

F3,

s

II

T__-_-X-+-F--,.,

]+-;.,_

-,+

_,

:1,,

-

i .... _L::+-:

' "+"+++

't

14

t

,

-

+1

_, ;1'

t '+_:1 ....

I

A ND

-

_.

_

q:i

,_

'

1,.

" ,,

;

I

h :'

t

+

i

1_

w) :__b_

-I

,hl

o

I_

1

q

(; +,

o

rl

J

L;

+, ,,,,+

.

I,I

I

+J

i,

u

l

l++

, I

.

Ii

1+,,_ :+1;

,+.l_

......

I

.----+u

u_ ut

w)

i

ii

41

I

H

, _ I

I

++ ,

<+ .:t,

+j +

+l

<+ r'r

i+ : + '

:H.

.

m_l

+

ilL+

ii

II'+_

=IF

-I I

lt_

-I

l,,,

.........

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

"_

+_+,

.

.'i+,

t

_

'I'+

i

-

;1, +

i

I

'+J

_

......... }L+:I

t

:,,,i

t _

I

:!++

i

h(,,,): ,.+,-,(4

,l

,1,i

-I

I,

r,,l.

J ,+uJ,

I

,

,+

i,

t

_.,1

_1 ZT:<

111

. ........ LI +

,

+I _

WJ

,

+

;I

.it

.

t

h

_

+ -i

,+

,F

I,

l

+;

1,,

I

,,,'+

_+

-(_

I

I

U

+ .

+

o

I

.............. ......i :+, I

,_

H,,

-+I

II+l

0

If;

I

H I

[

+1

. ........

p

:,

I I

.... = i

.

I

/'L

,, Ir,t

,'

+,

+

I

l

I

,11'1

+-'"

-+ ..... +.... LI

I

+-

'2Z'_

,l+

......... -+'''

'+

,, .+_"

-i

,+,

1

i,

+_

-I

7!,

1

+

:,;.. _

,

'_'

-.

+

++'

............ -..',-

I

-,+

,

,

.

IF I

_l i,i

lh_IPll,,+h,,,ll

+--

N i<,,

' "+; _,....+"' -I

,,I+,(

.....

t

Tit

t'+ :

........ o

-.

---

('

o,

Notes :

lc

,,

4<

--

+

ii

_.

,

-if+,.

I

'+'

: 1

.

[ i

I

?

_ [ ,H<, f---i I

+i

l'''r

'+

.

.: %.,.

.

i

55 F4

15, 1970

Section

D

October Page 6_EF1 q (q+l) (q+2)L

V°z = -

M

=

2_EF| (q+1) (q+2)

z

CASE

F(i -

b: MT

(xl) Z

EI

=

(xl)

Z

v(x)

= o

M0z

=

constant

,

-M T

, Z

V0z

0

Mz

=

-M T

= constant. Z

III. Fixed-Hin_d

Beam.

Y

t

B

A

Id

"l

A.

Boundary

Conditions:

dv

v-

dx-0,

v=

d2v -_x

@ x=0 MT +

z

EI

-0, Z

(_x=L

15, 56

1970

Section

D

October Page

v

= -

I

L2[ ( Ix 1+

z0

F4

)]

Ix 2 -Ix

3

where

F4 =

I1 (I3 - IJ

= _

_

V°z

M0z

(refer

to Table

E F_ F 4 L

3.0-4

for

values

of

'

= c_ E F 1 F 4 x

FQ L

{-I j 0

=

Uav(X)

A°

(r xx

=-a

ET+

PT -_

f(xl)

dx I

z +

y y+

I

I

z

If the

end

of

B

Uav(X)

is

: 0

hinged,

, MT

:-aET+ xx

z z

I

a:

EI

= constant Z

T Y+

z

CASE

y

My Y

z

I y

F4) ,

15, 57

1970

Section

D

October 15, Page 58

+ t Io z

3_ EF 1 (q+l) (q+2)

V°z =

M

--

'

-

z

CASE

>3t}

-

L

3_EFI (q+l) (q+2)

M°z -

(q+l)

(q+2)

b: MT Z

(x)

EI

= constant

,

Z

v(x)

aftI

=

_xZ

(

1--y x)

Z

3 o_EF t L

V°z =

MOz = -

IV. the

_EF1

_ (,-_) o_,

Z

From

3 __2

Deflection previous A.

Plots.

cases

Simply

v<_,: _

I0 z

the

Supported

deflection

expressions

Beam:

(_ _,-_,x)

1970

were

found

to be

Section

D

October Page B.

Fixed-Fixed

v(x)

:-

These

F'L2 (Ilx+

F212x+

Fixed-Hinged

expressions

depth, made

59

Beam:

I0z

C.

1970

15,

have

F313x)

Beam:

been

and degree of thermal in nondimensional form

plotted

for

gradient with tile

parameter

variations

along the following

(ff beam

beam length. designation:

The

width,

plots

are

and

FF:_-

Therefore FS,

(Ilx

the

or

FF

+ F212x

deflection,

+ F313x

)

v(x),

for

any

3.0-5

show

the

case

can

be

found

by multiplyinK,

SS,

by Ia z

Figures function

of lengthwise

stant the

3.0-4

cross

of the

and

variation x=

FF

N

variation

gradient.

equals

the

N=

0

In the

exponent

means

of

FF

and

figure,

of the

constant

FS G

thermal

as

a

II _- 0 (convariation

variation,

N

along 1

means

etc. 3.0-6 for

in width

O) (refer

and

beam;e.g.,

variation, Figures

FS,

temperature

section)

leng_h

linear

and

to

through

variation of beam Paragraph

3.0-13

show

of parameters along

the

3.0.3.1).

length,

the

three

G,

It, G

deflection and

equals

N, variation

parameters where

II in depth

SS, equals at

Section D October Page

15, 1970

60

ORIGINAL PAGE ;_ OF POOR QUALITY

,ql

[*I It

O

J:

,.

* .OO| --4---4

.--t--i * .O01

i •

s i

_:,

1

,

'L'k i

I

I ,

[ i i

* o00ql

:

--

I

:

----_--i

° ._1'

• _J I i

'

' .

I i

.

÷

i

N-2 N=3 N=4 N'5 N=6 N=7 N=8

i

•

I

" ._

r--*

*

'

.

.

.........

:'.,4

l

jk

" 4"-- f

Figure

3.0-4.

Deflection parameter variation of lengthwise

FF versus temperature

distance along beam gradient.

for

Section

D

October Page ORIGI_,7_.

F,-,_

OF POOR

QUALITY

15,

1970

61

_

.01_4

.ors

Figure

3.0-5.

Deflection parameter variation of lengthwise

FS versus temperature

distance gradient.

along

beam

for

Section

D

October

15,

Page

1970

62

095

g_$

,n,8

* 0 +

FF FS SS

P L o T

_iiiii

0 F

_i

ill

s $ f S *0t F t _-

%t--t-

=

P4 i

•

l

.ll

O.I

Figure

3.0-6.

Values for

of SS, H=0,

FS, G=

FF 0,

versus N= 1.

values

of

T(x/L)

1,0

Section D October

15, 1970

Page 63

OF

POOR

F_L!,_LI, V"

f---

{)[}4

I

i

I

III

[IA_I /llll

ql ....

!!!!

r

I VI

.o$

i

!

004

:

)

t

I

t

I

1

i

iiii iiii

iJJi

!

iii

P k 0

....

T

0

d!!!

....

i :

F

$

.OZ

q

$

F 4

*-

4-

-4---

+ S -+

-i

r F

&

o

-+

N

:

:

:

:

:

..

+

•

:

.0% :

I --+

+--

-t-

+

:

:

1,/ !/

:

441

'r

Z

_-

it

i ioi,

loi,

4-.

If'l--'

l

_

+

_

_

i

V_LUE$

Figure 3.0-7.

OK

T

Values of SS, FS, FF versus values of T for II = 0, G-_ 0, N= 2.

Section

D

October Page

15,

1970

64

_5 _ Z _5 9_J6 996

VALU(S

Figure

3.0-8.

Values for

of H=

OF

SS, 0,

T

FS, G=

0.5,

FF

versus N=

1.

values

of

T

Section

D

October Page ORIGINAL

Ps,,:._. • '_':: _:3

OF POOR

QUALETY

15,

1970

65

_., I ? %q

J

I

I ] i

'I

/ K

.DO4

.oolt

.0 Q .....

t

j

i

i - .004

Figure

:_.0-9.

Values of SS, FS, FF versus for tI = 0, G- 0.5, N=: 2.

values

of

T

Section October Page

D 15, 1970

66

_9Z2_

.084

.D|I

.Q$4

.850

.OO4

.0051

.OO4

V*LU(S

Figure

3.0-10.

Values for

H=

of SS, 0,

G=

(:_

I

FS, 1.0,

FF N=

versus i.

values

of T

Section

D

October Page

ORIGINAL

P:_,',;_. ""_:

OF POOR

QUALET't'

15,

1970

(;7

{L ....

919

VALUe'S*

Figure

3.0-11.

Values for

of It :

T

FS,

SS, O,

OF

G

1.0,

FF

versus N =- 2.

vulues

of

T

919

Section

D

October Page

15,

1970

68

3(_1Z_

i

(

,

b,,

__

-"

.Ol4

J

!

/

i

i

FI

\

-\

[

:

i

\

LLi- ....

--j

.Oil

iii

i

!ii

i

/!

!II

I

,_

iii .OIO

[

i

/

i

i

i

[

!

i

I

/ r _

J

\

f" /-

\

\

/

.0OQ

1 i

i

\

i i [

I

\ ii .OOq

i

i i [ ______._ ._ _ i

:

iii !_

ii i!

_-II ,OQ(

:!

\ (

;

\

,*

\

I

1

\

-t-"

A

/ f

m__ g

/

J Liii.-_- l/i

!,

r

F._-. !

r-iF J'-t-- ,2... ,-iW/

_.::

-t- " "

-4

-I

on

_" '--t _ ........... _ ._'_,l,,d-- _ _

"_

-___

......

_

i

! ......

i

q:: .,I

i :

I

1_

I i

j

Figure

3.0-12.

Values of SS, FS, FF versus for H= 0, G= 2.0, N= 2.

values

of

T

I I )

Section

D

October Page

15,

1970

69

ORIGINAt -OF POOR PAQ_ IS QUALITy

_33

°O0(

.001

o0

- .001

Figure

3.0-13.

Values for

of H=

0,

VALU[$

Of

SS,

FS, G=

I

2.0,

FF

versus N=

2.

values

of

T

'51)

Section

D

October Page 3.0.2.3

Representation

of Temperature

A temperature approximated

profile

N

M

n=l

m=l

T=

V

Accuracy

uniquely

is equal

to

coefficients

al,

ll

a2,11

a

y

of the

om z +T

approximation

of the polynomial; in the process

of the resulting number of grid

The

VIN

mn

of this

number of terms is to be integrated accuracy The total

obtained

by a polynomial

N, 11

1970

can

be

by Polynomial.

analytically

or experimentally

form

O

increases

with

the

increase

in the

but, since the temperature distribution of obtaining the stress distribution, the

distribution using a low-order points required to determine

the

polynomial coefficients

improves. V

nl n

(M + N). V

are

mn

al , 12 al,

obtained

13" " . a 1,1N

a2,12a2,13"

a

Gradient

15, 70

from

the

a 1,21 .....

" " a2,1Na2,21

N, 12 ....................

.....

following

relations.

a 1, MN

T I-T0

a2, MN

T2-T

•

aN, MN

0

TN-T o

V21 V22

V2N

VMN

a(M+N)

II a(M+N)

12 ..........

a(M+N)

MN

M+N)-T0

Section

D

October n

where point

a. 1, mn = Yi

15,

1970

Page

71

of the

ith grid

rn

z.1

at temperature

The T.

1

values

and

(Yi' z.)t = coordinates

T o = temperature

at the

centroid

of the

cross

section. If the about

the

temperature

z-axis,

then

and antisymmetric differences and

difference Vml=

Vm2

that

are

If the

Vm3

= Vm4

symmetric

if it is antisymmetrie

about

temperature

along the y-direction reduces to

= Vm5

= Vm6 about the

varies only)

(T.-T0) t

along

is continuous

= . . . = 0,

= . . . 0. the

Vln

V2n

only

in one

either

the

and

V4n

direction y- or z-axis,

symmetric

if it is continuous

Similarly,

y-axis,

y-axis,

and

for

temperature

= V:I n = Vsn = . . . 0, _

V6n

:

(e.g., the

for

.

•

•

_ 0.

w_riation

polynomial

N

)2 V y n

T-T0=

n--1

"all

V1

n

a12

......

aln a2n

V2

a21

a22

......

Vn

anl

an2

......

ann

/ T1-T

0

T 2.T 0

T

n

-T

0

n

aln

Yi

As before, if the temperature difference is continuous V l= V 3= V 5= . . . 0; and for continuous antisymmetrictemperature ferences V2 = V4 = V6= • • • 0.

distance

If the dl,

grid points are equally spaced on the cross section, then the polynomial can be written in nondimensional

T 1 - TO=

.

n=l

V

di n n

dI

:md symmetric, dif-

say at form

as

Section

D

October Page

15,

1970

72

EXAMPLES: 1.

Continuous

Symmetric a.

Y

Nine

Temperature Points:

T4

n

T. -To= l

( Vn di n)

(i)

n=2, 4, 6

2 -1

T_-_

%,o!

V2 dt 2

12

V 4 dl4

22 24 2 e 28

V6 di e

32

3 4 36

V 8 di s

42

44

_,

V 2 dl 2

O. 16000+01

V 4 dl4

-0. 67778+00

V 6 die

0. 80556-01

-0.20000+00

14

16 18"

T 1 - TO T2 - TO

38

4648"

0.25397-01

T3

To

T4

To

-0.17857-02" TI-To 1

0.23472+00

-0.33333-01

0.24306-02 T2-To

.-0.

V 8 dis b.

-0.36111-01

27778-02

Seven

0.13889-02

0.83333-02

/

-0.69444-03

-0.39683-03

0.49603-04

Points:

,v2,21 [iil oooloo olo.11111_o 1oo T2-T o

V4dl41

IV

=

dl eJ

c.

-

54167+00

41667-01

Five

0.16667+00

-0.16667-01

-0.13889-01

0.27778-02J

Points: [TI-To

:

T2-T

TI-T o T3-T o

Section

D

October

I

Page 2.

Continuous

Antisymmetric

a.

Nine

15,

1970

73

Temperature

Points:

T4

-I

1

13

15

17-

Ti-T o

2

23

25

27

T2-T

V s dl s

3

33

35

37

T3-T o

V 7 dl 7

4

43

45 47

V1 dl -l-d1

/ 0

V3 dl3 / =

!-

T4-T

o

#--

0.15003+01

-0. 30031+00

0.33466-01

-0.22126-04"

TI-To

-0.54209+00

0.33375+00

-0.41847-01

0.30116-04

T2-To

V s di s

0.41787-01

-0. 33454-01

0.83850-02

-0.86045-05

V 7 dl7

-0.86045-05

0. 86045-05

-0.36876-05

Vi dl V 3 dl3'

b.

Vld

Seven

T3-T 0

0.61461-06

T4-T 0

Points: -0.30000+00

0.33333-01

54167+00

0.33333+00

-0.41667-01

41667-01

-0.33333-01

:} [_!.i oo+o +L++0/ T1-T°

=

V3d

V sd c.

I'd

]

d.

Five

0.83333-02

Points:

T 2-T 0)

Three

Points:

Vi dl = T i - T O

[

T2-T

o

T3-T

0 [

Section

D

October Page 3.

Arbitrary

Temperature

a.

15,

1970

74

Distribution

Five

Points:

Y

1

-1, T2

1 2

!

/

T1

r

Vi dl

21

22

V 2 dl 2

11

12

V3 dl 3

-1) i

! 4 iV4 dl

3

-2)

23

24

T2-T

0

13

14

T1-T

0

(-1) 2 (-1) 3 (-1) 4 (-2)

1

(-2) 3

2

(-2)

4

T-i-T°

]

T-2-T0

I

4

VI dl

-0.

83333-01

0. 66667+00

V2 dl 2

-0.

41667-01

0. 66667+00

V 3 dl3

0. 83333-01

-0.

16667+00

V 4 dl 4

0. 41667-01

-0.

16667+00

b.

,v 17'lI dl

Once 3.0-3

F 4 to be the beam.

constant to

80°F

to find

Example

The

beam

in the on the

the

coefficients Refer

fixed-end

Problem

x-direction

0 0

0. 66667+00

-0.41667-01

T1-T

0. 16667+00

-0.83333-01

T_l-To

0.41667-01

T_2-T0

16667+00

are

toTable

determined,

3.0-4

moments,

for reactions,

refer the

to Table

constants and

F2,

deflections

3.0-2 F3,

and

of

1.

of rectangular

bottom

T2-T

Ii;] /

(-1)

F 1.

0.83333-01"

Points: -1

polynomial

constant

used

I. Given:

the

for

1

(-I)

66667+00

-0.

it

dl 2

or

Three

-0.

and

cross

section

varying

linearly

with from

a temperature 500 ° F on the

distribution top

surface

surface.

v"

Section

f

D

October Page

15,

1970

75

T 1 = 500 OF Y

T 1 = 5N0 OF

a=

6x

10 -6

E = 30 x 106 L = 100 IN.

TO =_

,I

80 OF

5 Find:

Maximum

stresses

and

maximum

deflection

v(x)

in center

of beam.

Solution:

1.

Find

distribution

the

in the

Antisymmetric

polynomial

expression

y-direction.

Refer

Temperature,

Three

representing

the

to Paragraph

given

3.0.2.3,

temperature

Continuous

Points

vt dt = T i - T O Therefore

Vl(5)

= 500

- 290

v 1 = 42 Therefore

Refer

expression

for

2. Find wllues to Table 3.0-2.

of

T-

v 0+

v ly-

F0,

F1,

F2,

F 0:= 290 F0= F1

Refer

to Table

(q-:

290+ and

42y

F 3 for

× b0d 0 = 2')0

> 10(5)

each

term

=--50

× 290

0 (42) 12

b0d30 --

42 1-_

(5)

(10)

F 2 = -1.

It= 000

0,

G-

0)

f(x) since

500

415.67(42)

(--_) q

const_lnt

in x-direction

q=0 F3=

3=

3.0-4:

0,

0

of polynomi:ll.

constant constant

depth width

G - 0 II - 0

Fl=

0

Section

D

October Page 3.

Refer

3.0.2.2

II (Case

76

b)

= _ EFIF 2

M0 z

M0 z

xx

to Paragraph

15,

= 6×

10 -6× 30×

= 3.15×

106

= - o_ET

+

106× 416.67×

40×

(-i.00)

Mo z Y I Z

Top

a

Fiber:

a Deflection: Therefore

× 106×

30 × 106×

500 +

(-3.15x

106 ) × 5 416.67

= -127 800 psi

XX

v(x) the

II.

= -6 XX

= 0

beam

Example

remains

straight.

Problem

2.

Given: The I-beam shown below with a linear varying x-direction and varying as shown in the y-direction.

temperature

in the

Y Y t

_'-_

1'00°

|

!_

Find:

L-

Stress

100 IN.

xx

and deflection

350o

X !

q

4.5

-*

v(x).

Solution: 1. distribution Temperature

Find

the

polynomial

expression

representing

in the y-direction. Refer to Paragraph Distribution, Five Points.

the 3.0.2.3,

given

temperature

Arbitrary

1970

r

Section

D

October Page %0.08333

0.6667

-0.6667

V 2 dl2

-0.04167

0.6667

0.6667

V 3 dl 3

0.08333

-0.16667

0.16667

V 4 dl4

0.04167

-0.1667

V i d_ fr

62.5 1.156

T = 150+

T = 150

F4

for

+ 54.1y

2. Refer each term

-0.

+ 7.81y

-0.04167

225-150

10.42

-0.08333

100-150

y2

y

and

3.0-4

V0 = 150

F 0 = 150(1.

n=

1

V l=

54.1

F 0=

54.1(0)

n=

2

V2 = 7.81

F 0=

7.81(2.774)=

V 4 = 1. 168

F 0 = 1. 168(

2.083 (1.156)

4

for

values

of

715)

= 257.3

=

0.0 21.6

6. 745)

-

Z F 0=

,....L/

f

350

q=

0

F 2=

-1.000

q=

1

F 2=

0.0

F 2=

-1.000

3.

Refer

M0z

M

Z

to Section

= -3/2

= M0z (1

2. 083

y4

7.9 286.8

F0,

F1,

F 3= Fj

3.0.2.2-III

54.1(2.774)=150.0

F l=

0

F1 = 0 F 1=

-1.502

= -1.0

F 4=

-0.502

-1.0

F 4=

-2.004

(180)

(Case

(150)=

I Z = 10.55

b).

- 40,500

in 4.

F3, and

F 1=

F 4=

F 3=

0.0

F2,

F 1= 0

- 250

oL EF 1 = - 3/2

-x/L)

: I 0

100-150

+

1970

4

n = 0

n = 4

62.5

0.04167

2 + 1.168y

to Tables 3.0-2 of polynomial.

350-150

t6_67

10.42 (1.156)

Y +

0.0833_

15, 77

150.0

Section

D

October Page

15,

1970

78

Moz Y a

(@x

= 0) = - t_ET+

I

XX

Z

= -_ou,..,ou, &

(7

t'_

I't

I

¢3t

_"

t',t _,

_

F

XX

XX

= -71

40 000(2.312) 10.55

860 psi

Deflection:

v(x)

= _

x2/4(1

I

-x/L)

: 21

35 × 10 -6 x2( 1 - x/L)

Z

x

v(x)

0 10

0 90 × 21.35

20

Refer

320 ×

"

40 50

960 1250

" "

60 80

1440 1280

" "

90

810

"

100

0

also

3.0.3

to Fig.

3.0-6

Indeterminate

-6

× 10

for

the

Beams

deflection. and

Rigid

Frames.

Continuous beams or frames can be analyzed by the method of moment distribution described in Section B5.0, Frames. In this method a continuous beam

is fixed

beam fixed

is continuous; against rotation

against

rotation

at all the

and, in the case and displacement.

intermediate

supports

over

which

the

of a frame, all the rigid joints are also For each beam segment the moment

required to keep the slope at each of the two supports unchanged is called a fixed-end moment. According to the convention of the method of moment distribution, the fixed-end moment, which is an externally applied moment, is positive vention

if it is clockwise

should

not be confused

and

negative

with

the

if it is counterclockwise. convention

internal movement (refer to Paragraph 3.0.2). nated by the subscript, F, and is abbreviated

This

con-

of strength-of-materials Fixed-end F.E.M.

moment

for is desig-

Section D

f-

15, 1970

October Page

A

B

A

B

MFAB

MFBA

The beam M0z must

magnitude

FIXED

A

M0 z

V0 z

A

B

C

MFAB

beam

is obtained

cases

MFBc

MFcB

by analyzing

of Paragraph

so, to convert

D

MFcD

the fixed-fixed

3.0.2.

FIXED

=M 0

+L Z

- HINGED

B

A

VLz

MFAB

In that paragraph,

M0z

to F.E.M.,

one

- END

MOMENTS B

= -M 0 z

MFBA = M L

V0 z

BEAM

FIXED-END

B

A

T-.

)

VL z

A

MFBA

BEAM

V °z

Z

C

D

by -l.

- FIXED

ML

C

counterclockwise;

A

M0z

B

of F. E. M.

is positive when

FIXED

A

N O F.E.M.

MFBc

or the fixed-hinged

multiply

C

79

MFAB

B

MOMENTS

= -M 0 •

A

B

= Mo V0

With this information, performed

•

MFBA

a normal

to solve any continuous

beam

moment

distribution technique

or fr_ime.

can be

Section D October 15, 1970 Page 80 3.0.4

Curved For

Beams.

a free

curved

beam

of arbitrary,

constant

centerline of which is an arc of a circle planes under a temperature distribution be made.

lying T(r,

1. For a small depth-to-radius instead of the curved-beam theory.

ratio,

used

2.

cross

section,

the

in one of the centroidal principal 0), the following comments can

straight-beam

theory

can be

theresultsof

Forlineartemperaturevariation(T=Ti-_-),

curved-beam theory compare well with the solution obtained by exact thermoelasticity method; e. g., for a rectangular beam where the ratio of outside radius to inside

radius

is equal

to three,

the

maximum

beam theory differs from the maximum only 2.4 percent. Straight-beam theory, considerably

stress

obtained

by the

curved-

stress obtained by exact solution by however, gives _ = 0, which is 00

erroneous. z 2

3.

For

quadratic

temperature

in maximum stress obtained 4.9 percent for a rectangular is equal to 3.0. For

solutions

other

distribution

by curved-beam section where

than those

T = T2 "_

,

the

difference

theory and exact solution is only the ratio of outside to inside radius

mentioned

above,

see

Ref.

1.

3.0.5

fore

A ring may be regarded be studied by the methods

3.0.6

Trusses.

3.0.6.1

Statically

elongate

as a closed, given in 3.0.4

is zero.

regarded analyzed

as simply accordingly.

curved in Ref.

beam, t.

and

can there-

Determinate.

Every member of a pin-jointed because of thermal loading.

member

thin, and

Therefore, supported

members beams

determinate As a result,

truss is not constrained to the net axial force in each

of a pin-jointed under

Deflections of joints of a truss can "dummy-load" method described in Section

thermal

be obtained B4.2.2.

determinate loading

by the

and

truss

should

conventional

be

are

J

Section July

D 1, 1972

Page

3.0.6.2

Statically The

81

indetermin'lte.

forces

in the

bars

of a statically

indeterminate

pin-jointed

truss

F

are

not zero,

but are

easily

determined

from

the

results

of the

previous

is based

on the

paragraph. 3.0.7

Plates. The

following

other

analysis

classical The

material

2.

The

constant

Plane

of plates

in this

thickness

section

assumptions:

is isotropic,

of the

sections,

plate, The

homogeneous, of the plate

remain

which plane

deflections

and

is small

linearly

when

elastic.

compared

with

before and

bending

normal

of the plate

are

are

to the of the

normal

median order

to the plane

median

after

of magnitude

bending. of the

thickness. Solutions

various

are

boundary

F.

3.0.7.1

Circular

A.

circular

,_nd temperature for

plates

made

and

rectangular

plates

with

distributions. of composite

materials

are

given

Gradient

Through

the Thickness.

Configuration.

design

plates

for

Plates.

Temperature

The

herein

techniques

in Section

I.

given

conditions

Analysis

circular

presented

dimensions.

4. plate

theory

1.

3. plane

of plates

curves

having

and

central

equations circular

presented holes

(Fig.

here

apply

3.0-14).

only The

to flat, plates

must

its

Section D July 1, 1972 Page 82 be of constant thickness, be made of an isotropic material, and obey t]ooke_s law. Curves are given which deal with bending phenomenafor a/b r._tios of 0.2, 0.4, 0.6, and0. San(! are basedon the assumption that plots

cover

only

with

bending

membrane valid the

behavior. solutions,

for

arbitrary

plate.

When

identified

n portion

of the The

are values

the

ratio

boundary

remainder

given

of these

t s ratio

approaches

considered

conditions,

as closed-form

of Poisson a/b

conditions

algebraic and unity,

v = 0.30.

the the

in connection

as well equations

inner

and

member

These

as all

the

which

arc

radii

of

outer

is more

properLy

as a ring.

N8

Nr

NI

N8 POSITIVE DIRECTIONS OF THE STRESS RESULTANTS

I. _'_._....

MIDDLE

NOTE: r, 8, z, u, v, and w are positiveasshown.

z,w Figure

SUR--FACE

3.0-14.

Circular

flat

plate.

Section D July 1, 1972 Page B.

Boundary

Solutions

are

83

Conditions.

given

for

various

combinations

of the following

boundary

F conditions

for

Bending

bending

and

membrane

behavior,

respectively.

Phenomena. 1.

Clamped,

that

is,

0w

-

w

2.

-

Or

Simply

w

= M

3.

Free,

M

0

at

supported,

= 0

r

that

at

r

Or

Membrane

that

is,

and/or

= b

r

and/or

= a

r

= a

;

rO

:

;)o

I

o

at

r

:

b

and/or

Phenomena. 1.

Radially

= _

= 0

;

is,

= 0

OM

= b

r

r

aM

r

fixed,

that

at

r

2.

Radially

free,

Nr

= Nr0

= 0

that

at

is,

= b

and/or

r

= a

;

is,

r

= b

and/or

r

= a

r

a

.

Section D july

1,

Page C.

Temperature

Arbitrary thickness. Hence,

= T(Z)

Design 2,

the foregoing

requirements.

through

be no gradients

can be expressed

the plate

over

the surface.

in the form

and

and Equations. Forray

stresses

present

5 by the same

This

number

of equations

authors. and design

the simple

and deflections technique

and is an extension

through

which

Curves

Newman

the thermal

theory

be present

that there

distributions

to determine

deflection

may

.

D. In Ref.

variations

it is required

the permissible

T

Distribution.

t_mperaturc

However,

for stable

is based

curves.

plates

here

that satisfy small-

published

in Refs.

the analysis,

These

given

on classical

of the procedures

To perform

method

use

are provided

is made

3

of a

in the summary

follows. It is assumed

by temperature

that Young's

variations.

of the thermal-expansion it is the

product

distribution

which

can

bc suitably

E.

Summary

w t a2M T W

_

Db(1-

v2)

modulus

On the other coefficient

_T

1972

84

and Poisson' hand,

that

modified

to compensate

of Equations

is,

and

the

are

unaffected

the temperature-dependence

can be accounted

governs;

s ratio

for by recognizing

actual

Curves.

for

temperature variations

,:_ ,_.

that

Section Juiy Page

G r

I Nr + (i-.) NT -- i-

+ -v12z t

r +_

(_--

M0 +

_

v)

and

% =

1

NO +

+7_

where

Et 3 Db

-- 12 (1- p2)

t/2 MT

Tz dz

= F_ -t/2

t/2 NT

= E_

Tdz

I-t/2

Mr

= -12

M_

= -12 MTM

Nr

MTM" r

= NTN" r

and

NO =

NTN 0'

0'

,

(i-.)

D 1, 1972 85

Section

D

July

1972

1,

Page The

values

for

w',

or Figures

3.0-15

ratio

must

be taken

When

Table

that,

for

is employed,

there

thickness

1_

the

obtained

from are

analysis

include from

Table

Table

used,

on

v .

multiplication 3.0-6

3.0-5

Poisson'

of bending

are no restrictions

the parameters are

either

s

phenomena. Also

note

factors.

and

are also

valid

gradient

through

the

The "or

s ratio.

Linear

the special

for

obtained

When the figures

throughout

of Poisson'

For

are

to be 0.3

and

F.

M0t

3.0-19.

of the plots,

N t r

any value

and

through

3.0-5

in most

values

Mr' '

86

a solid

Gradient. case

plate

of a linear represented

temperature by

7

T(z)

-- TO ÷ 2

the following

solutions

Unrestrained

Solid

0

=(7

r

The

Circular

becomes

to the

difference

proportional

to the

thickness.

Plate.

w

=

0

T°

- T1 2

Z

apply.

proportional

Clamped

+

Plate.

= 0

0

plate

Tl

curved

and fits

in surface

a sphere temperatures

_f radius and

inversely direct:y

Section D July I, 1972 Page 87

cTABLE

Boundary

3.0-5.

NONDIMENSIONAL PARAMETERS BENDING I)IIENOMENA

FOR

Conditions M

Outer

P

Edge

Inner

Free

Edge

W !

Free

Clamped

1(__

1)

(Relative to Inner Boundary)

0

Clamped

Simply Supported

12(1-

2 Simply Supported

TABLE

Boundary

3.0-6.

0

NONDIMENSIONAL PARAMETERS MEMBRANE PHENOMENA

Edge

FOR

Conditions N

Inner

Edge

Radially

Free

Radially

Free

Radially

Fixed

Radially

Free

No,

! r

'

1

-_(1-v_

(i_-.)

•-

Radially

i.o(i - .)

v)

l,'rec

Free

Outer

Mo,

t r

Free

Radially

Fixed

. (i- .)

+

V(I L

Radially

Fixed

R,_dially

Fixed

(i-,)

Sectio_

D

July

1972

Page

0

1, 88

-_j

Section July Page

i r

D 1, 1972 89

Section

D

July 1, 1972 Page 90

Q,I

i_

i,m

i1,,,

I..i

0

II

[:1 ! I I/I ..V-i3;

J"H"5 '41" il

[,,'1 1 1 1"--4-J1:

i::l

0 _ i::l v ..c:l 11)

J::l

l::l,,

I11II1!11: t,,.l

I::l., . ,,--,4

I_'! I_1 .,f"I il I",V'I 1 !1 J"l I I I--YII I I

_i_L,. _-_r_,_=.. _ ",k_

.!i

I:::I 0

z_

A

"--!!

i

il II

pl"q

•

ql'

°_,.,.i

Section

D

July 1, 1972 Page 91

t--and

f

rl

%1 max

max

Simply

Supported

Plate.

Crr

= o

= _0

and

-ol (T,2t

W

II.

For thickness,

where

A.

Plate.

the

Clamped variation

and

c

:

A r K

K

are

constants.

,

K

TD(a)..

= AKa

with

+c

K

I(

as a Function

of temperature,

variation

Z

TD

r2 )

Difference

= ---Ar h K

A

(a2

Temperature

and the

T

T,,)

r

,assumed given

by the

of the

Radial

to be linear monomial,

Coordinates.

through

the

Section

D

July

1972

1,

Page

w =

(K÷ 2)2

Mr

=

12(K + 2)

M0

-

Eh2OtTD(a) 12(K+2)

TD

is the

-

+

+2

92

'

+

and

where

Curves Figures

used

for

shown

through

TD

describing

temperature

difference

of nondtmenslonal

3.0-20

given

the

3.0-22

for

variation

in Paragraph

between

deflection

by polynomials

radial

K +l---'_J l+ul

I ( K+ 1)(r)

of

the surfaces.

and moments

K = 1,2,...

5

in

determination

TD

r.

The

.

are

presented

Superposition

can be obtained

in the

in

may

then

be

of a polynomial same

manner

as

3.0.2.3.

B.

Radially

Fixed

or Radially

Free

Plate.

Configuration. The design which

may

constant Formulas

or may

thickness, are

0 _

equations

provided

not have

a central

be made

given

a

b

<

which

1

here circular

of an isotropic

cover

the range

apply

only

hole. material,

to flat,

circular

The plate

must

and obey

plates be of

Hooket

s law.

A

Section July Page

I

w'-3.2

2.8

2.4

2.0

1.6

1.2

0.8

0.4

0

Figure

3.0-20.

Nondimensional

deflection.

D

1, 93

1972

Section July Page

D

1,

1972

94

x'0 1.4

1.2

1.0

0.8

0.6

0.4

0.2

ii I

l

I

I

0.2

0.4

0.6

0.8

r

Figure

3.0-21.

Nondimensional

radial

moment.

1.0

Section July

F

Page

D 1, 1972 95

f x_0 1,4

1.2

1.0

0.8

0.S

0.4

0.2

|

!

I

O_

0.4

0.6

I

0.8

f II

Figure

3.0-22.

Nondtmensional

tangential

moment.

1.0

Section

D

July

1972

Page As this

ratio

approaches

unity,

the member

is more

properly

identifiod

1, 96 as

a ring. Botmdar_

Conditions. Solutions

are

given

for

each

of the following

types

of boundary

conditions. 1.

Radially

free;

that is,

ar

= ar8

= 0

at

r

= b

(rr

_: (r r 0

= 0

at

r

= a

and

if hole

is present. 2.

boundary

Cuter

boundary

is radially

u =v

free)

= 0

radially

fixed

(if hole

is present,

the inner

; that is,

at

r

= b

and

ar

_: ar^u

=

0

at

r

a

ifhole is present. Temperature

Distribution.

The supposition is made thickness.

However,

that the temperature

the plate may

is uniform through the

be subjected to a surface distrioution

r

Section

D

July

1972

I,

Page

rwhich

has an arbitrary

Hence,

the permissible

radial

gradient

but no circumferential

distributions

can be expressed

97

variations.

in the form

(

T Design

= T(r)

Equations.

f-

In this ratio

.

are

single

section,

unaffected

effective

averaging

values

of this

product

_T

be suitably

which modified

radially

were

fixed

formulas to external

expansion.

to compensate

taken

are

or by numerical

the for

given

hand, for

actual

The

boundaries

were

derived

by superposition

given

in Ref.

6 for

the required procedures.

integrations

upon may

to the

that

follows

expressions

Ref.

Depending

of

it is the

distribution

which

from

pressure.

select

can

_.

The equations

1.

type

regard

by recognizing

in

must

the temperature-

summary

theory.

some

with

temperature

variations in the

by using

be taken

Poissonts

the user

directly

and the relationships

distribution,

may

On the other

that is,

and

Therefore,

can b_; accounted

governs;

modulus

properties

approach

small-deflection

outer

radial

changes.

of these

The same

equations

on classical plates

each

property

The design

free

for

of thermal

dependence

that Youngts

by temperature

technique.

coefficient

based

it is assumed

cylinders

the complexity

be performed

for

and

are

the radially

for plates

with

of the free-plate subjected

solely

of the temperature either

analytically

Section July Page Summary Solid

of Equations.

Plates

(No Central

1.

Radially

0"r

= o_E

Iiole).

frec

boundary:

(lb lr ) _'_

f

Trdr-r--

_ f

O

a0

= c_E

Trdr

O

fTrdr+_ b

I_ T+_ I

fTrd r

I

O

z_

0

and

u

= --

1 + P)

r

Tr fir

! (1 - v

Tr

o

These

three

equations

application

are

of I t Hospital

(ar)

= (aO) r=O

and

U r=O

---"0

.

indetcrminate

t s rule,

= orE r:O

d

o

at

r : 0 .

it is found

that [1]

1

f o

Tr

dr-

However,

_(T)r=O

by the

D

1, 98

1972

Section July

D 1,

Page

2.

l_adially

fixed

=_E

=

aE

boundary:

f b Trdr-_

1

Po

r

1972

99

f T+_-_

1

f

° b

f

1

r

_

Trdr-

b2(1 - v)

f

O

_ _] Tr

O

Trdr+_-_

f

1

r

2

Wrdr-b2(l_v/

f o

O

b ] Tr

dr

and

=

U

ot

1 +v)

Tr

r

These

three

equations

of 1' Hospital'

dr-

(1 +v)

f O

o

are

s rule,

(ar)

:

indeterminate

it is found

(a0)

r=0

at

r=0

b ] Tr

.

dr

However,

2

1 _-_

-- aE r=0

=

0

f b Tr

dr -%2(1

_ v)

o

•

r=[}

Plates

with 1.

Central Both

O" r

= -_y

Hole.

boundaries

radially

.,

f ,q

application

that

and

U

by the

Tr

free:

dr -

f a

r ) Tr

dr

e

f O

b 11 T rdr-_Tr=

0

8ec tim Jtfly Page

#0

ffi r'_

Trdr+

f

a

Trdr-

Tr 2

D

1, 1972 100

,

a

and

_°[( i

U = _

l+p)

2.

Inner

Cr

= _

boundary

(1-

a

aE_

b2 - a2

r U

r a [(1 +_)af

free

and outer

Trdr-

fbTrdr+

+ (err) r=b

a2

Tr dr +

Tr dr

radially

+(O'r)r= b bS__ a

a/Trd

/Trdr_Tr2

]

+ r7

(1- u)r 2+(l+v) (b2- a_

a2

b f a

b

1

b]

boundary

where

o,

f a

radially

Lb-t--_-f

v) r 2+(l+u) b 2 _ a2

Trdr+

I

+ _)

f a

Trdr

Tr dr

fixed:

I-_-_

,

Section

D

July 1, 1972 Page 101

÷ (1- _,)b_÷ (1 ÷ v)a_ f (b2- a2) a

b Trd

f

III.

Disk

With

Boundary and

Central

conditions

=

0

for

Shaft.

this

plane-stress

problem

are

u[ r=a

•

Crrr [ r=a

r

T

-

a_ I1

= T b \b - a/

arr -

'

1

\_-v- v + r-2

2

(l+v _+_

T*

_) - _

[

a2

_-7

a'/•_ (_-_)-_ T*

¢rO0 ET b 2[Tb

2T

rr

a ET b

r=a

°00I

aET

b

r=a

=

-

=

_

-_

(i+_) +a _ 2 (i-_)

'

+ a2

I_(1 _)[ (,_/a'-'__

r=b

Tb

1 +

and

r

T*

="

2 r 2 _ a2

f a

Tr dr

-

2Tb (ra_n r + a \b-_-a]

[(n+l) r +a) -(n+l)'(n+2)

=0

Sectioo

D

July

1972

1,

Pag e 102 where

2 T:

= _-_ r

b _

f

Trdr

-

2T b a+ b

[(n+l) b + a] (n+l) (n+2)

a

Curves

showing

the variations

of

rr

and

ETb and

a/b

are given

in Figures

3.0-23

through

with

r=a

n

r=b

3.0-25.

_.ll!

a/b 0.8 0.6

-0.7 0.2 0.1 0.4 "0-6 1 -0.5

0

Figure 3.0-23.

Additional 1.

Circular

2

3

Variation of tangential stress at outer boundary with n and a/b for a disk on a shaft. cases

that

plate

with

may

be obtained

asymmetrical

from

Refs.

temperature

7 and 8 are

as follows:

distribution

2. Circular disk with concentric hole subjected to asymmetrical temperature 3.

distribution Circular

plate

with

a central

hot spot.

Section

D

July

1972

Pagc

q

0.7

0.2

0.6

0.5

0.4

o_

P

a/b +-0

0.3

0.1

0.1 0.2 0.2

_

0.4

_

0.6

0.1 0.8

0

Figure

1

3.0-24.

boundary

Variation of disk

with

2

of radial n

and

3

and tangential a/b

for

a disk

stress

at inner

on a shaft.

-1.0 zlb -0.9

(ic

_

.o.e

_

•0.7

C

-0.5

0.8

_

o.6

o, O.2 -

/.

.o.,. /Z _ -0.3

-0.2

_

-0.1

.... 0

Figure

3.0-25. of disk

I

2

3

Variation of tangential stress at outer with n and a/b for a disk on a shaft.

boundary

1,

103

Section

D

July 1, 1972 Page 104 3.0.7.2

Rectangular I.

Temperature A.

plates

material.

that

The

that

The equal

(Fig.

These

of constant edges

both the

cover

aspect

Boundary

edges

x = 0

and

thickness

and the

elastic

limit

ratios

b/a

here

apply

only

are

made

of isotropic

and

are

supported

support

beams

in either of 1.0,

by flexible are

free

of these 1.5,

to flat,

rect-

beams.

of holes

members.

and

The

and 3.0.

Conditions. and

sttffnesses

3.0-26)

members

the Thickness.

provided

of the plate

plate

the

B.

flexural

ends

are

exceed

curves

Through

and equations

two long

no stresses

design

curves

which

It is assumed

Gradient

Configuration.

The design angular

Plates.

x = a

EbI b . are

frec

are Both

elastically beams

to undergo

supported

are

axial

simply expansions

by beams

supported

having

at their

or contractions.

offer no constraint to cnch of the following pl'itedeformations:

I. Edge- Hotntion 2. In-Place Edge-Displacements The beams and

y =b

w

along

these

two

v.

resist only transverse deflections w . The edges

are simply supported; that is,

=My

u and

-- 0

boundaries.

y = 0

Section

D

July

1972

1,

Page a

_1

L

105

ARBITRARY ONE-DIMENSIONAL TEMPERATURE DISTRIBUTION

t L r

X

,f

I

k

z

- 4------

y

' [

-V

r

o

i

l

NoTE x,

y,

z positive

m shown.

Figure

3.0-26. Rectangular flat temperature distribution C.

Temperature

Separate distributions

coverage through

1. Linear

that

to the

surface D.

In Ref. compute

thermal

there

each

of the

following

temperature

T = aJ + a{z

are

T = f(z)

.

no temperature

variations

in directions

parallel

of the plate.

Design 9,

for

the thickness:

gradient

It is assumed middle

Distribution.

is provided

gradient

2. Arbitrary

plate with one-dimensional over surface.

Forray, stresses

Curves et al.,

and Equations. present

and deflections

the

simple

at virtually

methods any

point

given

here

in fiat,

to

Section duly

D

1,

Page rectangular

plates

techniques

which

consist

of a varicty

on the conventional

by employing taken

and Poisson

the user

with

must

some regard

select

type

single

linear

is taken,

of the product Linear For

T

must

distribution

to compensate adopt

gradient

= a_ + a_ z

and with

TD

= T(z=t/2

) - T(z__t//2 )

E t_ Db

= 12(1 p_ v2)

expressed

for

each

is,

the

for

referring

by

actual

for

in any

other

be

hand,

by recogllizing

_.

mention

to a straight-line

properties may

temperature

variations that

variations.

approach On the

arc based that

of these

can be accounted

Gradient.

temperature

It is assumed

expansion.

the viewpoint

is actually

of which

The same

a T .

Temperature a linear

user

that

These

by temperature

values

technique.

governs;

modified

the

temperature

which

are unaffected

property

all

of plates.

of thermal

of this

specifications.

and curves,

effective

of averaging

T

can he suitably

approach

theory

to the coefficient

it is the product

buti(m

of equations

t s ratio

the temperature-dependence that

with the foregoing

small-deflection

Young t s modulus Hence,

comply

1972

11)6

distriWhen

this

of a variation

Section

D

July

1972

1,

Page

;P v

107

and

F

4Db_ M

The

TD(1

- u 2)

=

transverse

w

deflections

= :(x,y)

are

expressed

+wA(x,y)

by

+wA(a-x,y)

where

sin

osh

n__: a Y_

W(x,y) n=l,

wA(x,y)

=

re=l,3

_,

....

G ,[__Xcoshmrrx m

(

---_-

nacosh

3, . . .

2 mTra 1--_+---_coth---_---

mTra)

sinh

n7r____bb. 2a

_._]

sinm----_b

and

4M - _

x_

mira G

k=l,3

b \ma/J

component The

final

mr a

1 +

* 121

Db_._)

The

sinh

....

deflection

sinhmTra

N(x, y) bending

is the

moments

relation

into

deflection M

x

and

the following

(l_u)m_.a]

where M y

all

÷

2

edges

can be obtained

equations:

(Eblb_

are

simply

supported.

by substituting

the

Section

D

July 1, 1972 Page 108

F a2w Mx

a2w

: -DbL_+P

aT D

"]

J

_+-7(1+P)

and

[02W Y

[(9.--7 I v _3x----7t

Forray generate

the

assumed

that

of the plate these

et ",!. 191 used design

v _: 0.30 x/a

plotting

to the

curves

condition

necessary 3.0-35.

For used

and any

bending-moment

for

where

solutions. wouhl

since

the

are

the

most plots

However,

appear the

of the covers

prove

plate a wide

practical

near

the

,15. breaks

in Ref. latter

where

reference

This

corners

,',own at

10,

--- ¢0

(x = a/2 in Figures range

problems

a

includes

correspon(Is

Once

the_e

y = t)/2)

of values of this the

amount

the necessary

,

_;. 0-27

to be inadequate,

a considerable

form.

it is

edges.

centerlines

in making

in series

th.

to

where

discontinued

EbIb/Dbb

on :dl four

of curves

the

also

ex3_ressions

3.0-35,

conventional

where

quadrant

bc required

equations

curves

about the

one

mom(;nt

through

In addition,

c:::;c

support

and

3.0-27

results

used.

accommodate

situations

mathematics

was

only

(h;fh.ction

of the

of these

assortment

should

to obtain

_ 0

of symmetry

to show The

y/t)

of simple

Because

(1 + v J

of l.'il,_ures Some

format

t

these

.

Certain

supplementary

ables

--- 0,

locations.

different

eurw;s

]

OZTl)

O_w

-hr,_ugh in the vari-

p,_rticular equations

of rather substitutions

were

, it was

available,

class. c;m be

coutine to obtain it might

Section OF POOR

July

QUALITY

Page

f_

D 1,

1972

109

0.24

0.20

0.16

'f i

x ,, 0.50

+ 0

0.1

0.2

0.3

0.4

0

0.5

0.1

0.2

V/b

0.3

0.4

0.3

0.4

0.3

0.4

0.5

y/b

0.24

0.24

,r?"' 0.20_

0.12_

0.__

0

0.1

0.2

0.3

0.4

0

0.5

0.1

0.2

0.5

y/b

Y_

0.24

0.20 _

,L? 0.04 0.__

-=

" •

0

0.1

0.2

0.3

0.4

"0_ 0

0.5

0.1

Figure

3.0-27. opposite two edges

0.2 y/b

y/b

Nondimensional edges elastically simply

supported;

deflections supported b/a

for a plate with two and the other

-- 1.0,

v = _). 30.

0.5

Section D July Page

Figure 3.0-28.

Nondimensional

bending moments

Mx/M

1, 1972 110

for a plate

with two opposite edges elasticallysupported and the other two edges simply supported; b/a = 1.0, v = 0.30 .

Section ORIGi;_A;. OF POOR

July

"',_ ""

0.28

0.28

-I_b'b .o1-

0.26

I-

0.24 0.22

1

!

1972

111

1

-IEb'b i _1-__ -006-

0.24

O.2O

1,

Page

QUALITY

D

0.20

0.18 0.16

0.16

7"1,

;\

0.12

0.12

x

r-_=O__

0.10 _. \_

0.08 0.08

,---

X-o at

_

0.08

I I "_''-- I

0.04

0.04

- x.

0.20

- 0.20 --

0.02 0

0

:

_ _-o6o: L '

0.2

0.1

-0'.02 0

0.1

0.2

0.3

0.4

0.5

0.3 y/b

0.4

0.5

y/b

0.28

o28

:[Eb'b!

0.24

_

0.20

0.16

0.16

.

0.12

0.08

0.04 -

x

"a 0

0.1

0.2

0.3

0.4

0

0.5

" 0.50 i 1 0.4 0.5

J

0 0.1

0.2

0.3 y/b

y/b

o28_l%'b

0.28 Eb I b

5ool7--

0.24

0.24

0.20

_

"

"

- 2.00 0.20 0.16

0.16

_

o.12 _i

0.12

0.08

•

0.08

x

---

0.04

0 o

0.1

0.2

0.3

0.4

0

0.5

0.1

0.2

Y_

Figure

3.0-29. with

Nondimensional

two opposite two edges

edges simply

bending elastically

supported;

moments supported b/a

-- 1.0,

M /M Y and the v = 0.30

0.3 y/b

for other .

0.4

a plate

0.5

Section

D

July

1972

Page

1,

112

0.10 O.W

.1

0.2

0.3

0.4

O.I

0.1

0.2

y/b

0.21

0.4

0.6

0.211

0.14

0._14

0.2O

0._

,+I+

Figure

0.3 y/b

,F+o,. 0.1:

3.0-30. long

Nandtmensional

deflections

for a plate

edges etasticaUy supported and the short simply supported; b/a = 1.5, v = 0.30.

with both

edges

+

Section _ :._'_ .OF POOR

F

D

July 1, 1972 Page 113

_'-

QUAL_y

O.28

O.28

0.24

O.24

O.2O

O.20

0.16

,,i z o.1e 0.12

0.12

O.(Z

0.08

O,O4

0.04

0 0

6.1

0.2

0.3

0.4

0

_5

0

0.1

0.2

ylb

0.3

0.4

0.3

0.4

O.6

y/b

O.28

6.24

0.24

O.2O

0.20 0.16

:F"I" 0.12

o.'._.o.2o4_ 0.04 60

0.1

6.2

0.3

0.4

0.§

_ 00

O. 1

0.2

y/b

0.5

y/b

0.28

0.24

0.24

6.20

0.20

0.16

0.16

0.12

0,04

6o._,_

_-_ _ y_

Figure

3.0-31. with

both edges

Nondimensional long edges simply

bending

elastically

supported;

moments

supported b/a

= 1.5,

Mx/M and the

v = 0.30.

for short

a plate

oJ

Section

D

July 1, 1972 Page tl4

LM

- 6.OO

Figure

3.0-32.

Nondimensional

bending

with both l(mg edges elastically edges simply supported;

moments

My/M

for a plate

supported and the short b/a = 1.5, v = 0.30.

Section Jaly

ORIGI._,AL _:-.', ,:.,._EtS OF POOR QUALITY

C-

Page

D

1,

1972

115

IJII f

1.0

/

O.2O 0

O. 1

0.2

0.3

0.4

O0

0.5

0.1

0.2

y/b

0.3

0.4

0.5

y/b

o.,-o.1 l-

Eblb

0.38

"

0.50

----

o_o.'_. 0._-,_

O.6

0.6

0.20 0.3

0.10

0.1

0.05 0.10

O0

O. 1

0.2y/bO.3

0.4

_

00

0.5

0.1

0.2

0.3 y/b

0,4

0.5

0.1

0.2

0.3 V/b

0.4

0.5

0.28

0._

O.2O

0.2O *_" 0.10

i

0.12_ 0._ 0._

0.04

0.1

0.2

0.3

0.4

O0

0.5

y_

Figure

3.0-33.

Nondtmensional

deflections

for

a plate

long edges elastically supported and the short simply supported; b/a = 3.0, v = 0.30.

with both

edges

Section July Page

•

Figure with

X

3.0-34. both

Nondimensional long

edges

simply

bending

elastically supported;

moments

supported b/a

=

3.0,

and

M the

v = 0.30.

x

/M

short

for

a plate

edges

D 1, 116

1972

ORIGI_IAL

F,_CE

IS

OF POOR

QUALITY

Section

D

July

1972

t,

Page

117

0.28

0.=,

O.24 O.20 0.16

0.16

0.12

0.00 O.O4

0

0.1

0.2

0.3

0,4

0.5

y/t) 4.

0

0.1

0.2

0.3

0.4

0.5

y/b 0..28

0.28[-I%,b m

O.24

I_E]Z Z

.

0.24 O.2O 0.16

11.16

0.12

o.,2

0.08 0.04

0.04 0

0.1

0.2

0.3

0.4

0.5

O0

0.1

0.2 y/bO.3

0.4

0.6

o.4

0.6

Y_ 0.32, ,J 1

O.28

O.28

0.24

O.24

O.20

O.20

0.16

0.16

0.13

0.12

O.lU

0.00

0.04

O.O4

[Ebl b

]

i

0

Figure with

0.1

3.0-35. both

0.2

0.3 yro

0.4

Nondimensional long edges simply

oi

0.6

0.2

0.3 y_

bending

elastically supported;

o.1

moments

supported b/a

= 3.0,

My/M

and the v = 0.30.

short

for

a plate

edges

SectionD July 1, 1972 Page 118 prove profitable to develop a simple digital computer program to perforn" the summations embodied both in the deflection and moment expressions. It is import:mt to note th:lt the peak moments always occur ;it the simply supported boundaries :rod .re'

oricnt(_d

for the

:_ct p:_r_lllel

can

stresses

be computed

M

These along

vertical

from

supported

suppresses planes

which

arbitrary

pnss

following

The from

gradient,

the

through

procedures

w

cquntions lhat

TI)

12 tz

-t/2

t/2 ; !_

imposes

induced these

The

peak

valtlcs

maximun_

m,)ments

which

demands

a straightness

tendency

that

w - 0

constraint

to dcvelo|)

that

curvatures

in

edges.

[T =- [(z)]. be used

through

,_md figures is repl:lced

Tz dz

edges.

condition,

for the

and bending

following:

corresponding

t_T D E t2 p 12

This

may

distributions

deflections

provided

T*

to the

boundary

therm:dly

Gradient

temper.'|ture 1.

the

the

[11] :

edges.

the

Temperature The

obtained

tile following

result

simply

Arbitrary

¢r y

C_Tl)(l - v2) D b .... t

moments the

and

x

from

x(max)

completely

the

tr

s,_ that

the

thickness

moments given by

for T*

;malysis

M the

,

x

having

T = f(z) : and

linear

which

of plates

may

M y

may

be

temperature be computed

troln

_ J

Section July

1,

Page

2. relations

The

normal

stresses

x

and

y

may

then

be established

D 1972

119

from

the

hips

r

x _,"_-/

x

(l-v)

+"_-

and

r

+

Cry = My _'_-7

P (1-

v)

-T+--t

T

where

-T

-

d/z ? J

1 d

Tdz

-d/z

II.

Temperature A.

Variation

Edges

Free

Over

the

Surface.

or Constrained

Against

In-Plane

Bending.

Configuration. The which

are

assumed limit. a/b

equations

of constant that

The .

design

the plate equations

thickness is free

provided and

here are

of holes

are applicable

apply

made

of isotropic

and that only for

only to flat,

no stresses

large

values

rectangular

material. exceed of the

plates

It is the

elastic

aspect

ratio

Sectionl) July 1, 1972 Page 120 Boundary Conditions. Consideration

is given

to each

of the

following

two types

of boundary

conditions: 1. All 2.

Plate

completely

is fully

constrained

supposition

thickness

surface;

is made

against

that

in-plane

no thermal

but a one-dimensional,

that

is,

the

temperature

bending

but is otherwise

gradients

arbitrary

exist

variation

is a function

through

occurs

only of either

x

over or

the the

y .

Equations. It is assumed

changes. value

free.

Distribution.

The

Design

are

free.

Temperature

plate

edges

here

Therefore, must

On the

the user

may

thermal

expansion The

problem

fully

plate

with

The

results

for

other

the

contents

property the

for

s modulus

is unaffected of this

by using

results

arc

by temperature

section,

some

presented

temperature-dependence

type

a single

effective

of averaging

in a form of the

such

that

coefficient

of

_.

was

stress illustrated

a temperature may

this

hand,

account

appropriate

which

Young'

in applying

be selected

technique.

that

formulation in Figure

distribution

be obtained

by first

is developed 3.0-26,

T(y) imposing

and

as follows

which free

shows

a rectangular

of any external

a fictitious

stress

for the

constraints. distribution

J

Section July

1,

Page

aA

on the edges

suppressed.

_A

such

x=4-a/2

It follows

that

= - sET(y)

.

These

stresses

may

arrive

at the force

that

be integrated

all thermal

over

deformations

are

1972

121

entirely

b and the thickness

the width

D

t

to

b/2 PA

= - Et

f

sT(y)

dy

-b/2

and

the moment

about

the

z

axis

b/2 = - Et

MA

if

sT(y)

ydy

-b/2

Since,

at this

present,

point

the actual

To restore

the plate

force

equal

opposite

in the

PB to

PB

MA .

plate

derivation, must

to such

and opposite Hence,

= - PA

and

M B = -M A

•

it is assumed

be free

a state, to

PA

of forces

that no constraints

and moments

it is necessary and a moment

are

on all edges.

to superimpose M B which

both a

is equal

and

Section July

1, 1972

Page

The

stress

corresponding

qPB

The

stress

PB

PB

=: _A

: _bd

corresponding

is easily

I' B

to

122

to be

b/2

E b

::

found

D

f

c_'l'(y)

dy

-b/2

to

M B is

b/2 a

-

MB

It should

MBY I z

be noted

Saint-Venant

Is

-

12MBY, tb _

thnt the

12y b3

procedure

principle.

Hence,

E

-b/

I)cing the

!"

used

c_W(y)ydy

constitutes

stresses

an apl)lic:ation

arid °'p B

accurate Subject plate

representations to this may

only

limitation,

be computed

at sufficient

the from

actual the

distances

thermal

will

i)c

ffM B from

stresses

the

edges

at various

x = _ a/2

points

relationship

= qA + CrP B + _MB

or

I)/2

F a

:

-o_l':T(y)I

x

The foregoing temperature

f

t)/2 (_'l'(y)

dy +1b-_3 E

b- -b/2

discussion varies

only

has in the

f

(_T(y)y


-b/2

been

restricted

y

direction.

c_[

to those How(:,:er,

cases the

where same

tne method

in the

.

Section D July I, 1972 Page can be used of

to arrive

at the following

when

T

is a function

only

x:

tr

--

-or ET(x)

y

Complex which

E +-a -a/2

one-dimensional

make

it difficult

equations.

In such

integral

signs

arc

restraints.

is fully

completely

free.

oIT(x) dx+7

replaced were these

constrained This

12x

a/2 f

E

aT(x) xdx

-a/2

temperature

instances,

However,

the plate

2

to perform

The equations

each

expression

123

distributions

the

integrations

numerical

for

is achieved

can

be encountered

by the preceding

be used

whereby

having

no edge

rectangular can easily

in-plane simply

plates

bending

by deleting

be modified

to apply

but is otherwise the final

terms

from

equation.

Summary

of Equations. 1. All

T

ax

Edges

= T(y)

:

Free.

,

E + b-

-aET(y)

fb/2 -b/2

T

= T(x)

,

aT(y)

the

symbols.

relationships against

often

required

techniques

by summation derived

may

dy + 12y b3 E

_2 -b/2

crT(y)y

dy

when

SectionD July Page

1,

1972

124

and

o" y

a/2 E +-f a -a/2

= -aET(x)

2.

Plate

Completely

Fully

a/2

12x aT(x)

dx+--_

E

f

¢rT(x)

xdx

-a/2

Constrained

Against

In-Plane

Bending

But Otherwise

Free.

T = T(y)

, b/2 l,:

f

,v'l'(y)

x : - a'ET(y). _ -b -b/2 T

= T(x)

dy

,

,

and

a/2 O_

--o_H'(,,) , t._: f

.-

Y

_'l'(×),I.,,

n -a/Z

1L

l,:_l_cs

Fixed.

Configuration. The rectangular material

plates (Fig.

no stresses aspect

equations

ratio

and sample which

3.0--36).

exceed a/b

the .

are

solution

of constant

It is assumed elastic

limit.

provided thickness th, t the

The

here

apply

.and are plate

_ c:uatians

is free are

only

made

of isotropic

of holes

applicable

to flat,

and that to any

SecUon July Page

v

FIXED

X,U

r, uJ x m

LL

r

_=

'

IXED y,v

ill

ii

I

I

ii

l

NOTE:

Figure

3.0-36.

Rectangular against

Boundarlr

fiat in-plane

plate:

x, y, z, u, v, andw are positivem shown.

all sides

fully

constrained

displacements.

Conditions.

Consideration restrained

against

following

conditions

is given in-plane must

only

to plates

displacements be satisfied

having (fixed)

by each

all sides

; that is,

edge:

completely both of the

D

1, 1972 125

U =

0

V

0

Section

D

July

t972

1,

Page

t26

through

the

and

"

Temperature

Distribution.

The plate

.

supposition

thickness,

is made

tfowever,

that

no thermal

temperature

gradients

exist

variations

over

the

body-force

analogy

surface

may

bc

arbitrary. Design

Equations. As

certain which

noted

previously,

isothermal experience

in a number

problems

of different analogy.

problem

treated

The

being form

series

and

no transverse

as Duhamelts

in series

a so-called

were

thermal

In Ref. here.

for

w .

[12, 13, 14] and 15,

this

Assuming

obtained can

problems

displacements

references

coefficients

stress

the

be obtained

This

stresses

by solving

plates is derived referred

to solve

does q

between

flat

method

is used

buckling

in-plane

for

is frequently

approach that

exists

the

not occur,

x

,

the

following

_

\_ /' q=l

a

y

solutions and

+

4

b

=

+ v) f 0

--_

_ 2(1 • v)

p-1

b f_T _x sin _mnx a 0

sin nny b

dx dy

r

simultaneous

equations:

mn

to

mnpq Bpq(p2-m2)(n2-(l:')

xy

Section July

D

1, 1972

• Page

127

and co

co

p=l

q=l

_2ab r

b = -_(1

where

to the

which

0f

m,

sin--a

n, p,

restriction

preceding

any given equations

perform

these

q may

and

This

set

even

be assumed

involving

at any

point

of

from

Once

I

_ n=l

p and

q must

the right-hand

In many

cases

procedures. unity

1,2,3,... be deleted

this

it will

to any value Anm

has been

side

for

N.

and

of the

be desirable

The integers

simultaneously

This

n, p,

will

result

N 2 coefficients

to determine accomplished,

B

appropriate the stresses

m_x cos -sin a

n=l

n_ -'_ Smn

sin--

m_x a

cos

n__h_ ../

-

E c_ _

to

m,

from

m___A a mn

I-

,

numbers.

N 2 coefficients

may be computed

N + V Z m=l

values

distribution,

can be solved

for the coefficients.

dxdy

take on the values

be integrated.

to vary

b

each

by numerical

values

°'x -

q

temperature must

of equations

sin

that those

operations

2N z equations

m_rx

_y

(m 4. p) and (n ± q) are For

in

0f

the indices

subject

and

+v)

a aT

pq (P2-m2)(n2-qZ)

T(x,y)

'

mn

.

Section July Page

E 1 - v2

{7

Y

n=l

\m=l N +

m_x

n_ -_-B

sin mn

a

I,

1972

128

b

\

N

P

_

m=l

COS

D

n=l

a

mrx a

COS

mn

sin n-n--_]

Ea l-p

T(x,y)

,

and

NN(

"rxy

m=l

Then

the strains

f

1 E

_

y

cos

at any point

| --!': (Or x

x

n=l

m a

mn

mn _

(IT

y

-

_

a

sin

mTrx a

COS

b

sin

can be determined

from

PO" ) y

-

IP°'x)

'

and

T

_ Txy

-

x3, G'

"

Example. Let a rectangular be subjected

to a temperature

plate

having gradient

ali

[our

edges

fixed

(refer

to Fig.

3.0-36)

Sec tlc_

D

July

1972

Page

T(x,y)

= T.

-:,

0T 0x

3'" and a

0T 0y

1,

129

Then

Substituting the

these

following

A

O

expressions

is obtained

_2ab mn--4

into the

after

2

+ u) b_T 0

right-hand

side

of the design

equation,

integration:

1 - v ÷---_

mn _2 14(1

•

u

-t 2(1 +u)

_ p=l

q=l

mnpq Bpq (p2_m2) (n2_q2)

if

m

and

n are

odd numbers

if

m

and

n are

even

numbers

,

and

mn _2ab[(b)2 4

1-v(-_) 2 N

+ 2 (1 + v)

Let

N = 2.

_ p=l

N _ q=l

nm pq Apq (p2_m2)(n2_q2)

Then the preceding

_2ab4 A11

2]

+--_

equation

= 0

becomes

+ 2(1 + _) B22 (22_1) ( 1_22)

= 4

_2

,

Section July

D 1,

Page

130

2×2

1,:

1972

0

t

2(1 _ v) B2,_(22-I)

l

2x2

=

0

A21 0

,

and

(li_1

Solving these

equations

=

and while

=

0

+

7r2ab Bi2 4

At2

2x2

A2t

=

gives the results,

A22

B22

-= Bt I =

earl be obtained

At_

_¢2a_b (1

'

1 - v_

8

B2t

=

B12

=

0

,

[rorn the following relationships:

,

Section July Page

D

1,

1972

131

and

8 (i + p) Air- _Zab B22 9

For

any plate

determined

dimensions

by hsing

-

O" x

-- cos All a

v2

= 0

and material

the appropriate

1

d

properties, design

-a

Sin

the

stresses

equations.

+ v

This

sln--

may then

be

gives

cos

-b- B22

a

-1?_

-

y

1 -E p--_ (_

-

(1-

B22

v) T.

S in 2____XX

a

1--a

cos 2___ b + p__a A11cos___X a sin b__)

'

and

Txy

il_sin?

R can be seen

that,

values

of

N,

efficiency

simple

digital

computer

3.0.8

more

÷ B22 a cOS--a

complicated

temperature

considerations program

would

in applying

a

distributions

dictate this

sin

the use

method

"

and higher

of a relatively

of analysis.

Shells. The analysis

most

for

cos

part,

taken

of shells

the approach

subjected of treating

to temperature thermal

variations

loadings

has,

as equivalent

for the

Section D July 1, 1972 Page 132 mechanical

loadings

techniques Refs.

such

right

circular,

coordinate and the

The sections

mental

which

differences

Flat-plate

for 1.

stable

Except ness

for

the

(N T -- 0,

bending

which

middle

surface

that

is,

middle

the

are

temperatures

special M T -- 0), is,

only

surface

gradients

case the

of course,

analysis

parallel

flat

plates.

physical

behavior

a nature

and no _ut-of-plane

of revolution

that

through

the the

of thcsc

plate. occur

that

and

of

fundashells.

to group

the

categories:

thickness thickness. gradients

two cases

through

involves

by displacements In case

in directions bending

plates

it is helpful

following

from

is due to certain

of flat

into the

accompanied

displacements

curvilinear

different

This

of self-equilibrating

of the undeformed

in shells

of thin-walled,

to the axis

is somewhat

through

first

distributions

The middle-surface

taken

section

of such

constructions

Temperature

2. Uniform

the

in

rcsl)eetively.

isotropic

between

deformations

solutions

alw:lys

of this

cover

discussed

Shells.

shells.

direction,

organization

temperature

the thermostructural

(x and y) are

are

by

section.

cylindrical

circumferential

and displacements

approaches

common

Cylindrical

covers

isotropic

axes

These

of the more

Circular

section

the stresses

B7.3.

in the following

Isotropic This

the

Some

be discussed

3.0.8.1

solving

as in Section

7 and 8.

will

and hence

occurs.

1, the

the

out-of-plane normal

plate

thick-

to the

remains

parallel

to the

The

governing

flat;

original differential

Section

D

July

1972

1,

Page equations

in these

of the cases

two instances

is a logical

arc

format

quite

for the sections

However,

the situation

is not the same

For

components,

a single

these

phenomena these

related

two types

temperature boundaries,

will

wall

surface.

middle

with either

case

I.

which

are

assumed

Figure

in conjunction

Consequently,

is restricted thickness

that the shell

wall

3.0-37

depicts

3.0-38

shows

the sign

dw dx

of

supported

about

the shell-

which

in a single

three edge;

= 0

to thin-walled, made

of holes

the isotropic

Boundary

The following

--u

type

or simply

constructions

the

to isolate

either

and bending

given

includes

comply

grouping

as follows.

Problems.

and are

is free

Figure

w

is no need

structures.

Configuration.

of constant

1. Clamped

are

with Isothermal

discussion

B.

loading

shell

equation

with clamped

separation

with flat plates.

is because

for stable

1 or 2, the solutions

A. This

This

indicated

cylindrical

differential

lead to both membrane

Analogies

dealing

1 and 2 and there

conditions.

distribution,

and the

for circular

governing

to both cases

of thermal

different

133

convention

for

right

circular

of isotropic

cylinders

material.

It is

and that it obeys

Hooke t s law.

cylindrical

shell

configuration.

forces,

moments,

and pressure.

Conditions. types

of boundary

conditions

are

discussed:

that is,

at

x=0

and/or

x=

L

(1)

Section July Page

D 1,

1972

134

L

NOTE:

MIDDLE Figure

3.0-37.

u, v, w, x, y, z, and are positive as shown.

SURFACE

Isotropic

analogies

with

cylindrical

shell

configuration_

isothermal

problems.

for

P(P*) x

/_Mx(M

x_ I N X

" QxIQx*)

I

f

QxIQx*]

N x

NOTE:

Y Figure

3.0-38.

Sign for

N_

convention

analogies

All quantities are positive as shown. Internal pressure is positive N indicated.

with

for

forces,

isothermal

moments, problems,

and

pressure

Section

D

July

1972

1,

Page 135 2.

Simply

supported

w

=

= 0

M

edge;

that is,

at x--O

and/or

x=

(2)

L

x

3.

Free

Mx

edge;

= Qx

All possible

case,

direction

= 0

at x =0

combinations

it is not required every

that is,

of these

that those

it is assumed

at

and/or

boundary

x=

L

conditions

x = 0 be the same

that the

cylinder

(3)

.

Hence,

are permitted.

as those

is unrestrained

at

In

x = L.

in the axial

(N x = 0). C.

Temperature

The temperature gradients

distribution

may be present

direction.

Distribution. must

both through

The permissible

be axisymmetric

the wall

distributions

thickness

can therefore

but arbitrary and in the be expressed

axial in the

form

T

= T(x,z)

Any of the

special

the entire

shell

cases

for

this

equation

are acceptable,

including

that where

temperature.

Analogies.

It is helpful exist

(4)

is at constant

D.

analogies

.

for the user

between

problems

to recognize involving

that,

for

axisymmetric

circular

cylinders,

temperature

Section

D

July 1, 1972 Page 136 distributions thermal

and

effects

certain are

entirely

of correspondencc modulus other

and hand,

that

are

Poissont

is,

the

actual

compensate

for

discussed

variations

recognize

that

any

reference

to a straight-line It is also

complexity the

reference

helpful

distribution

to linear

the

gradient

can be resolved

1. A self-equilibrating

middle

2.

A uniform

3.

A nonuniform

When

user

linear

approach

to recognize

through

the

that

Young t s

sT

which

modified the

distribution

-t _/2

T dz

:

0

to user

that,

thickness,

regardless

of the

at any location

and/or

passing

L-c component

must

aT.

and/or

tempcr_t_

t/2

governs;

is actually

through

T = 0 at the

surface. A self-equilibrating

On the

into

component

types

thermal-

is taken,

of the product

component,

component,

this

various

of the

be suitably

temperature

variation for

of a thermal

c_.

can

but

The

changes.

it is the product

distribution in

it is assumed

dependence

that

is present

by temperature

temperature

(_ by observing

loading

problems).

where

un_dfectcd

for

temperature

mechanical

(isothermal herein

are

account

coefficient

where

absent

s ratio

one can

expansion

problems

is one

for

which

(x,y)

a

Section D July I, 1972 Page 137

and

t/2 (5) _t/i2 Tz dz

An example

= 0

of such

.

a distribution

is illustrated

in Figure

3.0-39.

j

j T=0

Figure From

arbitrary

crx

and

it may

from

is necessary

X

only

--

previously

0-@

of the

a

temperature

superimposed

self-equilibrating

viewpoint,

any deformation

stresses

to the

Sample

a practical

not cause

the

3.0-39.

.

If,

distribution a component to algebraically

= -

be assumed

cylinder.

for

temperature

example,

Their

add

which the

gradients

only

a solution

T = T(x, z) Tl(x,z)

that

, the

distribution. of this

type

will

be on

influence is available

effects

satisfies

for

any

can be easily equations

(5).

It

stresses

E_TI_ (1- V)

determined

will

(6)

values

for the

appropriate

locations

(x, z).

Section July

1,

Page In this 1.

section,

Uniform

axisymmetric,

longitudinal

= T(X)

2.

Nonuniform

through

longitudinal that

temperatures

T

passing

the following

two categories

through

gradients;

are

the thickness

treated with

D 1972

138

separately:

or without

that is,

.

(7)

linear

temperature

T = 0 at the

variations

and

middle

possibly

gradients surface

including

through

with

the

or without

thickness,

axisymmetric

self-equilibrating

components;

is,

Z

T

where

=

TI(X,Z ) + T2(x ) _-

T 1 satisfies

Uniform

equations

Temperatures In Ref.

equation

for the

convention

used

16,

Tsui

here,

the

presents shell. this

p +4_4w

(5).

Through

subject

d4w

(8)

= - Db

After

conversion

NT _b a-

= _]'4D_a

- _/

a't "_

governing to the notation

differential and

sign

becomes

1

d2M T

Db(_- - .) dx_"

where

fl

IT = T(x)].

the small-deflection

expression

_

Thickness

'

(9)

SecUon July Page

D

1,

1972

139

t/2 Tdz NT

= Ea

MT

=

-t/_2

t/j2 Tz dz

E_ -t/2

(10) and

Et 3 Db

-

An inspection

v 2)

of equation

the problem equation

12(1-

under

reduces

discussion.

is uniform

differential,

for the isothermal

applies

problem,

to this

where

through

pressure

the

wall

differentials

thickness

are

(M T = 0),

absent

and

one obtains

NT +4f14w

be treated

that,

which

(11)

problems

d4w

A comparison

Note

to an analogy

.E... = - Db

thermostructural

the temperature

the key

to

d4w _T +4_4w

For

(9) reveals

=

Dba

of equations by means p = p*,

(12)

(II)

and (12)

of an isothermal where

suggests

model

that the

that is loaded

latter

problem

by a pressure

may

Section D July 1, 1972 Page 140

p*(x)

With

regard

N'r

a

to the edges,

conditions

for the 1.

Clamped

w

dw = m dx

actual

-

3.

Free

d2w

and

equation

the

radial

can be expressed

x=0

at

-

and/or

x-=O

at

0

x=

and/or

x=-- 0

do not contain for the between

the

:malogy

desired When to note, deflections

this

any

corresponding

model

is important

since

MT = 0 ,

as follows

the boundary

[16] :

.

L

(14)

edge:

isothermal

(11).

that,

(15)

x--L

and/or

x = L

(16)

dx3

equivalence

ttence,

be noted

edge:

to those

the

at

-- 0

relationships

identical major

-

cylinder

supported

daw

dx2

"

it should

= 0

d_v dx 2

w

Tt

edge:

2. Simply

These

a

the

subject

is bound

however, and the

that a):ial

terms

isothermal

cases.

temperature

up in the

is achieved pressure

temperature

the

is complete

are

Therefore,

equation p*

radially

concerned;

the

[T = T(x)]

substituting it acts

stresses

therefore

differential

is positive, analogy

arc

distribution

governing

by simply

and

for

outward. only insofar that

is,

the

p

(9). in It as

thermal

Section

D

July 1, 1972 Page 141 deflections solely

w

and the thermal

to a pressure

boundary

p*

conditions

the thermal ing value

_@ ,

obtained

from

strains

_@ in the

growths

or shrinkages.

2.40

pressure-loaded

17,

which

cylindrical

shells

conditions

are

Temperature

Gradients

Through

subsection

nonuniform

linear

through

T : 0

at the middle

T

= Ti(x,z)

+ T2(x)

where

t/2

-t ff2

T l dz

= 0

geometry

hand,

be added

This

analogy,

to the correspond-

accounts

for the

should

for numerous of pressure

fact

that

thermal

the user

variety

and

to determine

with stress-free

solutions

(a wide

the

surfacc, including

z _-

Thickness

factors

temperature

and possibly

must

due

cases

refer

to

of

distributions

treated).

discusses

having

nal variations,

of this

includes

and boundary

This

associated

to those

the same

On the other

solution.

arc

be identical

having

(-E(_T)

the pressure

the application

of Ref.

will

x

structure.

the quantity

aT

_

on a cylinder

actual

amount

To facilitate Section

acting

as the

stress

stresses

[T = T(x, z)].

associated

gradients

with cylindrical

through

with or without self-equilibrating

the

thickness,

axisymmetric components;

shells passing longitudithat

is,

(17)

Section

D

July 1, 1972 Page 142 and

t/2 -t 1/_2

Here

again,

governing

Ttz

:

it is helpful differential

corresponding this

dz

(18)

0

for the

equation

formulation

user which

of Ref.

to study was

the following

obtained

16 to the notation

small-deflection

by converting and

sign

the o _l.vention

6f

section:

d4w dx--Y + 4 fl4w

_ _J2 NT

---

1

D b Dba-

Db(1 - v)

d2MT dx 2

(19)

where

4/ fl

Et

4/3(1-

= _/ 4Dba2

- 4

v 2) aZtZ

t/2

/

NT = Ea

T dz

-t/2 t/2 MT

=

Ec_

f -t/2

and

Et :_ Db

-

12(1-

z,2)

Tz dz

,

(201

Section July

1,

Page For

the

isothermal

problem,

d4w

t,/-'_

for the subject

NT

thermostructural

(19)

A comparison

becomes

it is clear with

o) dx T

not,

(21)

d'MT dx T-

be considered.

and

model

that the latter

that is loaded

problem

may

by a pressure

(24) "

provide

conditions

simplest

having

suggests

where

however,

that the

(23) "

and (23)

of an isothermal

1 (l-v)

-

a cylinder

deflection

= - Db({_

p = p*,

the boundary must

d2MT

by means

p*(x)

lation

1

of equations

differential,

since

problem,

(22)

_-T +4_ 4w

does

to

(21)

d_w

This

reduces

= 0

and equation

be treated

equation

1972

143

-P= - Db

+4,_V

while,

this

D

which From form

both ends

the related

a complete

slope

basis

are likewise a study

part

of conventional

of the subject clamped. must

for the desired

vanish

analogy

In this

case,

of the

problem

types is that both

at the boundaries;

analogy formu-

of boundaries, associated the radial that is,

Section D July I, 1972 Page 144

at

w °I _

x = 0

nnd

x-

L

.

(25)

0

dx

The

simplicity

of the analogy

relationships to those

do not contain for

the

equivalence loading

is bound

the

p

outward.

It is important

insofar

as the

deflections

radial

w

on a cylinder

will

structure.

On the

the

[-E_T/(1

quantity

obtained

from

the

_

,

in the

analogy

to note,

the other

are

and

boundary

be addcd

solution.

temperature

c_T ,

the

accounts

for

component

are

due

dolely

it acts

and

radially

is complete the

only

thermal p*

conditions

of the

for

substituting

to a pressure

to each

This

Hence

is,

thermal

major

(19).

by simply

That solely

the

the pressure

the analogy

due

to determine

- r,) ] must

amount

that

identical

and

is positive,

concerned.

geometry

hand,

equation

these

therefore

As a result,

pressure

to those

same

pressure

this

because

are

is achieved

however,

deflections

of a self-equilibrating

and

ends,

When

and

distribution

differential

the

about

problem.

temperature

be identical

having

terms

governing

(21).

comes

temperature

subject

at both

in equation

situation

isothermal

up in the

clamped

for

any

corresponding

between

cylinders p*

for this

acting

as the

stresses

a

x

actual and

corresponding the

for the

to stress-free

possible

fact

that

thermal

cr ,

values presence strains

c

x

growths

or shrinkages. Although both

boundaries

the

analogy

are

clamped,

under this

discussion general

takes method

on its need

not

simplest be ruled

form

where

out for

a

Section D July

1,

Page simply

supported

equation

,f

(19)

shell.

still

In the latter

holds

true.

case0

The a_lcd

the _malo_ complexity

with respect is introduced

1972

145

to only

thrtmgh

{

the boundary-condition

formulations.

M

as follows

X

may

be expressed

In this

connection,

the

bending

moment

[16i:

/-

M

This

x

-- - Db d2w MT _ - (i-v)

relationship

is applicable

around

the boundaries.

should

be recalled

'

(26)

anywhere

Only the latter

that the condition

in the shell, locations

of simple

need

including

positions

be considered

support

includes

the

here.

It

requirement

that

M

Suppose around

= 0

X

at

a uniformly such

The user

around --

-

(1-

wall

a simply

ccmdittons

external

as defined

v)

L

(27)

°

bending

moment

M * is applied X

by the equation

"

remember

bending

of the shell

x=

MT

must

a positive

and/or

distributed

a boundary,

, MX

x=0

become

that the sign

moment (refer supported

(28)

causes

to Fig. end,

convention

compressive

3.0-38). the

related

being stresses

By superimposing expressions

used

specifies

ou the

outer

moments for the

that surface M *

boundary

X

Section

D

July

1972

1,

Page

d2w mx

MT

= -Db_-

(1

MT

v)

÷(1-

v)

-

0

at

and/or

x =0

146

x = L (29)

or

Mx

=

-D b dx 2

Equation

(30)

is the

cylinder

which

does

circular, to the

same

can

1. The Identical subjected

bursting 2.

influences,, with

the

distributed

pressures The radial is identical

moments

both

de.flcctions

ends

p*

(17)

of the supported of any

and

of a

Hence,

for

and

subjected

(18),

thermal

supported

by equations

but is frec

boundary

influences.

simply

of a siml)ly

structure

a

following: cylinder

thermal

which

influences

is and

is

where

d2MT

- (1- v)

condition

having

(30)

x=: L

supported

any thermal

by superposition

actual

and/or

a simply

be obtained

1

and

for

defined

to a pressure

p*(x)

x = 0

distribution

radi,_l

to the

as that

shell

temperature w

at

not experience

cylindrical

deflections

0

dx =_

are

(31)

positive.

deflections to the

of a cylinder actual

of simple-support M * at X

each

which

structure, except end,

is free

and whose for the

where

of thermal boundaries

application

conform

of uniformly

f-

Section

D

July

1972

1,

Page

147

MT

Mx* - (l-v) The thermal

(32)

"

stresses

and

X

{-Ec_T/(1

- v)] to the algebraic

1 and 2.

To facilitate

to Section

2.40

be useful simple

remains

valid

greater

complexity

formulations.

to the temperature

distribution

w

can be obtained

1. The identical

radial

to the actual

subjected

to a pressure

p*(x)

and bursting

-

structure p*

1

d2MT

(1-v)

dxT

pressures

the user

17,

11,

with steps should

solutions

Hefs.

arises

when

with respect

case,

when

by equations

by superposLtion

deflections

associated

refer

that will

often

and 18 provide

of a cylinder but is free

both boundaries to equation

is introduced

In this defined

stresses

be accomplished.

the analogy

boundary-condition

deflections

may

the quantity

analogy,

In addition,

2

by adding

numerous

to the foregoing

As before,

but still

of this

includes

1.

step

similar

are free.

sum of these

which

step

by which

are found

application

17,

in performing methods

shell

the

of Ref.

A sLtuation the

cr_

through the

(17)

of

(19)

the applicable

cylinder and (18),

is subjected thermal

of the following: which

has both ends

of any thermal

influences

free

and is and is

where

are positive.

(33)

SectionD July 1, 1972 Page 148 2.

The

is identical except

radial

to the

deflection

actual

of a cylinder

structure,

for the application

which

is free

of thermal

and has the prescribed

of uniformly

distributed

influences,

boundary

moments

conditions

M * at each X

end,

where

MT M *

(34)

x - (1-.)

3.

The radial

is identical

except end,

deflection

to the actm,l

for

of a cylinder

structure,

the application

which

is free

of thermal

and has the prescribed

of uniformly

distributed

influences

boundary

shear

forces

conditions

Qx*

at each

where

1 dMT Qx* -- (1- .) "

The

thermal

[-Ee_T/(1 1 and

To facilitate

to Section

provide

2.40

and

a9

algebraic the

of Ref.

simple

which

in that

thermostructural

can

by which

behavior

help

of this includes

of step

of the types they

found

steps

and

user also

the

stresses

associated

analogy,

2

and

to develop

3

quantity

user

solutions

In addition,

in this

enable

the

numerous 1 .

presented the

by adding

of these

application

17,

methods

are

sum

in the accomplishment

Analogies function

Crx

- v) ] to the

2.

be useful

stresses

(35)

may

Refs.

with should that 17,

items refer

will

often

11,

and 18

be accomplished.

section

perform

a two-fold

some

physical

insight

him to solve

thermal

stress

into

,

Section

D

July

1972

1,

Page problems

by using

mechanical ends

loading.

easy

solutions

Although

of the cylinder

it relatively are

existing

have

members

the emphasis

the same

to apply

not identical

for

(for example,

has been

boundary

the same

basic

subjected on cases

conditions,

concepts

one end clamped

to

where

the user

when while

solely

149

both

should

find

the two boundaries

the other

is simply

supported). If.

Thermal

Stresses

Axisymmetrlc A.

made

equations

right

of isotropic

and that

configuration.

stress

resultants

circular

Hookets

Boundary

Gradient_

here

apply

which

are

only

to long

of constant

It is assumed

that the shell

Figure

depicts

law.

3.0--38

3.0-37

shows

the sign

(L -> 2_/_),

thickness wall

is free

the lsotropic

convention

and are

used

of holes cylindrical for the

types

Conditions. of boundary

conditions

are

discussed:

edges

2.

Simply

3.

Clamped

supported

edges

edges.

combinations

it is not required

Radial

of interest.

The following

All possible

cylinders

Figure

B.

1. Free

Linear

Gradlont.

provided

material.

it obeys

shell

Axial

--

Configuration.

The design thin-walled,

and Deflections

that those

of these at

boundary

conditions

x = 0 be the same

are permitted;

as those

at

x = L.

that

is,

However,

Section D July 1, 1972 Page 150 in every case, it direction

(N--

is assumed

Temperature

The following 1. A radial not vanish 2.

is unrestrained

in the axial

0). C.

need

that the cylinder

types

Distribution. of temperature

gradient

which

at the middle

Axisymmetric

The permissible

distributions

is linear

through

may

the wall

be present: thickness

and

surface

axial

gradients.

distributions

can therefore

be expressed

in the form

Z

T

--

TI(x ) +T2(Y ) _-

Certain

restrictions

functions

Tl(x )

These

conditions

cases

for

T2(x )

can be finite

equation

D. Throughout Poisson'

s ratio

must

select

some

type

regard

single

must

T2(x ) ,

are

explained are

this

depending

and either

complexities

that

of analysis

paragraph. is,

either

employed.

Any of the special

or both of

can be equal

of the

Tl(x )

and

to zero.

Equations. section,

effective

to the coefficient

on the

upon the method

in a subsequent

are unaffected

of averaging

be imposed

acceptable;

constants Design

(36)

sometimes

and

(36)

.

it is assumed by temperature

values

technique. of thermal

for The

each same

expansion.

that

Young ts modulus

changes. of these

Hence,

properties

approach

may

On the other

and

the user by employing

be taken hand,

with

SectionD Jaly 1, 1972 Page 151 the temperature-dependenceof this property can be accountedfor by recognizing that it temperature in

a.

is the product

distribution When this

reference straight-line

In addition,

the several

small-deflection

be aware

of this when

values

applying

classical

theory,

coupling

between

the

methods

thermal

modified

aT

types

user

must

cited

the

actual

for variations that any

a reference

because

presented

here

here

it is important

methods

is,

deflections

is,

to a

. are based for

to pressurized

and deflections

that

recognize

is actually

Therefore,

the given

that

to compensate

of solutions

stresses

to pressure;

governs;

distribution

theory.

the thermal

due solely

which

the

of the product

classical

superimposing

is taken,

temperature

variation

aT

can be suitably

approach

to a linear

of

upon

of their cannot

the

and the pressure-related

user

to

cylinders

by

corresponding

dependence

account

the

on

upon

for nonlinear meridtonal

loads. The small-deflection governing differentialequation for the subject cylindrical shell (refer to Fig. 3.0-37) can be written as follows [19]:

d4w

NT

1

dZMT (37)

+4_4w

where

=

-_-

Db(1-

v)

dx 2

Section D July 1, 1972 Page 152

' fl

ll-v'l

= 44Dba2

-

q

a2t2

t/2 =

NT

f

Ea

T dz

-t/2 (38) t/2 M T

= E__t/f

Tz dz

,

and

Et a Db

If

L >-2_/fl

ends

v2)

, then

of the

(37)

•

overlapping

cylinder

to equation CASE

12(1

-

effects

can be neglected

apply

to each

half

from and

of the

conditions

the following

at the

approximate

two

solutions

structure:

I (0 -< x -_L/2):

-fix w

CASE

boundary

= e

[I (L/2

(C 1 cos

fix+

C 2sin

fix)

-

aNT Et

_

a2 Et(1-

v)

d2MT dx 2

'

(39)

-
e -fl(L-x)[C

a cos_(L-

aN T Et

a2 Et(1

- v)

x)

+C 4 sinfl(L-

x)|

d2M T dx z

(40)

Section July

1.

Page The terms

e'_X(Cz

+ C 4 sin _(L

r

cos

- x)]

(37)..All

terms.

The remaining

the respective

portions and those

by Przemienieckl integration

Tx(x )

T2(x )

are truly

Ti(x ) =b 0+blx+b_

2 and

When the

temperature

of

T I and/or

soluUons

T 2 require

Truncate

than two and

2. the

actual

Either

of

Ignore

cases

higher

by using

either

where

from only

if the functions

or lower;

for

example.

z. that polynomial

expansions

of the

two procedures:

following

terms

as though

were

an

degree,

by eliminating

and

They

than the second

the

having

resulting

approximate

exponents series

greater

were

exact

T2(x ) .

restriction

on the polynomial

expansions

and use

formulations.

to determine

associated

those

terms

in these

the so-called

inexact.

degree

to

embodied are

results

Is such

been

the

exact

of second

dlstrllmtlon

are

1972

153

- x)

solutions

and (40)

somewhat

_L

approximations

give

will

have

(39)

are

two possibilities

no studies errors

influences

T2(x ) =d 0+dlx+d_

the stated

higher-degree

of these

and will

s cos

complementary

as first-order

the analysis T1(x )

here

polynomials

the series

perform

representations

given

process

can be obtained 1.

e-_(L-X){c

of equations

[20l

asymptotic and

and

of the boundary-condition

solutions

obtained

+ C 2 sin/3x)

comprise

equation

particular

_

D

performed

with various accurate

results

introduce

ranges are

inaccuracies the orders

and,

of magnitudes

of the parameters required

at present,

but equations

for

involved. (39)

For and

(40)

/

are

Section D July 1, 1972 Page 154

inexact,

these

the

user

expressions

retaining

can always

are

suitably

the foregoing

appropriate

solutions.

mathematical

operations

undetermined

coefficients

ships

any case,

must

from

can be accomplished but introducing

can be established of parameters

C 1 through

the boundary

the various

Free

or the

by more

by standard method

of

physical

C4 in the

conditions. possibilities

deflection

It therefore by means

relationbecomes

of the

edge:

Qx?M

=0

X

.

2.

Simply supported edge:

w

-- M

= 0

.

X

3. Clamped

W

where

as variation

whereby

•formulas: 1.

'°.

solutions The latter

the constants

to express

This

procedure

[21, 22].

be evaluated

necessary following

such

to an alternative

modified.

complementary

particular

For

resort

-

dw dx

-

edge:

0

.

(41)

Section July

F

D 1, 1972

Page

155

dM X

Qx

dx

and

(42)

dZw

MT

M x = -D b

The method having

to be used

a simply

the portion would

v) will

supported'edge

of the cylindcr

be inserted

identified

be illustrated at

as equation

x=

where

into equation (39a).

by using 0

and

0--x'_ (39)

Following

this,

example

a clamped

L/2,

to obtain

the

the

edge

values

a relationship equation

(39)

of a cylinder at

x=

x=:0, which must

L.

For

w =0 may

be

be substituted

into

d_v

MT

Db_-_+(1-

and

x

must

equation and and

(39b).

x = L, which

then be set

which

C 2 can

v)

may

then For

substituted

equal

(43)

to zero

be determined the

portion

be identified into

0

be identified

w = 0 would may

-

as (39b). by the

of the

be inserted as (40a).

in the resulting

constants

simultaneous

cylinder into

The

equation

Following

where (40) this,

formulation

of integration

solution L/2

to obtain C1

of equations

-
the

(39a)

values

to obtain

a relationship

equation

(40)

must

an

be

Section

D

July

1972

1,

Page

156

dw

-

dx

and

x

must

equation and

0

then

which

(44)

be set

may

equal

to

be identified

C4 can then be determined

L in the resulting as (40b).

formulation

The constants

by the simultaneous

to obtain

of integration

solution

an C3

of equations

(40a)

and (40h). Once cylinder,

the deflection

the bending

equations

moments

M

have

x

and

been M

found for both halves at any point

of the

can be established

from

d2w M x = -D b

MT

- (i

and

(45)

dZw M b = -uD b'_T-

Then

the stresses

X

-

MT (I- p)

at any location

are

given

by the following:

12z t3 M X

and

(46)

12z

Section D July 1, 1972

f-

Page 157 where

N¢

Example

.N T

(47)

wa Et

No. i. A thln-walled, right circular cylinder of length L(> 2_/fl) has both

ends simply supported and is subjected to the temperature

distribution

z

T

where

= d o_-

(48)

d o is a constant,

stresses

be found,

it is desired

assuming

For

the given

NT

= 0

that

temperature

that the

L = 40 in.

deflections and

distribution,

and extreme-fiber

/3 = 0.2.

equations

(38)

yield

and

(49)

MT

E c_t2dR 12

-

Then, for 0 _x

w

At x = 0,

-
-_x

= e

(CIcosflx+c

2sln_x)

.

the boundary condition of simple support requires that

Section

D

July 1, 1972 Page 158

w

=

M

= 0

x

(51)

where

Mx

From

d2w -D b "_

:

equations

(50),

MT - _-'--7-(1 v)

(51),

.

(52)

and (52),

the constants

C t and

C 2 arc

found

to

be

CI

=

0

and

(53)

C2

Hence,

ataz_2 (i(1v)

-

equation

w

-

(50)

function

_(_x),

may

now be introduced

at

-

may

now be rewritten

eet a2/_2 6(1 - Vi d° e-Bx

The

w

d0

sin

tabulated

fix

as follows:

.

by Timoshenko

(54)

and

Woinowsky-Krieger

[111

to obtain

a2_ 2

0(1 - v)

d,, ;(/ix)

(55)

where

;(_x)

= e-_x sin _x

.

(._.)

-_

Section D July I, 1972 Page 159 By substituting

equation

(55)

into (45),

the bending

moments

can be expressed

as

M

E°tt2d0

× - 12(1- v)

[00_x)

- lJ

and

(57)

E _ t2d II 12(1 - v) [v0(flx)

M

where [11]

the and

function

_)(flx)

is defined

is tabulated

by Timoshenko

and Woinowsky-Kricgcr

_,s follows:

0(_×) = o-_X cos _x

The

- 1]

extreme-fiber

stresses

.

(5s)

can then be determined

from

6M X O"

x

::

:F

t-V-

and

where

(5:))

the

equation

is obtained

upper (55)

for

signs

and

correspond

the first

N9 :

to the

of equations

outermost (49)

fibers.

into (47),

the

By substituting following

expression

Section July Page

D

1,

1972

160

E a_!a_2

N_ = - 6(1 ' _) do _(_x) For

the

which,

other

half

in view

x = L ,

w

-_(L-x)

the

(_o) (1,/2

(49),

" x :_ L),

equation

(40)

must

be used,

becomes

[Ca cos _(L- x) +C 4 sin/3(L-

boundary

M

=

cylinder

of equations

w = e

At

of the

.

condition

of simple

x)]

support

.

(61)

requires

that

-- 0

(62)

X

where

(1% Mx

From

MT

=- - Db (-_x2 -(1-

equations

(61),

v)

(fi2),

and

([;:l)

*

(63),

tile

constants

C 3 and

C 4 are found

to

be

C3 = 0

and

(64)

ata2_ 2 C4

Then,

6(1-p)

proceeding

following fiber

-

expressions

stresses:

do

"

in the same are found

manner for the

as for

the other

deflections,

half

moments,

of the

cylinder,

and extreme-

the

Section D July

1,

Page

1972

161

c_t a2_ 2

w -- (_(1-v)d,, _(l,-

Mx

E at2do 12(1-,,)

-

×)1 ,

{O[O(L-

x)l

((;._)

- 1} ((;6)

E_t2d° 12(1u)

M_b -

(u0[fl(L-

x)]

- 1}

6M X O'x

=

TT

'

and

0;7) N

6M

t

t_

where

Eo_t2,a_ 2 6(1u) do r_[_(L-

N@ =-

Here

again,

most

fibers.

used

the

upper

The

foregoing

to obtain

the

results

are

Example

clamped

T

for

(68)

formulas

the

solution

correspond

two halves listed

of the

in Table

to the outer-

cylinder

3.0-7.

wcrc These

3.0-40.

2.

A thin-walled, ends

relationships

in Figure

.

in the stress

nondimensionnl

plotted

No.

signs

x)]

right

and is subjected

= b 0+ blx

circular to the

cylinder temperature

of length

L(> 27_/_)

has

both

distribution

(G9)

Section July

1,

Page

0

I

1972

162

0

e_

!

D

I

O !

I

I

I

I

c_

0 u_

o I

Vl

"--

I

o

_4 t-,-

_D 0,1

0

o.

0

q_

_4

,-i

0

_4

_4 M

_4 o

,i,

I

_-

II

_4

_l 0 I

I

I

I

I

I

I

I

I C".I

.

i

I

' t"=

_

_

qO t"

_

0

...,4

_D C'.I

c; N e,l

H

n

Section July Page where

b 0 and

extreme-fiber bl/b o = 2

bI

are

constants.

stresses

be found,

It is desired assuming

that

that

the

deflections

L = 40 in.,

D 1,

and

fl = 0.2,

and

.

For

the

given

NT

= Eat(b

temperature

0+blx

distribution,

equations

(38)

yield

)

(70)

and

MT

]B

1972

163

P

=

0

°

0.4

/

\

J J /

/

\ \ \

/ /

/

\ /

\ \ L

/

\

I

L -f

Mx (E_ 2doI/12(l-v)

!

-1.2

0

Figure

10

3.0-40.

20 x(in.)

Nondimensional

moments

for

example

30

deflections problem

40

and axial No.

1.

bending

l

Section July •Page

Then for

0 -< x _ L/2

w

At

equation

the

clampc_l

dw

w ....dx equations

1, 1972 164

(4) becomes

= e-#X(clcosflx+C

x = 0 ,

From

,

D

2sin

condition

fix)

- acz(b 0+bix

rcquircs

)

.

(71)

that

o .

(71)

(7:_)

and (72),

the constants

C t and

C 2 are

found

to bc

C t = ezab o

and

(73)

C 2 = aab 0 +-==-=_-_

Hence,

equation

w

The

= e

functions

Krieger

[11],

w

(71)

may

now be rewritten

o_ab0(cosflx+sin_x)

9(fix) may

and now

= cxab09(flx)

_(flx)

,

as follows:

+_abt

tabulated

sinflx

by Timoshdnko

be introduced

to obtain

+_abt

- o_a(b 0 + blx )

_(flx)

l

-_a(b0_btx

) .

(74)

and Woinowsky-

(7_)

where

¢(#_)= e-_X(cos _x +_m _x)

(7s)

Section July

D 1,

Page

1972

165

and

;(Ox)

P

-- e -Sxsin_x

.

(76)

(Col_.) By substituting expressed

equation

(75)

into (45),

the

bending

moments

can

be

as

M×

Z {v:ll,,,J)tf_ [¢(fJ×)_ b-_Lb,,_ o(r_x)]

and

(77)

M

where

the

= vM

x

functions

¢(/3x)

Woinowsky-Krieger

[11]

and and

are

0(Bx)

are

defined

tabulated

by Timoshenko

and

as follows:

¢(_x) : e-_*_(co._l*x- sin _x) and

(7S)

0(_x)

The

: e -_x cos/_x

extreme-fiber

stresses

can

then

be determined

from

6M X (7 X

--

:1:-7.

(79)

SectionD July 1, 1972 Page 166 and

N --_ :_ t

a0

where

the

equations

upper

(75)

No

For

the

which,

correspond

(70)

half

the

-

equations

dw dx

of equations

(81)

the

- Etrblt_

cylinder

0

following

_(_x)

(L/2

(70),

fibers.

expression

By substituting

is obtained

for

condition

(so)

.

_ x -< L),

equation

(40)

must

be used,

x)

4 C 4sin/3(L-

requires

x)]

-_a(b

0+bix)

.

(Sl)

that

.

and

NO :

becomes

[C a cos_(L-

clamped

-

to the outermost

into (47),

of the

= e -_(L-x)

x = L ,

From

signs

(79) ( Con. )

= -Et_b0tO(13x)

in view

w

t2

and

other

w

At

(;M

(82)

(82),

the

constants

C a and

C 4 are

found

to be

C 3 = c_a(b o + blL)

and

(8:J)

C4 = _ _abt

+ aa(b 0 + btL)

.

Section

D

Juiy

1972

Page Then

proceeding

following fiber

in the

expressions

same are

manner

found

for

as for the

the

other

deflections,

half

of the

moments,

1,

167

cylinder, and

the

extreme-

stresses:

W

-"

_ab°(

1 + b--ll') o[_(L-x)]brJ

-_abfl+b-_x)fl _[_(L-x)]

-_ab°(

b:_

F

Mx

2a,_/t:"l,,I)l)

M

vM

{(l

I _-h lJ(, 10

¢[[_(1.-

x) J - t--h=0[_(l,-x) bfj_

l _I

,

x

6M X (y

×

.-_

-it-

and

(8(;)

N

GM

where

NO

Here

again,

most

fibers. The

used results

-F_bl,

the

are

upper

foregoing

to obtain

t

the

plotted

(1 + F-hbo I_/

signs

_l/3(l,-x)

in the

stress

relationships

nondimensional in Figure

for

_[/3(1,-

formulas

the

solution 3.0-41.

J + _fl

correspond

two halves shown

x)]

of the

in t_able

.

to the

cylinder

3.0-_.

outer-

were: These

(87)

BecUon D July 1, 19'/2 Pap TABLE

S. 0..8.

[_OIlDIMi_NSIONAI,

w

1

3

4

3

0

1.000

90LT_rYION

3

o

-I

0

7

A,"

,(_)

0

8

0

_ o(,e,0

1.0

10.0

Asb

11.0

0.5

0.1

0.9110'1'

o. 0o3

-8

-0.1063

0.61

9. 003

9.613

1.0

0.8

0.9651

1, 4:r?

-$

-O. 40'70

O. 6308

8. OM

8.6638

8.0

0.4

0.85'84

2,81o

-6

-1. 8118

O. 35414

6. 174

6.53o4

8.0

0.8

0.'7628

3.099

-7

-8. 1382

O. 1431

4. 530

4.6731

6.0

1.0

0.6O03

3.o!14

-11

-?.9067

1.086

1.8772

10.0i

2,0

0.0607

1.86o

-91

-19. 7033

-O. 1796

-0. 543

-0.7424

-0.0423

o.o71

-31

-80.0713

-O. ON3

-0.498

-0.5493

-o, 117

-41

-41.1438

O. 0019

-0. 120

-0.1181

15.0'3.0 20.0

4.0

-0. 036O

FOR

Mx (eXJLM1,LENO. 2)

jb. r.(ox) 0

TABULAR

168

-0.1108

,. ,, . (,,/a-%) - (3) +(4) ÷(8) b. X, - (,,,_'lo-'.,nb_') - (7) ."(S)

(,+L,,)b,

_b, boD X

X

x

(L..q

#(L-x)

_l,e(_,0!

_(p(L-x)]

1

2

3

4

5

X

-{1+_x)

-(4)uabo +(_)+(0) _,1 p(t,-x)I

6

7

x

elp(L,-x)l

M X

2 o_aboDbd_z = 18) - 10)

8

9

10

0.1539

-0. 120

0.0?39

20

20

4

-3.06O8

0.11'/

-41

-49, 9'/28

25

16

8

-8.4283

-0.071

-61

- 54.4073

-4.5603

30

10

2

5.4027

-1,230

-61

- 56. 8273

-14.5314

35

8

1.0

41.1';'1_

-8.0Oil

-71

- 32.995t7

-8.9748

1.988

- 10, 9628

37

8

0.8

61.7849

-8.099

-78

-16.3121

11.6Oll

4,530

7, 0611

88

8

0.4

71.1504

-2.010

-77

-8.4506

28.8084

0.174

22. 6944

H

LO

6.|

18.17:11

-1.037

-79

-3.4639

81.8238

8. 0P,4

43.7988

81, |

O. $

0.1

80. Me7

-0.908

-80

-0. 8668

68.610

9. 003

56. 007

40

0

0

81,00

0

-81

0

61.0

-0.49_ -0. 563

10.00

-4. 0078 -13.9684

71.00

Section July

1,

Page

?---_

/J

/f

00

I0 m

X

2__a boDb/32_

2O

0

!/

\ \

\

\

",,,_J/

\ \

\ \ _'°

/

1

\

/ 1

\ \ "411

Figure

I

I0

3.0-41.

10 _(o..)

Nondtmensional

moments

for

/

\

example

deflections problem

W

and No.

axial 2.

U

bending

D 1972

169

Section D July

1,

Page III.

Thermal

Stresses

Axisymmetric A. The walled,

of isotropic

equations

cylinders

which

the Isotropic for B.

The 1.

Qx

forces,

types

edge;

:

that

Gradient,

thickness

wall

is free

only when

configuration.

only

to thin-

and are of holes

h - _ .

Figure

made and

Figure

3.0-43

shows

3.0the

pressures.

Conditions. of boundary

conditions

are

discussed:

is,

(88)

supported

w

=

= 0

3.

Clamped

X

shell

is valid

and

apply

0

2. Simply

M

shell

here

of constant

that the

moments,

Boundary

= Mx

are

The method

cylinder

following Free

presented

It is assumed

Hooke v s law.

convention

Radial

Gradient.

and

material.

42 depicts sign

curves

circular

that it obeys

Axial

- Constant

Configuration.

design

right

and Deflections

1972

170

edge;

that is,

.

edge;

(89)

that

is,

dw

w

-

All possible itisnot

dx

0

.

combinations

required

that

(90)

of these those

at

x=

boundary

conditions

0

same

be the

are

as those

permitted. at

x=

L.

Hence, However,

Section July Page

NOTE:

D l,

1972

171

u,v,w,x,y,z, and _) are positive as shown.

t MIDDLE Figure

3.0-42.

SURFACE

Isotropic cylindrical thermal stresses and

shell deflections.

configuration

for

N_

Y _

Nx x

N

Mx_

J

NOTE:

All quantities positive as shown.

N_

Figure

3.0-43. pressures

Sign for

convention thermal

for

stresses

forces, and

moments, deflections.

and

Section

D

July

1972

1,

Page in every

case,

it is assumed

direction

(N x

the

wall

Temperature

supposition

thickness.

gradient

(or

lower)

permissible

apply

23,

shells

by equation this

the

product

no temperature

cylinder

tyT

in the

can

of the

variations

may

have

axial

subject

occur

through

any axisymmetrie

be adequately

Therefore,

are

I)esig'n

to compute

relationships

is unrcstr:dnetl

represented to that

surface by a fifth-

restriction,

the

form

(x) "r(×)

In Ref.

coefficients

the

that

polynomial.

I).

cylindrical

is made

distributions

,,'r :

below

cylinder

i)istribution.

However,

for which

degree

the

0) . C.

The

that

172

are

(.0

Newman the

which

method,

present

with

from

function

c_T

the practical

deflections, the

in series

product

this

Forray

stresses,

comply

expressed

the

and

thermal

can be obtained (91),

and I,:qu:ltions.

Curves

and

foregoing

form

and

the

3.0-44

will

be a function first

rotations

specifications.

Figures

must

methtxl

necessary

through of

in circular The

primary

term-by-term

3.0-49. x

given

and,

be approximated

As in order

indicated to

by the polynomial

Z (_T

where

= d 0+dz_

_ (12_ 2 _ °..

is a dimensionless

x

axial

dz z

coordinate

=

_ k=0

dk_k

defined

(:)2)

by the

relationship

(93)

! ! O

!

Section July

D 1, 1972

Page

173

0.48 I.l..

,.. 0.32 0

.m

0.16

I,,t..

0

Figure For

the purposes

of the

I

2

3.0-44.

technique

3 4 _;_ or _'X Functions

given

here,

5

6

F.. 1

the following

inequality

must

be satisfied:

(94)

z -<5 .

After and distortions conjunction

are

listed

coefficients

dk

can be determined with the design

the boundary boundary

the

conditions

conditions

at

in Table

3.0-9.

at

curves.

have

been

by using The

x = 0 (_ = 0)

x = L (_ = 1) .

established, equations

(96)

A 3 and

The formulas

for

thermal

through

A 1 and

constants while

the

stresses (98)

A 2 are

A 4 depend these

four

in

based on the values

on

Section D July i, 1972 Page 174 I

_.o_

,4 //

"I

i-" o.I l II1

/ /

/

..- o.s

/

.J

0 u 0.4

/

r. 0 w _

/

3.5

!

//

//

/

/

,11

,/ /

0.2

Ul

k'4/

/

i/

'_- 2.5 Z .J

Q w.

2

/

iu

0 1

A i H

o.e II1 m cJ --

.1

_

_

/2*

/f

1.5

.,<._'/I

0.6

'

IL Ut

0 ¢J

/ J

j// O.4

_'3

/

/

-_._//

/

3.5,

X> ,ss_

ul

0.2

o/-

W

Q

_._...._ _

0_.

"

t

I

r

A /! ///

i

i

-O.2 1 AXIAL

COORDINATE,

,n 0.8 I

A

Z _

6 I U.

0.6

,t

ul

_

Io

W .J Lk

z

I...3.2

0.4

i L [

I 3.5

!°

,

3.5'4-_ 15"-.

0.2

_> _5_

,//

/A / --..d_

0

0

/

0

-1 -0.2 (_

0.2

0.4

0.6

Figure 3.0-45. coefficients and coefficient

0.8

1

Deflection k = 5, the

depends

on

h .

0

Figure (for

0.2

3.0-46. k = 5 , depends

0.4

0.6

Slope coefficients the coefficient on

)_ ) .

0.8

1

Section July Page ZO

/ /

16

I ,,s/

J

/

U

I

|

/ /L

2

/

_L

/

J--'_1

_2 ! 10, I

o

02

04 •

I,'i_rc

AXIAL

115

Moment

3. O-47.

oil

COORDINATE._

coefficients.

$0

/

?

/

/.

u

j/

//r

0

0Z

0

04 AXIAL

Figure

05

Shear

3.0-48.

08

COORDINATE.

coefficients.

I

/

/

J

J 4q,

I 0,t,2,3_

0fi/ 0

02

04 AXIAL

Figure

3.0-49.

06

08

COORDINATE.

Hoop-force

coefficients.

/

D 1, 175

1972

Section July

D 1,

Page

TABLE

3.0-9.

Cylinder End

FORMULAS

FOR

THE

CONSTANTS

Boundary Condition

A 1

Constants

THROUGtt

At

l

Z A 1

i-'1ix'_I _ , dkmk(0

)

Free

z

A 2

" _1 2k_" k_=0 dk

i mk(O)

] %(0)"

4X

Z A I

2A_ k=0 _=0

(x = o)

Si mph' support

Z A:_

-

At

::

_' dkWk(0 k =0

_ _z dk k---O

)

Wk(O)

'

IOk(O }..

(!l;iml)ed Z dkWk(0) k -o

4=1 (x = L)

A3

Z 1 2_----_ _ k-0

A1 =

_

dkmk(1)

F re(,

,z

_ dk k---O

qk(1) - ink(l) r

Z

A_- 2)'2 _ k=0 E %%(_) 4=1

(x = L)

Simple support

Z

A, =- Z dk_k(1) k=0

A_

=

_Z d k i"1 kOk(1) k=0 -

_=1

(x - L)

Clampc_J Z

A, - - 2 _k (1) k=0

-Wk(1)

"I

1972

176

A 4

Section July Page The

f

solutions

Therefore,

to pressurized

deformations because cannot

upon

for

account

for

effective

values

averaging

upon

classical

coupling

it is assumed

for

each

shell

of this

when

the thermal

due

solely

theory,

the

thermal

Youngt

s modulus

theory.

the

method

stresses

to pressure; method

between

1, 1972 177

and

that

presented

deflections

is

is, here

and

the

loads.

by temperature

values

to be aware

the corresponding

mcridional

unaffected

the user

small-deflection

by superimposing

nonlinear

In addition,

on cl:lssical

cylinders

of the dependence

pressure-related

are

b:ised

it is important

applied

f

arc

D

that changes.

of these

Hence,

properties

the

and

user

by employing

Poissonl

must some

s ratio

select type

single of

technique. E.

Summary

of Equations

and

Nondimensional

Coefficients.

Z .T

:: d o_ d_

_ d_ :+...

d z _z

=

_ k=0

_k

,

(95)

Z w a

_ AlI,,2(/;_. )

i A2F4(_h. )

, A:lF._(t_'X ) + A4F4(_'_. ) + _ k=0

dkW k

Z L

0

=

XIA1F3(_,

)

-

A2Fi(_)C

)

-

A3F3(_'_.

)

+

a

h4Fl(f'X)l

+ k=0

dk0 k

Z \, '

L2M x

aD b

_

2;tZl-AiIe4(_2t)

_ A.,F2(_X ) - A3F4(_l)t) k

diem k , 0

Section July

1,

Page

MO

= VMx

D 1972

178

(96)

'

Z 2)_3[AIFI(_k)

aD b

+ A2F3(_)_)

- A3FI(_'_,)

- A4F3(_'k)]+

_ dkq k k=0

and

Z :: 4k4[AiF2(_2t)

+A2F4(_k ) +A3F2(_tk ) 4 A4F4(_'X)] -

a2D b

dkn k k=O

where

h

Et 3 Db

-

12(t-

(97)

v 2)

x L

'

and

It = 1-4

The

stresses

•

at any location

12z

% -- -V- M

are

given

by the following:

(98)

Section July

f.-

D

1,

Page

!972

179

and

F

'79

:: Not - -712z M 9

.

(98)

(Con. )

3.0.8.2

Isotropic This

right

section

circular,

different This

Conical concerns

isotropic

from

that

plates

and

it is helpful

the

conical

to group

solutions

which

cover

here

isotropic

between

deformations for

stable

of thin-walled,

organization

differences

Flat-plate

the

analysis

The

sections

fundamental

shells.

thermostructural shells.

of previous

is due tG certain

of flat

Shells.

the

are

is somewhat flat

plates.

physical

of such

constructions

behavior

a nature

into the

that

following

categories: 1.

Temperature

2.

Uniform

Except thickness

which

middle

surface

is,

middle of cases However, ponents,

the

for the

is,

surface

there

case

of these

occur

a logical

is no need

to isolate

two cases

In case

bending for

same

involves

2, the

in directions

format

is not the

gradients

by displacements

plate.

and no out-of-plane

situation

the thickness.

accompanied

undeformed

the thickness

of self-equilibrating

the first

displacements

is therefore the

through

special

of course,

of the

only

through

teml)eraturcs

(N T -: M T : 0),

bending

that

gradients

the

The

sections

for shell

the foregoing

plate

structures. types

to the

remains to the

indicated

dealing

the

out-of-plane normal

parallel

occurs.

through

with For

of thermal

flat;

original separation flat these

pl:_tes. com-

conditions.

Section

D

July

1972

1,

Page This

is because

clamped and

are

or simply

bending

shell

either

type

of temperature

supported

I)ounti:_ries,

about the

constructions given

shell-wall

which

as follows

middle

comply

as a single

with

distribution, will

load

surface. either

in conjunction

to both

with

membrane

Consequently,

case

180

i or 2, the

loading

for

stablc

analysis

methods

grouping.

Configuration. The

design

thin-walled, are

equations

truncated,

made

right

of isotropic

Figure tion

conventions

satisfy

which the

to long

are

(L -> 21r/hB),

of constant

thickness,

inequality

wall

the

is free

subject

of holes

configuration,

and obeys as well

Hooke' as most

s law. of the

nota-

of interest.

Conditions.

boundary

method

presented

conditions 1.

Free

2.

Simply

3.

Clamped

All possible

case,

cones

only

(9.'))

the shell

sign

The

required

that

depicts

Boundary

and

apply

.

3.0-50

and

here

circular

material,

xA>atcot

It is assumed

provided

are

can

be applied

where

any of the following

present:

edges supported

edgcs

edges.

combinations

that

herr,

those

it is assumed

at

of these

xA

that

the

be the cone

boundaries

same

as those

is unrestrained

are

permitted;

at

xB .

in the

that

is,

However, axial

direction.

it is not

in every

Section July Page

,

x

1972

181

HA

R2

HB

(R2)

NOTE:

1. s* = L-s 2. H A , HB, M A, M B, V, and W are axisymmetric. 3. All coordinates, forces, moments, and deformations are positive as shown. a.

D 1,

Overall

truncated

cone.

Me

Ne

N

b. Figure

Positive 3.0-50.

x

directions Configuration,

for

the

stress

notation,

resultants and

sign

and convention

coordinates. for

conical

shell.

Section July

l,

Page Temperature

1972

18?

Distribution.

The

following

1. A linear

types

through

change

Axtsymmetric

The permissible

of temperature

gradient

that the temperature 2.

i)

T

distributions

the wall

need

meridional

distributions

may

thickness

not vanish

be present:

subject

to the provision

at the middle

surface.

gradients.

can therefore

be expressed

in the form

Z

T

Naturally, either equal Design

= TI(s ) _ T2(s ) _-

any

of the

or both

of

here, the

and

T2(s )

for this can

equation

be finite

are

applicable;

constants

that

and either

is,

may

bc

Equations. of methods

including particular

manner

those

suggested are

accuracy

is desired,

Orange

Throughout

the

27.

governing

[16].

As

subject

exact

problem

In the

in Refs.

complementary

have

approach

differential

by an equivalent-cylinder the

may

solving 24 through

to the

by Tsui

obtained

[29]

for

of Refs.

solutions

solutions

and

cases

to zero.

A number lished,

special

Tl(s)

(100)

.

equations

25 and

28,

pub-

presented are

found

in

the complementary

approximation. solutions

been

When

published

greater

by Johns

be used.

this section It is assumed

Polsson' s ratio are unaffected by temperature

that Young' s modulus changes.

and

Hence, the user must

select single effective values for each of these properties by employing type of averaging technique.

The same

approach may

some

be taken with regard to

Section

D

July 1, 1972 Page 183 the coefficient dependence

of thermal of this

expansion.

property

may

On the

other

be accounted

hand,

for

the temperature-

by recognizing

that

it is the

f-

(

product

sT

be suitably is taken,

which

governs;

modified any

that

is,

to compensate

mention

the

actual

temperature

for variations

of a linear

temperature

in

distribution

_.

When

distribution

can

this

approach

is actually

making

/

reference

to a straight-line In addition,

deflection

theory.

pressurized upon

method

for nonlinear meridional

values

classical

coupling

tbe

due

between

the

thermal

o_T

is based

this

.

on classical

in mind

thermal

solely

theory,

product

here

to keep

by superimposing

the corresponding upon

of the

outlined

It is important

cones

dependence

when

stresses

to pressure;

method

that

of this

deflections

manual

small-

applying

to

and

deformations

is,

because cannot

of the account

and pressure-related

loads. The

Tsui

the

variation

[16]

governing

differential

equations

for

the

subject

cone

are

given

by

as follows:

dN T L'(U)

- VEt tan

dx

and

(101)

1 L'(V)

where

, U_b b cotO-

1 -Db

cot__ (1-

dMT _)

dx

Section July Page

D

1, 19"/2 184

Et 3 - 12 (I- v"_)

Db

t/2 MT

=

E_

Tz dz

f

-t/2 (102) t/2 NT

=

E(_

Tdz

f -t/2

and

v -xQ

and

L'

x

is the operator,

...,, =oot,[. To obtain lined

the desired

solution,

a three-step

(103)

procedure

is employed

as out-

below:

step 1,

Find

a particular

Step

Find

a solution

L'(U)

2.

solution to the

to equations

homogeneous

(101).

equations,

- VEt tan q) = 0

(lO4)

and

L'(V)

+U_bbCOtib

--

0

,

Section D July I, 1972 Page 185

such

that superposition

boundary

conditions

Free

results

Note

that

equations (]01)

Step

UP

and

VP

are are

= M

W

= M

W

= V

referred

1 satisfies

= 0

X

.

= 0

X

(105)

= 0

.

to as the complementary

obtained

the

by setting

the

solution.

right-hand

sides

of

to zero.

Superimpose

the

approximated

step

(104)

equal

3.

To accomplish

edge:

of Step

as follows:

"X

this

those

can be expressed

edge:

from

equations

upon

Q

supported

Clamped

The

which

results

edge:

Simply

f-

of these

first

the particular

of these

steps,

the

as polynomials.

It is then

can

in the

be expressed

and

complementary

functions

assumed

NT

solutions.

and

that

the

.

C

MT

are

particular

first

solutions

form

P

n

U

=

C_lx-l+

C 0÷

C1x+

C2x2+

C3x3+

.

.

+

x n

and

(I0,;)

VP

where

n

required formulations

= d_ix -i + d o t dlx

is an integer for

a sufficiently for

N T,

whose

_

value

accurate M T,

U P,

d2x 2

+

d3x 3 +

is a function representation and

VP

are

.

.

.

+ d xn n

of the of

polynomial NT

substituted

and into

degree MT .

If these

equations

(101)

Section

D

July

1972

Page and like powers obtained

where

equations

x

are equated,

the unknowns

can be solved

associated mined

of

radial

for

deflection

a system

are

of simultaneous

the various

C( )

and

polynomial

d( )

and stress

and hence

resultants

equations

coefficients. UP

of interest

and

1,

186 is These

VP .

The

can then be deter-

from

--P W

-

c°s2 P Et stn¢

Qx P

-

UP

P NX

= Qx cot ¢

Nop

X

x

_dU P -

vU

+ aRT

P)

m

t

(107)

d (R 2Qx)

=cotO

dU dx

])

P

M

Db

X

(d_+

v V P cot0 "_2

)

MT - (1:v)

and

P

M0

V

= Db

P cot_b

dVP) + v--'-_-

- (;-MT _)

where

t/2 T dz Tm

=1 t

-t _

(108)

Section July

1,

Page The complementary Figure

3.0-51

solutions

are given

to the edge-loaded

cone

corresponding

as equations of Figure

to the edge-loaded

(109)

3.0-52

and (110). are

given

cone

Those

V:

-

of

as equations

(111)

and (112).

A

Figure

=

1972

187

corresponding

HA

_C

D

3.0-51.

Truncated

cone

edge-loaded

at top.

sin 9 2A/ D b (_'A MA + ItA sin _b)

2?,/

1

Db

(2_,A MA d HA sin q)

m

--c W

-

sin9

2A/D

[xA

b

-

MA¢(AA

s)

+ HA (sin _) 0 (XAs)] (109)

VC

_

2_:2D

b

[2AAM AO(XAs)

+ HA (sin

q) 9 ()tAS)

C

Qx

= [2k AM A_

(AAS)

- H A (sin _b) _ (),AS)]

]

_ctlon

D

July

1972

Page

1,

188

C Nx

= Qx cot

NC

-

WEt R

+vN

X

(109) (Co..) C

1

Mx

= -_2k

C M0

= vM x

A

"[2hA

MA ¢ (XAs) + 2 HA (sin ¢) _(XAs) J

and

where

RA (R2) A

-

(110)

sin9

and

Et 3 Db

E

and which

-

12(1-

v 2)

m

9,

3, are

O, tabulated

and

_ are on pages

the

functions

472-473

9,

of Ref.

@, 11.

0,

and

_,

respectively,

Section

D

July

1972

1,

Page

MB

189

}4B

HB

Figure

3.0-52.

=

V:

_C

vC

_c

=-

Truncated

cone

edge-loaded

(ABM B + HB sin _)

_

= s___

(2ABMB

b

1 = - _kB-'_Db

.

+ H B sln_)

[ ABMB_(KBS,)

,

+ HB (sin _b) O (ABS')J

[2ABMBS

at bottom.

(ABe*)

+HB(ehl

_. _) ,,', (7,Be,*);

:

. . [2AsMs _ (AS,,}. "B (sm_) _'(Ass*)j

= Q_, cot

w

Nc

wzt ---'_-+VNx

'

(_i)

Section

D

July 1, 1972 Page 190

= _ 1

I2ABMB_

(ABS,)

+ 2 H B (Sin ¢) "_ (ABS*)I

and

Mc

= vM x

(111) (Con.)

where

BB (B2)B

= s-_n¢

'

(112)

and

Et 3 Db

and

9",

which

-

_',

are

0",

v 2)

and

tabulated

After imposed,

12(1-

_" are the on pages

the particular

the final

thermal

functions

472-473

¢,

_,

of Ref.

11.

and complementary stresses

8,

and

solutions

can be computed

_,

have

from

respectively,

been

super-

the following

formulas:

12z cr = X _

M

Nx +-X t

12z and

or8 -

ta

N8 MS+-_-..

'

(113)

Section

D

July

1,

Page 3.0.8.3

Isotropic The

discussion

deflection

solutions

arbitrary

shapes

along

Shells

presented for

and

with pertinent

of Revolution here

thin-walled

made

of Arbitrary

of isotropic

notation

and

Shape.

is concerned

shells

with

sign

approximate

of revolution material.

1972

191

having

small-

otherwise

A typical

configuration,

is shown

in Figure

conventions,

3.0-53.

.f--

It is assumed

that

temperature

the

shell

distribution

be present

both

To determine discussion,

must

through

the the

thermal

following

sets

2.

Strain-displacement

3.

Stress-strain

In principle,

together

with

a sufficient

difficult,

if not

numerical

integration

another

cited mations

avoid

the

are

gradients

meridional for

s law.

the

The may

direction. structures

under

available:

used may in Ref. need

for

boundary

for

subject

the

impossible,

to achieve

be taken

the

desired

by using

30 or Christensen sophisticated

these

formulations

of closed-form,

However,

in conjunction

to achieve

conditions,

development

problems.

procedures

approach

by Fitzgerald

and deformations

prescribed basis

extremely

frequently

but arbitrary

and in the

of equations

Hooke'

relationships.

to the

still

thickness

obeys

relationships

solutions

are

and

equations

deflection

gram

of holes

be axisymmetric

stresses

Equilibrium

provide

is free

thc wall

1.

should

wall

it will

such with

mathematical

bc

Thcrcfore,

a digital

computer

On the other

approximations in Ref.

often

solutions.

solution.

small-

31.

such Since

and/or

prohand,

as those these

numerical

approxi-

Section July

1,

Page

MIDDLE SURFACE

rl, -

f3

*

r 2,

positive

....

=

a.

Overall

shell

and

•

of revolution.

NOTE:

Positive moments,

directions pressures,

for

are

as shown.

MIDDLE SURFACE

be

_

c.

forces,

de

x, y, and z are positive as shown.

Element

of shell

wall.

and

coordinates.

Figure

3.0-53.

Configuration, arbitrary

notation, shell

and

of revolution.

_ign

convention

D

for

192

1972

Section July Page operations,

they

are

well

suited

to a manual

be desirable

to prepare

a section

these

However,

from

which

of this

outlines

type.

It would

detailed

D

1,

1972

193

therefore

procedures

along

f"

lines.

that

they

made.

should

a brief

be thoroughly

Consequently,

study

explored

in the

of Refs.

before

following

30 and 31,

specific

paragraphs

only

it was

concluded

recommendations

arc

the

con-

related

broad

/

cepts

are

presented.

The equations,

method

of Ref.

which,

except

d__ (N_r0)

30 relies for

_ N0rl

heavily

the term

on the

involving

following

m_ ,

cos _ - r0Q_b _ r0r 1 Y

are

set

of equilibrium

derived

in Ref.

11:

= 0

d N br 0 +N0r

1 sin_

+_

(Q_br0)

+Zrlr

0 :

0

,

(114)

and

d -_(M_r

These

O) -Mor

expressions 1.

First

are the

N O = N0

and

used

assumption

= NR

bending moments

lc°sO-%rlrO+m_brlr

in the

0

following

is made

that

= 0

manner: membrane

forces

(115)

Section D July

1,

Page

M0 = M

ar_: present

putations

--- M R

which

These

1972

194

(116)

completely

forces

arrest

all thermal

and moments

and do not represent

the

simply actual

displacements.

furnish values

a starting which

will

point

for the com-

be determined

later

in the procedure. It follows

that

E =

NR

(1-

t/2 f v) _ti../_

_T

dz

(117)

and

t/2

MR = __1E - V) 2.

In general,

be in equilibrium

_Tz

the above

unless

dz

(118)

.

-t

type

one or more

of force

and moment

of the following

distribution

will

not

is applied:

Q_ = (%). , Y = YB

'

Z = ZR

,

and

n1_

--

m r

.

(,19)

Section D July i, 1972 Page 195 At this point, in order to achieve an approximate makes

the assumption that

%

and

= (Q_)R

justifies

this

error

introduced.

(115)

through

arrive

loading,

= 0

practice

are

proceeding

for that

it is necessary

the

an order-of-magnitude with

substituted

formulas

Recognizing

(120)

by performing

Then,

(118)

at simple 3.

of the

solution, Fitzgerald [30]

into the

the

analysis,

equilibrium

YR '

ZR '

and

actual

shell

is free

to restore

the

study

equations

of the

(120)

and

relationships

(114)

to

of the

types

of

mR . of any

structure

to this

as outlined

below.

state

above

by application

following:

-YR; -ZR; -mR

This

is done

in a two-step

4.

The

Y

= - YR

procedure

expressions

and

(121)

Z

are

inserted

assumption

= -Z R

into the that

first

two of the

equilibrium

equations

(114)

while

the

Section D July

1,

Page

196

Qo = o

is retained.

(122)

The resulting

the stress-strain After

related

this,

equationB

relationships,

The bending

and deflections

moments

M8

and

are then solved

the corresponding

the strain-displacement

rotations

1972

formulations

of the shell _

for strains

may

wall

N0

N0 .

From

can be determined.

be used

in terms

and

of

can then be established

to express

N0 from

and

the

N0 .

the equations

M 0 = _ Db(X 0 +vX O)

and

(123)

M0

= - Db(X 0 +vX 0)

where

Et a Vb

while

Xe

-- 12(1-

and

X0

v 2)

are

'

(124)

the curvature

changes

of the hoop and

meridional

fibers,

respectively. 5. One may now proceed

m0

into the third

to substitute

= - mR

of the equilibrium

(125)

equations

(114),

along

with the assumption

that

_o = Mo = o .

(126)

f-

Section

D

July

1972

1,

Page

Simple

transformation

equations

(126)

N O and

N_

and the first

in terms

the development magnitude ships,

practical

ments

associated

priate loads with

Using

with final

and any

bending

The use

values

1, 4,

moments

self-equilibrating

and

must

together

in this

on the basis

for

phase

the

and

displacc-

in this

loads,

bending

by superposition

stresses

be augmented

relation-

obtained

membrane

found

distributions

membrane

stresses

which

step.

of appro-

due to these

by those

of

of an error-

the rotations

N_

with

expressions

(126)

N O and

for

The

temperature

to simple

for

are 5.

which,

and strain-displacement

loads

and displacements steps

[30]

can bc derived

membrane

Q_

of equations

by Fitzgerald

approximate

from

for leads

the stress-strain

the

rotations,

values

MR .

formulations

The

moments,

of

a formula

two of (114),

is justified

study.

6.

thon yields

197

associated

exist

through

the

thickness. To focus

attention

approach,

no mention

prescribed

boundary

now be helpful results Following enforce

this,

edge

the required

similar

in the

conditions that,

forces

concepts

foregoing

in the

for this

the assumption

The general very

is made

to note

under

on the general

that no external and/or

conditions philosophy

to that of Fitzgerald,

steps

problem

method,

moments

involved

of the need

solution.

constraints

best are

s [30}

to satisfy

Therefore,

it is probably

may

in Fitzgeraldl

it might

first

to obtain

present.

be superimposed

which

at the boundaries. behind

the approach

although

of Christensen

the details

are quite

[31] different.

is

Section D July 1, 1972 Page 198 Chrimtmmen

also

relies

pure

thermostructural

and

m#.

entire

these which

1. First

upon the equilibrium

problems,

Hence,

analysis,

heavily

he makea

quantities

the assumption

no use

are taken

is performed

equations

equal

of the loadings

to zero

in the following

is made

(114)

but, Y,

throughout

for Z ,

the

manner:

that

= M b = MB

Me

(127)

where

t/2

E

f MB

Here and

= - (1-

again,

these

procedure. The third

tions

the

These of these

By using

two equations

tribution measured surface.

and

simply

actual

moments

.

(128)

N0

furnish

values

are

equations

in the unknowns 2.

the

moments

do not represent

c_Tz dz

v) -t/2

and

N¢

step

the middle-surface

in the meridional

1 are

will

point

be determined

into the equilibrium

combined

for the computations

with the

other

later

in the

equations two

(114).

and two equa-

are obtained.

the stress-strain from

which

inserted

is then

a starting

and strain-displacement rewritten

in terms

displacements direction

and

w

v

relationships,

of the temperature and

is taken

w, normal

where

v

disis

to the middle

Section D July I, 1972 Page 199 3. The two equations from step 2 are combined formulation in terms of v

and the temperature

to arrive at a single

distribution.

f--

4. The equation from step 3 is then solved subject to the boundary conditions at the shell apex. expressed

This is accomplished

by assuming

as a polynomial and then calling upon the method

coefficients. The resulting expression for v

that v

car be

of undetermined

must then be substituted into the

appropriate equation from step 2 to obtain a solution for the displacement

5. From

Timoshenko

[11], the bending moments

are associated with the displacements

v

and w

M 0 and

M_

and, if they are not

small with respect to M R , an iterativeprocess must be used whereby moments

are successively revised.

reported in Ref. 31 seems

which

can be determined.

Christensen [31] refers to these as corrective moments

initiallyassumed

w .

However,

the

the study

to indicate that the first cycle will oRen be suffi-

ciently accurate for most engineering applications. 6. From

membrane

the stress-strain and strain-displacement relationships, the

loads N O and

N_

due to v and w

7. The final approximate loads, and displacements

can now be found.

values for the bending moments,

membrane

are found as follows:

a.

Final

M 0--M

b.

Final

Me = M R + corrective

M¢ .

c.

Final

N0

from

d.

Final

v

and and

R+corrective

N¢ = obtained w = obtained

M0 .

from

(129) step

step

4.

6.

Section July

1,

Page The total those

approximate

associated

values

with the final

any self-equilibrating To focus [33],

attention is made

scribed

boundary

it might

now be helpful

present

under

at such

positions.

ueed

for

sophisticated

solutions many

can also

c _which

programs tions.

are

the

efforts

include

work

this,

edge

the required are

by the use

best

along having

circular these

numerical of existing

for

feel

lines

to arrive shapes.

first

are moments

may

at the boundaries. of possibilities values

digital

without

accurate

it would

It is recommended at the equivalent

the

programs,

methsds.

mechanical

for

However,

computer

rapid,

for the problem,

into equivalent

best

operations.

obtaining

pre-

Therefore,

and/or

approximate

cylinders.

arbitrary

forces

or finite-difference

approach

a physical

to satisfy

constraints

only two of a number

and/or

by Christensen

it is probably

conditions

to obtain

distributions

done for isotropic

method,

and

thickness.

from'theapex.

that no external

discrete-element

to retain

temperature

removed

for this

can be used

either

probably

of revolution

that,

enforce

be obtained

as was

shells

of the need

mathematical

use

However,

to convert

steps

at locations

loads,

the wall

in the foregoing

Following

and

through

1972

200

by superimposing

membrane

proposed

approaches

problem

final

distributions

the assumption

which

The foregoing

moments,

concepts

to note

results

subject

bending

are obtained

on the general

conditions

be superimposed

the

the stresses

temperature

no mention

to obtain

for

D

Such solu-

be helpful

loadings,

such

that future pressures

for

Section

D

July

1972

Page

I.

Sphere A.

Under

Hollow

Radial

Temperature

Variation.

Sphere.

Inside radius = a. Outside radius = b.

cr

rr

2+F r3a3b

-

1- _ (b_- _r_

f

1 Tr2 dr - _-_

a

f

aO0

U

= _

olE [ a3 + 2r3 1- u [(b 3- a3)r 3 a

-

1

a3 3 I "_

T(r)

:

to :

= a00

constant,

:

and

u = _Tor

I+

f I

_5 +

b fTr2dr+ri r

2(12v)r (i +p)

I

Tr 2 dr

--

1 + p_ _'__p/b3_a

_rr

r

f It2 dr a

.

_

:: o

b3

f a

b

r fTr2dr a

Tr2d r

Tr 2 dr a

-

1,

201

Section

D

July

1972

Page B.

°'00

Solid

= °'¢_b -

Sphere.

aS 1-v

/l+v_[ U = "_-_.v/

Crrr(0 )

= _r0e(0)

T(r)

= To

Crrr

= _00

1 af r-'i

= _

•

f Trldr+._ 0b

Trldr + (1

= _(0)

= constant

and

U = _Tor

2 l_'l

-- 0

-

,

,

i-_"

f Tridr_ 1 0r

+ v)

1b

T/

Tri

0

-

3

'

1,

202

Secti(_n

l)

July

1972

1,

Page 4.0

THERMOELASTIC The

203

STABILITY.

thermoelastic

problems

considered

in the

previous

paragraphs

¢_---i.

have

followed

thus

excluded

depends

and

some

of the

of the linear

questions

on the

tions,

the

formulations

of buckling,

deformations

other

similar

principal

solutions

are

formulation.

(as

in the

of this

nature

from

of these

in which

case

It is the

problems

of thermoelasticity;

problems

effects.

approximate

The

theory

effect

purpose

of this

It should

viewpoint

large paragraph

approximations

was

loading deflec-

to discuss

be remembered

of an exact

have

of the

of beam-columns),

type.

the

the

they

that

thermoelastic

treated

in the

previous

subsections, 4.0.1

Heated

Beam

Columns.

If a beam-column temperature cases slightly

must,

in which

direction. former

The

Ends The

beam-columns

possesses

ends

latter

shape

that case

buckling

no axes

used will

into

are

when

action

of heat,

account,

restrained

the

ends

be considered

the

The in the

are

free

influence

analysis axial

in the

direction

to displace

in paragraph

of

is

in that

4.0.1.1

while

the

4.0.1.2.

Unrestrained.

behavior

under of the

beam

in paragraph

Axially

to the

be taken

of the

from

is considered

4.0.1.1

on the

in general,

the

different

is subjected

any cross

of beams

combination section:

of symmetry

(and

therefore

of transverse For

example,

can buckle

only

also and

a beam

their

axial whose

by a combination

behavior

loads) cross

as

depends section

of twisting

Section

D

July

1972

1,

Page and bending, possible.

whereas

The general

in Ref.

1,

beams,

distributed

load

such

some

following

moment

in the

that MT

= 0.

of this

analysis

principal

acting

of the uncoupled

and solution

the

with least

p = p(x)

distribution

cases

lormulation

but for simplicity,

symmetrical

ture

in other

xy

problem

of inertia and

will

arc

are

is restricted

plane,

The beam

modes

also

discussed

to doubly

under

subjected

thus bend

204

a transverse to a tempera-

in the xy plane

Y without

twisting

and with w = 0.

The governing

differential

equation

is

d_M T

(12 It is convenient temperature beam

would

(transverse

The

loads

quantity

and the

therefore

to obtain

_r

d'v

solution

and of transverse

load•

undergo

(E

if only absent);

in two parts, For

it therefore

(12V'l' % l_.,-;_T_ /

axial

z

temperature

, I,

Vp is the dcl'lcction

satisfies

dz

-, j, the

loads

dZ _7

d_v._

(_:

load' P were

satisfies

(IZv'['

dx _.

present

d2vp

+ P ,_--T--

load

would

p

of the

present

equation

•

undergo

•

effects

deflection

P were

differential

dx 2 z

(temperature

d2vp_

/

the

the

vT is the

d2MT

-

th(, Ix,,am

equation

\,:Iz_

purpose

and the axial

the differential

(_

this

by separating

(Z)

if only

effects

transvers(:

omitted);

it

(3)

Section

D

July

1972

1,

Page With

the

are

definitions,

acting

the

solutions

of the

combined

problem

in which

205

all loa(Is

is

v=

v T + vp

The

.

component

deflection

Vp represents

the

solution

of the

ordinary

!

isothermal

beam-column

Section

B4.4).

The

problem

and

determination

can

often

of v T musjt,

be found

in the literature

(see

in general,

be carried

out

of uniform

beam

t

anew

for each

under

new

problem,

a temperature

higher

than

the

MT

=

distribution

third

a 0 +

lIowcver,

in the

alx

÷

in the

of the

spanwise

a2 X2

form

special

of a polynomial

direction,

+

a3 X3

clx

+

case

that

is,

of a degree

when

,

z

then

MT Z

vT = _

where

+

p

k = _

and

%

the

+

constants

c 2 sin

kx

('0, ct,

+

c 3 cos

c2,

and

kx

% are

determined

Z

from

the

examples

boundary

for

MT

which

= a o + alx Z

are

given

conditions.

as follows.

Solutions

for v T for

three

important

special

not

Section

D

July 1, 1972 Page 206 I.

Both Ends

Fixed.

_--_

i- _ _. L L.. F

-I -!

4_ 2 EI Z

vT II.

0for

=

Both

P
-

L2

Simply

Supported.

y

I

P •---.--e.

X

P ---b_, t.

VT/co.kL-1 =-

\" si_kJ_

P

sinkx+l-

coskx

)

- P

sinkL

and 7r2EI z

p cr

-

L[

III. Cantilever.

P

a0 VT = - _

and

1 - cos kx cos kL

al[
"_

cos

kL

(1 - cos

kx) + kx - sin kx

Section July

D 1, 1972

Page

207

lr_ EI Z Pcr

The

=

axial

{7 f

stress

is given

xx =-_ET

by

+ _ A

+-

MT Z

In these axial

analyses,

load.

the beam-column

If the

replacing

the

load

quantity

a compressive

load:

(k 2) by (-k2),

(sin

by (i tanh kx),

etc.

k=

has

is tensile, P by (-P)

_ '4 EI

assumed

i = x/L'i-

the

quantity

kx),

(cos

and

to carry

results

in the corresponding

by (i sinh

Here

been

the appropriate

accordingly, kx)

+M z + P v Z

be obtained

expressions

(k) kx)

may

a compressive

must

valid

be replaced

by (cosh

kx),

the symbol

k

to be solved

is still

(tan

by for

by (ik), kx)

denotes

. Z

4.0°

1 ° _.

Ends

In this 4.0°

1.1,

from

Axially case,

but the

an additional

If both

ends

stipulated: unchanged mathematically,

are The and

Restrained. the

basic

magnitude condition rigidly axial

fixed distance

temperatures

equation

of the load concerning in the

axial

between must

remain

P

is unknown

the axial direction, the

ends constant

and

equation must

displacements these of the

bar

along

of the

must

o[

be determined

conditions

the

(1)

ends.

shall

be

remain

span.

Expressed

Section

D

July 1, 1972 Page 208

(P-

L A-'_

PT )

+A

=0

(4)

whe re

A

The bar are

-

f

2

0

=

dx

analogous

elastically

example,

the ends

takes

form

the

P

condition

by the

consequence, direction mined

the is quite

by the

equations

simultaneous

itated

use

iteration within

the 1.

concentrated

following

since

with

of graphs technique

the unknown (1)

have

ends

P and

v

and

(6).

A

equation

must

be

(4. 0. t. 1) ; as a fixed must

in the

However, are

available

axial

be deterif the

greatly

with a rapidly are

for

(6)

calculations

results

of the

If,

K,

quantity

in conjunction and its

derived.

modulus

paragraph

which

the

the ends

.

in the

of equations

in which

is easily

preceding

as a polynomial,

This

facil-

convergent in Ref.

7

limits:

Distributed loads

springs

appearing

in the

solution

of a series

procedure.

direction

of beam-columns

cumbersome,

is expressed

v

given

analysis

to the case

A-E ÷ A -- o

deflection

temperature by the

to linear

-

The transverse calculated

in the axial

attached

÷

(5)

appropriate

restrained are

.

are

transverse at the

midspan.

loads

are

uniform

over

the

span,

whereas

(4)

SectionD July

1,

Page

2.

The temperature

in a spanwise

The

and

cases

tables

beam

through

and

be used

to be simply

restrained

of numerical

of zero

may

is assumed

elastically

Tables the

linearly

the

depth

and is constant

direction.

3. bending

varies

1972

209

Thermal

Buckling

4.0.2.1

Circular

Plates.

in nondimensional

end

to determine

4.0.2

at its ends

for

axially.

results

full axial

supported

restraint

form

are

in rectangular

maximum

deflections

presented

beams.

and

for

These

bending

moments.

of Plates.

General. This havior

section

and involving

stresses

and

The

pression

structures

complete

collapse

deflections.

stress

and

function

in the

following

demonstration

and,

allowable

background,

in the

can

for

in the

this

as developed

based

some

seen

stress

of the methods

along

fact

with

involved.

accept

for

specific

the

Forray solution

in Refs.

be-

is because

that

plate

major certain

com-

loading

before

increased

stresses

of the plate

materials,

Particular

and

elastic

the

of additional

by an inspection

(a/t_.

by Newman

paragraphs,

well-known

increment

This

represent

of so doing,

be readily

parameter

on nonlinear,

theory. state

is the

process

compression

crippling

data

post-buckled

support

can

and

of large-deflection

basis

This

of the

curves

the use

deflections

considerations.

and

contains

taken

buckling as a

considerations 32 and

of an example

33,

and are

problem

given for

Section

D

July 1, 1972 Page 210

Confl_,uration. The design circular

plates

material.

curves

which

and equations

are

of constant

It is assumed

and that

Poisson_s

Boundary

ratio

that

provided

thickness

the plate

is equal

is free

here

apply

and are made of holes,

obeys

only

to fiat,

of an isotropic Hooke'

s law,

to 0.3.

Conditions. The solution

is valid

only where

both of the following

conditions

are

satisfied: 1.

The boundary

w=M

2.

r=0atr=b

0 at

Temperature

that is,

surface

(7)

of the plate

r = b

is radially

fixed;

that is,

.

_8)

Distribution.

The

plate

may

the distribution

required

supported;

.

The middle

_=_'=

that

is simple

that

permissible

the

have

a thermal

is symmetrical temperature

distributions

gradient about

be uniform

can be expressed

through

the

middle

over

the thickness, surface.

the surface.

in the

However, Therefore,

restriction

T(+z)

= T(-z)

these

the

the

(9)

to the

Obviously,

it is

form

T = T(z) subject

provided

special

specifications.

that . case

(10) of a plate

at uniform

temperature

complies

with

Section July

D

1,

Page Design

Curves

and

If a heated

1972

211

Equations. plate

is constrained

against

free,

in-plane

plate.

Initial

expansion,

com-

F pressive

stresses

occurs

at very

strained ness

are

it is able In view

manner.

plate

However,

of this,

some cases

may

arise

and

deflections

stresses

the

and deflections

deflection

at that

stage

theory may

In Ref.

as a result

load

after

of a plate

must

be used

be several

33 Newman

times and

Forray

initial

buckling

analysis

since

thickness present

mechanical

and

solution

of the

basic

differential

equations

computer

solution.

temperature Figures moments and

(16).

resultants, 4.0-1 for

through various

up the

Numerical

results

NT'. 4.0-5

These for

temperature

buckled

state. of the

To determine has the

occurred, actual

deflection

analysis

Since

differential

results

are radial

for

presented and

as defined

they

used

equations

obtained

of a

a closed-form

not possible,

then

gradients,

since

of the plate.

were

deflections,

useful-

magnitudes

helpful.

loading. was

governing

the

a large-deflection

thermal

con-

buckling

the

of the

after

under

to set

reached

be most

plate

procedure

initial

would

circular

difference

of thin plates

of this

it has

buckling

situations,

a knowledge

in the the

case

structural

lost

in which

out-of-plane

in the

in many

additional

post-buckled

large

increments

is not completely

to carry

stresses

in the

low temperature

in this

of the

developed

for digital

a wide

in curve

tangential in equations

a finite-

range form

of in

forces,

and

(9),

(10),

Section

D

July

1972

1,

Page

212

Ip - 0.3.-- _

NT' " 100

1.0

_'_

\\

U

0

Figure i_ven buckling an Ref. oxceedm

4.0-.1.

though

analysis

and stresses,

34 that initial

buckling

eigenvalue ihows

0.4

Nondimennional

nonlinear

deflectlonJ

the lowest

0.2

Timoahmflco

[ 35]

plate wtthv

- O. 3 is given by

methods

can occur of the small critical

0,8

deflection

these

that this

O,I

1.0

parameter.

are used

solutions

hold

to find the postsince

it has been

when the edge compressive deflection compressive

buckling stress

shown

stress

prohtem. for a circular

Section July Page

D 1,

1972

213

q ill,,.,,.-

i

I _t 7 /" lIII1#/_

,,hn

f

00

¢o

¢i ¢J

/lldl 1_! '/! 1/I J Jl ,_N

!

c_

..Q

N 0

0

o

fii

©

z_ c_

°l,,

'I.I::II3WYI:IVd

° T

3_)UO_l

-

,!I,

3NVU_]I_]IAI

° °fl

8

°<"

I

I

g a.

lVI.I.N39NV.I.

O

i

'/

_<_-ii --

-

i

i/'il t

!'

,

i

O

C

r

/

Z

1 ID

N

0

'_

_D p. !

,iN 'H:I1]INVHYd

",-_

i1[

/'11 _i

.

_ -iA"

i

_1 ....

I

"dOHOd - _NVHSVd:_W "IYIQVI:I

I

Section July Page

\

\

¢$

\

i l

\

II

214

%.

k

"t-

197

\

\

tl

1,

o.

t\ \

D

jt

J

/

f

/t

o

_

I I"

I #

/

i

I

I

!

,0W '_3i31NVUVd

J.NgWOW'ONIGN38

!

_° t_0

I

"IYIJ.N3DNYJ,

o.

\'\

\<

---_ _¢_

:\

:k ">

'j_ \

/,,i

,,

//!"

i

if'

ci

V o I

•Jw 'H3131NYHYd

I

1N3WOW'ONION38

"lVl(]VH

I

I

Section

D

July I, 1972 Page 2 1 5 Db

(11)

= (2.05) 2

(ar) cr

F For

the thermal

given

problem,

the compressive

stress

just

before

buckling

is

by -N

f--

r

= __T__

(12)

t(1-v)

where t/2 N T = Ea

From

equations

buckling

f -t/2

T dz

(11)

is given

and

.

(12),

(13)

the

critical

nondimensional

value

for

N T at

by b 2 R

(NT')

Equation

N T'

(14)

= 2.94,

remain

must

fiat

equal

Figures by the

= (NT)

cr

shows

zero.

4.0-4

and 4.0-5

The

deflection

In actual

which

practice,

_ = 0.3.

w,

can occur direction

.

Hence,

as well

buckled

in either will

the

plate

when

will

(17)

and

requirement

plate upward

be determined

occur

when

N T' < 2.94,

as the bending

that this

of the

(14)

of the

of equations

reveals

pattern

the

= 2.94

buckling

An inspection

method.

bulge

that

deflection

given

metric

that initial

provided

and the

Db

cr

the

moments

(18)

together

consist

or downward by the

type

will

M r and

is essentially

will

plate

M0

with satisfied

of an axisymdirection. of initial

Section D July 1, 1972 Page216 imperfections present in the plate, The formulas

and design

moments

are

based

the

hand,

other

at any point

on the

the

sense

taken

for both types

that

_T

can be suitably

which

the

contents

of this

section

Example

Problems.

gradient

the

with Figures

first

fully

supported is given

other

that

that

have

example

4.0-2

On

versus

Hence,

tensile)

the expressions

and 4.0-3,

there

are

been

corrected

against

valid

The plate

!

.

unaffected

temperature-dependence

temperature

of that it

distribution

in _. error

be changed

in equation

to (1 - v2).

(5) The

accordingly.

consider radius

in-plane

are

for by recognizing

the actual

(1 - v) z should

by + G.0 (z/t)

the

t s ratio

is a typographical

problem,

otherwise.

Poisson

for variations

and has an outer

restrained

and hand,

is,

to compensate

quantity

in thickness

T ffi 3.0

downward.

(compressive

direction.

etc.

and the bending

deflects

can be accounted

governs;

be noted

32 where

simply

of the bulge

coefficient

of Ref.

are

forces

disturbances,

deflections

that the plate

membrane

On the

modified

It should

edges

with the

Young t s modulus

variations.

thermal-expansion

is 0. t0 in.

of the

of external

of deformation.

by temperature

For

assumption

in conjunction

R is assumed

is the product

nature

associated

will be independent

for N r and N_,

the

curves

the

a circular

steel

a

in.

of 10.0

expansion;

is heated

such

however, that the

plate

which

The outer they

are

temperature

Section D July

1,

Page

Using

a = 6.0 × I0 "s in./(in.) (*F),

determine

whether

buckling

has

1972

217

P = 0.30, and E = 30.0 × 106 psi,

occurred

and find the stresses

and deflections.

P

From

equations

(16)

NT,=

12(1-Et 3V2) b 2

F_

-

[3.0

ttf/2 2

+6.0(z/t)

12(I.0-0.09)(i0.0)2 (30.0× (30.0 × I0_)(0.I0_)

NT '

6.0(z3)1 3.0(z) + (-iVS'J

2] dz

106)(6.0×

,

i0 -6)

t/2 -t/2

N

T

' = 6.55

3.0

[

+

(t

+ ,_-':_, ,t-)2"0 (__

t)

6.55(3.5)

(t)

and

NT'

= 2.29

Since

this is less

plate

has not buckled

M r and M 0,

are

However,

knowing

that,

be obtained may

be easily For

i

.

apply

than

the

and thus

equal

for NT'= Figures

calculated a second

a temperature

buckling

the

value

deflection,

given

as well

by equation as the

(14),

bending

the

moments

to zero.

from

from

critical

an inspection

2.29,

of equations

Mr'

4.0-2

= M 0' = 0 and the

and

4.0-3

by inserting

the

example gradient

T= 33.0+66.0(z/t)

use

the

.

for the

values

as a function appropriate

same

to the plate

2

(17)

plate

of values

as for

as follows:

the

stresses

of Nr' (r/b), in the first;

and

and the

N 0' can

stresses

equations. however,

Section

D

July

1972

1,

Page

For

this

thermal NT'=

Since

this

equation 4.0-5

218

loading

6.55

[33.0(t)

is greater (15),

+ 5.5(t)]

than

the plate

the

= 6.55(38.5)(0.10)=

critical

buckling

has buckled.

the deflections,

stress

value

Then,

resultants

and

25.2

of 2.94,

using

as given

Figures

moments

.

4.0-1

may

in

through

be obtained

for" the

4

calculated

(17)

v'.due

to obtain

stresses

on the needed

and

basis

of NT

the

.

These

stresses

and

deflections

of NT'

in the

stress

are

= 25.2, and

values

needed

following

deflection

0.30

1.75

0.6O

1.2O

Setting

4.0-2)

( Fig.

-1.70

(17)

this

of (r/b)

gives

for

into equations

problem

assume

= 0.3

values

M

N0 '

1.40

up equations

table

be substituted

and

calculation

4.0-3)

5

(Fig.

4.0-4)

-1.20

-12.7O

-10.00

-12.0o

of the

stresses

Letting

I -12z (M ')r t_j 6(1_v2)

z = t/2;

T = [33.0

+ -(NT') (l-v) + 66.0(0.5)

Then,

- 12(1+v)

=] = 49.5

M 0'

(bt_T./

(Fig.

and deflections

and

=

the

terms

r

w= w' (t) : o.10(w')

r

= 0.6.

for the

problem,

ff

that

calculations.

!

(Fig.

For

for values

r

r/b

then

deflections.

the

i

may

c_T * (Nr')

4.0-5)

lot

this

Section D July 1, 1972 Page 219

Db

30. 0 x 10 _ x (0.10)

b_ = z2(z.o-

3

0.09) (]0.0)' (o.zo) = 27_.0

Thela,

-12(0.a)(Mr') r

= 2750.0

_/6(z.o - 0.09)

25.2

+ (z.o- 0.3) Z

(io.o

- z2(z.o + 0.3) \o-T_/ r = l-

7060

(Mr')

+ 99000-

(e.o x zo-'

127 320+2750

(N r')]

(49. 5) + (N r'

,

1

and

r

= [-7060(Mr'

) + 2750

(Nr')

- 28 320]

Also,

a 0 --- [-7060

Then,

for (r/b)

(M0')

+ 27_)

(No')

- 28 320]

= 0.30,

w = o.zo (w') = o.zo (z.75) = 0.z75 in. and

r

-- 70eo (-lz. 70) + _750 (z.40) - _8 3_0 = + 80 660 4 3850

- 28 320 = + 65 190 psi

Similarly,

¢0 for

(r/b)

=+

55 220 psi

= 0. 60

(Note

that these

are

tension

stresses.

)

SectionD July t, 1972 Page

22O

As before, w ffi 0.120

r

in.,

= + 51 720 psi,

and _0 = ÷ 31 720 psi

The previously Critical term.

calculated

compression

.

stresses

stresses

represent

are

found

the critical

by letting

tension

z = - t/2

values.

in the first

stress

Then

_r = [ 7o60 (Mr')

÷ 2750

(N r' ) - 28 320]

and cre = [ 7060 The following deflections

table

(Me') gives

at the plate

+2750

(Ns')

the critical

tensile

.

and compressive

stresses

and

Stress

(psi)

stations. Tensile

w

- 28 3201

Stress

(psi)

Compressive ff

0"

(r/b)

(in.)

cr

0.30

0.175

+65 190

+55 220

-114

130

-118

460

0.60

0.125

+51 720

+31 720

-117

720

-143

360

Summary Critical

of Ecluations Condition

(NT')

r

0

and Curves.

for Buckling.

(i5)

= (2.94) cr

where

r

v=

0.3

_etion

D

July

1,

Page N t

T dz

..

T

221

b2 ,|,

N

1972

I}

b

_

(16) and

t/2

f

N T = Ecx

Postbuckling

T dz.

f -t/2

Deflections

W=

and

Stresses.

wVt

12zM

r

t

N

-

T

'

I

r

td'_(1- ,,_)

T + Nr

(1 - ,,)

t]

bZ--_l)b

(17) and b2 -

o'0=

+

12 z M 0' t46(1 - ,,_)

(1N T- ' v)

where b2

M M

'=

-

r Dbt

r

46(1

- v 2)

d .(1I}bt

M 0' = -

bz

N r

r

Db

and NOb 2 NOt -

Db

.

-

12(1+

v)

_

Db _T

+N 0'

b2--_-

Section July

1,

Page

The values 4.0-1

for w',

through For

Mr,'

4.0-5

M0' , Nr,'

for the

values

case

of u other

and N 0' are

of

obtained

from

D 1972

222

Figures

u = 0.3.

than 0.3,

the

values

of Mr'

and M 0' may

be

found by using equations (18) as shown:

= 1.049 v

=

0.3

_(1-

v2)(M

') v=0.3

l"

(

Similarly,

(Mo')v-- 1.o49 J (1- v2)

(M 0' ) v= 0.3

N r ' and NOt are 4.0.2.2

independent

Rectangular I.

Heated

Consider

Plates

the

plate

heat

input Q, and reinforced

present thickness

because purposes

distributed

in the

plate

of the heat the

and of the

T=T0_TI

Loaded strip,

by a uniformly

the edges

v.

Plates

ends

The temperature

of

in Plane

shown stress

along will sink

temperature

on the _0,

the edges be higher provided will

-- Ed[_es

Unrestrained

following

subjected (y = 0,

page,

loaded

to a uniformly

Plane.

at the distributed

y = b) by longitudinals.

in the center

of the panel

by the longitudinals.

be taken

in the

to be uniform

For across

than near the the

form

co s

(._)

(19)

.

Section

D

July

1972

1,

Page

in the plane

of the

the available

data.

plate,

where

T o and

T 1 are constants

223

to be adjusted

to fit

I

a

GO

The from

of the

(note:

the

solution

lower

buckling

by the

interaction alone

the

solution is,

when

may

prevented

It was

that

that

to (1)

is the

T 1 = 0,

the

problem

cases.

These

T is acting standard

strip,

extending

distance

from

to be independent

combination by obtaining

the

a 0 = 0 and

of the

of

at x = 0, a.

is found

and

be taken

1 the critical

shown

panel

to be at a sufficient

on buckling)

load

(2)

stresses

are

of two special and

to a single

is assumed

in Ref.

equations.

to the

1; that

panel

so that

no effect

simultaneous

The

pertains

displacements

T O has

is acting

This

strip,

Transverse From

Ref.

solution

x = 0 to x = a.

the ends x.

following

plate

symmetric

of a and 0

the case

T

1

determinant corresponds

can be solved

approximately

cases

T 1 = 0 and

are

(1)

of

a

0

alone. buckling

expression

found

in

Oo

Section July

(7

1,

1972

224

(20)

_

_

k_'lE 12(1 - v z)

1

=_ cr 0

[

Page

D

where a

k =4 for

b

and

k=

+

for

The solution

to (2)

•

b

when

¢r0 = 0 is as follows.

The critical

value

of

T 1 (= Tcr0 ).

o Tor0 2

where

-

12(1-

k I is determined

v _)

from

(21)

Figure

4.0-6.

12

10

0

1

Figure Then,

for the

4.0-6•

general

2

Values case

3

4

of the coefficient

in which

both heat

5

k t.

and edge

stresses

are __jr

acting,

the following

interaction

curve

is used:

Section July

1,

Page T

D !972

225

ff cr

cr

T

-

Gr cr 0

1

.

(22)

cr 0

F

IL

Heated A.

which

are

assumed limit.

from they

curves

of constant

The edge

must

is free

support

as the

1 through

in Plane

plate

presented

thickness

that the plste

expansion

Loaded

-- Edges

Restrained

in the Plane.

Configuration.

The design

f t

Plates

and are of holes

members

proper.

4.

tlowcvcr,

be used

to obtain

here

apply made

and

must

have

these

of isotropic

the

same

curves plots

appro×imate

to fiat,

rectangular material.

that no stresses

The design since

only

results

coefficient

cover

become

exceed

aspect quite

for aspect

plates It is

the

elastic

of thermal ratios

fiat

a/b

at a/b = 4,

ratios

greater

than

4. B.

Boundary

Solutions

arc

given

Conditions. for each

of the

following

two types

of boundary

conditions: 1.

such

that

contractions) 2.

Type

I -- The boundaries

a.

All edges

of the

b.

The edge

Supports

these

displacements

satisfy

plate

are

fully

are equal

both the simply

restrain to the

free

following

conditions:

supported. in-plane thermal

plate

displacements

expansions

(or

of the supports. Type a.

[I -- The boundaries All edges

of the plate

satisfy are

both of the following

clamped.

conditions:

Section D July I, 1972 Page 226 b.

The supports fully restrain in-plane edge displacements of the

plate such that these displacements are equal to the free thermal expansions .. (or

contractions)

of the supports. C.

Temperature

Distribution.

It is assttmed that no thermal gradients exist through the plate thickness. The following three types of temperature

distributionsover the surface are

considered and are iDustrated in Figure 4.0-7: 1.

Sinusoldal distributionswhich can be expressed

T = T

2.

o

+ T 1 sin 7r--_x_ sin a

Parabolic

T=To+T

3.

1

Tent-like

_y b

distributions

1-

-_'-1

distributions

a. Sinusoidal or parabolic distribution. Figure

4.0-7.

Selected surface

mathematically

as

(23)

" which

can be expressed

1-

which

2Yb

-

1

can be expressed

b.

mathematically

.

• (24)

mathematically

Tent-like distribution.

temperature

distributions

of a rectangtflar

plb.te.

over

as

the

as

Section July

1) 1,

Page

D. In Ref.

Design

1972

227

Curves.

36 Forray

and

Newman

present

the simple

means

given

here

to

fcompute

the

rectangular tions

and

plates.

Curves

were

were

buckling The

temperature

temperature

variations panel

critical

results

chosen

it will usually

of Forray The curves

ties

as those

the

material

for _

and

TABLE

4.0-1.

BoundatT Temperature Over the

for

the

the

heating. cause

Newman

in Table

variations

some

type

would

must

of averaging

condi-

The temperature be expec;tc(I

is conducive hotter

in Figure

4.0-8

because

of fiat,

of boundary

4.0-1.

nonuniformities

the user

buckling

to be much

plotted

for

thermal

condition

the plate

that arise

Hence,

of what This

are

initial

combinations

tabulated

do not account

behavior.

by employing

for

to Ix; representative t(_ vapid

v = 0.3. such

are given

distributions

subjected

since

values

to thcrm:d

than the supports. for

plates

having

material

proper-

of temperature-dependence

of

select

in the

if the

a single

effective

value

technique.

" COMBINATIONS OF BOUNDARY T E M I)E ItATU RE DISTRIBUTIONS

Type

Conditions

CONDITIONS

I

Type

ANI)

lI

Sinusoidal

Distributions urface

Parabolic

Parabolic Tent-Like

A nondimensional plates

of various

4.0-9

where

aspect

plot ratios

of the deflection against

at the center

temperature

level

of rectangular is presented

in Figlare

Section July Page POOR

SIMPLY SUPPORTED PLATE, TENT DI|TRIBUTION

1.20

_

1972

228

|IMPLY fd.JPPORTEDPLATE,

uo

_

L

+o. )x__.= .To_ o.\ 0,410+

1,

QUALITY

1.20

L

D

To--

0.40

_

0.20,

0

$

2

3

4

0

1

2

(I/hi

4

(a/b)

liMPLY KIRIOIPlTED PLATE. 81NUIQIDAL DIITRIBUTION

1.20

3

&m l_,U._m'"' ""PED PLATE, ' " I PARAB.gLIC DISTFUBUTION

2.10_

tOO '_ 2.40

.\

,,To

.._

_

0

O.Sl To--"---

.._

---.,+_...

0.61-I

_._

._

-

_o I..P 1.20 _,.+

2

2

OJlO 0.40 ¸

1

_1

3

4

0i 1

2

Is/b)

Figure

4.0-8.

Critical

3 (fro)

temperature

parameter

for

rectangular

plates.

4

Section D July 1, 1972 Page229 1.0

"

i//

o.e

_j- 1_.///

0.2

_

0

Figure

0.2

Deflection

4.0-9.

In plane

" __ 0.4

w at the center of plate

(Poisson'

J

. 0.6

0.8

_.U

of a rectangular s ratio

plate

for loads

v=l).

t/2 NT=a

E

f

T dz,

(26)

-t/2 t/2 Tz dz

,

(27)

-t/_. and

Et s Dh" In this

Is

12(1-

figure,

NT/NTc

vl)

the noadimensional

r)

, where

N T cr

parametric

,

the value

indicating

of N T at which

the

temperature

buckling

occurs,

level

is

SectionD July

1,

Page

NT

=

(1-

u)

1 + b-_ aZ )

_X2Db

1972

230

"

(28)

cr

Plots

showing

quadrant

of a square

different

temperature

such

a plot

in nondimensional plate

and presented

levels.

is sufficient

form

in Figures

Because

to determine

the variation

of the the

of M

4.0-10

double

in one

and 4.0-11

symmetry

distribution

x

for

of the plate,

of both

M

and M x

throughout

the

entire

moment

occurs,

for

showing

the

aspect

ratios

are

in Figure that the

nonlinear

terms

neglected.

Therefore,

that

y

the maximum

in the center

bending

of the plate.

with temperature

level

Curves

for various

4.0-12. preceding in the

as is usual

for values

indicate

moment

be noted

that

plots

considered,

of center

shown

assumption

only

These

the cases

variation

It should

valid

plate.

two

results

obtained

strain-displacement

in problems

of NT sufficiently

were

small

relations

of this

relative

on the

type,

the

to N T

could

results

be

are

. cr

III.

Post-Buckling A.

are that

limit.

The edge

expansion of 1,

2, 3,

curves

of constant

sumed

the

is free

support plate a/b

presented

thickness

plate

as the and

With All Ed_es

Simpl_,

Supported.

Configuration.

The design which

Deflections

-

proper. 5.

and are

of holes

members

here

and must

apply made that

of isotropic no stresses

have

The design

only to fiat,

the

curves

same cover

rectangular material.

exceed coefficient aspect

plates It is as-

the

elastic

of thermal ratios

a/b

Section

f..-

0

0.1

0.2

OR|_II_L

t_._,_G_

OF POOR

QUALITY

0.3

,

0.4

[$

July

D 1,

1972

Page 231

0.5

"0.25

-o.61' 2,T _":'_I

Figure

I

I

Distribution

4.0-10.

NT

-

I

.

of the

l

]

bending

moment

(

0.25

Poissonts

ratio

M

x

in a square

plate

for

v =

cr

B. The conditions

such

Boundary

solution are

applies

AU edges

2.

Supports these

contractions)

the

C.

Temperature 4.0-13)

are fully

where

both

the

following

boundary

restrain

be

in-plane

arc

Temperature

assumed

can

supported.

equal

edge to the

displacement

free

thermal

of the

plate

expansions

(or

SUl)lmrts,

that

distribution and

to cases

simply

displacements

of

It is

only

satisfied:

1.

that

Conditions.

expressed

Distribution. no thermal

over

the

gradients

surface

mathematically

exist

is

taken

through

to be

the

parabolic

plate

thickness.

(Fig.

as

(29)

Section

D

July

1972

Page

xw_,£

_I,LL

1,

232

J

Section July

1,

Page

D. In many

F

Design aerospace

applications,

can be tolerated

if the

losses

of aerodynamic

efficiency,

given

etc.

as follows

deflections

In Ref.

such

representative

of what

heating.

condition

cause

This the plate

deflections

Newman

values

be expected

is conducive

to be much

hotter

of flat,

of the

rectangular

do not cause

destructive

and Forray

The temperature

would

buckling

do not produce

absolute

plates.

thermal

post-buckling

37,

to compute

for

1972

233

Curves.

plates

disturbances,

D

present maximum

variation

when

the panel

to thermal

buckling

was

excessive

aerodynamic the simple

means

post-buckling chosen

to be

is subjected since

to rapid

it will

usually

than the supports.

/

Figure

4.0-13.

Parabolic

the

surface

temperature

of a rectangular

distribution

over

plate.

/ /

The

maximum

can

be calculated

J

post-buckling from

the

deflection relationship

occurs

at the

center

of the

plate

and

Section July

D

1,

1972 J

Page

K2+_

(3-.5) l+--_.

+4v

bZ/] a-" _

234

6z "_

T

(_0)

+ (3-_r) _z, where b2 1 + _

K--

(31)

and

/_ -

a (I-,,_)

•

(:_2)

Solutions to equation (30) arc plotted in Figures 4.0-14 and 4.0-15 for plates having v = 0.30.

It is useful to note that, for given values of T O/Tl, initiai

thermal buckling occurs at T1/fl values corresponding to 6/t = 0. The curves do not account for nonuniformiUes

in the material properties such as

those variations that arise because of temperature-dependence behavior,

Hence,

employing

some

4.0.3

Thermal

of the material

the user must select a single effectivevalue for _ by

type of averaging technique. Buckling of Cylinders.

Configuration. The design curves and equations provided here apply only to thinwalled, right circular cylinders which satisfy the relationship

L/R

_

3.2

(33)

OF POOR QUALITY

Section

D

July

1972

1,

Psg8 235 6.k 5.6 #Ib,l

_.8 k.0

/

e/b ,2

s

_"1-

J

3.* 2.h

.//./

o.8

__

/

,0

/// .0.5

/

02

- /

0.4

0.6

05

1.1

1.0

02

0

03

0.4

( 6/0 4.0-14.

0.11

10

1,

(6f0

__

oammNsrs

br

heated

rectangular

plates.

_.8 k.o _/#,3

j/,02

3.2

,l

2.b

Jj

1.6

/

0.8 lJ

,5

m m

04

Ol

011

tO

I.a

0

oJt

Figure

.o

4. O-.1S.

Post-buckling

04

)JL

OJ

I.O

6/t)

(d/t) parameters

for heated

rectangutar

prates.

12

Section

D

July 1, 1972 Page 236

and

are

made

of holes,

obeys

depicts sign

of isotropic

the

Hooket

isotropic

convention

Boundamj

forces,

following

w:-

is of constant shell

that

the

sheU

thickness.

configuration.

moments,

M

w=

types

Siml)lv

wall

Figure

Figure

is frcc

4.0-16

4.0-17

shows

the

and pressures.

Clamped

-= Ox

It is not required

to external

axial

edge;

atx=

that

the that

conditions

c(Ige;

0 and/or

0

it is assumed

of boundary

supported

:_ 0 :tt x

X

2.

that

x

that

is, (:_4)

is,

conditions

L

at the

cylinder

constraints

covered:

l,.

0and/orx=

the

are

at any

.

two ends

(including location

(35)

any

around

be the end the

same.

rings)

In ever), is not subjected

boundaries

at x = 0 and

L.

Temperature

Distribtttion.

The wall

and

cylindrical

for

1.

x=

s law,

It is assumed

Conditions. The

case,

material.

supposition

is made

that

and

axial

direction.

thickness

in the

variations

may

expressed

in the form T=T

be present.

(¢)

.

The

no thermal

permissible

gradients

However,

exist

arbitrary

distributions

through

the

circumferential can therefore

be

(36)

Section July

D 1,

Page

1972

237

/.

F

. NOTE:

P

x, y, z, _, and ax are positive as shown. In addition, P, M y, lind Mz are fictitious Ioldings used in derivation ond are likewise

MIDDLE

Figure

4.0-16.

Figure

Isotropic

4.0-17.

Sign

SURFACE

poeitive as shown.

cylindrical

convention for

shell

for

thermal

configuration

forces, buckling.

moments,

for

thermal

and

pressure

buckling.

Section

D

July1,

1972

Page238

Hoop membrane becaus_

of e_ernal

with this axial

compression

may

radial

condition

develop

constraint.

However,

is not considered.

constraints,

the

special

case

in regions

Because

the of this

of a uniform

adjacent

to the two ends

buckling

mode

and the lack

temperature

associated of external

is of no interest

here. Design

Curves

and Ec_uaUons.

It is assumed by temperature user

must

changes.

select

engineering effective

that Young'

moduli

in using

the

values

for each

of these

It will

in each

and Poisson'

Hence,

effective

judgement.

s modulus

sometimes

of the following

1.

Computation

of the stresses

2.

Computation

of the critical

s ratio

contents

of this

properties

be desirable

are

unaffected

manual,

the

by applying

to employ

different

operations_ _

x

present

buckling

in the

stress

cylinder

(_x)

. cr

On the other

hand,

to fully

account

cient

.

a

the

results

such

that

presented

for temperature-dependence

The appropriate a fictitious

are

stress all axial

formulation

distribution thermal

deformations

of the

for _

_A around

in a form

x

which

thermal-expansion

can be obtained

the boundaries are entirely

enables

by first

the user coeffi-

imposing

at x = 0 and x = L suppressed.

R follows

that

(r A=-_

ET (9)

•

(37)

Section D July 1, 1972 Page 239

fThese wall

stresses

may

thickness

be integrated

to arrive

"PA =-

EtR

at the

f 0

around

the circumference

and

through

the

force

aT(_)

d_

(38)

and the moments 2_

aT(_) A

sin _ dcb

0

and

(39)

aT A Since

(_b) cos _ d_b

0

it is assumed

that the

shell

is free

of external

axial

constraints,

the

conditions

y must

=g = o

be satisfied

it is necessary as moments

(_z)

P

(40)

z

at x = 0 and x = L.

to superimpose (_,)y

respectively.

B

and

(_z)

B

To restore

a force

PB equal

, which

are

equal

the

shell

and opposite and opposite

to such

a state,

to to (_)z

as well

A

and

Hence,

A

(41)

Section July

D

1,

Page

1972

240

and

B

The

stress

.

A

to "PB is easily

corresponding

E 2_

The stresses

due to (My)

\ y/B and

thoJe

B

to be

aT(_)

de

.

(42)

are

I y

due to (M)

j" 0

found

:

-_R_t

=

_r 0

(43)

are

Z

B

B

The procedure Hence,

shorter

being

the stresses

sentations rings

(44)

only

are

thermal relationship

from

equations

the

be this

stresses

constitutes

at sufficient

present,

will

used

greater

distance.

at various

an application (42)

distances

from

their

the

resistance

Subject points

through

to these

in the

shell

of Saint-Venant' (44)

ends

will

s principle.

be accurate

repre-

x = 0 and x = L.

If end

to out-of-plane conditions,

bending, the

may be computed

actual from

the longitudinal the

Section

D

July

1972

1,

Page 241

_r X

+

O'

÷

+

B

F

(45)

O

B

B

f, or 2z r

O.x=- aET(¢)

+ ;_-._

aT(c)

2_

E sin _

4-

f 0

aT(_)

sin_

d_

cos

# d@

27

E cos 9 f

÷

a T (_)

.

(46)

0 Complex

distrihuUons

the required techniques

may

Integrations. whereby

stress

In such

the integral

To investigate tudinal

be encountered

(Crx)

must

instances,

signs

the stability

which

are

use

against

it difficult

can be made

replaced

of a particular

be compared

make

by summation shell,

the

the critical

to perform of numerical symbols.

maximum value

longiwhich

can

max

be obtained

from

the

formula

Et (or x) er For

the

•

= y

design

(%)

RJ

3(1

to be satisfactory,

< (%) max

It is required

• cr

(47)

- v z) that

(4s)

Section

D

July

1972

1,

Page

The quantity

_/ appearing

in equation

which

accounts

the detrimental

mainly

Note

that equation

circular,

results

bility

(47)

is identical

cylindrical

on the basis

shells.

in these

references,

when

effects

to that used

studies

of the circumferential is reached

is a so-caUed

Its application

of small-deflection

given

nature

for

(47)

the peak

from

axial

initial

for uniformly

to the present

reported

in Refs.

it can be concluded

stress

knockdown

that,

distribution,

classical

compressive

stress

factor,

imperfections. compressed

problem

is justified

38 and 39.

From

regardless

of the

theoretical

insta-

satisfies

the

expres-

sion

(O.x)

_

Et R_] 3(1 - v 2)

max In view of this, cent

probability

as reported for

the

L/R

Summary

used

(confidence

in Ref.

ratios

values

40.

= 0. 95) The resulting

of 0.25,

of Equations

here

(49) " for

_, were

determined

data for uniformly T values

are

from

the 99 per-

compressed

plotted

in Figure

cylinders, 4.0-18

i. 0, and 4.0.

and

Curves.

E + _

¢r X =-aET(¢)

2_

f 0

aT(C)

d
2_ ÷

E sin

[

_.

J

_T(¢)

sin _ d¢

0 27r ÷

E

cos

f 0

_T(_)cos¢

d_

-j

242

(50)

the

Section

f---

D

July

1, 1972

Page

243/244

and Et ((7X)

(51)

= T

R"]-3 (1 - v2)

cr

When

v = O. 3 this

(Orx)

gives

= - 0.606T

Et _"

(52)

cr

The

knockdown

factor

y is obtained

from

Figure

4.0-18.

1.0

_,- 0.1

0.25 1.0

4.0

,,.,1 I , o.v 102

,

, , _

Figure

, , ,,,i

4o 0-18.

101 R Y

,

,

Knockdown

.....

factor.

i ,,.

104

Section D July 1, 1972 Page 245 5.0 F

INELASTIC EFFECTS. In the preceding paragraphs, elastic behavior was assumed. This

assumption is sufficiently accurate for large classes of materials at relatively low temperature and stress levels. At higher temperature and for higher stress levels, however, the divergence betweenthe behavior of real solids and that of the ideal elastic solid increases and the elastic idealization becomes inadequate; the behavior of the real solid is then said to be inelastic.

To predict the inelastic behavior

of a solid under given thermal andloading conditions, it is necessary to generalize the stress-strain relationship.

There are three types of approaches

to this generalization, although the borderlines betweenthem are not well defined. 1.

The most basic studies of this problem make use of the concepts

and methods of solid-state physics. In this approach, the microstructure of the material is taken into consideration and it is attempted to predict the mechanical behavior of materials from this information. 2.

It is also possible to disregard the microstructure of the material

and to regard it as a continuum; the general principles of mechanics and thermodynamics as applied to continua are then usedto determine the forms of stress-strain relations which are compatible with these principles. 3.

The most direct procedure is to postulate simple inelastic stress-

strain relations; these define various ideal inelastic bodies which, though not

Section

D

July I, 1972 Page representing binations

any actual some

relaxation,

of the different

plastic

Although thus the

far only stress

types

of inelastic

or work-hardening,

considerable

progress

the third

from

nevertheless

flow,

analyst.

discussed

materials,

approach

In this

the third

has been

has yielded

paragraph,

of these

incorporate

in simple

phenomena,

such

made

in methods

information

of direct

inelastic

stress-strain

246 com-

as creep,

1 and

2,

utility

to

relations

are

viewpoints.

s0.1 Creep Creep

is the

is normally

measuring

showing

Creep

curves

taneous,

extension

4. time

(tertiary 5.

B to C, the

From

(secondary

under

and

temperature.

The

is known as a creep

temperatures, in Figure

undergoes

stress.

on a specimen

elevated

of time

illustrated

A to B, the specimen

stage

3.

are

load

at a constant

materials,

occurs

and 5.0-1.

an initial,

curve.

stresses These

almost

have are:

instan-

on loading.

From

(primary

which

that

a constant

as a function

for various

features,

From

2.

stant

with time

obtained

common

deformation

by placing

the deformation

1.

time

observed

its deformation

curve

certain

time-dependent

or transient C to D,

stage

From stage At E,

specimen

at a rate

that

decreases

creeps

at a rate

that

is nearly

creeps

at a rate

that increases

or viscous

con-

creep). specimen

or accelerating specimen

with

creep).

the specimen

D to E, the

the

creeps

creep). fractures.

with

Section

D

July I, 1972 Page

PRIMARY

SECONDARY

CREEP

CREEP

247

ii

F

0

C

B INITIAL

ELONGATION

ON LOADING

A TIME AT CONSTANT

Figure The During

rate

the

of creep

secondary

is approximately curve

during

(refer

primary

which

the

vlrture

ly

decreases.

tween

to Fig.

of its

stage,

the

strengthening

as

The

idealized

in the rate

creep

manner

of creep

shown

by the

TEMPERATURE

curve.

shown drops

in

Figure

5.0-2.

to a minimum

essentially

straight

value

line

that

of the

5. 0--1). stage

resistance

own

The

changes

constant,

The

by

5. 0-1.

LOAD AND CONSTANT

of creep

of the

deformation.

secondary by

is

material

For

stage

work-hardening

a work-

or

strain-hardening

to further

this

reason,

of creep

(C-D)

and

weakening

creep

the

rate

stage,

is

being

of creep

represents by thermal

built

up

continual-

a balance softening.

be-

Section July

D

1, 1972

Page

248

i 8

TIME AT CONSTANT LOAD AND CONSTANT TEMPERATURE

Figure

5.0-2.

The tertiary fracture,

caused

This

decrease

decrease

of the

by an increase

cases,

structure,

temperatures

of the

creep

rate

specimen

load-carrying

These

of creep

immediately

that accompanies during

area

may

creep

can also

or to the

act

before the de-

under

be due either

as it elongates

cracks rate

with time.

of creep

in stress

of the specimen

the accelerating

a constant to the formation

as stress-raisers.

is due to a change

in the

In

metaUurgical

as recrystallization.

Design For

in the

cracks.

such

5. 0.1.1

area

diameter

of intercrystalline

in rate

or accelerating

in cross-sectional

load.

other

stage,

is often

crease

Variation

Curves.

engineering are

purposes,

summarized

the results in more

of tests

convenient

at various

form.

Figure

stresses 5.0-3

and shows

Sectlolt D July 1, 1972 Page 249 a plot that is fixed

most

by the eoaditions

so that

the part

service. can

where

stress

quently gives

is the

load divided on the

elongation,

and there

rupture

for a number

are

curve

of temperatures

can

curves

show

is usually

the variation

several

eotmtaat

curves

in Figure

temperatures,

the

intersec-

the maximum

quantity

less

a suitable

on logarithmic

cross-sectional

scales,

area.

Fre-

with a number

at fracture;

thus,

that

some

is provided. that

amount

is needed.

a part

must

not

of tolerable In such

hc collected

cases,

on a single

to fracture

and they are

used

fracture

the diagram,

as a function same

only

rupture

curves

as in

diagram."

in the

in

deformation,

km_wn as a"stress-rupture

in the time

during

the designer

it upwards;

are identified

:ire

be determined

amount

determines

In area,

simply

on the

5. 0-3

5.(_-4, which

stress

no limits

in Figure

Figure

There

are

must

a certain

plotted

curves

material

of a part

temperature,

be this

usually

or reduction

requirements

than

then

by the original

of the

stress

proper

curve will

are

lifetime

and project

stress

rupture

of the ductility

When the

of the

the abscissa

and stress

more

at the

with the proper

Time

percent

meuuremeat

curves

and the design

the data points the

service,

line

stress,

of safety.

along

and the

and an allowable or deform

a set of these

vertical

factor

the temperature

not fracture

set off the lifetime

allowable

when

of operation,

will

Having

tion of this

the

useful

of stress manner

These at as the

5.0-3.

are various versus

ways

temperature

of cross-plotting for lines

the prev/ous

of coutant

rupture

figures. life

Plots

or constant

SectionD July 1, 1972 Page250

T- CON$T. e3 > e2 > el LOG t

Figure 5.0-3. of stress

Schematic at constant

minimum

creep

the

diagram

same

as in Figure ical

rate

behavior

are

with

5.0-5.

presentation temperature amounts common,

curves

These

showing

Such diagrams

over

wide

creep

is unimportant,

tests;

at higher

below

the tensile

temperatures, test

are

based

designs

the

are

from

complete

ranges.

where

curves,

curves

results

give

temperature designs

of creep-rupture data showing effect on the time to rupture or specific of strain. sometimes short-time descriptions

tensile

results

of short-time

creep-rupture be based

strength

on tests,

of the mechan-

At low temperatures,

on the

must

plotted

where tenlile curves

are

on the creep-rupture

behavior. Curves MIL-HDBK

of this -5

[4i].

type

are

readily

available

for most

common

metals

in

Section D July

l,

Page

1972

251

b

LeQ tA

Figure 5.0-4. Schematic presentation of creep-rupture effect of stress on the time to rupture at various

5.0.1.2

Stress Creep

jected

Relaxation.

assumes

to a constant

a constant

elongation

with the elongation

being

this

will

be observed

mode

of inelastic

elongation

5. 0-6).

This

An important that time of this

is required

type

will

data showing temperatures.

force;

constant,

to decrease behavior

characteristic

be unimportant

on the other

and the temperature

maintained

for their

if,

the

of both creep

action.

is raised force

continuously is known

Thus,

for processes

hand,

a bar

is sub-

to a high level,

required

to produce

with time

(Fig.

as stress

relaxation.

and stress

relaxation

it may

be expected

of relatively

short

that

is effects

duration.

Sectic_lD July 1, 1972 Page252

ULTIMATE

YIELD

TENSILE

STRENGTH

STRENGTH

/

m

w E

TEWERATURERANGE FOR

CONVENTIONAL

DESIGN

100 - hr RUPTURE

10000-hr

LIFE

RUPTURE

LIFE

TEMPERATURE

Figure

5. 0-5.

Schematic creep-rupture

presentation properties.

of tensile

and

SectionD July Page

,

f

LOW

1,

1972

253

TEMPERATURE

P !

Figure 5. 0.2

Stress

idealized

and stress

mechanical defines

the

stress-strain

are

under

defined

These

composed

constant

deformation.

which

idealized

of springs

relationship

exhibit

bodies

take

and dashpots, for the given

the characteristics the form

whose

of

of simple

deformation

material.

This

approach

viscoelasticity. To represent

models

bodies

relaxation.

models

is called

the creep

can be formulated

dashpots.

Some

5. 0.3

Creep Consider

column

relaxation

ViscoelasUcit_: Often,

creep

5.0-6.

composed

of the more

of a material,

of different

common

ones

many

various

combinations

can be found

mechanical

of springs

in Refs.

and

1 and 42.

Buckling. a column

is not perfectly

unavoidable

behavior

manufacturing

under

straight

a constant initially

inaccuracies),

axial (as

compressive

is always

then some

the

load; case

bending

if the

because

will

occur.

of The

Section D July 1, 1972 Page254 bending

stresses

increasing excited rialto

are

accompanied

dellcctions; or unstable

critical

these situation

time

usual

higher

arises.

process

quite

buckling

deflection

different load

plot,

is possible;

as creep

from

beyond

#m_'kiing

buckling

the usual

which

which

stresses

irnjdies

so that a self-

leads

to creep

to collapse

buckling is,

type

a "point

rate,

at a

buckling.

apply

of creep

represents

a point

_recp

This

observations

The phenomenon

matically,

strain

in turn _.aust:

and is known

The following 1.

by a certain

of buckling

than

is char:tctcrizcd

1) :

both physically

of bifurcation"

more

(Ref.

and mathe-

phenomenon. on a load

one equilibrium

by deflections

The

versus configuration

increasing

bcy,_n_!

all bounds. 2.

Mathematic'_lly,

the material

follows

3. pressive 4. fecUonm,

load,

magnitude changes

buckling

load;

in the former 6.

immediatc

The

buckling

creep undergo

can occur

at a finite

time

only

of t

creep

buckling

at any

value

of the com-

how small. will

cr

occur

depends

it has been but very

small-deflection

neighborho_l

if

law.

whenever

the

column

has initial

imper-

then.

The value of the

will

no matter

Creep and only

5.

a nonlinear

The column axial

creep

of the

found

on the initial

to be not too strongly

sensitive

to changes

analysis critical

deflection

time

is,

on the

affected

by

in the latter.

of course,

because

and

not valid

the deflections

in the are

then

Secti,m D July 1, 1972 Page 255 large,

tlowever,

up to times range

close

of practical

5.0.3.1

column

based

on a small-deflection

to the critical;

they

thus

theory

cover,

in effect,

arc the

vali,I entire

interest.

Column The

each,

very

calculations

of Idealized

critical

time

of idealized

for creep

H-cross

at a distance

h

H-Cross

buckling

section

apart]

Section.

[ two

is given

to occur concentrated

for

a simply flanges,

supported of area

(A/2)

by:

where

k-

24

L = length

of column

P0 = column

,

load,

a 0 = maximum

wdue

of initial

imperfection,

and k = constant

in the

strain-stress

relationship.

n

5.0.3.2

Rectangular Analysis

difficult.

A way

Column.

of the critical of circumventing

buckling this

time difficulty

for

rectangular has

been

columns established,

is very whereby

Section

D

July i, i972

Page 256 and lower

upper

bounds

lot

the critical

time

arc

obtained

(t*)

and arc

s,

1•

desigm_ed

by the use Per

o[ an as_rlsk.

a column

b, subjected

with a rcctana_ar

to an average

stress

cross

section

(r 0 = P0/bk

of height

and obeying

h and wi_h

a stress-strain

law of

= ( Ix) upper

and lower

in Figure height

5.0-7

of the

bounds

for the nondimensional

against

the ratio

For

the values

ratio

bound

will

upper

bound

critical

of the

The for the followiJN;

on t*

cr

creep

of this

will

Plates

method

c

listed

the bounds

from

initial

ratio

time

t*

cr

are

plotted

center-deflection

(say

z0/h

to the

< 0.015)

these

expression

presented

buckling

requirements:

of fiat

5.0-1.

may be seen

from

I,

approximation

buckling

stress

Figure

5.0-7

to vary

if cr o/or E > 0. 8, then the lower

for t*cr, whereas

be a good

and 8hells

in Table

Ref.

approximation

GrE = Euler

Flat

of the

the asymptotic

coefficient

However,

be a good

5. 0.3.3

values

from

between

z0/h.

time.

(z0/h)

critical

h _ -,'_ z0

The spread

with the

small

be determined

t* --log cr with

,

bounds

bar.

may

n

to the

if _ 0/_ E < 0.2,

actual

= _/

value

the

of the

.

\

of Revolution.

here plates

may

be used

and shells

to predict of revolution

critical which

conditions satisfy

the

Section

D

July

1972

Page

f-

For

n

TABLE

5.0-I.

VALUES

Lower

Bound

1,

257

OF C

For

Upper

Bound

,,

3

0.987

0.26

4

1.57

O. 56

5

1.95

O. 81

6

2.26

I. 02

7

2.45

1.16

./-- n = 3

5,0 /

'_

"

4°1" I

_''

UPPER BOUND ON tcr

I

--

I

--£2_

_

/

1.0

_0_ 0.006

0.01

0.02

0.04

0.06

0.10

0,20

=.___0 h

Figure

5.0-7.

Upper creep

the

1.

The

member

2.

The

stress

structure.

and

lower

bounds

for the critical

buckling

of a rectangular

column.

is made

of an isotropic

mat,_11al.

intensity

a i [see

equations

(1)]

time

is uniform

t':-" for cr

throughout

Section D July 1, 1972 Page 258 3. such

The configuration,

that appropriate

both the

critical

duction

boundary

formulas

stress

are

intensity

conditions,

available

_l [ see

for

and type of loading

room-temperature

equations

(1)]

are

values

and the plasticity

of re-

factor.

Temperature

Distribution.

R is assumed

that the member

is at a uniform

elevated

temperature.

Method. The Gerard

in Ref.

concepts

set

Field

show

data.

The

published curves label

method

forth

theory

were

llcf.

Jahsman

developed

mentioned

The test

the

seem

predictions. 1.

there

and Field

comparisons

stability

given

to indicate

However, data were

basis.

[4.r_] identify

this

in this

conclusion

corrected

based 45,

46,

technique

to eliminate

on the

test was

hand,

the

Rabotnov-Shcsterik_)v

paragraph.

could

by

Jahsman

which

On the other

with the

that this

published

with column

is that of Ref.

a different

method

In Ref.

predictions

to Gerard

was

approach

[44].

theoretical

43 and has

from

that which

and Shesterikov

of various

attributed

is essentially a classical

by Rabotnov

comparisons

which

here

43 and constitutes

before

servative

presented

The aforewill

possibly the

effects

give

con-

be because: of initial

imperfections. 2. the

experimental

The analysis data arc

is concerned for final

with the

collapse.

onset

of instability,

whereas

and

Section

D

July

1, 1972

Page The

presentation

different

by Jahsman

theoretical

and

approaches

Field [43,

includes 46,

plots

47].

obtained

Of these,

from

the

259

each

approach

of three

of Ref.

f? 43 gave view

the

most

of the

creep

uncertainties

buckling

the

detrimental

for

this

would

the

duction

effects

not fall

should

imperfection

thing,

one

initial

this

stability

general

imperfections.

of this

an attempt

in shells

would

at least

knockdown

factors.

account and these

Therefore,

This

ignores

to properly be required

handbook.

of revolution.

room-temperature

to

concept

However,

is made

in

approaches

theoretical

methods

scope

situation

in the

to partially

is done

account

by the

intro-

Procedure. from

curves

the literature,

for specimens

loading

or

made

at the

appropriate

be o[ the type

shown

in Figure

and st ress

---:

strain

intensity,

a suitable

of the desired

while

O',

is a desirable

classical

complicated

below, effects

of available

intensity plane-

For

intended

recommended

Obtain

unaxial

with

more

the

This

associated

from

much

within

Recommended

creep

predictions,

problems.

influence,

procedure for

conservative

service 5.0-8

material

and

a family

and

a. are

1

and

defined

These E. are 1

of

subjected

temperature.

where

respectively,

program,

to

the

curves

as follows

stress for

conditions:

O"

+

l

0 .2

y

-

or

+

O"

x

y

and

3T 2

xy

1/2

¢i-

test

2 ,,f-3"

+

E2 +¢ £ + y x y

(i)

Section

Using

data obtained

those

of Figure

for a state

from

5.0-9

of plane

where

indicate

a£

x=

The

results

family, priate critical

of curves

intensity

260

similar

defined

to

as follows

+

.

•

(2)

y with

respect

to time;

for example,

•

in Figure

is illustrated

formulations stress

a family

1, 1972

Page

X

embodied

which

_ x

differentiation

C /' k-_-/

_2

create

_i is the strain-rate

+_2+_ y

_/-3 dots

plots,

J_y

stress:

ci -

The

these

D

(3) 5.0-9

in Figure

for conventional

intensity

are

(cri)

then used

5.0-10.

to develop

Following

room-temperature

and the plasticity

this, values

reduction

still

another

select

appro-

of both the

factor

q.

For

cr

example,

in the

case

of an axially

compressed,

moderate-length,

circular

cylinder, one obtains

(_i)

= _ r cr

Eh RJ 3(1 - V_e)

(4)

and 1/2

--'_ _ where

1

(5)

E

I" is the room-temperature

knockdown

factor and

E

v=o.5o-

_E

(0.50 - re)

(6)

=

Section D July 1, 1972 Pagc 2fiI

f-

o."

_"

t

Figure

u_-

A."

5.0-8.

Constant-load

,

°i

STANT

t

creep

curves.

" A3"

T -

. ,,

CONSTANT

A2" i

LOG SCALE

iiii

,,J

Figure Values In addition,

On the those

other

for

use

associated

F may

the

hand,

5.0-9.

Curves be obtained

short-time

the tangent with the

curves

derived from

from Ref.

Figure

48 or other

elevated-temperature

and scant illustrated

moduli

5. 0-8.

values

(E t and Es,

in Figure

5.0-10.

suitable for

sources.

E and v . e

respectively) Therefore,

_re

Section D July 1, 1972 Page262

(=

"" TANT

Figure working

with this

responding the

strain

to the

5.0-10.

Curves

figure,

proceed

applied

load

intensity

derived

from

from

left

Figure

to right

5.0-9.

along

and use a trial-and-error

the _.z line procedure

corto determine

c i at which

Eh appliedcri=

_ F

FoUowing

sociated

onset

this,

with this

of creep

return

combination

buckling

Although

R_3(1-

the preceding

compressed

circular

constitutes

a general

approach

loading (and

in various conditions.

columns).

(7)

to Figure

5.0-8

of _i and

c i.

and can be denoted

an axially

buckling

V2e)

types

which

of plates

It should

This

and

be noted,

is the

the time

predicted

t as-

time

to the

as tcr.

presentation cylinder,

and establish

has dealt it should

can be used shells

be obvious

specific that this

for the analysis

subjected

however,

with the

that

of

method

of creep

to an assortment F = 1.0

case

of

for fiat plates

Section July

D 1,

Page

6.0

TIIERMAI,

6.0.1

General. Thermal

sudden ternal

SII(}{:K.

shock

changes

rclers

environment. from

varying

temperatures.

the

more

such

of candidate

be accomplished

by means

shock

resistance,

which

methods.

The

indices,

as well

which may

is suddenly be of similar

as the true

actual

shock-type

scope.

accurate are

termine

the

temperature

during

to consider established.

the

used

in a hot In the

which

for

and

by both

include

stress

behavior.

The

the

mass pertinent problems,

distributions As a rule,

strain-rate-sensitivity

experimental

of thermal-shock such

of such

models

that

a

must

of time it will

allowable

it

by the

one

at a number however,

as well

Generally,

stresses

except

slab

to perform

effects.

thermal

as a flat investigation

unnecessary

when

selection

thermal-

experimental

inertia

aero-

shouhl

and

models

treatment

it is usually

involves

type

analytical

of simple

medium.

The

considers

comparisons

theoretical

steady-state

that

slowly

by reentry

of this

ex-

distinguishes and

variety.

structures

analysis

to compute

transient

the material

may

that

experienced

process

be ewduatcd

configurations,

analysis

as

space-shuttle

theoretical

immersed

methods

of the

method

the

complex

is sufficiently

ments

may

analytical as

are

of a screening

in the

of steady-state

transients

for

undergoes

change

transients

conditions

materials

a body

by a rapid

of severe

common

as those

and comparison

whereby

caused

presence

Severe

structures,

phenomenon

usually

It is the

shock

space

to the

in temperature,

thermal

1972

263

same

deincre-

be necessary

stresses

are

Section

I)

July

1972

Page

6.0o 2

Stresses

1,

264

and Deformations.

Configuration. The equations as illustrated

in Figure

1. are

free

In all

and tables

Flat

cover

each

of the

following,

6.0-1:

slabs,

of infinite

2.

Solid

cylinders

3.

Solid

spheres.

cases,

extent,

which

are

of uniform

thickness

and

length

that Hooke'

s law applies.

Conditions.

Temperature

are

free

in surface

always

Equations

of any external

constraint.

Distribution.

The supposition change

of infinite

it is assumed

All bodies

is made

temperature.

subjected

that

the subject

The upper

to identical

bodies

and lower

experience surfaces

a sudden of the fiat

slab

temperatures.

and Tables. This

section

Poisson

t s ratio,

pansion

are

single

here

of holes

Boundary

are

provided

the thermal

unaffected

effective

of averaging

is based

values technique.

on the assumption diffusivity,

by temperature for

each

of these

that Young'

and the changes. properties

coefficient Hence,

s modulus, of thermal

the user

by employing

must some

exselect type

Section D July i, 1972 Page 265

The method presented here was published by Adams

F

49.

It can be used to determine

the temperature

ing the subject transient phenomena.

and Waxler

in ReL

and stress distributions dur-

To accomplish

this, several simple

formulas must be used in conjunction with appropriate tabulated values. f

of these are given in the summary

All

which follows. In applying this method,

care should be taken to ensure that the units specified under NOTATION

are

used.

The called

the

method

makes

fractional

use

of a temperature

temperature

T-

Tf

excess

and

function, is defined

_j,

which

may

be

as follows:

= I

j-

The

T i-

subscripts

Wf

1,

2, and

an infinite-length

solid

lations

of _bj published

of this

section

associated the

werc

with the

temperature

Infinite

Solid

@]

=

1,

'

_1

Solid

•

that

the

or a solid

by Adams

and

developed subject

d

3

cylindcr,

from

Waxier a study

configurations.

(a)

Cylinder.

(1)

_

value

is for an infinite

sphere,

and

rcspectively.

appearing

transfer

The

also

tables are

slab,

tabu-

summary

phenomena

include

defined

solid The

in the

of the heat

_I,_, _I,_, and ,I_ _, which

Slab.

f

2,

3 denote

parameters

0 Infinite-Length

j

values

as follows:

for

Section July

D

1, 1972

zse

, r;-. ,,d(_). ¢' " _"_ o

(2)

r

•

o

a.

b.

_egment solid

Segment

of infinite

o! infinite-length

solid

slab.

d.

Solid

sphere.

cylinder. Figure

6.0-1.

Configurations.

..J

Section July Page Summary

of Equations The

/:

and

parameters

D

1, 1972 267

Tables.

q and

fl, which

appear

in the

tables,

are

defined

as

follows:

t

q : _

(:3)

and

where k

Infinite

t

K-

Cpp

Solid

Slab,

T = _I (Ti

- Tf)

*

(4)

+ Tf

(5)

and E(_ (T i -Tf)

x The

values

:_

y

=

for

Infinite-Length

_ i and ,I, [ are Solid

(*I-_)

(I - v)

obtained

J

"

from

Table

(_;) 6.0-1.

Cylinder.

T = q_2 (T i - Tf)

+ Tf

,

(7)

Err (T i - Tf) (rr=

(1 - v)

(q_v2 - @2)

,

(8)

(_I'2 ÷ _2 - _2)

(9)

and

Err

(T i - Tf) t

crt :

(1 - v)

Section

D

July

1972

1,

Page

where

_

is the value

from

Tables

Solid

Sphe re.

T=

6.0-2

_r =

z at r/R

and 6. o-3,

_b:j (T i-

2E_

of_

Tf)

+Tf

= 1.

The values

for #z andS2

are

268

obtained

respectively.

(lO)

,

(Ti-Tf)

(,;-,_)

C1- v)

,

(11)

and

E_(T

_twhere from

i - Tf)

(1 - _)

_I,_ is the value Tables

6.0-4

(2_ + _:,- _3)

of4_ 3 at r/R

and 6.0-5,

= 1.

respectively.

The values

(1_) for ¢3 and _s are obtained

TABLE

f

ORIGIN_£

FS:/i

OF.. POOR

QUALITY

6.0-1.

i:

Section

July 1, 1972 Page 269

SLAB-PARAMETERS

_bl AND

_t'

0.3000

0.3200

0,3400

0.3600

0.3_0o

0,400o

o. 4200

0.4400

0.4800

0.4800

0.

0. 0013

0. 0031

0. 0061

q. 0109

0. 017't

0. 0270

0. 0386

0. 0526

0. 0690

0. 0875

0.10 O. 20

0. 0013 O. 0013

o. 0030 O. 0029

0. 0061 0. 0058

0. ol08 0. 0104

o. o176 O. 0 t_;9

(). 0266 (). o256

o. 0381 O. 0367

0. 0520 O. 0500

o. o682 0 0656

0. 0665 O. 0932

O. 30

O. 0012

O. 0027

o, 0n55

O. 0097

O. ()158

O. 024 q

(}. o344

O. 0469

O. 0615

O. 0780

O. 40

O. 0011

0. 0025

O, 0050

O. 0o88

O. 0144

O. 0218

0.03

O. 0428

O. 0558

O. 0708

0.50

O. 0010

O, 0022

0.0043

O. 0r,77

0.0126

O. 0191

O. 0273

O. 0372

O. O488

O. 0619

0, 60

o. 0008

0. 0018

0. 0036

o. 0o64

o. ol o.t

0. 0158

o, 0227

0. 0309

o. 0406

0.0515

O. 70

O. 0006

O. 0014

O. 0028

_. 0050

o. oo_1

o. o122

o. o175

O. 0239

O. 0313

O. 0397

0.80 0, 90

0. 0004 0. 0002

O. 0010 (). 0005

0. 0019 0. 0010

0. 0034 0. o017

0. 005:, o. (Io28

o. q)08:1 i) t1012

0. (H19 0. oo60

0. 0163 (i. 0082

0. 0213 0. 0106

0. 0270 0. 0137

1.00

0. 0000

0, 00(10

0.0oo0

0. olin0

i). (l[)oo

i). (if}0()

0, 0000

(). 0()0o

0. 0000

0. 0000

0. o009

0.0(120

o. 0039

0. o009

(}. (_113

0. Ol72

_). 0246

(). 0335

O. 0,439

0. 0557

O. 5000

O, 5200

O,54o0

O. 5(iq)()

O, %qO(l

(I. ,,Llr)(I

H. G21)0

o. )14oo

o. 6600

o, 6800

O,

o. lO_0

')

ql. 2559 o. 2527

0.31}90

0. 3354

o. 2789

0, 3052

0. 3313

0.20

O. lO27

O. 1237

0. 1516 o. 1,160

0.22!!5 ,_. 22¢;7

0. 282,t

o. 1066

o. 13Ol o. 1285

O. 1525

0.10

,i. _'I _')

O. 2 i:(:l

f). 26_6

0. 291t8

0. 3190

O. 30

o. 0962

O. 2:?_(I

o. 2516

O. 2753

O. 2998

o. 0674

o. 13_16 O. 1242

'). '-)(145

0.4O

O. 1159 O. 1052

I_. It; 16

". 1857

O. 2¢)7(,

0.22_5

O. 2500

0.2713

0.50

O. 0764

O. 0920

0. 10_6

O. 1023

O, I',(l!)

II. 1997

9. 2185

O. flO

o. O635

O. 150.1

O. 0490

0. 0902 o. f}GH7

(). 17_49

0.70

O. 0765 0. o591

O. 1_16 0, 14011

0.80

0. 0334

o. n4()2

o. o47,1

O. 90

0. 0169

O. 0203

1. O0

O. 0000

O, I)0(IO

o. 0240 O. {)(ffl0

O, 0687

o, 0828

O. o977

O. 7000

o. 7200

O. 7400

.7(,(.)

O. 3616 0.3571

O. 3H74

0.4127 O. 4077

0. ,I :',';'(;

I). ,tfi ] !)

0.10 0.1t9

0. 3439

O. 11084

O. 3925

('. ,1:122 O. 41112

O. 45(;2 O. 4393

_)..i '79)i (I. 'I_; 1;q

0.30

0. 3222

O. 3452

o. 3678

O. :1_!)9

O..I 1 IG

O. 1:!27

0.40 0.50

O. 2925 O. 2557

0.31:H

0. 3339

O. 3541

o. :17:I7

O. :_92')

0. 2739

0.2919

o. 31}95

o. :_267

0. '14:15

0.80 0,70

0.2125

0. 2277

O. 2426

O. 257:_

(I. '_7IG

O.2855

0. 1642 0. 1117

0. 1759 0. 1197

0. 1H74 0. 1276

O. 19_7

II. 21)!1"(

O. Y20G

0.90

0. 1353

0. 142_

(). 1501

O. 90

O. 0566

0. o6(16

O. 0646

o, 06_5

o. O723

O. 07,;0

1. O0

O. 000o

O. 0000

o. ()0rl(I

0. t,0(}q

0. 000()

(}. _q()(){i

O, 2302

O. 2466

0. 2628

o. 27_(i

o. 29,t 1

O. 9o0o

O. 9200

o. 9400

0.96o0

O.

0.5941

O, 8137

O, 6327

o.6509

0.10

0.5868

O. 8062

0.20

o. 5652

O, 5_39

0. 6250 O. 6020

0.30

O. 5297

O. 5473

0.40

0.4911

0.50

O.

2(1:(5

l). 1759

v.

:Ui)O

O.

q).

l'/'_l

I(_9.1

t,. 15_7

19:15

_). I_(

o.

I i_

I

o.

]_59

O.

)I. I()17

O.

o.

o. (P9_,I

0_o9

_

I I:_!) I190

12

1). !042

t}. Jifi2

o. 1660 O, 12_2

q). "7(19

O. 2372 . O. 1971 O. 1523

t,..791

O. ()'_73

O. 0955

O. 1036

(I. 0279

0.031

(L 07,59

I). 0400

o. 0"142

O. 0.483

O. 0525

O,

()(i()()

(i. Of)l)((

O. O000

l). 61000

O. 0000

0. 0000

O. 0000

0.

I134

t). 1295

0. I,iC_I

14.162!)

o. 17!)_

o. 1967

o, 2135

I,. _21)0

l). _400

O. 8600

O. 8800

O. 50U6 o. 5o'24

I). 5310 O. 5245

0. 3527 O. 5459

O. 5738 O. 5667

0.4 H:I_

o. 5051

0. 5258

G. 5456

0.,15

0.4733

0..1927

0.5115

0.4i16

0.4298

0.4474

0.4645

o, :(59_

0. 3757

0. 3912

0.40fl2

¢). 2991

0.3124

0.11252

O. 3377

(). 2311

0.2413

0.2513

0. 2609

o. 1.572

0. 1643

O. 1711

O. 1777

0. H796

0. 0U32

0. 0866

0. 0900

0. O()I_O

O. 0000

O. 0000

O. 0000

O. 309'*

o. :_2:g9

0.3382

0.3521

0.3655

0, 9_(10

I, OO0o

I. 02¢,)

1.04[)0

i, 0,_00

1. 0800

0.66_5

0. 6654

O. 7017

0. 71711

O, 7323

O. 7467

0, 64:(0 O. 61115

0.6(i(_4 0. 6363

0. 6772 O. (;525

0. *)',):_:; (,.);(;_ t

0. 7087 O. 11831

O. 7936 O. 6976

O. 7378 O. 7114

O. 5643

o. ,5_o7

o. 5966

o. G119

0.62(i7

o. _,41o

o. 6547

O. 6680

O. 4972

O, 5127

o. 527_

0.5423

O. 55114

o. 57o_)

0.5831

0.5958

0.6081

0.4207

0.4346

O. 44h5

0,4617

0,4746

0.4870

0.4901

0.51o_

0.5221

0.5331

0.80

O. 3499

O. 3617

O. 3731

O. 38,t 2

0.3950

0.4054

o.4156

0.4255

0.4351

0.4445

O, 70

O. 2704

O. 2795

O. 2884

O. 297o

O. 3054

0. 3136

o. 3215

o. 3293

O. 3368

0._ 3442

O, 80

O. 1841

O. 190',1

O. 1964

O. 2023

O. 2081

0. 2137

0.2 t!)l

O. 2245

O. 2297

O. 90

O. 0932

O. 0964

O. 0995

0.1025

0,1054

0.1082

o. 111o

0.1137

O, 1164

1. O0

O. 00o0

O. 0000

o. 0000

O. 0o00

[). 0000

O, DO0()

0. 000(I

O. ((01)0

0. 0000

0. 0000

O. 3786

O. 3912

O. 4 034

0.4153

0.4267

0.4378

0.4485

0.4588

0,4688

0.4784

%,

0.

)_,

(I. ()_511

(I, (i(;2fl

':FL_

D

13

O. 2348 .0.1190

ORIGINAL

PAGE

POOR

IS

Section

QUALITY

TABLE

6.0-1.

(CONTINUED)

1. 1000

1. 1200

1. 1400

1. 1600

1. 1800

1. 2000

1. 2200

1. 2400

1. 2600

1. 2800

0.

0. 7804

0. 7736

0. 7662

0, 7982

0. 8097

0. 8206

O. 8311

0.8410

0. 8505

0. 8595

0. 10

0.7518

O. 7645

0. 7771

0. 7890

0. 8005

0.8114

0.8218

0.8318

0.8412

0. 8603

0.90

0.7248

0.7376

0.7498

0.7616

0.7729

0.7837

0.7941

0.8040

0.8135

0.8228

0.30

O. 680_

0.6930

0.7048

0.7162

0.7272

0.7378

0.7480

0.7578

0.7672

0.7783

0.40

0.8200

0.6315

0.6426

0.6534

O. 6638

0,6739

0.6837

0.6932

0.7024

0.7113

0.60

0.5438

0.5542

0.5843

0.6741

0.5837

0.5930

0.6020

0.6109

0.6195

0.8280

0.60 0.70

0.4536 0.3814

0.4625 0.3585

0.4712 0.3655

0.4797 0.372:]

0.4880 0.3789

0.4962 0.3855

O, 5042 0. :_920

0,5120 0, 3984

0.5197 0.4046

0.5272 0.4109

0.80

O, 2398

0,2447

0.2496

0.2543

0.2590

0.2836

0.2682

0,2727

0.2772

0.2816

0. _)

0. 1218

0. 1241

0. 1266

0. 1291

0. 1315

0. 1339

0. 1362

0. 1386

0. 1408

0. 1432

1. O0

O. 0000

O. 0000

O. 0000

O. 0000

O. O00O

O. 0000

O. OOO0

O. OOOO

O. 0000

O. O00O

0.4878

0.4968

0.5055

0,5140

0.5221

0.5301

0.5377

0.5451

0.5523

0,5593

1.3000

1.3200

1.3400

1.3600

1.3800

1.4000

1.4200

1.4400

1.4600

1.4800

_. _gSO

_t

0.

O. 8761

O. 8838

O. 8911

O. 898o

o. 9046

9.91o_

O. 9166

O, 9221

O. 9273

0. _.0 0.|0

0:9689 0.8313

0. 8670 0,8396

0.8748 0.8475

0. 8822 0.8551

0. 8892 0.8624

O. 8958 (,. 8693

0. 9021 (I.8759

O. 9081 0,8822

0, 9137 0.8882

0. 9191 0.8939

0.$0

0.7880

0.7935

0.8016

0.8094

0.8189

0.8242

0.8:]12

0.8379

0.8444

0, 8506

0. 40 0.60

O. 7200 0.6362

O. 7283 0.6443

O. 7365 0.6522

O. 7444 9.6599

O, 7521 9.6C74

O. 759(; 0.6748

0.7668 0.682!

O. 7739 0.6892

0. 7808 0.5962

O. 7874 0.7030

0. 60

O. 5846

O. 5419

O. 5491

O, 5562

O. 5632

O. 5701

0.57fi9

O. 5836

O. 5902

O. 5967

0.70

0,4170

0.4231

0.4291

0.4350

0.4409

0.4487

O. 452F,

0.4582

0.4839

O. 4696

0.80

0.2860

0.2904

0.2947

0.2990

0.303:!

0.3075

11.311_

0.3160

0.3202

0.3243

O. gO

0. 1455

O. 1478

O. 1501

O. 1523

O. 1546

O. 1588

0, 1590

O. 1613

o. 1635

O. 1657

1.00

O. 0000

O. 0000

O. OOOO

O. O00O

O. 0000

O. 0000

O, 0000

O. 0000

O. 0000

O. O000

0.8681

0.5726

0.5790

0.5852

0.5912

0.5970

0.8027

0.8082

0.6136

0.8188

1, 5000

1.5200

1.5400

1.5600

1.5800

1.6000

1.6200

1.6400

1.6600

116800

O.

O. g322

O. 9368

O. 9412

O. 9453

O. 9491

O, 9527

0.9581

O. 9592

0. 9622

O. 9650

O. 10

O. 9241

O. 9289

O. 9334

O. 9377

O. 9417

O. 9485

0.9491

O. 9524

o. 9556

O. 9685

O. 20

O. 8994

O. 9046

O. 9096

O. 9143

O. 9188

0.9231

O. 9272

O. 9311

O. 9348

O. 9383

0.30

0.8566

0.8624

9.8680

0.8734

0.8785

0. 8835

0.8883

0.8930

0.8974

0.9017

O. 40

O. 7939

O, 8002

O. 8064

O. 8124

O. 8182

0.8239

0, 8294

O. 8:_48

O, 8400

O. 8451

O. 80

O. 7097

O. 7163

O. 7227

O. 7291

o. 7353

O. 7414

o. 7474

o. 7533

O. 7591

O. 7648

O. $0

O. 6032

O. 6095

O. 6158

O. 6221

O, 6282

O. 6343

O. 8403

O, 6462

O. 8521

O. 6579

0.70

0.4752

0.4807

0.4863

0,4918

0.4972

0.5026

o. 5n80

0.5123

0.5187

0.5240

O. 80

O. 3288

O. 3326

O. 3368

O. 3409

O. 3450

O. 3491

O. 3532

O. 3572

O. 3613

O. 3853

0. 90

0. 1679

0, 1702

O. 1724

0. 1746

O. 1768

0, 1790

0. 1812

0.1834

0. 1856

O. 1878

1. O0

O. O00O

O. 0000

o. OOO0

O. O0OO

O. 0o00

O. 0000

O. 0000

O. oo00

O. O00O

O. 0000

0.6239

0.6288

0.6336

0.8383

0.8429

0.6474

0.6517

0.8560

0.6601

0.6642

1.7000

1.7200

1.74 O0

1, 7600

1. 7800

1.8000

1. 8200

1.84 oo

1.8600

1.8800

91t

Q'l_

_a_

O.

0. 9676

O. 9700

O. 8"/23

O. 9744

O. 9763

O. 9782

0. 9799

O. 9815

O, 9829

O, 9843

O. 10

O. 9613

O, 9640

O. 9664

O. 9687

O. 9709

O. 9729

O. 9748

O. 9766

0.9783

O. 9798

O. 30

O. 9416

O. 9448

O. M,79

O. 9507

O. 9535

O. 9560

O. 9585

O. 9608

O. 9631

O. 9652

O. 30

O. 9068

O. 9098

O. 9136

O. 9173

O. 9209

O. 9243

O. 9276

O. 9308

O. 9336

O, 9367

0.40

O. 8601

O. 8849

O. 8696

O. 8642

O. 8686

O. 8730

O. 8772

O. 8813

O. 8853

O, 8891

0.50

0.7704

0.7758

0.7812

0.7865

0.7917

0.7968

0.8018

0.8067

0.8115

0,8162

O. 80

O. 0837

O. 8693

O. 6749

O. 6808

O. 6860

O. 6914

O. 6967

O. 7020

O. 7073

O, 7124

0.70

0. 8292

O. 5344

O. 5396

O. 5447

O. 5498

O. 5549

O. 5600

O. 5650

O. 5700

O. 5749

0.80

0.3693

0.3734

0.3774

0,3814

0.3854

0.3893

0.3933

0,3972

0.4912

0.4061

O. I)0

0. 1BOO

O. 1922

O. 1944

O. 1966

O. 1987

O. 2009

O, 2031

O. 2053

O. 2075

O. 209_

1. O0

O. 0000

O. OOOO

O. O00O

O. O00O

O. 0000

O. O000

O. 0000

O. OOOO

O. 0000

O. 0000

O. 6881

O. 6720

O. 6758

0.6794

O. 8830

O. 6868

0.6900

O. 6934

O. 6967

O. 6999

tlt

D

July 1, 1972 Page 270

v

_

TABLE

1.

9000

1.

9200"

1. 9400

1.

9600

ORIGINAL

p_c_.

OF POOR

QUALITY

6.0-1.

1.

9800

Section

_,

July I, 1972 Page 271

(CONTINUED)

2.

O000

2.

0200

2.

0400

2.

0600

2.

0800

O.

0.9856

0.9868

0.9678

0.9889

0.9898

0.9906

0.9914

0.9922

0.9928

0.9935

0.10

0.9613

0.9826

0.9639

0.9851

0.9862

9.9872

0.9882

0.9891

0.9999

0.9907

0.20

0.9672

0.9690

O. 9706

O. 9725

0.9741

6.9757

0.9771

O. 9785

0.9798

O. 9810

0.30

O. 9395

0.9422

0.9448

0.9473

0.9497

O. 9520

O. 9543

O. 9564

O. 9584

0.9604

O. 40

0. 8929

O. 8966

O. 9001

O. 9036

0.

0.

0.

O. 9165

0.

0.

0.50

0.8208

0.8254

0.8298

0.8342

0.8385

0.8427

0.8468

0.850_

0.8548

0.8586

9070

9102

9134

9195

9224

0.60

0,7175

0.7226

0.7275

0.7324

0.7373

0.7421

0.7468

0.7515

0.7561

0.7607

0.70

0.5798

0.5847

0.5895

0.5943

0.5991

0.6039

0.6086

0.6132

0.6179

0.6225

0.80

0.4090

0.4129

0.4168

0,4207

0.4245

0.4284

0.4322

0.4361

0.4399

0.4437

0.90

O. 2118

O. 2140

O. 2162

O, 2184

O. 2205

0.2227

O. 2249

O. 2270

O. 2292

O. 2_I14

1.00

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.7031

0.7062

0.7092

0.7121

0.7151

0.

0.

0.7234

0.7261

0.7288

2.1000

2.1200

2.1400

2.1600

2.1900

2.2000

2.2200

2.2400

2.2600

2.2900

O.

O. 9940

O. 9946

O. 9951

O. 9955

O, 9959

O.

O. 9966

O. 9969

O. 9972

O. 9975

0.10

0.9914

0.9921

0.9927

0.9932

0.9938

0.9943

0.9947

0.9951

0.9955

0.9059

0.20

0.

O. 9832

O. 9842

O. 9852

O. 9861

O. 9870

O. 9878

O. 9686

O. 9893

O. 9900

0.30

0.9623

0.9641

0.9658

0.9674

0.9690

0.9705

0.9720

0.9734

0.9747

0.9760

O. 40

O. 9252

O. 9279

O. 9306

O. 9332

0.9356

O. 9381

0.9404

0.9426

0.9448

0.9470

@_

x/a_

9821

7179

9963

7207

0.50

0.8624

0.8661

0.8698

0.9733

0.8768

0.8802

0.8835

0.8866

0.6900

0.6931

0.60

O.

0.7696

0.7739

0.7782

O. 7825

0.7867

0.7908

0.7949

0.7989

0.8028

0.70

0.6270

0.6316

0.6361

0.6405

0.6450

0.6494

0.6537

0.6581

0.6624

0.6666

0.80

0.4475

0.4512

0.4550

0.4588

0.4625

0,4662

0.4899

0.4736

0.4773

0.4810

0,90

0.2335

0.2357

0.2378

0.2400

0.2421

0.2443

0.2464

0.2496

0.2507

0.2529

1.00

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0,7313

0.7339

0.7364

0.7388

0.7412

0.7436

0.7459

0.7491

0.7504

0.7525

2.3000

2.3200

2.3500

2.3600

2.3800

2.400O

2.42O0

2.4400

2.4600

2.4800

O.

0.9977

0.9979

0.9981

0.9983

0.9985

0.9986

0.9988

0.9989

0.9990

0.9991

O. 10

O. 9962

O. 9965

O. 9968

O. 9971

O. 9973

O. 9976

O. 9978

O. 9980

O. 9981

O. 9083

0.20

0.9906

0.9912

0.9918

0.9924

0.9929

0.9933

0.9938

0.9942

0.9946

0.9950

0.30

0.9772

0.9783

O. 9795

O. 9805

O. 9815

O. 9825

O. 9834

O. 9843

O. 9851

O. 9859

0.40

0.9490

0.9510

0.9529

0.954_

0.9566

0.9583

0.9600

0.9616

0.9631

0.9647

O. 50

O. 8961

O. 6991

O. 9020

O. 9048

O. 9076

O. 9103

O. 9130

O. 9155

O, 9180

O. 9205

0.60

0.8060

0.8106

0.8144

0.8181

0.8218

0.8254

0.8290

0.8326

0.8360

0.9394

O. 70

O. 6708

O. 6750

O. 6792

O. 6833

O. 6874

O. 6914

O. 6954

O. 6994

O. 7034

O. 7073

0.80

0.4847

0.4883

0.4919

0.4956

0.4992

0.5027

0.5063

0,5099

0.5134

0.5170

0.90

0.2550

0.2572

0.2593

0.2614

0.2636

0.2657

0.2678

0,2700

0.2721

0.2742

1.00

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.7547

0.7568

0.75H9

0.7609

0.7629

0.7649

0.7669

0.7688

0.7707

0.7725

2.5000

2.5200

2.5400

2.5600

2.5800

2,6000

2.6200

2.6400

2.6600

2.6800

• _

_a_

• _

7851

O.

O. 9992

O. 9993

O. 9993

O. 9994

O. 9995

O. 9995

O. 9996

O. 9996

O. 9997

O. 9997

O. 10

O. 9964

O. 9986

O.

99_7

O. 9988

O. 9989

0.

9990

O. 9991

O. 9992

0.

9993

O. 9993

0.24)

O. 9953

O. 9956

O. 9959

O. 9962

O. 9965

O. 9967

O. 9970

O. 9972

0.

9974

O. 9976

0.30

O. 9867

0.9874

O. 9881

0.9887

O. 9894

O. 9899

O. 9905

O. 9910

O. 9915

O. 9920

0.40

0,9661

0.9675

0.9689

0.9702

0.9714

O. 9726

O. 9738

O. 9749

O. 9760

O. 9770

0.50

0.9229

0.9252

0.9275

0.9297

0.9319

0.9340

0.9361

0.9381

0.9400

0.9419

O. 60

0.8427

0.8460

0.8492

0.9524

0,8556

0.8566

0.8617

0.8647

0.8676

0.8705

0.70

0.7112

0.7150

0.7188

0.7226

0.7263

0.7300

0.7337

0,7373

0.7409

0.7445

0.80

0.5205

0.5240

0,5275

0.5310

0.5344

0.5379

0.5413

0.5446

0.5482

0.5516

0.90

O. 2763

0.2784

0.2806

O. 2827

0.2848

O. 2869

O. 2890

0.2911

O. 2932

O. 2953

I., O0

O. OOOO

0.0000

0.0000

O. O00O

0.0000

O. O00O

0.0000

0.0000

0.0000

0.0000

O. 7743

O. 7761

O. 7779

O. 7796

O. 7813

0.

0.

O. 7863

O. 7879

O. 7895

7830

7847

D

ORIGINAL

F/;3E

OF POOR

QUALITY

TABLE

,

_

6,,.0-1.

Section

;_

July

D

1,

Page

(CONTINUED)

q" I. 7000

9.7100

2.7400

2.7800

2.7800

2.8000

2.8200

2.8400

2.8600

2. 8800

O.

O. Mr97

O. 9988

O. MHI@

O. M_,8

O. IHH)8

O. 9898

O. 9999

O. 9999

O. 9999

O. 9999

O. 10

o. 2M14

O. 08'94

O. _

O. _

O. _

O. 9996

O. 9997

O. 9997

O. 9997

O. 9997

0.|0

0,9_7

O. MM2 O. 9937

O. 9983 O. 9941

O. 9985 O. 9944

O. 9988 O. 9948

O. 9987 O. 9951

O. 9989

0.11125

O. 9981 O. 9933

O. 9988

0.90

O. 99'79 O. m9

0. 9954

O. 9958

O. 40

o. lffso

O. V/90

O. 9799

0. 9806

0.9817

O. 9825

0.9833

0.9840

O. 9848

O. 9855

0.00

O. NM

O. 9410 O. 8816

O. 9507 O. 8842

O. 9923 O. 8888

O. 9539 O. 8823

O. 9654 O. 8918

O. 9683

0.8"5'33

O. 9473 O. 8789

O. 9589

0.10

O. 9454 O. 8781

O. 8943

O. 8987

O. 70

O. 7480

O. 751.15

O. 7650

O. 7584

O. 7818

O. 7651

O. 7685

O. 7718

O. 7750

O. 7782

O.S@

0.9841

O. 544583

O. 5417

O. 6060

O. 5403

O. 5716

O. 5749

O. 5782

O. I0 1.00

O. 91fl'4 O. 0000

O. 2998

O. 3010

O. 3037

O. 3068

O. 30'/9

O. 3100

O. 3120

0.5814 0.3141

0.5847 0.3162

O. G_O0

O. 0000

O. 0000

O. 0000

O. 0000

O. 0000

O. 0000

O. 0000

O, 0000

o. 7010

O. 7986

O. 7941

O. 7958

O. 7971

O. 7985

O. 7999

O. 8013

O. 8027

O. 8041

2.0000

2.9800

8.9400

2.9900

2.9800

3.0000

3.0200

3.0400

3.0800

3,0800

O.

O. _

O. 9899

0.9999

0._9

0.9909

1.0000

I. 0000

1. 0000

1.0000

1.0000

O. 10

O. _

O. 9898

0.9998

0,_

0.9998

0.9999

0.9999

0.9999

0.9999

O. 9999

0.|0

O. IIHIO

O. 9890

0.9991

O.M_2

0.9_3

0.9999

0.9994

0.9994

0.9995

0.9995

0.30

0.9810

0.9M2

0.40 0.10

O. 8_11 0.982'?

0.9880 O. MII

0.9964 0.9874

0.9066 0.9880

0.2988 0.9805

0.9970 0.9891

0.9972 0.9896

0.9974 O. 9901

0.9975 O. 9906

0.9977 O. 9910

0. M24

0.9837

0.9849

0.9661

0. M73

0.9884

0.9895

0.9706

O. @0 0.70

O. 8991 0.7814

O, 9014 0.7840

0.9037

0.9080

O. 8082

O. 9103

O. 9124

O. 9145

O. 9185

O. 9185

0.7877

0.7908

0.7939

0.7989

0.7998

O. 8029

O. 8058

O. 8087

O. M@

O. 5870

O. 5911

0.5943

0.5075

0.6007

0.6038

O. 6070

O. 6101

0.6132

0.6163

O. 00

O. 3183

O. 3204

O. 3224

O. 3945

O. 3286

O. 3280

O. 3307

O. 3327

O. 3348

O. 3369

1. O0

O. 0000

O. 0000

0.0000

0.0000

0.0000

0.0000

0.0000

O. 0000

O. 0000

O. 0000

O. 0065

O. 8068

O. 8081

O. 8094

0.8107

0.8119

0.8132

0. 8144

0.8150

0.8168

3. 1900

3. 2000

3. 3000

3,4000

3.5000

3.8000

3.7000

3.8000

3. 9000

4.0000

O.

1. 0000

I. 0000

I. 0000

1,0000

1,0000

1. 0000

1. 0000

1. 0000

1. 0000

1. 0000

O. I0

O. 9999

1. 0000

1. 0000

O. 80

O. Mi,98

O. 9897

O, 9898

1.0000 0.9990

1.0000 0.9989

1.0000 1.0000

1. 0000 1.0000

1. 0000 1.0000

1. 0000 1.0000

1.0000 1.0000

0.3@

O. MY/9

O. 9985

O. 9989

O. 92tMI

0.9898

O. 99_

O. 9998

O. 9998

0.9999

O. 9999

O. 40

O. 9815

O. 9934

O. 9949

0.00

O. V/10

O. ff/63

0.9804

O. 9991 O. 9838

O. 99'/0 O. 9867

O, 9977 O. 9891

O. 9983 O, 9911

O. 9987 O. 9928

O. 9991 O. 9942

O. 9993 O. 9953

O. lO

0. U98

0.9297

0.9381

O, 9466

O. 9923

O. 9883

O. 9837

O. 9684

O. 9'728

O, 9';'63

O. 70

O. 8116

O. 8254

O. 8385

O. 8508

O. 8624

O. 8733

O. 8835

O. 8931

O. 9020

O. 9103

O. 80

O. 8194

O. 634a

O. 8494

0. M38

O. 8778

0,6914

0.7047

0.7175

0.7300

O. 7421

0.20

0.3388

0.3491

0.34599

1.98

O. 0000

O. 0000

O. 0000

0.3094 0.0000

O. 3794 0.0000

O. 3893 0.0000

0.3992 0.0000

0.4090 0.0000

0.4187 0.0000

0.4284 0.0000

O. 0180

O. 8237

O. 8290

0.8341

0.8388

0.8433

0.8475

0.8515

0.8553

0.8590

4. I000

4. 2000

4. 3000

4.4000

4. 5000

4. 9000

4. 7000

4.8000

4.9000

5.0000

O.

1.0000

1. 0000

1.0000

1.0000

1. 0000

1. 0000

1. 0000

1.0000

1.0000

1.0000

O. 10

1. 0000

1. 0000

1. 0000

1.0000

I. 0000

I. 0000

I. 0000

O. 20

1.0000

1. O00O

1.0000

1.0000

1.0000

1. 0000

1. 0000

1.0000 I. 0000

1.0000 1.0000

1. 0000 1.0000

_a_

t 0'

t_

_a_

q'/'

_a_

O. S@

1.0000

1. O00O

1. O00O

1.0000

I. 0000

I. 0000

I. 0000

0.40

O. 9M16

O.l_M

0,999'7

0._9

O. 9999

O. 9902

O. 9999

1.0000 1.0000

1.0000 1. 0000

1,0000 1,0000

O. IHI

0._3

0.99'70

0.99"/0

0.9981

O. I)985

O. MMI2

O. 9991

0.9993

0.9996

O. 9998

O. 80

O. 1_98

O. M20

O. 9880

O. 987|

O. 9891

O. _

O. 9922

0.9934

O. 9944

O, 9953

0.70

0.0181

0.9312

0.9981

O. N38

O. 9490

O. 9639

0.9583

0.9824

0.9861

O. 80

O. 75M

O. 7661

O. 7701

O. 7845'/

O. 7M9

O. 8068

O. 8163

0.8254

0,8342

0.8427

O. tO

O. 4.t80

O. 4478

O. 45011

O. 48S2

O. 4750

O. 4847

O. 4997

0.5028

0.5117

0.5205

1.00

O. 0000

O. 0000

O. 0000

O. 0000

O. 0000

O. 0000

O. 0000

0,0000

0.0000

O. 0000

0. 8614

O. M67

O. 8@80

O. 8710

O. 8748

O. 8774

O. 8800

0.8825

0.8849

O. 8872

tiV

" O. 925|

1972

272

O_U_.?,5"i' POOR

,-r ..... QUALI Ey

Section

D

July 1, 1972 Page 273

TABLE

x/a_

5.

O.

1.

1. 0000

1. 0000

1. 0000

1. 0000

I. 0000

I. 0o00

I. 0000

1. 0000

I. 0o00

O. 10

1.0000

1. oooo

1. O000

1. 0000

1. 0000

I. 0000

I. 0000

I. [)000

I. 0000

I- 00o0

O. 20

1.

0000

1, 0000

1.

0000

1. 0000

1. 0000

1.

0000

1.

0000

1.

000o

1.

0000

1. 0000

O. 30

1. 0000

1. 0000

1.

0000

1. 0000

1.

0000

1.

0000

1.

0000

1,

0000

1. 0000

1. ooo0

O. 40

1.

0000

l. 0000

1.

0000

1. OOO0

1. 0000

1.

O00O

I.

0000

1.

0000

1. 0000

1. 0000

O. 50

O.

9997

O. 9998

O. 9998

O. 9999

0.

9999

O. 9999

O. 9999

I,

(101}(I

I.

01109

l. 0000

O. 00

O. 9961

O. 9967

O. 9973

O. 9977

O. 9981

O. 9985

O. 9987

O. 9990

o,

9992

o.

O. 70

O. 9695

O. 9726

O. 9755

O. 9780

O. 9804

O. 9825

0.

9844

O. 9861

O. 9877

O. 9891

O. 80

O. 8508

0.

8587

O. 8661

O. 8733

O. 8802

0. 8868

O. 8931

O. 8991

(}. 9048

O. 9103

O. 90

O. 5292

o.

5379

O. 5465

O. 5549

O. 5633

O. 5716

O. 5978

9. 5879

0. 5959

I1. 6039

1.

O. 0000

O. 0000

O. O00O

O, 01)00

O. 0000

O. 0000

O- 0000

O.

[). 0000

O. 0000

O. 8894

O. 8915

11. 8935

O. _955

0. 8974

O. 8993

O. 9010

O. 9027

o.

o.

6.

1000

6.

2000

6.

:10o0

6.

4000

6.

6.

6000

6.

7000

6.8(}(1()

6.9000

7.0000

0.

1.

0000

1.

0000

1.

0000

1.

0000

I. 0000

1. 0000

1.

0000

1. 0090

1. 0000

1.

0000

O. 10

1.

0000

1.

0000

1.

9000

1.

0000

1. 0000

1. 0000

1.

001;.0

1. 0000

1. OOOO

1.

0000

O. 20

I. 0000

I. 0000

I. 0000

1.

0000

1. 0000

1. 0000

1.

0000

1. 0000

1 . 0000

1.

OOOO

O. 30

I.

1.

1.

I.

0000

1. o000

1. 0000

I.

0000

1. 0000

I . 0000

I.

0000

O. 40

I. 0000

I. 0000

1. 0000

1. 0000

1. 0000

1.

0000

1 • 000(;

1 . OOO(}

1 . 0000

O. 50

1.

0000

1.

0000

1.

0000

I. 0000

I. I)000

i. 0000

1.0o00

1 . 0000

I. 0000

I . 0000

O. 80

O.

9994

O. 9995

O.

9996

O. 9997

O. 9998

O. 9998

o.

9998

O. 9999

O. 9999

¢1. 9999

O. 70

O. 9903

O. 9915

O.

9925

0.

O. 9942

O. 9949

I). 9955

O. 9961

I}. 9966

(I. 9970

O. 80

O. 9155

O. 9205

O. 9252

0.9297

0.9340

0.9381

0.9419

(}. 9450

O. 9490

O. 9523

0.90

0.6117

0.6194

0.6270

0.6346

O. B420

0.6494

0.6566

O. 66218

O. 6708

9. 6778

1. O0

O. 0000

O. 0000

O. 0000

O. 0000

0.

0000

O. 0000

O. 0090

O. 001)0

O. 0o00

{}. 0000

0.90'15

0.9090

0.9104

0.9118

0. 9132

O. 9145

O. 9158

O. 917o

O. 9182

O. 9194

7.1000

7.2000

7.3000

7.4000

7. 5000

7. 6000

7. 7000

7. 800O

7,

9O00

8.

1.

01100

O0

'i'tt

_a_

$'1'

0000

0000

5.

3000

o000

I. 0090

5.

4000

9934

5.

5000

(CONTINUED)

5.1000

0000

2000

6.0-1.

5000

5.

6000

7000

5.

HO00

5.

O000

9044

6.

0000

9993

9060

I)000

O.

I. 0000

I. 0000

I. 0o00

I. 0000

1. [)000

1.

1,0000

I.

I. 0000

i, 0000

I. oo00

1. 0000

I. 0900

I. 0000

I. 0000

1.00(io

1. 0000

I. 0o00

O. 20

I. 0000

i. 0000

I. o0o0

I. 0000

I. 0000

I. 0000

I. 0000

I. 0000

I. 0000

I. 0000

O. 30

1. 0000

1.

0000

1. I)11(10

1.

I)1100

1.

1.

1. 0000

1.0o00

I.

0000

1. 00110

O. 40

1.

1.

0000

1.

1.

0000

1.0o1}0

1. 0000

1. 0000

1.

1.

O000

1.

0.50

1. 0000

I. (10o0

i. 0000

I. 0000

1. 9000

I. 0000

I. 01)00

O. 60

0.

9999

0.70

0.

9974

O. 80

O. 9554

0.

O. 90 1.

O0

_

0000

OOOO

1.01)00

0000

i. 0000

I. 0060

I. 0000

1. 0000

1.

0000

1. 0000

1.

1. 0000

I. 0(Io0

I. 001}0

I. 0000

0.

9977

0.

9980

O. 9983

O. ,(1985

O. 9987

O. 9989

O. 9991

O. 9992

0.

9583

O. 9611

O. 9037

O. 9661

O. 9684

O. 9706

0.

O. 9745

O. 976,'1

O. 6847

O. 6914

O. 6981

O. 7047

0.7112

0.7175

0.7238

0.7300

0.7361

0.7421

O. 0000

O. 0000

O. 0000

0.

O. 0000

O. 0000

O. 0000

O. 000(I

O. 0000

0.

0.9205

0.9216

0.9227

0.9238

O. 9248

O. 9258

O. 9267

O. 9277

O. 9286

O. 9295

8.1000

8.2000

8.3000

8.4000

8.5000

8.6000

8.7000

8.8000

8.9000

9.0000

1.0000

0000

0000

9993

0000

O.

1.

1.0000

1.0000

1.0000

1.0000

1.

1.0000

1.0000

1.0000

1.0000

1,0000

1.0000

1.0000

1.

0000

1.0000

1.0000

1. OOO0

1.0000

1. 0000

0.20

1.0000

1. 0000

1.0000

1.0000

1.

0000

1.0000

1.0000

1.0000

1.0000

1.0000

0.30

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

0.40

1. 0000

1.

0000

1.0000

1.

0000

1.0000

1.

0000

1.

0000

1.0000

1-0000

1. OOOO

0.50

1.0000

1.

0000

1.

1.

0000

1.

1.

0000

1.

0000

1.0000

1.

1. OOOO

0,60

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

0.70

0.9994

O. 9995

0.

9996

0.

9996

0.

0.9997

0,9998

0.9998

0.9998

0.9999

0.80

0.9"/80

0.9796

0.

9811

0.

9825

0.9836

0.

0.9861

0.9872

0.9882

0.

0._

0.

0.

0.

7595

0.

7651

0.

0.7781

0.7814

0.7867

0.7916

0.7969

1.00

0.0000

0,0000

0.000O

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.9303

0.

0.9320

0.9328

0.9336

0.9344

0.9351

0.9359

0.

7480

7538

9312

0000

0000

9997

7707

0000

9726

0.10

_

OOOO

1. O00O

0000

00110

0(}00

9000

O. I0

0000

D000

5.

9850

0000

9386

0.

9891

9373

ORIGINAL

F&_E

pOOR

L'_

QUALITY

Section

D

July

1972

1,

Page

TABLE

6.0-1.

274

(CONCLUDED)

9.

9.7000

9.

i. O000

1.0000

l. O000

1.0000

1.0000

1.0000

1.0000

|.0000

1.0000

1.0000

I. 0000

I. OOOO

I. 0000

i. 0000

I. OOO0

I. 0000

i. 0000

I. 0000

i. O000

1.0000

1.0000

1.0000

1.0000

i. 0000

1. 0000

i. 0000

I. 0000

I. 0000

I. 0000

I. 0000

i. 0000

1.0000

1.0000

l. O000

i. O000

I. 0000

I. 0000

I. OOO0

I. 0000

I. 0000

1. 0000

I. 0000

1. 0000

i. O000

1. 0000

1.0000

I. 0000

1. O00O

i. O000

I. 0000

0.70

0.9999

0.9999

0.9999

0.9999

0.9999

1.0000

1.0000

1.0000

1.0000

1.0000

0.80

0.9899

0.9907

0.9915

0,9922

0.0928

O. 9934

O. 9939

O. 9944

O. 9949

O. 9953

0.90

0.9019

0.8068

0.81t6

0.8163

0.8209

0.

0.

0.

0.

0.

1. O0

O. OOO0

O. 0000

O. 0000

O. OOO0

O. OOO0

0.0000

0.0000

0.0000

0.0000

0.0000

0,9380

0.9387

0. 9393

0.9400

0.9406

0.9412

0.94i8

0.9424

0.9430

0.9436

9. I000

9. 2000

9. 3000

9. 4000

9. 5000

O.

i. O000

1.0000

1.0000

1.0000

i. OOOO

0. I0

I. O000

1.0000

1.0000

1.0000

i. O000

0_20

I. 0000

i. 0000

I. 0000

I. 0000

0.30

i. 0000

i. 0000

1.0000

0.40

I. 0000

i. 0000

0.50

i. O000

O. 60

6000

8254

8299

8000

8342

9.

§000

8385

10.

0000

8427

I0, _000

10. 2000

|0. 3000

i0. 4000

I0. 5000

10. 6000

I0. 7000

iO. 8000

I0. 9000

1 i_ 0000

O.

I. 0000

I. 0000

I. OOOO

i. 0000

I. 0000

I. 0000

I. O000

I. 0000

I. O000

i. 0000

O. I0

I. 0000

I. 0000

I. O00O

i. 0000

I. 0000

I. 0000

i. 0000

I. 0000

i. O00O

i. 0000

O. 20

I. 0000

I. OOO0

I. OOOO

I. O00O

i. 0009

I. 0000

I. 0000

I. 0000

J. 0000

I. 0000

O. 30

i. 0000

I. 0000

I. 0000

I. 0000

i. 0000

I. 0000

I. oo00

I. 0000

I. 0000

i. 0000

O. 40

1. 0000

1. 0000

I. 0000

i. 0000

1.

1. 0000

1. 0000

I.

1.

1. 0000

O. 50

I. 0000

I. O000

I. 0000

I. 0000

I. 0000

I. 0000

I. 0000

I. OOO0

I. 0000

I. O00O

O. 60

t. 0000

t. 0000

1. 0000

1. 0000

1.

OOOO

i. 0000

1. 0000

i.

0000

1. 0000

1. 0000

O. 70

1. 0000

1. 0000

1. 0000

1.

1.

0000

1. 0000

i. 00o0

1.

0000

1. 0000

1. 0000

O. 80

O. 9957

O. 996

O. 9964

O. 9967

O. 9970

O. 9973

O. 9975

O. 9977

O. 9979

O. 9981

O. 90

O. 8468

O. 8508

O. 8548

O. 8586

O. 8624

O. 8661

O. _698

O. 8733

0.

6768

O. 8802

1. O0

O. OOOO

O. 0000

O. 0000

O. 0000

O. 0000

9. I)000

O. 0000

O. 0000

O. 0000

_1. 0000

0.

0.

0.9452

0.

0.94_3

0.

o.

0.

0.

0.9487

_1,t

9441

9447

i

OOOO

9457

0000

946

_

9473

0000

9478

0000

!)482

Section

D

July 1, 1972 Page 275

TABLE

6.0-2.

CYLINDER-PARAMETER

q)2

0.4800

0.5000

0.5200

0.5400

0.5600

0.5800

0.4000

0.4200

0.4400

0.4500

0.

0.0'002

0.

0004

0. 0009

0.

0017

0.0030

0.0049

0.0076

0.0113

0.0159

0.0218

0. I0

0.0002

0.0004

0.0009

0.0017

0.0030

0.0049

0.0075

0.0111

0.0157

0.0215

0.20

0.0002

0.0004

0.0009

0.0016

0.0028

0.0047

0.0072

0.0100

0.0150

0.0205

0.30

0.0002

0.0004

0.0008

0.0015

0.0026

0.0043

0.0067

0.0098

0.0139

0.0190

0.40

0.0001

0.0003

0.0007

0,0013

0.0024

0;0039

0.0000

0.0088

0.0125

0.0170

0.50

O. 0001

0.0003

0.0000

0.0012

0.0020

_

0.

0.0O75

0. 0107

0. 0146

0. 60

0. 0001

0. 0002

0. 0005

0. 0009

0.0016

0.0027

0.0041

0,0061

0.0087

0.

011_

O. 70

O. O00i

O. 0002

O. 0004

O. 0007

0.0012

0.0020

0.0031

O. O0_i

0.

0.

0089

O. 80

O. 0001

O. O00i

O. 0002

O, 0005

0.0008

0.0013

0.0020

0.0030

0.0043

0.0r)58

0.90

0.0000

0.0001

0.0001

0.0002

0.0004

0.0006

0.0010

0.0o15

0.0021

0.0028

1.00

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.

t}.0000

0.7000

0.7200

0.7400

O. 7600

O. 7800

0.6000

0.6200

0.6400

0.

0.0289

0.0373

0.0470

0.058(I

0.0703

0.0838

0.09_5

0.1143

O. 1311

O. 14_8

0. i0

0.0285

0.0367

0.0463

0.0571

0.0693

0.0826

0.0971

0.1126

O. 1292

O. 14(;7

0.

0.0272

0. 0351

0.

0.

0.

0.0790

0.0929

0,1078

O. 1236

O. 1403

0.30

0.0252

0.0326

0.0410

0.0507

0.0614

0.0732

0.0861

0.0999

O. 1146

0.

1301

0. 40

0.0226

0. 0291

0. 0367

0. 0453

0. 0549

0.0655

0.0770

0.0893

0. 1025

0.

1163

0.50

0.0193

0. O250

0.0315

0.0388

0.0471

0.0561

0.0660

0.0766

0.

0878

0. 0997

0.60

0.0157

0.0202

0.0255

0.0315

0.0382

0.0455

0.0535

o.

0.0712

0.0_09

0.70

0.0|18

0.0132

0.0191

0.0236

0.0286

0.0342

0.0401

0.0466

0.0534

0. 0606

O. 80

O. 0077

0,0100

O. 0126

O, 0155

0.0188

0.0225

0.0264

0.0306

0.

0.

0.90

0.0038

0.0049

0.0061

0.0076

0.0092

0. o109

0.0128

0.0149

0.0171

0.0194

0.0000

0.0000

0.0000

O. 0000

0.

O.

9000

O. 9200

O. 9400

O. 9600

O. 9800

0.6600

20

1.00

0443

0033

0051

0065

0000

0.6800

0547

0653

0621

(1351

0399

0o00

0.0000

0.0000

0.0000

0.0000

0.0000

0.8000

0.8200

0.8400

0.8600

0.8800

0.

0.1673

0.1866

0.

0.

0.2476

0.

2687

0.

2901

0.3117

o. 3334

0.

3551

0. 10

0. 1649

0. 1839

0. 2304

0. 2235

0. 2440

0.

2649

0.

2860

0.

0.

3286

0.

3500

0.20

0.1578

0.1759

0.1947

0.2139

0. 2335

0. 2534

0. 2736

0, 2940

0.

:1145

0.

3350

0.30

0.1463

0.|631

0.1804

0.1983

0.21(;4

O. 2349

O. 2537

O. 2725

O, 2915

0.

3106

0.40

0.1308

0.1458

0.1614

0.1773

0. I n_c

O. 2'_01

O. 2269

0,

2431_

O. 200_

O. 2779

0.50

0.1t21

0.1250

0.1383

0.1520

0.1630

0. 1801

0.

1945

0.

2090

0. 2236

0.23_2

0, 60

0.0909

0. i014

0. 1122

0. 1233

0. 1346

0.1401

0.

1578

t). 1696

0.

0.

0.70

0.0682

0.0760

0.0841

0,0924

0.1010

0.

1096

0.

1184

O.

1272

O. 1361

+}. 1451

0.80

0.0448

0.0500

0.0553

0.0608

0.0664

0.

0721

0. (1778

0.

0836

0. O895

0.

I}!)54

0.90

0.0218

0.0243

0.0269

0.0295

0.0323

O. 0350

0.

0.

0407

0.

0435

0.

041;4

1.00

0.0000

0.0000

0.0000

0,0000

0.0000

O. 0000

O. 0000

O. 0000

O. 000o

0.

(I0o0

1.0200

1.0400

1.0600

1.0800

1.1000

1.

1.14o0

1.16o0

I. ISO0

2064

2268

0378

1200

3072

1_14

1933

1.0000 O.

0.3768

0.3984

0.4199

0,4412

0.4(;22

0.4830

0.5035

0.5236

I).5433

0.5627

0.|0

0.3714

0.3927

0.4139

0.4349

0.4557

0.4762

0.4964

0.5162

0.5357

0.5548

0.20

0.3555

0.3759

0.3962

0,4163

0.4362

0.4559

0.4753

0.

0.

5132

(}. 5316

0.30

0.3296

0.3486

0.3674

0,3862

0.4047

0.4231

0.4412

0.4590

0.4766

0.4038

0.40

0.2949

0.3119

0.3289

0.3457

0.3624

0.3789

0.3952

0.4114

0.4272

0.4429

0.50

0.2529

0.2675

0.2821

0.2966

0.3110

(I. 3252

0.3394

0.3533

0.3671

0.3807

0.60

0.2053

0.2172

0.2290

0.2408

0.2526

0.2642

0.2758

0.2872

0.2986

0.3098

0.70

0.1540

0.1630

0.1719

0.1808

0.1897

0.1905

0.

0.2159

0,2245

0.2330

0.80

0,10t3

0.1072

0.1t31

0.1190

0.1248

0.1306

0.1364

0.1421

0.1478

0.1535

0.9O

0.0493

0.0521

0.0550

0.0579

0.0607

0.0636

0.

0.

0.

0.

0.0000

O. 0000

1.00

0.0000

0.0000

0.0000

0,0000

0.0000

2072

0654

4944

0(;92

O. OOOO

I

i I

0720

O. 0000

0747

O. 0000

GRIGiNAL

PAGE

OF POOR

QUAUTY

IS Section

D

July 1, 1972 Page 276

TABLE

6.0-2.

(CONTINUED)

t. 2000

1. 2200

1. 2400

1. 2600

1. 2800

1. 300

1. 3300

1. 3400

1. 3600

t. 3800

O.

O. 5816

O. 6001

O. 6181

O. 6356

O. 6527

O. 6692

O. 6853

O. 7008

O. 7158

O. 7303

O. 10

O. 6735

O. 5918

O. 6096

O. 6270

O. 6439

O. 6603

O. 6762

O. 69t6

O. 7066

O. 7210

O. lO

0.5497

0.5673

0.5845

0.6014

0.6178

0.6337

0.6492

0,6643

0.6789

0.6931

O. 30 O. 40

O. 5108 0. 4583

O. 5274 0. 4734

O. 5436 0. 4882

O. 5595 0. 5028

O. 5751 0. 5171

O. 5902 0. 531 !

O. 6050 0. 5448

O. 6194 0. 5582

O. 6335 0. 5713

O. 6471 0. 5841

O. 50

O. 3941

O. 4073

O. 4204

O. 4332

O. 4458

O. 4582

O. 4703

O. 4823

O. 4941

O. 5056

O. 00

O. 3208

O. 3318

O. 3428

O. 3532

O. 3638

O. 3741

O. 3844

O. 3945

O. 4044

O. 4142

0. 70

O. 2414

0. 2497

0. 2580

0. 2662

0. 2743

0. 2823

0. 2902

0. 2980

0. 3058

0. 3 t35

O. 80

O. 1591

O. 1647

O. 1702

Oo 1756

O. 1811

O. 1864

O. 1915

O. 1971

O. 2023

O. 2075

0.0802

0.0829

0.0856

0.0883

0.0909

0.0935

0.0962

0.0988

0. 1013

0.0000

0. 0000

0. 0000

O. 0000

0.00OO

0. 0000

0.0000

0.0000

0.0000

0.0000

0.90 i. O0

0.0775

1.4000

I. 4200

i. 4400

1. 4600

I. 4800

I. 500

i.5200

I. 5400

i. 5600

I. 5800

0. 0. t0

0. 7443 0. 7349

0. 7578 0. 7484

0. 7708 0.76t3

0. 7833 0. 7738

0. 7952 0. 7857

0. 8067 0. 7972

0. 8177 0. 8082

0. 8282 0. 8188

0. 8382 0. 8289

0. 84700 0. 83868

0. 20

0. 7068

0. 7201

0. 7329

0, 7453

0. 7572

0. 7687

0. 7798

0. 7905

0. 8007

0. 8106

O. 30

0. 6604

0. 6733

0. 6858

0. 6980

0. 7097

0. 7212

0. 7322

0. 7429

0. 7533

0. 7633

0. 40

0. 5966

0. 6088

0. 6207

0. 6323

0. 6437

0. 6547

0. 6655

0. 6760

0. 6863

0. 6962

0, 50

0.5169

0. 5280

0. 5389

0. 5496

0. 560 t

0. 5704

0. 5805

0. 5905

0. 6002

0. 6097

0. 60

0. 4239

0. 4335

0. 4429

0. 4522

0. 46 i 3

0. 4704

0. 4793

0. 4881

0. 4967

0. 5053

0. 70

O. 3210

0. 3286

0. 3360

0. 3434

0. 3507

0. 3579

0. 3651

0. 3722

0. 3793

0. 3862

0.80

0. 2127

0. 2178

C. 2229

0. 2280

0. 2330

0. 2380

0. 2430

0. 2480

0. 2529

0. 2578

0.90

0.1039

0.1065

0.1090

0,1116

0.1141

0.1166

0.1191

0.12/6

0.1240

0.1265

I. O0

O. 0000

O. O00O

O. 0000

O. 0000

O. 0000

O. 0000

O. 0000

O. 0000

O. 0000

O. 0000

1.6000

1.6200

t. 6400

1,6600

1.6800

1.700

1.7200

1.7400

1.7600

1.7600

0,

0. 8570

0. 8657

0. 8740

0. 8818

0. 8893

0. 8964

0. 9031

0. 9095

0. 9155

0, 9212

0.10

0. 8478

0. 5566

0. 8650

0. 8730

0. 8806

0. 8879

0. 8947

0. 9013

0. 9075

0. 9133

0. 20

0. 8200

0. 829!

0. 8378

0. 8461

0. 8541

0. 86|7

0. 8690

0. 8760

0. 8827

0. 8890

0. 30 0. 40

0. 7730 0. 7060

0. 7823 0. 7154

0. 7914 0. 7248

0. 8001 O. 7336

0. 8085 0. 7423

0. 8166 0. 7508

0. 8244 0. 7590

0. 8320 0. 7670

0. 8393 0. 7748

0. 8463 .0. 7824

0.50

0.6191

0.6282

0.6372

0.6460

0.6547

0.6631

0.6714

0.6796

0.6876

0.6954

0.60

0. 5137

0. 5220

0. 5302

0. 5383

O. 5463

0. 5542

0. 5620

0. 5697

0. 5772

0. 5847

0. 70

0. 593t

O. 4000

O. 4068

0. 4135

0. 4202

0. 4268

0. 4334

0. 4399

0. 4464

0. 4528

0.80

0. 2626

0. 2674

0. 2723

0. 2770

0. 2818

0. 2865

0. 2913

0. 2960

0. 3006

0. 3053

0o 90

0. 1290

O. 1314

0. 1339

0. 1363

0. 1387

0. 1412

0. 1436

0. 1460

0. 1484

0. 1506

1. O0

O. 0000

O. 0000

O. 0000

O. 0000

O. 0000

O. 0000

O. 0000

O. 0000

O. 0000

O. 0000

1. 8000

1. 8200

1. 8400

t. 8600

1. 8800

1. 9000

1. 9200

1. 9400

1. 9600

t. 9800

0.

0. 9265

0. 9316

0. 9363

0. 9408

0. 9450

0. 9489

0. 9527

0. 9561

0. 9594

0. 9624

O. 10

O. 9t89

O. 9241

O. 9291

O. 9337

O. 9361

O. 9423

O. 9462

O. 9499

O. 9534

O. 9566

O. 20

0, 6951

0. 9006

O. 9063

0.9116

O. 916_

0. 9213

0, 9258

O. 9301

0. 9341

0. 9380

0.50

0. 8530

0. 8595

0. 8657

0. 8717

0. 8775

0. 8830

O. 8884

0. 8935

0. 8984

0. 9031

0.40

0,7696

0,7969

0.8039

0.8106

0.8t72

0,8236

0,8297

0,8357

0.6416

0.8472

0.50 0.60

0.7030 0.6921

0.7105 0.5994

0.7179 0.6066

0.7251 0.6t36

0.7322 0.6206

0. 739| 0.6275

0.7458 0.6343

0.7525 0.64t0

0.7690 0.6476

0.7653 0.6542

0.70

0.4592

0.4655

0.4717

O. 4779

0.484t

0.4903

0.4963

0.5023

0.5082

0.5141

O. 80

O. 3099

O. 3145

O. 3191

O. 3237

O. 3282

O. 3328

O. 3373

O. 34t8

O. 3463

O, 3507

0. 90

0.1532

0. /556

0. 1580

0. 1604

0. 1628

0. 1651

0. 1675

0. 1699

0.1722

0. t746

i. O0

O. 0000

O. 0000

O. 0000

O. 0000

O. OOOO

O. 0000

O. 0000

O. 0000

O. 0000

O. 0000

ORIGI_!AL

_. _,:.... ,_....

OF POOR

QU/4L_T't'

Section July

/

l,

Page TABLE

f

6.0-2,

(CONTINUED)

2. 0000

2. 0200

2. 0400

_0600

2. 0800

2, 1000

2. 1200

0.

0.9653

0.9679

0.9704

0.9727

0,9748

0.9766

0,9787

0.10

0.9597

0.9625

0.9652

0.9677

O, 970J

O. 9723

0.9743

O. 20

0.9416

0,9451

O. 9484

0.9515

0.9544

O. 9572

O. 9598

0.30

0,9077

0,9120

0.0t62

0°9202

0°9240

0.9276

0.40

0,8527

0.8580

0.8632

0.8682

0.8730

0.50

0.7716

0.7776

0.7836

0.7894

0.80

0,6606

0.6669

0.6732

0.70

0.5200

0.5258

0.5316

O. 80

O. 3552

O. 3596

O, 3640

0.90

0.

0.1793

1.00

0.0000

0.0000

O. O. t0

2. 1600

2. 1800

0.9820

0,9835

0,9762

O. 9780

0.9797

O. 9623

O. 9646

0.9669

0.9311

0.9345

0.9377

0.9408

0.8777

0.8822

0.6866

0.8909

0.8950

0,7951

0.8007

0.6062

0.81t5

0.6i67

0.82t8

0.6794

0.6654

0.6914

0.6973

0.7032

0.7089

0.7146

0.5373

0.5430

0.5486

0,5542

0.5597

0.5652

0.5707

O, 3684

O. 3728

O. 3771

O. 3815

O. 3858

O. 3901

O, 3944

0.1616

0.1840

0,1663

0.1987

O. 19tO

0.1933

0.1956

0.1980

0.0000

0.0000

0.0000

0.0000

0.0000

0,0000

0.0000

0.0000

2.2000

2.2200

2.2400

2.2600

2,

2.3000

2.3200

2.3400

2.3600

2.3800

O. 9649

O. 966i

0.

_'9884

O. 9894

O. 9903

O, 9912

O. 9920

O. 9927

O. 9933

9873

2800

O. 98i2

O. 9827

O. 9840

O. 9853

O, 9864

O, 9875

O. 9885

O. 9894

O. 9903

O. 9911

0.20

0.9690

0.9709

0.9728

0.9745

0.9762

0.9777

0,9792

0.9805

0.9816

0.9830

O. 30

0.9437

0.9466

O. 9492

0.95t8

O. 9543

O, 9566

O. 9588

0.9610

0.9630

O. 9650

O. 40

0.6990

0,9029

0.9066

0,9102

0,9137

O. 9171

0.9204

0,9235

0.9266

0.9295

0.50

0.8268

O, 8317

0.8365

O. 84tl

O. 8457

O. 8501

O. 8544

O. 6587

0,6628

O. 8669

0.60

0.7202

0,7256

0.7311

0.7364

0,7416

0.7468

0.7519

0.7569

0.7619

0.7667

O. 70

O. 5760

O. 5814

O. 5867

O. 5920

O. 5972

O. 6023

O. 6075

O. 6125

O. 6176

O. 6226

0.80

0.3986

0.4029

0.407t

0.4113

0.4t55

0.4t97

0.4239

0.4260

0.4321

0.4362

0.90

0.2003

0.2026

0.2049

0.2072

0.2095

0.2118

0.2141

0.2164

0.2187

0.2210

1.00

0.0000

0.0000

0.0000

0.0000

0,0000

0.0000

0.0000

0.0000

0.0000

0.0000

2.4000

2.4200

2.4400

2.4600

2.4600

2.5000

2.5200

2.5400

2.5600

2.5800

0. 9939

O. 9945

0. 9950

O, 9955

0, 9959

_), 9963

O. 9966

O. 9970

O. 9972

O. 9975

0.10

0.9918

0.9925

0.9931

0,9937

0,9942

0,9947

0,9952

0.9956

0.9960

0.9963

O. 20

0.9842

0,9653

0.9863

0.9672

0.9681

0.9889

0.9897

0.9904

0.9911

0.9917

0.30

0.9668

0.9686

0.9703

0,9718

0,9734

0.9748

0.9762

0.9775

0.9787

0.9799

0.40

0.9324

0.9351

0.9377

0.9403

0°9428

0.9451

0.9474

0.9496

0.9517

0.9538

0.50

0,8709

0.8746

0.8784

0.8820

0,8856

0.8891

0.8925

0.8958

0.9990

0.9021

0.60

0.7715

0.7762

0,7809

0.7854

O. 7899

0.7943

0.7987

0.

0.8072

0.8113

0,70

0,6275

0,6324

0.6373

O. 6421

0°6468

0.6515

0.6562

0.6608

0.6654

O. 6700

0.

0.

4403

0.

0.

0.

0.

4565

0.

4605

0,

0.

4685

0.

4724

0. 4764

0,90

0.

2233

O. 2256

O, 2279

O. 2301

O. 2324

0.

2347

O. 2370

O. 2392

0.

2415

0.

i,

0.0000

0.0000

0,0000

0.0000

0.0000

0,0000

0.0000

0.0000

0.0000

0.0000

2.

_

_7600

2, 7800

80

O0

-

t769

2. 1400 '0,9604

4444

4485

2. 6200

_

2. 6800

2.7000

2,

0.9980

0.9982

0.9984

0.9985

0.9967

0.9986

0.9989

0.999t

0.9992

0.9967

0,9970

0.9972

0.9975

0.9977

0.9979

0.9981

0.9983

0.9984

0.9986

0.20

0.9923

0.9928

0.9934

0.9938

0.9943

0°9947

0.9951

0.9955

0,9958

0.9961

0.30

0.9810

0.9621

0.9831

0.9841

0.9850

0.9859

0.9867

0.9875

0.9882

0.9689

O, 40

O. 9558

O. 9577

O. 9595

O. 9613

O. 9629

O. 9646

O. 9661

O. 9676

O. 9691

O. 9705

O, 50

O. 9052

O. 9082

O. 9111

O. 9139

O. 9166

O, 9193

O, 9219

O, 9244

O. 9269

O, 9293

0.60

O. 9|54

0.8193

0.6233

0.8271

0.8309

0.8346

O. 8383

O. 8419

0.8454

O. 8489

0.

O, 6745

O, 6789

0.6633

O, 6677

O. 6920

O. 6963

O. 7006

0.7048

O, 7089

O. 7131

0.80

0.4803

0.4842

0,468t

0.4919

0.4956

0.4996

0.5034

0.5072

0.5t10

0.5148

O. 90

O. 2460

O. 2483

O. 2505

O. 2528

O. 2550

O. 2572

O. 2595

O. 2617

0.2639

O. 2662

1.00

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

t0

70

7200

7400

2437

0.9978

0.

6600

4645

2. 6000 O.

6400

4525

8029

D 1972

277

Section

TABLE

2. 80O0

2. 8200

2. 640O

0.

0. 9993

0. 9993

0. 9994

0. 10

0. 9987

0. 9988

0. 20

0. 9964

0. 9967

0. 30

0. 9895

0. 40

6.0-2.

D

July

1, 1972

Page

278

(CONTINUED)

2. 8600

2. 8800

2. 9000

2. 9200

2. 9400

2. 9600

2. 9800

0. 9995

0. 9995

0. 9996

0. 9996

9. 9997

0. 9997

0. 9997

0. 9990

0. 9991

0. 9991

0. 9992

0. 9993

0. 9994

0. 9994

0. 9995

0. 9969

0. 9972

0. 9974

0. 9976

0. 9978

0. 9979

0. 998t

0. 9983

0. 9902

0. 994)8

0. 9913

0. 9918

0. 9923

0. 9928

0. 9932

0. 9937

0. 9941

0. 9718

0. 9731

0. 9743

0. 9755

0. 9766

0. 9777

0. 9787

0. 9797

0. 9807

0. 9816

0. 50

0. 93t0

0. 9336

0. 9360

0. 9382

0. 9402

0. 9423

0. 9442

0. 946 J

0. 9480

0. 9497

0. 60

0. 8523

0. 8557

0. 8589

0. 8622

0. 8653

0. 8684

0. 8715

0. 8745

0. 8774

0. 8803

0.70 0. 80

0.7171 0. 5t85

0.7212 O. 5223

0.7251 0. 5260

0.7291 0. 5297

0.7330 0. 5333

0.7369 0. 5370

0.7407 0. 5406

0.7445 0. 5442

0.7482 0. 5478

0.7519 0. 5514

0. 90

0. 2684

0. 2706

0. 2728

0. 2751

0. 2773

0. 2795

0. 2817

0. 2839

0. 2861

0. 2883

I.O0

O. 0000

O. 0000

O. 0000

O. 0000

O. 0000

O.0000

O. 0000

O. 0000

O.0000

O,0000

3. 0000

3.1000

3. 2000

3. 3000

3. 4000

3. 5000

3.6000

3. 7000

3.8000

3. 9000

O.

0.9998

0.9999

0.9999

0.9999

1.0000

1. ooo0

1.0000

i. O000

1.0000

1.0000

0. t0 O. 20

0.9998 O. 9984

0.9997 O. 9990

0.9998 O. 9993

0.9999 O. 9996

0.9999 O. 9997

1.0000 O. 9998

1.0000 O. 9999

l. OOOO O. 9999

|.0000 1. 0000

1.0000 1. 0000

0. 30

0. 9944

0.9960

0. 9971

0. 9980

0. 9986

0. 9990

0. 9993

0. 9995

0. 9997

0. 9998

0.40

0.9825

0.9863

0.9894

0.9918

0.9937

0.9952

0.9964

0.9973

0.9980

0.9985

0. 50

0. 9515

0. 9594

0. 9662

0. 9720

0. 9769

0. 9810

0. 9844

0. 9873

0. 9897

0. 9917

0.60

0.8831

0.8965

0.9086

0.9194

0.9292

0.9380

0.9458

0.9528

0.9590

0. 9645

0.70

0.7556

0.7733

0.790t

0.8058

0.8207

0. 8347

0.8478

0.8001

0.8716

0.8823

0. 60

0.5550

0. 5726

0. 5897

0.6064

0.6228

0.6384

0.6538

0.0687

0.6832

0.8972

0.90

0. 2905

0. 30t5

0. 3123

0. 323t

0. 3338

0. 3445

0. 3550

0. 3655

0. 3759

0. 3862

1.00

0.0000

0.0000

0.0000

0.0000

0.000O

0.0000

0.0000

0.0000

0.0000

0.0000

4.0000

4.1000

4. 2000

4.3000

4.4000

4.5000

4.6000

4.7000

4.8000

4.9000

O.

1.0000

1.0000

1.0000

l. OOOO

1.0o00

1.0000

1.0000

1.0000

1.o000

1.0000

0,10 0.20

t. 0000 1.0000

1.0000 1.0000

1.0000 1.0000

i. O000 1.0000

1.0000 1.0000

1.0000 1.0000

1.0000 1.0000

1.0000 1.0000

1.0000 1.00o0

1.000o 1.0000

0.30

0.9999

0.9999

0.9999

1.0000

1.0000

1.0000

1.0000

1.0000

1.0oo0

1.0ooo

0.40

0.9989

0.9992

0.9994

0.9996

0.9997

0.9998

0.9999

0.9999

0.9999

i.0000

0.50

0.9933

0.9947

0.9958

0.9967

0.9973

0.9979

0.9984

0.9987

0.9990

0.9992

0.60

0.9693

0.9736

0.9773

0.9806

0.9834

0.9859

0. 9880

0.9899

0.9914

0.9928

0.70

0.8923

0.9016

0.9t03

0.9t83

0.9257

0.9325

0.9388

0.9446

0.9500

0.9549

0. 60 0.90

0. 7108 O. 3964

0. 7240 0.4066

0. 7367 O. 4166

0. 7490 O. 4266

0. 7609 O. 4366

0. 7724 O. 4463

0. 7834 O. 4560

0. 7941 0.4656

0. 8044 O. 4751

0. 8143 O. 4846

1. O0

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

5.0000

5.1000

5.2000

5.3000

5.4000

5.5000

5.6000

5.7000

5.8000

5.9000

O.

t, 0000

t. 0000

t. 0000

t. 0000

J. 0000

i. 0000

|. 0000

1. 0000

t.0000

1. 0000

0.10

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

0.20

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

O. 30

I. 0000

I. 0000

I.0000

I.0000

I. 0000

I. 0000

I.0000

I.0000

I. 0000

i.0000

0.40

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

i.O000

O. 50

O. 9994

O. 9996

O. 9997

0,9997

O, 9998

O. 9998

O. 9999

O. 9999

1, 0000

1. 0000

0.60

0. 9939

0.9949

0. 9958

0.9965

0,9971

0. 9976

0.9980

0. 9984

0. 9986

0. 9989

0,70 0.80

0. 9594 0.8238

0.9635 0.8329

0.9672 0.8416

0.9706 0.8501

0.9737 0.858i

0. 9765 0.8658

0.9790 0.8732

0. 9813 0.8803

0.9834 0.8870

0. 9853 0.8934

0,90

0.4939

0.5032

0.5t23

0.52t4

0.5303

0.5392

0.5480

0.5566

0.5662

0.5737

1.00

0.0000

_ 0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

OF.. POOR

OUAi..IT_

Section

D

Jl_

1972

I,

279 TABLE

6.0-2.

(CONTINUED)

/-

F

o. o. 1o o.lo o.lo o.41o o.6o 0.so o.lo O. 90 O. 0O L99

0. 0.io 0.m 0.N o.4o 0.99 0.99 0. ?o 0.11 41.1o s, ol

5,0000

S. I000

4. 3000

It,.3000

9. 4000

6. 5000

4. 0000

G. 7000

i, OOO0 |. 0000 1.0000 1.0099 1.0_@ i. 00@0 O. 11901 O. 98?0 0. 8994 O. 11010 0,O12O0

1. O00O 1.0000 |. 0000 1.0000 t. 0000 i. 0000 O. 9993 O. 9864 0. 9054 O. 5_103 o. oooo

t. 0000 t. OOOO 1. O00O 1.0000 1. 00040 I. 0000 0, t994 O. 9998 0. 9t l0 O. 6986 O. OO9O

1. 0900 1. OOOO ..... 1.000o 1. OOC)O 1. O000 I. 0000 0, 9996 O. 9910 0, 9183 O. 41065 O. 0000

i. 00._., 1. O00O t. O000 1. 0000 1. O000 L0000 0, 091M O. 9921 O. 9213 0.01M 0. O000

1.0_00 |. 0000 I. 0000 |. 0000 |. 0000 i. 0000 0. 9997 O. 9930 O. 9_Jl 0. 0224 @.0000

1. 0000 1. oooo L O00O 1. 0000 1. 0000 I. 0000 O, 9998 O. 9939 O. 9:107 O. I_301 O. 0000

t. 0000 _. 0000 i. 0900 1. O00O 1. O000 i0000 0. 9998 O. 9946 O. 9350 0.4378 O. 0000

1. uOOO

t. oot_u

1. _-ooo 1. _,000 1. oooo I. 0000 I. oooo 0. 9999 O. 9953 O. 0390 O. 9453 O. O000

i. oooo 1. 0009 i. o_oo 1. 0000 i. oooo O. 9999 O. 995_ O.942_ O.662_ O. 0000

7.0000

7. iO00

7.JOOG

7.3000

7.4000

7. 6000

7. 8000

7. 7000

7. 8000

7. 9000

i. O0_ 1.0000 LOO00 I.O00e i. 00@@ i. 0000 1. oooo O. MN;O 0.9_ O. SSM o. ooo0

t. 0000 I. 0000 1. 0000 1._00 |. 0000 1.0400 1.9940 0.14M10 0.141/ 0.lllJ 0.0010

1. 0000 i. 0<300 LO000 1.0000 I.000_9 1.0000 J, 0000 O. 9966 0. MM 0.7021 0. 0000

i. O000 i. 0000 LEO00 I. 0000 I. 0000 1.0000 1.0000 O. 99_/ 0. MTi 0.7087 0.0000

1.0000 J. 0000 I. 0000 i. 0000 I.0000 i. 0000 1. 0000 0. 9989 0. M93 0. 718_ 0. 000_

i. 0000 f, 0000 1. 0000 1.0000 1.0000 1.0000 i. 0000 O,9990 O.W/IS 0.7017 O. 0000

I. 0000 i. 0000 1. 0000 I. 0000 t. 0000 1.0400 0, M199 0.1144 0.1411 0. IMl01 0.OOOO

1.00o0 LO000 LO000 1.0000 I. 0000 1.0000 O. 19111 O. 9999 0.9600 0. 44174 0.OOOO

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i omo 1.0000 I._ 1. M99 i. 0010 i. 0000 i.0_ O. _ O.IN4 0,7_ 0.11040

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i. ooeo i._o@ 1.0@04 I, 0000 i. 0000 t. _'..,000 O. 999"/ O. 9832 O. 7430 O. 0000

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s. SOO0

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_._ l.oem i. 0099 1.0990 I._ l.mm LION L_ S.99N I,,_ 41.IIII 0. 0990

9.(1000

9.7000

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0.9000

i. 0000 i. ooeo i. 0000 1. oooe I. oooe LOOm _._ l.m_ 0.NN 0. 9/19 o.oo99

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i oooo i. oooo t, 0000 t. oooo i. oooo /.moo 1.oo<)o i.0000 0.99_ o. lJl_ o.oo_

Section

D

July

1972

1,

Page

TABLE

10. 0000

I0. I000

10.2000

10.3000

6.0-2.

10.4000

280

(CONCLUDED)

10.5000

I0.8000

10.7000

I I0.8000

[ 10.90o0

I 0.

1.0000

i. 0000

i.0000

I.0000

i. 0000

I. 0000

1.0000

1. 0000

1.0000

I.0000

0. t0

1.0000

I. 0000

i. 0000

I.0000

I. 0000

I. 0000

I. 0000

I. 0000

I.0060

I.0000

O. 20 0.30

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

l. O000

1.0000

1.0000

1.0000 1.0000

1.0000

1.0000

1.0000

t.0000

1.0000

1.0000

1.0000

1. o000

1. o0_0

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.000o

I. 0000

1.0000

I.0000

1.0000

1.0000

1.0000

1.0000

I.0000

i.0000

0.00

1.0000 1.0000

i. 0000

I. 0000

1.0000

I. 0000

I. 0000

I. 0oo0

I.O00e

I.0000

1.0000

O. 70

1.00oo

1.00o0

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1. o00o

1. i_000

o. 80

0.9948

O. 90

0.8341

o. 9952 0. 8385

o.9956 0.8427

0. 9960 0.8469

o. 9964 0.8510

0. 9967 o, 8549

o. 9970 O. 8588

0. 9972 O. 8627

o. 9975 O. 8664

o. 9977 o. 87oj

1. O0

0.0000

0.0000

0.0000

0.0000

0.000_

0.0000

0.0000

0.0000

0.0000

O. 40 O. 50

_

I

U. O000

ORt_._I,.._L F.",Gi-'. _L; /1_ POOR QUALITY

Section

D

July

1972

1,

Page

TABLE

6.0-3.

CY_NDER-PARAMETER

281

_b2

0.4000

0.4200

0.4400

0.4800

0.4800

0.5000

0.5200

0.5400

0.5600

O. 5800

O.

0.0001

0.0002

0.0005

0.0009

0.0015

0.0025

0.0038

0.0056

0.0080

0.0109

0.10

0.0001

0.0002

0.0005

0.0009

0.0015

0.0024

0.0038

0.0056

0.0079

0.0108

0.20

0.0001

0.0002

0.0004

0.0009

0.0015

0.0024

0.0037

0.0055

0.0077

0.0106

0.30

0.0001

0.0002

0.0004

0.0008

0,0014

0.0023

0,0036

0.0053

0.0075

0.0102

0.40

0.0001

0.0002

0.0004

0.0008

0.0013

0.0022

0.0034

0.0050

0.0071

0.0097

0.50

0.0001

0.0002

0.0004

0.0007

0.0013

0.0020

0.0032

0.0047

0.0066

0.0090

0.60

0.0001

0.0002

0.0003

0.0007

0.0011

0.0019

0.0029

0.0043

0.0061

0.0063

0.70

0.0001

0.0002

0.0003

0.0000

0.0010

0.0017

0.0026

0.0039

0.0055

0.0075

0.60

0.0001

0.0001

0.0003

0.0005

0.0009

0.0015

0.0023

0.0034

0.0048

0.0066

0.90

0.0000

0.0001

0.0002

0.0004

0.0008

0.0013

0.0020

0.0029

0.0041

0.0056

1.00

0.0000

0.0001

0.0002

0.0004

0.0007

0.0011

0.0016

0.0024

0.0034

0.0047

O. 6000

0.6200

O. 6400

O. 6600

O. 6800

O. 7000

O. 7200

O. 7400

O. 7600

0.7800

0.0144

0.0186

0.0235

0.0200

0.0351

0.0419

0.0492

0.0571

0.0655

0.0744

0.0143

0.0185

0.0233

0,0288

0.0349

0.0416

0.0489

0.0567

0.0651

O, 0739

0.20

0.0140

0.0181

0.0228

0.0282

0.0341

0.0407

0.0478

0.0555

0.0637

0.0723

0.30

0.0135

0.0174

0.0220

0.¢)271

0.0329

0.0392

0.0461

0,0535

0.0814

0.0997

0.40

0.0128

0.0166

0.0209

0.0258

0.0312

0.0372

0.0438

0.0508

0.0582

0.0661

O. 50

O. 0120

O. 0J55

0.1)195

O. 0211

0.

0.0348

O. 0409

O. 0474

O. 0544

O. 0617

0.60

0.0110

0.0142

0.0179

0.0221

0.0267

0.0319

0.0375

0.0435

0.0499

0.0566

O. 70

O. 0099

O. 0128

Oo 0181

O. 0199

O. 0241

O. 0287

O. 0337

O, 0392

Oo 0449

O. 0510

0.

0.

0.

0.

O. 0175

0.

0.

0.

0.

0.0396

0.0449

0, 0.

t0

80

0087

0112

0142

0292

0212

0253

0297

0345

O. 90

O. 0075

O, 0097

0.0122

0.0150

Oo 0182

O. 0217

0.0255

O, 0296

O. 0340

O. 0386

t.00

0.0062

0.0080

0.0101

0.0125

0.0152

O. O1RI

0.0213

0,0247

0.0289

0.0321

0.8000

0.8200

0.8400

0.8600

0,8800

0.9000

0.9200

0,9400

0.9600

0.9800

0.

0.0837

0.0933

0.1032

0.1134

0.1238

0.1344

0.1451

0.1559

0.1667

0.1776

0.

I0

0.0831

0.0926

0.1025

0.1126

0.1229

0.1334

0.]440

0.1547

0.1655

0.1763

0,

20

0.0813

0.0906

0.1002

0.1101

0.1203

(1.1305

0.1409

0,1514

0.1619

0.1725

0.30

0°0783

0.0873

0.0966

0.1062

0.1159

0.1258

0,1359

0,1460

0,1561

0.1663

0,40

0.0749

0.0829

0.0917

0.1008

0.1100

0.1194

0.1290

0.1386

0.1482

0,1579

0.50

0.0694

0.0774

0._56

0.0941

O. 1027

0.1115

0.1204

0,1294

0.1384

O. 1475

0.60

0.0637

0.0710

0.0786

0.0863

0.0943

0.1023

0.1105

0.1187

0.1270

0.1353

0.70

0.0573

0.0639

0.0707

0.0777

0.0848

0.0921

0.0995

0.1069

0.1t43

0.1218

0.80

0.0505

0.0563

0.0623

0.0684

0.0747

0.0811

0.0876

_

0.1007

0.1073

0.90

0.0434

0.0483

0.0535

0.0588

0.0642

0.0697

0.0752

0,0908

0.0865

0.0921

1.

O. 0301

O. 0403

O. 0446

O. 0490

O. 0535

O. 0580

0.0627

O. 0673

0.0721

O. 0768

1.

1.

1.

1.

1.

1.

1.

O0

1. 0000

1.

O,

0.1884

0.1992

0200

O. 2100

0400

O. 2206

0600

0900

1.

1000

1200

0941

1400

JOOO

O. 2311

O. 2415

O. 2517

O, 2618

O. 2717

1900

O. 20t3

0°10

0.1871

0.1978

0.2085

9.2190

0.2295

0.2398

0.2500

0,2600

O. 2698

0.20

O, 1831

O. 1936

O. 2040

O. 2144

O. 2246

O. 2347

O. 2447

0.2545

0.2641

0.2735

0.30

0.1765

0.

0.

0.2067

0.2166

O. 2264

0.2360

0.2455

0.2549

0.2640

i866

i967

0.2794

O. 40

0.1675

0.1772

0.1868

0.1963

O. 2057

O. 2150

O. 2242

O. 2332

0.2422

O. 2509

O. 50

O. 150

O. 1655

O. 1745

O. 1834

O. 1922

O, 2009

O. 2095

O° 2180

O. 2204

O. 2346

O. 6C

0.143,

O. t519

0,1602

0.1683

0.1765

0.1845

0.1924

O, 2003

0.2080

O. 2155

O. 70

O,_?)J

O. 1368

O. 1442

O, 1516

0.1589

O. t662

O. 1733

O. t804

O. t874

O. 1943

O. flO

0.2139

O. 1205

O. 1270

O. 1335

O.

O.

0.

O. 1590

0.1652

O. 1713

0.90

O. 0978

0.1035

0,1091

0.1147

0.1203

0.1258

O. 1313

O. 1367

O. 1420

0.1472

1. O0

O. 0815

O. 0862

O. 0909

O. 0956

O. 1002

O. t048

O. 1094

O, 1139

O. tt83

O. 1277

1400

1464

1528

Section July Page

TABLE

6.0-3.

(CONTINUED)

I. 2000

i. 2200

I. 2400

I. 2600

I. 2800

I.3000

i. 3200

i. 3400

1.3600

i.3800

O. 0.10

0.2905 O. 2888

0.3009 0.2980

0.3090 0.3069

0.3179 0.3157

0.3263 0.3241

0.3346 0.3324

0.3426 0.3404

0.3504 0.3481

0.3579 0.3556

0.3652 0.3628

0.20

O. 2828

O. 29i8

0.3006

0.3092

0.3176

0.3257

0.3336

0.3412

0.3487

0.3558

0. 30

0.2729

0. 2817

0. 2903

0. 2986

0. 3068

0. 3147

0, 3224

0. 3299

0. 3372

0. 3443

0.40

0. 2595

0. 2679

0.2761

0. 2841

0. 2920

0. 2996

0. 3071

0. 3144

0. 32t4

0. 3283

0.50

0° 2427

0. 2507

0. 2584

O, 2660

O. 2735

0.2807

0. 2878

O. 2948

0. 30t5

0. 3081

0.60 0°70

0.223t 0.201t

0.2305 0.2078

0.2377 0.2144

0.3447 0,2208

0.25t7 0.2271

0.2585 0.2333

0. 2651 0.2394

0.2716 0.2454

0.2760 0.2512

0. 2842 0.2570

0. 80

0. t774

0. 1833

0. 189t

0. 1948

O. 2005

0. 2060

0.2i14

0. 2168

0. 2220

0. 277t

0. 90

0. 1524

0.1576

0.1626

0,1675

0.1724

0.1772

0. 18t9

0. 1865

0.1911

0.1955

1.00

0,1270

0.13|8

0.1355

0.1397

0,1437

0.1477

0. t517

0.1555

0.1593

0.1631

t. 4000

1. 4200

1. 4400

1. 4600

1. 4800

1. 5000

1. 5200

1. 5400

1. 5600

1. 5800

0.

0.3722

0.5789

0,3854

0.3916

0.3976

0.4033

0. 4088

0.4141

0.4191

0.4239

0.10

0.3698

0.3765

0.3530

0.3993

0.3952

0.40t0

0.4065

0.4117

0.4168

0.4216

0.20

0, 3628

0.3695

0. 3759

0. 3821

O. 3881

0. 3939

0. 3994

0. 4047

0. 4098

0. 4146

0.30

0.5611

0.3577

0.3641

0.3703

0.3762

0.3820

0.3875

0.3928

0.3980

0.4029

0,40

0. 3349

0. 3414

0. 3477

0. 3538

0. 3596

0. 3653

0. 3708

0. 3762

0. 3813

0. 3863

0.50

0.3145

0.3208

0.3268

9.3328

0.3385

0.3441

0.3495

0.3547

0.3598

0.3648

0. 60

0. 2902

0. 2961

0.3019

0. 3076

0. 3130

0. 3184

0. 3236

0. 3287

0. 3336

0. 3384

0.70 0. 80

0.2626 0. 2322

0. 2680 0. 2371

0,2734 0. 2420

0. 2786 0, 2467

0. 2838 0. 2513

0. 2888 0. 2559

0. 2937 0. 2604

0. 2985 0. 2647

0. 3032 0. 2690

0.3077 0. 2732

0.90 t. 00

O. 1999 0.1667

0. 2042 0.1703

0. 2085 0.1739

0. 2126 0.1773

O. 2167 0. t808

0. 2207 0.1841

0. 2246 0. t874

0. 2284 0.1906

0. 2322 0.1938

0. 2359 0.1969

t. 6000

1. 6200

|. 6400

1. 6600

t. 6800

1. 7000

t. 7200

1. 7400

1. 7600

1.7800

O.

O. 4285

O° 4328

O. 4370

O. 4409

O. 4446

O. 4482

O. 4516

O. 4547

O. 4577

O. 4606

0. 10

0. 4262

0.4306

0, 4347

o. 4387

0. 4425

0. 4461

0.4495

0. 4527

o. 4557

0. 4586

O. 20

O. 4193

O. 4237

O. 4280

O. 4320

O. 4359

O. 4396

0.4431

o. 4464

o. 4496

O. 4526

0.30

0.4076

0.4122

0.4165

0.4207

0.4247

0.4285

0.4322

0.4357

0.439o

0.4422

O. 40

O. 39tt

O. 3957

0.4d)Ol

o. 4044

O. 4085

O. 4125

o. 4164

O. 4200

o. 4236

o. 4270

0.50

0.3695

0.3742

0.3787

0.3830

0.3872

0.3913

0.3953

0.3991

0.4028

0.4063

0. 60

0. 343t

0.3477

0.352t

0. 3564

O. 3606

0. 3647

0. 3687

0. 3726

o. 3763

o. 3800

0.70

0.3122

0.3165

0.3208

0.3250

0.3290

0.3330

0.3369

0.3407

0.3444

0.3480

0.80

0.2773

0.28t4

0.2853

0.2892

0. 2930

o. 2967

0.3003

0.3039

0.3074

o. 3108

0. 90

0, 2395

0.243t

0. 2466

0. 2500

0. 2534

0. 2567

0. 2599

0. 263i

0. 2663

0. 2693

1.00

0. 2000

O. 2030

0. 2059

0. 2088

0. 2117

o. 2144

0. 2172

0. 2199

o. 2225

o. 2251

1.8000

|.8200

1.8400

1.8600

1.8800

1.9000

1.9200

1.9400

1.9600

1.9800

0.

0.4633

0. 4658

0. 4682

0. 4704

0. 4725

0. 4745

0. 4763

0. 4781

0. 4797

0. 4812

0. l0

0.4613

0,4639

0,4663

0°4686

0. 4708

0. 4728

0. 4747

0. 4765

0. 4782

0. 4798

0. 20

0.4555

0. 4582

0.4608

0.4632

0. 4655

0. 4677

0.4697

0.4717

0. 4735

0. 4752

0. 30

0.4453

0. 4482

0. 4510

0. 4536

0. 456t

0. 4585

0. 4608

0. 4630

0. 4651

0.4670

0.40

0.4302

0.4334

0.4364

0.4393

0.4421

0.4447

0.4473

0.4497

0.4520

0.4543

0.50 0.60

0.4098 0. 3835

0.4131 0. 3870

0.4t93 O. 3904

0.4195 0. 3936

0.4225 0. 3968

0.4254 0. 3999

0.4282 0.4029

0.4309 0. 4058

0.4335 0. 4086

0.4360 0. 4113

0.70

0.3515

0.3550

0.3583

0.3616

0,3649

0. 3679

0.3710

0.3740

0.3769

0.3798

0.80

0.3141

0.3174

0.2206

0.3238

0.3269

0.3299

0.3328

0.3359

0.3386

0.3414

0.90

0. 2723

_2753

0. 2782

0. 2811

0. 2839

0. 2866

0. 2893

0. 2920

0. 2946

0. 2972

1.00

0. 2277

0. 2302

0. 2327

0. 2351

0. 2375

0. 2396

0. 2421

0. 2444

0. 2466

0. 2488

D

1,

1972

282

OF Poor

TABLE

Section

Qu u..i'IT

6.0-3.

July 1, 1972 Page 283

(CONTINUED)

2.0000

2.0200

2.04{)(}

2.0600

2.0800

2.1000

2.1200

2.1400

2.1600

2.1800

O.

0.4826

0.4840

0.4852

0.4864

0.4874

0.4884

0.4894

0.4902

0.4910

0.4918

0.10

0.4812

0.

0.4839

0.

0.

0.4873

0.

0.4892

0.

0. 4908

0.

0.4769

0.4784

0.47!}8

0.4812

0.4825

0.4837

0.4848

0.4858

0.4868

0.4877

0.30

0.4689

0.4707

0.4723

0.4739

0.4754

0.4769

0.4782

0.4795

0.4807

0.4819

O. 40

O. 4564

0.

O. 4_o5

O. 4623

0.

O. 4659

0. 4675

O. 4691

O. 4706

O. 4720

O. 50

O. 4385

O. 44(}8

0. 4431

O. 4453

0. 4474

0. 4494

O. 4514

O. 4533

O. 4551

O. 4568

0.60

0,414o

0,416(;

0.4101

0. 42111

0.4239

0.4263

0.4285

0.4307

0.4328

0.4348

O. 70

O. 3825

O. 3853

O. 3879

0.

O. 3931

O. 3955

O. 3979

O. 4003

O. 4026

O. 4049

0.80

0.3441

0.3468

0.3495

0.3521

0.3546

0.3571

0.3595

0.3619

0,3643

0.3666

O. 90

O. 2997

0.3022

0.3046

0.3070

0.3094

0.3117

0.3140

0.3162

0.3|84

0.3206

1.00

0.2510

0.2531

0.2552

0.2572

0.2592

0.2612

0.2632

0.2651

O. 2670

O. 2689

20

4826

45_5

2200

4851

3905

4862

4641

4883

4900

2.2000

2.

2.2400

2.2600

2.2_00

2.30o0

2.3200

2.3400

2.3600

2.3800

O.

0.4924

0.4931

(L4937

0.4942

0.4947

0.4952

0.4956

0.4960

0,4963

0.4967

0. I0

0. 4!)15

0,4922

0,4928

0,4934

0,4940

0.4945

O. 4949

0,4954

0,4957

0.4961

0.20

0.4886

0.4894

0.4902

0.4909

0.4915

0.4922

0.4927

0.4933

0.4938

0.4942

0.30

0.4829

0.4840

0,4849

[).4859

0,4867

0.4875

0.4883

0.4890

0.4897

0.4904

0.40

0.4734

0.4747

0.4760

0,4772

0.47H3

0.4794

0,4804

0.4814

0.4824

0.4833

0.50

0.4585

0.4602

0.4617

0.4633

0.4647

0.4661

0.4875

0.4688

0.4700

0.4712

0.60

0.4368

0.4388

0.4406

O. 4425

0.4442

0.4459

0.4476

0.4492

0.4508

0.4523

0.70

0.4071

0.4092

0.4113

0.4134

0.4154

o.

4173

0.4192

0.4211

0,4229

0.4247

0.80

O. 3688

O. 3710

O. 3732

0.3753

O, 3774

0.

3795

0.3815

0.3_35

O. 3854

0.3873

0.90

0.3227

0.3248

0.321;9

0.3289

0.3309

0.3329

0.3348

0.3368

0.3386

0.3405

1.00

O. 27[)7

0.2725

0.2743

0.27¢;0

O. 2778

0.2795

0.2812

0.2828

0.2844

0.2860

4800

2.4000

2.4200

2.4400

2.460o

_

2.5000

2.5200

2.5400

2.5600

_ 5800

O.

0.4970

0.4972

0.4075

0.4977

0.4979

0.49_I

0.4983

0.4985

0.4986

0.4988

0. I0

0.4964

0.4967

0.4970

0.4973

0.4975

0.4978

0.4980

0.4081

0.4983

0.4985

O. 20

0.4947

0.4951

0.4954

0.4958

0.4961

0.4964

0. 4967

0.4970

0.4972

0.4974

0.30

0.4!)10

0.4915

0.4921

0.492(;

0.4931

0.4935

0.4939

0.4043

0.4947

0.4950

0.40

0.4841

0.4849

0.4857

0.4864

0.4871

0.4878

0.4884

0.4891

0.4896

0.4902

0.50

0.4724

0.4735

0.4746

0.4756

0.4766

0.4776

0.4785

0.4794

0.4802

0.4811

0.60

0.4538

0.4553

0.45(;6

0.4580

0.4593

0.4(;0(;

0.4618

0.4630

O. 4642

0.4653

0.70

0.4265

0.4282

0.4298

0.4315

0.4331

0.4346

0.4361

0.4376

0.4391

0.4405

O. 80

O. 3892

O. 3911

O. 3029

O. 3946

O. 3964

0. 3981

0. 3998

O. 4014

O. 4030

O. 4046

O. 90

O. 3423

O. 3441

O. 3459

O. 3476

O. 3493

0.

3510

O. 3527

O. 3543

O. 3559

O. 3575

1.00

0.2876

0.2892

0.2907

O. 2922

O. 2937

O. 2952

O. 2987

0.2981

O. 2995

0.3009

I 2. O. 0.

6000

2. 6200

2. 6400

2.

7000

2. 7200

_

O. 4989

0,

0.

O. 4992

0.4!)93

O. 4993

O. 4994

O. 4995

O. 4995

O. 4996 0.4994

499(}

4991

6600

2.

6800

2.

D

7400

_

7600

_

7800

t0

0,4986

0.4987

0.4989

0.4990

0.499{

0.4992

0.4992

0.4993

0.4994

O. 20

0.4976

0.4978

0.4980

0.4982

0.4983

0.4984

0.4986

0.4987

0.4988

0.4989

0.30

0.4954

0.4957

0.4960

0.4062

0.4965

0.4967

0,4970

0.4972

0.4973

0.4975

0. 40

0,4907

0.49i2

0.4917

0.4921

0.4925

0.4930

0.4933

0,4037

0.4940

0.4944

0.50

0.4818

0.4826

0.4833

0.4840

0.4847

0.4H53

0.4860

0.4866

0.4871

0.4877

0,60

0.4664

0.4675

0.4685

0.4695

0.4705

0.4714

0.4723

0.4732

0.4741

0.4749

0.70

0.4419

0.4432

0.4445

0.4458

0.4471

0.4483

0.4495

0.4507

0.4519

0.4530

0.00

0.40(;2

0.4077

0.4092

0.4107

0.4122

0.4i36

0.4150

0.4|64

0.4178

0.4191

0.90

0.3591

0,3606

0.3621

0.3636

0.3651

0.36(;6

0.3680

0.3694

0.3708

0.3722

1. O0

O. 3023

O. 3036

O. 3050

O. 3063

O. 3076

O. 3089

O. 3iOi

O. 3114

O. 3126

O. 3138

OF POOR

Section

qUALiTY

July Page

TABLE

6. 0-3.

1, 1972 284

(CONTINUED)

3.9000

2.8100

3.9400

3.0600

2.9800

2.9000

3.9200

2,0400

2.9600

2.9000

O.

0. 4964

0. 4097

0. 4997

O. 4M_

O. 4000

O. 4009

O. 4909

O. 4009

O. 4998

O. 4M_

0. i0

0.4969

0.4998

0.4996

0.4994

0.499?

0. 4N't

0,4997

O. 4968

0.4996

0.4_8

0, 00

9. 4M0

0. 499|

0. 4992

0. 4992

O. 4993

0. 4964

0. 4M)4

0. 4965

0. 4M_

0. 4996

0. 30

0. 4977

0. 40?9

0. 4260

0. 4081

O. 4983

0. 4264

0. 4985

O. _

0. 4987

0. 4988

0.40

0. 494/

0. 4960

0. 4962

0. 4966

0. 4960

O. 4940

9. 4982

0. 4964

0. 4986

o. 4968

0. 50

0. 4882

5. 4887

0. 4803

0. 4897

o. 4801

0. 4999

0. 4909

o. 4913

0. 49/7

o. 4921

0. 60

O. 4757

0. 4765

O. 4775

0. 4780

O. 479?

O. 4794

O. 4a01

0. 480?

O. 4614

O. 4830

0o70

0.4641

0.4582

0.4562

0.4572

0.4582

0.4592

0.4803

O. 4_1/

O. 4699

O. 4_20

0. 80 0. 90

0. 4204 0. 9795

0. 4217 0. 3749

O. 4260 0. 376 '_

0. 4240 0. 3770

0. 4264 O. 3788

0.4266 0. 3800

0. 4278 0. 3813

O. 4399 0. 3026

0. 4601 0. 5837

0. 4312 O. 3849

I. 00

0. 0151

0. 5109

0. 3174

0. 3189

0. 319'7

0. 3208

0. 5230

0. $23!

0. 3242

0. 3252

3. 0000

3. 1000

6. Z000

3. 3000

3. 4000

3. 0000

6. 6000

3. 7000

3. 0000

3. 9000

0. 0. 10

0. 499| 0.4969

0. 401HI 0. 4990

0. 6000 0. 4990

0. §000 O. 9000

0. 5000 0. 6000

0. 6000 0. 5000

0. 500Q 0. 5000

0. 5000 0, 0000

0. 5000 O. S000

0. 5000 0. 6000

0. I0

O. 4998

0. _

0. 496@

0. 4969

O. 4099

0. 9000

0. 6000

O. 0000

O. 5000

0. 0000

O. 90

O. 4080

O. 41)93

O. 4996

O. 4997

0. 4998

O. 4998

O. 41199

0. 4999

O. 5000

O. 6000

O. 40

O. 411/0

O. 49?9

0. 4994

O. 4998

O. 4991

0. 4994

O. 4996

0. 499?

O. 4908

O. 4999

O. 60

0, 41HI4

O. 41140

0. 49611

0.4982

O. 49?0

O. 4977

0. 4962

0. 4084

0. 4909

O. 4991

O. 00

O, 4826

O. 4853

O. 49T8

O. 482 II

O. 491 |

O. 4028

O. 4981

O. 4948

O. 4967

O. 4964

0.70 0.80

O, 4400 0.489

0, 44'/9 6.44'/9

0. 47/6 0.4434

0. 4747 0.4449

0. 4778 9,46/0

0. 4fi02 0.4847

0. 4091 0.4292

O. 4046 5.4614

0. 4863 0.4444

0, 4979 0.4471

0.00

6.1194|

0.110|11

0.09?/

0.4090

0.4000

6,4113

0.4164

0.4/94

0.426/

0.4N4

I. O0

O. 1110I

O. 91119

O. I9111

O. 1140I

O. 8400

5. 84t9

O. 09119

0. 0009

0. 9104

O. M110

4.0000

4.1000

4.1000

4.8000

4.4000

4.11000

4.9000

4.7000

4.0000

4.9900

0.6000

0, Ii000

0. 5000

O. iO O. 90

0. 8000

0. S000

0.6000

O. 9000 0.6000

0. BOO0 0.9000

O. 9000 0.8000

O. 9000 5.6000

O. |000 0.5000

O. 6000 0.8000

0. 0000 0.0000

0. 6000

0.0000

0. 8OOO

0,9000

0.0000

0.6000

0.0000

0.6000

0.6000

0. SO00

O. $0

0. 5000

O. lSO00

0. 6000

0. 0000

O. 6000

O. 9990

O. 9000

0.4099

0. S000 0, 4009

O. 6000

0.40

0. 9000 0. 4990

0. 0000

O. 5000

O. 0000

O. 6000

0. 6000

0, 6000

O, 0000

0,00

O. ¢_ml 0. 4971

0. MIN

0.90

0. 4R11 0. 49?0

0o 4U0

0. 40117 0. 4009

O. 4960 9. 4964

,. O. 44911 "_t.

O. 44111 0, 9601

O. 41_1 O, 4002

O. _ 0. 4964

O. 4996 0. 4091

O. _0

0. 4894

O, 41NM

O. 4917

O. 4097

O. 4926

O, 4944

O. 491|

O. 409?

O. 4NJ

0. 4267

O. 00

0. 4494

O. 4730

0, 4741

0. 478/

0. 4700

0. 4?07

0. 481J

0, 44:ff

0. 4440

0. 4098

9.80

0. 4009

O. 48111

0. Ui

0. 4389

0. 4418

5. 4441

0. 444#

0. 4489

0. 41)10

0. 4091

1.00

0. NT0

0. 9700

O. 11799

O. 0767

O. 1704

O. 9000

O. 5914

O. 0897

O. 0910

O. 900|

6. 0000

6.1000

6. I000

LJ000

L4000

&8990

L9000

0, 7000

6. I000

6._00

O,

O. 6000

0,10g0

0. _

O. 6000

0. _

0. 8000

0. S000

0, S000

0. I_00

O. 9900

0. J9

0. S000

0. 0000

0. 8000

0. 9000

0.1000

(). 0000

0. 9000

0.11000

5. 6000

9. J000

O. 110

O, 6000

O. 9000

O. 8000

O, 9000

O, 6000

O. 0000

O. 0000

O. 6000

O. 6000

O. 0000

0.90

0.9000

O. I000

0. S000

0.0000

0.6000

0.1_00

0.1000

0. _000

0,1000

0. S000

0.40

O. 9000

0.9000

0.9000

0.6000

5.6000

0.9000

0.0000

0,11000

0,6000

0,6000

0. I_ 5. _0

0. 0000 0. _

0. 9000 9. 409?

0. _000 0, 4_

0. 0000 0. 4969

5. 8000 0. 4968

0. _000 0. 4196

0. 6000 0. 49_

0. _00 0. 49_

0. _000 0. 49_

0. 9000 0. 4009

6,

D

0, T0

0. 4_'_I

0. 49?0

O. 49?9

0. iNS

0. 4964

0. 41HM

0. 4N0

0. 4910

0. 4961

0.

O, IO

O, 4_4

O, 4670

O. 4_il

O. 4964

O, 4990

O. 4915

O. 4917

O. 44_4

O, 40110

O. 4_91

0.90

0, M00

0.4_10

0.449?

0.4409

0.440t

0.4417

0,41011

0.44414

0.44110

0,4091

1.000

0.91111

0.9141

9.111419

O, 9961

0.8960

0.4019

9, 41011

0.4040

0.410411

0.4010

+

Section

TABLE

6.0-3.

July

1, 1972

Page

285

(CONTINUED)

6.0000

6.1000

6. 2000

6.3000

6.4000

6.5000

6.6000

6.7000

6.8000

6.9000

O.

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0. t0

0.5000

0.5000

0.5000

0.5000

0.5000

O. 5000

0.5000

0.5000

0.5000

0.5000

O. 20

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0,30

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

O. 40

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0,5000

0.5000

0.50

0,5000

0.5000

0.5000

0.5000

0.5000

0.5000

0,5000

0.5000

0.5000

0,5000

0,60

0.4999

0.5000

0.5000

0,5000

0,5000

0.5000

0.5000

0,5000

0.5000

0,5000

0.70

0.4993

0.4994

0.4995

0.4996

0.4997

0,4997

0.4997

0.4998

0.4998

0.4998

O. 80

O, 4941

O. 4946

O. 4950

O. 4954

O, 4958

O. 4962

O, 4965

O, 4968

O. 90

0.4705

0.4717

O. 4728

O. 4739

0.4750

O. 4760

O. 4769

1.00

0.4095

O. 4109

0.4123

O. 4136

0.4_49

0o4162

O,

4970

O. 4973

O. 4779

O. 4788

O. 4796

O. 4174

0,4196

0.4199

0.4209

7.0000

7.1000

7.2000

7.3000

7.4000

7.5000

7.6000

7.7000

7.8000

7.9000

O.

0.5000

0.5000

0.5000

0.5000

0.5000

O. 5000

0.5000

0.5000

0.5000

0.5000

0. I0

0.5000

0.5000

0.5000

0,5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.20

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

9.5000

0.30

0.5000

0.5000

0.5000

0.5000

0,5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.40

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.50

0.501)0

0.5001)

0.50O0

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.60

0.5000

0.501)0

0.5000

0.50O0

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.70

0.4999

0.4999

0.4909

0.4999

0.4999

0.4999

0.5000

0,5000

0.5000

0.5000

0.80

0.4975

0.4977

0.4979

0.4981

0.4983

0.4984

0.4985

0.4987

0.4988

0.4989

0.90

0.4804

0,4812

0.4819

0.4827

0.4834

0.4840

0.4847

0.4853

0.4859

0.4864

1.00

0.4220

0,4230

0.4241

0.4251

0.4261

0.4270

0.4279

0.4298

0.4297

0,4306

8,0000

8.1000

8.2000

8.3000

8.4000

8.5000

8.6000

8.7000

8.8000

O.

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0. i0

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.20

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.30

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0,5000

O. 5000

0.5000

0.5000

0.40

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0,5000

0.50

0.5000

0,5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.60

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.70

0.5000

0.5000

0.5000

0.5000

0.5000

0.5001)

0.5000

0.5000

0.5000

O. 5000

0.80

0.4990

0.4991

0.4992

0.4993

0.4993

0.4994

0.4994

0.4995

0.4995

0.4996

0.90

0.4869

0.

0.4880

O, 4884

O. 4999

O.

O. 4898

O. 4902

0.4906

O. 4909

t.00

0.4315

0.4323

0.4331

0.4339

0.4346

0.4354

0.4361

O. 4368

O. 4375

0.4352

9.9000

4875

4893

8.9000 -

0.5000

9.0000

9.1000

9.2000

9.3000

9.4000

9.5000

9.6000

9.7000

9.8000

O.

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.10

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

O. 20

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

O. 5000

0.30

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

O, 40

O. 5000

O. 5000

O. 5000

O. 5000

O. 5000

O, 5000

O. 5000

O. 5000

O. 5000

O. 5000

O. 50

O, 5000

0.5000

0.5000

O. 5000

O. 5000

O. 5000

O. 5000

O. 5000

0.5000

O. 5000

0.60

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

0.5000

O. 70

O. 5000

O. 5000

O. 5000

O. 5000

O. 5000

O, 5000

O. 5000

O. 5000

O. 5000

O. 5000

0.80

0.4996

0.4997

0.4997

0.4997

0.4998

0°4998

0.4998

0.4998

0.4999

0.4999

O. 90

O. 4913

O. 4916

O. 4920

O. 4923

O. 4926

O. 4929

O.

O. 4935

O. 4937

O. 4940

1. O0

O. 4389

O. 4395

O. 4401

O. 4408

O. 4414

O. 4420

O. 4426

O. 4431

0.4437

O, 4443

4932

D

Section

D

July 1, 1972 Page 286

TABLE

10.0000

10.1000

10, 3000

t0.3000

6.0-3.

10.4000

(CONCLUDED)

10, 5000

I0. 5000

iO. 7000

J.O. 5000

_0. 9000

O.

O. 5000

O. 5000

O. 5000

O. 5000

O. 5000

O. 5000

O. 5000

O. 5000

O. 5000

O. 5000

0,10

O. 5000

0. 5000

O. 5000

O. 5000

O. 8000

O. 5000

O. 5000

O. 5000

O. 5000

O. 5000

0.20

0. 5000

0. 5000

0. 5000

0. 5000

0. 5000

0. 5000

0. 5000

0. 5000

0. 5000

0. 5000

0, 50

0. 5000

0. 5000

0. 5000

0. 5000

0. 5000

0. 5000

0. 5000

0. 8000

0. 5000

0. 5000

0. 40 0, 50

O. 5000

O, 5000

O. 5000

O. 5000

O. 5000

O. 5000

O. 5000

O. 5000

O. 5000

O. 5000

O, 5000

O. 5000

O. 5000

O. 5000

O. 5000

O. 5000

O. 5000

O. 5000

O. 5000

O. 5000

0. 60

O. 5000

O. 5000

O. 5000

O, 5000

O. 5000

O. 5000

O. 5000

O. 5000

O. 5000

O. 5000

0, 70

O. 5000

O. 5000

O. 5000

O. 5000

O. 5000

O. 5000

O. 5000

O. 5000

O. 5000

O. 5000

0. 80 0. 90

O. 4999 O. 4942

O. 4999 O. 4945

O. 4999 O. 4947

O. 4999 O. 4949

O. 4999 O. 4951

O. 5000 O. 4953

O. 5000 O. 4955

O. 5000 O. 4957

O. 5000 O. 4958

O. 5000 O. 4960

1o 00

O, 4448

O. 44_4

O. 4459

O, 4464

O, 4469

O. 4474

O. 4479

O. 4484

O. 4455

O. 4493

OF POOR

QUALITY

Section

D

July 1, 1972 Page 287

TABLE

6.0-4.

03

SPHERE-PARAMETER

O. 5000

O. 5200

O. 5400

O. 5600

O. 5800

O. 6000

O. 6200

O. 6400

O. 6600

O. 6800

0.

0.0001

0.0002

0.0004

0.0008

0.0013

0.002t

0.0033

0.0048

0.0069

0.0096

0.10

0,0001

0.0002

0.0004

0.0008

0,0013

0.0021

0.0032

0.0048

0.0088

0,0095

0.20

0.0001

0.0002

0.0004

0.0007

0.0012

0.0020

0,0031

0.0045

0.0065

0.0090

0,30

0.0001

0.0002

0.0004

0.0007

0.0011

0.0018

0.0028

0.0042

0.0060

0.0083

O. 40

0,0001

0.0002

0.0003

0.0006

0.0010

0.0016

0.0025

0.0037

O. 0052

O. 0073

0.

0.0001

0.0001

0.0003

0.

0.0008

0.00t3

0. 0021

0, 0031

0,

0.

0061

0.60

0.0001

0.0001

0.0002

0,0004

0.0007

0.00t1

0.0018

0.0024

0.0035

0.

0049

0.70

0.0000

0.0001

0.0002

0.0003

0.0005

0.0008

0.0012

0.0018

0.0026

0.0035

0.80

O. 0000

O. 0001

O. 0061

0.0002

0.0003

O. 0005

O.

O. 0011

O. 0018

O. 0023

0.90

0.0000

0.0000

0.00o0

0.0001

0.0001

0.0002

0.0004

0.0005

0.0008

0.0011

1.00

0.0000

0.0000

0,0000

0.0000

O. O00O

0.0000

0.0000

0.0000

0.0000

0.0000

0.7000

0.7200

0.7400

0.7600

0.7800

0.8000

0.8200

0.8400

0.8600

0.8800

0.0130

0.0171

0.0221

0.0279

0.0347

0.0423

0.0510

0.0606

0.0711

0.0827

0.0128

0.0169

0.0217

0.0275

0.0341

0.0416

0.0501

0.0596

0.0700

0.0813

0.20

0.0122

0.0160

0.0207

0.0261

0.0324

0.0396

0,0477

0.0567

0.0666

0.0773

0.30

0.0112

0,0147

0.0190

0.0240

0.0297

0.0363

0.0438

0.0520

0.06tl

0.0709

0.40

0.0098

0.0130

0.0167

0.0211

0.0262

0,0320

0.0386

0.0459

0.0536

0.0626

O. 50

O. 0083

O. 0103

O. 0141

O. 0178

O, 0221

O, 0270

O. 0325

O. 0386

0.0453

O. 0526

0.60

0.0066

0.0086

0.0111

0.0141

0.0175

0.0214

0.0257

0.0306

0.0359

0.0417

0.

0.

0. 0063

0.

0.

0.

0.

0.

0. 0223

0,

0.

0.0030

0.0040

0.0052

0.0065

0.0061

0.0009

0.0119

0.0t42

0,0166

0.0193

0.90

0.0014

0.0019

0.0024

0.0031

0.0038

0.0046

0.0056

0.0066

0.0078

0.0990

1.00

0.0000

0.0000

O. OOO0

0.0000

0.0000

0,0000

0,0000

0.0000

0.0000

0.0000

50

O. 0.

t0

70

0.80

0048

0081

0005

0103

0127

0156

0008

0188

0044

0262

0304

0.9000

0.9200

0.9400

0.9600

0.9800

1.0000

1.0200

1.0400

1.0600

1.0800

O.

0.0951

0.1084

0.1225

0.1375

0.1531

0.1695

0.1865

0.2041

0,2222

0.2407

0.10

0.0935

0.1066

0.1205

0.1352

0.1506

0.1667

0,1835

0.21)08

0.2186

0.2368

0.29

0.0899

0.1014

0.1146

0.1286

0.1433

0.1586

0.1745

0.1910

0.2079

O. 2253

0.30

0.0816

0.0930

0.1052

0.11_0

0.

t315

0.1455

0.1602

0.1753

0.1908

0.2068

0.40

0.

0.

0.

0.

0.

t159

0.1283

0.

0.

0.

0.

0,50

0.0605

0.0690

0.0790

0.0875

0.0975

0.1080

0.1188

0.130t

0.1416

0.60

0.0480

0.0547

0.0619

0.0694

0.0773

0.0856

0.0942

0.1031

0.

0.70

0.0350

0.0309

0.0451

0.0506

0.0564

0.0624

0.0687

0.0752

0.0818

0.0888

0.80

0.0222

0.0254

0.0287

0.0322

0.0358

0.0397

0.0437

0.0478

0.0521

0.0565

0.

0.

0.

0.

0. 0150

0,

0.

0.

0. 0224

0,

0,

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

90

1.00

0720

0104

0820

0118

0.0000

0.0000

0927

0134

0.0000

1041

0168

0185

1412

0204

1546

1683

t123

0243

1824

0.1535 0,|217

0264

i. 1000

I. 1200

I. 1400

I. 1600

I. 1800

I. 2000

I. 2200

i. 2400

I. 2600

I. 2800

O.

O. 2597

0.2790

O. 2986

0.3183

O. 3383

O. 3584

O. 3785

0.3986

0,4187

O. 4388

0.10

0.2555

0.2745

0.2937

0.3132

0.3328

0.3526

0.3724

0.3923

0.4121

0.4318

O. 20

O. 2430

O. 2611

O. 2795

O. 2980

O. 3168

O. 3356

O. 3545

O. 3735

O. 3924

O. 4113

0.30

0.2231

0.2398

O. 2566

O. 2737

O. 2910

0.3084

0.3218

0.3433

0.3809

0.3783

O. 40

O. 1968

O. 2115

O. 2265

O. 2416

O. 2569

O,

O.

2875

O. 3034

O. 3t90

O. 3346

O. 50

O. 1657

O.

1781

O. 1907

O. 2035

O. 2164

O. 2295

O. 2426

O. 2559

O. 2691

O. 2624

O.

O. 1314

O. 1413

O. 1513

O. 1615

O. 1718

O. t822

O.

O. 2032

0.

O. 224_

0.70

0.0!J59

O. t031

0.

0.

0.

0.

t331

0.1408

O. i496

0,1564

0.

O. 80

0.0610

0.0656

0.0702

0.0750

0.0798

O. 0847

0.0896

0.0946

O. 0998

O. 1047

O. 90

O. 0285

O. 0307

O. 0328

O. 0351

O. 0373

O. 0396

O. 0419

O. 0443

O. 0466

O. 0490

1.00

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0,0000

0.0000

0.0000

60

t104

t179

t254

2723

1927

2139

t642

Section D Jtdy I, 1972 Page 288

TABLE

6.0-4.

(CONTINUED)

I. 3500

I. 3800

I. 4000

I. 4200

I. 4400

I. 4600

i. 4800

O. 4979 O. 4902

O. 5172 O. 5099

O. 5362 O. 8280

O. 5850 O. 5465

O. 5734 O. 5647

O. 89t4 O. 5825

O. 6091 O. 6000

O. 6260 0. 8171

O. 4488

O. 4673

O. 4856

O. 5036

O. 5215

O. 6390

O. 5563

O. 5732

O. 6899

O. 4131

O. 4303

O. 4474

O. 4648

O. 4810

O. 4974

O. 5137

O. 5297

O. 5454

0.3502

0.3657

0.3811

0.3965

0.4117

0.4268

0.4417

0.4566

0.4711

0.4855

O. 30

O. 2957

O, 3090

O. 3222

O. 3354

O. 3485

O. 3616

O. 8748

O. 3874

O. 4001

O. 4128

O. 60

O. 2353

O. 2460

O. 2566

O. 2673

O. 2780

O. 2886

O. 2992

O. 3097

O. 3202

O. 3306

O. 70

O. 1721

O. 1800

O. 1880

O. t959

O. 3038

O. 2117

O. 2197

O. 2776

O. 2354

O. 2433

0. 80

O. 1097

O. 1148

O. 1199

O. 1250

O. 1302

O. i353

O. 1404

O. 1486

O. 1507

O. 1568

O. 90

O. 0614

O. 0538

O. 0563

O, 0586

O, 0610

O. 0634

O. 0659

O. 0688

O. 0707

O. 0732

I. O0

O. O000

O. 0000

O. OOO0

O. OOO0

O. 0000

O. 0000

O. 0000

O. 0000

O. 0000

O. 0000

1. 8000

t. 6200

t. 5400

t. 8500

t. 5800

1. 5000

1. 6200

1. 5400

t. 6600

1. 6800

t. 3000

1. 3200

1. 3400

O. 4587 O. 4514

O. 4784 O. 4709

O. 20

O. 4301

O. 30

O. 3958

0.40

O. O. I0

'0.

_

0,5432

0.6595

0.6756

0.69t2

O. 7083

0.7209

0.7350

0,7487

0.7618

O. 7746

0. t0 O. 20

0.6339 O. 6061

0.8502 O. 6220

0.6661 O. 6375

0.6815 O. 6627

0.8965 O. 5674

0.7111 O. 6818

O. 7259 O. 6957

0.7388 O. 7093

0.7620 O. 7224

0.7647 O. 7381

O. 90 0.40

O. 5608 0.4997

O. 5760 0.5137

O. 8908 0.5275

O. 8054 0.54t0

O. 6196 0.5543

O. 6335 0.5674

O. 6471 0.6802

O. 6603 0.5928

O. 6732 0.6051

O. 6857 O. 6i72

0.50

0.4252

0.4376

0.4498

0.4619

0.4738

0.4856

0.4973

0.5086

0.5199

0.5310

O. 60

O. 3409

O. 3512

O. 3614

O. 37t5

O. 88t6

O. 3915

O. 40t3

O. 41tt

O. 4208

O. 4304

O. 70

0. 2511

O. 2889

O. 2667

O. 2744

O. 2821

O. 2898

O. 2974

O. 3050

O. 3126

O. 3201

O. 80

O. 1609

O. 1661

O. 1712

O. 1753

O. 1814

O. 1868

O. 1915

O. 1966

O. 2017

O. 2057

O. gO I. O0

O. 0756 e. 0000

O. 0780 O. 0000

O. 0508 O. OOO0

O. 0829 O. OOO0

O. 0854 O. OOO0

O. 0878 O. 0009

O. 0902 O. OOOO

O. 0927 O. OOO0

O. 0951 O. OOO0

O. 0976 O. 0000

t. 7000

1. 7200

1. 7400

t. 7600

t. 7800

1. 8000

1. 8200

i. 8400

1. 8600

1. 8800

O.

O. 7868

O. 7985

O. 8098

O. 8206

O. 8310

O. 8409

O. 8504

O. 8fi94

O. 8680

O. 8782

O. 10

O. 7770

O. 7888

O. 8001

O. 81t0

O. 8214

O. 5315

O. 8410

O. 8602

O. 8590

O. 8673

O. 20

O. 7474

O. 7593

O. 7707

O. 7818

O. 7928

O. 802"/

O. 8t26

O. 8221

O. 8312

O. 8400

O. 50

O. 6979

O. 7098

O. 7213

O. 732.5

O. 7434

O. 7539

O. 7641

O. 7740

O. 7835

O. 7927

O. 40

O. 6290

O. 6406

O. 6519

O. 6629

O. 6737

O. 6842

O. 6945

O. 7045

O. 7143

O. 7238

O. 50

O. 5420

O. 5527

O. 5633

O. 5738

O. 5840

O. 5941

O. 6040

O. 6137

O. 8233

O. 6327

O. 60

O. 4398

O. 4492

O. 4585

O, 46"/7

O. 4768

O. 4868

O. 4947

O. 5035

O. 5122

O. 5208

O. 70

O. 32'76

O. 3350

O. 3424

O. 3497

O. 3570

O. 3642

O. 3714

O. 3786

O. 3857

O. 3927

O. 80

O. 2117

O. 2167

O. 2317

O. 2267

O. 2317

O. 2367

O. 2416

O. 2465

O. 2515

O. 2564

0.90 i. O0

O. lOOO O. 0000

0.1024 O. O00O

0.1049 O. 0000

0.1073 O. 0000

0,1097 O. OOOO

0.1122 O. O000

0.1146 O. 0000

0.1170 O. 0000

O. i194 O. 0000

0.1218 O. 0000

1. 9000

1. 9200

1. 9400

t. 9600

t. 9800

2. O000

2. 0200

2. 0400

2. 0600

2. 0800

O,

O. 8840

O. 89t4

O. 8984

O. 9051

O. 9114

O. 9173

O. 9230

O. 9283

O. 9333

O. 9380

O. 10

O. 8783

O. 8828

O. 8900

O. 8969

O. 9034

O. 9095

O. 9154

O. 9209

O. 9261

O. 9310

O. 20 O. 30

O. 8484 O. 8016

O. 8564 O. 8102

O. 8641 O. 8186

O. 8715 O. 8266

O. 8786 O. 8343

O. 8852 O. 8417

O. 8916 O. 8489

O. 89/7 O. 8558

O, 9035 O. 862,4

O. 9090 O. 8688

O. 40

O. 7331

O. 7422

O. 7510

O. 7696

O. 7679

O, 7760

O. 7838

O. 7915

O. 7989

O. 8062

O. 50

O. 6419

O. 6509

O. 6598

O. 6686

O. 6771

O. 6854

O. 8937

O. 7017

O. 7096

O. 7173

O. 60

O. fi293

O. 5376

O. 5459

O. 5541

O. 5623

O. 5702

O. 6781

O. 5858

O. 8935

O. 6011

O. 70

O. 3997

O. 4067

O. 4136

O. 4205

O. 4273

O. 4341

O. 4408

O. 4475

O. 4541

O. 4607

O. 80

O. 2613

O. 2661

O. 2'710

O. 2/88

O. 2807

O. 2855

O. 2903

O. 2951

O. 2998

O. 3046

O. 90

O. 1243

O. 1267

O. 1291

O. 1315

O. 1339

O. t368

O. t387

O. 1412

O. 1436

O. 1460

I. O0

O. 0000

O. O00O

O. O00O

O. 0000

O. O000

O. O000

O. 0000

O. 0000

O. 0000

O. 0000

ORIGiI'_/_L

P,._:;

POOR

_o

Section July

QUALITY

1,

Page

TABLE

6.0-4.

D

(CONTINUED)

2.1200

2.1400

2. 1600

2.1800

2. 2000

2. 2200

2. 2400

2.2600

2. 2800

2. 1000 0.

0.9424

0. 9466

0. 9505

0. 9541

0. 9575

0. 9607

0.9637

0.9665

0.9691

0.9716

9.10 0. 20

0.9357 0.9143

0. 9400

0.9442

0.948i

0.9517

0.9551

0.9583

0.9613

0.9642

0.9668

0. 9193

0. 9240

0. 9286

0. 9328

0. 9369

0.9407

0.9444

0.9476

O. 9511

0.30

0.8750

0. 8808

0. 8865

0. 8919

0. 8971

0. 9021

0.9069

0.9115

0.9159

0.9201

0.40

0.8132

0.8200

0.6266

0,8330

0.8392

0.8452

O. 8510

0.8567

O. 8621

O. 8674

0.50

0.7249

0.7323

0.7396

0.7467

0.7536

0.7604

0.7671

0.7736

-0.7799

O. 60

0.6160

0.6232

0.6304

0.6375

0.6445

0.6514

0.6582

0.6648

0.6714

O.70

O. 6086 0.4672

0.4737

0.4801

0.4865

0.4928

0.4991

0.5053

0.5115

0.5177

0.5237

O. 80

0.3093

0.3141

0.3186

0.3235

0.3281

0.3328

O. 3374

O. 3420

O. 3467

O. 3512

O. 90

0.1484

0.1508

0.1532

0.1555

0.1579

0.1603

0. 1627

O. 1651

0. 1675

0. 1699

I. O0

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

2.3400

2.3600

2. 3800

2. 4000

2. 4200

2. 4400

2.4600

2, 4800

_ 3000

2.320o

O.

0.9738

0.9759

0.9779

0.9797

0.98i4

0.9829

0.9844

0.9857

0.9869

0.9881

0.10

0.9693

o. 9716

0,9794

0.9810

0.9825

0.9839

0.9852

0.9542

O. 9571

0.9757 0.9624

O. 9776

0.20

0.9737 0.9598

0.9648

0.9671

0.9693

0.9713

0.9732

0.9750

0.30

0.9241

0.9279

0.8316

0.935i

0.9385

0.9417

0.9447

0.9477

0.9504

0.9531

0.40

0.8726

0.8775

0.8823

0.8869

0.50

0.7922

0.7982

0 8040

0.8097

0.8914 O. 8152

0.8957 O. 8206

0.8999 O. 8259

0.9040 O. 8311

0.9079 0.8361

0.9116 0.8410 0.7323

fl_

O. 7862

0.60

0.6779

0.6844

0.6907

0.6969

0.7030

0.7090

0,7150

0.7208

0.7266

0.70

0.5298

0.5358

0.5417

0.5476

0.5534

0.5592

0.5649

0.5706

0.5763

0.5818

0.80

0.3558

0.3604

0.3649

0.3694

0.3739

0.3784

0.3829

0.3874

0.3916

0.3962

0.1722

0.1746

0.1770

0.1794

0. t817

0.1841

0.1865

0.1888

0.1912

0.1936

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

2.5400

_5600

2. 5800

2.6000

2. 6200

2.6400

2.6600

2.6800

2. 5000

_ 5200

O.

0.9891

0.9901

0.9910

0.9918

0.9925

0.9932

0.9938

0.9944

0.9949

0.9954

0.10

0.9864

0.9875

0.9885

0.9895

0.9904

0.9912

0.9919

0,20

0.9767

0.9783

0.9798

O. 98J2

0.9825

0.9837

0.9849

0.9926 0.9859

0.9932 0.9869

0.9938 0.9679

0.30

0.9556

0.9580

O. 9645 0.9285

0.9665 O. 9316

O. 9684 O. 9345

0.9734

0.9t88

0.9624 0.9254

0.9718

0.9153

O. 9603 0.922i

0.9701

0.40 _50

0.8458

0.8505

0.8550

0.8595

0.8638

0.8680

0.872t

0.9373 0.8761

0.9400 0.8800

0.9426 0.8638

0.00

0.7376

0.7433

O. 7487

O. 7540

O. 7593

O. 7644

O. 7695

O. 7744

O. 7793

O. 7841

0.70

0.5874

0.5929

0.5983

0.6037

0.6090

0.6143

0.6t95

0.6247

0.6299

0.6350

O, 80

O. 4006

0.4050

O. 4094

O. 4137

O. 418t

O. 4224

O. 4267

0.4309

0.4352

0.4395

0.90

0.1959

0. t983

O. 2006

O. 2030

O. 2053

O. 2077

O. 2100

O. 2t23

O. 2147

O. 2170

1. O0

O. 0000

0. 0000

0.0000

0.0000

O. O000

0.0000

0.0000

O. O00O

O. 0000

O. 0000

2. 7600

2.7800

2. 9000

2.8200

2. 8400

_8600

2.9800

2. 7000

2. 7200

2. 7400 0.9975

0.99/8

0.9980

0.9982

0.9984

0.9965

O. 9968

0.9971

0.9974

0.9976

O. 9923

O. 9929

O. 9934

O. 9939

O. 9944

O. 9814

_9825

O. 9836

O. 984,5

O. 9855

90 1.00

1972

289

O.

O. 9958

O. 9962

O. 9966

O. 9969

O. 9972

O. 10

O. 9944

O. 9949

O. 9953

O. 9957

O. 9961

O. 20 O. 50

O. 9888 O. 9749

O. 9896 O. 9764

O. 9903 O. 9777

O. 9910 O. 9790

O. 9917 O. 9803

O. 40

O. 9451

O. 9475

O. 9498

O. 9520

O. 9542

O. 9562

O. 9582

0.9601

0.9619

0.9637

O. 50

O. 8875

O. 89t I

O. 8946

O. 8980

O. 9013

0.9046

0.9077

0.9t08

0.9t37

0.9166

O. 60

O. 7889

O. 7935

O. 7991

O. 802_

O. 8070

0.8113

0.8t_

0.8197

0.8218

0.82'/9

O. 70

O. 5400

O. 6460

O. 5499

O. 6548

O. 6591

O. 90

O. 4437

O. 4479

O. 452t

O. 4562

O. 4604

O. 6645 0.4645

0.6692 0.4686

O. 6739 _ 4727

0.6786 0.4768

O. 6852 0.4809

O. 90

O. 2194

O. 23t7

O. 2240

O. 2263

O. 2287

0.3310

O. 2333

0.2386

0.3379

0.24[03

Section July

D 1, 1972

Page

TABLE

6.0-4.

(CONTINUED)

2. 9000

2. 0100

2. 9400

3. 9600

3. 9800

3. 0000

3. 0200

3. 0400

3. 04100

3. 0800

O.

O, 9966

0. 998/

O. 9988

O. 9990

O. 9991

O. 9993

O. 9993

O. 9993

O. 1_94

O. 9995

O. t0

O. 9978

O. 9980

0. 9982

O. 9984

O. 9985

O. 8987

O. 9988

O. 9989

O. 8990

O. 999|

0. BO

O. 9948

O. 9952

O. 9956

O. 9959

9. 9919

9.1966

O. 9968

8, 997|

0. 99'/5

O. 9975

O, 30

O. 1964

O. 9872

O. 9880

O. 9987

O. 9894

O. 9901

O. 990"/

O. 9913

O. 9918

O. 9923

0,40 O. 50

0.N53 O. 9|94

0.9609 O. 922!

0.9609 O. 9247

0.9700 O. 9273

0.9714 O. 9298

0,9727 O. 9325

0.9'740 O. 9346

0.9753 O. 9388

0.9"/95 O. 9390

0.9770 O. 9412:

O, 90

O. 8318

0. 8357

O. 8396

O. 9432

0. 8469

O. 8505

O. 8540

0. 8575

O. 8609

O. 8642

0,70

0.6878

0.692,3

O, 6968

0.7012

0.7055

0.7099

0.7141

0.7184

0.7226

0.7267

0. 80 O. 90

O. 4849 O. 2425

O. 4889 O. _t48

O. 4929 O. 2471

O. 4969 O. 2494

O. 5009 O. 3917

O. 5048 O. 2840

O. 0088 O. 31919

O. 512/ O. 3988

O. 5155 O. _09

O. 5204 O. 2632

1. O0

O. OOO0

O. 0000

O. 0000

O. 0000

O. O00O

O. O000

O. 0000

O. O000

O. 0000

O. 0009

3. 0000

3. iO00

3. 2000

3.3000

3. 4000

3, 5000

9. 41000

5. 7000

3. 8000

3. 9000

O. O. I0

O. 9992 O. 998"/

O. 9996 O. 9992

O. 999'/ O. I;992

O. 9999 O. 9997

O. 9998 O. 9991)

L 0000 O. 9999

f. 00_ i. 0000

11.0000 i. 0000

|. 0000 #. 0000

|. 0000 I. 0000

0. IO

O, 9995

O. 9977

O. 9986

O. 999t

O. 9994

O. 99941

O. 9900

O. 91H5tl

O. 9999

O. 99118

O. 30 O. 40

O. 9901 O. 9"/2'7

O. 9939 O. 9'787

O. 9949 O. 8834

O. 9964 O. 9872

O. 99"/5 O. 9903

O. 9982 O. 9925

O. 9988 O. 91)44

O. 9992 O. 91)6"/

O. 9994 O. 91)69

O. 90_ O. 9977

O. 50

O. 9321

0. 9432

O. 9527

O. 191608

O. 94'/6

O. 9734

0. 9752

O. 11825

0. 911511

O. 99114

O. 90

O. 8500

O. 84175

O. 0819

O. 8968

O. 90t53

O. 9205

O. 9806

O. 1994

O. 9474

O. 9544

O. 70

O. 7099

O. 7308

O. 7508

O. 7893

O. 78419

O. 80315

O. 8190

O. 838

O. 6472

0. 84500

O. 00

O. 5048

O. 53,43

O. 6432

O. 6617

0. 57V/

O. 5975

O. 5/43

O. 5309

0. 5469

0. 9628

0. 90

O. 2540

O. 24_58

O, 2768

O. 2981

O. 2999

O. 3|04

O. 53|5

O. 3324

O. 3433

O. 31543

1.00

0.0000

0.0000

0.0000

O. 0000

O. O000

0.0000

0, 0000

0. 0000

0. 0000

0. 0000

4.0000

4.1000

4.2000

4.3000

4.4000

4.5000

4.8000

4.1'000

4.8000

4.1)000

O.

i. 0000

1. 0000

1. O000

1. O000

LO000

1.0000

LO000

I. 0000

LO000

i. 0000

O. 10

I. 0000

1.0000

1.0000

I. O00O

1. O000

I. 0000

$. 0000

I. 0000

i. 0000

1. 0000

0.10 0. 30

1. 0000 O. 999"1

i. 0000 O° 9998

I. 0000 0. 9999

1. 0000 O. 9999

1. 0000 O. 9999

t. 0000 1. OOOO

1. 0000 1. 0000

1. 0000 1. 0000

1. 0000 1. 0000

1. 0000 1, 0000

O. 40

O. 9983

O. 9987

O. 9991

0. 9993

O. 9995

O. 9997

O. 9919

O. 9999

O. 9199

O. 99911

0.60

0.9906

0.9925

0.9940

0.9953

0.9963

0.997|

0.99'/'/

0.9192

0.9906

0.89511

0. 80

O. 9_,06

O. 9'660

O. 9708

O. 9'750

O. 9787

O. 9818

O. 9846

O. 9849

O. 9890

O. 8907

0.70

0.8719

0.8839

0.5932

O. 9Q27

0.9t15

0.9197

0.92'/2

0.9341

0.9404

O. 94d2

O. 80

0, 8776

O. 5923

O. 7064

O. 7201

O. 7334

O. 74181

O. 7586

O. 7703

O. 7816

O. 7928

O. 90

O. 3849

O. 3755

O. 3861

O. 385

O. 4088

O. 4173

O. 4P14

0. 4375

O. 4475

O. 4674

i. O0

O. O000

O. 0000

O. 0000

O, 0000

O. 0000

O. 0000

O. 0000

O. 0000

O. 0000

O. 0000

5. O00O

5. iO00

6. 5000

5. 3000

S. 4000

5. 6000

5. 4000

6. 7000

6. 8000

5. 9000

O.

i. 0000

1. 0000

i. 0000

1. O000

1. 0000

1. 0000

1. CO00

1. 0000

i. 0000

1. 0000

0.|0

t. 0000

1.0000

i. 0000

i. 0000

1. 0000

1. 0000

J.. 0000

1. 0000

LO000

LO000

0.19

1. 0000

1. 0000

t. 0000

I. 0000

1. 0000

I. 0000

1. 0000

I. 0000

t. 0000

1. 0000

0. $0

i. O00O

t. 0000

1. 0000

1. 0000

1. 0000

I. 0000

1. 0000

!. 0000

L 0000

1. O000

0.40

0.91911

t. O000

1, 0000

J, 0000

I. 0000

1, 0000

1.0000

I. 0000

|. 0000

1. O00Q

O. 60 0. 60

O. 9992 O. 9n2

O. 9994 O. 9935

O. 9996 O. 5944

O. 9996 O. 1955

O. 9997 O. 11115

O. 9998 9. 9189

O. _ O. 99'74

9. 9999 O. 90'/5

O. 0999 O. 9983

O. 9999 O. 9MM

0. 70 O. 50

O. 9616 O. 9034

0. 1555 O. 8t28

O. 19011 O. 8233

0. 9649 O. 531r'/

O. 1986 O. 8417

O. 9"/39 O. 9603

O. 9760 O. 96111[,

O. 87Y7 O. 8603

O. 96011 O. 0'/39

O. 9824 9. 8810

0.19

0.4471

0.4759

0.486

0.4N!

0.5055

0.5148

O. SNO

O. 6_11

0.6431

O. 56i0

LO0

0. 0000

O. 0000

O. 0000

0.0000

O. 0000

0.0000

0.0000

0.1900

0. 0000

O. 0000

290

ORIGINAL OE POOR

P2:.%}:-'._3 QUALITY

Section

D

July

1972

Page

TABLE

6. 0-4,

(CONTINUED)

6.0000

6.1000

6.2000

6.3000

6.4000

6.5000

6.6000

6.7000

6.

O.

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1,0000

0.10

1.0000

1.0000

1.0000

1.0000

1.0000

t.0000

1.0000

1.0000

t. 0000

1,0000

0.20

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

t.0000

1.0000

0.30

1.000o

1.0000

1.0000

1.000o

1.0000

1.0000

1.0000

1.0000

1.0000

i. O000

0.40

1.0000

1.0000

1.0000

1.0000

l.

1.0000

1.0000

l.

1.0000

1.0000

0.50

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

O. 60

O. 9989

O. 9991

O. 9992

O. 9994

O. 9995

O. 9996

O. 9997

O. 9998

O. 9998

O. 9999

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

70

9944

0. 9951

0.8879

0.8944

0.9006

0.9065

0.9122

0.9175

O. 9226

0.9274

0.9319

0.9363

0.90

0.5598

0.5685

0.5771

0.5866

0.5940

0.6023

0.6104

0.6185

0.6264

0.6343

i.

0.0000

0.0000

0.0000

0.0000

O. O00O

0.0000

0.0000

0.0000

0,0000

0.0000

7.0000

7.1000

7.2000

7.3000

7.4000

7.5000

7.6000

7.7000

7.8000

7.9000

i. O000

1.0000

1.0000

t.

0000

t.0000

1.0000

J. O0OO

1.0000

1.0000

1.0000

1. 0000

1.00o0

1.

1.

0000

1.

1.

1. 0000

1. 0000

1. 0000

1.

O. 20

I. 0000

i. 0000

I. 000n

I. 0000

I. 0000

i. 0000

I. O000

I. 0000

I. 0000

I. 0000

O. 30

1. 0000

1. 0000

1, 9000

1.

0000

1.0o00

J.

0000

1. 0000

1. 0000

t. O00O

J. 0000

O. 40

1.

1. 0000

I. 0000

1.

0000

1.

OOO0

I.

0000

1.

1.

0000

1. 0000

1. 0000

1.0000

1.0000

1.0000

J.

0000

1.

0000

1.0000

1.0000

1.

0000

1,0000

1,0000

0.60

0.9999

0.9999

0.9999

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

0.70

0.9957

0.9963

0.9968

0.9972

0.9976

0.9979

0.9982

0.9984

0.9987

0.9988

0.80

0.9403

0.9442

0.9478

0.9513

0.9546

O. 9576

0.9605

0.9632

0.9658

0.9682

0.90

0.6420

0.6496

0.6572

0.6646

0.6719

0.6791

0.6862

0.6931

0.7000

0.7068

1.00

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

O. O.

t0

50

0000

9862

0878

0000

9893

9905

0000

9917

0000

9927

O000

OOO0

9936

6.9000

0.80

O0

9844

O000

8000

O000

8.0000

8.1000

8.2000

8.3000

8.4000

8.5000

8.6000

8.7000

8.8000

8.9000

O.

|.0000

1.0000

1.0000

1.0000

1.0000

1.0000

i. O000

1.0000

1.0000

1.0000

0. I0

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

0.20

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

t.0000

1.0000

t. O00O

1.0000

O. 30

1. 0000

1. 0000

1.

t.

1. 0000

1. 0000

1. 0000

t. 0000

1. 0000

t.

0.40

1.0000

1.0000

t.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

0.50

1.0000

1.0000

1.0000

1.0000

1.0000

l. O000

1.0000

1.0000

1.0000

1.0000

0.80

t. 0000

1.0000

1.0000

1.0000

1.0000

1.0000

t.0000

i. O000

t.0000

1.0000

0.70

0.9990

0.9991

0.9993

0.9994

0.9995

O. 9996

0.9996

0.9997

0.9997

0.9998

_80

O. 9704

O. 9726

O. 9745

O. 9764

O. 9781

O. 9797

O. 9813

O. 9827

O. 9840

O. 0852

0.90

O. 7135

O. 7200

0.7265

O. 7328

0.7391

O.

7452

O. 7512

O. 7572

O. 7630

O. 7687

i. O0

_

_

0.0000

0.0000

0.0000

_

0000

_

_ 000_

_0000

0.0000

0000

0600

0000

0000

0000

0000

9.0000

9.1000

9.2000

9.3000

9.4000

9.5000

9.6000

9.7000

9.8000

9. 9000

O.

|.0000

1.0000

1.0000

1.0000

1.0000

1.0000

|.0000

i. O000

i. O000

1.0000

0.10

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

0.20

t.0000

1.0000

t.0000

1.0000

1.0000

1.0000

t.0000

1.0000

1.0000

t.0000

0.30

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

_40

1. 0000

1.

1,

1,

1,

1.

1,

t,

1.

1.

0.50

1.0000

1.0000

1.0000

1.0000

0.60

1.0000

1.0000

1.0000

1.0000

0.70

0.9998

0.9998

0.9999

0.9999

0.80

0.9864

0.9875

0.9885

0.9894

O. 90

O. 7743

O. 7799

O. 7853

I. O0

O. 0000

O. 0000

O.

0000

0000

0000

0000

0000

1.0000

0000

0000

0000

0000

0000

1.0000

Io0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

0.9999

0.9999

1.0000

1.0000

1.0000

1.0000

0.9902

0.9910

0.9918

0.9924

0.9930

0.9936

O. 7906

O. 7959

O. 80t0

O. 8060

O. 8tt0

O. 6t58

O. 8206

O. 0000

O. 0000

O. 0000

O. 0000

O. 0000

O. 0000

O. 0000

1,

291

Section

D

July

1972

1,

Page TABLE

'

q!

iO. O00O

10.1000

6,. 0-4.

292

(CONCLUDED)

10. _000

10. 3000

10. 4000

10. SO00

10. 6000

10. '7000

t0. BOO0

10.9000

O.

1. 0000

I. 0000

1. 0000

I. 0000

1. 0000

1. O00O

1. O00O

1. O000

1. 0000

1.0000

O. 10

t. OOO0

t. OOOO

t. 0000

1. OOOO

1. OOOO

1. O00O

t. O000

1. O00O

t. OOOO

1.0000

0.20

1.00O0

1. O000

1. OOO0

1.0000

1. O000

I. OOOO

I. OOOO

1. O00O

I. O00O

1, 0000

0. 30 0.40

1. O000 1.0000

1.0000 1. OOOO

1. O000 1. O00O

I. 0000 1. 0000

1. 0000 1. 0000

1. O000

I. O000

1. 0000

1. OOO0

1.0000

1.0000

t. 0000

1, OOO0

1. 0000

1.0000

O. 50

1. OOO0

1. 0000

1. O000

1. O000

1. O00O

1. 0000

1. 0000

1. 0000

1. 0000

1.0000

0.6O

I. O000

I. O00O

1. OOO0

I. O00O

1. 0000

t. 0000

1. O00O

1. OOO0

1. 0000

1. 0000

O. 70

I. 0000

I. OOO0

1.0000

I. 0000

I. 0000

1. 0000

1. O000

1. 0000

1. O00O

1.0000

O, 80

O. 9942

O. 9947

O. 995t

O. 9955

O. 9959

0.90

O. 8252

O. 8299

O. 8343

0. 9387

O. 8429

O. 9963 O. 9472

O. 9966 O. 8513

0. 9969 O. 8553

O. 9972 O. 8593

0.9974 o. 863t

1. O0

0. 0000

0. 0000

0. 0(300

O. 0009

0. 0000

0. 0000

0. 0000

0, 0000

0. 0000

0.0000

ORIGINAL POOR

PAGE. 13 QUALITY

Section

D

Jttly

1972

Page

TABLE

6.0-5.

SPHERE-PARAMETER

_b3

O. 5000

O. 5200

O. 5400

O. 5600

O. 5800

O. 6000

O. 6200

O. 6400

O, 0600

O, 6800

O,

0.0000

0_0001

0.0001

0,0003

0.0004

0.0007

O. OOit

0,0016

0,0023

0.0032

0.10

0.0000

0.0001

0.000t

0.0003

0.0004

0.0007

0,0011

0,0016

0,0023

0.0032

0.20

0.0000

0.0001

0.0001

0.0002

0.0004

0.0007

O. OOtO

0.0016

0.0022

0,0031

0.30

0,0000

O. O00t

0,0001

0.0002

0.0004

0.0006

0.0010

0.0015

0.0021

0.0029

0.40

0.0000

0.0001

O. O00i

0,0002

0.0004

0.0006

0.0009

0.0014

0.0020

0.0027

0.50

0,0000

0.0001

O. O00i

0,0002

0.0003

0,0005

0.0008

0.0012

0.0018

0,0025

0,60

0.0000

0.0000

0.0001

O. 0002

0.0003

O. 0005

0.0007

0.0011

0.0016

O, 0022

0.70

0.0000

0.0000

0.0001

0.0002

0.0003

0.0004

0.0006

0.0010

0.0014

0,0019

0.80

0.0000

0.0000

0.0001

0.0001

0.0002

0.0003

0.0005

0.0008

0.0011

0.0016

0.90

0.0o00

0.0000

0.0001

0.0001

0.0002

0.0003

0.0004

0.0006

0.0009

0.0013

1,00

0,0000

0,0000

0.0000

0,0001

0,0001

0.0002

0.0003

0.0005

0.0007

0.0010

0.7000

0.7200

0.7400

0.7600

0.7800

0.8000

0.8200

0.8400

0.8600

0,8800

O. 0043

O. 0557

O. 0074

O. 0093

O. 0116

O. 0t41

O. 0170

O. 0202

O.

0237

O, 0276

O. 10

O. 0043

O, 0057

O, 0073

O. 0092

O. 0114

O. 0140

O. 0168

O, 0200

O, 0235

O. 0273

0.20

0.0042

0,0055

0.0071

0.0089

0.0111

0,0136

0o0163

0.0194

0.0228

0.0265

0.30

0,01_40

0.0092

0.0067

0.0085

0.0106

0.0129

0.0155

0.0185

0.0217

0.0252

0.40

0.0(137

0.0049

0.0063

0.0079

0.0098

0,0120

0.0145

0.0172

0.0202

0.0234

O. 50

0,0(134

0.0044

0.

O. 0072

0.0089

O, 0109

O. 0132

0.0i56

0.0184

O. 0213

0,60

0.0030

0,0039

0,0051

0.0064

0,0079

0,0097

0.0117

0.0130

0.0163

0.0189

0.70

0,0o26

0.0034

0.0044

0.0055

0.0068

0.0084

0.0101

0.0120

0.014t

0.0163

O, 80

O. 0021

O. 0028

O. 0036

0.0046

O. 0057

O. 0070

O. 0084

O. 0100

O. 0117

O. 0136

O, 90

O, 0017

O. 0023

O. 0029

0.0037

O. 0046

O. 0056

O. 0068

O. 0080

O. 0094

O. 0110

1.00

0.0013

0.00t7

0.0022

0.0028

0.0035

0.0043

0,0052

0.0061

0.0072

0,0084

0.9000

0.0200

0.9400

0.9600

0.9800

1,0000

1,0200

1,0400

1.0600

1.0800

0.0317

0.0361

0.0408

0.0458

0,0510

0.0565

0.0622

0.0680

0.074t

0.0802

i0

0.0314

0.0358

0.0404

0.0454

0.0505

0.0559

0.0616

0.0674

0.0733

0,0795

O. 20

0.0305

0.0347

0.0392

0,0440

0.0491

0.0543

0.0598

0.0654

0.0712

0.0771

0.30

0.0290

0.0330

0.0373

0.0419

0.0467

0.0516

0,0568

0.0622

0.0677

0.0734

0.40

0.0270

0.0307

0.0347

0.0390

0,0434

0.0481

0.0529

0.0579

0.0630

0.0683

O. 50

O, 0246

O. 0280

O. 0316

O, 0355

_0395

_0438

O. 0481

O. 0527

O. 0574

O, 0622

0.60

0.0218

0,0240

0.0281

0.0315

0.0351

0.0388

0.0427

0.0468

0.0509

0.0552

0,70

0.018_

0.0214

0.0242

0.0272

0.0303

0.0335

0.0369

0.0404

0.0440

O. 0477

O, 80

O. 0157

O. 0179

O, 0202

O. 0227

O. 0253

O, 0280

O, 0308

O. 0337

O. 0367

O, 0398

0.90

0,0126

0.0t44

0.0163

0.0i82

0.0203

0.0225

0.0248

0.027t

0.0295

0.0320

1.00

0,0096

0.0110

0.0124

0.0139

0.0155

0.0172

0.0189

0.0207

0.0226

0.0244

1.

i,

1.

t.

1.

t.

O.

O. 0.

1900

2200

1.

2400

1,2600

1.

O.

O. 0866

O. 0930

O, 0995

0.1061

0,1128

0.1195

O. 1262

O.

1329

0.1_96

O, 1463

0.10

0.0857

0.0921

0.0985

0.1051

0.1117

0.1183

0.1249

0.1316

0.1382

0.

O. 20

O. 0832

O, 0894

O. 0967

O, 1020

O. 1084

O, 1149

O. 1214

O. 1278

O. 1343

O. 1407

0.30

0.0792

0.0851

0.0910

0.097t

0.1032

0,1093

0.1155

0.1217

0.1279

0,1340

O. 40

O, 0737

O, 0792

O. 0848

O. 0904

O, 0961

O, 1019

O. 1076

O. 1134

O, 1192

O, 1250

0.50

0.0671

0.0721

0.0772

0.0823

0.0875

0.0928

0.0981

0.1034

O. 1087

0.1140

0.60

0.0596

0.0640

0.0685

0.0731

0.0778

0.0824

0.0871

0.0919

0.0966

0.10t4

0.70

0.0514

0,0553

0.0592

0,0632

0.06?2

0,0712

0.0753

0.0794

0.0835

0,0876

0.80

0.

0.

0.

0,

0.

0562

0.

0.

0.

0.

O, 0733

0.90

0.0345

0.0371

0,0398

0.0424

_

0451

0.0479

0.0506

0.0534

0.0562

0,0590

1. O0

O. 0264

O. 0284

O. 0304

O. 0324

O. 0345

O. 0366

O. 0387

O. 0408

O. 0429

0.0451

0430

1200

0462

1.

0057

1400

0495

1600

0528

1800

2000

0596

0630

0664

0699

2800

t449

1,

293

Section

D

July

1972

1,

Page

TABLE

6.0-5.

(CONTINUED)

294

1. 3000

1. 3200

1. 3400

1. 3600

1. 3800

1. 4000

t. 4200

1. 4400

1. 4800

t. 4800

O.

O. 1529

O. t595

O. t680

O. 1724

O, 1787

O. 1850

O. 1911

O. 1971

O. 2030

O. 2088

0. 10

0. 1514

0. 1580

0. t644

0. 1708

0. t771

0. 1833

0. t894

0. 1953

0. 2012

0. 2069

0. 20

0. 147t

0. 1535

0. t598

0. 1660

0. 1722

0. 1783

0. t842

0. 190t

0. t958

0. 20t4

O, 30

0. 1402

0. 1463

0. t523

0. 1583

0. 1642

0. 1700

0. 1788

0. 1815

0. 1870

0. 1925

O. 40

O. t308

O. 1365

O. t422

O. t478

O. 1534

O. 1589

O. 1644

O. 1897

O. 1750

O. t802

O. 80

O. 1193

O. 1245

O. 1298

O. t350

O. 1401

O. t452

O. 1503

O. 1553

O. 1602

0. 1650

O• 80 O. 70

O. 1061 O. 0918

O• 1108 O, 0959

O. 1155 O. 1000

O. t202 O. 1041

O. 1248 O. 1081

O. 1294 O. 1122

O. 1340 O. 1162

O. 1385 O. 1202

O. 1430 O. 1241

O. 1474 O. 1280

0. 80

0. 0768

0. 0803

0. 0837

0• 0872

0• 0906

0. 0940

0. 0974

0. 1008

0. t042

0. 1075

0. 90

0. 0618

0. 0646

0. 0674

0. 0702

0. 0729

0. 0757

0. 0785

0. 08t2

0. 0839

0. 0866

I.O0

O. 0472

O. 0492

O. 0515

O. 0536

O. 0558

O.0579

O. 0600

O. 0621

O. 0642

O. 0663

1. 5000 0.2144

1. 5200 0.2199

1. 5400 0. 2252

1.5600 0.2304

1.5800 0.2354

1, 6000 0.2403

1. 6200 0.2450

1. 8400 0.2496

1. 6800 0.2539

t. 6800 0.2582

0. 10

0.2t25

0. 2180

0. 2233

0. 2285

0. 2335

0.2383

0.2430

0.2476

0. 2520

0. 2562

0.20

0. 2070

0. 2123

0. 2176

0. 2227

0. 2276

0. 2325

0.2371

0.2417

0.2461

0. 2503

0. 30

0. 1978

0. 2030

0.2082

0. 2132

0. 2180

0.2228

0. 2274

0. 23t8

0. 2362

0.2404

_40

0. 1853

0. 1903

0.1952

0,2000

0. 2047

0. 2093

O. 2138

0, 2182

0. 2225

0. 2266

0. 50

0. 1698

0. 1745

0. 1792

0. 1837

0. 1882

0. 1925

0. 1968

0. 2010

0. 2051

0.2091

0.80

0. t518

0. t561

0.1604

0.1646

0.1687

0.1727

0. J767

O. 1806

0.1845

O. t882

0. 70

0. 1319

0.1357

0. 1395

0. 1432

0. 1489

0. 1505

0. 1541

0. t577

0. 1612

0. 1646

0.80

O. t108

0. 1t40

0. tt73

O. 1205

O. 1237

0. 1268

0. 1299

0. 132_

0. 1380

0. 1390

0.90

0. 0893

0. 0920

0. 0946

0. 0972

0. 0998

0, 1024

0, 1049

0, 1074

0. 1099

0. 1124

1.00

0. 0683

0. 0704

0. 0724

0. 0744

0. 0764

0. 0784

0. 0803

0. 0823

0. 0842

0. 0861

•

1. 7000

I. 7200

t. 7400

t. 7600

1. 7800

1. 8000

1. 8200

1. 8400

1. 8600

1. 8800

0.

0. 2623

0.2662

0. 2699

0.2735

0. 2770

0. 2803

0. 2835

0. 2865

0. 2893

0. 2921

0. 10

0. 2603

0. 2642

0. 2680

0.2716

0. 275t

0.2784

0. 2816

0. 2846

0. 2875

0. 2903

0.20

0.2544

0. 2583

0. 2621

0. 2658

0.2693

0.2727

0.2759

0.2790

0. 2820

0. 2849

0. 30

0.2445

0. 2485

0. 2523

0. 2560

0. 2596

0.2630

0.2663

0.2695

0. 2726

0.2756

0.40 0. 50

0.2307 0. 2131

0,2346 0. 2J69

0.2384 0. 2206

0.2421 0. 2243

0.2457 0. 2279

0.2492 0. 2313

0.2526 0. 2347

0.2559 0. 2380

O. 2591 0.2412

0.262t 0. 2443

O. 60

O. 1920

0. t956

O. t99t

0.2026

O. 2061

O. 2094

O. 2127

O. 2t59

O. 2190

O. 2221

O. 70

O. 1680

O. 1713

O. 1746

O. 1778

O. 1809

O. 1840

O. 1871

O. 1901

O. 1931

O. 1960

O. 80

O. 1419

O. 1448

O. 1477

O. 1505

O. 1533

O. 1561

O. 1588

O. 1615

O. 1641

O. 1667

0.90

O. 1148

O. 1172

O. 1196

O. 1220

O. 1243

O. 1266

O. 1289

O. 1311

O. 1333

O. 1355

1.00

0.0880

0.0898

0.0916

0.0935

0.0953

0.0971

0.0988

0.1006

0.1023

0. i040

1.9000

I. 9200

1.9400

1.9600

1. 9800

_000

_0200

_0400

2.0600

2. 0800

0.

0. 2947

0. 2971

0.2995

0. 3017

0. 3038

0. 3058

0. 3077

0. 3094

0. 3t11

0.3t27

0.10 0.20

0.2929 0.2876

0.2954 0.2902

0.2978 0.2927

0.3001 0. 2950

0.3022 0. 2973

0.3042 0. 2994

0.3061 0.3015

0.3079 0.3034

0.3097 0.3052

0.3113 0.3070

0,30

0.2784

0. 2811

0. 2838

0. 2863

0. 2887

0. 2910

0.2932

0.2953

0. 2973

0. 2992

0.40 0.50

0. 2651 0,2473

0.2679 0.2503

0. 2707 0. 2531

0.2734 0. 2559

0,2759 0. 2586

0.2784 0.2812

0. 2808 0.2638

0. 283t 0.2662

0. 2853 0. 2686

0. 2875 0.2709

0. 80

0. 2251

0. 2280

0. 2309

0. 2337

0. 2364

0. 2391

0. 24t7

0.2443

0.2468

0. 2492

0. 70

0. 1988

0.2016

0.2044

O. 207t

0. 2097

0. 2123

0. 2149

O. 2t74

0. 2198

0. 2222

O. 80

O. t693

0. t7t8

O. 1743

O. 1767

O. 1791

O. 18t5

O. 1839

O. 1862

O. 1884

O. 1907

O. 90

O. 1377

O. 1398

O. 1419

O. 1440

O. 1460

O. 1481

O. 1501

O. 1520

O. 1540

O. 1559

t.00

O. 1056

0. t073

0.1089

O. 1t06

O. 1122

O. 1137

O. 1153

O. i168

0.1184

O. t199

OF POORQu JTY

Section

D

July

1972

1,

Page TABLE

6.0-5.

(CONTINUED}

2.1000

2.1200

2.1400

2,1600

2.1800

2.,2000

2.2200

2.2400

2.

O.

0.3141

0.3t55

0.3168

0.3180

0.3t92

0.3202

0.3212

0.3222

0.3230

0.3239

0.10

0.3128

0.3142

0.3156

0.3168

0.3180

0.3191

0.3202

0.3211

0.3221

0.3229

0.20

0.3086

0.3102

0.3116

0.3130

0.3143

0.3156

0.3168

0.3179

0.3189

0.3199

0.30

0.3011

9.3028

0.3045

0.3061

0.3076

O. 3090

0.3104

0.3117

0.3129

0.3141

0.40

O. 2895

0.2915

0.2934

O. 2952

0.2969

O. 2986

0.3002

0.3018

0.3033

0.3047

O. 50

O. 2732

O. 2753

O. 2775

O. 2795

O. 2815

O. 2834

O. 2852

O. 2870

O. 2888

O. 2904

O. 60

O. 2515

O. 2539

O. 2561

O. 2583

O. 2604

O. 2625

O. 2645

O. 2665

O. 2685

0.2703

0.70

0,

0.

0,

0,

2314

O. 2335

0.2357

O. 2378

0.2399

0.

O. 2438

0.80

0.1929

0.1950

0.1972

0.

1993

0.

0.

0.

0.2074

0.2093

0.2112

O. 90

O.

O.

O.

6.

1634

O. 1652

O. 1670

O. 1687

O. 1704

O. 1722

O. 1738

1.00

0.1214

6,1228

0,1243

0.1257

0.1271

0,1285

0.1299

0.1313

0.1326

0.1340

2.

2.

2.

2.

2.

2246

1578

3000

2269

1597

1615

3400

2014

2034

2419

2800

3600

2. 3800

_

4200

2. 4400

2. 4600

2.

O. 3246

O. 3253

O. 3260

O. 3266

O. 327t

O. 3276

O. 3281

O. 3286

O.

O. 3294

0.10

0.3237

0.3244

0.3251

0.3258

0.3264

0.3269

0.3274

0.3279

0.3284

0.3288

0.20

0,3208

0.3216

0.3225

0.3232

0.3239

0.3246

0.3252

0.3258

0.3263

0.3269

0.30

O. 3152

0.3163

0,3173

0.3162

0.3191

0.320o

0.3208

O. 3215

0.3223

0.3229

0.40

0.3060

o.

3073

0.3086

0.3098

0.3109

0.3120

0.3131

0.3141

0.3150

0.3159

0.50

0.2920

0.

2936

O. 2951

O. 2966

0.

O. 2994

0.3007

0.30t9

0.3032

0.3043

0.60

O. 2722

O. 2739

0.2757

0.

0.2790

O. 2806

O. 2822

O. 2837

O.

O. 2866

0.70

O. 2458

O.

2477

O.

2495

O. 2514

O. 2531

O. 2549

O. 2566

O. 2583

O. 2599

0.80

O.

O,

2150

O. 2168

O. 2186

0.2204

0.2221

O. 2238

0.2255

O. 2272

0.2288

0.90

0.1755

0.1772

0.1788

0.1804

0.1820

0.1835

0.1851

0.1866

0.1881

0.1896

1.00

0.1353

0.1366

I).1379

0.1392

0.1404

0.1417

0,1429

0.1441

0.1453

0.1465

2774

2980

4000

2054

2.

O.

2131

3200

2292

2600

3290

2852

4800

0.2615

2. 5000

2.5200

2.5400

2.5600

2.5800

2.6000

2.6200

2.6400

2.6600

2.6800

O.

0.3297

0.3300

0.3303

0.3306

0.3309

0.3311

0.3313

0.3315

0.3316

0.3318

0. I0

0.3292

0.3295

0.3298

0.3301

0.3304

O. 3307

0.3309

0.3311

0.3313

0.3315

O. 20

O. 3273

O. 3279

O. 3282

O. 3286

O. 3289

O. 3293

O. 3296

O. 3299

O. 3301

O. 3304

0,30

0.3236

0.3242

0.3248

0.3253

0.3258

0.3263

0.3267

0.3272

0.3276

0.3279

0.40

O. 3168

0.3176

0.3184

0.3192

0.3199

0.3206

0.3212

0,3219

0.3225

0.3230

0.50

0.3055

0.3066

0.3076

0.3087

0.3(}96

0.3106

0,3115

O. 3124

0.3132

0.3141

0,60

O. 2880

0,2894

O. 2907

0,2920

0.2932

0.2944

_

O. 2968

O. 2979

O, 2990

0.70

O. 2631

O. 2646

O. 2662

O.

O. 2691

O. 2705

O. 2719

0.

0.2746

0.2759

O. 80

O. 2304

O. 2320

O. 2336

0.2351

0.2366

0. 2381

O. 2396

O. 2410

0. 2424

O. 2438

O. 90

0.1911

0.1925

0.1940

0.1954

0.1908

0.1981

0.1995

O. 2008

O. 2022

O. 2035

1.00

0.1477

0.1488

0.

0.1511

0.1522

0.1533

0.1544

0.1555

0.1566

O. 1576

2.7000

2.7200

2.7400

2.7600

2.7800

2.8000

2.8200

2.9400

2.

2.

O.

O. 3319

O. 3321

O, 3322

O. 3323

O. 3324

O. 3325

O. 3326

O. 3327

O. 3327

O. 3328

0.10

O. 3317

0.3318

0.3320

0.3321

0.3322

0.3323

0.3324

0.3325

O. 3326

0.3326

0.20

0.3306

0.3308

0.3310

0.3312

0.3314

0.3315

0.3317

0.3318

0.3320

0.3321

O. 30

O. 3283

0.3286

O. 3289

O. 3292

0.3295

O. 3398

O. 3300

O. 3302

O. 3304

0.3306

0.40

O. 3236

O. 3241

O. 3246

O. 3250

O. 3255

O. 3259

O. 3263

O. 3267

O. 3270

O. 3274

O. 50

O. 3148

O. 3156

O. 3163

O. 3170

O. 3177

O. 3184

O. 3190

O. 3196

O. 3202

O. 3207

0.60

0.3000

0.3011

0o302!

0.3030

0.3040

0.3049

0,3058

0.3066

0.3075

0.3083

0.70

0.2772

O. 2784

O. 2797

O. 2809

O. 2820

O. 2832

O. 2843

O. 2854

O. 2805

O. 2876

O. 80

O. 2452

O. 2466

O. 2479

O. 2492

O. 2505

O. 2518

0.2530

0.2543

O. 2555

O. 2567

O. 90

O. 2048

O. 2061

O. 2073

O. 2086

O. 2098

O. 2110

O. 2122

O. 2134

O. 2146

O. 2157

1.

D. 1587

0.

O. 1507

O. t617

0.

O. 1637

O. 1647

O. 1657

O. 1666

O. 1676

O0

1597

t500

2676

1627

2956

2733

8600

8800

295

Section July

1,

Page TABLE

6.0-5.

(CONTINUE

D)

2.9000

2.9200

2.9400

_9600

_ 9800

3.0000

3.0200

3.0400

3.0800

3.0800

O.

O. 3328

O. 332_

O. 3329

O. 3330

O. 3330

O. 3331

O. 3331

O. 3331

O. 3331

O. 3332

0.10

0.3327

0.3328

0.3328

0. 3329

O. 3329

0.3330

O. 3330

O. 3330

O. 3331

O. 3331

0.20 0.30

O. 3322 0.3308

O. 3323 0.3310

O, 3324 0.3312

O. 3324 0.3313

O. 3325 0.3315

O. 3326 0.3316

O. 3327 0.3317

O. 3327 0.3318

O. 3328 0.3319

O. 3328 0.3320

0.40

0.3277

0.3280

0.3283

O. 3286

0.3259

0.3291

0.3293

0.329_

0.3298

0.3300

0. 50

O. 32t2

0. 3218

0. 3223

0. 3227

0. 3232

0. 3238

0. 3240

0. 3244

0. 3248

0. 3252

O. 60

O. 3091

O. 3098

O. 3106

0.3113

O. 3120

O. 3127

O. 3t34

O. 3140

O. 3t46

O. 3152

O. 70

O. 2886

O. 2896

0.2906

O. 2916

O. 2925

O. 2934

O. 2944

O. 2952

O. 296i

O. 2970

O. 80

O. 2578

O. 2590

0.2602

0.2613

0.2624

0.2635

0.2645

0.2656

0.2666

0.2677

0.90 1.00

O. 2169 0.1685

0.2180 0.1694

O. 219t 0.1703

O. 2203 0.17t3

O. 22t3 0.1722

O. 2224 0.1730

0.2235 0.1739

O. 2245 0.1745

0.2256 0.1756

O. 2268 0.1765

3.0000

3. 1000

3. 2000

3. 3000

3.4000

3. 5000

3. 6000

3. 7000

3. 8000

3. 9000

O.

0.3331

0.3332

0,3332

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

O. 10

O. 3330

O. 3331

O. 3332

O. 3333

O. 3333

O. 3333

O. 3333

O. 3333

O. 3333

O. 3333

0._0

O. 3326

O. 3329

0.3330

0. 3332

O. 3332

O. 3333

O. 3333

O. 3333

O. 3333

O. 3333

0.30

0.3316

0.3321

0.3325

0.3328

0.3330

0.3331

0.3332

0.3332

0.3333

0.3333

0.40

O. 3291

O. 3302

O. 3310

O. 33t6

O. 3321

O. 3324

O. 3327

O. 3328

0. 3330

O. 3331

O. 50

O. 3236

O. 3256

O. 3271

O. 3284

O. 3294

O. 3303

O. 3309

O. 3314

O. 3319

O. 3322

O. 80

O. 3127

O. 3t58

O. 3185

O. 3208

O. 3228

O. 3244

O. 3259

O. 3271

O. 3281

O. 3290

O. 70 O. 80

O. 2934 0.2635

O. 2978 O. 2687

O. 30t7 0.2735

O. 3052 0.2779

O. 3083 O. 2820

O. 31t2 O. 2859

O. 3137 O. 2894

O. 3159 0.2927

O. 3179 0.2957

O. 3197 O. 2935

0.90

O. 2224

O. 2276

0.2326

O. 2372

O. 2416

O. 2457

0.2497

0.2634

O. 2569

0.2603

1.00

0. t730

0.1773

0.18t4

0.1853

0.1890

0.1925

0.1959

0.1991

O. 2022

O. 2051

4. 0000

4. 1000

4.2000

_3000

4.4000

4. 5000

4. 6000

4. 7000

4. 8000

_9000

O.

0.3333

0.3333

0.3333

O. 3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.10

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.20

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.30

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.40

O. 3331

0. 3332

0.3 332

O. 3333

O. 3333

O. 3333

O. 3333

O. 3333

O. 3333

0. 3333

O. 50 O. 60

O. 3325 O. 3297

O. 3327 O. 3303

O. 3328 O. 3308

O. 3329 O. 3313

O. 3330 O. 3316

O. 3331 O. 3319

O. 3332 O. 3332

O. 3332 O. 3324

O. 3332 O. 3326

O. 3333 O. 3327

O. 70

O. 3213

O. 3227

O. 3239

O. 3251

O. 3261

O. 3270

O. 3277

O. 3284

O. 3290

O. 3296

O. 80

0.301t

0.3036

0.3058

O. 3079

O. 3098

0.31t6

O. 3t33

O. 3148

0.3162

O. 3175

0.90

0.2634

0.2665

0.2693

0.272t

0.2747

0.2771

0.2795

0.2818

0. 2839

0. 2859

1. O0

O. 2079

O. 2106

O. 2132

O. 2156

O. 2180

O. 2203

O. 2225

O. 2246

O. 2266

O. 2286

5.0000

5. 1000

5. 2000

5. 3000

_4000

5. 5000

5.6000

5.7000

5. 8000

5. 9000

O.

O. 3333

O. 3333

O. 3333

0.3333

O. 3333

O. 3333

_3333

O. 3333

O. 3333

O. 3333

0.10

0.3333

0.3333

0.3333

0.3333

0.3333

O. 3333

0.3333

0.3333

0.3333

0.3333

0.20

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.30

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

O. 40 0.50

0.3333 0.3333

0.3333 0.3333

0.3333 0.3333

0.3333 0.3333

0.3333 0.3333

0.3333 0.3333

0.3333 0.3333

0.3333 0.3333

0,3333 0.3333

0.3333 0.3333

O. 60

O. 3328

O. 3329

0. 3330

O. 3331

O. 3331

O. 3332

O. 3332

O. 3332

O. 3332

O. 3333

0.70 0.80

0.3300 0.3183

0.3305 0.3199

0.3303 0.3209

0.3312 0.3219

0.33t4 0.3228

0.33t7 0.3236

0.3319 0.3244

0.3321 0.3251

0.3323 0.3257

0,3324 0.3263

0.90

O. 2879

O. 2897

0.2915

O. 2932

O, 2949

O. 2964

O. 2979

O. 2993

O. 3007

O. 3020

1.00

0.2303

O. 2323

O. 2341

0.2368

O. 2374

O. 2390

0.2406

O. 2420

0.2435

0.2449

D 1972

296

J

Section

D

July

1972

1,

Page

6.0-5.

TABLE

(CONTINUED)

,f-

6.0000

6.1000

6.

6.3000

6.4000

6.5000

6.6000

6.7000

6.8000

6. 9000

0.3333

0.3333

0,3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.10

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.

0.3333

0.20

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.30

0.3333

0,3333

0.3333

0.3333

0.3333

0,3333

0.3333

0.3333

0.3333

0.3333

0.40

0.3333

0.3333

0,3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.50

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.60

0.3333

0.3333

0.3333

0.3333

0.3333

0.

0.

3333

0.3333

0.3333

0.3333

0.

0.

0.

0.

0.

0.

0.3330

0.

3330

0.

0.

0.

O.

70

3326

3327

2000

3328

3329

3329

3333

333i

3333

3331

3331

0.80

0,3269

0.3274

0.3279

0.3283

0.3288

0.3291

0.3295

0.3298

0,3301

0.3304

0.90

0.3033

0.3045

0.3056

0.3067

0.3077

0.3088

0.3097

0.3107

0.31t6

0.3124

0,

2462

0.2475

0.2488

0.

2501

0.

2513

0.

2525

0.2536

0.

2547

0.

2558

0.

2568

7.

0000

7.

3000

7.

4000

7.

5000

7.

7.

7000

7.

8000

7.

9000

1.00

7.2000

7.

O.

0.3333

0.3333

0.3333

0,3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.10

0.3333

0.3333

O.

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.20

0.3333

0.3333

0,3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.30

0.3

0.3333

0.3333

0.3333

0.3333

0.3333

0,3333

0.3333

0.3333

0.3333

0.40

O. 3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0,3333

0.50

0.3333

0.3333

O. 3333

0.3333

0.3333

0.3333

0.3333

0.3333

O. 3333

0.3333

0.60

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

O. 3333

0.3333

0.70

0.3332

0.3332

0.3332

0.3332

0.3333

0.3333

0,3333

0.3333

0.3333

0,3333

0.80

O. 3306

0.3309

0.3311

0.3313

0.3314

0.3316

0.3317

0.3319

0.3320

0.332t

O. 90

0.3i32

0,3140

0.3148

0.3155

0.3162

O. 3169

0.3175

0.3182

0,3188

0,3193

l.

O. 2578

O. 2588

O. 2598

O. 2607

O. 2616

O,

O. 2634

O.

0.2651

0.2659

8.

O0

333

iO00

i000

3333

2000

2625

6000

2643

8.0000

8.

8.3000

8.4000

8.5000

8.6000

8.7000

8.8000

8.

O.

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.10

0.3333

0.3333

O. 3333

0.3333

0.3333

0.3333

0,3333

0.3333

0.3333

0.3333

0.20

0.3333

0,3333

0,3333

0,3333

0,3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.30

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.40

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.50

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

O. 3333

O. 3333

9.3333

O. 3333

0.60

0.3333

0.3333

O. 3333

O. 3333

0.33:13

0.3333

0.3333

0.3333

0.3333

0.3333

0.70

0.3333

0.3333

0.3333

0.3333

0.3333

0,3333

0.3333

0.3333

0.3333

0.3333

O. 80

O. 3322

O. 3323

O. 3324

O. 3325

O. 3326

O. 3327

O. 3327

O. 3328

O. 3328

O. 3329

O. 90

0.3199

0.3204

0.3209

0.3214

0.3219

0.3223

O. 3227

0.3232

O. 3236

0.3240

1.00

O. 2667

0.2675

0.2682

0.2690

0.2697

0.2704

O. 2711

O. 2718

0.2724

0.2731

9. 9000

2000

8000

9000

9. 0000

9.1000

9.

9. 3000

9.4000

9. 5000

9.6000

9.7000

9.

O.

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

O. iO

0.3333

0.3333

0.3333

0.3333

0,3333

O. 3333

0.3333

0.3333

0.3333

O. 3333

0.20

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.3333

0.30

0.3333

0.3333

0.3333

0.3333

O. 3333

0.3333

0.3333

0,3333

0.3333

0.3333

0.40

0.3333

0.3333

0.3333

0.3333

0,3333

0.3333

0.3333

0.3333

0.3333

0.3333

O. 60

O. 3333

O. 3333

O. 3333

O. 3333

O. 3333

O. 3333

O. 3333

O. 3333

O. 3333

O. 3333

0.60

0.3333

0.3333

0.3333

0.3333

0.3333

O. 3333

0,3333

0.3333

O. 3333

0.3333

0.70

0.3333

0.3333

0.3333

0.3333

0.3333

O. 3333

0.3333

0.3333

O. 3333

0.3333

0.80

0.3329

0.3330

0.3330

0.3330

0.333i

0.3331

0.3331

0,3331

0,3332

0,3332

0.90

O. 3243

O. 3247

O. 3250

O. 3254

O. 3257

O. 3260

O. 3263

O. 3266

O. 3268

O. 327|

t.00

0.2737

0.2743

0.2749

0.2755

0.276t

0.2767

0.2773

0.2778

0,2783

0.2789

297

_ctton July

D 1,

page

0°

O, t0 0.10 O.N O.4O O.SO 0.60 o.70 o._) O, lO i.oO

10, 00o0

10. lOOO

O. 1833 O. 3835 O. 38M o. 1,181 0.8383 O. 3838 0.8,188 o, mill O. U$| O. 33'P3 O. I"/N

o. uu o. uu o. 3333 o. 3333 o. 3831 O, 8,538 O. 3338 O. 3305 o. 3331 o. 815'11 o. 2781

to. moo o. 33u o. 3333 o. 3333 o. 11,538 o. 3838 O. 3_38 O. 81138 O. 8881 o, _l| o. 335'8 o. 3804

TABLE

6.0-5.

to, 3oo0

to. 4oo0

O. 3333 O. 9338 O. 3333 O. 3038 O. M338 O. 3838 O. 1338 O. 3333 O, 2323 O. 38110 o. Moll

O. 3533 O. 3333 O. 3333 O. 3338 O. 3833 O, 3383 O. 3238 O, 1238 O. 1338 O. 3338 O. 1614

298

(C_)NCLUDED)

to. sooo O. 2838 O. 3838 O. 3833 O. 3333 O. 3338 O.JI3AB O, 3838 0. _ O. 32_12 O. 3384 o. 3811

10. eooo O. 0533 O. 3333 O. 33138 O. 3333 O. 31333 O.3833 O. 3838 O.33:12 O.3833 O. 3388 o, 1818

' 10. 7000 !+ O. 3338 O. 3383 O, 3333 O. 3333 O. 3333 O. 3338 O. Sa_ O. 3333 O. 3333 O. 3288 o. 118211

10, 80oo O. 3333 O. 3333 O. 3333 O. 3333 O. 3333 0,, 3+533 O. 3333 O, 3333 0_.3333 O. 3290 o. 11832

10ogooo ' O, a38$" O. 3338 O. 3303 O. 3333 0. 3383 O. ,5,533 O. 3333 o. 3333 O. 3333 O. 3893 op2837

1972

Section

D

July

1972

I,

Page

299

REFERENCES

0

Boley,

B. A. ; and Weiner,

John Wiley .

Newman, Machine

0

o

o

.

and Sons, M.

Newman,

M.

in Hollow

Circular 1960.

Forray,

M.

Inc.,

New

and Forray,

Design,

October

J. H. : Theory

Nov.

8,

November

1960.

Newman,

M.

in Hollow

Circular

December

1960.

Plates,

Plates,

and Forray,

Timoshenko,

Plates,

S. : Strength

and Problems.

Third

Stresses

in Circular

Rings.

1962.

and Newman,

Circular

Stresses.

1962.

M. : Thermal

and Forray,

in Hollow

York,

of Thermal

ed.,

M. : Bending Part

M. : Part

I.

Journal

Bending II.

III.

Due to Temperature

of the Aerospace

Stresses

Journal

M. : Bending Part

Stresses

Due to Temperature

of the

Stresses

Journal

of Materials. D. Van Nostrand

Aerospace

Sciences,

Due to Temperature

of the Aerospace

Part

Sciences,

II, Advanced Co.,

Inc.,

New

Sciences,

Theory York,

1956.

.

Thermo-Structural TR-60-517,

.

vol.

Thermo-Structural TR-60-517,

Vol.

Analysis I, August Analysis II, October

Manual.

Technical

Report

No.

WADD-

Technical

Report

No.

WADD-

1962. Manual. 1962.

Section

D

July 1, 1972 Page 300 REFERENCES .

Forray,

M. J. ; Newman,

Deflections

Panels

Technical

Report

Laboratory,

11o

on Beam

Supports°

Tlmoshenko,

McGraw-Hill

Book

Co.,

York,

16.

Y.

Publishers,

New

J.N.

Flight

Air

Base, sad

: Theory

of

Systems December

1962.

in Heated

Plates

1962.

S. : Theory York,

and Test

the Thickness.

Force

Ohio,

15,

and

Dynamics

Deflection

Feb.

New

York,

: Thermal

J.

of Plates

and

1959. of Elasticity.

McGraw-

1951.

Stresses.

A. : A Proof

Journal

Dharmarajan,

Tsui,

Co.,

Through

Stresses

McGraw-Hill

Book

Co.,

Inc.,

1957. A.

Dynamics,

Design,

Book

Inc.,

B.E.

Stresses. 15.

Force

M. : Stress

S. ; and Goodier,

Gatewood,

Morgan,

Air

Machine

1,

Division,

S° ; and Woinowsky-Krieger,

Tlmoshenko,

New 14.

Wright-Patterson

1, The Analysis

Part

Systems

Thermal

Gradients

ASD-TR-61-537

M. ; and Newman,

Hill 13.

No.

J. :

Part

with Temperature

Forray,

Shells. 12o

Plates.

Aeronautical

Command, 10.

M. ; and Kossar,

in Rectangular

Rectangular

(Continued)

of the Aerospace

S. : In-Plane Convair

Park,

Memo

in Shells Calif.,

s Analogy

Sciences,

July

Stress

Analysis.

Thermal

Division,

W. : Stresses Menlo

of Duhamel'

No.

SA-70-13,

of Revolution. t968.

for Thermal 1958. General July

Pacific

17, Coast

1970.

Section

D

July

1972

1,

Page

f_

REFERENCES 17.

18.

Baker,

E. H. ; Cappelli,

Verette,

R.M.

; Shed

Hetenyi,

M. : Beams

A.

19.

Dharmarajan, General

(Continued)

P. ; Kovalevsky,

Analysis

Manual.

on Elastic

Michigan Press, Ann Arbor,

L. ; Rish,

NASA CR-912,

Foundation.

Mich.,

S. N. : Thermal

Dynamics,

301

F.

L. ;and

April

1968.

The University

of

1946.

Stresses in Circular Cylindrical Shells.

Convair Division, Stress Analysis Memo

SA-70-

16, Aug. 28, 1970. 20.

Przemieniecki,

J. S. : Introduction to Structural Problems

Reactor Engineering. 21.

Chapter 8, Pergamon

Wylie, C. R. Jr. : Advanced Hill Book Co., Inc., New

22.

Press,

New

Engineering Mathematics.

York,

Engineering.

York,

1962.

McGraw-

1951.

Sokolnikoff, I. S. ;and Redheffer, R. M. : Mathematics and Modern

in Nuclear

McGraw-Hill

of Physics

Book Co., Inc., New

York,

1958. 23.

Newman, Machine

24.

25.

M. ; and Forray,

M. : Thermal

Stresses in Cylindrical Shells.

Design, June 6, 1963.

Huth, J. H. : Thermal

Stresses in Conical Shells. Journal of the

Aeronautical Sciences, September

19_3.

Nowacki,

Addison-Wesley

W. : Thermoelasticity.

Reading, Mass.,

1962.

Publishing

Co.,

Inc.,

Section D July I, 1972 Page 302

REFERENCES 26.

Kraus,

H. : Thin Elastic Shells. John Wiley

York, 27.

28.

29.

Switzky,

H. ; Forray,

Analysis

Manual.

Smith,

Dynamics,

Convair

Division,

R. H. ;and

butions

Arising

Orange,

from

Structures.

Fitzgerald,

Inc.,

New

and

Report

No.

Thermal

Load.

of

General

63-0044, Elastic

and Edge

July

31,

Stress

1963.

Distri-

Loads

in Several

Shell-

Shells

of Revolution

1961.

Stresses

in Thin

Distributions.

Systems

1962. in Shells

GD/A

T. W. ; Theoretical

Temperature Space

August

of Meridional

Discontinuities

A. :

M. ; Thermo-Structural

Discontinuities

Effects

NASA TR R-103,

E.

Engineering,

Report

No.

Douglas

for

SM-42009,

Aircraft

Co.,

Inc.,

1962.

Christensen, No.

R.

58-A-274,

York,

33.

of Multiple

Coupled

Johns,

Newman,

I, WADD-TR-60-517,

Including

Missiles

32.

Vol.

G. W. : Analysis

Axisymmetric

31.

M. ; and

Revolution

June

and Sons,

1967.

Type 30.

(Continued)

N. Y.,

M.:

Thermal

presented Nov.

M. : Thermal

Machine

Design,

Newman, Circular

at the Annual

30-

Forray,

Stresses

Dec.

M. ; and Forray,

12,

Meeting

Nose

Cones.

Paper

of the ASME,

New

5, 1958.

Buckling

March

in Missile

and

Postbuckling

of Circular

Bulkheads.

1964. M. : Axisymmetric

Plates

Subjected

to Thermal

of the Aerospace

Sciences,

September

Large

and Mechanical 1962.

Deflections Loads.

of Journal

Section

D

July

1972

1,

Page

REFERENCES 4.

35.

36.

37.

Friedrichs,

of the

vol.

1941,

63,

Book

P. ; and Co.,

Machine

Design,

June

Bijlaard,

Journal

Abir,

D. ; and

Shells

Under

of Thin Dynamics,

Vehicle

Under

Arbitrary

of the

Aerospace

Structures.

p.

Stability.

391.

of Heated

Rectangular

Sciences,

Plates.

Buckling

1963. Instability

Report

U

1960.

of Circular

Curves

Circular No. and

8,

of Axial

Cylindrical

Journal

of

1959.

Unpressurized

February

of a

Variation

Gradients.

of Design

Materials

of Heated

November

Temperature

Division,

5,

Circumferential

December

Metallic

Deflection

Dec.

S. V. : Thermal

and

Convair

1961,

R. H. : Elastic

G. : Development

Pressurized

MIL-HDBK-5A,

Design,

Gallagher,

Sciences, J.

Value

of Mathematics,

of Elastic

M. ; Post-Buckling

Circumferential

Schumacher,

New York,

Machine

Nardo,

Aerospace

M. : Theory

M. : Buckling

Forray,

P. ; and Shell

Journal

Boundary

7, 1962.

Plates. P.

J.

Inc.,

Newman,

M. ; and

J. : The Nonlinear

American

Gere,

M. ; and

the

41.

Plate.

Forray,

Newman,

J.

839-888.

McGraw-Hill

Stress.

40.

pp. S.

Cylindrical

39.

Buckled

Timoshenko,

Rectangular 38.

(Continued)

K. O. ; and Stoker,

Problem

303

for

the Stability

Cylinders.

AZS-27-275, Elements

for

May

General 8,

1959.

Aerospace

1966.

S GOVERNMENT

PRINTING

OFFICE:

1972

-

746-624/4751

REGION

NO,

4

SECTION E

SECTION El FATIGUE

TABLE

1_1 FATIGUE I.i

OF

CONTENTS

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

INTRODUC

TION

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

I I

I. I. 1 General Considerations for a Fatigue A_llysi8 ..... I. I.I. I Life Cycle Determination .... 0 . . ° ..... 1.1. i.2 Material Fatigue Data ............. ,.. i.I. 1.3 Fatiguc Analysis .......... , ........ I. i. 2 General 1.2

DE FINITIONS

1.2.2

Fatigue Testing Techniques .................. Presentation of Test Results .................

Mechanism

1.2.3.1 I. 2.3.2 FACTORS

•

•

•

8

on Fatigue ................

BASIC 1.2.1 1.2.3

1.3

Background

•

•

•

•

.

,

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

of Failure ......................

S-N Diagrams ..................... Goodman Diagrams .............

INFLUENCING

FATIGUE

STRENGTH

I 4 7

I1 12 16 19

0...

........

1.3. I Metallurgical Factors ...................... 1.3. i. I Surface Defects ....................

19 20 23 23 23

I. 3. I. 2 Subsurface and Core Defects, Inhomogeneity

1.3.2

1.3.3

1.3.1.3

and Anisotropy Heat Treatment

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

i. 3.1.4 I.3.1.5

Localized Overheating ............... Corrosion Fatigue ..................

].3.1.6

Fretting Corrosion

.....

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

1.3.1.7 Reworking ..................... ,. Processing Factors ................. , ...... 1.3.2.1 Hardness ........ ..... , ..........

24 28 29 29 32 32 33 33 34

1.3.2.2 I. 3.2.3 1.3.2.4

Forming ........................ Heat Treatment ............. , ....... Surface Finish ............... , .....

35

I. 3.2.5

Cladding, Plating, Chemlcll Conversion Coatings, and Anodizing ..............

37

35

1.3.2.6 Cold Working ...... . . . o ........... Environment Effects ....... . ...............

39 39

1.3.3.1

Irradiation .......................

41

1.3.3.2

Vacuum

41

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

El-iii

TABLE

OF

CONTENTS

(Continued)

Page 1.3.3.3

Meteoroid

1.3.3.4

Solar

1.3.3.5

1.4

Design

1.3.5

Welding

E fleets

1.3.6

Size

Shape

1.3.7

Speed

LOW

CYCLE

1.4..1

Below

Effects

and

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

Effects

45

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

FATIGUE

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

47

Range

Practical

In Creep

Range

1.4.2.1

Ductility

1.4.2.2

Procedure

....................... Problem

Versus for

Estimating

.........

53

HighFatigue

53

.......

54

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

Two-Slope

Thermal

Cycling

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

1.4.3.1

Idealized

Thermal-Cycle

1.4.3.2

Effect

of Strain

of Creep

Comparison

59

Partitioning

Fatigue

...........

Low

62 63

Model

.........

65

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

Fatigue

Temperature

60

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

66

of Thermal-Stress at

Fatigue

With

Constant

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

Summary

68

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

CUMULATIVE

FATIG1TE

1.5.1

Theory

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

1.5.2

Analysis l

Strength

Low-Cycle ........................

Method

of Data

50 51

Creep

Method

1.,t.3.4

...........

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

1.4.2.,t

;3.3

49

Solutions

1.4.2.3

Examph,

46 46

Mechanical

1.5.3

43

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

Creep

1.5.2. 1.5.2.2

42

of Testing

II.

1.5

42

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

I. 4.1.1

1.4.

42

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

Temperature, I. Basis

1.4.:3

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

Temperature

1.3.4

1.4.2

Damage

Irradiation

DAMAGF

70

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

71 71

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

72 74

P{,ak Counting Techniques ............. Statistical Methods of Random Load Analysis

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

Problem

(Paired

El -iv

Range

74 Count

Method)

....

77

TABLE

OF CONTENTS

(Concluded)

Page 1.6

MATERIAL 1.6.1 1.6.2

1.7 DESIGN REFERENCES

SELECTION

High Cycle Low Cycle GUIDES

TO RESIST

FATIGUE

..........

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

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

El-v

81 81 81 87 89

1..J

Section OF_ POOR

f

E1

FATIGUE.

1.1

IN TRODUC

1.1.1

General Before

design,

Considerations

detailing

the

analysis. there are

cvaluatect,

engineer

materials,

writing Figure As

analysis.

1972

Analysis.

involved

in a fatigue

of the

general

analysis

and.

considerations

that

Often when a new vehicle, structure, etc., many concepts, designs, and configurations must

provide

ready

test

requirements

assistance

in the

and

flow diagram figure, there

the

life

of information are two primary

necessary as input to a fatigue analysis test requirements for a given structural

is to be

selection

of

evaluation. for'a fatigue groups of

and/or determinacomponent.

One group of information is the necessary data on the given material. data include curvcs, graphs, etc., of laboratory tests to determine versus cycles to failure (S-N) diagrams, Modified Goodman diagrams

These stress

other

other the

a Fatigue

factors

of preliminary

are

I November

an overview

El-1 shows a general can be seen from this

information that tion of preliminary

and

for

many

to give

comprise a fatigue being considered, the

E1

TION.

it is desirable

and

QUALITY

factors

which

information, life

cycle

presented pressures,

would

which

or service

affect

is usually history

the

of the

in terms of stress-environment temperatures, and other

life

the

first

of the

component

information

structural

component.

versus environmental

Life

Cycle

life,

The important mission profile

Fig.

El-l).

spectra tural

The

are

l)csi_n ments,

or

VAA

[lights,

landings,

elements in the formation of life (or condition lists), and significant or anticipated

considerations

of a flight life and

This

is

is usually show In general,

and/or the preliminary Actually, the prefrom the life cycle

Determination.

design

primary

component

The

is required,

time curves which considerations.

with these two groups of information a fatigue analysis test requirements can be determined for the component. liminary test requirements can usually be determined information alone. 1.1.1.1

in question.

that

operational

life

in evaluating

effects

cycle data are the fatigue loadings and

service

of fatigue

design (see

loading on the

struc-

vehicle.

is generally

specified

by contractual

Military

_;pecifieations,

usually

pressurizations,

flight

hours,

etc.

or managerial in terms

of number

agreeof

Section

ORIGI_IAL OF POOR

/ L

" '__ ;_

LIFE

CYCLE

Irr R IFJ;S LEVkt. TEMP. PRESSURE ENVIRONMENT

VS V$ VS VS

1)age

QUALITY

DESIGN LIFE (NUMaER OF FLIGHTS, PRESSURIZATIONS. J MliStON PROFILE/CONDITION LIST (ACTIVITY VS TIME) "1 SIGNIFICANT FATIGUE LOADINGS (MOST IMPORTANTI I AERODYNAMICS (HEATING, PRESSURES, LOADS _ DYNAMIC LOADS)

HOURS.

REQUIREMENTS

1972

2

ETC.)

MATERIAL

TIME TIME TIME TOME

1,:1

1 November

TEST

DATA

$-N CURVES CRACK PROPAGATION RATES BIAXIAL STRESS DATA FLAW DATA. WELD DATA CREEP-RUPTURE. DUCTILITY TOUGHNESS

-ENV. &'TRESS

N (CYCLE|| TIME

FATIGUE

ANALYSIS

LIMITATIONS

:[,,/N MARGINS TEST

OF SAFETY

-4k------

REQUIREMENTS

FAIL-SAFE INSPECTIONS REPAIRS TURNAROUND

FOR COMPONENTS TEST LOAD SPECTRUM RESULTS NEEDED

l

FIGURE

CUMULATED DAMAGE REPLACEMENT SCHEDULE INSPECTION METHODS DESIGN ALLOWABLE STRESS SAFE LIFE MATERIAL SELECTION

E]-I.

PROCEDURE

FOR

TYPICAL

FATIGUE

} ANALYSIS

TIME

Section E I 1 November

1972

Page 3

F basis

Mission profiles, or condition for the selection of design stress

ponents to be tested, of the service life. craft.

These

description of the flight.

lists, are generally levels, the choice

the definition of the Figure E1-2 contains

mission

profile

and/or

due to fuel the fatigue

usage, loading

experimental tional loading

altitudes, spectra

fatigue test spectra, and evaluation a typical mission profile for an air-

condition

of the activity versus time (or Also, the flight configurations speeds, and are required.

used to provide a of structural com-

lists

must

contain

a detailed

distance) applicable to each portion and anticipated changes in weight any other parameters that could The spectra used for analytical

evaluation of fatigue should realistically history deseribed by the mission profile

represent and/or

affect and

the total operacondition lists.

For preliminary considerations, all of the detailed data on mission profiles may

not be available. It is then necessary to consider what are the most

significantconsiderations for the component

in question.

For example,

in

TIME CRUISE

CLIMB

| --

TAKEAT TAXI OFF WEIGHT

FIGURE

]

_.,

E1-2.

TYPICAL

MISSION

OR

FLIGHT

PROFILE

FOR

AIRCRAFT

Scction

E1

1 Novembcr Page

tankage, wings,

the most important the most important Now,

given

all the

factors may be pressure and factor may be only temperature. information

above,

1972

4

temperature;

it is required

for

to evaluate

the

stress versus time curves for a given component. Ordinarily, the aerodynamic data and dylmmie loads must be considered as input to aid in the evaluation of load and stress levels. In conceptual guess" data. The stress-versus-time perature

versus

mission El-3.

profile.

time,

and

other

An example

A

designs, curves

environmental

of a life

•

I p_._F TEST

however, these must, of course,

-

cycle

factors

present

for a component

C

D

. I-

- I-

may be "bestreflect temover

part

is given

75

HOURS

•

•320 ° F

of a

in Fig.

Ir '

TIME

FIGURE Of c,)ur_e, many random

tain

the actual type loads

El-3.

in Fig.

l. 1.1.'2

Material The

that

can

1:1-4

fatigue

be applit'd

for

ina

Many techniques later section.

an aircraft

flight

Fatigue

Data.

strength

of a material

repeatedly

CURVE

lift, cycle history for a given compom'nt may conwhich must be condensed into a form which can

be used in fatigue life evaluation. this, and thcywillbc discussed shown

LIFI':-IIISTORY

to the

is simplified

may

matcrial

are For

available example, to a more.

be defined without

as the

eausin_

to accomplish the spectrum usable,

form.

maximum

failure

in less

stress

Section

E1

1 November Page

1972

5

f

STORM

ieg

TAKE-OFF

LANDING

a.

Actual Aircraft Spectrum

2

2o0,0o0

2 ooo

l_l

1

i oo

1

t_ }-

I -I

,

o

IIS

b. Simplified Spectrum FIGURE than

a certain

E1-4. definite

SIMPLIFICATION number

of cycles.

imum stress which can be applied number of cycles without causing repeatedly presented

applied stress in the graphical

discussed

indc'tail

of the

resented.

factors

Some

The

endurance

fatigue

1.2.3. life,

influencing

SPECTRUM strength

repeatedly to a material failure, The relationship

to the number of cycles form known as the S-N

in Paragraph

a re'presentation

OF A FLIGHT

The

is that

max-

for an indefinite of the magnitude

to failure is conventionally curve. The S-N curves

shape

of theS-N

will

vary

according

the

shape

of the

to the S-N

curve

curve,

are

which

conditions are

of

rep-

as follows:

is

Section

E1

1 November Page

1.

Type of load a.

Tension

b.

Compression

c.

Torsion

d.

Combined Loads

2.

Relationship of maximum to minimum loads

3.

Manner of load application a.

Axial

b.

Flexural

c.

Torsional

4.

Rate of load application

5.

Frequeney of repetition of loads

6.

Temperature of the material under load

7.

Environment

Q

a.

Corrosive

b.

Abrasive

c.

Inert

Material condition a.

Prior heat treatment

b.

Prior cold work

Design of part or structure Fabrication techniques

6

1972

Section E1 1 November 1972 Page 7 Thus, it can be seen that the mission profile or load spectra can dictate what type of S-N data are required for a fatigue analysis. Other material information which may be necessary for input into a fatigue analysis are low-cycle fatigue data, fracture toughness, crack propagation rates, biaxial stress field data, welding, flaw size effects, and creep rupture data. 1.1.1.3

Fatigue Analysis.

Referring again to Fig. El-l, with the input as discussed above, a fatigue analysis can be performed on the structural component or element in question. inspection

Other information which techniques, determination

reliability

and

analysis

design phase, factor-of-safety

is usually

concerning

on structural for The

paragraphs;but, a structural will specify and/or

test

reuse details

the

life

of the data,

and

material

properties;

thus, levels

analysis is the results obtained

Also,

the

on subsequent of the

stress

vehicle

first

determination, only preliminary

fatigue of the

elements.

and data best

and

interpretation, from fatigue

is assessed

is the

for

are

estimates

requirements

in terms tests condamage

and

missions.

fatigue

analysis

will

be discussed

in the

following

requirements. the

reevaluation in light other cnvironm('ntal

assurance

cycle

The

_cncrally, the fatigue analysis will determine the safe life for compon¢'nt, will evaluate accumulated damage during service life, inspection requirements, and will determine allowable stresses

Therefore, and and

in two phases.

From this design phase, design fatigue tests are determined.

The second phase of actual flight-measured qualified

performed

which is the structural sizing, material study phase. Usually, in this phase

must be made. for developmental

ducted

on and

safety.

Fatigue

available

may be required includes limitations of a design (fail-safe or safe-life),

that

exist

analysis

of fatigue

of newly data can

in the

acquired provide

evaluation

life

is a continuing

process

of study

data. Thc acquisition of new loading additional insight into the levels of

of the

service

life

of structural

on flight vehicles. Any significant variation between service-recorded histories (and related environmental conditions) and the spectra used mine fatigue life would require a reevaluation of anticipated structural

components loading to deterlife.

• I_°

I

_

_..,

OF POOR

-

_1

Section E1 1 November

_

Q._i;,_Lii i

Page 1.1.2

General A rapid

spread

use

Background growth

in many

types

tt'nsile

strength

considered was soon

with

of machines

and

it'on

and

steel

structures

of the

materials

involved,

the failures that most

and,

coming

into wide-

occurred

During this time, engineers at calculated nominal stresses

ductile, discovered

B

on Fatigue.

of metallurgy,

dle of the ]9thcentury. failures that occurred

1972

during

the

mid-

were confronted with considerably below the

although

generally exhibited of the brittle fractures

the

materials

were

little or no ductility. It developed only after the

structures had been subjected to many cycles of loading. Therefore, it came to bc supposed that the metal degenerated and became fatigued under the action of cyclical stresses and that its ductile behavior turned brittle, tlowt'ver, later experiments showed that no degeneration or fatigue of the metal o(:currt,d as a result

of cyclical

H¢'nee

the

stressing

t(wm

and,

"fatigue"

therefore,

is still

widely

the used

idea

of metal

although

it has

fatigue quite

is false.

a different

nleatlJllg.

Generally a part

under

repeated,

Fati;_uc when a fatigue failure occurs. fatigue effects

crack cause

three

fatigue

cyclic,

failures

can be defined

or fluctuating

may

be simple

failure starts A compound

The

and

initial

from a single fatigue failure

propagates,

damage and

occurs the

final

as a progressive

failure

of

loads.

or compound.

originates from two or more total failure. The sequence

parts:

initiates

speaking,

Simple

failures

result

crack and propagates until ultimate results when the origin of the locations in which

and propagates; the joint failure occurs consists of

in a submicroscopic rupture

takes

scale,

place.

The

the rate

crack

at which

the crack propagates varies considerably, depending upon the intensityof stress and other related factors. Nevertheless, the rate is always much lower than that observed for low-tcmpcrature brittlefractures in steel. It has proved during its life;

part can

cause

removed) the

catastrophic

failures.

do not lead

fatigue The

stress, fatigue

difficult, however, to detect progressive failures in the hence, fatigt, e _.ailure8 can occur with little warning and

damage criterion

tensile is also

to a recovery

Also, from

periods the

of rest effects

(the

fatigue

of stress;

stress

in other

words,

is cumulative. for

fatigue

stress, and eliminated.

plastic

failure strain.

is the

simultaneous

If any

one

action

of these

of cyclic

is eliminated,

Section

E1

1 November Page Rarely constant from

is a mechanical

loads

throughout

vibration,

repeated a few.

its

variations

temperature

component entire

or structural

service

history.

in atmospheric

changes,

and

elt'ment Cyclic

b,:ust pattern,

repetition

9/10

sub ieeted

loading

variable

of design

load,

1972

can

wind

to result

loadings,

to mention

only

Many fatigue failures that occur in service are only minor, but others, such as those which result in the loss of a wing, propeller, or wheel, constitute a serious threat to life or property. Such failures have become more prevalent in recent

years 1.

The

2.

More stress.

3.

The

Good ance

because

under

of the

continuing

trend

refined

use

static repeated

resistance.

fatigue strength

design

of materials design

higher

strength/weight

techniques

and the

of ultra-high

does

loading,

factors:

toward

static

even higher than, the yield in failure after a relatively that the limitin;_

following

not necessarily and

point short

the

choice

the

use

of higher

working

strength.

result

in satisfactory

of a design

stress

perform-

close

to,

or

of the material would almost inevitably result period of service. There is every possibility

problem will become more criterion in many future

Thcr(-fore,

static

ratios.

likelihood

acute designs

of a fatigue

and that, consequently, the will be adequate fatigue failure

should

be an early

des ign consideration. A major problem in designing to prevent fatigue failure lies in the identification of all the factors that affect the life of a part. Even with the knowledge

of which

exact determination sections to enable

problems

to consider,

of fatigue the engineer

one

is still

short

of the

goal:

life. Techniques are furnished in the to develop fatigue data for a particular

the

following part.

All the basic definitions used in fatigue testing and failure are discussed in Paragraph 1.2. The many types of effects which influence fatigue strength are prt'sented in Paragraph 1.3. A separate section (Paragraph 1.4) is de_oted

to low-cycle

fatigue

Vari()us Paragraph mat(:rials

cumulative fatigue 1.6 is a special to resist fatigue

because damage section failures.

of its unique

application

and

terminology.

theories are discussed in Paragraph 1.5. which will be uscful in selecting certain Design guides are given in Paragraph 1.7.

Section

E1

1 November Page

1.2

BASIC

This the

methods

The discussion

DEFINITIONS.

paragraph

general

describes used

following of fatigue

Stress

the

in fatigue

definitions

basic

mechanism

testing

are

and

some

Stress

of the

Tiw

smallest

tion

that

is

failure

of the

terms

frequently

division

of the

Obtained

repeated.

from

bending, and discontinuities. Maximum

of a fatigue

documenting

and

results. used

in this

analysis:

Cycle

Nominal

1972

11

Stress

The

largest

stress S

the

simple

torsion,

or

stress-time

(See

Fig. theory

in tension,

neglecting

highest

ill a stress

geometric

algebraic

cycle.

func-

E1-5.)

value

Positive

of a

for

tension.

max

Minimum

Stress

The

smallest

stress S rain Mean

Stress

Stress

Stress

Range

Amplitude

algebraic

mum

stress

The

algebraic

difference

and

minimum

stresses

Half the between

ratio

or

value

Positive

of

the

S

of a

for

maximum

cycle.

tension.

and

mini-

m

between in one

value of the the maximum

S

The

mean in one

cycle,

algebraic

cycle.

The

range. Ratio

lowcst

in a stress

in one

Stress

or

the

maximum

cycle.

S

r

algebraic difference and minimum stresses

half

the

valuc

of the

stress

a of minimum

stress

of stress

cycles

to

maximum

stress.

Fatigue

Life

The

number

tained

Fatigue

Strength

for

The

greatest

can

be

number

a given number

sustained of

test

stress

by

which

of stress a member

cycles

can

be

condition.

without

cycles for

which

a given fracture.

sus-

Section

E1

1 November Page

Fatigue

Limit

"l'h('

highest

withstan(I without Fatigue

IAfe

for

Percent

Survival,

P

The --

stress

for failure.

fatigue

level

an

life

that

infinite

for

Np

expected

to

which

survive

a inember

number

sample has a longer life; the fatigue life for which and

1972

12

of

can

load

I) pt'rccnt

cycles

of the

lot example, Ng0 is 90 percent will be 10 percent

to fail.

I_tt

l TIME

FIGURE 1.2.1

Mechanism

In zones zone

:

El-5.

investigating

or

a fracture

instantaneous,

information

for

of loading,

and

sufficiently

apparent

the

sizes

relatiw;

degree

categorized

is

very

if the area

small size

the

Highly with

of both

instantam.'ous

the

zon(>._ zone

rubbed,

The following f('atures velvrty al)pcarance;

oyster granular

shells, trace

concave

with

.stop marks, which shows to the

instantaneous

zone

type

and

structure.

are

origin

beach oriuiin

two

the

the

following

material,

damage

pattern

of loading.

In

zone

The

degree

of overstress

can

of the

zone

rul)ture equal,

zol,e, and

low

fatigue

medium

be

addition,

fatigue

area

relate

type will

the

the

overstress

ovt'rstress

if the

small.

arc characteristic a presence of waw.s and the

provides

and

if the,

of the nearly

is very

fatigur,

of the

direction

and

overstresscd area

from

(hlctility

distortion

zorn.

to the

follows:

resulted

specimen:

instantaneous

applied

respect

('UI_VI'}

and a rul)turc zone,. The fatigue the area of final failure is called

The

to designate of the

area

TIME

which

The

a failed of loading.

compar(M

or

of the

zonr.

direction

an

surface

a fatigue zone propagation;

investigating

ofoverstrcss

bc

VERSUS

of Failur('.

are evident, namely, is the area of the (,rack

rupture,

STRESS

marks; of the

of the

crack

of the known

an(I crack. but

t:ltigue zone: as clam shells

a herringbone Most clam can

also

bc

a smooth, or

pattern or shell marks convex,

are

depending

Section E1 1 November Page on the

brittleness

of stress

of the

material,

concentrations.

in the rate of crack application varying exhibit these waves

degree

In general,

propagation with time. but instead

of overstressing,

the

stop

marks

due to variations in stress There arc some aluminum have a smooth appearance.

__RUPTURE

ORIGI_4 FATIGUE

CRACK--_

7_

/

./._k_

V

if:

El-6.

of a brittle

A TYPICAL

FATIGUE

fracture,

whether

fracture.

Not

the

SHELLS

Most fatigue fatigue

fatigue

strains,

failures failure

bending mations Torsional

/

of the

tension

in a cyclic may not El-6.)

cracks

MARKS

PATTERN

OR GRANULAR

TRACE

FAILURE

SECTION

SHOWING

MARKS material

all brittle

is ductile

failures

discussed

and tension

OR STOP

HERRINGBONE

are

stress.

above

were

Typical

or brittle,

fatigue

The most recognizable features of a fatigue failure are pattern and the existence of a singular plane of fracture, cross section.

loads,

amplitude alloys that (See Fig.

|/_....._._CONC:AVEMARKSKNOWNAS

IDENTIFYING

that

influence

variations

ZONE

CLAM

A fatigue

the

FATIGUE_ _.

OF

FIGURE

13

and the

indicate

1972

follows

failures,

however.

lack of deformation usually a 90-degree

caused

fracture

by tension appearances

in bending and torsion are shown in Fig. El-7. can be divided into three classifications according

of

Bending to the

type

of

load, namely, one-way, two-way, and rotary. The fatigue crack forassociated with the type of bending load are shown in Fig. El-6. fatigue failures occur in two modes: (1) Longitudinal or transverse

along

planes

shaft

and

of maximum

along

planes

shear

and

of maximum

monly associated with a smooth characteristic that can be used

(2)

helical

tension.

at 45 degrees Transverse

to the

fractures

surface because of the rubbing to identify this type of fracture.

axis are

of both

of the com-

sides,

a

Section E1 1 November 1972 Page 14

0

0 Z

o m

m

.._

q)

0 °_

_

7_ m

_

r.D

o

"0 r.)

0

_

o

,< 0 L£ r.J .:.:./..:.

r-

©

[..,

./

i!

,-1

3

bO

b_

0

¢-

0

,.a _

"D e.,

t_

I

/

0

!

o

o

Section E1 1 November

1972

Page 15 However,

a statement

of the

signs

and

features

of fatigue

not explain the true nature of the physical changes which metals under cyclical stress to cause their breakdown. To understand mechanism of fatigue

these changes, behavior in the

it is necessary whole volume

take

fractures

place

does

inside

to study the internal of the metal; but this

sub-

ject has yet to be thoroughly investigated. A considerable amount of theory has been written about fatigue fracture, and there are many interpretations to the process of metal fatigue. (See Rcf. 1.) Fatigue

is basically

a property

of crystalline

of fatigue cracking is a problem in dislocation motion and interaction of dislocations activated description

of the Stage

1.

inally present An irregular The thin

mechanism During

of fatigue

the

early

in the crystal and disoriented

fine slip lines that and faint, according

appear at first in some favorably to the maximum resolved shear

may

be fully

reversed

movements are generated tural features are not the annihilation relief zation,

stages:

dislocations

orig-

increases sharply. starts to form.

oriented stress

grains are law. As the

slip lines become more numerous. Some broaden, and the very pronounced ones

with

only same

of dislocations,

the

in some in both

or other

stress.

New

local slip directions

dislocation

are

considered

to be secondary,

Stage 2. After the persistent like protrusions, called extrusions, face, and fissures, called intrusions, slip

planes.

Several

proposed to explain 1.) In some of the be a critical

and

mechanisms,

or side

may

how the proposed

extrusions models,

intrusion

is the

models

lead

to

softening, local recrystalliother thermal activation effects.

slip bands are fully matured, of metal are emitted from the appear. Both develop along

dislocation

their

or mechanisms

thin ribbonfree surthe per-

have

been

and intrusions are formed. (See dislocation cross slip is considered

Ref. to

process.

Because slip

dislocations

are disin one

zones in which microstrucof motion. Sometimes

of lattice strains or strain softening. Strain overaging, clustering of point defects, and

processes

along

result of the A simple

in three

the

initiation

the so-called persistent slip bands. Meanwhile, the crystals and strain-hardened to saturation. Then, dislocation motion

direction

sistcnt

the

It is the stress.

is given

of stressing,

and

grains multiply and their density cell wall, or subgrain boundary,

number of stress cycles increases, are localized, some continuously become torted

physics. by cyclic

cracking

cycles

solids,

as

planes

the

according

to the

embryo

maximum

of a crack, resolved

the shear.

crack

initiates

Sometimes

cracks

Section

E 1

1 November Page

may initiate the surface

at cell walls of a member.

Stage slip

planes

The

crack

planes,

perpendicular

of the

microscopic

lift;

fatigue

and

grain

maximum

of a member

cracks,

affcetin,_

finally

properties

is

those

the

spent

and

path

As

ensues.

mainly

influence

along

as

of

99 per-

fissures

Many the

at

a general

much

development

fracture

start

maintains

stress. in the

that

majority

transgranular

to grain,

tensile

complete

arc

although

in a zigzag4

from

tothc

fatigue

boundaries,

propagates

andeleava_e

direction cent

3.

or grain

1972

16

into

factors

rate

of crack

propagation. 1. '2.2

I,'ati_ue

The carry

out

methods have

Testing

only

way

fatigue

small

which the is out,

under

the or

the arc

controlled

such

most

m|:asure

of fati_4ue

conditions.

tests,

widely

and

beams at

Thert,

numerous

any

types

but

sl)e('inlt_rls

in Fig.

results that

in detc'rmining

are what

comparatively reasonably might

is the

strength

ar(,

many

of testing

without

is to different

equipment

opposite

called

FLEXIBLE

As upper

in siKn,

the

A possible

El-8.

This

rapidly the

It inherent

-,,,-7

loaded

lower

plane

and

either

limits

direction

suitable

strength

MAIN

L

1

3 GI,:NEtLAL

(CANTILEVEII

ARRANGEMENT

TYPE)

FATIGUE-TESTING

OF

ROTATING MACIIINE

and for

of materials,

ER

I.:! --8.

a test

to carry

of equipment

k.,

FIGURI':

of

of such

MOTOR CYCLE COUNT

in

is

is simple

use

particularly fati_/ue

test,

specimen

and

o[ test

mak[,s

is

are the

arranKement

type and

bending

notche,_;,

between

int;xpensive. be

rotating

loading.

in it varies

throughout.

diagran_matieally

or

four-point

point

constant

method with

under

in magnitude,

remaining

generates

used

specimens,

as

strc'ss

equal

loads shown

out

cylindrical

('antilevt'rs

rotated,

a quantitative

developed.

Probably as

obtain

tests

of carrying b,,en

which

to

Techniques.

I_ENDING

use

Section E 1 1 November

1972

Page 17

since in such work one is interested in the material itself,i.e., itscomposition, microstructure, etc., rather than its form in the engineering sense. However,

to provide

data

for design

purposes,

such

tcsts

are

not of

great value since designs can simplicity that it is necessary material. To provide specific

rarely, if ever, be reduced to such a degree of to know only the basic fatigue strength of the information for the designer, tests must be

carried

forms.

out on the

actual

joint

any means, but is particularly process cannot satisfactorily some of its effects.

use

Therefore, of equipment

tigation.

The

fatigue of much method

times

to reproduce

occur

in service.

fatigue testing three types:

of loading

With

this

Axial

load

2.

Tests

in bending,

3.

Pulsating

All the

of welded components capacity than that used is also

different,

in view, and

the

the

fabricated

the

objective of loading

loading

conditions

can

by

welding altering

normally involves the for fundamental inves-

type

structures

mainly

pressure

numerous

load

of structures

being

at all

that

is likely

used

be reduced

to

in the

essentially

to

testing

testing

on specimens

in the

of pressure

vessels

testing

machines

that

are

form

of beams

and

available

pipework.

and

suitable

tests will not be described here in detail, but itmay of the essential features since, to some extent, the

of fatigue-testing carried out. Axial

is true

in the case of welding. The down without simultaneously

as possible

end

joints

1.

carrying out such to describe some teristics has been

testing larger

as faithfully

of welded

This

relevant be scaled

machines

fatigue-testing

have

influenced

machines

may

the

research

be divided

for

be useful charac-

work

essentially

that

into

three types according to the method by which they are driven, i.e., hydraulically, mechanically, or electromagnetically. Hydraulic machines which give higher loads than those operated either mechanically or electromagnetically are available, but testing speeds are limited. A mechanically fatigue oped

testing in the

machine

of welded

United

is shown

States

opcrated

machine

components at the

diagrammatically

which

has

is the walking

University in Fig.

of Illinois. El-9.

been

used

beam

machine,

The It consists

extcnsively first

arrangement of a simple

in develof this lever

Section

E 1

1 November Page

I I

I

FIGURE

El-9.

with

the

able

uptoa

upper

through the The

for

actuated throw

machine, constant

be

the

used

amplitude

to the

its

resonant

frequency

the:

compon('nt

Special fatigue

testing

hydrauli(,

or

rigs.

at

its

by

several

will

attaching

million

techniques

by

l)e discussed

are

excite, each

usually to thermal

in the

following

stress

amplitude which

industry

jacks

supply

that

In one

method,

in specially

a structuraleomponcnt and

text

of a large-scale

on('

aircraft

testing

sections.

near

oscillator

fatigue only

a fairly

arrangement

tested.

of hydrauli(,

is to

it to a mechanical

dollars,

the

aircraft

mounte(l

require

t() above

aircraft,

more

satisfactory

strain

referred

means

B('cause

relating

shows

in the

entire

method

points.

!.:1-8

magnitude

even

specimen

that

it is a constant

machines

sp(_'cimen Anoth(,r

node

that

since

specimen.

is particularly

in bending)

Figure

noted

()f such

t(_st

costs

be

components, appli('d

beams

loads.

should

to the

constructe(I

aircraft

lower

as

the

beam

it is

to the

with

which

but

bearings,

attached

lever,

lever,

(such

in the

gages

vari-

to the

n_('asurcment,

loads

strain

MACtlINE

is continuously

is transmitted

load

a first-order

is a problem

full-scale is

as

for

BEAM

which

load

frictional

a second-order at

It

The

by using

WALKING

eccentric

1)o us('d

any

load

specimens

in contrast loads.

load

can

either

as

case.

a driven

measures the

or

Fatigue often

by

OF

of 4 inches.

which

also

flexible

latter

ARRANGEMENT

beam

grips,

strain the

ICC_NTmC--..,,- _

maximum

may

end

testing

large

_

to adjust

beam

in the for

I

GENERAL

dynan)onleter

18

JA

a dynamom(_ter,

satisfactory

1972

supporting

is tested. and

low

cycle

Section

E1

1 November Page

1.2.

?

P resentation

1.2.5.1

S-N

the

of fatigue

and versus

log

beginning data.

N denotes

neering.

19

Results.

Diagrams.

Since bone

of Test

1972

the

S denotes

number

N scale (See

of fatigue

is

Fig.

testing,

stress

amplitude

of stress

the

most

S-N

cycles

common

to and

curves or

have

the

the

maximum

complete is used

been

cyclic

fracture. almost

back-

The

stress, linear

exclusively

S

in engi-

El-10.)

f /

I

4OOOO

| m

Io I

m

I

NUMBER

FIGURE

Several the

relation

express will

the

S-N

standing

becomes

GENERAL

been

made

and

life,

and

more

S-N

certain

or

less

in a mathematical

of curve-fitting

of the For

curve

data

CYCLES

have load

relations

the

standardization

I

l

10 3

OF STRESS

El-10.

attempts between

embody

,,

10 2

10

methods.

TO

FAILURE

FORM

(LOG

OF

to find several

S-N

general

for

CURVE

mathematical

It may

have Use

data also

10 5

SCALE)

equations

empirically. form

-

10 4

laws

been

of these

reduction, provide

metals

and

alloys,

to a horizontal

including line.

the

ferrous

The

stress

some

group, value

to

equations analysis,

and

under-

relations.

asymptotic

for

proposed

the

S-N

corresponding

Section

E I

1 November Page

to this asymptote, of cycles,

the

fatigue

limit

The than

that

of the

loading

the

stress

same

stress

that

fatigue

limit

flt,xural

or

energy cent

tensile

theory of tensile

1.2.3.2

Goodman

that

varies

the

types

material

whereas,

a large

in axial

loading,

a discontinuity indicate 58

is consistent properties

in the

that

the

torsional

percent

with

of the

the

of steels

distortion-

are

57.7

The

S-N

curves

about

a zero

static

value

and

fatigue

mean that

limits

stress. may

relate

general

the

manner.

decreases

as

the

range

All mean

the

has

But

be

Goodman is

lines,

a

str('ss zero,

value, stresses.

stress

or

prediction

of operating diagrams

stress

indicate

approaches

that

some

S

it is

e

is the

diagram

was

the

most

[orm

easy

the

first

type

commonly

to construct.

The

proposed,

and

the

modified

used.

Because

it consists

Goodman

equation

is

fatigue

strength

in terms

of the

stress

amplitude,

S

is m

superimposed is the

of

(y

a

T

dealt

the

positive,

about a nonzero:etatie of static and varying

diagrams

ina

range

diagram

straight

the

per-

value.

Goodman

where

shear

stress varies the combination

of failure

stress

maximum

a mean

properties

allowable

on

alternated

about

Whena cyclic must consider

Several to the

discussion

cycles

usually

negative. of failure

figure

the

In axial

Diagrams.

prct'eding

stress

cycle

at

loading

lower

bending.

Thus,

is approximately

This

that

is usually

sectmn;

occur

shear

specimens

limit.

predicts

cross is applied.

will

or

loading

(rotating)

properties.

The with

fatigue

which

load

stress

steel

l in axial

the

bending

in torsion

of polished

limit.

in reverse

throughout

maximum

tests

teste,

tested

20

to failure at an infinite number

endurance)

of a material

whena

the

Fatigue

corresponding (or

material

exists

probable

material.

fatigue

is uniform

gradient

it is

or the stress

is called

1972

ultimate

mcanstress, tensile

S strength.

e

is the This

endurance equation

limit is

plotted

when in Fig.

S

m

= 0, EI-ll.

and

Section E1 1 November Page

1972

21

8 I

0 S

T.S. RI

FIGURE In the modified operating stress,

stresses and

El-If.

Goodman

is described

minimum

GOODMAN

DIAGRAM

failure

diagrams

(Fig.

by three

values:

mean

El-12), stress,

the

range

of

maximum

stress.

I \..-,.o

\

.,o_

i

...___I

" 0 .60

-40

-30

-20

-10

MINIMUM

FIGURE

El-12.

MODIFIED

0

10

20

30

STR ESS.ksi

GOODMAN

DIAGRAM

40

60

60

Section

E1

1 November Page In the

maximum-minimum

form

cycle is plotted as a point on the diagram diagram is advantageous for it requires values

of a half In the

they

diagrams.

the

mean form

Goodman

stress limit for stress

axis. equivalent reversed instead

Although reversed stress of being

these stress

diagram,

instead of as a lint;. only the determination cycle

of diagram,

points

a stress

This form of of the maximum

is not required. a stress

is plotted as a point on the line of zero mean from zero stress to a tensile value is plotted

represent

fatigue reversed

finding

maximum-minimum

stress is zero a stress cycle maximum

cycle;

of motlified

1972

22

have

cycles.

different In this

cycle

in which

stress. Similarly, as a point on the mean form

stress of diagram,

is plotted as a line of constant equivalent a point as in the other forms of failure

values, the

Section E 1 1 November Page 1.3

FACTORS

INFLUENCING

ideal

Fatigue properties environment of the

wide

variety

of factors

affect

the behavior

geneity surface

level of the structure, and metallurgical of materials, finish often

factors, however, of the structure, fatigue section:

strength

will

be classified

2.

Processing

3.

Environmental

4.

Design

the ambient service that determine the

into

1.3.1.1

which

to fatigue

Surface Primary

and

Those homo-

stresses, and the and metallurgical

groups

for

discussion

in this

Factors Factors

Factors. between

processing

adversely

affect

arise from melting practices or primary or may be characteristic of a particular detriment

temperature. cleanness

influence on the fatigue performance The factors which influence

four main

is not always clear. In fact, it is rather section, however, the focus is on regions

the

under

those that deal and material

Factors

distinction

or core,

or assembly

in the A

Factors

Metallurgical

surface

of a member

specimen in practice.

obvious parameters are of loading; the geometry

may have an overriding to its benefit or detriment.

Metallurgical

The

and factors

polished achieved

the sign and distribution of residual are not considered. These processing

1.

1.3.1

The most frequency

23

STRENGTH.

obtained from a carefully test laboratory are rarely

conditions of fatigue loading. with the sign, magnitude, and strength processing

FATIGUE

1972

properties

results

factors

and metallurgical

factors

arbitrary in some areas. In this within the material, either at the fatigue

properties.

or secondary alloy system. from

a local

These

regions

may

working of the material, In nearly every instance stress-raising

effect.

Defects. and secondary working are often responsible for a variety of

surface defects that occur during the hot plastic working of material when ping, folding, or turbulent flow is experienced.

lap-

The resultant surface defects

bear such names as laps, seams, cold shuts, or metal flow through. Similar defects are also noted in cold working, such as filletand thread rolling, in which the terms lap and crest cracks apply. Other surface defects develop

Section

E1

1 November Pagt, from the (,mbe(Iding working process. sionally

roiled

o1 foreign luaterial under high pr('ssures during Oxides, slivers, or chips ofthebase matt, rialare

or forged

into the

surface.

The

surface

defects

intensity

which

properties. and

arc

aforementioned

acts

as a §tress-raiser

Because open

surface

most

to the

under

of these

surface,

defects

defects

standard

produce

load are

of varying

detriment

present

nondestructive

in the extrusion are not

a notch

to the

the occa-

in castings

might include entrapped die material, porosity, or shrinkage; or drawing processes such surface defects as tears and seams uncommon. A 11 of the

prior

of fatigue

to final

testing

1972

24

processing

procedures

such

as penetrant and magnetic particle inspection will readily reveal thcir presence. If they are not detected, however, the defects may serve as a site for corrosion or crack initiations during processing (in heat treating, cleaning, etc.

),

further

1.3.1.2

compounding

Subsurface Subsurface

and and

in the as-cast ingot. improper metal fill

nation

The

core

Defects,

defects

effect

on fatigue

Inhomogeneity,

considered

here

internal and

defects

normally

involved

homogeneous

weld

in the

product.

wrought to this

such as unhealed porosity these defects existed before

defects

product direction

Terms Since

the major of plastic

diameter of the deformation.

I.'atigue testing of high-strength of the type discussed in this I .

2.

Stressing parallel fatigue strength, surface.

normaltothe

direction

reduced, removed

shut

contaminated, defective area

oblate

or

on the (that

originate and In the portion and dis-

under

the

of the

combi-

ingot,

when

the

sur-

healing (welding) is retained in the

and laminations arc applied working, in the final rod-shaped

flaw

ah, minum alloy specimens section revealed the following

to the defect plane provided the defect

The effect of defect size verse direction of testing

which

(porosity) materials.

Occasionally,

oxidized or otherwise is precluded and the

wrought with the

those

reduction

faces of the defects are of the opposite surfaces product. condition.

are

from gas entrapment not uncommon in cast

pressure

in a continuous,

strength.

and Anisotropy.

are to be subsequently hot and cold the preponderance of voids is often

remaining

of temperature

resulting

Core

deleterious

Voids resulting (shrinkage) arc

castings (ingots)that of the ingot containing carded.

the

is parallel

containing trends:

has a small effe('t does not intersect

on the a free

fatigue strength in the short transis, _ith the plane of the grain flow

of loading)

is shown

in Fig.

E1-13.

Section

E 1

1 November Page

1972

25

IM

N F

¸

3o |

g

10

i

I

I

I

0._

"_

MINIMUM

3.

An

.

OISTANCE

CENTER

defect

adversely

With

to fatigue

respect

Inasmuch

the area.

approximately should to the as

I

most

morc

fatigue

and

reducing

d('t'ccts

might be used, is preferred.

whereas,

the

of the

do For

not

by

2.40

OIAGONAL

load

edge

center

defect include int('rsect

wrought for

introducing

the

when

diameters

difficult.

I

OF DEFECT,

C/D

affects

properties,

stLbsurface

I 2,_

TO SURFACE/LARGEST

S VERSUS

material

two

I

1._

be considered as one large extreme distance which will

is somewhat testing inspection

I

OF DEFECT

E1-13.

into

inspection

or eddy-current radiographic

FROM

concentrator cross-sectional

these equal

A 1._

FIGURE

internal

within

part,

i

0._

castings,

a stress

resisting

of one

defect

of another having both

defect,

a diameter defects.

a surface pro(Ita-ts,

is

of a ultrasonic

fluoroscopic

or

Section

E1

1 November Page There

are

two types

of inclusions

in metals,

metallic. The amount and distribution the chemical composition of the alloy, the final

heat

complex carbon,

compounds phosphorus,

important Figure

treatment

parameter E1-14

for

in assessing 4340

Although this relation gested that a separate inclusion.

steel

heat

26

and

inter-

of these inclusions is determined the melting and working practice

of the material.

of the metallic sulphur, and

nonmetallic

Nonmetallic

alloying silicon.

elements The size

its

on fatigue

effect

treated

to the

inclusions

1972

are

by and

usually

with oxygen, nitrogen, of the inclusion is an properties,

as shown

260 to 310 ksi tensile

in

range.

does not apply to all inclusion types, it has been sugcurve exists for each predominant type of nonmetallic

_e

o.B

L2

I

I

loo

2OO

I

I

3oo

4oo

MEAN DIAMETER.

FIGURE PERCENT

second uent

E1-14. UTS

Intermetallic phases with is believed

CORRELATION AND

AVERAGE MEAN

I

I

640o

qlO0

14IN.

BETWEEN

ENDURANCE

LARGE INCLUSION DIAMETER

LIMIT

AS

ARITHMETIC

inclusions may be either complex metallic compounds or variable compositions. The type of intermetallic constit-

to be an important

consideration

in determining

the

effect

on

fatigue life, although the mechanism is not clearly understood. The site of such an inclusion, however, is a discontinuous region with physical and mechanical properties different from those areas would serve as stress-raisers.

of the

matrix

phase.

Under

load

these

Section

E1

1 November Page

Some an adverse local the

alloy._ effect

chemical alloy

at room

banding

austenite and

in others

it is not.

steels. The

steel

the

severity

is shown

lm

of prior

and

matrix

presence

in Fig.

has by

present

in fatigue

properties

maximum working)

phases. seen

of ferrite

properties

often prc_luced

normally

loss

occasionally

in fatigue

not

27

which

is usually

to the

direction

are The

loss

of the relative

banded

ferrite

banding a phase

banding

in the

between

stainless

stainless

The

delta

banding

The

stabilizes

of the

is always

and

low-alloy in 431

which

direction

of compatibility

retained

to miero._tructural

properties.

temperature.

on the (the

degree

subject

segregration

is dependent direction

ar(:

on [atiguc

1972

stress and

on the

Banded

in a large in these

produced

in

by

is

number

of

intentional;

ferrite

stringers

E1-15.

|

!

m

dlk V

N-

I

Jm

I

L

1os

te'

Tol CYCLE|

FIGURE STEEL

El-15.

NOTCHED

HEAT-TREATED FERRITE

THE

WITH

I0 T

FAILURE

FATIGUE TO

AND

TO

STRENGTH 180

to 200

5 PERCENT

OF ksi

431

RANGE

FERRITE

STAINLESS WITH

NO

Section k

Page

Finally, tial

E1

1 November

the grain

alignment.

short

and

subgrain

As previoiasly

transverse

grain

structure

indicated,

direction.

It has been

num alloy forgings that the endurance limit percent when testing in the short transverse tudinal direction. For loading

many

normal

material to the

perties in this direction however, directionality 1.3.1.3

Heat The

i material _trolled

such

transverse

reflect

is most

shown

in tests

is reduced direction

a preferen-

pronounced

in the

on 7075-T6

alumi-

by approximately as opposed to the

as sheet,

light

direction

is low such

plate,

20 longi-

and extrusion, that

the

fatigue

pro-

are not critical. For heavy plate, bar, and forgings, or anisotropy can be a crucial design consideration.

heat-treatment

not properly

also

28

Treatment.

because mechanisms

chemical

forms

short

may

aniqotropy

1972

processes

at the arc

elevated operative

controlled.

If the

composition

a low strength

of the

or brittle

are

potentially

temperatures that could furnace

surface

surface

skin.

of hazard

to a

encountered many diffusion conharm the integrity of the alloy if

atmosphere layer

a source

might

The

is not controlled, be altered

diffusion

and,

the

thus,

of hydrogen

produce

into alloys

during heat treatment has long been recognized as a serious problem. Hydrogen embrittlement of low-alloy steels and titanium alloys can produce disastrous results in subsequent processing or in service. Hydrogen is also suspect

in the

blistering

mechanism

in aluminum

alloys.

With

respect

specifi-

cally to fatigue properties, a brittle case will render an alloy susceptible surface cracking. The introduction of a shallow crack produces a notch so that the detriment to fatigue (life) is essentially one of a high surface stress

raiser

in a layer

If the coarsening

of material

heat-treating

may

occur

with

temperature which

heating of high-strength most of these alloys are

lowers

The

a temperature

fatigue

sequent

quench In order

alloys

must

liquid

medium.

with

associated

and

age

There

of some

grain alloys.

full

cooled are

treatment temperature. of the alloy coupled

austcnitizing with

or temper

to develop

be rapidly

properties

controlled,

Over-

aluminum alloys is particularly disastrous, subject to eutectic melting at temperatures

difficulties are

toughness.

is not properly

marginally higher than the solution heat molting results in a gross embrittlement _trength.

low fracture

a lack

to effect,

or solution of hardening

heat

with

since only Eutectic reduction

treating

potential

in

at too low

for the

sub-

treatments. strength,

most

martensitic

from

high temperatures

at least

two considerations

and

age

by quenching in the

hardening into

quenching

a process

Section E1 1 November 1972 Page 29 that could affect fatigue prol)erties, lligh residual quench stresses are built up in most materials and, if the geometry of the part bt_ing quenched is highly irregular, the tensile high stresses resulting

strength in the

hand, if the quenching tion may occur which

rate is for some adversely affects

1.3.1.4

Localized

occasionally

are

may be exceeded quench cracks.

reason fatigue

retarded, properties.

at points of On'the other

preferential

precipita-

Overheating.

There are some temperatures,

surface

of the material not too uncommon

processes that the consequences

responsible

for

are capable of developing high, localized of which are often difficult to detect and

a failure

in service.

Grinding

is one

of these

processes. The steel below

effect

of severe

grinding

fatigue

properties

is shown in Fig. El-16. The rapid quenching the grinding wheel by the large mass of cold

If actual cracking might result or,

does not result, brittle, with lower temperatures,

High-strength steels sensitive to grinding

(for which techniques.

generally

produces

a larger

Electrical employs

zone

is most

heat-affected

often characterized by evidence the substrate is similar to that

highly which

of the material immediately metal can produce cracks.

crack-prone, softened,

grinding

of high-strength

untempered overtempered

often

used)

are

In the electroplating processes a plating burn sometimes result of arcing between the anode and the work piece.

as the

that

on the

discharge

of surface discussed

machining

a spark-erosion

localized are swept

zone

surface

principle.

cracking

melting. relative

(EDM)

improper

Corrosion Corrosion

ment

intermittent

fatigue

with an alternating

the

corrosive

damage

of metal spark

and

is to

removal

produces

of the workpiecc and metal fragments coolant. Although the heat-affected

and

untempered

martensite

are

sometimes

other evidences controlled.

Fatigue. is that peculiar stress

and propagation, possibly alone would be sufficient term,

is observed Such a burn

The potential to grinding.

observed on martensitic alloys along with eutcctie melting and of overheating in aluminum alloys if the process is not properly 1.3.1.5

particularly

grinding

is a process

The

melting on the surface away by the dielectric

is shallow,

than

martensite martensite.

field

interaction

which

causes

where neither the to produce a crack.

environment

usually

of a corrosive accelerated

environment nor In the practical

serves

to introduce

crack

environinitiation

the stress application stress

acting of the

raisers

Section

E 1

1 November Page

1972

30

©

e,D

.<, e/3 Z

-...._eo -I "

-1

m _

_m z_ 5

•

©

_

m :5

i I

I

M

t

1

I

I

I

It

8

II

Ii

!81 "Nlltii

ImNnil_

>

_,.-.-r._,,,;_.

PAGE

],_

OF POOR

QLIAUTY

Section

E1

1 November Page

1972

31

in the surface in the form of corrosive attack. The irregular surface, in turn, is detrimental to the fatigue properties of the part in a mechanical or geometric sense. For materials susceptible to embrittlement by hydrogen or for parts which are exposed to a fairly continuous corrosive environment with intermittent

applications

complex. which pcrties

of loading,

An example

the cicacking

of corrosion

mechanism

fatigue

illustrates the effect of a corrosive of precipitation-hardened stainless

testing

may be somewhat is presented

test environment steels.

more

in Fig.

E1-17,

on the fatigue

pro-

140

Im

lm

110

W

\\ % •

•

I

l

|

•

I

J|

|

I

l

t

J

t

Jl|

i

I

I

I

L

CYCLI[8 TO PAILUNIE

FIGURE E1-17. CORROSION FATIGUE AND AIR FATIGUE S-N CURVES FOR PRECIPITATION HARDENING STAINLESS STEEL TESTED AT ROOM TEMPERA TURE

1 ,/

I

A

i

Section E1 1 November Page 1.3.1.6

damage

Fretting The

fretting

that

arises

32

-N

Corrosion. corrosion when

is potentially

phenomem,n

two surfaces

relative periodic motion. eomplett'ly mechanical, Fretting

1972

has

in contact

been

defined

and

as that

normally

at rest

In vacuum or inert atmospheres the but in ordinary atmospheres oxidation

dangerous

because

it can

result

from

form

of

undergo

process is also

is involved.

extremely

small

surface monuments that often cannot be anticipated or even prevented. with amplitudes as low as 5 × 10-9 inch are sufficient for this mechanism

Motions to be

operatiw'.

metals. cycles, oxidized

Soft metals Fretting

a higher susceptibility to fretting increases with load-amplitude,

contact pressure, and an increase particles that accumulate between

chemical fatigue

exhibit corrosion

and crack

mechanical

surface

initiation.

The presence

fatigue number

than hard of load

of oxygen in the environment. The the fretting surfaces lead to both

disintegrations

which

of fretting

may

generate reduce

nuclei fatigue

for strength

by 25 to 30 percent, (tependin._ on loading conditions. When a part or assembly is known to be critical in fretting, one or a combination of the following factors will be b('neficial in reducing or eliminating fretting corrosion:

I .3.1.7

1.

l':lectroplating

critical

2.

Case-hardening

3.

Lubricating.

4.

l.:liminating

5.

Increasing

_;.

Bonding

7.

l':xcluding

surfaces.

w(,aring

surfaces.

or dampening fastener elastic

load

material

vibration. or closeness

of fit.

to surface.

atmosphere.

Reworkin_. 'l'h,. success

of any

repair

or rework

(l¢'pend_'nt cm the analysis of the degrading _mderstanding of the cause of failure can _l_'COmldishe, I. In the failure, ,,rengineering

procedure

mechanism. a satisfactory

area of service damage caused test failure of a part usually

ix necessarily Only with permanent

closely a proper r_'work

b3 fatigue, in-service provides tht: impetus

be to

Section

E1

1 November Page

rework

procedures.

In general,

categories: those to have undergone Usually,

these

parts that contain fatigue damage. cracked

procedures

actual

structural

by means that

of doublers,

new

sites

straps,

of fatigue

cracks

parts

part. Occasionally, however, because circumstances, such a part is repaired. crack or blunting its root and supporting etc.

cracking

can

are

be separated

and those

scrapped

and

those

such

or buffing are high,

frequently

Care are

as increasing

a sharp

must

avoided.

areas

surfaces, Residual of fatigue

in doubler

Factors

such

by shot

crack

the depth is difficult

alloy-forming-heat,treating data indicate that the aluminum it may

alloy be many

1.3.2

coraccess, method.

concentrators radius,

and

When fretting may be inserted

be experimentally

are

grinding

is contribbetween the

or eliminate into the critical

or below

the

determined

and the load spectrum. damage beneath cracks

0.003

so

However,

for

tip of

for

all

Preliminary for 7()75-T6 high

strength

steel

inch.

Factors. initiates

particularly

strength

stress

on a surface

0. 003 inch. than

at a surface

since

higher stresses effect of processing

effect on the ma te r ia 1. 1. "3.2.1

greater

usually

there,

substantially beneficial)

level

or

most

because

parts

undergo

stresses

are

normally

ben[ling

loads

resulting

in the outermost fibers. on fatigue properties resi, lual

stress

The detrimental is usually manifest

condition,

or both,

of the

in

(or in its surface

tlardness. Str_'ngth

high_,r

is approximately times

design

the

operations.

damage

should

conditions of fatigue

a new

as fretting

be tightened to reduce are often introduced

or coining

and

with

scratches. If assembly stresses might be planed or mechanically

could be provided. strip or lubricant

of fatigue

dcpth

Processing Fatigue

higher

peening

believed

redistribution, such a rework

or fillet

nicks, and surfaces

clearance a wear

are

replaced

be taken

minor

corner,

or the fasteners may compressive stresses

Estimating a fatigue

to remove edge,

out coarse tool marks, a joint having mismatched

realigned, or improved uting to fatigue cracking, working motion.

used

into two

of the location of the crack or other Repair would consist of removing or strengthening the damaged area

rosion, dissimilar metal corrosion, detrimental stress and practicality are prime considerations in establishing Procedures

that

1972

33

with

of metals

increased

commonly

hardness,

for

enlzincering

up to a point.

us,.'d

In steel,

purposes for

example,

is generally inereas(xl

Section E1 1 November Page hardness does not necessarily fatigue limit is also affected limit these

values curves

for a range represent

effect

(larger

34

indicate a higher fatigue limit because by the surface finish. Curves of average

of surface finishes are shown in Fig. El-18. average values, allowance should be made for

size

generally

0

10

means

lower

fatigue

1972

the fatigue Because size

limit).

HAMONEll ROCKWELL

C

IIMIIMELL

100

30

100

26

30

240

36

280

100

3,?0

•J

310

HED

.,,

400

440

4410

/"

. -,OROEO SOR,ACE,. U,ERL,,,T •"FORGED

SURFACES,

J

o M

tN

126

1_ TENSILE

E1-18.

EFFECT

FATIGUE

LIMIT

OF STEEL

1N

STRENGTH

(1031hi|

LIM,

T

i 200

2_

AND SURFACE

IN REVERSED ETER

LOWER

I

1_

OF HARDNESS

( 0.3-INCH-DIAM

FINISH

34O

ON

BENDING

SPECIMEN)

Forming. By definition,

residual

150

_LLED:--I

26

1.3.2.2

4S

1 POLl

FIGURE

4O

stresses)

Occasionally these there is some loss duced in forming for materia Is.

the ina

forming

part

residual in fatigue (and

their

process

to achieve stresses life. effect

produces

plastic

deformation

a permanent

change

in configuration.

may prove Consequently, on fatigue)

beneficial; howew'r, the residual stresses

often

dictate

the

forming

(and usually prolimits

Section E1 1 November 1972 Page 35 Residual

forming

stresses

in th(_' ('ompl('tcd

least three additional factors: The essing, the temper of the material, and

subsequently

completely

Parts formed and stress upon the stress relieving rial temper, e.g., AQ, magnitude of forming of the material at the strength when forming 1.3.2.3

Heat

treat

ing rates produces

for both

stress occurs treatments. between residual

surface surface

face compressive stresses higher fatigue strengths. Aging

treated

arc

/

free

of prior

forming

stresses.

to the extent that they affect the yield strength temperature. In general, the lower the yield the weaker the residual stress field generated.

temperatures

are

both

ferrous

produced and

of heat

treatment,

and

and core. compression are

For

for aluminum

alloys

are alloys,

are

such

as

principal

common source or cool-

to produce

too low to produce

before heat treatment, use of less relief/equalization by cold working stretch-stress relief tempers).

1.3.2.4

Surface

any

tempered at temperaConsequently, for

in machining,

increased

detrimental effects on fatigue alloys, special processing

reducing

machining and stress example,

slightly

not recognized as a detrimental however, persist after com-

by distortion

techniques

developed,

The

most steels are quench stresses.

and possible in aluminum

been

alloys.

of the

high temperature solutioning are built up by nonuniform

magnitude

susceptibility to stress corrosion life. To minimize these effects have

in many

aluminum alloys, differential cooling and core tensile stresses. These sur-

of sufficient

as indicated

relieved

nonferrous

in quenching from Residual stresses

after tempering, quenching stresses Quenching stresses in aluminum

pletion

on at

hcat-treatm('nt-forming sequence in procand tile forming teml)eraturc. Parts formed

appreciable stress-relieving; however, tures sufficiently high to affect residual steels factor.

(l(,i)endent

relieved contain reduced forming stresses, depending temperature. The forming temperature and the mateT-4, or T-6 for aluminum alloys, also influence the

stresses forming occurs,

stresses

cycles

of residual austcnitizing

at'('

Treatment.

Residual heat

heat

part

section

sizes

by rough

severe quenches where of quenched materials

possible, (for

Finish.

A given surface-finishing process influences the fatigue properties of a part by affecting at least one of the following surface characteristics: smoothness, residual stress level, and metallurgical structure. The effects of surface finish on fatigue litre it can bc seen that,

life for 7075-T6 extrusions are shown in Fig. El-19. in general, fatigue life increases as the magnitude

Section

1,:1

l November Page

1972

36 O

m N u

Z

© ,_a

[--,

-

[..,

--

t u_

_

C" [-..

©

-

_

<

2:

<:

I

;D

l

,1

J

l

%

0

IIM (NOIIN! 1} SS:IilJLI

"_

Section E1 1 November Page

of surface roughness decreases. Decreasing method of minimizing local stress raisers. Aside

from

effect

on surface

surface roughness

roughness,

the final

1972

37

is seen as a

surface

finishing

process will be beneficial to fatigue life when it increases the depth and intensity of the compressively stressed layer and detrimental when it decreases or removes this desirable layer. Thus, sandblasting, glass-bead peening, and other

similar

operations

generally

processes, machining

such as electropolishing, ( ECM), which rcmove

point

reduce

may

Many

fatigue

local

life

surface

improve

fatigue

of accidental minor

fatigue

of structural

life

from

design

defects

and

irregularities

parts

as

large

depending

Conversely,

in fabrication

and

Stress consmooth surface

scratches, corrosion pits, or as effective in reducing the

scale

on the

occur

for, or control. on an otherwise

tool marks, grinding damage are occasionally

deficiencies,

properties.

properties.

service that are difficult to anticipate, inspect centrations resulting from small indentations in the form service-related

life

chem-milling, and electrical-chemical metal without plastic deformation at the tool

stress

concentrations

material,

resulting

heat-treat

range,

design

margins, etc. Such unintentional stress-raiscrs are damaging in a structure only if thcir notch effect is more severe than the most severe stress concentration arising from design, unless they are located so as to intensify the stress-raising design stress 1.3.2.5

Cladding, Clad

lifetime

sheets

become

Chemical

the

similar

sprayed reduction

to that

weaker

reduction

to the same overshadow

for

the effect of by surface finish.

Coatings, than

in fatigue

and

the

bare

life

for

Anodizing.

sheets plain

as the fatigue

degree in built-up assemblies, where the the effect of claddding on fatigue strength.

cladding.

applied to extrusions and effect on fatigue properties However,

it has

been

found

zinc finishes on aluminum alloys do not produce any measurable in fatigue life. Also, it is generally agreed that neither zinc

cadmium

plating

has

Chemical alloys

reduction

observed

Usually produced

Conversion

progressively

tlowever,

is not present stress raisers

feature. the effect

Aluminum metal spraying is sometimes for added corrosion protection. The

forgings

num

critical design far outweigh

Plating,

increases,

specimens fabrication

quite

effect of the concentrations

for

any

appreciable

conversion corrosion

in fatigue

life

effect

and anodic protection

ranging

from

on fatigue

coatings

or wear

nor

properties.

which

are

resistance

a negligible

is that

amount

applied

usually

to alumi-

produce

up to 10 or

a 15 percent

Section E1 1 November Page

of the

enduranc(_

limit.

The methcxl

of producing

fatigue properties. 7075-Tfi indicated

As an example, fatigue tests that anodizing with a 5-percent

slight

lowering

but definite

in fatigue

life

metal, whereas, a sulfuric type anodizing reduction in fatigue life as shown in Fig.

the

coating

further

al[ects

run on chromic acid anodized dichromate seal offered a

as compared treatment E1-20.

to that of unanc_lized resulted

in a substantial

E

I. UNANOOI2ED 2. SULFURIC ACID |. EULIFURIC ACID

Ie

ANODIZED ANODIZED

- BARE |DQEJI - ALL EURFACEII

-

I

FIGURE

1_:1-20.

PROPERTIES:

show_.'d

EFFECT 7075-T6

Fatigue

tests

of 7075-T6

no loss

in fatigue

OF SULFURIC

ACID

ALCLADSHEET R= +0.2 bare

properties

sheet compared

ANODIZE

0.090IN.

with

1972

118

an A lodine to untreated

1

ON FATIGUE THICKNESS,

#1200

coating

material.

Section

E1

1 November Page 1.3.2.6

C old Working. Cold

has

been

simple coining

working

found

of parts

to induce

to be an effective

and complicated of holes, thread

shapes. rolling,

tumbling,

grit

stressing,

and

tool

residual for

stresses,

as

beneficial

to the

For cold-working subsequent controlled

in bending.

the

fatigue

improving

compressive

of axially

thermal treatment when it i8 essential.

Perhaps

the

most

the

fatigue

stresses

life

of both

cold work include hole expansion,

surface compressive which produce high

must

widely

material relatively

stresses,

loaded

desirable effect of surface process must be accomplished

stresses in surface of its low cost and

compressive

pre-

blasting.

Residual life

surface

Methods of imparting fillet rolling, peening,

It should be realized that residual beneficial under loading conditions

most

1972

39

however,

are

also

specimens. cold working to be maintained, the in the final heat-treated condition;

be eliminated

used

stresses are surface tensile

process

when

for

is shot peening. easy application

feasible,

inducing

but closely

residual

compressive

Its use is widespread because to a wide variety of materials

and

parts of varied size or configuration. As an example, peening might be used before chrome plating or hard anodizing, or in special situations, after a heat treatment which produced decarburization of a steel part.

in the

When a part undergoes loads opposite direction, prestresssing

that

in the direction of major service loading surface compressive stresses and work

are much higher in one direction the part by applying an excessive will often hardening

produce that will

beneficial significantly

fatigue life properties. In reverse to fatigue life may result. In the

loading aircraft

this effect is lost and industry, prestressing

for

to name

two examples.

torsion

in some increased

bars

and

Techniques 1.3.3

Environment The

of the

bomb

specimens that the endurance

fatigue

environment,

frequencies.

Even

hooks,

tensile limit

involving

prestressing to 90 percent about 100 percent. other

types

of cold

working

than load

residual improve

a detriment is often used

It has

been

of the

tensile

can be found

shown strength

in Ref.

2.

Effects. properties

of a metal

especially normal

in long-time air

has

can be greatly tests

a deleterious

at low effect

influenced

by the

nature

stresses

and

at low

on fatigue

life

properties,

Section

E1

1 November Page

and

recent

adverse

indications

show

components.

aluminum,

magnesium,

influence

of

aluminum

and

environmental

alloy

that

moisture

and

decreases

the

Moisture

is given

l

iron air

by

5 to

moisture

in Fig.

oxygen fatigue

15

;)erct'nt.

on

the

are An

fatigue

the

two

strength

1972

40

print'ipal

of copper,

('xample

properties

of the of an

El-21.

i

44 i I

i--

_DllY

-

al

INVINOfIMINT

\

i" E N MILAN Ti[MIIL_ I[(_JAL

i[TR|m

C_I|.THIRO

IITREa

_

OF

AMPLITUOi.

24

,

kit

I

lu'

J

FIGURE

E1-21.

ALLOY

It

S-N

6060-T6

is generally to fatigue

CURVES

life

FOR

IN HIGH-

agreed is

on the

]

100

CYCL|I

respect

I

10I

AND

that

the

TO FAILUlll

CLEAN

SPECIMENS

LOW-HUMIDITY

main

propagation

10)

effect

OF

ALUMINUM

ENVIRONMENTS

of a normal

of cracks

rather

atmosphere than

in their

with

Section

E1

I November Page initiation. clean

The

metal

principal

exposed

nonpropagativ.g ture; thereby,

effect

at the

of the

environment

tip of a fatigue

crack.

will

be in its

Cracks

that

vari,::tj in vacuum propagate in the presence they lessen the fatigue resistance.

1972

41

reaction

would

to the

be of a

of oxygen

or

mois-

In environments that are more corrosive than air and that substantially reduce the resistance of the surface layers to fatigue crack initiation, both initiation as vJc!l as propagation may be affected. Environments that lead to stress-corrosion cracking may (Refer to Paragraph i. 3.1.5.) 1.3.3.1

irradiation effects

adversely

affect

fatigue

properties.

irradiation. Some

changes

also

investigations on mechanicaI

are

usually

have

been

loss in ductility effects evidenced

have fatigue

referred

been made properties

to as radiation

detrimental

in one

have been increased

with respect of metals.

way

damage

to the effect of nuclear Radiation-induced

since

or another.

in many

Damaging

cases

the

effects

noted for many metals. On the other yield and ultimate strengths, fatigue

such

as

hand, beneficial strength, and

surface hardening also have been noted. Rotating beam fatigue tests on 7075T6 aluminum alloy (Ref. 3) indicate an improvement in life due to the effects Gf a total integrated amount of irradmtion ficient

to alter

flux of 2 x 1018 fast neutrons received by the specimens

mechanical

properties,

icant for the characteristics straining and irradiation. simultaneous 1.3.3.2

action

been

direct

results

are

sufsignif-

the conjoint action of fatigue need to be undertaken for the

mechanical

which

may

properties.

experimentally

of an aluminum

straining

This

result in-air greater

strongly

that

alloy

pressure. trends toward

eventually

growth rates obtained from held in vacuum for periods fatigue

and

demonstrated

stressing

readily as within atmospheric Experiments have disclosed vacuum

during studies

if the

at low temperatures.

Vacuum. It has

uniaxial

it is not known

of metals Additional

of irradiation

per cm 2. Although the in these tests is believed

occurs

cracking

in hard

This is contrary increased crack

in characteristics tests. than

fatigue

the

more

time

under almost

as

to general belief. growth rates in

Results on aluminum a week yielded drastic

indicates

vacuum

dependency

critical

than

continuously reductions of the

in

phenom-

enon. Results from short-time test exposures in vacuum have shown improved properties. This is the usually accepted belief. However, for many cases an extension of the vacuum outgassing time provides a more realistic environment than

do the

short-time

test

exposures

previously

investigated.

Because

of this

Section E1 1 November 1972 Page 42 anomaly, prolonging procedure

at present

the vacuum exposure for evaluating metals

1.3.3,3

bleteoroid

Damage.

is suggested for service

as the only reliable in space environment.

In the environment of space, tiny meteoroids and micrometeoroids may strike the surfaces of a space vehicle. At present, no information is available as to this effect on fatigue properties. However, it can be reasoned that indentations caused by these meteoroids fatigue cracks by reason of the increased 1.3.3.4

Solar Solar

irradiation

Elevated

will

of time, problem.

In general, the intempcrature

differential

thermal

The and

frequency

may

fatigue (Fig.

decreases,

the

increases. fatigue

cycling

of cyclic

mechanism responsible

and loads

of creep for this

heating

on orbiting

and

resulting

the

spacecraft. fatigue

of stress cycles, to temperature

cycling, damaging,

tIowever, thermally lives

and

of load

metals

could

crack

nucleation

period

strength;

(or

propagation as affected is an additional parameter

is decreased.

low tem-

time

with an that as the to produce

as tt'mperaturedecreases, decreases.

it is known that rate increases The damaging,

an the'

by rate of cyclic to be considered. the number of cycles as the speed or thermally

activated

cracking, acting conjointly with fatigue, is In general, this is true because, in the accu-

slower rates of load cycling result in exposure of for longer periods of time than do fast rates of load

from cryogenic temperatures activated mechanisms arc

crack-growth frequency

to lose

notch sensitive and fail by britbe common at normal (standard)

life of most metals will decrease I':1-22). Studies have indicated

fatigue testing, the crack-growth

or creep behavior.

mulation the metal

cause

to become that would

Moreover, crack growth

rate of fatigue-crack test-load frequency

In elevated-temperature to fracture decreases

pendent

the

temperatures

observable crack} I)eriod of observable

fatigue

cause

may cause some metals under load conditions

temperature

loading,

initiation

Temperature.

peratures tle fracture temperature.

increase

ideal sites for the concentration.

Irradiation.

Over long periods become a serious 1.3.3.5

provide stress

rates for

most

at cryogenic metals.

to room temperature, active. It is believed temperatures

will

no that be inde-

of

Section E1 1 November 1972 Page 43

I

N,

FIGURE

E1-22.

1.3.4

Additional 1.4.3.

AS A FUNCTION FOR

on thermal

parts holes,

and

often include and keyways;

of elastic strains and stresses. change in cross section is greater tion with either the direct stress, Th,} geometric

MOST creep

OF

TEST

METALS) effects

will

be given

in

maximum

stress

stress-concentration in the

geometric

in fatigue loading. higher than would

abrupt these

section

This phenomenon is called factor results in a fatigue

factor, to the

K,

nominal,

is defined

scction such distribution

factor

as the

or average,

net remaining section. the basis upon which concentration factor.

stress-concentration

The actual be expected

changes in cross cause a nonuniform

The maximum elastic stress at an abrupt than the nominal stress obtained by calculaflexural, or torsional formulas.

Nominal stress is usually based on the exceptions, it is necessary to ascertain is established before applying a stress 2'he full

LIFE

Effects.

Manufactured as grooves, fillcts,

of the

FRACTURE

(TkPICAL

informatio_._

Design

TO

FATIGUE

TEMPERATURE

Paragraph

CYCI.EI

ratio

stress.

Because of the nominal stress

is not always

applicable

fatigue limit of a notched member is frequently from the geometric stress-concentration factor.

notch sensitivity. stress-concentration

Use of the notch-sensitivity factor that is determined

from:

Sectioa E1 1 November 1972 Page 44 .K 1"

where

Kf

bending;

q (K t-t)+l

is the Kt

.

fatigue

'); .

stress-concentratioa

is the geometric

stress

modified geometric stress-concentration is the notch sensitivity. Typical values of various

strengths

are

shown

in Fig.

factor

concentration

•

.

for

direct

factor

for

factor for bending of q versus notch

tension direct

(Ref. radius

or tension,

4) ; and q for steels

E1-23.

U

f

J

NOTCH

FIGURE

E1-23.

NOTCH-SENSITIVITY SPECIMENS

The

fatigue-stress

IIIAOIU$.

r |i_.)

CURVES

OF VARIOUS

concentration

FOR

POLISHED

STEEL

STRENGTHS

factor,

Kf,

represents

the

extent

to which a notch can be expected actually to reduce the fatigue limit of a part. It is the ratio of the fatigue limit of the specimen without the notch to its fatigue limit with the notch. Specimens must have the same effective section when

Kf

determined

is evaluated without

experimentally including

an effect

so that

only

which

is due

the

effect

to reduction

of the

notch

in the

is section.

or

Section

E1

1 November Page When

one

notch

is located

in the

region

notch, the resulting stress-concentration the two individual stress-concentration ence of one notch the stress patterns When posed other,

encounters developed

of maximum

factor factors.

influence

is determined Whether the

1972

45 of another

by multiplying maximum influ-

the maximum influence of the other depends by the two notches and the extent of overlap.

on

stress concentrations are adjacent to each other, but are not superimto the extent that one is placed in the region of maximum influence of the the resulting stress concentration factor lies between the value of the

larger

factor

members

and

with

applicable

the

product

numerous

in most

of the

notches

cases,

and

two stress-concentration

and

factors.

discontinuities,

a value

of

Kf = 4

a value is the

of

In steel

Kf=

maximum

3

is

likely

to be

encountered. 1.3.5

Welding

I.:ffects.

There exists a considerable of various welding effects

perties

information primarily

on welding effects because of the late

a cons(_uence, been limited;

the use however,

amount of information on the fatigue proin steel (Ref. 5). tIowevcr, the amount of

in nonferrous development

materials of welding

of welded aluminum it is becoming more

behavior

been

found that

of welded

joints

of the joint as a whole relatively unimportant The than weld

fatigue

the

most

in aluminum

and of the variable.

limit

for low-strength bead is removed,

by far

of welded

bead.

high-strength

steels if the weld the fatigue limit

These

arc

average

made. Removal aluminum.

values, of weld

and beads

fatigue in zero

is the

Parent

improves

implies

that

in the

fatigue

factor geometrical

material

steels

shape,

seems

is only

As

slightly

surface. Even partial limit considerably.

for a lower fatigue

both

to be a

higher

in place, ttowever, if the the amount of increase

limits of approximately to maximum tension

an allowance also

frequently

important

bead is left is increased;

depends upon the smoothness of the finished the weld reinforcement increases the fatigue steel with the weld undisturbed, reversed bending and 23 000 psi

This

an(I steel

weld

scarce, alloys.

in fatigue-loaded structures has extensive, particularly for applica-

tions where weight ,_avings is of importance. fatigue strength is the design criterion. It has

is relatively of aluminum

removal of For welded

12 000 psi arc common.

failure

characteristics

rate

should of

in be

Section E1 1 November Page

1.3.6

Size Usual

diameter. has

been

sizes region the

larger

ably

Shape

observed.

This

reduction

is generally

in bending or torsion stress. This increases

stress

region,

leading,

attributed

have a larger volume the probability that

in turn,

to a higher

of 0.2to 0.5-inch size of specimen to the

fact

Fatigue tests in axial loading show because of the uniform stress across

size

has

correction

applied

little

effect

upon

is neglected

only to the

larger

probability

of failure

in

in

1000

1.3.7

Speed The

less

stress

cycle

point

little under at 1000 of the

size effect, proban axial load. cycles. S-N

curve

has some of circular

effect on the flexural cross section have

increase

than

round

Square beams loaded of the square exhibited

effect

in strength Of course,

involved.

of the

speed

so that fatigue

of of

the limits

beams.

of testing

was the

were was

has

been

studied

by a number

effects

encountered. noted, while

of

there temperature

At very low speeds for very high speeds

a slight some

found.

speed

stress-versus-life These

limit limits

of Testing.

in the specimens in fatigue strength

of the

a

and

fatigue fatigue

investigators. In the range of from 200 to 7000 cycles per minute appeared to be little effect on the fatigue properties, except when increases decrease

Thus,

limit.

8 to 10 percent higher than square beams. maximum stress occurred at the corners 4 to 8 percent

relatively a section

the failure

at the

fatigue

Shape of cross section materials. Steel beams

some

side

that

of material in the a defect will occur

sizes.

Size

from

46

Effects.

specimens for fatigue tests arc in the range Some reduction in fatigue limit with increasing

stressed of high

high

and

1972

of testing curve

arc

covered

is of great and also

when

in Paragraphs

importance thermal 1.4.2

at the cycling and

1.4.3.

low-cycle or creep

is

Section E1 1 November 1972 Page 1.4

LOW-CYC

47

LE FA TIGUE.

range two. cycle

A complete S-N curve may be divided and the high-cycle range. There is no Wc might arbitrarily say that from 0 to and from about 103 or 104 cycles to 107

m_u,

Until World War II little attention was paid to the low-cycle range, and o.e t'_ c .... _l,lg results were for high cycles only. Then it was realized

ti_.at ;or some pressure vessels, missiles, etc., only a short fatigue life was required. fatigue

phenomenon

began

T, ,i_c low-cycle

to gain range

into two portions: the low-cycle sharp dividing line between the about 103 or 104 cycles is low or higher is high cycle.

spaceship launching equipment, Consequently, the low-cycle

attention.

of fatigue

life

below

approximately

10 000 cycles,

the primary parameter governing fatigue life appears to be plastic strain cycle as measured on a gross scale. For higher (cyclic) lives, elastic

per strain

also

strain

per

assumes cycle,

importance; but

perhaps

the plastic

strain

the

governing

is highly

variable

localized

structure and is difficult to measure or compute. 10 000 cycles and continuing upward, it becomes total strain (elastic plus plastic) as the primary

is still

plastic

at imperfections

in the

Beginning at approximately more appropriate to regard variable. Alternatively, the

fatigue life can be regarded as being governed by stress range, and the ensuing simple relationship between fatigue life and stress range appears to be valid for most of the materials that have been tested to date, over a very large range of life lems

on both the

sides

stress

for practical range. The mechanical The

of 10 000 cycles.

range

purposes relationships

is less

likely

to be known

it is desirable as applied

loading

are

discussed

failure

mechanism

However, to express to problems

for than

most is the

thermal-stress strain

prob-

range;

life in terms of total of both thermal and

hence,

strain

below. in the

low-cycle

range

is close

to that

in static

tension, but the be termed "true

failure mechanism in the high-cycle range is different and may fatigue." A comparison of the failure mechanisms in the two

ranges

in Table

is made

El-1.

Section E1 1 November

1972

Page 48

Table El-1.

Comparison

of Low Cycle

Low Internal

stresses

stra

in hardening

Net

sum

and

of plastic

Gross

sum

X-ray

disorientation

and

High

Cycle

Micro

of plastic

flow

Fatigue

liigh

tIigh

flow

Cycle

Cycle

Low

size

Micro

size

Small

Large

Large

Small o

Slip

Coarse

Slip

plane

N orma

distortion

Crack

origin

Crack

path

F inc

(103-10'IA) 1

Persistent

Interior Along

maximum

(1 0A )

Surface shear

Cross maximum tensile stress

Fracture

Delayed static

Structural deterioration

The Figure

terms

used

in this

section

are

defined

below

and are

shown

in

E1-24. E

=

modulus

N

=

cycles

t e

E

P C

f

total

of elasticity

to failure strain

range

elastic

portion

of strain

range

plastic

portion

of strain

range

a material

constant

true

at fracture,

strain

commonly

known

as fracture

ductility

Section E 1 1 November Page

SAS

stress _ E E t amplitude

=

Se

endurance

Su

ultimate strength tensile

Sy

yield

Sut

true

STRESS

49

///'

limit tensile in ordinary test

strength stress

/ at frac-

L_

.., !

ture

1972

in tensile

test

L

j

'P

"

W"

Below

Creep

In recent

years,

has been conditions

studied where

1.4.1

fatigue under perature

effects.

subject ditions the

Range.

"

extensively little attention

There

now exists

obtained from laboratory (Ref. 6). The general

strain

between

fixed

cycles while

strain. cycles

to failure. the latter

behavior.

has been a fairly

E1-24. STRESS-STRAIN C Y C LE paid

large

to temperature

body

limits

and,

The results to failure,

if desired,

to measure

the

is determined. Such tests strain, longitudinal strain,

of the

more

on the

stress,

while

may be under bending strain,

can be represented in terms of total or, if desired, of plastic strain range

The former is most often used for direct is applied when one is primarily interested

One

and tem-

of information

experiments under room temperature conprocedure in such investigations is to control

cycling the specimen until failure conditions of controlled diametral or torsional range versus

FIGURE

low-cycle

fascinating

features

the form of plastic-strain range versus cycles that the curve (on logarithmic scales for both

of representing to failure A_ and P

design in real the

strain versus

procedures, material results

is the observation is of constant Nf)

slope of value 0.5 to 0.65 (depending on the experimenter) and that the vertical position of the curve relates to the tensile ductilityof the material. A plot of this curve is shown expressed by the equation

Ae

q-_ P

= C

in Fig. EI-25.

The line in this figure can be

in

Section

E I

1 November Page

1.0

E •

m

0.01

1

!

I

I

i

!

10

10 2

10 3

10 4

CYCLES

FIGURE

E1-25.

PLASTIC

the, constant test.

C

I 2

=

A fair temperature

C

degree

rclated

creep

Problem

fatigue curve stress

or, conversely,

culat(.(lstress.

The

and

reduction

CYCLES

of area

understanding low-cycle

questions

RA

in static

yet

now fatigue

to be

exists

problem,

for

the

low-

although

answered.

Solutions.

or equation versus

factors to givc safe allowable cycles;

to the

range),

Practical

shows

VERSUS

FAILURE

of information the

of unresolved

The

RANGE

100-RA

a number

1.4.1.1

one which

be

..

I00 In

(below

are

may

,

TO FAILURE

STRAIN

TO

ther('

50

0.1

i

where tension

1972

cycles design

allowable

stress

values

shoul(l be directly comparable culates

using his usual methods

stress,

stress

concentration,

needed

and one that contains stresses

operating

of analysis

or designer

cycles

values

is

sufficient safety

for a given number

of operating

for a given value of cal-

on the fatigue curves

to the stress

etc.

by the engineer

which

for pressure

or in the equations the designer stress,

cal-

thermal

Section E1 1 November Page For equation

fatigue

for

below

the

the

fatigue

plotting

S = --

E

creep

100

In

rang(',

curve

to Fig. For

E1-24

for

a design

recommended

the

(Ref.

7) has

mean

stress

has has

little

an

,

of terms).

to represent

equation

been

proposed

material:

above

a lower be used

bound

with

on the

a safety

shown

that

in the

or no effect

low cycle

on fatigue

(below

thus,

it is of either

2

at each point. It the effects of size,

range

life;

data,

factor

on stress or 10 on cycles, whichever is more conservative is believed that these safety factors arc sufficient to cover environment, surface finish, and scatter of data. It also

51

e

definitions

stress

that

any

+ S

100-RA

(refer

Langer

for

1972

this

10 000 cycles),

effect

can

be

ignored. Some Table half

typical

El-2. the

values

of

If information

value

of the

E

and

on the

ultimate

RA

endurance

strength

In Creep The The

state. vated

high-temperature, material behavior because

At temperatures

low-cycle becomes of the

in the

dependent. low-cycle

the

is important.

if one stress

rate

of the

test

introduces a hold time is a maximum. While

relaxes

for

cycling

show

conditions is raised

some that

(Se)

are

given

is not available

in

one-

used.

is required

the

period the

into the

of time.

fatigue

life

fatigue problem very much more

occurrence

creep

be very strongly frequency tant in high-temperature, strain

materials

techniques

proposed

is in quite complicated

a different at ele-

Range.

temperatures

esses.

various limit

is often

If a more detailed fatigue analysis in References 6 and 8 should be used. 1.4.2

for

range

of creep

and

of a material,

other the

diffusion

test

results

procwill

Citing the fact that frequency is imporfatigue is the same thing as saying that A somewhat

related

the cycle, say, at the strain is held constant, Results decreases

of prolonged with

point the hold

increasing

beingthe same. A common observation is that, and the rate decreased (or hold time increased),

situation

occurs

where the tensile peak stress times

during

hold times,

strain other

as the temperature the character of

Sectio

n E1

1 November Page

__IC

"3

CCO_O

o3 ×

C O tD =9

...d

_D 0

0

M __I

_

_

c'l b_ °_,_

c,; !

,.Q > I

.-.:.-_ c c

<

t_

0000_

¢0

CO

_

¢_

C':,

--

52

1972

ORI,G;,"L:'L _""" :: ' Section

OF. POOR q

E1

1 November Page

the is

fracture

changes

largely

bctween

these

when

dealing

taM'

strain

that

known,

for

amount

of strain

at

th(,

that

for

l. 4.2.1 As

notc(i

that,

rc(tu(,e(I

at

additions

in the

desirable which anal

behavior

of metals,

introduces

a dilemma

It

is well

a fixed

strain

is applied

is the

can

creep

be

strain

van

I)e aecomplishe¢l Ilowcvc'r,

tlwse

for

it should

sam_'

regions

be

show

heat

treatments

this

will

fatigue

material

in as

so the

as

relationship

scl_'ction

to develop

ductile

fatigue

a condition

Itowever,

it is apparent

that

one' the

as wants

best

that

between

of materials

the

ductility,

to optimize

strength).

and

the

to optimize

lower and,

invc_rse

into

necessary,

but

creep

the

strength, the

precipitate

l)uctility

is of

prime

resistance, possible can

it is (a

both

condition

creep be

for

to

strength

achieved

is

condition.

proct_dure for

is

and

resistane(',

Procedure The

i)roc('(lure fatigue

the

in low

compromise

1.4.2.2

describes

advantage

involved

strengthening

strengthening,

metal,

to have

fatigue

can which

It

applying if the

creep,

of tcmperaturc's.

with

in low-cycle

results

some'

creep

service.

c.onsidcration

hardening"

by

for

strain

strain effects

Strength.

rcgions

ductility,

alloy

._tructtlre

alloy

in metallurgi-

plastic

greatcr

appre-

ductility.

hi_h-temt)erature _,mploy

the

above, various

well-known

and

provided

is

be

strength.

reached

ages

must

strengthening

additional

of strength strain

Crcep

simultaneously

This

stress.

an

\.'crsus

fraetur('

slrenlzth

change

and

"strain-ag(,

to

seen

design.

mentioned

alloys

is the

term

is,

be

rupture.

fatigue

cyclic

which

problem

stress

temperature

and

one

can

which

In designing

hardening

the

Ductility

,_l)_'ei[ic

level

the

effect

devc_loping

which

and

instance,

The on

the

temperature.

at

that

53

to

resemblance

in creep

precipitation

structure.

to a metal

strongly

related,

present

the

strainhas

of strain-age

sclectcd

time

in the on

metal

example,

aging

taken

an(l,

plastic

found

certainly

with

in nature

A close

high-temperature

bearing

in the

eflcct

vc,cy

those

and

occurs

important

place

the

suggests

and

with

which

is tr_,nsgranular

innatur(,.

important,

Plastic an

This

observations

structure

have

which

metallurgical

A second cal

one

intergranular.

a largcclegree,

ciated

from

1972

estimating

characteristics

for

Estimating

given the

h('re

Iligh-Tcmperature, is

taken

from

high-temperature,

of laboratory

specimens.

I,ow-Cycle

Rcfe.renees

9 and

Fatigue. 10.

It

is

a

reversed-strain-cycling It shoul(I

bt'

emphasized

that

Section

E1

1 November Page

the

method

material actual

is intended selection.

fatigue

possible I.

Final

data

those

to give

only

first-approximation

design

generated

of important

under

to be encountered

s which

E1-26

illustrates

is divided

into

strain

range

poncnt

is plotted

materials.

against

the

method

its elastic

cyclic

life

The

plastic

and

of universal plastic

on log-log

components-are

-0.6, and the elastic components should be emphasized that these materials; values betwecn -0.4

be based

on

as closely

arc chosen obtained.

r

for the

plastic

and

as

The

components,

and

total

each

Straight

The method of universal assumed to be the same to have

an average

com-

lines slopes for all

slope

value

an average slope value of -0.12. Again, it values arc not always -0.6 and -0.12 for all and -0.8 have been obtained for various mate-

elastic

values between -0.08 and -0.16 When average values of -0.6 components,

reasonable

have and -0.12

results

are

FSTIMA

TING

00"I

!

1

1

CYCLES

](l':

slopes.

coordinates.

taken

for the plastic component, and found for the elastic components.

,(,)I

should

simulate

to

in service.

usually result for each of these components. prescribes that the slopes of these lines arc

'

as a guide

Basis.

Figure

rials been

estimates

equipment

conditior,

1972

54

1':1--26.

M E THOD

TO FAILURE.

Nf (LOG

SCALE)

OF UNW ERSA L S I.OPES

AXIAL

FATIGUE

LIFE

FOR

of

Section E1 1 November 1972 Page 55 Having decided on the slopes, cepts

of the

which

most

D,

where

where

two

straight

lines.

significantly 100 D= in IO0-RA

For

the

a

is the

ultimate

range,

given

u

total strain becomes

elastic

In general,

governs

line,

it is necessary

the

the

governing

tensile

strength

as the

it has

intercept

sum

to determine been

of the

property and

of the

for

E

elastic

found

plastic

that line

the

the

inter-

the

property

is the

intercept

ductility

is

the

elastic

modulus.

and

plastic

components,

a

/E, u The thus

3.5(7 _

Act as shown

u

E

in Fig.

-o. 12

(Nf)

temperatures found that than those results

within

the

is,

range

of course,

very

low-cycle

crack-initiation period; only the crack-propagation This

concept

initiation and it is assumed by the within

thus,

has

yielded

method of universal the creep range. it has been

are

higher the

than fact

that

of taking

those

actually

the

10-percent

into

account

this formula one use this

material?

the

for many procedure

In general,

relationship

intercrystalline

the total period.

between cracking

cracking bypasses

fatigue

life

10-percent

slopes

found

will

that

rule frequency

it has

been

high-

is approximately

actually

In this

to a lower 10-percent

in test. effects,

This

yields

effect

equal

method,

be realized

a computation

approach

important.

due to a creep the transcrystalline

rule.

such

it leads even the

achieved

materials at high

is very

regarded as functions of fatigue only 10 percent of the fatigue

tive results; that is, in most cases, ttowevcr, there are cases in which with

)-0.6

yields fatigue life predictions higher The explanation for the unconservative The

and

propagation are not that, on the average,

Generally,

sibility

of the

complex.

fatigue

with Can

It has been shown that intercrystalline early in the fatigue life, and thereby

occurs

puted tures

creep

obtained arises:

the procedure almost always actually obtained by testing.

temperature,

(Nf

E1-26.

Good agreement has been 9). However, the question

(Ref.

DO.6

+

to

crack

life; rather life comat tempera-

conserva-

bound on fatigue life. rule predicts lives that observation

inherently hold-time

excludes effects,

(together the

pos-

mean

load,

Section E 1 1 November Page

etc. ) causes one to seek not as limited. The

simplified

analysis

following. The creep spent at stress to the Since

the

stress

further

and

number

that

the method tion between analytical description

of universal the time expression is shown

(C reep-Rupture

that has

temperature

would

simple

damage effect time required

be obtained directly from the damage effect is taken as the the

for

are

slopes. of the test

been

adopted as the rupture

presumably

in the

for the number in the following

Damage)

of cycles derivation:

+ (Fatigue

-0.12/m A

= A(Nf)..

hence,

= 1 AF(Nf)-0"12/'n

+ \Nf/

OF

N N'

, 1 _

this

f

k/AF(Nf)__ (m

+

.12)/m

by the

rupture

of creep

effect

Since the test frequency yields and number of cycles sustained,

k --f Nf'

:

be explained

ratio of tile time actually at that stress value.

known,

absence

1/m tr

can

that art,

curve of the material. number of cycles actually

where

t'=

of estimation

is taken to cause

creep-rupture ratio of the

be sustained

methods

1972

56

to failure

Damage)

can

=

time

according

to

a definite relaa closed-form

be obtained.

I

can

The fatigue applied to

This

Section

E1

1 November Page

1972

57

Here

Nf'

=

Nf

number

of cycles

to failure

under

combined

number

of cycles

to failure

in fatigue,

fatigue

based

and

on method

universal slopes, using ductility, ultimate tensile elastic modulus from uniaxial tests at strain rates to that undergone by the metal priate data are not available, conventional tensile tests. k

empirical information slope

constant, becomes

of straight

nates,

that

as shown A

=

coefficient

F

=

frequency

Table El-3 contains conditions (Ref. 9).

in the

of cycling, of the

_-

>o.3o

L°ae'

insert

*''A

strength and comparable

but adjustable

curve

curve

in Fig.

in creep-rupture

some

_" o.x

assumed to be 0.3, available.

representing

of

in the fatigue test. Where approuse may be made of data from

line-creep-rupture

is,

creep

by

as more

on log-log

ar = 1.75

au

coordi(tr/A)m

E1-27. relation

cycles

information

per

unit time.

above,

obtained

from

l._,,u

various

/

_

test

.

/

LOG t r SCHEMATIC

ITRES4i

IqUPTUR!

CUR'VI

I_

/-

0.24

:

_

:o,. o.,,

, . _.

10"2

J,t,J.I

10 "1 PRODUCT

FIGURE

EI-27.

CRITERION

CRE EP-RUPTURE

USE

--

10_

N

FOR

LowER,o'u,,o

:,:..

J . I._,z.I i , I,l,_.l

10 0 OF CREEP.RUP'rURE

_

101 COEFFICIENT

10 2 AND

TO ESTABLISH CORREC

I , ,,l,l,i

TION

FREQUENCY.

NEED

113 AF

FOR

ORIG|NAL O(:

' ......

POOR

Section

QUALHY

Page Table

E 1-3.

Alloys,

Test 'l't,mperatu

Test

Conditions,

Test rt_

Frequency

,tA

E1

1 November

and Pertinent

1972

58

Properties

C rcep-Ruptu

'l't.nsile

E la ,tic

Strength

Modulus

S hipt' (-m}

re

Intercept,

A

h Ih,v I)t'nignat A-2_(i

-

ion C11 114f_)

(F)

( _:pm )

(<_)

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(mini

1000

5 to 50

31.

1_12.

26.R

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167.

211. 1

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18.

115.

23.0

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57.6

75.

19.0

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16.I

104.

20.8

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0,028

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42.0

179.

22.0

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6.6 _/q.

12 45. t40.

Section E1 1 November Page II.

life

have

for

1.

The

universal

and

this

By the

10-percent

to determine By the

The interpret clusions

rule.

cyclic

combined

problem

life,

creep

been

life

values. Figure the computations

the

the

lower

bound

curve, use

use the

the

and

representing equation

use

the

the

10-percent

the

for

and

how to

following

lower

con-

of the

two

plot that which value

the product, at the test coordinates

Nf'.

If it lies

AF, lies

below

determined

in a.

serves

as a lower

life.

As an estimate of the lower bound determined

it is to conclude

upper bound in b. cases

that

probable

on life,

encountered

the

life,

method

use

in the

gives

use

twice

10 times

the

laboratory,

results

the

the

good

engineering uses, it is perhaps of greater importance to of the cautions involved in its use. It must first be emphasized

analyzed relate to constant amplitude-strain cycling The method appears to place the high-temperature,

behavior

us_,l

of 10.

rule.

d.

as

and

E1-27 is an auxiliary needed to determine

of the two values

on estimated

is first

by a factor

to use,

study

As an estimate of average or most lower bound determined in b.

many some

equation

equations

c.

As tempting

fatigue

of the

If the point

on fatigue

effect.

by both methods,

It has been found that, in most rule is the applicable one.

that the data temperature.

slopes

bound

It is merely necessary to determine slope, m, of the creep-rupture curve

curve,

The

lower

is divided

considerable

calculated minimizes

above

the

life

fatigue

which

given

temperature.

enough for emphasize

computing

Determine

to use. and the

b.

and

of determining

the results has can be drawn:

a.

10-percent

59

Method.

In summary, two procedures been outlined. Proceed:

2.

1972

in the

proper

range.

Thus,

it would

seem

under constant low-cycle

that while

the

Section

E1

1 November Page

method

may

material more

be

regarded

ehoict's

shouhl

promisint_

stress

and

test

as

very

be

goo(I

made

materials.

on

And,

temperature

history

for

the

,_;creening

baqis

the

in service

important

tests

wheneve:'possible, expected

60

materials,

o[ actual

1972

from

amon_

the

complexiti_'s

should

be

of

included

in the

evaluation.

1.4.2.

"]

Method

of Strain-Partitioning.

In their continuing Manson

and co-workers

attempts

to analyze

life (Ref.

11).

dependent

nature

depend,

and whether

was

developed

it is reversed

method, four

rule) have

the

parts.

because

where

fatigue,

a method

which

of the highly time-

the life-reducing

effects of

within a cycle the creep

is intro-

by plastic flow or by creep.

completely

These

low-cycle

deveb_pcd

and plastic flow on cyclic

effect and because

to a large extent, upon

In this into

method

of the creep

duced

of high-temperatur(,,

directly the effects of creep

This

creep

divided

examination

(the 10-percent

reversed

are

(1)

cyc!ir

A¢

inelastic

-- t,'_ile

strain

plastic

flow

is revers,',!

PP by

compressive

sire

plastic

plastic

flow,

flow,

(3)

AE

(2)

Ae

-- tensile

cp

-- tensile

plastic

creep

flow

reversed

reversed

by

by

compr,'_-

compressive

pc creep,

and

(4)

A¢

-- tensile

creep

rcversud

by

compressive

creep.

CC

In any tensile

arbitrary

inelastic

iCl))

hysteresis

strain

components.

also

tensile

into

neither

components

can

Likewise,

b,_ separated

zeneral,

(AD)

loop,

(C-D inelastic

its

the

two

and

B-A)

strain

bc

such

an

separated

into

theeempr,'ssive plastic

(]_i_)

plastic (A-'D)

be

and

be

equal.

plastic

ert'cp

(t_--A)

(AC It is

equal

in Fig.

to the

and

entire

strain

the

and

('reM_

(DA)

can

components.

DB)

nt_cessary

E1-28,

(AC)

inel,_¢tic

compon('nts

will

.qhown

nor only

In

the

two

creup

that

the

entir¢_

compressive

inelastic

I

_train

(DA)

tiont_d

strain

reversed

since

we

are

ranges

plastic

_:_mt_ts and ._'rain range,

are

strain

dealing obtained

closed

in the

range,

in this _,xample AE

witha

Ae

PP

,

hysteresis

following is the

is equal to DB.

, is equal to the smaller

loop.

manner. smaller

The

The

The of the

two

complet¢qy

completely plastic

reversed

of the two creep

parti-

comcr,'ep

component._

cc

and becomes

CD.

plastic components components, or

A(

As

or AC in accordance

cp

can be seen graphically,

must - DB

the difference

be equal to the difference = BA with

- CD. the

notation

between

This difference just

stated.

between

is then equal to For

the two

the two creep

this

example,

Ae pc it i,_

Section

E1

1 November Page

equal

to

plastic

Aepc

since

strain.

partitioned range

It strain

or

the

width

the

tensile

follows

from

ranges

will

of the

plastic the

strain

preceding

than

procedure

necessarily

hysteresis

is greater

be

equal

that

to the

the the

total

1972

61

compressiw_ sum

of the

inelastic

strain

loop.

STRESS CREEP

CREEP

PLASTIC FLOW

FIGURE

The

effect

of each

Ref. 11, and the damage relationship,

total

N N pp

cyclic

N

is

the

strain

life

LOOP

ranges

is determined

on

cyclic

using

life

a linear

is noted cumulative

N pc

cyclic

cc

life

associated

with

the

strain

range

of

Ac

PP N cp is the

or

N

, PP

is the

cyclic

life

associated

with

pc cyclic

in

N

or N cp

HYSTERESIS

of these

N

N

where

E1-28.

A_

or cp

life

associated

with

AE

cc

.

AE

, and pc

N cc

Section E1 1 November Page

This deal

method

directly

with

fatigue

rather

fatigue

findings.

at

temperatures

high

life

rather

ation the

differs the

than

10-perc('nt

occurring

estimate

base.1

from

other

all

relations

in that

of that

and

versus

cyclic

life

requires

only

static

well

abasis

for

that

with

the

effects

and

hohl-time,

10-percent

rule.

rule

This

method

effects.

for

Finally,

low-cycle

low-temperature,

low-cycle

which are

is

the

Coffin

Fatigue

and

Manson

partition

strain versus

damage

range time

required

in this

materials useful

to quote

and

may

of this

many 11).

where

mt'thod prove

frequency

authors

and (i(cl.

gener-

method

in explaining the

cyclic

The

This tested

effects

versus life.

properties.

is in an early stage of development, before its merits canbe evaluated"

Two-Slope

to

high-temperature,

tensile

is also

it attempts

on

stress

data

in that

62

at

use(I

rule

agrees

rule

metno(Is

range

10-percent

1.4.2.4

an

a combination

of strain

the'

phenomena

u,_ing It differs

than

"The method be answered

from

1972

method,

questions

must

Law.

have'

proposed

the

follc)win_*

fatigu,'

law

for

plastic

strain:

NfI_' A,-

= c

P

wh('r_'

N[

in the

C

the

constants.

arc

( 1)

,

cycles,

to

failure,

This

in hi
has

low-cycle

N[ v k- 1)I_

A< P

:

A¢11 been

is thc,

modified

i)lastic to inclu(h_

strain

range,

the

effect

and of

fl ,

frequency

fatigue,

C2

,

(2)

k-I wh('re as

the

further

v

is th('

frequency,

fr(_qucney-modifit_d to

include

lhe

I< , C_ fatigue

stress

A(7 ::: A (_()'_v P

are life.

the The

eonstants, frequency

anti

Nfv

cffectwas

is defined ext('nde(t

range,

Iq

(a)

Section E1 1 November Page

63

and

(3)

1972

and

Aa

where

=

A, rl,

generalizing

and

range

k arc

for the

AE

where

stress

constants.

total

= (AC27?/E)Nf

E is

the elastic

Coffin

reports

many

at

Nf

mode due

(Fig.

of crack

E1-29).

Fig.

E1-30,

accurate

the two-slope of time in the of the equations if experimental

mode

phenomenon environment

Thermal

physical emphasizes (frequency

in slope

(4)

the

the

to low lives

tcnsile

is due

ductility

to a change

this

from

exponent

well with the (1) reported

ductility. Because in the curve of the

materials

resulting

increases

tcmpcraturc

through

ductile

change in slope to transgranular

operative

are data

and

,

C correlates less for 13 in equation

the tensile be a break

change

In less

as temperature

of the

(1-k)fl

extrapolation

pass

change

environmental

could this Ac

P

value in the

in slope

change

in slope

is

interaction.

at high temperatures results from fracture and as can be seen in the

The main Benefit of the two-slope description of the occurrences

recognition

1.4.3

it will

This

fracture

In ductile materials this change from intergranular

increasing

temperature,

so that

in the

+ C2Nf-flv

with

exceeding there must

propagation.

to a change

12 that

at elevated

relationship

Nf = 1/4

(1-k)fl_

one and the constant of the high exponent

lead to plastic strain ranges situation is highly unlikely, versus

(2)

modulus. in Rcf.

materials

equations

range

-flrlkl+

/3 increases approaching tensile ductility. Because for

Combining

strain

is more

a

drastic.

fatigue law is that it gives a more in plastic strain and leads to a clearer

process.

A recognition

the importance and hold time),

highly material dependent are not available, these

of the

causes

of

of the environment and tIowever, the constants

and temperature constants cannot

dependent and be determined.

Cycling.

The prcvious sections have considered mechanical strain or stress cycling at a constant temperature; this section will consider mechanical

Section

F 1

1 November Page

I

I

I

1972

64

I

600°C

_= o.5

1010 STEEL

111

Z < e¢ Z

0.1

< er

,,°'_-_'_

¢t)

1/8 in. 1/4 in. FREQUENCY

m

_

-

"_,

o _

V-

[ -X-

0.2 cpm

.,

,o°.,

'_,;x_=o.,,-

¢0

<

® x

-t

0.01

I _a

<1

I 1

0.001 0.1

FIGURE

E1-29.

MODIFIED

;=..

,

i 100

FREQUENCY

MODIFIED

PLASTIC

STRAIN

FATIGUE

' '-,2 ' ' i

I 10

....

LIFE

r

i 1000

FATIGUE

RANGE

LIFE

=

....

I

....

600°C

I

....

A,S,- 3o4ss _

'

_

TEMP.

CODE

430 °

0

_o °

_=o.,L-

o.o11-

0.1

FREQUENCY

STEEL,

_"0_ '

Nf v k'l

VERSUS

--AISI CI010

....

--

.

1

10 FREQUENCY

' 100 MODIFIED

FATIGUE

1000 LIFE

--

10 000 Nf vk'l

FIGURE E 1-30. PLASTIC STRAIN RANGE VERSUS FREQUENCY MODIFIED FATIGUE LIFE -- AISI 304 STAINLESS STEEL

100 000

Section

E1

1 November Page strains cation

induced bycyclic of thermal stress The

thermal has been

stresses. termed

whenever

activity in which analysts have, problems were encountered

of elastic

stresses.

The

treatment

65

Failure under repetitive thermal stress fatigue.

principal thermal-stress

of this

subject

until recently, has been the

may

be found

1972

appli-

engaged computation

in Section

D,

Thermal Stresses. While such computations constitute a necessary and desirable first step in any practical analysis, they unfortunately do not provide s_ffficient Thus, occur

information

for

it is important when the yield

problem.

for taking start has

evaluation

not only to take point is exceeded

might change during Recent contributions this

a final

progressive have made

Not only

have

when

dealing

with

materials.

detcrmine

suitable

1.4.3.1

and load cycling of the material. start toward an understanding

new computational

From working

hlealizcd A model

techniques

been

an engineering

point

of view,

the

R(efcrring

bc chosen

[n which

all

E1-31,

the bar

hardening. At F the occurs (relaxation),

the

thermal

strain

the

stress

is converted

is assumed

that

clamped

creep

and

anclasticity

relation

at which

occurs

when

to

negligibh,.

proceeds

yielding

will

alton

Upon I'GE,

during

hot,

so

cooling. Plastic to F because of

specimen is held for a dwell period, since the temperg_turc is i)resumed

time there is a much lower stress than effect. (The Bauschinger cflect states rcduccs

is to

The case is shown in Fig. I::1-31 in which a bar is fixed two immovable plates so that the length of the bar must To approach the problem realistically, the model should

to Fig.

stress-strain

problem

le Model.

that tensile stress is developed along OAF during the first flow is initiated at A, but the stress continues to increase

to make

but a

st res ses.

show strain hardening, and the effects of stress relaxation any hold period that may be imposed at high temperature.

the

of

developed

in terms of material behavior. The failis, of course, different from the criterion

Thermal-Cyc

will

mc'chanieal strqin. at its ends between remain constant.

men,

that flow

into account inelastic, effects such as creep and plastic flow, been made toward inc'ludim_ cyclic effects both in computational

for ductile

enough

materials.

into account plastic-flow effects but to consider how such plastic

thermal a valuable

procedure and in the interpretation ure criterion for brittle materials

strain change

ductile

but no stress to be low

reheating

yielding

atG,

of the speciat which

there is at A because of the Bauschingcr that plastic flow in one direction occur

in the

opposite

direction.)

Section E1 1 November Page When the restored,

t/H// IHl iH I ll//l// l ll

--.----"

66

initial high temperature the state of the material

at p(_int E, where strain is necessary

eA. WiT. CONSTRAINED ENOSI_

is is

a compressive to offset the

elastic tensile

plastic flow that occurred during and, thereby, return the specimen a net strain of zero. This elastic

COOLING HEATING

1972

AF to

I

strain introduces the stress OE. hold period at the high temperature

I F

stress

may

convert

this

elastic

to inelastic strain-creep ticity, thereby reducing

A

Thus,

point

E moves

Any and strain

and anelasthe stress. to point

E'

by

the time the specimen starts to cool again. The cooling causes the path E' F' to be traversed. Reheating results in the path F' E". The hold period e"toE''',

TOTAL

at the

high temperature converts etc. After a fcwcycles,

MECHANICAL

the

STRAIN

stress-strain

path

to an essentially

may

loop FIGURF 1':1-31. SCHEMATIC STRESSSTRAIN RELATIONSItIP FOR FIRST FEW CYCLES OF TIIERMAL CYCLING 1.4.:3.2

OF CONSTRAINED Effect

In the

fatigue

strain

per

strain strain

is determined range, which

rlastic

cycle

modulus.

the cycle, improperly

and takes

For

loop may be This correhysteresis

into account

not only

all

the hardening or softening characteristics but also creep and anelasticity effects as well as effects due temperature and metallurgy.

BAR

to changing

of Creep.

experiments

thermal-stress

to an asymptotic

down

loop.

illustrative purposes this taken as E' ' ' F' l':"I,:' ' '. sponds

settle

unchanging

to the

that behavior number

have has

been

of cycles

by subtracting the in turn is computed

When

creep

performed

been

occurs

made

to date,

largely

to failure.

Normally,

elastic-strain by divi(ling

during

the

high

of cyclic the effect

life because of slip-type

the

of the the

plastic

plastic

range l rom the totalthe stress range by the temperature

this procedure produces inaccuracies. Not only computed if creep is neglected, but the omission

s_,riously affect the computation on life differs appreciably from

analysis

by relating

portion

is the plastic of the creep

the effect of creep plastic flow.

of flow can strain

Section

I£ 1

1 Noven_bcr

F"

Page

Consider two

idealized

Fig.

E1-32,

cases.

In

Fig.

which

shows

the

1,',1-32

(a)

no

E

is

stress-strain

creel)

1972

67

lor

rclationshil)s

is assumed

to take

place.

Then,

A_

where modulus

eP

= ff T O -

Aa

is the

stress

is assumed

range

to be

constant

and

over

the

the

elastic

entire

modulus. temperature

(a) FIGURE

E1-32.

RE LA TIONSHIP DURING

(b) PLASTIC-STRAIN FOR (a) NO HOLD PERIOD

RELAXATION

DURING

AND

STRESS-RANGE

CREEP RE LA.KATION AND (b) CREEP HOLD

PERIOD

(The range;

elastic

Section E1 1 November 1972 Page 68 assumption

of a temperature-dependent

problem

even

further.

tlowever, plastic the

when

strain

tensile

per

cree

cycle

portion

elastic

modulus

complicates

the

)

l) occurs

is not

of the

during

directly

cycle,

the

a hold

related plastic

period

to the strain

l:1-32(b)

],

stress

range.

For

total

I Fig.

¢

'

the

is

P

T

£ P

where

=

A(_l

cycle

only.

thus

reducing

plastic

O_ T O

is

the

In

stress

the

range

heating

the

strain

_IE

direct

¢

"

developed

portion

of the

compressive

also

during cycle,

the

plastic

includes

the

creep

strain

fore,

equal

£

vt =

q'vl

£

the

strains

1.4.3.3

13 and tures

strain

not

dirc'ctly

compared

14).

Fatigue

loading

TO

were

fatigue

tests

range.

The

in terms

of cyclic

life

are

shown

the

specimen

cycles

to

mechanically even

though

failure

500 °C, figure, was

cycled no part

so at

much at

that

creep

and

is,

there-

p

in tension,

Mechanical

less

but

Fatigue

specimen

was mean the

even

500 °,

compared

naturally

varied

in Fig.

E1-33

plastic

strain.

600 °C,

whereas

constrained

and

and

temperature

for

(Refs.

hight('mperawere

shown

completely

thermally

in the

are

temper-

conducted

constant tests

measured

of cyclic

various

been

temperature test

350 °,

values for

several

the

at

have of the

era

at

the

equal

:;50 ° C and

of the

at

at a given

tests

the

This

range.

fatigue

results

in which

tests

200 ° and

with

fatigue

results

mechanical-fatigue

from

Fatigue

mechanical

the

the

thermal-stress

the

to that

stress

conducted and

an

seen

total

period.

E

total

thermal-stress

tests

appreciable

during

is equal

to the

in which

with

thermal-stress

be

hoht

F

of Thermal-Stress

mechanically,

between

the

the

Temperature.

over

analyzed

is larger,

ttowever,

-

related

with are

relaxed

in compression

experiments

was by

'"

P during

rang(, '"

of the

Thus,

__ _

plastic

Constant

Several

_

E

Comparison

atures

strain

P

are

at

elastic

vvvl'

+ E

P

Therefore, these

the

(Aa 2 - A%)/E.

P

stress

portion

P

replaces

to

cooling

strain E

P creep

the

was

350 °C,

plastic

strain

cycled

specimen

one'

mechanically

200"

to 500 °C

the cycled

th['rmal-strcss

and The in

cycled As

can

number than at

for

of one

000 ° C,

Section 1':1 l Novcmbcr

19__ e-r,)

PaK_' 69

0.1

----,--

CONSTANT

-,, --

CYCLIC

TEMPERATURE

FATIGUE

TEMPERATURE

FATIGUE

THERMAL-STRESS

F

%

0.001021

103

104

CYCLES

FIGURE

E1-33.

CYCLES

COMPARED

fatigue

test

ever

reached

traversed For

TO FAILURE

600 ° C.

Nf

IN TIIERMAL-STRESS

it would

problem, the temperature as the highest temperature lurgical phenomena will life will be indicated; conducted to date.

FATIGUE

this

IN

discrepancy

have

been

along the specimen to temperature in the

tests.

appear

that,

until

further

tests

clarify

the

for the mechanical-fatigue tests should be taken of the thermal-stress test. In this way, metalnot be overlooked and pessimistically low values of

this

(Ref.

for

of temperature of the materials

thermal-cycling

reason

based

Reasons

to nonuniformities of the properties by the

this

Spera

TO FAILURE

10 li

WITH CYCLES AT CONSTANT TEMPERATURE SIMILAR PLASTIC STRAIN RANGE

generally attributed and the sensitivity range

108

15)

has

appears

to be necessary

presented

a method

for

calculation

of thermal-

life can be determined from the basic mechanical properties of a by calculating lines for each of two distinct and independent failure (1) cyclic creep-rupture, using a modification of the well-known lifeusing

developed detail

proposed the

by Manson

to define

by Robinson

empirical (Refs.

completely

and

equations 9 and the

Taira, of the

10).

analytical

He proposes

tests

fatigue material modes:

role

damage.

of most

life

fatigue,

creep

basis

fatigue

fraction

on accumulated

on the

and Method

Equations procedure.

(2)

conventional

of Universal are

that

presented

thermal-

low-cycle Slopes in sufficient

HeetionEi I Noveml_er1972 Pag,' 70 1.4.:¢.4

Summary.

When plastic strains are introduced by constraint of thermal expansion, l atigue

ultimately

results.

The

number

of cycles

on the plastic strain and the temperatures Whether tension or compression occurs little

effect

on fatigue

life.

maximum temperature metallurgical effects

Probably

a given temperature range will cycles to failure than increasing and

maintaining

the

same

imum temperature smaller magnitude. depemling

on the

maximum

material,

Limited

data

most

that

be withstood

a much range

temperature.

fatigue

single

depends

are induced. seems to have variable

greater by the

The

time

reduction in same amount at which

the

and

life

life

in a thermal-stress

for

at a constant temperature fatigue test.

The effect stress rupture

of prior depends

cycles. In general, some cases it may

thermal greatly

rang(-'.

has been andthe

to the

on specimens material and

the effect can be expected actually be beneficial.

Fach of the materials that the cobalt-base alloyS-816,

steel,

equal

cycling on the

fatigu¢'

average

that

of another

is difficult.

Hence,

investigated before a reasonably hehavi()r of materials, in general, tion

on behavior

of the

l(;,

17,

and

18,

19,

20.

various

complete: canbe

types

test

strainspeci-

temperature

of

subsequently evaluated the number of prior

to be detrim('ntal;

howevcr,

in

studied to date -- type :_47 stainless nickel-base alloys Inconcland

Inconel 550 -- displays characteristics considerably different from the in thermal-stress fatigue, and prediction of distinctions in the behavior l rom

max-

an effect on fatigue life, but of or detrimental to fatigue life,

is sometimes considerably less than the fatigue life of a mechanically cycled specimen having the same total strain as the thermally fatigued men and tested the thermal-stress

is the

if it is higil enough to cause the maximum temperature for

can also have can be beneficial

temperature,

indicate

important

particularly Increasing

generally cause the temperature

is maintained The effect

can

at which these strains at the high temperature the

of the cycle, to take place.

that

more

classes

of materials

others of one

must

be

i)icture of the thcrmal-fatilgue understood. For specific irfforma-

of materials,

investigate

Refs.

13,

14,

Section E 1 1 November

1972

Page 71 1.5

CUMULATIVE

1.5.1

Theory. One

important

FATIGUE

problem

DAMAGE.

in fatigue

analysis

is how to calculate

fatigue

life. From the S-N diagram, we know that the higher the alternating stress, the lower the number of cycles a part will endure before failure. Also, at stresses

belbw

the

In most an allowable an S-N curve

fatigue

limit

the

weight

cases

an infinite penalty

number imposed

stress cannot be tolerated. can be used to determine

stresses

are will

constantly

account

changing.

for the

Thus,

damage

by using

If stress the number

the life of the part can be predicted. Most subjected to irregular fluctuating stresses which

of cycles

caused

can the

be sustained.

fatigue

limit

as

cycling is at a known level, of cycles to failure and thus

structural in which

the

components maximum

it is necessary by different

or parts are and minimum

to use

a fatigue

magnitudes

theory

of stress

cycles. Of the

several

cumulative

widely used and best pendently, by Miner. damage applied cause cycle

fatigue

theories

known,

the

one most

known is the one suggested by Palmgren and later, The Palmgren-Miner hypothesis is that the fatigue

incurred at a given at that stress level failure at the same ratio or cumulative

at the same When fatigue

damage

stress divided level. damage

level until failure loading involves

damage is a sum of the different when the cycle ratio sum equals

level is proportional by the total number

to the number of cycles of cycles required to

This damage is usually ratio. If the repeated

occurs, the many levels cycle one.

referred loads are

to as the continued

cycle ratio will be equal of stress amplitude, the

ratios

and

inde-

failure

should

to one. total

still

occur

n.

2. 1

This same

time

, = Lt. + n,2 +

Ni

+ ....

N,

= 1.0

1

equation

has

it is under

1.

In many

2.

The cycles

been

criticism test

fatigue

results damage

or the

cycle

used

by designers

for

by researchers. the

summation

is not linearly ratio

nl/N 1.

many

years,

It is found of

n/N

proportional

but

at the

that:

is far

from

to the

one. number

of

Section E1 1 Nt)veml)(,r 1972 Page72 :3. Thcru is interaction in the fatigue levels

which

sequence the

effect

high

load

the damage next. Aware

of its

preliminary guide racy, commensurate What loading and the

which

first,

are

tha;

the

the

of its simplicity, the data currently

According In some

and

0.60.

to Freudenthal cases the sum

The

Freudenthal, a mplitudes.

primary

is the

On the other hand, a summation value greater

ratios

was

1.4

(Ref.

ttowever, is recommended 1.5.2

for

for

it has been than one.

and lo

more

(,quation

load

as a

and sufficient accufor this type of analysis.

equation

is used

discrepancy,

damages

for

found Since

according

between

that notched practically

of notch,

the

random-

to

various

load

parts generally give all of the structural

question

and Spac 9 Administration airplane win_s and their

preliminary

of what

conducted summation

value

to

full-scale of cycle

becomes

available,

Miner's

equation

analysis.

assumptions

Mainly, these The following

secondary

are

often

needed

to speed

up the

assumptions have to do with primary and four conventions serve to establish and define

cycles:

A mean

value

mat(.d.

Although

yield

this

information

simplifying

of data. cycles.

primary

versatility, available

from

high

of I)ata.

Numerical analyzing secondary

Miner's

the

22).

until

Analysis

be different with

from

testing is time-consuming and costly, a general clear-cut answer to this

of fatigue

The National Aeronautics tests on C-46 transport

use

Miner's

fatigue failures originate in some form use in place of 1.0 has been asked.

fatigue

will

first,

resulting

(Ref. 21), the sum g n/N is always less is as low as 0.13 but it is mostly between

reason

interaction

damage

next,

still

when

fatigue

low load

designers

consequences

the

low load

from

limitations,

the

means

with

resulting

because with

(lamages l)etwecn various stress In the interaction there is alsoa

neglected.

conditions ? Random-load fatigue scant data- available do not permit

question. than one. 0.20

Miner

different

of stress, a time

average

S

mean

average values,

for

the entire

and

a peak

the

two may

record

point

is approxi-

average

be assumed

generally to be equal.

Section E1 I November 1972 Page 73 .

The

maximum

{positive

in slope}

is called

S

as

e

S

The

occurrinl_

All other

other

stress

minimum

S

slope)

S

(negative

maximum

occurring

stress

The

secondary

between

positive stress

combination

cycle.

values

_'rossing

mean

crossing

mean

are

values

are

Smin

and

of an

combination

a negative

crossing

of an

S

S

is designated

Smi the

points.

next

designated

See

next

crossing

mean

as

S max

and the

mi

Smi n.

Fig.

All

E1-34.

form

a primary

S

form

ma

a

cycle. Accurate usually

damage I

a positiw'

ma

a corresponding

The

between

and a negative

.

max

minimum

and

1

stress

mary

__ $mu

cycles

considered, all secondary

eva luation of fatigue requires that all pri-

within

a given

period

be

but it rarely requires cycles be considered.

that

The effect of neglecting secondary stresses in random data will be inves-

t

tigated

in terms

of the

following

propositions: Proposition and troughs

mums) Y

TI MI

FIGURE

E1-34. STRESS

A PARTOF

Proposition

II -- If peaks

last

it lies

established, mean

stress

above and

the

mean

when

crossing.

the damage cycles which

mean

stress

it reaches

are

line

about

troughs are a secondary when the

a mean

can be neglected.

normally

distributed

about which

normally distributed about cycle needs to be considered

it occurs

lowest

(maxiare

per cycle caused do not cross the

per cycle caused by all cycles stress line can be neglected.

and time

stress,

distributed

stress, by all

and troughs

the damage the mean

Proposition III -- If peaks mean stress, the only

is when been

A

RECORD

a positive mean stress, neither cross nor exceed

a positive

normally

Snt_n

I -- If peaks (minimums)

S

just min

after

established

S

max

has

since

the

Section E1 1 November 1972 Page74 Usually valid.

Fig.

some than base

for random data, on(; of these propositions E1-35 illustrates several more or less useful

of which

are

the half-cycle line is used

1.5.2.1

Peak Figure

methods

generally

all

,ff which

rearrangement described to indicate a mean stress

above. line.

Counting E1-36

among

valid,

and

are

can be considered approximations,

usually

In each

more case,

valid a heavy

Techniques.

shows

the

three

a considerably

used. Some of the others the first one is frequently

most

greater

commonly

number

arc modifications modified so that

used

which

have

of the ones positive and

are counted and tabulated separately. The second positive and negative amplitudes so that each cycle

cycle been

counting proposed

shown. negative

For half

and

example, cycles

method pairs off succec(ling mean can be calculated.

The third method, without refinement, is not very accurate unless one frequency is present. If a low and a high frequency appear with equal amplitudes, the high-frequency, low-range (amplitude) activity will mask low-frequency, a high-frequency, of Fig.

high-range component; the result high-amplitude loading which

will be in(listinguishablc is very damaging. See

only the

from Case 3

E1-36.

Ilowever, an important refinement to th(, third m(.thod of Fig. E1-36 is the use of zones and the counting of passages from one, zone to another, i)articularly to a nonadjacent zone. This method, which is listed as method ,_a in the

figure,

sel)arations

are

1.5.2.2

is equivalent used

Statistical

Most cycle-counti|N random

M(,thods

Direct

form

accurate Ilowever, acceptable.

Load

of this

in evaluating their

use

means. cumulative

with

large

and

zone

Analysis.

type,

Two such indirect methods or alternative measurements

by statistical

of zones

r('cor(led.

of Random

techniques

numb¢'r

indicate that some typ(_ of fatigue damage caused by such

as peak

simple and accurate but often quite cumbersome methods are sometimes sought to estimate the

less laboriously. in which sampling useful

prop('rly

2 if a large

of the fundamental concepts of fatigue technique is nee(led to evaluate the

loading.

('ounts art, less direct

and

to metho(1

are

In general, fatigue

quantities

damage of data

counts

methods

as the makes

have

more the

range

to apply. Other, same information

presented in Rcfs. are ust,d andconverted

these

and

loss

direct

2 and 3, into a not been methods.

of accuracy

as

Scdion

1':1

1 Novcnlb_,r Palzc I'lrREM

VERIU|

TIME

AND

HYPOTHETICAL

EOUIVALEIIT

1972

75 |UPPO

R TII_0

PROIl_el

T ION

A l, II,

iU

V

^^AA, VVvv

AA^ VV

NONE IINVALIO)

NONE ilNVALIO|

FIGURE

E1-35.

FIVE

DATA-REDUCING DEGRE

[':S OF

APPROXIMATIONS VALIDITY

OF

VARYING

li|

Section

EI

1 November Page

76

H

i

K

L

-a_@@ TIME

COUNTING

QUANTITIES

METHOD

OR

MEASURED

EQUIVALENT

COUNTED

WAVEFORM D

1.

MEAN PEAK

ClIO&SING COUNT

H SM 2.

PAIRED

RANGE

n,

COUNT IMIIH IMj

3.

LEVEL

AND

FH

L AND

JL

n4000

-

nZO00

" 3

-y

1

CROImlINO

COUNT

o, /

n0 " Z

/

MG.

3L

]'ONE

PASEING

COUNT

SAME

AS

WHEN

PRECEDED

METHO0:3

EXCEPT BY

VARIOUSLY SEPARATED TO SEPARATION

NOTE:

nlEANt.RAIIER CROSSINGS

FIGURE TO

El-36.

A TYPICAL

OF (IF

APPLICATION STRESS

POSITIVE

WITHOUT

OF

RECORD

in Terms

of an

EACH

CROSSING

A CI_O_ING LEVELS

SLOPE

StJ6SCRIIrlr

SOME (Lower Equivalent

AT

EACH

AND

Table

ACCORDING

LEVEL

DESIGNATED

CYCLE

ONLY

SEVERAL

I£ CATEGORIZED

DESIGNATED THE

IS A COUNT OF

LEVEL

COUNTING Illustrates

Record)

JS Sme4m)

METItODS Each

Method

1972

Section

E1

1 November Page 1.5.3

Example

Problem

necessary

To evaluate to correct

stress

alternaticns,

(Paired

Range

the damage caused for the difference S • a

Four

Count

The which

by complex stress-time in mean stress, S

c c:astantamplitudeS-N

first step is to determine are crosshatched in Fig.

approximately

20ksiand,

as

alternating stress of 4ksior than 20 ksi, the alternating any alternating DE, GIt, JK,

second

IL _nd RU

less stress

_or

in Fig.

3tep iL,

by RU inthe

'_trcss-time variationJ

the

alternating

and r('sult 10 -4 this

S

third

= 5 ksi.

113

step

the

for

this

Interpolating

in 3.5x 10 a cycles, tlence, The last step is to sum the

1/2.2

in Fig.

between

Thus, The

remaining

S

rn

the

x 104=

stress less so that

variations stress-time

mean

BC,

variations

stress

4.55x

variations

stress

10 -5

with

cycle,

= (t and

an

is 10

failure would result in Calculating the damage

El-aS(e),

remaining

for

stress

evaluated

is 5 ksi and

n/N=

the

mean

previously

smallest

stress

is to evaluate

hisotry

no damage. At an) mean causes no damage increases

the

reptotted

for different

caused by the smaller variaThe largest mean stress is

under these conditions is n/N = 1/107 = 10 -7.

mannqr

is again

cycles, it is for the various

stress-time

E1-37,

with

is to evaluate

same

hiutory removed.

The

causes which

E1-38(b),

ksi. Referring to Fig. F1-37, l() 7 cycles. !tence the damage caused

in Fig

m

,

curves

stress of 4 ksi or le._s can be omitted. NO, PQ, and _T cau_,;e he fatigue damage.

hi_tory is repl¢),_ted removed. The

shown

the damage E1-38(a).

77

M(,tho(l).

mean stresses are shown in Fig. E i-37. A complex an identical specimen is shown in Fig. 17:1-38.

tions

1972

S

FM, = 10,

m

the

The

evaluated

with

S

failure

the damage caused is 1/3.5x 103= damage caused by all the variations.

a

= 20 ksi

would 2.86 In

x

ease,

10 .7

+ 4.5

Using the cumulative con be repeated 3000

x

10 -5

4

2.86

damage theory times (1.0/3.316

x 10 -4

=

with E n/N x 10-4=

3.316

= 1.0, 3000).

x

the

10 -4

stress-time

history

Section

E1

1 November Page

1972

78

,%

I

C'1

cg ©

8 |

E

)

©

c) >_ L)

5C

_2

Z

Z

"o

<

I

/

_3 O

tilt

'0 9

Section

E1

1 November Page

N

O

J

L

1972

79/_J0

li

(a)

M R

i '° 10

v 0 4 .10 -11 M

U

(b)

ii

'It/

-

\

v

(o)

FIGURE

EI-38.

STRESS-TIME

HISTORY

CURVES

-.,!,4

-.....J

Section

E 1

l Novemi)er

1972

l)a_,( ' 81

1 .(;

MATERIAL

1.6.1

High

The nium,

and

in Fig.

selected data,

strength

of all far,

i.e.,

limits.

It should

For

the

AZSOA-F

The

equal.

limits.

For

Only

a result mens.

anal

one

the

of

selecting

The

notch

the

had

and which

accurate

sometimes

selection

with similar

1.6.2

Low

didate task

materials

straining their

and

materials

material,

are

have

well-defined changed and

approxifatigue alloys

notched

as

speci-

of approximately

2.8.

usually give different results than axial gradients

and stress

may repeated

properties

differences,

of extensive

aluminum, approxi-

best

aluminum,

produced

Different

in the bending

forms

different fatigue strengths.

form,

fatigue

aluminum, ksi,

factor

on

well-defined

is the

in unnotched

alloy

based

should

concentration

(e.g.,

sheet,

Therefore,

be based

tests bar,

an

on fatigue data

factor as the particular

Cycle.

Different

retain

four

strength

in axial tests.

cause

loading,

in question.

of these

all

are

Each

titanium

are

2024-T4

concentration

specimens

have

260-280

of alloy for fatigue strength

structure

repeated

fatigue a stress

greatest

ksi

titanium

material,

class,

7075-T6

steel

heat-treated

due to the large stress

arc not present

rod, etc.)

and

tita-

data

Ti-155A,

materials,

specimen,

best

used

the

260-280

Ti-155A,

steel

has

steel,

El-40.

in its

materials,

titanium

other

of structural

Rotating-bending load specimens

4340

notched

class

other

also, three

ksi

of the

are fatigue

specimen,

heat-treated

that

specimen

El-40.

magnesium,

mately

three

steel

selection

in Fig.

strength

260-280

The noted

shown

fatigue

unnotched

is

materials

rotating-beam

are

the

4340

be

notched

in Fig.

data

highest

For

and

used

Unnotched

heat-treated

material.

fatigue

shown

4340

design

commonly

notched the

steels.

best

equal.

as

and

magnesium,

mately

I'ATIGUI:.

in fatigue

most

magnesium. it has

the

the

AZS0A-F

I{I:SIST

considered four

E1-39

because

available by

to be The

aluminum,

was

is,

step

material.

presented

TO

Cycle.

first

structural

SELECTION

when the

should fatigue

subjected rated,

testing

substantially

stressing;

relative be

show

to large

low-cycle preferably (Ref.

different

also,

23).

they cyclic

fatigue without

have plastic resistance going

abilities various

to resist abilities

strains.

Because

of several through

to

the

canarduous

Section E1 1 November 1972 Page82 % /t /1

11 I!

II I

/

/

t 1 ,

Z ©

I

v--4

I !

IJ

©

U

_

/,

,

ti

-

!

©

I

I I I

t,

<

0

l

/ //_/

/

/,

i

/ ....

© Z

I

/

L_

/

/

/

0

l

s

8 pOI, X _'u_lqt

"ALI|N|O

/i_'u!/q!

'_]WIS|

%

Section

E1

1 November

OF POOR

Ql.:,c_.,,-i-y

Page

1972

83

,,d IJ

=-

L_ < D..

C Z

I

b4

Section

E 1

1 November Page

A distinguishing is that in the large plastic response

differ(race'

latter strains

of most

in deformation

the plastic can induce mate,

fixed

limits

ical

hysteresis

loop

The, is one

of strain,

rather

most

suita_Jle

in which

the

than

develops

high-cycle

an(I

84

low-cycle

fatigue

strain component is consi(lerably larger. significant ( _mngcs in the stress-strain

rials.

resistance

between

1972

stress.

(Fig.

test

for

_tu(lying

specimen

During

E1-41). total

is

strain

range

of an

ponent.

a mechan-

parameter Al,;t,

elastic

is

which

and

Plastic

changes

between

straining,

controlled

posed

these

cycled

cyclic

The

These

is

plastic

com-

com-

component

AE

is P

the

width

of the

whereas, loop

is

hysteresis

the

height

2(Ya,

wh¢'re

loop;

of the a a

hysteresis is the

stress

amplitude.

During

Z# b

rials

/ I

either

ing

ITRAIN

upon

6at

i-

"

E1-41.

cycles.

On

worked

materia

Changes

*IE('HANICAL

SCtlI;:MATIC

but

OF

IIYSTERESIS

to After attaine,l

this

during

shape

until

the,:e

st:abiliz."d

aml)litudcs vidos

which

prior )

ferent

variables

the

in the

description Thus,

on

cyclic

the can

be

(,_btained

of the diagram,

represented

Fig.

rally response

drawn

stable

cyclic

way.

level

or

about

condition

10

is

constant the

tested

at

tips

It pro-

stress-strain

_trcss.strain the

of

different

E1-42(a).

so that

life

life.

through

Fig.

occur of the

after

_p,'cin_t,ns

cyclic

soften.

portion

saturation

E1-42(1)),

in a concise

an

cold-

l_; gent

of the

,:urve,

and

by

stress

an essentially

Curve

st',*ady-:;tat_

(static)

hardening

hand,

condition

maintain fr_m

other

a reasonably

or

utre:;_-sirain

monotonic same

reach

The

For will

indicated

early

20 p('rcent

loops

fract_lrc. lo%r-'

the

in lhe

a steady-state

hy,_;lercsis

hV:_tcr,'qis

a conv('ni_'nt

displayed

sta:,_',

to c,_mplo_-

is c:_lled

of a-material. be

transient

is

the

stt,ady-:-:tate

LOOP

depend-

history.

in th,'_ stress required to the strain limit on successive

rapidly FIGURE

soften,

a cyclic

which

increase enforce

mate-

materials

undergo

process;,

.

or

previous

annealed

generally

"%p

straining,

harden

t h(_'ir

example,

_

cyclic

response curves

effect

may

of dif-

Section 1':1 1 November 1972 Page85

MONOTONIC-COLD CYCLIC

WORKED

ITRi#

ITRAIN

CURVE

\

/

_9 Lu _r I--

I P

/ _NIC-ANNEALED STAIILE

LOOPS

STRAIN

(a)

FIGtTI{I,:

(b)

E1-42.

FROM

CYCLIC

SEVI.;RAL

S'I'RESS-STIUkIN

IIYSTI.:RESIS

LOOPS,

MONOTONIC

Although h'm

oi softening

rial

(lurin_

from

the

metals

low-eych' static

important:

fatigue

the

A component,

load

straining

or

environment.

others

show

whether

gnod

The

relation Nf,

A,

c

a critical

repcatcc[

stress

Two

location

resistance.

It

total

^,

2 P

is

the

fatigu('

--

is,

strain

therefore, or

strain

_-[ ' (2N[)C

ductility

If so,

good

a strain-resistant

by

exponent.

than

one

how

much

the

resistance, necessary

stress-resistant

amplitude,

expect

seem

may

upon

the prob-

of the mate-

would

therefore,

depending

show

Ac '2 t

WITIt

properties

in a structure,

stressing,

the

less

questions,

soften?

materials

between is given

COMPARISON

b(' appreciably

cyclically

requires

I)I:TERMINEI)

(b)

:_incc the strength

curve.

Some

a design

to failur(',

where

or

AS

un(lergo cyclic har(lening or softening,

may

material

(a).

CURVE,

important,

stress-strain

Will

repeated

may

is more

CURVE,

Act/2,

will

tobe it soften?

undergo

either

geometry

and

whereas to decide material.

and

the

cycles

Section

1':1

1 Nero.tuber Page Thus, upon

the

values

for the

Ef t` and

sarily

fatigue

a small

the

short-life ductility

strain

of

c

has

The fact that a material imply that it has a good

resistance

in Ref.

fourth in ability in cyclic strain

ef v ,

it is apparent

importance of deciding whether strains or repeated stresses, was best

resistance

coefficient,

constants, value

strain

23.

to resist resistance

thai.

the best

the material seven materials material

and

c.

,_ material

resistance

shows good cyclic stress

The

of materials

with

typical a large

to repeated

to be selected were tested the

highest

m:Jinly experimental value

of

straining.

strain resistance does resistance. To point

with

86

depends

Using

1972

not necesout the

must resist repeated for both stress and stress

cyclic strain. Likewise, the mat( rial dropped to sixth in stress resistance.

resistance third

Section

E1

1 Noveml)er

j.

Page 1.7

I)I,:SIGN

At tatively The

the

the

level

limiting

or' knowlvdge,

fatigue

in missions,

design

rules stresses,

suggests

this

to be

tatively,

however,

the

practices

strueturc'can

practices stage

is

thcdt, be

sign

taken.

When regulate

space

vehicle

followed

structural

environments

if not

impossible,

in the

design

task. of

to these

problems

can

quanti-

configurations.

and

adherence

fatigue

may

procedures design,

Quali-

fatigue

resist-

established

be

reduced

in the

a few

Keep

2.

Provide

3.

Give

4.

Apply

may

the

for extra

of the

be

design

though

of "design

initial

some

learnin_

apply from

precautions

in the

should

environment.

they

o[ten

they

may

rules"

will

become be

are

more

pertinent

paths

when

general

rules

inappropriate. far

more

guides

which

Recomappropriate.

for

the

design

of

listed:

simple.

multiple

load

consideration

fitting

industry and

although to a new

proposed, even

airframe

experience

other,

unsuited

sense

systems

1.

be

in the Past

to the

are

in the

used

systems.

interpretation,

vehicle

to establish

materials,

if strict

rules

over

designs

used

this

to be and

basic

carried

structural

mendations With

be

Old

specific

impracticable,

potential

the

o[ space can

for

impractical

design.

general,

field

an

defined,

maintained,

of structural In

one

be

it is

loads,

certainly ant

B7

GI TII)I,',S.

))resent

diversity

1972

feasible.

to tension-loaded

factors

of safety

to net

fittings

stresses

anti

around

components. holes

and

cutouts.

5.

Laboratory test tried" structures.

6.

Utilize

longitudinal

(particularly 7.

Provide

8.

Break

for generous

all

sharp

all

newly

designed

joints

grain

direction

of materials

aluminum fillets cdl_es;

and and

polish

steel

and

compare

with

whenever

"time-

possible

alloys).

radii. critical

regions

if it is

considered

necessary.

t

Reduce bearing minimums.

stresses

in riveted

and

bolted

members

to design

Section

E1

1 November Page 10.

Take

precautions

11.

Whenever

12.

Ensure

possible, that

gradual, Provide

easy

14.

Provide structure.

inspection

15.

When rather

16.

Design

service

procedures

inspection during

parts

for

minimum

residual

and

superposition

18.

Make ables,

a proper selection of materials fabricability, and environmental

19.

Pay close attention of components.

20.

Establish strengths.

21.

Construct rigid duction parts.

to the by the

list

above.

and reliability manufacturing. i'hese

welding

and

are

and

assembly

Many

such

guides

for

recommended

fatigue

are

unwritten

results

practices

for

development

allow-

optimum

forming

of joints

manufacturer

guides and

which only

Excellence

experienced

for the sole purpose of reducing the overall desig_ l:lv_JL_t to assembly in production.

this

strength in mind.

for

the

useful

specialists

as those

material

reproducibility

for

specialists.

metallurgists,

as well

and

techniques

tooling

guides

design

with cost, effects

by the designer alone. It requires to perform complex dynamic stress

engineers, analysis,

precision

installation;

of

strains.

techniques

additional

experienced

is not accomplished of specialists able vibration

section.

in design.

to fabrication

reliable

there

most

of "notches"

in

of structure.

or

tensile

Avoid

added

result

fabrication

mismatch

preload

17.

known

in cross

fittings.

oractical, produce parts and fittings from forged than from extrusions or machined plate stock.

Undoubtedly,

ticians,

for

and

reinforcements

changes

88

corrosion. of joints

structural

abrupt,

access

from

e(eentricity

and

than

13.

parts

reduce

doublers

rather

in lower

ever, ation

to protect

1972

of pro-

could

intuitively

in design, the close analyses:

in structural in tooling

be how-

cooperacoustesting

and

designers

are

time

from

suggested preliminary

Section

E1

1 November Page

1972

89

REFERENCES

.

1

Kennedy,

A.

and

Inc.,

Sons,

Madayag,

J.:

Processes New York,

York,

Fatigue

Design

Missile

and Space

Peterson,

4,

R.

and Sons,

o

Manson, Book

o

.

B.

New

Fatigue

R. W.,

Q

Manson,

John Wiley and

No.

Design

Structures. 3290,

Douglas

January

Factors.

1965.

John Wiley

1953.

Stress

Structures.

and

Cambridge

Low-Cycle

of Pressure

Hirschberg, NASA

Vessels of ASME,

M. H., under

and

Strain

TN D-1574,

Fatigue.

for

University

McGraw-Hill

Low-Cycle

September

Manson,

Cycling

April

Manson,

June

Fatigue.

1962.

S. S. : Fatigue

in Low and

Intermediate

1963.

1967.

Fatigue.

Procedure

Experimental

May

for Estimating Mechanics,

High-Temperature,

August 1968.

S. S., Halford, G. F., and Hirschberg,

Fatigue Analysis by Strain-Range

12.

Spacecraft

Paper

of Welded

S. S. : A Simple

Low-Cycle 11.

Wiley

Manson, S. S., and tialford, G. : h Method of Estimating High Temperature Low-Cycle Fatigue Behavior of Metals. NASA TM X-52270,

10.

and

Concentration

Transactions

of Materials

Range.

John

1966.

Engr.,

Behavior

Launch Division,

York,

F. : Design

of Basic

Life

for

E. : Stress

Inc.,

Company,

Smith

in Metals.

1963.

Systems

S. S. : Thermal

Langer, J.

Fatigue

1969.

Criteria

Gurney, M. A.: Press, 1968.

.

and

A. F. : Metal Fatigue: Theory and Design.

Sons, Inc., New

.

of Creep

M.

Partitioning. NASA

H. : CreepTM

X-67838,

1971.

Coffin, L. F., Jr. : A Note on Low Cycle Fatigue Laws. Materials, JMLSA,

Journal of

Vol. 6, No. 2, June 1971, pp. 388-402.

Section

E1

1 November Page REFERENCES

13.

Baldwin,

E.

E.,

Sokol,

G.

Fatigue Studies on AISI 57, 1957, pp. 567-586. 14.

Clauss, Ductile

15.

Spera,

D.

lated 16.

Francis Materials,

Damage.

Itarry,

Titanium.

(:offin,

347 Stainless

TN D-5489,

: Thermal Am.

James Tech.

L.

F.

Soc.

Jr.

Steel.

Trans.

Life October

and

Mechanical

Metals,

Vol.

51,

1959,

18.

Coffin, L. F., Jr,: The Problem of Thermal-Stress Austenitic Steels at Elevated Temperatures; ASTM 165,

1954,

pp.

Fatigue 1969.

ASTM,

of Ductile

Based

Fatigue

Clauss, Tech.

J. : Thermal D-69, October

Strain Vol.

on Accumu-

1969.

17.

F. Note

: Cyclic

W. : Thermal Fatigue of Notes 4160 and 4165, 1958.

of Thermal-Fatigue

NASA

Jr.

Trans.

and

and Freeman, I and II. NASA

A. : Calculation

Creep

Majors,

J.,

(Concluded)

J.,

Type

1972

90

of Nickel pp.

Materials,

III.

and

421-437. NASA

Fatigue in Spec. Tech.

Publ.

31-49.

19.

Liu, S. I., Lynch, J. T., Ripling, E. J., and Sachs, G.: Low Cycle Fatigue of Aluminum AHoy 24ST in Direct Stress. Trans. AIME, Vol. 175, 1958, p. 469.

20.

Avery, Strain

It. S.: Discussion in Heat-Resisting

1942,

pp.

of Cyclic TemperaturcAccelerationof Alloys. Trans. Am. Soc. Metals,

Vet.

30,

1130-1133.

2].

Freudenthal, A. M.: Some R('marks on Cumulative Damage in Fatigue Testing and Fatigue Design. Welding in the World, Vol. 6, No. 4, 1968.

22.

Bruhn,

E.

H. : Analysis

Tri-State 23.

Feltner, Low

_"

U.

Offset

II.

GOVERNMF.NT

C.

Cycle

PRINTING

E.

Company, and

Fatigue.

OFFICIrl

and

Landgraf, ASME

1973--746_Z7/4779

Design

of Flight

Cincinnati,

Ohio,

Vehicle 1965.

R. W. : Selecting Paper

No.

Structures.

Materials

to Resist

69-DE-59.

"_"_

SECTION E2 FRACTURE MECHANICS

_'_

TABLE

OF

CONTENTS

Page

E2

FRACTURE

_

EC

HANICS

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

,f-

2.1

GENERAL 2.1.1

2.2

................................. Comparison

STRESS-INTENSITY 2.2.1

Plane

2.2.2

Plane

Strain Stress

GROWTtI

2.3.1

Sustained 2.3.1.1

2.3.2

Cyclic 2.3.2.1

2.3.2.2

2,3.2.3

2.4

2.4.1

Load

Load

Surface

Flaws

10

......

16 C racks

2O

........

21

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

Flaw

23

Growth

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

Effects

FlawGrowth

Theories

23

Paris

II.

Foreman

25

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

25 27

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

29

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

III. Tiffany Crack Growth

32

..................... Retardation

Wheeler's

Retardation

II.

The Signi[icancc of Fatigue Closure .....................

Transition

Parameter

from

Partial-Thickness

Through-Thickness

Cracks

and

Sustained

FRACTURE

32

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

I.

Cyclic

24

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

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

I.

Flaw

35 Cracks

Cyclic

Growth

39/40

41

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

44

Problem

A

orSustaine(I

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

Loading

..........

Example

l)roblem

A

II.

Example

Problem

B .............

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

E2-iii

PAGE

......

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

1.

PRECEDING

39,/40

.........

41

Loading Example

32

MECIIANICS

of Materials Static

....

Crack

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

I. 2.4.1.2

6

Deep

Environmental

Selection 2.4.1.1

5

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

OF

TECHNOLOGY

for

Determination

Combined

APPLICATION

. . .

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

to 2.3.3

Mechanics

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

Through-the-Thickness

Experimental

FLAW

Fracture

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

Correction

2.2.2.1 2.2.3

and

FACTORS

2.2.1.1

2.3

of Fatigue

,$

_',

2

_,.,_k_,_('_.O'f FtcJ_r..l.)

45 49 49 57

TABLE

OF CONTENTS

(Concluded)

_J

Page 2.4.2

Predicting 2.4.2.1

Critical Flaw Surface Cracks

2.4.2.2

I. Example Problem A ............. II. Example Problem B ............. Embedded Flaws ..................

2.4.2.3

Through-the-Thickness I.

2.4.3

Structure 2.4.3.1

2.4.3.2

59 60

C racks

Problem

A

61 62 64 ........

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

Design ........................ Service Life Requirements Predictions .....................

69

Problem

II.

Vessel) Example

..................... Problem B (Thin-Walled

A (Thick-Walled

Vessel) ..................... Allowable Initial Flaw Size Problem

Nondestructive Inspection Limits ........................

2.4.3.4

Proof-Test I.

Example

...........

A ..............

2.4.3.3

Factor

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

A

76 80 87 88

Acceptance 89

Selection

Problem

66 69

Example

Example

66

and

I.

I.

REFERENCES

Example

Sizes ............... ...................

..........

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

94 97 100

E2-iv

Section E2 1 November E2

FRACTURE

2.1

GENERAL. Structures

been

MECHANICS.

subjected

designed

primarily

of the

material.

stresses

below

the yield

failures

suspension

have

defect

of such

or flaw

was

strength,

with

in such

aircraft

landing

failures

indicated

the

found

effects

of flaws

influence

effect.

To be useful

must

be translated

to the type

into

metal

producer

of failure

has

Basic near

the

flaw

geometry.

Intensity

of flaw

types

to fracture

cracks

concepts

the are

motor

tanks,

cases.

An

A small

origin. control

and

lies and

body

in understanding those

sense,

structural

as fracture

and

at

These

fcature:

in an engineering

The

prematurely

as storage

rocket

in metals

is the

failed

predominant

failure

have

ultimate

consequences.

and

one

and/or

structures

fracture

mechanics

These

have

of tests

known

crack

structures

diverse

designer.

become

strength

temperatures

factors this

that

understanding

mechanics

of knowledge

both

familiar

concerning

this

mechanics. understanding

relationship discussed

of the

between

state

gross

in subsection

2.2,

of stress

stress

and

Stress-

Factors. growth

or crack

is handled

best

growth

is given

Finally, relates

and

tip of a sharp

Flaw which

the

and

yield

gears,

key to brittle

at moderate

disastrous

at the

the weakening this

of the

of these

usually

Therefore,

loads

Many

occurred

bridges,

examination

to constant

on the basis

strength

brittle

1972

by fracture

subsection

stress-intensity

analysis

of structures.

because

of its

propagation

importance

mechanics

in subsection 2.4, factors Particular in the

under

cyclic

concepts.

loads

is a basic

A thorough

problem

discussion

2.3.

Application and flaw

of Fracture growth

attention acrospace

to the

is given industry.

Mechanics engineering to pressure-vessel

Technology, design

and design

Section E2 1 November Page 2.1.1

Comparison Similarities

are

summarized

primarily

of Fatigue and

concept

makes

manner

and

has

E2-1.

tests;

to handle

greater

fatigue

fracture

applicability

2

Mechanics.

between

Both

of laboratory

it possible shown

Fracture

dissimilarities

in Table

on results

and

1972

fatigue

and fracture

and fracture

however,

the

to fatigue

mechanics

fracture

considerations crack

mechanics depend

mechanics in a quantitative propagation.

Section

E2

1 November Page

1972

3/4

.,.._

o o

o

0 0

o

_-_

f.)

-_

0,.._

[/1 o

,_

.,._

• tm

o

0

_ r/l

0

0

_

o _

-_

_

o

o

"_

_

bD

-_

o

o

r/l • ,"_

0

o

0

2o

I

_o

0

_

_

o

_

o o

o

_

o

o

o

.,_

o

_

o

_:_

m

b_

• ,-i

_

>

_._

,x_

_x2

[/1

m

2"d

o

<

_ 0

g_

°_.._

0,.._

%

0_

_d

d

0

""'_

0 _

o

_

m

o

o

0,-.4

o

o

o

E °,.,_

.n

o >

0

0

"o o

z'_

.,-_

tm

o m

_d

.,_

0

_S

_,

"_

•

b_

_=_

bl

.,.-i

°,.._

°,-t O9 °_._

I Oq

o

0

• 0

._ m

_

I_,

m

m

•,_

0

0,...I

_

_

_

-,_

b.O "_

_

_ _

_

_,_ .,._ _

>

o •

0

<

O

< ._

.g

_o

Section E2 1 November Page 2.2

S TRE SS-INTENSITY To understand

first

to learn The

attempts

precise

crack,

The vicinity

are

The

for

crack

stress

fields

can be divided

types,

each

mode Fig.

into

of deformation, E2-1.

The

associated

with

in which

the

directly

apart.

mode,

II,

placements

crack The

slide

mode

III,

faces

slide

with

parallel

Mode

I is the

is the

only

I,

and the

and

manner

focuses

takes

in the

place.

.

__.,_/

a.

Mode

I.

b.

by dis-

the

crack

Mode

II.

sur-

respect to the

J

sur-

the crack

c.

to one leading

critical

edge.

mode

one to be discussed For

of the in which

on the conditions

and

Mode

III.

FIGURE E2-1. THREE MENT MODES FOR SURFACES

section.

shape

edge-sliding

In

most

crack,

size

is

move

one another.

tearing,

of the

crack

displacement

is characterized

over

to unstable

in

mode,

surfaces

in which

faces

another

a local

It

a local

as shown

opening

value

basic

with

the

concisely:

crack

three

associated

it is helpful

part.

fracture

near

of resistance

containing

a quantitative

tip where

can be stated

be independent

to the

in design,

it is based.

mechanics

must

applied

is used

measure

of the part

search

of the

of fracture

measure

the geometry loads

on which

a quantitative

This

external

mechanics

theory

goal

to provide

propagation.

tips

of the

5

FAC TORS.

how fracture

some

1972

information

in this

on modes

II and III,

see

Ref.

1.

DISPLACECRACK

Section

E2

1 November Page 2.2.1

Plane The

stress

tip for

mode

These

equations

stress)

I are

in terms

pendicular)

crack

1972

6

Strain. conditions, defined give

the

of the

or plane-strain

by the

expressions

components polar

surface

elastic shown

of stress

coordinates

r

displacements.

(a

in Fig. = normal

and Only

stress

_

for

the

first

field, E2-2

at the crack (Ref.

stress,

r = shear

opening-mode term

2).

of each

(perequation

y

m w ee I-

I CRACK

Ox _-_---_ r

Ox

O!

HORIZONTAL

DISTANCE

FROM

Oy - (2 lr r)1/2

Ox"

(21rr)1/2

",,v" FIGURE KI,

E2-2.

RELATIONSHIP

AND STRESS

2

FRONT

2 /

(_

costs' BETWEEN

COMPONENTS

CRACK

cosT

"'"

STRESS-INTENSITY

IN THE

XrICINITY

FACTOR,

OF A CRACK

Section

E2

1 November Page is shown. crack

The

complete

half-length).

For

equations

are

power

practical

purposes,

series all

in

terms

r/a

7

(crack

beyond

1972

tip radius/

the

first

are

negligible. All that

has

the

been

three designated

independent

of

intensity

stress

r

the

and

_

at any point

which,

components

provides

The

assumption

proportional

stress-intensity

factor,

and therefore

near

if known,

are

the

crack

gives tip.

complete

to a scalar K I.

This

quantity

factor

is

a single

description

of the

It is a purely

numerical

quantity

knowledge

of the

stress

field

at the

stress

crack

tip.

ture

basic

occurs

called

when

fracture

between

KI

KI

in fracture

reaches

in an equation

Kic.

The

describing

unstable

propagation

of the

reflects

a material's

ability

and

Kic

strength Irwin

and Sneddon

KI

for a body (Ref.

crack.

stresses

free

(Ref.

This

to withstand

and Kic

designated

factor

is a particular

3) used

analysis

elastic

is that

the

K

in the

vicinity

of

value

is a material

is analogous

KI

difference a coefficient

of a crack

corresponding

stress

frac-

commonly

is simply

value

a given

an unstable Kic,

to appreciate

stress-intensity

the

toughness

between

value

It is important

Fracture

difference

a critical

toughness. and

mechanics

tip. to

property

at a crack

to the difference

and tip.

The

between

stress

with

the Green

of discontinuities. the

expressions

4) to show

shown

that

the

in Fig.

expression

E2-2 for

the

stress

t /

intensity E2-3)

around

the

crack

periphery

for

the

embedded

is

KI

= T

a_a

[a 2 cos 2 _b + c 2 sin 2 9]

elliptical

flaw

(Fig.

Section E2 1 November Page where

_

equations is the and

is the uniform of the flaw

semimajor •

stress

periphery

axis

k=

=

are

of the

is the complete

the modulus

perpendicular g - c cos

ellipse,

elliptical

a

integral

[(c 2 -a2)/c_]t/2;

_2

1 -

_

and

The parametric

y = a cos

semtminor

of the second

8

axis kind

_

, where

of the

c

ellipse,

corresponding

to

\

i.e.,

"

0

is the

to the crack.

1972

sln2 q

d@

c2 i

or

• = 1 + 4.593(a/2c)

of

a/2c

from

the graph

In seeking surface

flaw

t"_s.

Values

shown

of

in Fig.

an expression

4) can be obtained for various values E2-4.

for the stress

in a finite-thickness

plate,

Irwin

intensity assumed

for

a semielliptical

that

i/4 KI =

where

a

a T

cr_

[a2 cos_ _b +

is a correction factor to account for the effect on stress intensityof

the stress-free surface from which the flaw emanates,

and _

is a correction

factor to account for the effect on stress intensityof the plastic yielding around the flaw periphery. Values of a surface flaws with

and _/ were a/c

estimated by Irwin and considered valid for

ratios less than one and flaw depths not exceeding 50

percent of the plate thickness.

The resulting expression for the stress inten-

sity was

KI =

1.1 _

cr(a/Q)I/2 I _-1

[a2 cos2 @ + c2 sin 2 _1 It/4

v

Section

E2

1 November Page

vf I

1972

9

/"

\\\\

/

FIGURE

E2-3.

/

/

I I I I I EMBEDDI':D

UNIFORM

ELLII)TICAL-SIIAPI':D

TFNSILI':

STRESS

CRACK

UNDER

IN y-DIRECTION

1 1 1

I

I

"1 J

J 0.1

I

l l ! 1

.,_,

1 I

jr

I

l

f

0.01 0.001

J

/"

l 0.01

0.1

1

_2.1

FIGURE

E2-4.

SIIAPE

FACTOR

VALUES

Section

E2

1 November

where of the

Q = _2

_ 0.212

material.

Figure

depth-to-width

ratio.

The

maximum

of the ellipse

and

rapidly. called

The the

stress and

_

value

critical

=

Figure

E2-6

values

of

conditions,

of

1.1

_a

For the

crack

Mk

= 1.1

is the

and

becomes

at the

fracture

for

and the

semiminor

flaw

axis

Surface

that

are the

are

Thus,

Kic

plane-strain

is is the

conditions Thus,

equation.

shown

shapes

Some

in Table

of cracks,

in Table

typical

E2-2.

different

loading

E2-3.

Flaws. with

opposite

respect

to plate

thickness,

surface,

Irwin's

equation

5) as follows:

(a/Q)1/2

factor

propagates

instability

toughness.

of this

given

deep

and

.

other

for Deep

magnification

Q

of this

under

materials

are

M k _-a

between

KIc.

fracture

l/2

location

(Ref.

strength

of the

inception

representation

approaches

yield

unstable

is designated

plane-strain

shuttle

flaws

by Kobayashi

KI

where

surface

at the end

size

to cause

factors

crack

flaw

(a/Qcr)

for space

relationship

uniaxial

1/2

KI

the

is the

10

of

computed

of

a ys

K I occurs

the

KI

called

Correction

modified

of

a

necessary

and

when

the

of

Stress-intensity

is,

shows

is a graphical

Kic

2.2.1.1

E2-5

a (a/Q)

value

is commonly

Kic

and

value

value

intensity

YS)2,

has a value

K I = 1.1

At some

(a/a

1972

Page

for

deep

flaw

effects.

that has

been

Section

E2

1 November

f-

Page

1972

11

Ot: pOOR QUA_._TY

,F "o

< 2;

('Nt

0

o

_=._o i

@ [

< I

(3 _.-

,o

ua i" UJ

•

_9 < m ©

o.

, • _

,,d

i i -

>

"1-

I

0_.

j

M

r_

_d

ol

I

M

e_ 0Zl.

ci

o

Section E2 1 November Page

1972

12 _j

i

!

FLAW SIZE RATIO (a/Q)

FIGURE

E2-6.

APPLIED

Experimental and

shapes

E2-7

2.2.2

and

curves

Plane

or simply

strains.

In general, and

For do not vary prevail. and

thin

consideration the

directions

the

strains sheet the

such,

strain

considerable

of substantiation

RATIO

varying of the

investigations

two different

plastic

in fracture and

flaw

sizes

Kobayashi are

materials

of stress

exist

in all three

being

are

shown

"state

of

in the

thickness

flow attends

the

external

of plane

cracking

stresses

and that

directions.

to in-plane

direction

applied

is three-dimensional,

principal

a condition

is the

of the

in a body

subjected

thickness,

mechanics

magnitudes

state

specimens

through

As

for

with

SIZE

Stress.

stress,"

stresses

materials

experimental

Mk

FLAW

E2-8.

An important

is,

degree

more for

CRITICAL

on several

a fair

however,

Typical

VERSUS

obtained

to provide

factor;

performed. Figs.

data

appear

magnification

STRESS

stress

is virtually process.

loads

which

is thought

to

unsuppressed

in

Section

E2

1 November Page Table

E2-2.

Properties for

of Typical Use

on Space

Ftu

f_-k

Alloy

4340

(High

4340

(Low

Strength) Strength)

Materials

13

Considered

Shuttle F ty

(ksi)

(ksi)

260

217

52

180

158

100

D6AC

(High

Strength)

275

231

61

D6AC

(Low

Strength)

218

203

112

18 Ni

(250)

263

253

76

18 Ni

(200)

206

198

100

190

180

226

190

180

160

150

140

250

115

100

180

2014-T6

66

60

23

2024-T4

62

47

28

2219-T87

63

51

27

6061-T6

42

36

71

7075-T6

76

69

26

169

158

51

125

118

120

12

Ni

9Ni-4

HY

Cr

-

150

T-1

6AI-4V 5A1-2.5

(STA) Sn

1972

Section

E2

1 November Page Table

E2-3.

Stress-Intensity

normal

stress

T 1

xI = o.,/;T

__.if,

KII

= Kll I =

CIII3

i

at

infinity

_:_ •

"

IT

_

infinity in'plane

shear

at

•

Kll I =

0

sheet with

tunnel

crack =u_ect

to out.of.plane

_'.]

KI =

Infinite

ahem

at infinity

KII

f)

14

Factors

Cam 2 Infinite cracked sheet with uniform

Infinite cracked Iheet with uniform

1972

(.9

(9

_

r K I ffi KII

=

0

r r

r

"

o

<>

_

-.: .

"---"--C_

L Case 5

array of cracks along a line uniform stress at infinity

KI = o _

KII

= Kil I =

_eo_

.-teor-,_

r

4

Periodic a sheet,

--dzo_--

tan

2h

in

na

cl_e

Periodic

array

a sheet, infinity

uniform

_ff cracks in-plaue

% = ,,,_7 _

0

KI =

Ktl

along shear

a line stress

in

Periodic array of cracks along a sheet, unitorm trot-of-plane at infinity

at

,an :_/

Kll I =

I = 0

KI =

¢ _Vr_'l

KII

=

/2h 7r--_- tan

a line in shear

_¢ra )

_A

0

Y

r

p

f

I

i

Case 7

Case 8

Concentrated slirlace infinite

KI

KII

K =

=

force

of a crack sheet

on the

=

)½

' (:/

2 nx/_'

3.4 v (for

i

in an

P (a+ 2v_\a - hb

+

plane

II

strain)

a

I

.

2,/;7 \, , ,/

2,/E\.

- _/

Curved crack m equal axial stress licld

hi-

t_

(,,oo(, •

¢

Section

E2

1 November Page

Table

CaN

E2-3.

tension

15

(Continued)

Ca,,e 12

g

Inclined

1972

crack

in uniform

in infinite

sheet

Edge

crack

m

finite shear

body

subjecte(t

a semi-inh>

K I = o sin 2 _ KI Kll

= KII

Kll |

CaN

= 0

= o sin_cosCv"_a = r x/_'_"

10

Crack

in infinite

subject

sheet

to arbitrary

and couple point

:%-"

torce

at a remote

_L

Case 13

L--2o:..t

Central crack in strip snbiecl to tension (finite widlhl

right end

At

K=

I

2 x/;_ (t +,O

I [ (P + iO)

(a + zo) (_02 _a2), A

2

•---------

b

+_)M

(_o- a)(_2 _ a' )'_ =

tt zn

(3-v)/(l+v)forplanesereas =

go

+ iYo

x _

= Xo-

a/b

7,

J

+ ai(I

= o _X/'_flX) =

X

"1 I +

a(P-iQ)(ro-Zo)

,I

KI a

_-a

= 3.4vforplanestrain

fO,)

0074 0.207

I (10 I 0_

0.275 0.337

1.05 I Oq

0.410

1.13

0.466 0.535 0.592

I. I 1.25 1.31

iyo

6M -- One L/r

Crack

--

--Two

EL/r) Uniaxial

Crack

-

f(L/r) Biaxial

Uniaxial

Biaxial

Case

Case 14

11

Cracks

from

infinite

sheet

hole

KI

in

Notched

beam

= (h

KII 0

Stress 3.39

Stress 2.26

Stress 3.39

Stress 2.26

0.1 0,2

2.73 2.30

1.98 1.82

2.73 2,4l

I.()8 1.83

0.3

2.04

1.67

2.15

1.70

0.4 0.6

1.86 1.64

1.58 1.42

I.q6 1.71

1.61 1£2

0.8

1.47

1.32

1,58

1.43

I.O

1.37

1.22

1.45

1.38

1.5 0.5 2.0

1.1_ 1.73 1.06

1.06 1.49 1.01

1.20 1.83 1.21

1.26 1.57 1.20

3.0

0.94

0.93

I. 14

I. 13

5.0

0.81

0.81

1.07

1.06

10.0

0.75

0.75

1.03

1.03

0.707

0.707

IO0

1.00

**

KI

=

ov,_-

a)*/2

g(a/h)

in bending = Kll

I =

[)

_- f(+)

= 0

a/h

g(a/h

().05

0.3t_

0.1 0.2

0.49 060

0.3 0.4 (}.5

0 t,6 ()6q 0.72

0.6

073

0.6

073

KII

e

w _-'_"_e

> l'__

_

Section

E2

1 November ORIGINAL

PAGE

IS

OF POOR

QUALITY

Page Table

f(alb) L/b

a/h

f(alh)

= I

L/h

flalb)

= 3

L/h

1.13

1.12

l

02

1.13

I.II

1.12

(14

l

I +Of+

I 14

0.5

1.14

1.02

115

06

I I0

1.01

1.22

07

IO2

1.00

134

O.K ()')

l.Ol I .IX)

I00 I .O0

I.<;7 2 0')

K I

=

kll

l.r

u

=

gll

L _ _

_

I

_2h

=

tan

_-_ na

15

I)_,.hle-edge

nolch

10

K I =

Stogie-edge

f(a/b)

KII

12

=

Kll I

=

O

_"

--2b---

f, _,r

OI O+2

115 I?O

114 I.Iq

0.3 0.4 a/h 0.5

1.2') I.t7 f(a/b) 1,51

1,29 1.37 Ra/h) 1.50

0.6

1 6M

I_+

07

I,_lg

I;'17

08 {) q

2.14 246

2.12 2 44

I O

2K6

• _.X.

O

LISt:

+ O lsin

um _

=

C=w r(mnd

19

wilh c,ack

c'lrC'lllnlerenlial

K I hal

m

=

One I V_

fid/DI

lenslt)n 0

= KII

=

KIll

-_

C4..

" V_

notch

'/-' I(:J/h)

)

16

(Concluded)

Corn

-o,,

OI

16

Cose

E2-3.

1972

diD

Rd/D)

d/D

lid/D)

OI

OJII

0.75

O2"_7

l)2 (); il4 o

0.I s.i ()IX'; O21_) o

o.80 0 x5 (I 7o (]ql)

(2 0 225 0 2()5 (; 24_l

It 5,

0227

(),15

0 1(+2

ILl+

()2IX

0')7

(1 IR)

0-_40

I IK)

(I

16

_+'lll+¢lllpllCal

Still'ace

crack

i11 plate

|o gC|leral

SLII')+_CI

exlensH)rl

,+[,.0.,+, I,-+>]

lib

(;

Case 20 KII

= ()

Ill

Kill

r d)'_ n_/_a

=

'b,

where

crauk b(,tly

('Irclll;tr inrinllc

(21 _kn-'-_-

isglVell

+r;,)

tan

Illlll_/lll

t

17

lwl)

Cqtlal

"_"

/

..---.

/2S

by

I-_

h_

]_i,_=++

:

Kill

:

')

,....

t'l_line;ir

in an

cracks

in_inill2

_heel

_uhlec{

h+ llnll(+llM

IL'n_IIHI

lhe

heal

Mlhlecl

ilqllrll

dL'ft'l K I =

ID lhhnHC

+"

I,+ tllI+10llll

_-++

/

\

I E'll%il III

<'l}d_

I'o1

-_

_ra_k

l'11iph_;d body

At

_*_"_"

_)TI klilt_

Illlllt'd by

k III!C

/lll_e

+I_ i h 2 _ ;,2 )'_

r

K

=

--

I h 2 _ KIp)

(3 t"

I%1II \

'1).

LOS 2

_ a J

KII = + (h _ -a2l A!

the

lar

kll

.

- Kll

ends

,,, KI

'T.

=

ov/_-£(._

pKIp) E(p)

gll = rvr_

)

#Kip)

1 E K(_)

K

anti of

are Ihe

Ihe

firs!

Ctilllplclc and

_cond

//,'/'/

5/"

do

0

Caa

o \--_.,

/ /"

'k

KII

'_,, = ./"

Ill +Ill _ublecl I('n_lon

elllplic kinds

inlegrals fespedwely

F4#)

and

I

=

()


\.__._.. /

_6 de

" " "'""

--

Section

E2

1 November

r

OF

POOR

QU,_J.iTY

Page

1972

17

2.2 al2c_ 0.06 2.0

I f I IBASED ON TILTS OF O.I_.in.-THICK BASE METAL (TRANSVERSE GRAIN) AND WELD METAL AT -_°F AND -42301:]

1.8

_lr

m

1.8

A

o/2c •

0.10

alZc = 0.20

• ad2c = 0.30

"1:i _*-o=, _

,,_ o/2c = 0.40

1.2

•

1.0 0

0.10

0.20

0.310

0.40

0.50

0.60

0.70

0.80

0.90

a/t

FIGURE

E2-7.

M k CURVES

FOR

5A1-2.5Sn

(ELI)

TITANIUM

ALLOY

2.0

I •,/2c - 0.o6

Jr 1.4

- 0.40

1.0 0

0.1

0.2

0.3

0.4

0.6

0.|

0.7

0.|

a/t

FIGURE

E2-8.

M k CURVES

FOR

2219-T87

ALUMINUM

ALLOY

O.O

Section E2 1 November Page

For

thick

considerably flow

specimens,

by the

is associated

very

the

plate

but

rather

the

there

are

no transverse

and tip

stress).

much

less

plastic

zone

The plastic the

zone

fracture

flow

in a plate

size

amount

ance

plastic

of the

of shear surface.

of the

of both.

stresses

occurs.

to restrain

is shown

will

vary

in either At the plastic

in Fig.

plane

stress

free

surfaces

flow

(a condition

of

conditions

representation

of the

prevail crack-

E2-9.

to

7-----_A/", THICKNESS __/

at the

the

plastic

SHADED AREA = CRACK SURFACE HEAVY SOLID LINE = CRACK FRONT//'_

stress

left

less

plane-strain

A schematic

specimen

Thus,

fracture

proportion

to be related tip

is suppressed

and noticeably

completely

at mid-thickness,

plane

is thought

material

is seldom

in some

In contrast,

direction

18

process.

specimen

strain

of plane

of the

cracking

or plane plate

in the thickness

thickness

with

A laboratory

strain

1972

.J_-fZ

//_//_

OF SPEC_/MEN /J//

appearaccord-

/

/f_'_ /

and plane

strain

thickness

of the plate. The

conditions

influence

through

of stress

(and

associated

plasticity)

ture

toughness

is illustrated

E2-10,

which

shows

the

appearance.

This

by low values

of toughness.

appearing)

fracture

figure

the

FIGURE E2-9. REPRESENTATION OF PLASTICALLY DEFORMED

state

on the

frac-

REGION

AT A CRACK

FRONT

in Fig.

effect

of plate

shows

appearance.

This

that

thickness the

larger

corresponds

on the toughness thicknesses to a completely

are

and

fracture

characterized square

(brittle-

Section

E2

1 November Page

1972

19

f-APPEARANCE OF FRACTURE SURFACE

I

|0

I

Kic

U

0

! 0.2

l 0.4

! 0.6

1 0.8

I 1.0

THICKNESS (in.)

FIGURE

E2-10.

TOUGHNESS

EFFECT AND

A reduction at

the

quently zone,

advancing raises in turn,

the

constraint.

toughness

increases in Fig.

E2-10.

In the

aerospace

enough

to

fall

testing

in this

in the area

tip.

fracture

relaxes

decreases

shown

PHYSICAL

in plate crack

OF

PLATE

APPEARANCE

thickness This

The rapidly

stress process

enlarges

region is being

The

in the is

of plane done.

the

the

local

ON THE

thicknesses stress

plastic

zone

However,

constr',int

and

conse-

of a larger

direction,

which and

of thickness

and

a determination

the

plastic further

fracture

variation,

of structures behavior

FRACTURE

of plastic

self-accelerating range

FRACTURE

degree

development

thickness

in a narrow

industry

OF

decreases

toughness.

the

TIIICKNESS

are as

as

usually

a result of plane

thin more stress

Section

E2

1 November Page intensity able

factors

research

is far is needed.

propagation

occurs

as crack

length

mixed

complicated

It is very

because

the

than

hard

was

first

unstable

to determine condition

20

supposed when

and consider-

unstable

is approached

crack

very

gradually

increases.

At present the

more

1972

there

mode

is no direct

fracture

method

condition

for

to useful

translating

numbers

for

laboratory designing

data

for

practical

hardware. 2.2.2.1

Through-the-Thickness In thin-walled

before exist

thickness

failure

any

load

cracks

the

_e

is the

stress

strength

length

normal (ksi),

The

and

critical

K

where

w

c

= a

e

plane crack

The basic

for plastic

+

plane

is the width

zone

of the

crack

is the critical stress

of the

through

the thickness crack

stress

in an infinitely

plane-stress

intensity

n c

panel

crack at failure

for

is

_r _

plane

through-thickness

{ [( wtan

grow

may

equation

for

through-

wide

plate

is

(in.)

t

1 Kc2

of the

to the

a through-thickness

K

c

_ T

may

or a through-the-thickness

is applied.

lr

c

cracks

occurs

corrected

K 2 =if2

where

structures,

catastrophic before

C racks.

+

II nff y2

(in.).

at failure (ksi), stress

a finite-width

a

a

is

is the yield Y intensity (ksi 4-_.) panel

containing

Section E2 1 November 1972 Page 21 2.2.3

Experimental Among

the

improved

the

design

were

specimen

values

tests

has

result

in

geometry

Kic and

several

but this

test

may

Table

E2-4

describes

how to analyze

uses

and

how to set data,

see

of test

Numerous

tests

are

specimen

mechanics gives

Refs.

test valid

instance,

for

data

ASTM

types

limitations. up and

for

1, {I, and

on

The and data

independent

of crack

used

No.

(E399-70T

detailed

to determine

materials;

committee

the tests, 7.

E2-4.

has

of the been

is proposed), materials.

specimens, information what

each E-24

high-toughness

For

to

performed

procedures,

all

of fracture

conduct

been

is universally

low-strength,

relates

in Table

are

is

loading.

out a standard

some

have

shown

that

mechanics

specimens

types,

of external

to bring

not be valid

and their

specimens,

manner

in fracture

behavior

determinations

For

years

progress

of which the

no one test

its limitations. for

some

no fracture

because

working

other

components.

to determine

At present KIc

recent

of how the

types,

designed which

obtained,

important

understanding

of specimen

analysis and

most

of structural

a variety tests

the

Determination.

data

the data on these to obtain,

and and

Section

E2

1 November Page Table

E2-4°

Seven

Specimen

Data

Uniaxial

tension,

induced

bending

Edge

Obtained

tension

tension or eyclie)

Breaking

stresses,

Kic

Breaking K

stresses,

C racks in size

Ic

Breaking flaw

stresses,

growth

Kie,_

I ,J

o

must

(cyclic

0

,1

rates,

K , c

Simulates penetration flaw in har(h_art,. K

e

tension or

Breaking

static)

stresses,

Kic

or rotating flexure fatigue Bar

Is width

(lept'ndt,nt.

C

Simulates

bolts

and

shafts. Difficult form concentric prccraek.

Notched

C racldine-Loaded or

Tension

with

induced

bending

KIc,

Kii

Compact

Wedge Compact

Uniaxial (static

tension or

Partial-Thickness

cyclic)

Breaking stresses, flaw sizes, apparent Kic

L_ o

Simulates flaws

Difficult

to analyze.

May

provide

not

valid

KIc

Kic

Only

standardized

test

for

applicable

Cracked

natural

in hardware.

values.

Crack

Three-point loading

ASTM

be equal

Through-Crack Uniaxial

Opening, Tension

Uses/Limitations

C rack

(statie

Round

Specimens

Crack

Uniaxial

Central

of Fracture

22

w

Uniaxial

Double-

Types

Loading _to

Single-Edge

Common

1972

Slow

Bend

Kic.

Not

to most

thin and tough materials.

to

Section

E2

1 November f--

Page 2.3

FLAW

2.3.1

Sustained

1972

23

GROWTH. Load

F law Growth.

//

One space

industry

sustained

most

is the

and the

surfaces

thickness

ensues.

time

leaked

under

To predict flaw

When

such

growth

failures

grew

one

can occur,

maximum

possible

initial

When

sustained

arise

vessels

loading.

to critical

happens, must

in the

caused

as

size

sizes

the

in the

before

growing

which

initial

when

through

fracture

under

actual

have small

catastrophic

the

vessel

by

cases,

conditions

either

aero-

cracks

In other

complete know

as well flaw

can

through-the-thickness

sustained

this

that

of pressure

cases,

flaws

shell.

problems

failure

In some

or embedded of the

subcritical

structural

delayed

vessels

cracks

or the

serious

pressurization.

formed

the

of the

flaw

it is placed

size into

service.

is often

the

termed The

probably

stress

found

flaw

initial

closely

and

it can be oriented A procedure

The using

surface Kic

for the

loads

observed,

e.g.,

not occur to obtain some

for

low-stress

the

the

induced,

laboratory

material

specimens

growth fatigue

vessels.

of flaws

evaluation

of sustained

each

1 and time

information cycles)

of 2,

(e.g.,

flawed

it

illustrated

in Fig.

specimens by "marking"

and

pulling

the

3 and the

this

in

specim in service

flaw

in Fig. (pull)

required E2-11. 4),

tests.

apart.

Then,

with

differ-

for failure If failure

it is still front

growth

E2-11.

is loaded

crack

specimen

growth

stress

specimen time

has

desired.

static

Kic ) and the

flaw

encountered

illustrated from

specimen

With

often

characteristics

established

fractions

stress

growth

specimens,

(various

pressure

is schematically

is first

of cracked

sustained

type

flaw

specimens

in a reasonable crack

aerospace

to suit

of flawed

ent initial

is environmentally

type

in evaluating

simulates

flawed

a batch

use

thin-walled"

the

growth

or "part-through"

the widest

and

flaw

corrosion.

surface-flawed

both "thick-

using

stress

does

possible

(applying

.,

Section

E2

1 November Page

1972

24 r

Ki

Kk_

KTH 5 6 • FAILED GROWTH. NO FAILURE NO GROWTH, NO FAILURE LOG TIME

FIGURE FOR

E2-11.

SCHEMATIC

LABORATORY GROWTH

USING

The

threshold

highest

stress

level of K

intensity,

OF

SURFACE

A point is finally reached occurs.

ILLUSTRATION

EVALUATION

FLAWED

at which

or

A

PROCEDURE FLAW

SPECIMENS

neither failure nor flaw growth

for which

KTH;

OF

SUSTAINED-STRESS

this condition occurs

Kiscc

if due to stress

is called the

corrosion

cracking. 2.3.1.1

Environmental

The in relatively than half of

discovery

Effects.

of a unique

inert environments;

can be 80 percent

hostile media

of

can reduce

Kic

or higher

the value to less

Kic.

Considerable under

evidcnce

most

severe

from

through-the-thickness

in specimen

KTH

conditions

thickness.

indicates that sustained of plane

cracked

strain with

specimen

KTH

load flaw growth values

tests increasing

is

determined with a decrease

Section

E2

1 November Page Studies indicate rate

of flaw

a monotonic

and design

materials, wide

relation

titanium,

on environmental

in Table

and

the

has

been

for

the life.

These

empirical

which

have

Load

In these

to failure,

obtained

on a number

tests

important

for

KTH ,

marked are

KTH

a

dependence encountered.

of sustained

of different

of some

and growth

critically

and very

amount

Flaw

load

material-

information

is given

stress

of fracture

are

then

growth

mechanics

relating enable

K

a designer

loads

to the

design

is a basic

for testing

to various

following

The

fatigue

laboratory

correlated

factor

cyclic

characteristics

dealing

rates.

under

the

The

of studies intensity

material

propagation

proposed.

A number

would

crack

data

been

crack

Growth.

through

Theories.

factor

the

a considerable

determined

2.3.2.1

ular

for

temperature)

flaw-growth

theories.

fatigue

steels.

environments

intensity

and

Subcritical

prominent

the

stress

determined

A summary

application

generally

that

been

(media

combinations.

Understanding

are

increasing

times

few years

in aggressive

E2-5. Cyclic

service

intensity

high-strength short

past

data

2.3.2

ment

between

characteristics

During

environment

stress

have

abnormally

growth

and

correlations

scatter,

flaw

growth

1972

25

require-

of structures various

for

materials

of flawed

specimens.

crack-propagation

is a discussion

theories

of some

of the

with

fatigue

crack

is the

most

important

variable

affecting

of a master

curve

for a partic-

availability

crack-growth to predict

rate growth

and

propagation

range

have

mor(,

shown

of stress-intensity

rates

for

any cracked

have

been

published

equations

that

body

configuration. Numerous the

last

10 years.

are

simply

the

"laws"

of fatigue

Basically, attempt

all

crack the

of an individual

growth

various

investigator

to obtain

have

been

a curve

during obtained that

will

Section

E2

1 November Page

Table

E2-5.

Typical Threshold Stress-Intensity Data Material-Environment Combinations

Temp. Material

a ys

(" 1")

6A I-4V

Titanium

Forging

- STA

(ksi)

Titanium (llcat-

Affected

Zones

5A1-2.5

Sn

Titanium

)

(ELI)

Plate

2219-TI47

Aluminum

Fluid

)

! 6O

44

Methanol

44

Freon

R.T.

16o

44

N204

R.T.

16o

44

N:O

160

44

[[_o

(distilled)

! 60

44

lt,O

(distilled)

T.

30f_o

NO)

4 (o.

6o%

NO)

16o

44

lh.lium,

It.

T.

16O

,14

Aerozine

It.

T.

16o

44

Fre,m

9O

16O

44

N:O

90

T.

I (ii.:lli',rO

44

Ai,rozine

0.75

R.T.

126

39

Mt_ttlanoi

It. T.

126

39

I,'l'con

R.T.

126

39

II,(J

(liistilled)

II. T.

126

39

II,O

(l)istilled)

-32o

1 St}

64

LN:

(¢r

," I'vol).

lAir,

-:12o

1 I_o

64

LN,

((r

". Prop.

Limit)

-423

210

52

Lll,

5t4

36

Air

-320

66

41

I,N;!

-423

72

44

LII,

Water

GTA

Welds 1:|0

Salt

(ll.i;ll
NO)

ti. 75

50

I,'.

I). 4 II O. I'16 _

Na,2CrO

I

0.

it)

_0.90 o.

II. 1,2 a -0.

W:itCI'

Steel

II.T.

200

(250)

Steel

It.T.

235

75

12

Ni-5

R.T.

170

1 55

Salt-water

R.T.

17(]

120

Salt-ware

R.T.

140

"_2011

Stilt-

R.T.

165

",l:lli

Gaseous

Alumimlm

R.T.

50

Aluminum

II. T.

65

Salt-water S;lit-v,

<0.

Slira ate

wattH"

_'

a.

No

failure

KTI l ; some

growth

observed

at

lower

values.

20

-0.70

r Spray

">l). 7h

Spray

-(I.

r Slirav

70

_0.70

Spray

tlydroileo

at

5o00

psig

o.

25

N '(it

O. 70

N.O_

0.35

Plate

5

_,_:[

O. 24

(200)

30.

l_2

ii. fill

Ni

?late

_2

-0.90

18

2021-T851

75

0.28 M.

1i'1 Ni

2219-T851

O. B0 0.7t

NO)

16(I

"-- 60

71 8

II, H2

l".

11 (I

-20o

lneonel

ft. 9o

50

o.

R.T.

Steel

GOX

Monomcthylhydrazinc

Steel

5 NI-Cr-Mo

or

N:()_

4340

Steel

O. _6

Air,

.14

90

Co-2.5C

O. _2

.t-i

205

9 Ni-4

4

i 6O

R.T.

Steel

O. 8:1

+ Na2CrO

160

Steel

Mo

0.74

105

4:;:10

CR-3

O. 58

F.

(o.

T.

KTII/KIc

0.24 M.

It.

26

for Various

Environment

16o

It. T.

Plate

_'i'_.

it.. T.

R.T.

Weldments

KIe

(ksi

R.T.

It.

6A I-4V

Typ.

1972

Section E2 1 November

r

Page best

fit his

order

data.

Some

polynominal

still

others

with

different

choice

accuracy

range

of interest.

results quite

divided

slopes

versus

over

curve-fitting

data,

others

the data

in each between

For

techniques

have

used

into regions

to obtain

a statistics

and

a high-

approach,

constructed

and

straight

lines

region. equations

of flaw-growth

range

may

be that

prediction

example,

a limited

given

an equation

of data,

of simplicity from

may

the

equation

be very

but out of this

of equation over

simple

range

the

argued

that

and

the give

equation

good

may

be

inaccurate. I.

Paris.

Paris

and Erdogan

should

be a function

factor

defines

large

body

da

(IN

where and

c n

stress

of data

could

be fitted

-

examph',

field

factor around

K on the

the

crack

by an expression

growth

rate

that

this

grounds

tip.

of the

the

They

found that

a

form

c (AK) n

is a material

is an exponent

the

is four,

AK

a typical

of Paris's

equation

plot,

which

by substitution

Separate

values ratio)

constant, having

obtained

(load

for

stress-intensity

elastic

On a log-log

R

8),

the

E2-12. line

(Ref.

of the

An example

of

used

to fit the

have

The

have

1972

27

of the

the

of data

of

into the

coefficients

because

value

equation

is the value

Paris's

is the

c

range

of strcss-intensity

of four

for

for a typical becomes n.

The

Paris and

equation

is shown

a straight

equation n

steel.

steel

constant

must does

factor,

line. e-

and

The

slope

5.(; x 10 .24 solving

be computed not have

in Fig.

R

for for each

of

is c. value

as a function.

Section

E 2

1 November POOR

W

--

Page

QUALITY

AIS! 1048 STEEL oys - 37 _ lid _*F AIR ENVIRONMENT 10 Wdes/_

1972

28

__

2

10

FIGURE

20 STRESS-INTENSITY

E2-12. INTENSITY

FATIGUE FACTOR

40 80 80 100 FACTOR RANGE, AK (ksi_'n.)

CRACK RANGE

GROWTH FOR

RATE AISI

1045

VERSUS

STRESS-

STEEL

P

+

Section

E2

1 November Page II.

Foreman rates

Paris

et al. tend

applied

fourth-power (Ref.

stress also

to explicitly

approaches

modified

the

for

the

c

and

AK

K

C

(1

-

n

are

= plane

data

fell

into an

also

obtained

(1 - R) for

the

does fit the

K

c

The straight-line

Kmin)

and

constant midrange

observed

crack

growth

maximum

material. behavior

. /K . mm max

reflex

The

and

Foreman

c

of the

It was

rates

by

shape

Foremants

equation

is determined

that

AK

is the

is the from

the

slope

the 7075-T6 curvature

is

approaching reason

Paris's

the

as ForemanVs

by Hudson

primary

equation.

consequently, as well

made

A reflex

it is induced intrinsic

was found

of curvature.

curvature;

conditions.

material.

aluminum.

by using

test

cycle.

equations

type

in Foreman's

and

Foreman's

This

data

and

a load

equation;

low growth n

and

or reflex

fit of the

at high or

R = K

toughness

7075-T6

denominator.

this

the

on material

during

fracture

shape

for

for

of the

by

is

dependent

Foreman's

in the

not provide data

S

that as thc

toughness

law to account ratio,

behavior instability

fracture

modified

AK

of Paris_s

from

excellent

_

stress

10 for 2024-T3

the

was

n C

-

equation

an apparent

of load

constants

= (Kmax

rate

the observed

conditions

R)K

A comparison in Ref.

effect

c (AK)

dN

for

Paris

plane-stress

growth

toward

intensity

da

where

rapidly

express

expression

crack

9) to account

to increase

Foreman

29

Foreman.

The F

1972

equation

equation

cannot

equations. of the

substitution

curve

in the

of data

bectlon

E2

1 November Page

value

into

ing

on the

and

da/dN

the

equation.

type

of plot

should

used.

be

values

c

and

that

n

a log-log

30

and

c

will

change,

plot

of

AK

in psi

depend_/-_m.

is used.

equation of

noted

Generally,

in microinches/inch

Foreman's typical

It

1972

2219-T87

for

other

n

for

is

shown

common

in Fig.

materials

E2-13 are

and

given

some

in Table

E2-6.

Table

E2-6.

Crack

Propagation

Coefficients

da

c (AK)

d"_"

:

(1

-

R)

for

Foreman's

Equation

n

K

_

AN

c

da/dN AK

in./cycle and

K

psi

x/_n.

C

KIc Material

psi

2219-T87

R°T.

:_00

1.4

x 10 -il

2.5

33,000

1.5

× 10 -li

2.47

31,

600

10 -13

2.7

36,

200

x 10 -t4

3.0

81,000

-320

9.0X

TI-6A1-4V

R.T.

7.8

2024-T3

R.T.

3.22

7075-T6

R.T.

2.13 x i0- 13

x 10-i4

3.38

3.21

517A(TI)

The value

problem

Runge-Kutta

solution and method.

of the can

Foreman

bc

solved

For

most

equation by

direct

practical

can

be

formulated

numerical

integration

problems,

an

initial

as

an

using crack

initialthe size

is

Section

E2

1 November

¢-

Page

1972

31

§ r i

,':

! _C

!

I: ¢d

0

_D IIi,

,

i

:!I (3 r..)

•

.<

r

o.

Z

g

2; I

,E

r_2 ..1

z

I&.

c

'-

_

_

:,,_'

m

#

II

le

U

,U

_

..zn

_c.u .I-

_g_'"

d I

I..,,4

0 0 F-

('ua_!"l))IV

'3ONW_I AJ.ISN31J.NI-Sl;]WJ.$

q i,-

Section E2 1 November Page known the

at an initial

crack

length

number

value (or

applied

N = 0.

The

problem

stress-intensity

on

data

flaw

is to determine

factor)

of the

material

particular

using

required

the

method

after

the

called

curves

for

industry

to-failure

curve

for 2219-T87

2.3.2.2

Crack

Growth

The of varying level Many

(Ref.

a given

12).

levels.

accurate

initial

stress

cyclic

life

curve.

Because

conditions The problems

observed

for use

of

rates

Kii/Kic

were

the

analysis

each

test,

Kii/Kic

is common a

were

combinations

Flaw-growth

shows

intensity

data

it was

material-cnvironment

E2-14

of speci-

the versus

in the

versus

cycles-

temperature.

Retardation.

of crack

papers

lives

where

curves.

been

to the plane-strain

graphs,

design

at room

retardation

of load

(KIi)

analysis.

Figure

has

the cyclic

Accordingly,

and final

practical

Retardation

but a computational and

initial

Wheeler's

load

.

life

an end-point

aerospace

• I.

cycle

and

cyclic

that

rates

of maximum

by a unique

of the

of only

cycles-to-failure

loading

conditions

slopes

was

ratio

(Kic)

represented

knowledge

Tiffany

first

noted

cycles-to-failure

test

be reasonably

computed

of the

the

versus

flaw-growth

Tiffany

during

Kii/Kic for

11).

a function

toughness

plotted

to plane-strain

(Ref.

primarily

to the

fracture

simple

the

approach

by Tiffany

were

a high

as

Tiffany.

presented

could

such

additionally

An alternate

that

N,

32

of cycles. III.

mens

of

1972

Parameter. growth

Retardation

followed have

has

by a lower discussed

technique to gain

has

is a phenomenon been level

crack

use

to occur

occurs particularly

because after

of load.

growth

not been

widespread

shown

which

retardation

presented (Ref.

which 13).

to some is sufficiently

extent

Section

E2

1 November Page

1972

33

/

b4 <

|

/

0 0

< oo

F_ o,1 C ..J I

0

L,_

O b..J t_ )-

8 0 0 r..) r_ r..)

4 I

R

q

w.

P

o

q. o

o

Section

E2

1 November Page Wheeler can be made equation, His

(Ref.

13)

suggests

by introducing

which

equation

serves

for

more

a retardation

to delay

crack

that

length

the

accurate

crack

parameter

crack

growth

in the after

growth

crack

a high

1972

34 predictions

growth

load

application.

is

r

r

where

a

o

pi

i=1

is the

crack

length

after

r

load

applications,

a

is the

r

crack the

length, change

eter

Cpi in the

is taken

C

where from

R

Y

crack

exponent

Ra')m

is the

,

extent

dependent

upon

parameter

in specimens

a

a

of the

+ R

Y

configurations,

that

approach

cumulative ana lysis.

crack

represents growth,

load.

load,

is

AK

and

1

The

retardation

- a

is the

param-

>- a

P

P

yield

interface and

been

and

ith


current

material has

ith

Y

+ R

subjected

physical this

,

and

at

at

form:

tip to elastic-plastic

This cracks

1

parameter factor

following

Y _

p

=

retardation

stress-intensity

=

P

is the

in the

P

C

initial

o

used

test

zone, (Fig.

which

can

E2-15),

successfully

spectra,

improvement be used

and

distance

m

to predict

of two materials

a useful

P

is the

shaping

data.

to six different made

a

with

the growth

having (Ref. on the

confidence

three

13). idea

of

different

It is believed of linear

in design

and

Section

E2

1 November

f-

Page

1972

35

4ira

/

\ =

/

J

\

J

My

\

/iNTERFACE

/ I i

":_ .l.\

!

FIGURE The in crack tion

computational

growth

The

perform the

This

any

II.

realistic

analysis.

The

work,

closed

the

usually

under

mination stress

alloy

stresses

are the

compressive

of the

crack

analysis

that

digital

(see

(Refs. close

the

crack

technique the

transmitted

loads closure

of a cracked

has and

stress

load

of a highly

been

applicanonlinear

required

incorporated

Utilization

to into

Manual).

Closure.

15)

has

all tensile across

the

been

made

must,

parameter

one

is obviously

has

Crack

open

structure.

be grown

Computer

before

retardation

linearization

14 and

assumption

ZONES this

computer

of Fatigue

by Elber

of aluminum

compressive vious

CRACKS

YIELD

incorporating

This

Significance

work

TIP

to a piecewise

of a high-speed

Recent in sheets

amounts

program

for

requires

use

computer

CRACK

scheme

predictions

at a time.

process.

E2-15.

under

shown load

crack

fatigue

is removed. at zero

implicitly

tensile

therefore,

that

loads.

Significant

load.

that

cracks

In pre-

a crack

The

be a necessary

is

deterstep

in

Section E2 1 November Page EIber stress

level

analysis the

(Ref. and uses

crack

closure

behind

a fatigue

crack

crack

crack

experiments

the

crack

produced

larger

to show

length zone

the

mations.

Figure

the

surrounded

been

subjected

mations. crack mations behind of the

which

During growth, are the

moving

surfaces.

residual

by the

shows

and

tensile

amplitude

existence strains.

loading

that

retardation

of a zone In Fig.

is shown

of

E2-16

at three

CRACK

by a

been defor-

represents

by the during

envelope

crack

to plastic a single

crack

zone

retained

to plastic

in the

stress

plastic

had

E2-16c

|YMBOLIC FLAITIC ZONE

at a

the

been

residual left

crack

because

material

previously

of all zones

loading

The

shows

surrounded

has

subjected

crack

opening

normally.

The

E2-16a that

for acceleration

by a plastic

the

is higher.

of Figure

crack

equation.

account

constant

E2-16a

shows

plastic

intensity

propagation

can be explained

under

as it is represented

crack

the

amplitude

tip having

tip surrounded

greater

a crack

for

on variable could

stress

Figure

E2-16b

for

relation

36

propagation.

lengths.

Figure

had

it as a basis

closure

material

an empirical

phenomenon

in crack Crack

zone

obtains

of qualitative

effects

the

15)

1972

ENVELOPE OF ALL PLASTIC ZONIEI

growth defor-

cycle

tensile

of

FIGURE

E2-16.

DEVELOPMENT

A PLASTIC ZONE AROUND FATIGUE CRACK

defor-

OF A

material front,

as only

elastic

recovery

occurs

after

separation

Section E2 1 November Page Crack cycle

propagation

in which

the

crack

to analytically

predict

crack

opening

stress

which

an effective

is defined

there

can

occur

is fully crack

open

should

that

crack

rates,

be used

range

during

at the

propagation

level stress

only

could

portion

tip;

of the

in attempting

stress

The

loading

reasonable

as a reference

be obtained.

37

therefore,

it seems

1972

that

level

effective

the

from

stress

range

as

,f

ASef f = S

where

S

is the

op

crack

An effective

C

S

op

opening

stress

range

stress. ratio

(Sma x

-

Sop)

ASef f

(S max

-

Smin)

AS

is then

dcfined

as

were

conducted

_.

Constant

amplitude

relationship

between

a significant

effect

stress

-

max

U on

loading and

three

tests

variables

U , namely,

which

stress-intensity

were

to establish

the

anticipated

to have

range,

crack

length,

and

ratio. For

significant

the

given

variable.

expressed

range The

of testing relation

conditions,

between

U

only the and

as

U =

for 2024-T3

0.5

+ 0.4R

aluminum

alloy.

where

-0.1

< R<

0.7

R

stress

is linear

ratio and

R

can be

is a

Section

E2

1 November Page One of the to predict

accurately

tude loading. amplitude

in 2024-T3

may

aluminum 3/2.

K

be a significant

is still

at

these

Kop

does

is the inability

under

variable

on the basis leading

ampli-

of constant

to errors

of

level

variable

Therefore,

changes.

amplitude observed

a high load application

been

termed

delayed

loading

followed

retardation.

left by the high-load this

compressive

residual

crack

tip,

this

crack

propagates

new fracture

cycle

plastic

zone

does

into the

This

plastic

zone,

clamping

zone,

requires

to open the crack;

hence,

the crack

this

come

to a standstill.

and may

tip.

will

for some

This after

builds

opening.

at a decreasing

plastic

of the As the

act on the

up as the

externally

a

the

is ahead

will

has

material

causing

the crack

action,

time

of the large

zone

action

propagate

until stress

growth

zone,

the clamping

a larger,

intensity,

The elastic

plastic

which

how-

magnitude.

of crack

not influence

level,

opening

to grow

on this

as this

Kop =

stress

the behavior

like a clamp

--

max

the new con-

not propagate

of smaller

of the crack

K

be investigated.

continue

by examining

As long

into the plastic

surfaces.

will

is at

opening

of the crack

Such retardation

acts

action

does

therefore

by loads

ahead

stresses.

clamping

must

that a crack

level

to the new peak

the crack

that a crack

is halved,

The crack

The behavior

high load can be explained

surrounding

, equal

interaction

R = 0 and

opening

suddenly

R = 0.

these

Assume

the conditions

the crack

and

in causing

example.

range

10 MN/m3/2

after

zone

rates

ignored,

factor

under

conditions

not open.

opening

under

propagates

structures

propagation crack

by the following

= 10 MN/m3/2

max

It has been

zone

crack

are usually

If the stress-intensity

are

the crack

single

in aircraft

these

is propagating

Under

so the crack

level

effects

can be shown

10 MN/m3P.

ever,

of fatigue

to calculate

interaction

closure

This

ditions

the rate

problems

38

magnitude.

Crack

20 MN/m

important

In attempts

data,

significant

effects.

most

1972

applied rate

crack stress into

Section

E2

1 November Page

2.3.2.3

Transition Cracks. It was

from

shown

partial-thickness

in Section

cracks

the

thickness this

in crack

crack

occurs,

value

of

and will

grow

when

a

the

(crack

at=

t

Combined Tiffany

threshold load)

have

limited

to the

Kii

to the

one

additional

stantiate

a large

effect.

number

initially

the

to a through

plate

When

must

crack

face

be a partial-

thickness.

be made.

is chosen

of the

is _iven

this

prediction.

(KTH), cyclic

to

material.

The

as

hypothesis the

flaw

growth

amount the

stress

the

data

below (or rate,

the

level,

the

cyclic initial

failure

at maximum

lifc stress

could

stress

KTH was intensity

occur

in

long.

do tend

threshold

time

but above

of experimental

effect

sustained

hold

minimum

sufficiently

of no significant

sustained

speed

KTH

were

is only a limited

that

to increase

the

time

However,

Growth.

cyclic

words,

required

above

hold

Flaw

hypothesized

In other

and

if the

there

below

l)

the

of cycles

value,

cycle

(Ref

not affect

To date

rates

and Sustained

value

would

KTH

Tiffany-Masters growth

Cyclic

stress-intensity

it could

through

occurs

a finite

for

13).

will

the back

Also,

for

expression

reaches

for

ys /

and Masters

probably

Case

a crack

flaw

different

_-

\

2.3.3

be made

stress-intensity

this

was

cracks.

E2-2,

it extends

for which

1

-

until

zone

length)

to Through-Thickness

intensity

must

(Table

a surface

plastic

stress

problems

in the from

the

corrections

propagation

transition

point

that

equation

corrections

The be the

intensity

C racks

for through-the-thickness

cracks,

stress

Often

2.2

and

through-the-thickness within

Partial-Thickness

1972

39/40

to support of cyclic

stress

data

to sub-

the

original

speed

intensity.

on flaw

Section E2 1 November Page

f-

APPLICATION Selection

FRACTURE

OF

MECHANICS

1972

41

TECHNOLOGY.

of Materials.

f_

In the

material

as a pressure 1.

parts 2.

the The

fracture material.

flaw

growth,

and

the

maximum

and

possible

generally

size

before

fracture

data

initial

depend

failure) levels

likely

in the

?

to exist

flaw

in the

crack The

through (KIe)

leading initial

the

flaw

2.2),

can

also

of flaws

flaws, embedded

edge

is high,

and

or may

depending

applied

stress

load. encountered

flaws,

flaws, the

plane-strain

not reach

upon levels,

in

to determine

embedded

and

may

used

subcritical

a [)roof

types

structural

are

be used

after

the

inherent

evaluate

surface

flaws

failure

of the

sizes,

as surface

the thickness, value,

the

specimens

in a structure

For

cause

upon

test

They

(Section

cracks.

prevail.

heavily

from

life.

and

?

critical

size

size

characteristics

derived

can be categorized

at the

toughness

sizes

flaw-growth

mentioned

growing

cause stress

to critical

to predict

through-the-thickness

ditions

flaw

structure

structural

structures

of constraint

of the

subcritical

estimate

degree

grow

questions

toughness

As previously in fabricated

life

analysis

and

flaws

to these

mechanics

which

operational

initial

such

be considered:

(sizes

at expected

initial service

Fracture

must

structure,

?

these

answers

of a tension-loaded

questions

maximum

service

toughness

fracture

the

expected

design

criticalflawsizes

structure

are

Will

and

following

the

of the

before 3.

the

are

What

structure

during

vessel, What

different

selection

con-

critical

the

plane-strain

and

the

material

respect

to the

p

thickness.

If the

wall

thickness,

ture

is not likely.

calculated

the

formation

critical

flaw

size

is small

of a through-the-thickness

with

crack

before

frae-

Section E2 1 November 1972 Page 42 For material,

through-the-thickness

stress

ness.

If the

material

predominate. from for

of full

fracture

the

been

The also

been

cance

(Kc)

common

and to show

common

degree been

that

fracture

values

values

and

should

thick-

generally changes

fracture. the

as the

be used.

Thus,

plane-stress

thickness The

is

theory

of this

2.2.2).

of fracture

specimens

2.2.3).

and

their

It is appropriate

application

and

fracture-toughness

material

requirements

to point

anisotropy

correlations

differ the

applied

have

out the

signifi-

on specimen

among

several

Kic

values

stresses

Considering

directions found

selec-

of the

more

vary.

of these

in a given thus

far

In a rolled

possible,

and

(Fig.

of these

directions

to some

basic

alloy,

on various plate

it

materials

or forging,

plane-strain

directions

in each

vary

six

toughness

E2-17).

The

depends

(Kic)

need

to

on the direction

in the hardware. the

that

banding the

and,

likewise,

to be the

case

in

are

generally

directions

performed

also

propagation

properties

grain

tests

values

in each

it appears

and

fracture

toughness

may

mechanical

forms

from

of flaw

determine

been

important,

a given

material

appearance

cracks,

(Section

various

found

directions

B

the fracture

through-the-thickness are

the

conditions

or plane-strain

as conventional

among

has

plates,

plane-stress

for

specimens. Just

of the

upon

flat

(Section

types

of fracture

depends

thickness,

in detail

discussed

thin,

(Kic)

of end-hardware

tion

temperature

values

plane-strain

discussed

the mode

to an essentially

containing

thoughness

increased has

increasing

shear

sections

and test is relatively

With

that thin

level,

cracks,

the

differences

KIc

bar

or single-edge-notched

Kic

and

values

the from

values

delamination

C

can be different

and

D

fracture

using specimens.

(Ref.

16)

This

thick

the has

and tends

surface-flawed The

in some

between

directions.

investigation obtained

problems

and

surface-flawed

A

and

actually to explain round-notchedspecimen

Section

E2

1 November Page

1972

43

LONG TRANSVERSE*

LONGITUDINAL-

r-" J E

/

F

FORWARD

"_

/

UP

/

, OUTBOARD

A UP

SHORT TRANSVERSE*

A - F: DIRECTIONS *GRAIN

OF FLAW PROPAGATION

DIRECTION

FIGURE E2-17. POSSIBLE

is normally the

used

that

its

in the

tions.

the For

same

toughness

In the

short

a significant

although

B

longitudinal

of either

for

to measure

single-edge-notched

toughness

there

A

STRESS FIELD, DIRECTIONS OF

or

center-cracked

D

directions.

B

material

is

where

should transverse

be

apparent

to the or

the

are

no

obtained

direction

difference is no

there

in

Kic

lower

A

experimental

or

C

notched

directions,

(removed

measures

of either

the

C

directional

of which there the

substantiation

E

so

the or

lower

D

direc-

effects,

the

specimen appears

and

measure

bar

surface)

between

AND

specimens

pronounced

materials, values

round plate

regardless of

the

(pop-in) The

parallel

directions

DIRECTIONS, PROPAGATION

in either

or

axis or

toughness

GRAIN FLAW

is used. to be

and of this.

F

no

reason

directions,

Section

E2

1 November Page For

weldments,

toughness

between

it is considered growth

it is known

the weld probable

establishment

centerline

that

characteristics

vary

of realistic

that

and

fracture within

there

heat-affected

toughness

the

allowable

the

In addition,

as subcritical

zone

sizes,

in fracture

zone.

as well

heat-affected

flaw

44

can be differences

the

1972

so that

minimum

flaw

for the

Kic

values

must

be determined. The use

discussion

of comparable

when the

foregoing

they

are

desired

valid

(i. e., wall

lower

to use the

the

specimen

toughness

in the

A

and

and

selection,

the

selected

a valid

comparison

growth

characteristics

2.4.1.1

Static

the and

flaw

growth

data

types

in either

valid

directions

While

because the

A

it may

Kic

to obtain

selection.

they

or

B direc-

not always limitations

exceeds

the hardware

specimen

and

be

thickness

single-edge-notched D

the

desirable

of material for

where is such

toughness

selection of the

the

might

surface-flawed

and material

that

there

toughness required. data

is no single data

needed

Of primary

for different

be representative as used

are

in the

"best for

material

importance

materials of toughness

hardware

fracture

provide and

application.

Loading.

An evaluation requires

for

directions,

because

appears

nor

specimen for

D

insuring

specimen

materials

values

or

for

directions.

in all situations

is that

fracture

C

B

it presently

to use

comparisons

a case,

subcritical

be considered

diameter

in the

In summary, specimen"

type

specimen In such

C

necessity

of proper

toughness

the

specimen

required

for

lower

in either

thickness).

be used

the

selection

might

the and

in comparing

specimens obtain

or the

possible

data,

clear

toughness

or the

directional

automatically tions

fracture

available,

round-notched-bar

makes

of the the

resistance

following

basic

of materials material

to catastrophic

properties:

brittle

flaw

Section E2 1 November 1972 Page 1.

Plane-strain

fracture

2.

Conventional

tensile

An evaluation specimens

Example

Three initially The

yield

ratios. and

The

design

yield

requirement

is chosen and

are

shown

Alloy

( [b/in.

3)

ys (ksi)

and

materials to attain

Kic

values

in the

a Density

accumulated

a hypothetical

a titanium,

candidate

strengths

ys

on the data

by using

-- a steel,

of each

(r

from

test

example.

A.

as potential

strength

Kic.

strength,

based

best

Problem

materials

selected

yield

of materials

can be illustrated I.

toughness,

45

an aluminum for

minimum

nearly

equivalent

obtained

from

following

-- are

weight

design.

strength/weight the

tested

specimens

table.

/Density ys × 1000

Applied 1/2

Kic,

(in.)

alloy

Stress ys

(ksi 4 in.)

(ksi) 125

Steel

0. 284

250

880

1oo

Alum inure

0. 098

85

870

30

42.5

Titanium

0. 163

140

860

80

7O

Assume

that

1.

The

defect

is a semielliptical

2.

The

defect

is located

To decide establish

which

which material

material requires

surface

flaw

plate

loaded

in a thick provides the

largest

the

most critical

with

a/2c

= 0.2.

in tension.

fracture flaw

resistance size

for

is to catastrophic

fracture. For "thick walled" structures criticalflaw sizes can be determined from the following equation:

Section

E2

1 November Page

(a/Q) er

1972

46

i.

or

cr

where

the

1.217r

shape

comparison,

factor

parameter

can

be

obtained

from

Fig.

E2-5.

For

this

Q = 1.26.

The

results

are

shown

in the

following

Depth,

table.

a Length,

cr

Alloy

2c

(in.)

(in.)

Steel

0.212

1.06

A luminum

O. 165

0.83

Titanium

0.432

2.16

Conclusion.

The the

largest

titanium critical

This ratios

for

the

alloy flaw

is

most

fracture

size

defect,

conclusion

could

have

various

materials

Alloy

a

been shown

resistant for

er'

catastrophic

reached in the

Kic/O'y

in terms

by following

s (_m.)

Steel

0.400

Aluminum

0. 353

Titanium

0. 572

of requiring fracture.

considering table.

the

Kic/Cr

ys

Section

E2

1 November

F

Page

The be the

titanium,

toughest

material

Tiffany materials, that

of the

unflawed E2-18a

The

ordinate

the

influence

Fig.

E2-18b

critical a 135-ksi

materials

example,

titanium,

and

a 70-ksi

make

the

vides

a somewhat

comparison

Based

for

shown lighter

also

Reevaluate detectable

flaw

(NDI)

the

Rearranging

in.

the

basis

equation

fracture

stresses

the

practical

flaw

of equal

a 200-ksi

that

steel,

might

the wish

titanium

pro-

of available to catastrophic

maximum

allowable

that

minimum

applJ_d

material. assuming in. results

the

long. in

Kic 2 (Q) 0-2

The

resulting

summarized

=

1.21 (a) critical in the

following

table.

and other

pertinent

to

size.

resistance the

From

one

capability the

placing

approximately

shows

in

E2-18b.

thus

basis

from

of weight,

of equal

shown

in Fig.

size,

have

percentage

data

perspective.

all

This

Recognizing

the

shown

upon

effect

E2-18c.

by 0.75

basic

E2-18a.

flaw

would

example,

deep

to

several

to a fixed

desi,_ned

by calculating

preceding

is 0.15

as

critical

techniques,

in each

be expected

screening

factor,

in better

the

of the

defects

safety

plotted

aluminum

be evaluated

equivalent

design

structures

on the

for

controlled

to the

in Fig.

tank

inspection

could

generally

Considering

on considerations

nondestructive

stress

size.

could

in Fig.

can be compared

For

flaw

that

shown

strength

materials

size.

critical

as

appropriately

of varying three

showed

by the

proportional

same

fracture

are

ratio,

s

47

application.

plotted

strength

be more

Kic/ay

17)

is directly

the

flaw

often levels

tensile

given

(Ref.

are

stress

might

highest

for the

data

operating

Fig.

the

and Masters

Kic

the

having

1972

information

are

Section

E 2

1 November Page

1972

48

160

i 0

1

50

100 ULTIMATE

150

20(

STRENGTH

(Ftu) (ksi)

250

300

aD

0

50

100

150

ULTIMATE

200

STRENGTH

250

300

(Ftu) (ksi)

be

1.5

_

,_STEEL

I

¢4

-"_

0.5

.......... 0

50

,

100

EQUIVALENT

150

ULTIMATE

200

STRENGTH

250

IN STEEL

Co

FIGURE

E2-18.

MATERIAL ROOM-TEMPERA

COMPARISONS TURE

TRENDS)

(BASE

METAL,

300

Section

E2

1 November

f--

Page

Alloy

ys (ksi)

Design Stress 0.5_ ys (ksi)

250

125

Steel Aluminum

140

From greatest

the

safety

2.4.1.2

factor

Cyclic

An

materials

(Rcf. is

that

ys

144

1.15

43

1.01

112

1.60

the

titanium

provided

the

to fracture.

Loading.

resistance

crack

growth

of data

18).

The

to evaluate

of materials rate

to fracture

characteristics

obtained

realistic

their

condition.

Let

crack

us

requires

in addition

consider

Problem

A.

1.

Materials

to be

considered

data

are

The

stresses 3.

material:

The ksi

The surface

in Figs.

component that

88

4.

given

vary

design for

worst flaw

from

zero

fixes steel

a

the

to other

and

possible with

32

ksi

type a/2c

hypothetical

are

the

and

steel

a given example.

aluminum

alloys

for

E2-20.

a thick

one-half

plate

cyclic

tension thc

loaded

during

yield

in tension

each

strength

cycle. for

aluminum.

of flaw

= 0.20.

E2-19

comparing

under

following

is

for

in Figs. for

characteristics

and

as

shown

approach

to maximum

max

are

the

E2-19

of interest

tests

practical

growth

Example

the

from and

I.

2.

elliptical

resistance

of the

examples

application

under

(ksi)

it is apparent

Sustained

Factor

propertics.

E2-20

which

or

of the

Some and

and

evaluation

consideration material

data

Safety _/0.5

70

above

49

Stress (7

42.5

85

Titanium

Fracture

1972

that

is envisioned

is a semi-

each

Section

E2

1 November Page 140,

1

I

I

! I

I

I

i

1972

50

1 I

HP-0-4-25 STEEL, 1.0 in. THICK 0.25 YJ. - 175 lui AVG. CRITICAL STRESS - INTENSITY FACTOR r

KIt - 144 ksi%/_. ROOM TEMPERATURE ,_

DATA (75°F)

I

100

I

c"

/

J

/

| W 1-

/

40 tt w mr

tM

20

0 1

2

4

6

S 10

20

40

SO

100

200

400 000

CRACK GROWTH RATE Zb4/AN (mils/1000 cydes) FIGURE

E2-19.

CRACK

t

I

GROWTH

I

I

7079-T6 ALUMINUM,

I

RATE

1

FOR

HP-9-4-25

1

1.0 in. THICK

CRITICAL STRESS- INTENSITY FACTOR K[c 34 ksiVi-n. 211 ROOM DATA (75°F) "__/ 32 -- 0.2 Y,$.-TEMPERATURE _51ui

I

I

i,. m

w me I,-

/" /

;1

2

4

IS 8 10

29

CRACK GROWTH RATE _a/_N FIGURE

/

///

I

II

STEEL

E2-20.

CRACK

GROWTH

RATE

40

IN)

100

200

400

Imils]1000 cycles) FOR

7079-T6

ALUMINUM

1000

Section E2 1 November Page

¢5. is 0.15

in.

contain

this

The

minimum

deep

by 0.72

size in.

flaw

that

long.

could

be detected

Therefore,

each

1972

51

by the NDI technique

material

is assumed

to

flaw.

f-.-. 11

Under

these

circumstances,

which

material

has

the

longest

of the

initial

life ?

Solution. Step intensity, and

1.

The

first

step

KIi,

for

each

material

stress.

component

The

appropriate

geometry

is

1.21

is to compute for

the

expression

the value prevailing

for

Kii

conditions for

the

stress

of defect

stipulated

defect

size and

_a a 2

KIi 2 -

Q

a

= 0.15

where

= crack = applied

steel {Y

ys

stress

= 88 ksi, =

in.

yield

-- specified,

(maximum aluminum

strength,

during

1.26

for

The

calculations

KIi

=

1.21r

specified

reveal

(0.15) 1.26

= 1/2

each ys

= 32 ksi,

steel

= 175 ksi,

and

Q =

cycle)

flaw

the

(88

geometry.

following:

000) 2

aluminum

= 65 ksi,

materia",

Section E2 1 November 1972 Page 52 and

Kii for steel,

=

59 000 psi

and

1.217r

(0.15) 1.26

KIi 2 =

(32

000) 2

and

Kii

for

=

aluminum. The

can KI

21 500 psi

crack

growth

now be determined values

for the

rates from

imposed

for

the two materials

Figs.

E2-19

conditions.

and The

at the beginning

E2-20

using

results

are

their

shown

of life

respective in the

following

table.

C rack Alloy 59 000

0.035

Aluminum

21 500

0.030

material. crack well

a knowledge

is not sufficient One growth as the

must

rates threshold

of the crack

to determine consider for

Rates

(mils/cycle)

Steel

However, life

Growth

each

stress

the the

respective

change

material intensity,

growth

life

in K I

as the KTH.

rates

crack

at the

beginning

expectancy

and the grows

associated during

of

of each change service

in as

Section E2 1 November

r

Page Step E2-21

f

and

2. E2-22

expectancy growth test

rates.

E2-21

for

Since

toughness

59 000

Kic

144 000

in the

form

a convenient

method

for

involved

E2-22

are

Figs.

that the

KI KIe

tests,

the

-

0.41

with

the

and

ratio

shown

in

from

Kii

Kii/Kic

for each ratios

readily

basic Figs.

The

previous

and 21.5 were

life

and

I

same

to Kic.

material

are

the

To utilize

is 59 ksi i_'_'_'_'_'_'_'_'mT, for steel values

K

the

E2-20.

of

in Figs.

c'valuating

changes

constructed

E2-19

to know

1 showed

KIi

steel,

and

to construct

aluminum.

illustrated

intimately

it is necessary

in Step

static

data

provide

E2-21

used

E2-22

for

18)

Figures

calculations

rate

becoming

as were

and

growth

(Ref.

without

data

from

The

1972

53

ksi

known

determined:

and

KIi

21 500

KIc

34 000

-

0.63

for aluminum.

The mined

cyclic

directly

aluminum,

life

from

4000

corresponding

Figs.

cycles,

E2-21 if the

to these

KIi/KIe

and E2-22

time

-- steel,

at maximum

values 1800

stress

may

cycles,

is short

be deterand

(luring

each

cycle. Thus,

for this

same

given

size

yield

strengths,

to generalize of initial

defect

and type the

It should the

specific

example

of defect

aluminum

has

bc emphasized relative sizes

that

behavior

and/or

applied

where

and both

both were

the greatest the

result

materials stressed

life

to one-half

the their

expectancy.

of this

example

of the two materials. stresses,

contained

it is possible

For

cannot other that

the

be used conditions steel

Section

E2

1 November Page

1.0

I

I

0.11-0.1

I I I I I I I I I'1P-1.4-215STEEL, 1.0 in. THICK Klc " 144 kj_'_. 0.2'1+ Y.I. = 17t kai ROOM TEMPERATURE

0.7 '_

i

i

i

1972

54

I

DATA (76°F|

i

t

0.| 0.4 O.3 O.2 0.1 0 0.1

0.2

0.4

0.60.81

2

4

6

810

N. NO. OF CY(:LES TO FAILURE

FIGURE

E2-21.

COMBINED

CYCLIC

HP-9-4-25

FLAW

STEEL

20 x 10.3

GROWTH

DATA

FOR

PLATE

I i i I I I I I 1.0 AT <_ 100 CYCLES 7079-T8 ALUMINUM, 1.0 in. THICK Klc= 34 ksi_/_. 0.2% Y.$. - 16 _IOON TEMPERATURE DATA |75°FI

4

S 8 10

20

40

e0

N, NO. OF CYCLES TO FAILURE FIGURE

E2-22.

COMBINED 7079-T6

CYCLIC ALUMINUM

FLAW PLATE

100

+

200

440

DATA

FOR

x I(Y _1

GROWTH

Section

E2

1 November Page

could

have

table,

I

the

which

of initial

greater shows

defect

the

sizes

life

expectancy.

This

life

expectancy

of the

and

for

a constant

is demonstrated two

in the

materials

applied

for

stress

of

1972

55

following

a wide

a

range

/2. ys

Initial

Initial

Stress-

Intensity

Defect

Factor

a

(ksi

i

)

( Life

Aluminum

Steel

19.6

7.2

0. 136

0. 210

>>300

0.07

27.5

10.1

0. 191

0. 297

>100

0.10

39.4

14.3

0.274

0. 420

0.15

59.0

21.5

0.410

0.632

0.20

78.8

28.7

O. 540

O. 845

0.25

98.4

35.9

O. 683

larger,

the

defect

Kic ,

catastrophic in

K

as

in Figs.

in the

maximum that

the

0.10

in.

the

steel

that

when

have

the

or

smaller,

therefore has

Failure

life the

the

larger

has

the

largest

a greater

the

differences

E2-23

and

E2-24

(Ref. the

300

x 103

× 10 '_

100

x

4x

x 10 a

defect

depth

value

critical

crack

growth

is

Therefore,

expectancies

0.15

the

in.

the

greater

lift _

of fracture size

rate

of the

10 a

when

have

absolute

103

1.SX

IIowever, will

10 a

21 x 103

× 103

0.25

steel

in slope

life

× 103

× 10 a

N.

crack

18).

Aluminum

0.37

initial

longer

Expectancy)

× 10 :_

1.8

the

has

from

between

30

> 1.0

seen

for

for

growth

a given

it is

rate

curves

possible

to have

of steel

and

aluminum,

reflect

short

time

table.

the

cyclic

seen

will

it also

situation

Again,

of time

and

failure,

a "crossover" noted

is

Steel

it is

aluminum

Although

toughness,

shown

table

depth

expectancy.

change

Aluminum

KIc

0.05

initial

as

i_'_n.

Steel

the

to N

(in.)

From or

Cycles

Kii

Kii

Depth

life

expectancies

stress.

If the

stress-intensity

in preceding time level

at

table

maximum is above

stress the

threshold

is

long, stress

the

at

portion intensities

Section

E2

1 November Page

1972

56

m

10"4

10-6

CRACK

FIGURE

E2-23.

CRACK

GROWTH

GROWTH

INTENSITY

M.

10"4

FOR

RATE

RATE

_N

(in./cycie)

AS A FUNCTION

HP-9-4-25

IO-S

OF STRESS

STEEL

'_-

i 1 104

10.1 CRACK

FIGURE

E2-24.

GROWTH

A_ RATE

CRACK GROWTH RATE INTENSITY FOR 7079-T6

_ ZIN

10-4

104

(in./cyde)

AS A FUNCTION ALUMINUM

OF STRESS

Section

E2

1 November Page for the

steel

different

initial The

data

and aluminum defect

could

also

in Figs.

E2-21

tolerate

the

material

could

not grow

to a critical

cause

reductions

in the

cyclic

57

lives

for the

sizes.

materials

provided

would

1972

be compared

and

E2-22

largest

size

to answer

initial

during

some

in another

defect given

manner

by using

the question (of a given

minimum

the

of which

type)

lifetime

that

would

for the

component. II.

Example

Known

Problem

B.

Information:

Plate

cyclic

Required Applied

loaded

life

(sinusoidal)

in tension.

-- 50 000 cycles.

stress

(maximum

stress

during

cycle)

one-half

yield

strength: steel

= 88 000 psi.

aluminum Type

of defect

Fracture

32 000 psi.

-- semiclliptical

toughness,

steel

=

aluminum

defect

=

surface

flaw

with

a/c

-- 0.4.

Kic:

144 000 psi = 34 000 psi

Unknown

Information:

Step

From

Which

material

can

tolerate

the

largest

?

Solution.

corresponding

1.

to the

Figs. desired

E2-21 life

and

of 50 000 cycles:

Kli at 50 000 cycles =

E2-22,

0.25

find the

Kii/Kic

ratio

initial

Section

E2

1 November Page

for steel,

at

cycles,

2.

solve

KIi

steel,

Knowing for

0.34

=

the

Kic

and

ratio

corresponding

to 50 000

Kic

= 0.25

(144

KIc

= 0.34

(34

000psi_m.)

= 36 000psi

and

= 0.34

000psi_/'_n.m.)

= 11 500psi

i_n.

aluminum.

Step 3. possible

Since

to solve

defects

with

a/c

for

Kii defect

= 0.4,

the

depends size

i

=

1.21

upon

knowing

following

(Q) a.

for

Kii/Kic

Kii:

= 0.25

Kii

for

50 000 cycles

aluminum. Step

for

58

and

Kii

for

1972

_ ¢2

steel,

(36000) 2 (1.26) a._= i.2i. (SS000)'2

stress stress.

expression

and defect For

size,

semielliptical

is appropriate:

it is now surface

Section E2 1 November

1972

Page 59 and

a.

=

O. 056 in.

1

when

the defect

a

i

is 0. 056 in.

(11

-

deep

by 0.28

in.

long;

for aluminum,

500) 2 (1.26)

1.21 7r(32 000)2

and

a

when

i

the

=

0.043 in.

defect Thus,

tolerate

larger

in the

ultimate

choice

the

that initial

for

factors, type

and

2.4.2

Predicting

Critical

mentioned

in Section

can be obtained

given

material

critical The

defect

flaw

form, sizes

engineering

diction

of critical

size,

from heat

flaw

sizes

imposed,

could

initial

the

defect

situation

may

defects

availability,

ease

Flaw

could

Since

more

the

heavily

capability

as related

the

on

of NDI

to the

of fabrication,

maximum

costs,

etc.

Sizes.

2.2.3,

plane-strain

several

types

treatment,

test for

of the basic and the

steel

is not great,

depend and

of insidious

the

aluminum.

size

the applicability

can be calculated usefulness

long.

condition than

this

i.e., size

the

defect

of a material

initial

values

for

allowable

allowable

As

by 0. 215 in.

maximum

comparative

techniques,

deep

it is apparent

a slightly

difference

other

is 0. 043 in.

use

stress

intensity

of specimens. temperature,

given

hardware

stress-intensity of

a/Q

With and

valid

data

for

environment,

operating concept

to describe

(Kic)

flaw

stresses. in the size

prehas

a

Section E2 1 November 1972 Page 60 been supported by a number of hardware correlations, some of which are shown in Refs. 17 and 19. Comparisons betweenmeasured critical flaw sizes on test hardware and predicted critical flaw sizes based on test specimen plane-strain toughnessdata have showngood correlation. From the equationshown in Fig. E2-6, it is apparent that critical flaw size is equally as dependenton applied stress as on the material fracture toughness. The following sections showapproachesfor calculating critical flaw sizes for the three basic types of initial flaws (surface, embedded,or through-the-thickness) based on the appropriate fracture toughnessvalues measured from valid specimen tests. 2.4.2.1

Surface Cracks. Calculations for surface flaws can be carried out by rearranging the

stress-intensity equationdevelopedby Irwin (Section 2.2.1),

(a/Q)cr

1.21rr

for a "thick-walled" structure ( i. thickness)

where

fracture

KIc

toughness

normal

to the

shape

parameter

is the

specimen

plane

of flaw,

(obtained

e.,

flaw

depth

plane-strain tests, a

cr

from

a is the Fig.

less

than

half

of the

material from

fracture

toughness

obtained

is the applied

stress

in structure

critical

E2-5),

flaw and

depth,

(a/Q)c

Q r

is the

flaw

is critical

flaw

size. Since a flaw

aspect

the

critical

and

Kic.

the

flaw

ratio, flaw

depth,

size

is an unknown

a/2c,

to determine

a

cr'

quantity, Q.

can be determined

it is necessary Using

the

to assume

preceding

for a specific

value

equation, of

a

Section

E2

1 November Page

/

I.

Example

Aluminum 20-in.-diam

Problem

alloy

be stored

A.

2219-T87

spherical

gas

is the

is selected

bottle.

in a liquid-nitrogen What

The

flaw

size

as the

bottle

propellant

critical

1972

61

material

is to operate

for

use

at 4000

in a psig

and

tank.

?

J

A.

Assumptions.

1.

The

defect

2.

The

operating

B.

Solution.

is a semielliptical stress

is

surface a = 80 percent

flaw

with

(yield

a/2c strength

= 0.2. of the

material).

The yield mens

are

strength

and

Iic

values

obtained

as follows:

a

= 60 ksi ys

and

Kic

The

operating

a

The

=

wall

=

37 ksi

stress

0.80

thickness

treq

is

(ays)

= 0.80

required

=

(60)

48 ksi

is

vR

(4000) (lO)

2a

(2)

(48

000)

=

0.417

in.

from

the

tested

speci-

Section

E2

1 November Page For

thick-walled

1972

62

structures,

a 0 cr

where

1.21

the shape

Q = 1.18;

parameter

Q

can be found

from

Fig.

E2-5.

For

this

problem

then

a

1.18

=

cr

/37_

2 =

1.21

O. 184 in.

and

2c

= a/0.20

For

surface

the flaw

magnification

critical

flaw

Use the

spherical

t

0.92

that are deep Mk,

in.

.

with respect

can be applied

to material

to give

a more

thickness, accurate

1

(KIc

_

2

structures. Example

the

Problem

same

design

diameter

_ req

factor,

---

thin-walled II.

flaws

=

size,

(a/Q)

for

= 0.184/0.2

Pa 2a

B. that

was

of the bottle

_

4000 (7.5) 2 (48 000)

shown

in Example

Problem

The wall

thickness

is 15 in.

=

0.313

in.

A except

that

required

is

Section

E2

1 November

(-.

Page

For

thin-walled

1972

63

structures,

f

cr

1.21, t,Mk<,/

Flaw

magnification

from

Fig.

error"

factors,

E2-8.

iterative

corresponding

Since

the

solution

is

to the

Without a/t

Take

an average

For

a/t

-

=

n

1"18137] 1.217r

1.15

average

a

Further

an

For

a/t

a cr

-1"18[ 1.21

reiteration

=

a

are

is unknown

cr' the

magnification

in.

(Example

a

= 0. 184 cr

M k = 1.21, 2

1

(48)

0.184

=

0.126

+

0.126

2

313

--

0.50,

.t48) .

7r

1.13 37 (48)

will

provide

Mk

2 --

+

=

0.147/0.313

available a "trial-andfactor

depth.

0.155 Take

aluminum

to determine

= 0.59,

1.21

0. 155/0.

2219-T87 depth,

factor,

[

a

flaw

flaw

= 0. 184/0.313

1.21

the

necessary

critical

cr

for

critical

a magnification

For

a cr

Mk,

O. 139

< 0.184

=

0.155

=

1.15,

in.

< O. 155

=

0.147

0.47,

M k

=

1.13,

=

0.144

_

0.147

] 2

more

accuracy

if desired.

in.

in.

0.139

2

=

in.

in.

in.

in.

Problem

A).

Section

E2

1 November Page If adequate material, tion

flaw

a reasonable

shown

result

in Fig.

unconservative

2.4.2.2

Embedded

flaw

Mk

effect

on stress

nated.

Thus

However,

for

answers for

not available

is the

it should for

more

embedded

for a particular

approximate

Kobayashi

be understood more

brittle

flaws

surface

flaws

except

embedded

flaw,

and

intensity the

are

64

ductile

that

solu-

its

use

materials

can

and

materials.

Flaws.

as for of the

for

answers

calculations

same

depth

estimate

conservative

perhaps

be the

values

E2-25.

in somewhat

The

magnification

1972

of the

equation

in thick-walled

that

is the

cr correction

the

stress-free

for one-half

a

one-haft

factor

surface

critical

structures

internal

critical

of 1.21

(Section

will

for

2.2.1)

flaw

size

for

the

is elimi-

is

'a'O'er Although flaws,

apparently

large are

hidden,

no similar

making

presents

used

for

embedded

might deep

flaw

effects

research

this

geometry surface

and (or

(Q _

other

size

has

assumption hand,

1.0).

that

studied

been

done

for

internal

The

fact

that

of internal

the

might there

same

flaw

be applied

flaw

surface

flaws

with

internal

flaws

to accurately

deter-

magnification

magnification to the

is no evidence

deep

effects. factors,

equation

for

critical

of how conservative

or

is.

to account

subsurface)

been

if not impossible

study

flaws

orientation,

barely

difficult

be made

However,

have

ratios.

in the

surface

sizes.

unconservative On the

their

a problem

assumption

M k,

depth

magnification

flaw-depth-to-material-thickness

mine, The

flaw

for the

lack

it can be conservatively flaws

and that

they

of knowledge assumed are

long

about that

flaw

flaws

in relation

are to

Scction E2 1 November 1972 Page 65

o.

if--

?:

% \ \ \ w.

\

O

\ 0

\

r_

0 E_ .< 0

Z i I

L_. Z b-..J

--3>'.

0

z

z

N

! _

U

v

o I

I

o O

Section E2 1 November Page 2.4.2.3

Through-the-Thickness To calculate

plane

stress

plate

(Section

can

,

_"

is the

plane

notched

or center-cracked

normal

to the

material,

plane

and I.

!

psig

What

fracture

crack,

Problem

alloy

room is the

A.

the

basic

in an infinitely

wide

to give

ys /

toughness

specimen,

a a

ys crack

obtained

is the

is the

applied

tensile

from stress

an edgein the

yield

strength

material

for

structure

of the

length.

A.

2219-T87

compressed

in ambient

\

is the critical

Example

diameter

stress

of the

cr

Aluminum in.

length,

1

cr

Kc

crack cracks

be rearranged

/2=1_

where

critical

for through-the-thickness

2.2.2.1)

66

Cracks.

through-the-thickness

equation

1972

air

is selected

cyclinder.

as the

The

cylinder

use

in a 15-

is to operate

at 1000

atmosphere.

critical

flaw

size

?

Assumptions. 1.

The

defect

is a semielliptical

2.

The

operating

surface

flaw

with

a/2c

=

0.2.

yield

is

a = 80 percent

of material

strength.

B.

Solution.

The mens

stress

are

yield

as follows: a

ys

=

50 ksi

strength

and

__KIc values

obtained

from

test

speci-

Section E2 1 November

¢--

1972

Page 67 and

Kic

.

=

32 ksi

An estimate of K

versus material thickness based on C

2219-T87

test specimens The

is shown operating

stress

= o.so (o The wall

in Fig.

req

thick-wa

cr

1.21

acr

Therefore, the

tank

dicted

the will

to be

critical

leak

before

Fig.

-

-

O. 188 in.

40 000

llcd

structures,

7r

E2-5,

1.18 1.21 _

flaw

21).

is

10oo(7.5)

_

From

required

--

For

20 and

is

thickness

--

(Refs.

) : o.8o (so) = 40 ksi

ys

PR t

E2-26

Q=

_¢32/2 \4-01

is apparently

failure.

The

1.18;

=

then

0. 199 > 0. 188 in.

a through-the-thickness critical

crack

length

crack of failure

and

is pre-

Section

E2

1 November Page

68

I IRWIN'S

I

APPROXIMATION

Kc'Kic

1+1.4

OF

K

2

TRANSITION

"

MODE

1/3

I DATA

POINT

FROM

REF.

21,

P. 112

lOO

v _t

so TA

K]c

" 32.0

POINT

FROM

REF.

20_TABLE

V

ksi_'n,

1.0

2.0

t (in.)

FIGURE

E2-26.

ESTIMATE (W=

OF 70'F,

K

C

KIe=

VERSUS 32.0ksi_)

t FOR

2219-T87

ALUMINUM

1972

Section

E2

1 November

f-

Page

1972

69

= 2_ cr

7r

\

The plane-stress

ys/

fracture

toughness

value,

K ,

from

C

Fig.

E2-26,

is 84 ksi

.... _

cr

2.4.3

Structure

2.4.3.1

Service With

structure ating

stress

in size

failure

to the

critical

depends

test

actually

in the

vessel.

level

and flaw

size

trated

in Fig.

E2-6.

Probably fatigue

growth

sustained severe

the

environmental size.

the

results

=

most

1.90

in.

growth.

the

effects

The the

initial

or defect

in a

applied

oper-

at the

potential

size.

The

potential

flaw

sizes

and

generally proof

inspection

procedures

maximum

possible

the functional

cyclic

critical

(in

life the

of the subcritical

initial

flaw

size

be considered

size

that

between

intensity

even

the

A successful

(Kic)

flaw

environmentally

occur

upon

can

flaw

of subcritical and

may

test

available. initial

stress

stress

relies

relationship

types

growth

if the

size

flaw-growth

conventional

predominant

Also,

critical

flaw

material.

by the

from

an initial

flaw-growth

initial

from

as defined

the

minus

this

of the

resulting

stress

size

positive

at stress,

result.

of the

defines

This

will

however,

of the most

proof

0.91

Predictions.

it attains

upon

determination

of NDI procedures;

to be one

until

characteristics

The

and

and time

and

directly

flaw-growth

flaw

Requirements cycles

level,

structure

use

Life

grow

is equal

-

Design.

pressure

will

inches)

2.81

in the

approaches

exists stress

and illus-

growth

are

induced absence the

of

critical

Section E2 1 November Page The stress

technique

flaw

intensity

growth

for predicting

makes

use

the subcritical

of fracture

specimen

cyclic

testing

70

or sustained

and the stress-

concept. It has been

a given

maximum

initial

stress

intensity, seen

used

1972

shown

(Refs.

applied

gross

intensity Kic

stress-intensity

stress

at the flaw

[that

that the ratio

6 and 17)

is,

cycles

of initial

ratio

tip,

level

depends

Kii,

compared

or time

flaw

size

that the time

to failure

to critical

or cycles

to failure

on the magnitude with

flaw

size

of the

the critical

= f (Kii/Kic)

].

stress

Also,

is related

at

it is

to the

as folIows:

a.

l

Qcr

Thus, obtain

if cyclic

or sustained

experimentally

the

cycles

can

be predicted.

the

or time

flaw

can be determined. The

ratio,

cyclic

KIi/Kic

By squaring critical

mental

number

or time

value,

versus

the

the

flaw

in

cycles,

of cycles. would it would

are

For

have have

the

flaw

grown

in

the

plot of the

A

(Fig.

of the

cycles,

to 0.8,

etc.

initial

size

initial

of stress-intensity in Fig. flaw

can

the

E2-27a.

size

to

be obtained.

after

flaw-size

increasing

a material,

is known

allowable

of initial

E2-27b)

to

to critical

schematically

ratio

for

structure

maximum

can be determined if the

are used

curves

to grow

in terms

as shown

example,

grown

life

plotted

size

or time flaw

required

log of cycles

that

initial

at stress,

data

specimens

cycles

given

log of cycles,

be recognized

0.40,

any

if the

flaw-growth

ordinate

size

also

cycles

versus

the

should

B

,

flaw

for

fracture

versus

Conversely,

of stress

size

KIi/KIc

required

in terms

stress

any

incre-

ratio

was

ratio

to 0.6;

It

Section

E2

1 November Page

1972

71

1.0

1.0

0.8

\ 0.6 u _g

0.4

X

0.2

0.2

10

100

1000

10

IO0

CYCLES

CYCLES

a.

FIGURE

b.

E2-27.

SCHEMATIC

REPRESENTATION

F LAW

Cyclic used and

in the

flaw-growth aerospace

have

industry.

application

sustained flaw

load

growth.

initial

been

Some

obtaincd

such

CYCLIC

on a number

data

stress

plotted

on

Plots

does

both

dry

men

data

are

shown

load

for

growth. high.

than

Kii/Kic

values

log

of time

to failure.

not

and

is applied

is less

Kii/

Kic

wet

occur.

that

versus

of materials

in Figs.

critical

are

computed

of time

and

aluminum

the

environment

In both

cases

shows Fig.

tested played the

same

for

apparent

most below

data

for

the

E2-28

which 17-7

effects

such

the

time

cyclic

that to

Kii/Kic

the

[ailure

ratio

is

indicate

sustained Ptt

steel

tested speci-

In neither in the

stress-intensity

the stress

surface-flawed

role

of

in defining

materials

nitrogen.

important threshold

used

and

shows

in liquid

the

specimen

and

E2-31

an

as

value

level

E2-30

to define

to a flawed

stress-intensity Figure

the

the

log

environments,

2219-T87

testing

is essentially

The

of

it seem

quite

growth

of a threshold

growth

stress

flaw

intensity

versus

existence

of fracture-specimen

A constant

is recorded.

are

data

OF

GROWTH

E2-29.

The

does

tOO0

case

sustained levels

in

Section

E2

1 November Page

1972

72

1.0 NOTES:

1!

• BASE METAL:

2.

SPECIMEN:

4.

0.B0

o

FRACTURE

k 1.0-in. THICK

2219-TB7

SURFACE LOAD

Omex

PLATE

FLAWED SPECTRUM

I"" l(minF-_

I-4( le

v 0.O0

0.70 10 NUMBER

FIGURE

E2-28.

BASE

METAL

( -320 °F,

1000

100 OF CYCLES

CYCLIC

LONGITUDINAL

TO FAILURE

FLAW-GROWTH

DATA

GRAIN)

100

-CYCLIC

..9O u

LO,

_-1

_g

Omx

,t N

SCATTE

R BAND

OF DATA

(min)._

Fl__/--l_

oJ

7O 10 NUMBER

FIGURE

E2-29.

CYCLIC

TITANIUM

1000

100 OF CYCLES

TO FAILURE

FLAW-GROWTH

PLATE

TESTED

DATA AT

-320°F

OF 6A1-4V

Section

E2

1 November

6-.

Page

1972

73

1.0

•.--

._.

DATA BAND 0.8 2 r z

NOTCHED BAR SPECIMENS PLATE ANO FORGING

0.6

0.4

0.1

1.0

10

100

1000

TIME AT LOAD (hr.)

FIGURE

E2-30.

SUSTAINED

STRESS

ROOM-TEMPERATURE

Let and

now

specifically

cyclic like is

us

life

the

To

at

Now

consider

Kii/Kic

the

in an

This

is

strcss

intensity

it is

initial

flaw

to critical

initial

flaw

would

have

threshold

seen

that

size

and

increased value

of

containing

initial

equal it would cause

the

flaw to 50

take

a total

enough = 0.80.

size

or

crack-

of the

K-N

curve

curve

to cuase With

applied

of the

However

is a horizontal

stress

and

of

total

crack

threshold

percent

failure.

in size KIi/KIc

to be

this

growth

cstimated

initial

representation

assumed

flaw

on the

on

FOR

STEEL

stress

an

superimposed

the

Ptt

concept

schematic but

where

the

the

E2-32,

situation

initial

curve,

the

17-7

DATA

of sustained

stress-intensity

this,

-- 0.80.

OF

significance

structure

in Fig.

From

rcach

the

threshold

illustrate

reconstructed

like

to

TESTS

of a tension-loaded

flaw.

result

consider

FLAW-GROWTtt

A

cyclic

critical

cycles

in

B

the additional

intensity. stress value. to grow

cycles, stress

this

the intensity

cycles

the

Section

E2

1 November Page

DATA BAND

1972

74

_m

0.S t_

NOTE: ,S

BASE METAL:

2219-T67 1.0-in.-THICK

PLATE

-'

0.01

FIGURE

0.1

E2-31.

1.0 TIME TO FAILURE

SUSTAINED 2219-T87

STRESS

10 (hr)

100

FLAW-GROWTH

ALUMINUM

AT

-320

DATA

FOR

°F

1.0____,__

COMBINED

I I

-_

AND I CYCLEGROWTH I

_

LI

A CYCLES

r I

TOTAL CYCLIC

FIGURE

E2-32.

COMBINED

GROWTH

CYCLIC

SCHEMATIC

LIFE

AND

TIME I

SUSTAINED

INTERPRETATION

STRESS

FLAW

Section

E2

1 November

r

Page stress

intensity

sufficiently .

would

long,

further

it appears

increase

and,

possible

that

if the

failure

stress could

were occur

1972

75

sustained on the

(B + 1)

cycle. If,

on the

other

maximum

cyclic

stress,

realized.

It is hypothesized

sustained

stress

value

there

within

the

stress

is held

will range

interaction task

has

(luring

more

cycles

it appears that

little

each

above

that below

A

the the

total

The

the threshold

of

A

could

the the

time

be a complex

may cyclic

at

threshold

the

and

maximum

time-cycle and

expensive importance.

the threshold-

intensityvalues and then verify (through prolonge(l-time specimen tests) that time at load is not of major

be time

not be of grcat data

at

anywhere

of the exact

wouhl

basic

time couht

occur

on the

development

the

cycles

Above

depending

structure,

little

K-value,

life.

failure

value

to determine

with

threshold

that

cycles,

tankage

applied

on cyclic

such

cycle.

to most

rational

were

or no effect

of (B ÷ 1) to

as applied

It appears

the

be an interaction

curvcs

and,

hand,

cyclic

significance below the threshold value.

In the application of the data to fatigue-lifeestimation, thc maximum stress intensitywould be limited to the threshold value as determincd material in question and for the applicable service environment.

allowable for the

If the

threshold is very low, steps should be taken to protect the material from the environment. The operational cyclic lifeof pressure vessels can be determined the

following

vessel

data

are

available:

1.

Proof-test

factor

2.

Maximum

design

3.

Fracture toughness

4.

Experimental

material.

a . operating

stress

a

op"

Kic.

cyclic and sustained stress flaw growth for the

if

Section E2 1 November Page If the

cycles

stress

to be applied

aop ,

¢alue

Kic

long

flaw

times

be obtained

the

pressure always

for

of the are

show

the

ratios),

flight

cyclic

could

KTH

a

critical

- a..

er

For

1

not be

and the allowable

stress-intensity

data

Vessel).

by utilizing

the

to failure

proof-test for

a cycle)

various

for the

factor

and

values

of

material-

based

structural

analysis vessel,

R = 0. cyclic

the

on

is conducted An excellent

for the

Since life

less

analysis

than

then

pressure

based

life

illustrative

actual

example

the

R = 0

on the

analysis

on the

analysis

is shown

unsatis-

analysis R

struc-

all

on

based

vessel

the prediction

on the

that

based

that

thick-

of the

assumed

of cyclic

If the

based

the

of the

assessment

it is always

prediction

R = 0,

integrity

values

abstracted

for

the

at which from

the

Ref.

12

as follows.

successfully

following

intensity

during

the

conservative.

life

applied.

Suppose

design

stress

and cycles

first

at

R

R = 0 is invariably

is given

is

threshold

stress

thick-walled

remaining

of

for

In the

the

R ¢ 0 (actual

are

the

potential

value

maximum

to reach

growth

threshold

be made

assessing

applied

of

cycles

at the

22.

Kii/Kic

follows.

cycles

remaining

the

Typical

to maximum

procedure

vessels

factory

flaw

A (Thick-Walled can

hold time

combination.

integrity

will

_ a..,

short

can be allowed

stress,

12 and

between

aop

stress

prediction

of minimum

The

tural

ath

Refs.

have

allowable

Problem

life

environment

walled

is

from

relationships

the

at

sustained

Example

Cyclic

R (ratio

intensity

maximum

potential

I.

the

at the

to exceed

growth

can

stress

and therefore

hold

allowed

the

to the vessel

1972

76

that a thick-waUed

proof

operating pressure

tested stress. cycles

6A1-4V

at a proof-test Suppose before

(STA) factor

titanium

of 1.50

that the proof-tested the flight,

as shown

times tank in Fig.

helium

tank is

the maximum is subjected E2-33:

to the

Section

E2

1 November Page

1972

77

1.5 CYCLE

ANTICIPATED

PREFLIGHT

SERVICE LIFE

f-/

200 CYCLES I,[email protected]

1.0

Oop

4300 CYCLES

260 CYCLES

@ Oop

D

R

40 CYCLES @ Oop

@ 0.95 0oi)

.

B

=0.,

=o

D

0.5

UJ

a. 0 CYCLES

FIGURE

E2-33.

CYCLIC

IHSTORY

( I,:XAMPLI.: 1. and

R=

200

loading

cycles

OF

A THICK-WAI,I,

PROBLI
with

the

ED

VESSEL

A)

maximum

stress

as

90

percent

of

O"

()1)

0.1.

2.

4300

3.

260

loading

cycles

with

the

maximum

stress

as

_

1l=

and

op

0.7.

loading

cycles

with

the

maximum

stress

as

95

percent

of

_r op

and

R=

0.4.

4.

40

loading

The

cyclic

life

of room-temperature R = 0.7

in Fig.

iK'i/KIe

for

R=

cycles

with

curves

for

air E2-34. 0

and

the

6Al-4V

are

reproduced

The

dif[erence

R-0.1

maximum

stress

(STA) for

titanium R = 0.0,

between

is negligible

as

the for

this

(r for

op the

R = 0.1, plots

of cyclic

and

It=

0.1.

environment R = 0.4, life

material-c'nvironment

and against

Section

E2

1 November Page

1972

78

m

_OE

_

,,"

©

..J

_

0

]_

y)

Lu •.J ).

@ _ _

o_ v

o

m

! O

8.

'1

Cp

q

_ 0

o 0

:JI)i/xmu (r[)i)

o 0

0

Section E2 1 November Page combination,

and

threshold

air

The after

the

plot

in Fig.

instead

a op'

point

is indicated

Crop

and of

R=

0.1

the

4300

Point

on the

change

R = 0.1 the

environment

could

exist

The

of room-

in the

It can be seen to failure

small.

are

If the

from

R = 0

600 at op on

is based

shows

of the vessel

vessel

the

about

analysis

history

cycles

Kii/ KIe

with

the

is given

by 0.90

curve.

The

KIi/KIe ratio

ratio

that

the

is marie

at the

maximum x 0.667=

200 loading

from

vessel

based

Point

stress 0.60.

on

as This

cycles

of 0.90

E to Point

end of 200 loading

at the

beginning

0.70.

This

loading

0.74,

cycles on the

stress

0.4

Sop

plot

of

is shown R = 0.4

plot,

where

at the cycles

and

R = 0.7

R=

0.7,

D on the

cycles

of

change its

the

value

on the Kii/Kic is 0.80.

of 201} cycles,

Sop

D on the

change

where

the

end

0.4

ratio

R = 0.7

plot

of

R=

ttence, is 0.7.

ratio

from

is 0.78.

of 4300

at 0.95 R=

and

KIi/KIe

its value

at the

of 260 cycles

by PointC

end

at

by Point

by 5 percent

at the beginning

which and

at

of 4300 is shown

is decreased

ratio

Sop R=

by 10 percent

ratio

Kii/Kic

at 0.95

on

is increased

DtoPointC

0.78=

E2-34.

R.

× 0.63=

The the

that

pressure-cycle

KIi/KIc

The

stress

Kii/Kic

(1.0/0.9) The

in the

ratio

are

of 200 loading

E

in Fig.

Kic.

the assessment

of

plot

79

is 0.63. The

the

the

following,

by

0.1.

material

stress

R,

maximum

R = 0.1 R=

that

beginning

0.90

for the

is 1/or = 0.667. op the maximum cycles

values

At the

same

a

of actual In the

by the

KIi/KIc

at maximum

appropriate

plot

at

shown

of

possible

test

E2-34

is critical. the

level

maximum

proof

are

is 90 percent

hold times

R = 0

both

stress-intensity

temperature

if the

hence

1972

(top plot.

from

cycles.

Hence

is (0.95/1.0) The

Point

×

260 cycles C to Point

B

Section

E2

1 November Page

The the

stress

Kii/Kic__

0.84,

ratio

which

aop

and

shown

is increased

at the beginning

is illustrated

R = 0.1

by Point Since

A in Fig.

the

at the

of 40 cycles

by Point

increase

the stress

the threshold

by 5 percent

B on the

liK'i/K'c

at

intensity

stress

intensity,

the vessel cycles

at

Kii/Kic

from

0.875

to 0.90

Thus,

the estimated

wall

for the vessel

thickness

prior

magnification thin-walled depth

and,

factor vessels

is (1.0/0.95)

plot. 0.84

is considered

20 loading

The 40 cycles to 0.875,

at

which

is

or

op

and

at

or

op to be safe

R = 0.1

is less

than

for the

to increase

minimum

cyclic

life

is 20 cycles. Problem

In thin-waUed

x 0.80

at the end of 40 cycles

take

Example

aop

E2-34.

It will

II.

Hence,

from

flight.

remaining

80

end of 260 cycles.

R = 0.1

ratio

1972

B (Thin-Walled

vessels

the flaw depth becomes

to reaching for

deep

the critical surface

it is assumed

consequently,

Vessel).

Q

flaws

size.

with

respect

Therefore,

M k must

that the flaws

is assumed

deep

are

to be equal

Kobayashits

be considered.

long with

to the

respect

In to

their

to unity

in the Kobayashi

vessel,

the following

equation. To determine relations

flaw

are

required

1.

Proof-test

2.

The

size,

equation:

Or

ai,

or acr,

the cyclic (Ref.

life 22).

factor, versus and

of a thin-walled

orop' a

curve,

aTh.

The

Kic, similar curve

and

KTH. to Fig.

is obtained

E2-35, from

to determine the

following

the

=

Section

E2

1 November Page

160

I

1972

81

I 37 kziV_.

150 --_

K]c

140

J

130 _

PROOF_

'_

\

;\

90-

MAXIMUM DESIGN -- "-" --

io . -

,-_,H o,_,_

I _OPERATING -I----_STRESS 1"-- --alk--" --

! i\ l ,

70

•

WALLTHICKNESS

_

MAX. POSSIBLE I

I a

I

[!i

0 0.010

_ _

aTH I

0.014

cr

1

i

I

0.018

0.022

FLAW DEPTH, a (in.)

FIGURE

E2-35.

DETERMINATION

OF

F LAW

3. flaw

growth

The

Kii/Kic

rate

at

by differentiating of Fig. a

cr

al, N

/t

(Ref.

is less the

curve

than

given by

any

the

E2-36

a/Q -

stress

flaw

22). half.

The

versus

cycles

This For versus

equation

1.21 _ \ _'1/

growth

level.

KIi/Kic

Kii/Kic the

versus

curve an

is

assumed N

INITIAL

AND

CRITICAL

SIZ ES

curve

rate flaw

growth

maximum be

to determine rates

to failure

obtained

can

da/dN

can

curve,

from

the

cyclic converted

be

the obtained

similar

specimens stress to an

to that where

level, a/Q

say versus

Section

E2

1 November Page

1972

82

CYCLIC LIFE CURVE (1 qlm) FOR 1.0 , 0.1 "--'"

J----'--

_1)

--

TITANIUM

"1....

LESTIMATED

_

THRESHOLD

.....

LEVEL

I

0.7

LOAD PROFILE O.I--

Omx+ 0

I

I

T,ME----I

,

I

I

II

1

I

I

J

I

l

I

I11

10

i

I

I

I

J

I111

100

1000

CYCLE$ TO FAILURE

FIGURE

E2-36. LIFE

COMBINED DATA

SUSTAINED

[5 A1 --

2 1/2

Sn

-320

°F]

AT

The

slope

d/dN

(a/Q)

equation at

of the

al

a/Q

versus for

a given

versus

N

Kii/Kic

curve

for

Kii ,

a/Q

the at the

AND (ELI)

gives

stress

the level

stress

CYCLIC

STRESS

TITANIUM

plot

for

a 1.

level

a2

the

From

flaw the

growth

rate

preceding

is related

with

a/Q

as

From

this

level

cr2

equation is

[d/dN

related

(a/Q)]

it can

be

to the

a2

concluded

growth

=

(al/a2)2

rate

that at

[d/dN

the a I

flaw as

(a/Q)]

growth

rate

follows:

a 1

.

at

any

stress

Section

E2

1 November Page The of the

prediction

thin-walled

from

Ref.

of the

vessel

22 and

Suppose

cycle

is demonstrated

is given

that

remaining

at room

at room

temperature

operating

stress,

Sop.

following

pressure

cycles

the

structural

by an illustrative

6AI-4V

temperature

N204

and

83 integrity

example

abstracted

as follows.

a thin-walled

containing

life

1972

titanium

(STA)

is successfully

to a proof-test Suppose before

the

proof

factor

of 1.41

the

proof-tested

that

propellant

times

tested the

tank

tank with water

maximum

design

is subjected

to the

flight:

1.

Twenty

loading

cycles

with

the

maximum

stress

as 90 percent

of

2.

Twelve

loading

cycles

with

the

maximum

stress

as 95 percent

of

3.

Five

op

(1

o

op loading

It is desired from

the

fracture

remaining

for

numbers ratio

according

maximum gage

the

of

Kic.

The

_

Fig.

E2-35.

from

Fig. Here

the

Sop"

The

stress,

of 37 ksi

E2-35

the that

it is assumed

has

thickness (7op,

and

of the

the

stress

maximum that

the

is 1.41 possible

depressurization

prcssure

with

tank

ksi.

threshold

cyclic

the

is 0.022

in.

material

has

stress

the

a/t The of this

minimum

intensity

KTH

life

specific

for

The

and

vessel

minimum

conditions

a plots are given for Kic proof

op

to be corrected

is 87.5

the

the

is treated

environmental _

as

of the

estimate

example

factor

E2-25.

stress

integrity

and

This

stress-intensity

operating

Since

maximum

standpoint

at

versus

the

structural

above-mentioned

toughness

percent

vessel

to Fig.

design

under

fracture

in.

the

with

to assess mechanics

the

since

cycles

= 0.80

of 80

KIc

in

× cr

= 123.6 ksi, it is clear op a that could exist is 0.0143 1

from

the

proof

pressure

is

Section E2 1 November

1972

Page 84 rapid

enough

ization.

so that

Also,

is 0.0196

as shown

in.

and

The

aTH

plot

of the

at room

temperature

percent

confidence

data

R = 0.0;

at

of R=

rates

in.

are

E2-35,

is 0.0160

in.

Kii/Kic

level

growth for

versus

is reproduced flaw

in this

during

stress

level

growth

in Fig. rate

occurs the

flaw

growth

it is assumed

of

aop.

level

of

a

The

rate

E2-37 curve

for

the depressurof

for

that

a

a

op'

6A L-4V

all the

cr

titanium

a = 100 ksi.

is obtained

example

the

acr

is 0.0208

the

stress

level

1

similar

in Fig.

in.

in.

to

to failure

a

a

from

depth

The

from

the

cycles

are

flaw

growth

99-

cyclic applied

= 0.208

is followed for the

stress

of the

proof

on the

a. = 0. 0143

in.

1

the cycles

against

stress

aTH op'

level

cycles

to

a

rates,

= 0. 0196

er

to failure

for

the

stress

to failure

for

the

stress

E2-39.

cyclic

and

cr

of stress

to calculate

of flaw

of 0.95

procedure

cycles

plot

maximum

but

effect

integrated

E2-38

is shown

op

When

a. = 0. 0143

the

arithmetically to Fig.

level

is 0.95

= 0.0167

in.

the

flaw

in.

to calculate

to obtain level

growth

the

a

from rates

Fig.

the

is still

i

E2-35.

are

Based

depth

These

plots

in. on

from

to failure.

of flaw

aop.

0 0143

integrated

cycles

relation

of 0.90

a

op'

A against

are

shown

in

E2-39. At the

maximum 0. 0143 with

in Fig.

into account

according

Fig.

flaw

0.0. Taking

the

no significant

cyclic in.

the

end

This

maximum

stress

cycle

of 0.90

is shown stress

a

and

the

op'

by Point

D in Fig.

as 0.90

a op

on the

plot

of 0.90

a

op

the beginning

(Fig.

E2-39).

maximum

change

E2-39. a

of the possible The from

first

cycle

flaw

20 loading

depth

at the is

cycles

Point D to Point

C

Section

E2

1 November Page

1972

85

f

0.4

0.2 (BASE

0.1 10

100 d(a/O)/dN

FIGURE

E2-37.

CYCLIC (FOR

1000

(/_ in./cycle)

FLAW-GROWTH

a

--- 100

RATES

ksi)

max

MEAN n

Aa

I/t

Kb

Mk I

MEAN K

c N

Z_N de/dN

I 01

aS/t

Mkl

KS

K1 + K.._____2 de 1-2/dN 2

a2/t

Mk 2

K2

K__.3 K2 +

a2 • as II2 • 3" '2

I

I

I

I

I

.,

I I I I I

2 I I I I

_Ns

de 2.3 IdN

AN 2

I

I I I I

I

I I

t

ANs

_NI+

_Nz÷ _N 3

1

a. Obtain from Kobayashi's solution of Mk vs sit b. K = 1.1q/_r'o(a)

1/2 M k

c. Obtain from beeic K vs de/dN

FIGURE

E2-38. DATA

curve.

ARITHMETIC (DEEP

FLAWS

INTEGRATION IN

TItIN-WALLED

OF

FLAW-GROWTH-RATE VESSELS)

Section

E2

1 November Page

1972

86

|

!

Cut) • 1,1,131(1MY'IJ

Section

E2

1 November

The cycles

tank-wall

with

the

stress

maximum

is increased stress

as

by

0.90

5 percent

a

at

The

flaw

by

Point

1972

Page

87

end

of 20

the size

loading

remains

the

op" same

during

0.95

a

the in Fig.

op The

a

from

12

At stress

of

_

the

end

cycles

Point

with

the

B on the

of 12

loading

increased

in Fig.

This

is

shown

C on

the

plot

of

E2-39.

C to

is

increase.

loading

Point

the

stress

by5

maximum

plot

stress

of 0.95

cycles percent.

a

with

the

This

is

as

0.95

in Fig.

op

maximum shown

by

a

change

op

E2-39. strc_ss

I)oint

as

0.95

B on

cr

the

op'

plot

E2-'19.

op The

live

loading

cycles

with

the

maximum

stress

as

_

change

a

op from at

Point A

is

the

vessel

will

take

to

B to Point 0.01534

in.

This

is considered seven

0.0160

vessel

Hence,

seven

2.4.3.2

Allowable life

of the

material

actual

initial

is smaller

tr

aTli,

the

flight.

for

to

increase

a

the

minimum

op

in Fig.

op

than

safe

at

Initial

initial

tire

estimated

Flaw

flaw

requirements

I':2-39.

The

which Also flaw

is 0.0160 from

depth

cyclic

flaw

life

in.

Fig.

from

del)th IIence

E2-39,

it

0. 01534

remaining

in.

for

the

flaw

sizes

initial

flaw

sizes.

initial

flaw

size

maximum

possible

determine

the

the

the

maximum

provides

can

initial

maximum ratio,

structure

prevention

A successful which

in a designed

The or

inspection

Size.

sizes

for

selected.

Nondestructive

stress-intensity

of

cycles.

Allowable

service

plot

to be

cycles

in. is

determined.

A on the

to critical allowable based

only

on the

test the

initial

means

service

flaw

flaw

test

that

either

size

be

the

maximum

and,

in turn,

ratio, size,

life

properties

of determining

specifies proof

on the

toughness

requires

stress-intensity initial

depend

fracture

possible

the

after

and

of failure

proof

exist

structure

the

initial

requirements,

the known.

actual possible provides

Kii/Kic. to

To

critical must

be

the

Section

E2

1 November Page The calculation following

initial

flaw

size

88

is demonstrated

by the

example. I.

Example

A cyclic following

Problem

loaded

design

A.

pressure

vessel

of aluminum

alloy

must

meet

the

conditions:

1.

Required

2.

Maximum

3.

Kic

4.

Semielliptical

What size

of allowable

1972

minimum stress

of weld

40 000 cycles.

in a cycle

metal=

1/2

o

ys

= 35 000 psi.

15 000 psi

surface

is the allowable

in 40 000 cycles

life,

defect

initial

flaw

(length size

4 x depth).

which

will

grow

to a critical

?

Solution.

From 40 000 cycles

Fig.

E2-40

of life

(Ref.

is 0.36.

18), the The initial

KIi/Kic stress

ratio intensity

corresponding

to

can now be

determined:

KIi/KIc

:

Kii 15 000

0.36

:

(15

000)

and

Kii

=

Knowing defect, 5400

0.36

the

design

it is now possible psi

_

:

= 5400psi

stress

_

.

of 35 000 and the

to find the defect

size

expression

corresponding

of the type to a

Kii

of of

Section

E2

1 November Page

1972

89

1.0

N \ 0.8

z , , [ DATAi HYPOTHETICAL 70.-kzi YIELD STRENGTH

0.11

-

WELD METAL, z

I

ALUMINUM

K]c,,15kzi

0.7 2

[

i_n.

I

I

0.6 0.S

_g

0.4, 03i

!

INITIAL

0.2

STRESS-INTENSITY

CRITICAL

FACTOR

STRESS-INTENSITY

FACTOR

0.1 2

4

6

N,

FIGURE

8 10

NO.

E2-40.

OF

20

CYCLES

CYCLIC

TO

KIi 2 Q

I

1.21

The and

_2

value

a /a

of

(5400)

-

1.21

Q -- 1.4

= 1/2=

60

FAILURE

_ (1

_ (:15

200

400

DATA

FOR

A LLOY

4) " 000)

is taken

80100 x 10 .3

FIAW-GROWTtI

A LUMINUM

a.

40

-

0.0088

in.

_

from

Fig.

1':2-5

a/2e

for

= 1/4

= 0.25

0.50.

ys

2e.

=

4a.

1

=

Therefore, size

in 40

2.4.3.3

cations

000

the cycles

size

=

NDI flaw

associated

of an

is 0. 0088

Nondestructive The

allowable

4 (0.0088)

0.0352

in.

1

They with

in.

deep

Inspection

requirements sizes.

initial

for are

a proof-test

flaw by

which 0.0352

Acceptance any

I_ivcn

limited

by

failure

will in.

just

grow

to a critical

long.

Limits. structure

any and

are

economic by

the

a function or

schedule

reliability

of the impliof the

Section

E2

1 November Page inspection for

any

there

is

techniques lack

of specific

a lack

orientation

or

sidered.

are

flaws

Ladish

D6AC

flaws.

The

ratio

(Kt/K_)

point

is that

flaws

shown

curves versus

surprisingly

has

are

is close

is,

Allowances

geometry

worst

and

possible

limits,

aligned

analytical

obtained

E2-41

along

should

be

orientation.

flaw

made

When

geometry

flaw

spacing

very

little

flaws

allowable

and

for

with

the

in terms ratio

of

coplanar

(Ref.

results

coplanar

con-

23).

surface

magnification the

most flaws

significant unless

together.

Kic

DATA POINTS -

Klc FROM COPLANAR FLAW SPEC. USING SINGLE FLAW EQUATION

FOR LADISH ' D@AC STEELS

1

o0

,/2¢ - 0.32 [ PLANE STRAIN STREI|S INTENSITIEI ! OAO

COPLANAR FLAWS _ _1 0.6

j 1.0

SEPARATION

FIGURE

E2-41.

STRESS-INTENSITY COP

LANAR

E LLIPTICA

(d/a)

MAGNIFICATION L F LAWS

The

on several

semielliptical

Probably

between

be

of elliptically

Hall

stress-intensity

(d/a).

interaction

and

for

must

interaction

experimental

two

spacing

in weldments)

by Kobayashi

containing

plotted

the

solution

been

specimens

there

the

(that

in Fig.

steel

of flaw

at acceptance

An approximate

results

flaws.

assumed.

surface

coplanar

initial

definition,

in arriving

shaped

are

be

detecting

knowledge

of flaw

might

Also, internal

for

1972

90

FOR

TWO

they

Section E2 1 November Page The ments

are

establishment known

hypothetical loading

might

of one

stress) stress

level

required

that

when

service

by an illustrative

is expected

terms

= 1.0).

life

example

to encounter

as shown

d are

rained

load

flaw

require_

involving

a rather

a

complex

approach

1.

The

of critical

service

life).

tained

the

stress 3. cycles

size

at time

maximum

size

backwards

relationship ordinate

cyclic

flaw

size,

, evaluating

growth.

at operating

of the

critical

of stress-to-

now is plotted

stress. load

Figures

flaw

growth

in E2-42e

and

sus-

is as follows:

critical

flaw

and

is the

size

at operating

maximum

allowed

maximum

flaw

stress

allowed

size

at time

flaw

size

allowable

is represented

at time

TC

at the

TD

(at

is shown start

of the

in Fig.

E2-42c

as 10O the

end of the

by Point 100-hr

C and sus-

period. The

1000

flaw

operating

standards

the

flaw

The

operating

respectively.

Maximum

represents

inspection

a dimensionless E2-6).

the

con-

at a constant

to determine

cause

requirement

times

both

minimum

can

(Fig.

_

2.4 " 2) and work

that

represents

life

of

100 hr,

the

representations

The

2.

profile

growth,

percent

then

(Section

of critical

schematic

level

it is necessary

previously

of percentage

service

a stress

To define

loading

E2-42b

assumed

and

end of service

of the

Figure

(at

cycles

to service,

cr , at the

the

cycle

by 1000

prior

size

shows

proof-test

of (a

portions

and

vessel

E2-42a

followed

flaw

be shown

limits

91

history.

sisting

all

best

pressure

Figure

(a/Q)

of NDI acceptance

1972

effect

from TB

allowable

of cyclic

loading

to

Point

TC

T B.

or at the size

start

before

B then

of the the

is shown

represents

1000-cycle

vessel

is placed

the

period.

by moving

maximum This

in service.

size

allowable is the

Section E2 1 l_ovember page

:_ZlS MV'ld

92

1972

Section

E2

1 November Page

4.

It can be shown

that the previous

has a negligible effect on flaw growth

one-cycle

compared

1972

93

proof test generally

with the chosen

service

life.

f-

Note

in this schematic

size is less than that which

illustration that the maximum

could have been

present

test, and thus the proof test could not guarantee service

life requirement.

as small

As

assurance

of service

be increased

to assure

that inspection

that (ai/Qi) max

of service

made

about flaw geometry

small

flaw depths.

Consequently, in order

(ai/Qi)

max

For

example,

detection)

tank

locations

been

illustrated

noted

to bc the minor

of detecting

flaws

requires

a required

and

a detailed

the

flaw

higher

Example size

toughness

service

life.

considered,

the

calculated

demensions

which

NDI

initial must

ordinate

sizes

be

be

in X-ray

of critical,"

capable

both

size

used.

of detecting.

length

various has the

is calculated

acceptance

and

for

This

in which

vessel

siz¢"

deter-

in the

2.4.3.2

NDI

depth

flaw

stresses

pressure

flaw

stress

the

smallest

materials

where

1.0).

flaw

Obviously,

(or

of the

case

I,- 2a, Q _

of Fig. E2-42b.

of applied

alloy

In the

might

In other words,

A in Section

aluminum

existing flaw

(and the operating

values.

toughness Problem

in an

axis (i.e.,

upthe

knowledge

fracture

factor must

of an in(lication seen

be larger

flaw

the

[or the inability to detect

of "percentage

allowable

of the

by

preclude

assumptions

life requirements.

in terms

stress

maximum

and

limitations

axis of an elliptical flaw where

to the minor

move

that

of actual

initial

being

minimum

be

of finite

on

be capable

to account

the length (L)

the service

(a/Q) cr

should

NDI

are

to meet

and

independent

based

proof

fulfillment of the

is the largest possible

the critical flaw size must

lower)

allowable

must

life, or conservative

axis is large with respect

mination

successful

technique

and orientation

inspection could be assumed

is

a successful

life reliability, then either the proof-test

size at the beginning

It

during

flaw

as (ai/Qi)max.

If it is determined

major

a result, NDI

allowable

limits are

the

tho

Section E2 1 November Page 2.4.3.4

Proof-Test

It has the

been

maximum

to the

noted

flaw

size

of service.

available

The

and

94

Selection.

previously

possible

beginning

presently

Factor

1972

offers

that

a successful

which

can

proof

test

the

most

exist

proof

after

is the

reliable

the

most

test proof

determines test

powerful

method

for

and

prior

inspection

guaranteed

test service

life.

Figure

E2-43

critical

flaw

viously

illustrated

initial applied

flaw

size,

size

stresses

shows

a schematic

(a/Q)cr'.

and

in Fig. (a/Q). below

E2-6,

and the

the

relationship

corresponding

along

stress yield

theoretical

with

level. strength

fracture

a similar

between stress,

relationship

The

relationships

of the

material.

hold For

as

al.0

w ii,

true stresses

PROOF

MAX-OPERATING

1.0

a w

ta/Q) i

I

s

Ktc- 1.1

o( Q)cr

(a/Qlcr

I I

] 1.0 FLAW SIZE, _O

FLAW GROWTH POTENTIAL (1-1/a 2)

FIGURE

E2-43.

TYPICAL

MATERIAL

RE LA TIONSHIP

STRESS-INTENSITY

pre-

between

°ult

]

the

for above

Section

E2

1 November Page the

yield

value

up

to the

the

relationships

ultimate

pressure,

follow

the

strength, critical

aul t.

flaw

(a/Q)

= Crproo

size

some

If proof

experimentally pressure

1972

95

determined

is

a

times

the

curve operating

is

max(a/Q)

_

f

iopcr

1 1.21

i. _ /K \ 2 .) \c_ oper/

_

and

_

Thus

the

and

critical

( ic)

1

(a/Q)croper

a

1.21----7

proof-test

factor,

flaw

size

\

a

for

the

oper/

, is a function operating

of the

pressure

maximum

initial

flaw

size

level

(a/Q). 1

1

oper _2

cr

Since

oper

subcritical

compared

flaw

with

the

growth

critical

1.1 max

Kii

KIc

is a function value,

_n"

_

the

a

proof-test

_'a

a

'oper (a/Q)!

oper

where

Kii

temperature,

is the and

initial K

Ie

stress is the

initial factor

stress can

intensity be

related

as as

(a/Q)!/2 oper

1.1

of the

intensity fracture

1

_ 1 oper

at the toughness

operating value

stress at

proof

level test

and

Section

E2

1 November Page temperature. lower

It should

proof

stresses)

temperature and,

can

where

the

consequently,

proof-test

equation selection the

difference

temperature.

For

proof

stress

In fact,

if

Kic

i

(ma)

minimum

a given by

were

Kic

maximum

were (max.)

proof

tested

Kic

and

way

temperature

the

risk

size,

of

a flaw level,

Figure

Kic (ma),

required

fabricated

the

l

in the

E2-44

minimum

slightly

in the

it results

(a/Q).

that

a component

lower

be employed

because

and

than

containing

at the

at a

at operating

event.

flaw

is less

and

is performed

In this

should

of maximum initial

(min.) used

than

(and therefore

as practical.

a conservative use

factors

test

to flaws.

or average

the

Kic

Kic

temperature

factor,

between

(rain.) of

insofar

test

proof

a lower

at proof

proof

required

characteristic (a/Q)

the

proof

if the

susceptibility

Kic

than

lower

has

is minimized

of a higher

trates

material

maximum

rather

that

be employed

greater

failure

The

be noted

1972

96

by from longer

component

illusat proof the

Kic

(max.).

material than would

b

i

Z""°°N" "'-

\

I

\

|

:/la/Olilmmz

ALLOW)

mr-

I

F LAWQSI ZE. alO

FIGURE

E2-44.

DETERMINATION AND MINIMUM

OF PROOF VALUES

STRESS

OF IK-c

BY MAXIMUM

pass

Section E2 1 November Page the

proof

Kic

test

(max)

successfully

P in the

It has

been

thickness,

the

allowable

Kii/Kic.

ance

against

required

vessel

illustrated

failure

mode.

that

Proof-factor

determination

in the

of structural Evaluation

components

including stress,

and

use

of

wall

is always

test

decreasing

structural 1 --

in providing

wall

as occurs

assur-

thickness with

in more

and/or

the

predicted

detail

inRef.

22

the

cyclic

and

necessary will

be applied

sustained

proof-test

meet

the

service

to at least

flaw

growth

factor

can be

life

requirements.

two general

problem

areas

components: modification

of proof-test

stress

and

mission

of components maximum

conditions are

intended operating

for

already for

fixed.

known

stress,

current

missions,

minimum

proof

temperature.

following

factor

for

a hypothetical

I.

Example

pressure

can

design

The

sample

Suppose

_

is discussed

the structure

of material,

proof

of the

proof

same

the

operating

Preliminary selection

This

obtained

and

for which

2.

the

consideration,

to assure

1.

with

The

this.

factor of the

97

E2-45.

under

design

value

Kic,

in service.

regardless

proof-test

changes

toughness,

fail

precludes

that

the

experimentally

a material

determined

by analysis

However,

failure

would

equation

minimum

in Fig.

ttaving for

shown

fracture

pressure and

proof-factor

service

increasing

but probably

1972

problem

where

the

pressure

vessel

has

already

pressure Problem

that

vessel

illustrates vessel

meet

liquid

a service

is sustained been

proper

selection

of a proof-test

design.

A.

a thick-walled

must

the

for

successfully

life

nitrogen requirement

a prolonged proof

5A1-2.5Sn

tested

period with

(ELI)

titanium

of 600 pressure during

each

cycles cycle.

LN 2 to a proof

factor

The

Section E2 1 November Page

!

1972

98

I

I

=

\-

_I! J

i_ " ,

.

0 0

0

< 2; r_ M

0

' _i--

_

u

i o

o_ o_

_ _

_

_._ -! uli'_

II

o

" I

_" ,.

0

Ik

•

•

t

t 0

,.,3 _=

_._ •

_ii_

Section E2 1 November Page = 1.25. service

Fig. stress

Fig.

Will life

this

and

proof

if not,

The

cyclic

E2-36.

The

flaw

growth

For

long hold

E2-36

for

factor

what

life

curve

estimated

no failure

occurs

factor

is required

?

for

5A1-2.5Sn

(ELI)

titanium

threshold

time,

that

proof

(KTH/KIc)

Kii/Kic

assure

stress-intensity

is approximately max.

= 0.80,

Kii/Kic N=

1972

99

during

the

is reproduced

value

for

in

sustained-

0.92.

= 1/_

300 cycles.

= 1/1.25 For

= 0.80.

KTH/KIc

From = 0.92,

N = 100 cycles. In 300 cycles would

have

growth. proof

reached Thus

factor For

minus the

estimated

the

predicted

of 1.25

will

100 cycles

KI i /K I c

100 cycles,

plus

= 0.70

threshold

minimum not assure

or 200 cycles,

life

value would

a service

600 cycles,

life

for

the

stress

intensity

sustained-stress

be only

200 cycles

flaw and

the

of 600 cycles.

or 700 cycles,

= 1(_

and

-

Thus, times

the

1 0.70

= 1.43

the 600-cycle

operating

pressure.

service

life

requires

a proof-test

factor

of 1.43

Section E2 1 November Page

1972

100

REFERENCES

Fracture Toughness Testing and itsApplications. ASTM Special Technical Publication No. 381, American Society for Testing and

.

Materials, April 1965. Irwin, G. R. : Fracture and Fracture Mechanics Report No. 202, T&AM Department of Theoretical and Applied Mechanics, University

.

of Illinois,October

'

Irwin, G. R. : Crack Extension Force

for a Part-Through

Plate. Journal of Applied Mechanics, December 1962.

Vol. 84 E, No. 4,

Green, A. E. ; and Sneddon, I. N. : The Distribution Neighborhood of a Flat Elliptical Crack in an Elastic

.

ings e

.

1

of the

Boeing

Co.,

December

Review of Developments ASTM Special Technical

Paris,

and P.

Materials, C. ; and

Propagation the ASME, .

Philosophical

Society,

Vol.

Crack

in a

of Stress in the Solid. Proceed46,

1959.

Factors of Deep Surface Memorandum No. 16,

1965.

Plane Strain Crack Toughness Publication No. 410, American December 1967.

Testing .

Cambridge

Kobayashi, A. S. : On the Magnification Flaws. Structural Development R_search The

10.

1961.

Testing. ASTM Special Technical Society for Testing and Materials,

in Plane Strain Fracture Toughness Testing. Publication No. 463, American Society for September

Erdogan,

F.:

1970. A Critical

Laws. Journal of Basic Series D, Vol. 85, 1963.

Engineering,

Foreman, Analysis

R. G. ; Kearney, V. E. ; and Engle, of Crack Propagation in Cyclic-Loaded

of Basic

Engineering,

Hudson,

M.

C. : Effect

7075-T6

and

2024-T3

August

1969.

Transactions of Stress Aluminum-Alloy

Analysis

Transactions

September

on Fatigue-Crack Specimens.

of

R. M. : Numerical Structures. Journal

of the ASME, Ratio

of Crack

NASA

1967. Growth

in

TN D-5390,

Section E2 1 November 1972 Page 101 REFERENCES(Continued) 11.

12.

Tiffany, C. Plane-Strain 1966. Fracture May

F. ; Lorenz, P. M. ; and ttall, Flaw Growth in Thick-Wall_d

Control

of Metallic

Pressure

Wheeler, O. of the ASME,

14.

Elber, Fracture

15.

Elber, Wolf: The Significance Tolerance in Aircraft Structures,

16.

18.

American

Crack Vol.

Society

Air

Tiffany,

C.

ASTM

Special

Testing

and

Wessel,

E.

Progress of Fracture

Vol.

Closure II, No.

for

Under Cyclic 1, Pergamon

of Fatigue ASTM

Testing

Tension. Engineering Press, July 1970.

Crack Special

Closure. Technical

and Materials,

Damage Publications

1971,

pp.

230-242.

Fracture

Base,

Ohio,

F.;andMasters,

J.

Technical Materials, T.,

N.:

Publication April

et al.

November

1963.

Applied No.

Steel Report

Fracture

381,

Mechanics.

American

Society

for

1965.

Engineering Against

Methods

Fracture.

for

the

ADS01005,

Design June

and 1966.

in the Measurement of Fracture Toughness and the Application Mechanics to Engineering Problems. American Society

4, No.

Pyle,

Force

of Materials

for Testing High-.Strength

20.

SP-8040,

Pellissier, G. E. : S/ome Microstructural Aspects of Maraging in Relation to Strength and Toughness. Technical Documentary RTD-TDR-63-4048. Air Force Materials Laboratory, Wright-

Selection 19.

NASA

E. : Spectrum Loading and Crack Growth. Transactions Journal of Basic Engineering, March 1972.

W. : Fatigue Mechanics,

Patterson 17.

Vessels.

1970.

13.

486,

L. R. : Investigation of Tanks. NASA CR-54837,

and Materials Special Metallic Materials, 3, March

R. ; Schillinger, Toughness

and 5AL-2.5Sn(ELI) 5AL-2.5Sn(ELI) of the Army,

1964, D.

p.

Committee Materials

on Fracture Testing of Research and Standards,

107.

E. ; and

and Mechanical

Carman,

C.

Properties

of 2219-T87

Titanium Alloy Weldments Titanium Alloy Plate. NASA

Frankfort

Arsenal,

September

M. : Plane

Strain Aluminum

and One Inch Thick CR-72154, Department

1968.

Section

E2

1 November page REFERENCES

21.

Wilhem,

D.

P. : Fracture

1972

102

(Concluded)

Mechanics

Guidelines

for Aircraft r_

Structural February 22.

Shah,

Applications. 1970.

R. C. : Fracture

AFFDL-TR-69_111,

Mechanics

Assessment

Vehicle and Spacecraft Pressure Vessels. The Boeing Co., November 1968. 23.

Kobayashi, Intensity Research

A.

S. ; and Hall,

Northrop

Corp.,

of Apollo Vol.

Launch

1, Report

L. R. : On the Correction

Factors for Two Embedded Cracks. Structural Memorandum No. 9, The Boeing Co., 1963.

D2-114248-1,

of Stress Development

SECTION F COMPOSITES

-,q...1

SECTION FI COMPOSITES CONCEPTS

,,..j,f

TABLE

OF

CONTENTS

Page F1.0

COMPOSITES

- BASIC

CONCEPTS

AND

NOTATIONS

. . . ....

1

f--, 1. i

1.2

Basic

........................... Strain

17

Law

20

1. 1.2

Generalized

Hooke's

1.1.2.1

Homogeneous

1.1.2.2

Elastic

1.1.2.3

Monoclinic

1.1.2.4

Orthotropic

1.1.2.5

Isotropic

1.1.2.6

Transformation

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

Isotropic

Linear

of Laminated

17

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

Stress

Mechanics

and

, •

1. 1. 1

Material

Anisotropic

Material

Material Material

.....

20

Material...

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

2.2

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

22

25

............... of

Stiffness

Composites

20

Matrix

....

27 33

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

1.2.

1

Micromechanics

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

33

1.2.

2

Macromechanics

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

34

1.2.3

"-_

Concepts

1.2.2.1

Lamina

Constitutive

1.2.2.2

Laminate

Example

Calculations

Relationship

Constitutive

Relationship

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

..... ....

35 41 50

1.2.

3. 1

Example

Problem

1 ..............

5O

1.2.

3.2

Example

Problem

2 ..............

54

FI -iii

TABLE

OF

CONTENTS

(Concluded)

Page 1.3

Laminate

Coding

i. 3. 1

Standard

1.3.2

Codes

Computer References

........................... Code

for

Programs

Elements

Various

Types

in Composite

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

F_ -iv

57 ................. of Laminates Analysis

57 ....... .........

59 61 63/64

DEFINITION OF SYMBOLS

Definition

Symbol

[Al

extensional

[a]

average laminate (F1. 2-28)

[A*]

laminate

[B]

coupling

[Cl

lamina

stiffness

[c*]

column

form

of the

stiffness

tc* ]

column

form

of the

transformed

rigidity

matrix

(Ft.

[C']

modified form (Ft. 2-I)

[c'l

transformed

11

matrix

- equation

- equations

stiffness

(F1.

(F1.

matrix

(F 1. 2-18)

- equation

(F1.

2- i 8) and

- equation

matrix

of lamina

matrix (Ft.

rigidity

matrix

matrix

- equation

2-30) ( F 1. 2-19j

1-6)

-equation

stiffness

(F 1. 1-27)

matrix

-

1-28)

transformed

equation

of laminate

extensional

compliance

equation

C°,

matrix

of [C]

stiffness

of the

- equation

matrix

modified

(F1.2-15)

- equation

stiffness

matrix

-

2-10)

components

of the

stiffness

matrix

components

of the

transformed

[C],

[C r]

m

oil

stiffness

matrix

[C],

[c'l

[D]

laminate

flexural

E

YoungVs

modulus

of elasticity

Young's

modulus

of laminae

Eij

>-_

[c'l,

- equation

rigidity

(F 1. 1-15)

F1 -v

matrix

- equation

in lamina

(F1.2-19)

principal

direction

DEFINITION

OF SYMBOLS

Definition

S_mbol

Gij J

shear

moduli

invariants (Ft.

in the

M

laminate

bending

N

laminate

forces

[sl

lamina

compliance

dA

index

components

stress (F1.

plane

- equation

matrix

(F 1. 1-15)

transformations

- equation

2-12)

lamina

[TI

i-j

of stiffness

K

sij

(Continued)

indicating

and twisting - equation

of the

and

position

strain

in laminate moments

(Ft.

- equation

2-16)

matrix

- equation

lamina

compliance

transformation

(F1.

1-7)

matrix

matrix

- equation

1-3)

differential

area

- equation

(F1.1-1)

distance

from

reference

laminae

K and

K+I

t

laminate

total

X

principal

longitudinal

Y

principal

transverse

Z

principal

normal

O_

principal longitudinal lamina axis (Fig. FI. 0-2)

hK

surface

- Fig.

thickness

- Fig. laminate laminate

laminate

to plane

separating

F1.2-3

axis

F1.2-3 axis axis (Fig.

(Fig. (Fig.

F1.0-1) F1.0-1)

F 1.0-1)

principal transverse lamina axis (Fig. FI. 0-2) T

shear

strain

Fl-vi

(F1.2-17)

DEFINITION

OF SYMBOLS

(Concluded)

Definition

Symbol 0

3/

shear strain of laminate middle surface - equation (F!. 2-13)

E

strain

F"

0 E

strain of laminate middle surface - equation (FI. 2-13)

[el

strain column

matrix - (FI. 1-10)

shear coupling ratios - equation (FI. 1-15) 0

angular orientation of a lamina in a laminate, i. e., the angle between the x and counterclockwise

a

axis - positive is

Poisson's ratio O"

st r es s

[cr]

stress

T

shear

column

matrix

- equation

(F1.

1-9)

stress

curvature

of laminate

arbitrary

local

middle

surface

- equation

SUBSCRIPTS 1,2,3

lamina

._

coordinate

principal

system

axes

x,y,z

laminate

reference

ij

the (i-j) between

position 1 and 3

axes in a sequence

F1 -vii

where

i and j vary

(F1.

2-13)

F1.0

COMPOSITES The

utilized

purpose

in the

of this

design

and

An attempt

was

made

elementary

as possible.

is more

complex

analysis

techniques In order

must

have

- BASIC

from

reference

for

stress

to keep

than

that

and

concepts

References section

analytical the

may the

of certain

stress

basic

but also

definitions.

materials

as a result,

the

composites,

These

intended

as a guide

as

involved.

of laminated

3, are

material

of composite and,

rather

structures.

and

analysis

materials, seem

techniques

composite

developments

mechanics

1, 2, and

state-of-the-art

of advanced

of conventional

to understand

this

the

AND NOTATIONS.

is to present analysis

However,

a knowledge

primarily

section

CONCEPTS

definitions,

to serve

to general

one obtained

not only as a

literature

on composite

materials.

AEOLOTROPY

See anisotropic.

ANGLEPLY

Any filamentary equal

laminate

numbers

symmetry

about

An alternate current

the

literature,

of laminae

coordinate

definition

is a laminate of layers

of pairs

constructed

used

having

with (x,

the

same

y)

frequently

but not in this

consisting

with

axis. in

section,

of an even

number

thickness,

and

i,

Section

F1.0

1 October Page the

ANISOTROPIC

orthotropic

ply are

alternately

+ e and

- 0 to the

Not isotropic;

COMI_SITE

of symmetry oriented

having

which

direction

relative

to natural material.

A composite

laminate

whose

symmetrical

with

of the

laminate

(Fig.

and/or

vary

in the

relation

layup to the

F1.0-1).

_3_

_.3.

_

FIGURE

F1.0-1.

hK

BALANCED

OR SYMMETRIC

with

reference

hK

z:

of

axes.

mechanical

properties

inherent

in each

at angles

laminate

physical

axes

BALANCED

axes

1971

2

COMPOSITE

is midplane

Section

F1.0

1 October Page BUCKLING

Buckling

is

generally

a mode

by

caused

by

structural

an unstable

element

local

COMPLIANCE

MATRIX

The

COMPOSITE

MATERIAL

but

may

take

also

the

the

form

not

instability

and

of a microinstability

fibers.

compliance

matrix

is defined

e. = S..a., 1 D J

where

components

of the

compliance

be

by

obtained

Composites

are

the

considered

or

constituents

retain

composite;

that

otherwise

merge

they

the components

form

is,

act

on

the

the

matrix;

may

stiffness

matrix.

to be

of materials

composition

by

S.. are U

inverting

combinations

although

on

In advanced

general

instability

equation

deflection

involved.

buckling

of individual

characterized

action

of conventional

3

lateral

compressive

composites, only

of failure

1971

differing

in

a macroscale.

their

they

identities

do

completely

in concert.

not

The in the

dissolve

into

each

or other

Normally,

can be physically

identified

Section

F1.0

1 October Page

1971

4

and lead to an interface between components (Fig. FI. 0-2). Z

MATRIX (1

FIBER FIGURE

CONSTITUENT

CONSTITUTIVE

F1.0-2.

PRINCIPAL CONSTITUENTS OF COMPOSITE LAMINA AND PRINCIPAL AXES

In general,

an element

in advanced

composites,

constituents

are

(refer

to Fig.

Refers

to the

grouping;

the principal

the fibers

and the matrix

F1.0-2).

stress-strain

(Hooke's

relationships

for a material

because

stress-strain

relations

the

CROSSPLY

of a larger

mechanical

Any filamentary equal

numbers

of 0 deg

and

actually of the

laminate

constructed

90 deg

of laminae

to the

laminate

the

describe

constitution

of pairs

Law)

material.

with at angles axes.

Section

F1.0

1 October Page An alternate current

definition

literature,

is a laminate of layers

all

of the

orthotropic

axes

alternately

oriented

The

separation

number

thickness

of symmetry at angles

in

section,

of an even

same

laminate

5

frequently

but not in this

consisting

90 deg to the

DE LA MINA TION

used

1971

with

the

in each

ply

of 0 deg

and

axes.

of the

layers

of material

in

a laminate.

FIBER

A single

homogeneous

essentially

constituent

CONTENT

The

sense,

axial

to Fig.

amount

fraction

used

as a principal

composites

strength

and

because

modulus

F1.0-2).

of fiber

reinforcement usually

in the

in advanced

of its high

FIBER

of material,

one-dimensional

macrobehavior

{refer

strand

present

as

in a composite.

expressed or weight

This

as a percentage fraction

of the

is

volume composite.

Section

F1.0

1 October Page FIBER

The

DIRECTION

orientation

longitudinal

or alignment

axis

to a stated

of the

reference

of the

fiber axis

6

with respect

(Fig.

F1.0-3).

a

X

FIGURE

FILAMENT

FI. 0-3.

LAMINA

AXIS

A variety extreme normally except Filaments

ORIENTATION

of fibers length,

characterized such

no filament

that

there

ends

within

at geometric

discontinuities.

are

in filamentary

used

by are a part

1971

Section F1.0 1 October 1971 Page 7 composites and are also used in filament winding processes, which require long, continuousstrands. FILAMENTARY COMPOSITES

Composite materials of laminae in which the continuousfilaments are in nonwovcn, parallel, uniaxial arrays.

Individual

uniaxial laminae are combined into specifically oriented multia×ial laminates for application to specific envelopes of stren_,_th and stiffness requirements (Fig. F1.0-4). Z

o o o o_ I oo ooooooo FIGURE

GENERALLY

F1.0-4.

ORTHOTROPIC

C'-,,'"."-_( o__ FILAMENTARY

Descriptive the

constitutive

COMPOSITE

term

for

a lamina

equation,

when

for

which

transformed

Section

F1.0

1 October Page to an arbitrary populated.

set

That

of axes,

is,

1971

8

is fully

for

u

-0.x 1 K

F_ll -E12 _16

'L

0.y

'

=

C12

• ] xy

C16

HENCKY-VON MISES DISTORTIONAL ENERGY THEORY

(al_

o.2)2+ (a 2-

HETEROGENEOUS

for

o.3)2+ (0.3-

C26

criterion

isotropic

using

distortional

energy

for

a material

contituents

properties

2o"02 .

consisting separated

not homogeneous.

Generalized

yon Mtses

for anisotropy:

consisting

separately

boundaries;

account

,

materials:

a medium

of unlike

YxyJ

_e O,

term

identifiable;

Ey

_66

°"1) 2+ 6(0" 122+0.232+0.sl 2)=

Descriptive

1 I

16 _26

of dissimilar

HILL

_26

°

_ O,

Yield

_22

Ex

yield

of regions by internal

criterion

to

Section

F1.0

1 October Page A!

((72

-

0"3) 2 +A

2 ((73 -

o"1) 2 + A 3 ((71

-

o-2) 2 +

1971

9

2A4o'232

/

+

2A5CT312

+ 2A6cT122

= 1 ,

where

2f(oij

)

2A 1

= the =

plastic

(F2)-2

+

potential, (F3)-2

_

(F1)-2

2A 2 =

(F3)-2+

(F1) -2_

(F2) -2

2A 3 =

(F1) -2+

(F2) -2_

(F3) -2

2A 4 =

(F23) -2

,

2A 5 = (F31) -2

,

2A6

= (F12) -2

,

and

F1,

F 2,

and

FI2,

and

F23,

and

F 3 are F31

are

determined determined

HOMOGENEOUS

from

unia×ial

from

pure

shear

composition

throughout;

for

physical

Descriptive no plane orthotropic

a material

of uniform

a medium

boundaries;

properties

at every

tests,

tests.

term

whose

ANISOTROPIC

or compression

Descriptive

no internal

HOMOGENEOUS

tension

are

constant

for

a material

which

has

a material and

isotropic

point.

term of material

material.

symmetry

which such

has as the

Section FI. 0 1 October 1971 Page I0

HOMOGENEOUSGENERALLY ORTHOTROPIC

Descriptive

term

in a manner

for

a lamina

similar

to the

which

behaves

anisotropic

lamina.

HOMOGENEOUSISOTROPIC

Descriptive

term

a constant

modulus

for

SHEAR

See interlaminar

The

INTERFACE

boundary

physically

which

of elasticity

C16 = C26 = 0 in its

HORIZONTAL

a lamina

constitutive

has

and the equation.

shear.

between

the

distinguishable

individual, constituents

of

a composite.

INTERLAMINAR

SHEAR

Shear

force

relative

which

tends

displacement

in a laminate

along

to produce

a

between

two

plane

of their

the

laminae

interface.

ISOTROPIC

Descriptive uniform

term material

for

a material

properties

which

has

in all

directions.

LAMINA

A single

ply or layer

a series

of layers

in a laminate

(Fig.

F1.0-5}.

made

of

Section

F1.0

1 October Page

1971

11

f

X

/ /-

/ / / Z 0

0 Z I

N

/

/

/ /

Z

_q L_ I

o

.J

/ r_ ©

I I I /

V i.l.l

l-

z --i

Section

F1.0

1 October Page

1971

12

LAMINAE

Plural of lamina (refer to Fig. FI. 0-5).

LAMINATE

A product more

layers

materials

LAMINATE

ORIE NTATION

The

made

by bonding

of laminae (refer

two or

of material

to Fig.

configuration

together

or

F1.0-5).

of crossplied

composite \

laminate

with

crossplying, angle,

LAYUP

MACRO

the

and the

to the

number

exact

angles

of

of laminate

sequence

at each

of the

individual

laminae.

A process

of fabrication

involving

placement

of successive

layers

In relation

to composites,

denotes

properties

of a composite

as a structural

element

but does

properties

MATRIX

regard

or identity

The essentially which are

the

not consider

fibers

imbedded

of the

homogeneous or filaments {refer

to Fig.

the of materials.

the

the

gross

individual

constituents.

material

in

of a composite F1.0-2).

Section

FI. 0

I October

W"

Page

In

MICRO

F-

relation

to composites,

properties

of the

denotes

constituents

effect on the composite

Descriptive

ORTHOTROPIC

three

term

for

mutually

1971

13

the

and

their

properties.

a material

which

perpendicular

planes

has

of

f-

elastic

PRINCIPAL

AXES

The

symmetry.

set

of axes

parallel

and

direction

axes

is

(refer

lamina

is

filament

principal

F1.0-2).

for

a laminate

which

stiffnesses

and

has

st rengeth.

Descriptive

the

Fig.

which

to the

the

isotropic

perhaps

ORTHOTROPIC

called

term

essentially

SPECIALLY

perpendicular

to

Descriptive

QUASI-ISOTROPIC

in a lamina

Cl6

term

: C26

for

0

lamina

in its

for

which

constitutive

equation.

TENSOR

A tensor

which relations.

is

obeys

a physical

certain There

entity

in nature

transformation are

different

orders

of

Section

F1.0

1 October Page tensors,

and

each

transformation

TRANSVERSELY

ISOTROPIC

term case

properties

for

a material

identical

but

in which

third;

in both

but

not the

Hencky-von

Mises

exhibiting

in two orthotropic

not in the

properties

directions

its own

of orthotropy

are

dimensions, identical

has

relations.

Descriptive a special

order

1971

14

having

transverse

longitudinal

direction.

VON-MISES ENERGY

DISTORTIONA

L

See

Distortional

Energy

THEORY Theory.

x-AXIS

An axis

in the

plane

of the

is used

as the

0 deg

reference

designating Fig.

y-AXIS

The

which

for

the

angle

of lamina

in the

plane

of the

(refer

to

F1.0-5).

axis

is perpendicular Fig.

laminate

FI.0-5).

to the

x-axis

laminate (refer

which to

Section

F1.0

1 October Page z-AXIS

The the

f

f

J

reference laminate

axis (refer

normal to

Fig.

to the F1.0-5).

1971

15/16 plane

of

Section

F1.0

1 October Page 1. 1

BASIC

Some

stress

CONCEPTS.

of the

composites,

are

and

i. I. 1

basic

concepts

presented

strain

at

STRESS

f-

1971

17

Following

in this

a point AND

applicable

and

subsection.

their

continuums,

These

transformation

in particular

include

the

concepts

of

relations.

STRAIN.

the guide of Ilefcrence

point will be reviewed

to all

to form

a firm

i, stress

base

and strain relations

at a

for the analytical development

for

composites.

A tensor,

which

obeys

as defined

certain transformation

tensor

of zero-th

known

that the components

altered

order,

or rotated.

establish

of a vector

transformation

Stress

physical

A scalar,

is a tensor

change

when

in the components

relations; therefore, entity is a tensor

entity in nature

for example,

of first order.

of the vector Each

and determine

system

is

is governed

order

it is necessary

is a

It is well

the coordinate

relations or transformations.

transformation

that a physical

is some

relations.

and a vector

This change

by certain mathematical

has its own

previously,

of tensors

only to

its order,

and the

relations are defined.

and strain are both second-order

relations are well known

circle).

These

relations

of a small

(the graphical

transformation

element.

relations

Consider

form

may

tensors

and their transformation

of the transformation

be derived

a two-dimensional

from

is the Mohr's

the equilibrium

problem

as shown

Section

FI.

i October Page in Fig.

FI.

equation

1-1.

By summing

forces

in the

"1"

direction,

1971

18

the following

results:

_ldA-_

-_

x

(cosSdA)

(sin0dA)

xy

(cos

(cos

e)

-

_

8) -

T (cos xy

y

(sinedA)

0dA)

(sine)

(sin0)

=

0

,

or

_I = _

cos29

x

+ o

sin 28+

y

r

xy

(2 sin#cos

8)

(FI.I-I)

2

o2

/'

Y 2

_r12/,_°1

o2

T12A O x

m

"rxylJ

X

Oy

o¥

FIGURE

determined. developments,

_dA

"T-

J In a similar

A

manner, These

FI.

the

I-i.

other

equations

a matrix

form.

STRESS

COORDINATE

transformed may

be written

Thus,

ov

ROTATION

stresses, in a form

cr2 and

rt2 ,

convenient

may

0

be

for later

Section

F1.0

1 October Page cos 2 0

[ lj

sin 2 0

_2

-sin

Tt

Using

a more

sin 2 0

compact

(2 sin

cos 2 0

0 cos

0)

(sin

notation,

¢r2

0 cos

we may

I =

(-2

[T]

is the

transformation space. any

Equation

symbol

for

(F1.

1-3)

two-dimensional With

transformed

a slight by the

stress

state

modification, same

0)

_y

- sin 20)

7

(F1.1-2)

as

matrix.

tensor

when

necessary from

one

set

Equation

reduced

relationship

. (F1.

(FI.

1-3)

(F1.

1-3)

to a two-dimensional required

of coordinates

the two-dimensional

strain

to transform

to another may

set.

be

transformation:

1 ¢2

_

0 cos

19

(F1. !

½":,2J

=

LT]

_y

2

1-2)

xy

the transformation the

0) 1

[TI

is the

stress

equation

L

for

relation

0) (cos20

write

T12 J

where

sin

0 cos

1971

xy

]

J

1-4)

is the

Section F1.0 1 October 1971 Page 20 1.1.2

GENERALIZED HOOKE'SLAW. In this paragraph the constants of proportionality between stress and

strain {Hooke's law constants} are shownto be componentsof a fourth-order tensor andtherefore have a set of transformation relations different from those for stress and strain.

Several forms of the Hooke's law relationships andthe

elastic constants will be shownfor the various material conditions. 1.1.2. 1

Homogeneous

For stress

state

the

Isotropic

familiar

[1],

the

Material.

homogeneous Hooke's

law

isotropic relationship

material

in a one-dimensional

is

= EE .

The

proportionality

of elasticity, 1.1.2.

2

(FI. i-5)

and

is a scalar

Elastic

Linear

Consider linearity,

which

constants.

The

constant

is the

is Young's

modulus,

or the

modulus

value.

Anisotropic

the most

(E)

Material.

general

material,

anisotropic

constitutive

material.

equation

(Hooke's

but require This law)

elasticity

material is

has

and 21 elastic

[2]

m

C11

C12

C13

C22 Cn [a]

=

C14

C15 C16

C24 C2s C2_

C33 C34 C3s C36 C44

C45 C46

Symmetric Css Cs6 C66 --

a

[_] .

(F1.1-6)

Section

FI.

1 October Page The

components

equation

may

C.. 11 be

are

written

called

components

of the

"stiffness"

matrix.

0 1971

21 The

as

m

Sit

S12

St3

$14

$15

$16

822

S23

824

S25

$26

$33 $34 $35 $36 Symmetric

[(_1

(F1.

1-7)

(F1.

1-8)

$44 $45 $46 Sss Ss6 S66

'where [C]

=

The

components

[ _]

conventionally

IS] -1

.

S.. 1]

are

called

symbolizes

components

of the

"compliance"

The

matrix.

[ 3]

- _rl

I [or]

=

°3

I

(F1.

1-9)

T23I ?13

I

I_ T12J

Similarly, ;--El

"7

e2

I I

[E]

=

_3

(FI. i-I0) I

_/23

_13 __/12J

Section

F1.0

1 October Page 1.1.

2. 3

Monoclinic

If a material monoclinic symmetry

and

is assumed

22

Material. possesses

material

1971

one

has

13 independent

to be the

x-y

plane,

0

0

Ct6"

C23

0

0

C26

C3s

0

0

C38

C4$

0

C55

0

CtICz2Ct3 C22

[ a]

plane

C44

Symmetric

of symmetry, elastic the

it is termed constants.

constitutive

a

If the equations

plane are

of [ 2]

[c]

(rl.

[a].

(FI. 1-12)

i-li)

C66

and

Sit St2 St3 0

0

$22S23 0 [c]

=

833

S16

0 $26

0

0

836

$44 S4s 0 Symmetric

Sss 0 866

m

1.1.2.

4

m

OrthotroDic

If the

anisotropic

assuming

x = 0 and

condition,

there

material symmetry

has

Material.

are

material z = 0, only

two orthogonal

exist.

The

nine

the

possesses material

independent planes

constitutive

two orthogonal is termed elastic

of symmetry, equations

of the

planes

orthotropic. constants. three

In this Note

orthogonal

orthotropic

of symmetry,

that planes

material

ff a of are

Section

F1.0

1 October Page -C11

C12

C13

0

0

0-

C22 C23 0

0

0

C33 0

0

0

C44 0

0

[=J

Symmetric

[(1

1971

23

(F1.1-13)

Css 0 C66

m

and Sll S12 S13 0

0

0

$22 $23 0

0

0

$33 0

0

0

$44 0

0

[_]

Symmetric

Sss

[el

•

(F1. 1-14)

0 S66

B

Since laminates, values

u

most this

which

composite

material

structures

is of particular

are

establishable

from

be equated

to the

components

of the

components

are 1

interest.

uniaxial

The

and

compliance

pure

of orthotropic engineering

shear

matrix.

l $22-

E22

The

tests,

constants, may

easily

compliance

1 '

S_3 = :L22_ S12 = El I

_-v_l__

,

:Lv__

S31 =-vaX_ E33

,

=:._V_I__

E33

1 S6_ = 2GI--' _ El I

E33

$33-

E22

E22

=

be constructed

then

$11 = 1El-- '

Sl 6

will

1

1

$55 = 3Gl_

$44- G23

S26 '

Ell

:

7726 E22

836 = '

E33

(FI. "

1-15)

Section FI. 0 1 October 1971 Page 24 where Eft,

E22,

E33 = Young's

moduli

directions,

in the

I,

2, and

are

used

3 (x,

y,

and

z)

respectively,

v.. = Poisson's ration lj strain in the j direction strain in the i direction caused

by a stress

in the i direction,

Gij

= shear

moduli

_ij

= shear

coupling

Note material.

that the From

be determined

SI6,

equation

in the i-j plane, ratios. S26, and (F1.

S36

1-8),

terms

the components

for a "monoclinic" of the

stiffness

matrix

as

Cll = (I

- v23 _32) VEIl

,

C22-= (I

- v31 v13) VE22

,

C33=

- vi2

,

(I

v21) VE33

C12 = (v21 + v23v31)

VEIl

= (VlZ + vI3 v32) VE_2 C13 = (v31 + v21v32)

VEil

= (vla + v23 v12) VE33

, , , ,

C23 = (v32 + v12 v31) VE22 , = (v23 + vZl v13) VE33 C44 = G23

,

C55 = G31

,

C66 = G12

,

and

,

(FI.

1-16)

may

Section

F1.0

1 October

1971

(

Page

25

where

V--

(1

For

equation

an

is

-

1_12

t_21

-

/"23

orthotropic

P32

-

1)31

/_13

material

-

2te12P23t_'31)

in a state

-1

of plane

stress,

the

constitutive

[ 1]

=

(_2 _r 1 712

icll c10111 C12

C22

0

(2

0

0

C66

3/12

(F1.

1-17)

(El.

1-18)

where

Ell Cll

=

(1-

_12v21)

C22

=

(1

-

C12

=

(i-

C66

=

G12

E22 -_12_21)

'

l' 12E22

21E11

71_,21)

=

(i - .12.21)

'

and

1. 1.2.5

Isotropic For

constants.

an The

Material.

isotropic constitutive

material, rel:_tions

there

are are

only [3]

two

independent

elastic

Section FI. 0 1 October

1971

Page 26 m

Ctl C12 Ct3 0

0

Ct2

0

0

0

Ctt

0

0

0

½ (ctl- ct2) 0

0

Ctt [_]

0

=

Symmetric

[c]

(F1.1-19)

[_]

(FI. 1-20)

- Ct2) 0

½ (Ctt

½ (c. - ci_) and i

SIt

S12

S12

0

O

0

0

0

0

0

0

0

2 (Slt- St2) 0

0

Sll $12

[_] Symmetric

Sit

2

(Sll-

S12 )

2 (Stl

0

- St2)

The constitutive constants may

be defined in terms

of the engineering

constants as

(l-u) E Cit =C22 =Csa = (1+ v) (I- 2v)

rE Ct2 =CI3 =C2s = (1 + v) (1 - 2v)

'

E C44=C55

=C66=G

=

1 Stt=S22=S3a=_

St2 --Sis = $23 = E

,

2(1+v)

'

(Ft. 1-21)

Section

F1.0

1 October Page

1971

27

and 1 $44

= $55

If the

isotropic

(plane

stress),

= $66

2(1

material

E

is assumed

equation

(Y2

+ v)

---G_-

(F1.

to be in a two-dimensional

1-19)

Cll

C12

0

C 12

C 22

0

may

be written

as

stress

state

[ 1]

1 E1

£2

(F1.

1-22)

(F1.

1-23)

I

0

T12

0

C66

. "F12

where E C11=C22-

(1-,2) E

(1 - v 2'J

'

and

-

1.1.2.6

Transformation In Reference

stated certain material, given

G

of Stiffness 1, the

to be components transformation in three in Reference

elastic

constants

of a fourth-order relations.

dimensions 2 as

Matrix.

The and

(stiffness) tensor

and

transformation

rotated

an angle

for

a material

consequently,

must

of a general 0 about

the

are

z-axis,

obey

anistropic is

Section F1.0 1 October 1971 Page 28 [_*1

= [TRI

-_iN

(F1.1-24)

[C*l

where T .... tl_n 2

"_

"-I" .......

I I tara!

m*+a4

I I-_an

rata t

2man-_tm ! m*.$man z -2rant 2m_

I n4 J._n

rant.man

!

2mat.

3manl.n

2rata

4

ennt-_'nan

{ I I ITR]=

man-ma

-m'n' I

t

2ma) 3mtaZ-n * m_n-mn$ m_ -2man m_ t 2mtn

ma-3m_

4m _nz

_

a-_rana

t

rand-man (m_-n:l)

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

I {

I

]

i [

J 1 I

-T_T--_

ma,--..-r -_

_ran l

}mln

m !

nj

mn 2

mn t

_nta

J,_,

_._

m'

-_

-_'n

roll I

Jn I

roll I

man

m ]

l-2m_

2ran 2

2m_n

-2ran

-2man

2ran t

2rain

-ran

[ I J

.mZn

!

] J

1

I [ I ; I I t-- ........... 1

I-2ran

2

t

mr-ran mtn-n

r r

_ nt-m t

mr-ran

ta lj -_'mt

r _ -_,

n !

mn

I

_t -ran; zma m2-o'I

I

-_ I i

I F

[ _J,

!

Ima m'._' -m_t I., zma n' I

I

m

m=

and

n terms

cos

(F1.

1-25)

(Fl.

1-26)

represent

0

and n = sin 0

The the

[C*]

transformed

and

[C':'I

equation.

are The

column [C*]

matrix is defined

forms as

of equation

[ I

t° ml__j

l.__

The

I ]

(F1.1-6)

and

Section

FI. 0

1 October Page

1971

29

m

Cll C12 2C16 C22 2C26 4C66 C14 C15 C24 [C*]

=

C25 2C46 2C56 C13 C23 2C36 C44 C45 C55 C34 C35 C33

(FI. i"-27)

Section

F1.0

1 October Page and the

[_'*]

is defined m

m

1971

30

as

-.-,,i

rl

!

_. 12

; I

2C16

I

C2_ I 2C2s

I

4C66

I J_

_24

I

_25

I

2C_

(FI.

1-28)

1

2Cs6 I -"-_13

,, I

'_23

I

2Cae 1244 C45 C65 C34 C35 C33 The in a plane applicability elastic axis,

transformation stress

state

to fibrous

constants

are

the transformed

of the

stiffness

is of particular composites. needed elastic

with

matrix interest

As shown respect

constants

for since

this

axis

material

will be of direct

in Reference

to some are

an orthotropic

other

1, when

the

than the material

Section

F1.0

1 October Page

f-

Cli

=

Cll

cos 4 0 +

2 [C12

+ 2C66 ] sin 2 0 cos 2 0

C22

=

Cll

sin 4 6

2 [C12

+ 2C66 ] sin 2 0 cos 2 0

C12

=

[Cll+C22-

4C66 ] sin 2 0 cos 2 0

C66

=

[Cll

2C12 - 2C66 ] sin 2 0 cos 2 0

C16

=

[CII-

C26

=

[Cli - C12 - 2C66] sin 3 0 cos

The

angle of rotation (0) is about the z-axis

+

÷ C22 -

C12-

+

2C66] sin 0 cos 3 0 +

C12 [sin +

[C12-

+

+

40

C2 2

1971

31/32

sin 4 0 ,

C22 COS 4 0

'

+

cos

4 0]

C66 [sin

4 0

+

(F1.

1-29)

, cos 4 O]

,

C22 + 2C66 ] sin 3 0 cos _)

and m

the counterclockwise

direction.

0

+

[C12 - C22 + 2C66 ] sin 0 cos 3 0 .

and assumed

positive in

Section F1.0 1 October 1971

¢.-

Page i. 2

MECHANICS Two types

the

OF LAMINATED

of mathematical

stress-strain

response and

macromodeis.

problem

the

individual

models

array

of filaments

lamina their

respective

of the

behavior

lamina

and models

1.2.

MICROMECttANICS.

directly the

(sheet)

strength

related

and

to the

micro-geometry

encompasses

the

as a result to predict

the from

perhaps

provide criteria With

composites whieh

the

the for

properties laminate

The

macroscopic

material

the

basis

from

the

and for

The

stress

field

understanding

state-of-the-art,

aerospace

application

stress

and

aerospace

of the

the matrix

the

homogeneous

composites and

are

matrix,

as well

as

of micromechanics

distribution

in the

fiber

and

of any mieromechanies material

properties failure

and

ignores

as a thin

of the modes

matrix

effort

properties

of a

constituents

and

and establishing

states. a mieromeehanies

does

a random,

stress.

fiber

objective

geometric

predicted

to design

fiber

of fibrous

[ 1].

to the

response

lamina

of the

composite

or possibly

of plane

to predict

approach

approach

(average)

present

can be utilized

of the

a state

utilized

stress-strain

behavior

loading.

intrinsic

mieromechanics

individual

of the internal

of external

laminate

failure

study

of a laminated

macromechanics

structural

of the

lamina

average

the

under

elastic

normally

constants

The

medium

[ 1] are

as a periodic,

The

elastic

orthotropic

The

The

geometries.

fiber-matrix

1

models

in a matrix.

is a function

COMPOSITES.

of a constituent

m mieromodels

33

not give structures.

the

approach designer

to

a design

It is feasible

that

tool at

is

Section

F1.0

1 October Page some

time

in the future,

fiber

and matrix

determine

his

composites the

level design

come

filament

establishing

and will

to the

and other

resulting

application

of micromechanics

I.2. 2

MACROMECHANICS.

aerospace

or stress

response

of an individual

data may

subsequently

laminate

composed

stage

at present

good

this

creation

at the to

the advanced tape

form

Of course,

analyses

to insure

34

of micromechanics

established.

however,

were

transverse

represents

some

utilized

with of

in

and shear

a limited

to designs.

the macromechanic

composites analyst.

lamina be used

his

in a preimpregnated

state-of-the-art,

of filamentary

designer

However,

begin

tools

micromechanic

laminate;

the present

will

the analytical

spacing

in the final

designer

parameters

from

properties

the mechanics

use

fabrication

a good filament

With

aerospace

parameters.

spacing

the predictions

the

1971

may

is the most The elastic be determined

to determine

of any orientation

of the

the

usable constants

approach technique

characterized

for the

and stress-strain

experimentally,

stress-strain

[1] to

response

and these of a

laminae.

\

Section FI. 0 1 October 1971

z-

Page 35

I. 2. 2. i

Lamina

Constitutive Relationship.

For a filamentary composite

(Fig. FI. 2-I), the constituent laminae have

three

of elastic previously, mutually

FIGURE

F1.2-1.

FILAMENTARY

COMPOSITE MUTUALLY PLANES

WITH THREE PERPENDICULAR

mutually symmetry

[1].

a material

was

therefore,

the the

As discussed

planes

termed

of

orthotropic;

possibility

lamina

planes

with three

perpendicular

symmetry

model

perpendicular

exists

to

as a homogeneous

OF SYMMETRY orthotropic

medium.

Since

the

thickness of an individual lamina is small relative to its other dimensions, may for

be considered the

K-th

to be in a state

lamina

is then

given

of plane by equation

K

stress. (FI.

The 1-17)

constitutive

t_

(Tp

Ctt

C12

0

C 12

C22

0

0

0

2C66

T

equation

or

K

(7

it

K £ o{

(F1.2-1)

L

The

lamina

rewritten

stiffness here

matrix

terms

were

defined

in equation

(F1.

1-18)

and

are

as

ElL

...

C,, - (i - v,2v_1) E 2?.

C22-

(1 - "12"2,) (F1.

2-2)

Section

FI. 0

1 October Page P21E11

=

c12= (i- v,2v21)

1971

36

v1_E_

(1 - v,2v2,)

and C66 = Gi2

As

shown

by inverting

the

in equation stiffness

(F1.1-8),

matrix.

This

the compliance would K

E] K

Sll

S12

0

S12

$22

0

result

Or

matrix

may

be determined

in

-K

(FI. 2-3) =

0

½_

]

½S66

where 1 m

$11

=

$22

--

El I

,

I E22

,

(F1.2-4) _v_la = $12 =

El I

v__ - E22

and 1

s_e = b-"

"

Since the lamina principal axes (c_,_ ) generally do not coincide with the laminate reference axes (x, y) (Fig. i.2-2), the stresses and strains for each lamina must be transformed

as discussed previously.

When

the constitutive relations for each lamina must also be transformed laminate reference axis system. paragraph

FI.

1.1,

are

The transformations,

this occurs, to the

as discussed in

Section

F1.0

1 October Page

\ \

a

\

1¢

FIGURE

F1.2-2. WITH

GENEllAL LAMINATE

LAMINAE REFERENCE

ORIENTATION AXIS

37

1971

Section FI. 0 1 October 1971 Page 38

'_ "K = [T]

(F1.2-5)

(7

Y T • xy.

and

-,K

,K E

E Ot

° X

:

cfi

[T]

(FI.2-6) Y

.½"Y_ where

[T]

is defined in equation (FI. i-3) and K denotes the K-th lamina.

Then

.

.K

.

O" x

a

Y

= IT] -t

aft

T

. xy

The transformation

IT]

,K (7 (_

.

(F1.2-7)

Ta,a

matrix,

T,

may

be written

"m 2

n2

2ran

n2

m2

-2mn

-mn

mn

m 2 - n2

in a shortened

form

as

(F1.2-S)

where m = C08 0 and n = sin 0

.

_

Section

F1.0

1 October Page

Note

that

substituting

the

for

Using

inverse

the

positive

equations

constitutive

of the

T

angle

matrix,

[T]

0 a negative

angle

(F1.2-1),

equation,

when

(F1.2-6),

transformed

and to

the

-1,

be

0 (refer

39

obtained

by

to Fig.

(F1.2-7),

laminate

-

may

the

1971

1.2-2).

lamina

reference

axes,

is

K

% :

[T]-'

[C'I K

tTI

ie

(FI. 2-9)

l

!Yl "rx:

The

transformed

lamina

stiffness matrix

[C'] is defined

as

16 [_,]K

where

the

=

IT]-

terms

1 [c,]K

_..

are

[T]

given

=

by

C12

C22

2C26

el6

C26

2C66

equation

(F1.

(FI. 2-10)

1-29).

The-C'

matrix,

E which

is

now

constants

which

independent In the

fully

as

govern

they

transformed

to the

C

matrix

lamina

is

said

is

to be

said

populated

for to be

the

the

are

(C16

¢ C26 ¢ 0),

lamina

linear

behavior;

system,

a fully

anisotropic

constitutive

C16

of the the

C '

lamina

orthotropic.

equation

to have

however,

combinations

coordinate

"generally"

appears

for

four

matrix (C16

a "generally"

elastic C26

elastic

similar

¢ 0 and

Therefore,

and

basic is

six

orthotropic

not

constants. in appearance

C26 ¢ 0), equation

are

and (F1.2-9)

lamina.

the

Section F1.0 1 October 1971 Page Equation

(Ft.

orthotropic

2-I)

is referred

lamina

equation

for a "specially"

(CIe = C2s = 0).

For convenience,

Clt

to as the constitutive

40

= (3J1

equation (FI. 1-29) may be written as

+ J2) + J! cos

2 8 + J4 cos

4 0,

- Jscos

20

40,

- J2) - J4 cos

4 0,

C66 = (J1 + J2) - J4 cos

4 8,

m

C22 = (3J1 +J2)

+J4cos

(Fi.2-ii) C12 = (Jl m

C16 = _ J3 sin 2 0 + J4 sin 4 0, and m

C26=½Jssin2

e-

J4sin4

0

where Jl=Jf

[Cll

+ C22 + 2C12]

J2 =

[c.-

J3 =_

[Cl1-C22]

,

J4=_

[C11+C22-

2C_-4Css]

clzl

,

,

(F1.2-12)

and

Note

that CII,

C_0

CIZ° and

.

Css

of the angle

of rotation

(0)

and a term

Therefore,

it is evident

that

there

which

are

only

dependent

are

on the material

are composed dependent certain being

of a term

on the angle

inherent used.

lamina

independent

of rotation. properties

Section

F1.0

1 October Page 1.2.

2.2

Laminate From

laminate

the

will

adjacent

normally

Kirchhoff-Love

be neglected, remain nor

1, for

the

and

the

laminae

dictate

line

used

must

that

of a mathematical

to the

assumptions

be made.

Since

relative

deformed and

surface

shell

theory

the

uses

other

dimensions,

middle

to the and

of a

practical

to the

normal

model regarding

to its

perpendicular

originally

in thin-plate

Essentially, these assumptions

certain

it be thin

(stresses

seg_ments

and normal

contractions)

formulation laminae,

hypothesis

straight

Relationship.

of orthotropic

between

a laminate

f

Reference

composed

interaction

Constitutive

1971

41

appear

surface

middle

suffer

of

may

surface

neither

extensions

reasonable.

reduce to

K E X

Xl

E

=

X

_

-

Z

(F1.

X

2-13)

Y 21o i V- ,

Txy

Y j 2Xxy

I

where of the

•

0

,

x

e

0

y

laminate,

transformed

,

and

and the lamina

0

Txy X's

are are

constitutive

the

strains

the

at the

middle

equation,

surface similar

geometric

middle

curvature. to equation

surface

The (F1.

2-9),

is

then

cy x

cr

] =

Y rxy

I

1

£

o x

[_1K

(

o Y

o

Tx

_z[Cl K

×

x

× Y 2× xy

(FI. 2-14)

Section

F1.0

1 October Page

1971

42

where -A (Fi. 2-15) C1e

C2s

CseJ

This equation then relates the stress in the K-th lamina, oriented to the laminate reference axis, to the laminate middle surface strains and curvatures. On an element of the laminate, the stress resultants and stress couples are

defined

as

N

t/2 f

= X

o dz

J

X

-t/2

(FI. 2-16) N

= Y

t/2 I

o dz

-t/2 J

Y

and

N

t/2 = [ xy -t/J 2

_

dz xy

and

M

= X

t/2 _

zo dz

_'

X

P

-t/2 (F1.2-17) t/2 M

= Y

f -t/2

zodz Y

\

Section

F1.0

1 October Page

1971

43

and

M

= xy

These the

t/2 ]

zT

-t/2 J

integrals results

dz xy

may

of each

be

evaluated

by

integration.

By

integrating

ov,:r

substituting

each

equation

lamina (F1.

and

2-14)

summing

into

,f-

equations

(F1.2-16)

and

(F1.2-17),

O

"

N

All

A12

A1G

A12

A22

A26

"

B12

B16

X

B12

]322

B26

X

B16

B26

B66

2×xy

"_Bll

E

X

X

X

O

N

Y

Y

(F1.2-18) Y

0

N

Ale)

A2_

Ac)6

7xy

xy

r

Bll

B12

I31_

B12

B22

B26

Big

B26

B6g

!"

0

°

£

DI_

DI2

DI6

X

])12

D22

D26

Xy

DI_

D26

DG_

X

X

O £

Y

( F1.

2-19)

O

Txy

where

]K

LAijl

=

n K=I

[Bij ]

=

n K ' E [C..] -2 K=1 lj

[C ij

(F1.2-20)

(h K - hK- 1)

(h2K

h2K_ I

)

(F1.

2-21)

(Ft.

2-22)

and = -_- n E ° K=I

Refer

to Figure

F1.

-- ]K (hBK[C.. U

2-3.

h3K_l )

Section

F1.0

1 October Page

1971

44

.¥

t

.._J_ kit

FIGURE

F1.2-3.

Equations

(F1.2-18)

and (FI.

laminated

composite.

They

LAMINATE

2-19) may

are

the

be written

ELEMENT

constitutive

equations

a

as

I:l I::1I:°J This is the general constitutiveequation for laminated composites is mathematically

for

(F1.2-23)

and

equivalent to the constitutive equation for a heterogeneous

anisotropic medium.

In this general form,

is coupling between extensional (membrane)

the significantpoint is that there deformation and bending

Section

F1.0

1 October

deformation

words,

caused

by

the

_.ithin

the

limits

even

existence

of the

of small

B matrix

deflection

induce

this type of coupling.

FIGURE

F1.

2-4.

COUPLING

being

F1-2.

theory,

within the laminate

neutral

(Fig.

forced

in-plane

This

45

4).

In other

curvatures

loads through

coupling

axis and the midplane

1971

Page

is caused

by the

of the laminate

not

coincident.

OF

DEFORI\LATION

DUE

TO

With

various

combinations

of laminae,

B MA TtlLX varying laminate will

is fabricated

be

identically

[N]

=

[hi]

:

of coupling

symmetric

about

and

constitutive

zero,

[a]

degrees

the

the

may

midl)lane

be

caused.

(balanced),

equation

reduces

the

If the B matrix

to

[e°]

(I.'1

2-24)

and

These

-[DI

equations

are

of a homogeneous referred and

type the

to as

(F1.2-25)

mathematically

anisotropic

are

of coupling strains

still

populated

exists. shear

For stress,

to the

llcnee,

anisotropic

fully

and

equivalent

material,

homogeneous

D matrices

normal

[xl

(Fig. and

the

this

or

the

type

F1.2-5).

anisotropie

A matrix

constitutive of laminate At this

in nature,

the shear

equations is

stage

and

A1G and

A26 terms

strains

and

the

a second

normal

couple stress.

A

Section

F1.0

1 October Page For

the

normal and vice the x-y FIGURE

of pairs

bending

the Dis and D_s terms moments

versa.

and twisting

If the laminate

(laminate)

axis

(Fig.

46

couple

the

curvatures,

is symmetric Fi.

2-6)',

about

this

coupling

FI. 2-5.

COUPLING EFFECTS A HOMOGENEOUS ANISOTROPIC

D matrix

1971

IN

may

be reduced.

LAMINATE

of laminae

When the laminate

symmetric

\ \ \_\\\

about the x-y

axis

has equal

numbers

angleply

laminate),

(termed

\ \\\

\\\\\ \\\ \\\\ \\\\ 8

"4 FIGUR E F 1.2-6.

LAMINATE

SYMMETRIC

ABOUT

+8

THE

x-y

AXIS

the A matrix is orthotropic in nature (Ale= A26 = 0) (Fig. FI. 2-7). D matrix is stillfully populated and anisotropic in nature. has equal numbers

When

The

the laminate

of pairs of laminae at angles of 0 deg and 90 deg to the

Section F1.0 1 October 1971 Page

x-y

axis

(termed

and the

A matrix

erossply are

laminate),

orthotropic

the

47

D matrix

in nature.

!

,_[_Y_

._)z'

Because

__;_;_

during

the

of the warpage

fab,ication

process

which

when

will

symmetry

occur does

,:r,_" not exist, FIGURE

the. laminates

symmetric

CHARACTERISTICS the midplane,

composites

and because

experience

reexamination

the

relatively

of equation

be written

(F1.2-24)

about

the

Since

most

midplane,

or nearly

laminates

are

so.

symmetric

majority

of applications

of advanced

low transverse

shear

a

is warranted.

strength, Equation

(F1.

2-24)

may

as

N1

i-£ X

X

N

y

:

[A]

occurs.

2-26)

Y

xyj

no bending

(F1.

E

i T

_ when

be designed

F1.2-7.

LAMINATE EXtIIBITING ORTHOTI{OPIC

about

should

xy

Dividing

both

sides

by the total

laminate

thickness

yields

O" X i (7

Y

1 t

[A]

li11 ¢-

3xy

(Ft.

2-27)

Section

F1.0

1 October Page "A n

AI2

"E

Ai6"

1971

48

-s

x

I

1 t

At2

A22

A26

Ai6

A_s

A66

A12

2A16"

"All

Y

"E

I

"1

!

1 t

E

A 12

A22

2A26

A16

A26

2A66

I

m

EX

=

[A]

(Ft.

Cy

2-28)

m

The as the

¢Ws ore

the

laminate

average stiffness

laminate

stresses,

matrix.

and the A

matrix

may

be defined

Thus,

i

o" x

x

=

Y

[A*]

(F1.2-29)

Y T

.

where

the laminate

compliance

xy

matrix Atl

[A*]

=

[A]

-1

=

is At=

½At_

A* 12

A22*

* ½A26

Ai6

A_

½A6_

(Ft. 2-30)

Section

F1.0

1 October Page

The f-

gross

or

laminate

average(,

laminnte

compliance

[A]

elastic

matrix.

For

A12

A22

0

0

0

A66

moduli

a balanced

'

[._]

may

be

obtained

angleply

1__ t

trom

49

the

laminate,

AI2

A22

0

0

0

2A66

and

ai

a_i

0 (F1.2-31)

[ A ::-_] =

L° Then,

comparing

laminate

,)

equation

elastic constants

I';

= XX

½,\ (F1.

2-29)

with

equation

(F1.2-3),

the

gross

are

1 -:":

1 YY

A 22 1

G xy

A _:r.

= _ !xaz xy

A n

'_}x yx

A2 2

(F1.2-32)

1971

Section

FI. 0

1 October Page i. 2. 3

EXAMPLE The

example

computations

problems

Example Calculate

for

CALCULATIONS.

involved

i. 2. 3. I

a three-ply

+45 deg with

the

which

when

working

Problem

I.

A,

laminate the

B,

and

with

laminate

with

demonstrate composite

D matrices

the

axis

follow

oriented

the following

of the

materials.

of the laminate

laminae

(see

some

constitutive

at - 45 deg, sketch).

equation

0 deg,

The

and

lamina

material

obeys

"30.

O"

i.

0."

E OL

i.

3.

0.

0.

0.

l.

-45 ° ]

0° ]

+45°I

I.

Example

Problem

The [ C] for the

to the laminate

orthotropic,

the

I

0. I in.

0.21n.

0.1

in.

1 Laminate.

- 45-deg

transformed

values

1971

50

axes.

lamina Since

of the transformed

and the

+ 45-deg

the lamina stiffness

using equation (FI. 1-29) or equation (FI. 2-11).

lamina

material matrix

must

be

is homogeneous may

be calculated

Section

F1.0

1 October Page

Using for

the

C-11=

45-deg

106

(F1.

lamina

[30.

= 9. 75

C26=

equation

-G.

75

the

C.. 1j

terms

of equation

sin 2 (-45)

cos 2 (-457

(F1.

cos 4 (-45)

+ 2(1.

+ 2.)

+ 3.

×

sin 3 (-45) cos

(-45)

+ (I.-3.+2.7

1106 [975 777] =

for

L_](37

For

the

7.75

9.75

O-deg

the

+45-deg

-6.75

-6.75

7.75

lamina

= 106

9.75

7.75

6.751

7.75

9.75

6.75

6.75

6.75

7.75

J

lamina

[_I(2) : [cl = i,) 6 30. 1. 0.

From

equation

i. 3. 0.

(FI. 2-20),

n

A

=

Z

44

"J

sin 4 (-45)]

sin (-457 cos 3 (-45) ]

106

-6.75

Similarly,

2-157

x 106

Then,

[C]

51

are

106 [ (30.-i.-2.7 =

1-29),

1971

K =i

(C..7

(K)

l]

0.1 [c..] D

(1)

(h K - hK_i7

+ 0.2

[C..J (2)-- " 1J

0. ] 0. 1.

Section F1.0 1 October

1971

Page 52

Thus

p

All =

A6e =

10e [9.75x0.

I0617.75x

I+

30. x0.2+9.75×0.

0. I+I.

x0.2+7.75×0.

I] =

I] =

7.95×106

1.75×106

.

x

[A] =

From

equation

I0s

(FI.

I. 75

2. 55

0.

7.95 0.

I.75 0.

0. i.75 1

2-21)

n

Bij =

_

=

Thus

2; (_ij)(K) K=I

0.015

(h2K_h2K_I)

[(Cij)(3),

(_ij)

(1) 1

t

Bil =

106×0.015

B66 =

0.

[9.75-

9.75] =

0

and

0. 2025" [B]

=

I0e

O. 0. O. 2025

O. O. O. 2025

0. 2025 0.

Section

F1.0

1 October Page

From

equatioa

(Ft.

1971

53

2-22)

n

Dij

=

_

Z K=I

(_..) (K) I]

O. 00233

[C-..]

(1)

(h3K

0. 00066

+

i)

h3K-

-

](2)

[C..

1J

+

0. 00233

1J

[_..](3) IJ

Thus,

Dll

:

106

[9.75

× 0. 00233

O. 06522

D86

0. 03676

Combining

× 0.00066

+ 9. 75×

0.00233]

× 106

tO _ [7.75×

[ D]

+ 30.

tO _

0.00233 ×

+ i.

× 0. 000(;6

+ 7.75×

0.00233]

10 6

0. 06522

O. 03675

O.

0. 03676

0.0474

0.

O.

0.

O. 03(;76

the results above,

the constitutive equation

may

be written

as

m

0

7. 95

1.75

O.

O.

0.

O. 2025 X 0

i. 75

2. 55

0.

O.

O.

O. 2025

E

Y

O.

O.

1.75

0.

O.

0. 2025'

O. 2025

O. 2025

0.06522

0.03676

0

O.

Yxy

O.

×

xy|= 106

M[

X

I I

M x

O.

O°

0.2025,

0.03676

0.0474

0.

Xy

I

MY _

xy]

I

I

O. 2025

0. 2025

O.

, O.

0.

0. 03676

2× _2

xy

Section FI. 0 1 October 1971 Page 54

1.2.

3. 2

Example

Calculate at + 45 deg, thickness

Problem the

constitutive

- 45 deg,

of 0. 4 in,

as in paragraph

1.2.

II. Example

2. matrix

- 45 deg,

{see

for a four-ply

laminate

and + 45 deg and with a total

the following

sketch).

Use, the same

with

laminate lamina

Problem

+ 45"

0. 1 in.

- 45"

0. 1 in.

- 45"

0. 1 in.

+ 45"

0.1

in.

2 Laminate. about the midplane and symmetric

about the x-y axis, the constitutiveequation will be in the form of

where

Ai2 All 0.

material

3. 1.

Since the laminate is symmetric

[A] =

laminae

A22 A12 0.

0. 0. ] A6s

._=i

\

Section

F1.0

1 October Page

and

[D]

Similar

=

DI2 Dil

D22 DI2

D26 Di6

L DI6

D26

D66

to paragraph

1.2.

[_l (I)

3. 1,

[_l (4)

1o 6

9.75

7.75

6.75

7.75

9.75

6.75

6.75

6.75

7.75

and

[_]

(2)

=

[_]

(3)

=

10 6

7.75 _9.75 6.75

Using

equations

[A]

(F1.2-20)

and

7.75

-6.75

9.75

-6:75

-6.75

(F1.2-22)

= 1() _; 3.1

3.9

0.

!1.9 ).

3.1 0.

0. 3.

and

[D] = 10 6

Therefore,

(). 05187

0. O4123

0. 027

0. 04123

0. _51S7

0. 027

0. 027

0. 027

0. 04123

7.75

55

1971'

Section F1.0 1 October 1971 Page 56 n

O

m

-3.9

N

3.1

O.

3.9

O.

m

E X

X

O

N

3.1

E

t

Y

Y

O

N

Yxy

Oo

xy = I0e I

M

0.05187

0.04123

0.027

0.04123

0.05187

O.027

0. 027

0.027

0.04123

XX

X

M

e

Xy

Y M

2Xxy

xy m

i

y-, -,.._j

Section

F1.0

1 October Page 57 1.3

LAMINATE

CODING.

In Reference composites of any

2 a laminate

which

provided

laminate.

to provide

presentations

and

discouraged internal 1.3.

that

this

code

must

STANDARD The

(2} the

(1)

the

number

by each

type

of coding

recommended

drawings

using

identification

in engineering

it is neither

on shop

filamentary

a positive

following

conciseness

however,

code

must

of laminae

of laminae

the

for

since

that

nor policy

is an

organization.

FLEMENTS.

of the angles

devised

and

reference,

be employed

CODE

was

reference

of achieving

be decided

formulation

as possible

in that

a means

code

a concise

communications;

one which

1

orientation

both

As expressed

is intended

1971

at each

be adequate

relative angle,

to specify

to a reference

and

(3)

the

as concisely axis

exact

(the

x-axis),

geometric

sequence

of laminae. The

basic

1.

Each

in degrees

laminate lamina

between

its

code

will

is denoted filament

adhere

to the

by a number

direction

following

gaaidelines

representing

(refer

to Fig.

its

FI.

0-3)

[2]:

orientation and

the

x-axis. 2. angles

are 3.

other, lamina F1.2-3).

with

Individual

adjacent

laminae

are

separated

by a slash

if their

different. The

laminae

brackets

should

be the

are

indicating most

listed

in sequence

the beginning

positive

lamina

from and

in the

end

one

laminate

of the

z direction

code. (refer

face The

to the first

to Fig.

Section FI. 0 1 October 1971 Page 58 4.

/Adjacent

laminae

of the same

angle

are deonoted

by a numerical

subscript. 5.

A subscript

T to the bracket

indicates

that the total

laminate

is

shown. When

adjacent

appropriate

use

represents which are

of

laminae + or

one lamina

is used

only when

assumed

-

and

are signs

of the

same

angle

but opposite

may be employed.

supersedes

the directions

the use are

Each

+ or

of the numerical

identical.

Note

in sign, -

the

sign

subscript,

that

positive

angles

to be counterclockwise.

Several

examples

are

shown

demonstrating

the basic

coding. _r

! 45 r----i 0

Code

m

[ 45/0/-602/30]

-60

T

u

-60 N

3O

+45 -45 -30 +30 0 +45 +45 -45 -45

[,45/_30/0/+(452)

]T

Section FI. 0 1 October Page

I. 3.2

CODES

As

FOR

discussed

arbitrarily

(balanced)

The

symmetric

S

more

but more

common

LAMINATES.

laminates

may

often the laminae

standard

59

be formed

assume

arrangements

of

standard

arc symmetric

and sets of laminae.

(balanced)

indicates

OF

sections,

to the total laminate

starts at the most

The

laminae,

laminates

Similar

TYPES

in previous

arranged

arrangements.

VARIOUS

code has

positive

z

as shown

an

S

lamina

that only one-half

in paragraph

as a subscript,

I. 3. l, the

and the coding

and stops at the plane

of the laminate

Laminate

sequence

of symmetry.

is shown:

Code

190/0_/45]

90

S

0 0 45 45 0 o i 90 If the the either

symmetric

center side

lamina of that

laminate is

1971

has

overlined,

lamina:

an

odd

number

indicating

that

of laminae, half

of the

the

code

laminate

denoting lies

on

Section

F1.0

1 October Page Laminate

1971

60

Code

0

10/45/90]

m

S

45 n

90 45 0 m

A repeating accordance

with the

sequence same

rules

of laminae which

apply

is termed to a single

a set.

A set

is coded

in

lamina:

Laminate m

u

m

45 0 9O

Set

45

=

u

0 9O 45 0

Set

[ (45/0/90)4]T

or

[45/0/9014T

or

[45/0/90]

Set

9O 45 m

0 9O

Set

Set

01

Set

Set 901 01 901

Set

[ (45/0/90)

2] S

2S

Secti(m

F1.0

1 October Page 1.4

COMPUTER

Several support programs 5.

The

F1.4-1.

personnel have program

PROGRAMS

computer

programs

to assist been

I or

names

IN COMPOSITE

ANALYSIS.

have

available

in analyzing are

currently

and

a brief

been

made

composite being)

comment

material

documented on

each

to

61

MSFC

elements.

sho_vn

in

and These

in Ileferences are

1971

Table

4 and

Section

F1.0

1 October Page 62 TABLE

Program CDLAMA

F1.4-1.

Computer

Programs

Name

Analysis

Comment Yield

analysis

orthotropic GDLAMT

in Composite

Yield

of composite lamina

analysis

orthotropic

of composite lamina

plates

with in-plane

with

composed loading.

plates

in-plane

composed

INTACT

Locates optimum strength envelopes under the action of combined loading.

for

LAMCHK

Laminate

check

- computes

of safety

specified laminates

0-deg, under

90-deg, + 45-deg boron combined loading.

skin/stringer/frame

Analysis of single overlap bonded 329 adhesive mechanical behavior.

LAPI

Analyzes

single

arbitrary applied basis correction

PANBUCKII

overlap loads factor.

compre.ssion

margins

LAP

bonded and

with Metlbond

and accepts

incorporates

critical layered,

for

epoxy

joints

joints

laminates

the

Calculates the buckling loads of radially inhomogeneous anisotropic, cylindrical shells wherein the effects of boundary conditions are considered. Panel buckling -- calculates and mode for orthotropically

of

loading.

Design panel.

composite

of

-

GREPPA

MMBCK

B

not

buckling loads rectangular,

anisotropic plates and honeycomb sandwich panels. Also computes local instability modes of failure for composite panels. STAB

1971

Stability analysis -- local instability analysis orthotropic honeycomb panels, columns, and -- failure-mode intercell

dimpling,

analysis and

for filament layer

instability.

rupture,

of beams

Section

F1.0

1 October Page 1.5

.

REFERENCES.

Petit, P. H.: Laminated

Basic

Concepts

Composites.

Structural Advanced

Systems

Ohio,

August

Ashton,

J.

Materials: Conn.

.

239,

of

Lockheed-Georgia

, Marietta,

Composite

Air Force

Analysis

Ga. , March

Applications.

Materials

Wright-Patterson

Air

Laboratory, Force

Base,

J.

C. ; and

Technomic

Petit,

P.

Publishing

lt.:

Primer

Co.

, Inc.

on ,

Composite

Stamford,

1969.

Analysis Division,

Computer

Flight Center

Analysis

Mechanics Space

for Advanced

Division,

C. ; Halpin,

Mechanics

Strength

SMN Corp.

and Stress

1969.

Analysis.

,

Aircraft

Command,

Structural

Space

.

Guide

Composites

Air Force

.

Design

in the Design

I{el)ortNo.

Co., division of Lockheed 19(;8.

.

1971

63/64

Flight

Division, Center

Utilization Manual.

Astronautics

Laboratory,

Analytical

George

C.

Marshall

(NASA).

Computer Astronautics (NASA).

Program

Description Laboratory,

Manual. George

C.

Analytical Marshall

._J

SECTION F2 LAMINATED COMPOSITES

TABLE

OF

CONTENTS

Page F2.0

STRENGTtI

OF

2. 1

Strength

Yield

LAMINATED

Distortional

2. 1. 2

Maximum

Ultimate

2. 3

lleferenees

..........

1 3

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

2. 1.1

2. 2

COMPOSITES

Strength

l':ner_w Strain

Criteria

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

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

F2-iii

Theory

4

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

6

11/12 13

F2. 0

STRENGTH

The

lamina

[ i].

degradation

general

lamina

ultimate

strength

This

LAMINATED

is because

test.

composites

COMPOSITES.

of laztiaa_ed con_,posites must

begin to occur

laminate

of advanced

OF

For

it is easier

to determine

on a unidirectional

this reason

test specimen

is to establish strength

of the laminate.

_here

nonlinearity

allowables

to predict

and

than on some

the trend in determining

and then to utilize analytical methods

strength

be related to the individual

the strength

for the orthotropic

the yield or the

Section F2.0 1 October 1971 Page 3 2. i

YIELD

STRENGTH.

The available yield strength theories for laminated composites best tentative at this time [2]. Only a minimal

amount

are at

of test data is

available to substantiate any of the yield theories for advanced composites. Ifthe yield point is defined as the onset of inelastic action, it is apparent that the prediction of the yield strength of an orthotropic lamina is a linear problem.

Several yield theories of failure have been hypothesized for

anisotropic materials, of which two will be discussed (the distortional energy theory and the maximum

strain theory).

Before the discussion of yield theories, the basic difference between the yield surfaces for an isotropic and for an orthotropic or anisotropic material must be explained.

state,

crx,

and

{Y

2 ' P

yield

Cry, TXy ,

will

figure

some

give

with

may

angle,

the

al

therefore, yield

criteria.

figure

with

a2

(Fig.

F2.1-1).

the

biaxial The

the

yield

a plot

of the

surface.

as axes.

The

When

principal

stresses,

principal

result

_1

stresses

P

that

and

cause

is a two-dimensional

considering

the

yield

surface

for

an

P

lamina, for

into two

0; therefore,

P orthotropic

be resolved

required and

For an isotropic material, any biaxial stress

stresses

must

be referred

stress

states

three

resulting

yield

surface

directions

of

_rl, or2, and

stress

to the components

will

appear

T12 as reference

_REcED|NG

lamina

PAGE

may

principal

axes;

appear

in the

as a three-dimension_ coordinates

[;LANi( NOT

F_

lip

Section

F2.0

1 October Page 2. 1.1

o_

1971

4

DISTORTIONAL

ENERGY

THEORY. Cira, independently

developed

generalizations lsotropic

Norris,

and Hill

their

of the yon Mises

distortional

energy

yield

criterion,

(_1 - _z)2+ (_ _ (_ 2 + (_3 - o'1)_ + 6 (_121 + rl31 + Tsi 2) :

1" 12

FIGURE

F2. 1-1.

DIMENSIONAL YIELD AN ORTHOTROPIC metals

to exhibit

involving Mises

SURFACE LAMINA

certain

severe

strains.

t "plastic

for application

criterion

has the

to account

for the anisotropy

respective

problems

concerned

with the tendency

properties

Hill claimed which

0

*

(F2. 1-1)

of their

[3].

Hill was

FOR

anlsotropic

potential,"

criterion

2f (_iJ)

THREE-

2or I

when

undergoing

he had a physical

allowed

to anisotropic

meted

working

interpretation

him to generalize metals.

of isotropic

The plastic

of yon

yon Mises' potential

yield or yield

form

= AI (cT2 - o3) = + A2 (_s

- (_I) 2 + A3 (_I - o2) 2 + p..A4T2S= (F2.

+

2._5"/'312

+

2AsT121

=

I

1-2)

Section

F2.0

1 October Page

1971

5

where 2f (a..) = the 1j

plastic +

potential, (F3)-2

_ (FI)-2

2A 1

=

(F2)-2

2A 2

:

(F3)-2+

(F1)-2_

(F2j-2

2A 3

:

(F1)-2+

(F2)-2_

(F3)-2

2A 4

=

(F23) -2

,

2A s

= (F31)-2

2A 6

=

,

and

F 1, F2, tests,

and and

F 3 are

based lamina with

[2]

on the

adapted

this

reference

an orthotropic

r

The

strengths

principal

to these

_

--r

since

material

uniaxial

determined

criterion

the

The

from

criterion

yield

tension pure

as a yield

individual

axes.

Crly

either

failure

of the

axes

2

from

F31 are

composites.

%y for

determined

F12 , F23 , and

Tsai laminated

(F12) -2

lamina are

criterion

Cr2y

%y

in plane

stress.

+

criterion

reduces

for

composite referred

established

T12y

Note

tests.

a laminated

orthotropie

Hill

shear

and failure

for

strengths

or compression

is

to the

experimentally

to

2

1

(F2.

1-3)

that

(F2. 1-4)

Section F2.0 1 October 1971 Page6 Also, and

and

aly

a2y

2 directions

stress. when

The

2. 1.2

the

the

orthotropic

for

yield

plotted

are

surface

The

materials

the

[2].

strain

components The

(F1.2-3)

with the

E1

principal

C2y

Y12y

Thus,

equation

yield

strains

made

if

(F2.1-5) in the

r12 = 0

lamina.

Sll°rl

yield

+

is the

in the

shear

yield

depending

on

r,

F2. 1-2).

yield

in the

for

orthotropic

not be

isotropic

lamina

it is possible

must for

be

three

criteria.

criteria

may

be developed

yield

S11

$12

0

0"1

S12

$22

0

o-2

0

0

S6e

T12

the

should

criteria

therefore,

yield

here

to the

gives

is assumed.

ely

axes;

equal

Y

Y

sphere,

presented

strain

in the

strain

strains

or

(Fig.

criteria

components

to appear

maximum

TI2

strengths

CRITERIA.

principal

strain

lamina

space

yield

and

an ellipsoid

yield

maximum The

to the

strain

lamina,

is either

STRAIN

maximum

with

referred

or compressive

in a three-dimensional MAXIMUM

confused

tensile

envelope

A two-dimensional

from

equation

strains:

of stresses plot

.

( F2. 1-5)

which of the

produce

equation

the

may

be

Then,

8|2(_ 2

(F2.

and

C2y

=

S|2o-I +

$22o'2

,

1-6)

Section /

F2.0

1 October Page

1971

7

o _/_J2 y

1.0

10

FIGURE

F2.1-2.

ItlLLYIELD

SURFACE

o july

Section

F2.0

1 October Page

1971

8

or

Si2

cri

Si2

and

(F2.

$22

Equations system

(F2. which

strengths

1-7) define

_2

$22

are the yield

1-7)

•

equations

of two lines

of an orthotropic

in the _1 - _2 coordinate

lamina.

After

defining

the yield

as

Ely Crly = $II

E11 = CZy

and

(F2. I-8)

_2y

$2 2

_2y

equations (F2.1-7) may

E22

be put in a form similar to the Hill equation:

E1 Cr2y

c2

y

$12

C2y

$12

and

O'ly (F2.

_2y

C2y

C2y

St1

_ly

1-9)

Section

F2.0

1 October Page Equations

(F2.

assumed appear

that as

1-7)

the

shown

may

shear in

Fig.

be

plotted

stress F2.

on

a

component,

_1 - _2

T12,

set

was

of

coordinates

zero.

Such

since

a plot

1971

9/10 it was

might

1-3.

(J,

i.

u

FIGURE

F2.

1-3.

= (}

LAMINA

YIELD

SURFACE

Ol

Section

F2.0

1 October Page 2. 2

ULTIMATE Until

laminate based

upon

to failure lamina

moduli

zero.

Recently,

simple

techniques,

specialized verify

their

upon

are

have

on linear

assumption

or that

laminates

attempts

based

the

yield

some

accuracy

theory; the

that

to some

developed.

for

programs

lamina,

These

programs test

use.

data

value

been

determine

the

ultimate methods

not available

a was

is linear

all)

of the

or set

written

and are

(or

for

load

response some

small

load

ultimate

stress-strain

have

that

or rupture

the laminate

arbitrarily

sufficient general

ultimate

is,

laminate

new methods

and

the

of a constituent

computer

and

laminates,

to predict

that

reduced

been

11/12

STRENGTH.

recently,

were

1971

equal

with

to

relatively stren_,_h

are

of

usually

at this

time

for to

Section

F2.0

1 October Page 2.3

REFERENCES.

i*

Ashton,

J. C; Halpin,

Materials: Conn.,

*

Petit,

Analysis.

P.

It.

Basic

Concepts

Composites.

Co. , a division 1968.

Kaminski,

of

B.

Anisotropic Report

Technomic

Publishing

Co.,

13

on Composite

Inc. , Stamford,

1969.

Laminated

*

J. C. ; and Petit, P. H. : Primer

1971

Lockheed

E. ; and

Lantz,

Materials. STP

4(;0,

in the

Report

American

Design

No.

SMN

Aircraft

I/.

Composite Society

and 239,

Corp.

B. :

Strength

Stress

Analysis

of

Lockheed-Georgia

, Marietta,

Ga.

Theories

of

Materials:

Testing

and

of Testing

Materials,

, March

Failure Design. 1969.

for

SECTION G ROTATINGMACHINERY

.......d

-.._4-

SECTIONGI.2 SOLID DISCS

".,--.4"

TABL£

OF

CONTENTS

Page Gi. 2

SOLID i

DISKS.

i. 2. i

CONSTANT

(UNIFORM)

THICKNESS

i. 2.2

VARIABLE

THICKNESS

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

1.2.3

EF_AMPLE

PROBLEMS

CIRCULAR

DISKS

r

ROTATING

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

2 3

SOLID 4

Example

Problem

I

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

4

If. Example

Problem

2 .....................

7

I.

REFERENCES

FOR

...........

10

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

GI. 2-iii

DEFINITIONOFSYMBOLS Symbol b

Rim

radius

g

Gravity

h (r)

Disk

r

Distance

thickness

PoissonTs P

r

%

Disk

constant at location

from

axis

of revolution

I

ratio

material

density

Stress in radial positive denotes

direction; tension

Stress in tangential direction; positive Constant

r

angular

(hoop) denotes velocity,

Gl. 2-iv

tension rad/sec

"-_

SectionGi. 2 1 May, i97t

G1.2

are

SOLID

DISKS.

In this

section

presented.

disk.

assume

the

are

quite

constant

methods

are The

are

The

Because

hardware

shown

some disks

rotate

methods involved,

stress

geometry, in Figure

Gi.2-i.

methods about

for a final only

across

preliminary,

FIGURE

of the

the

no modes coordinates,

of analyzing

an axis analysis

methods disk

for

stresses

circular

is perpendicular

to the

preliminary are will

analysis

considered.

be discussed for

a rotating

which Since

at this circular

G1.2-1.

CONFIGURATION

disks

of turbomachinery-type

thickness

of failure and

which

rotating

OF SOLID

CIRCULAR

DISK

the time. disk

Section Gi. 2 i May,

197i

Page 2

i.2.i

CONSTANT

(UNIFORM)

THICKNESS. -\

For solid circular disks rotating at a constant angular velocity with uniform thicknesses and temperature

fields, the radial and tangential

stresses (Ref. i) are

= _pv=) 13+_) (i-x _) r

(1)

8g

and

a0 = (Dv2)8g(3+/_)It

- (l+3#a) 3+/a x21

(2)

where r

x = -

(3)

b

and v =be0 The maximum

.

(4)

stress occurs at the center of the disk (r = 0) and is given by IP v2)_3 ÷/a)

(ao) max

= ( at)max =

H the disk is centrally clamped (Ref.

2) become

•

(5)

8g (Fig. Gi.2.i-i),

the in-plane stresses

SectionG1.2 I May 1971 Page 3

f--

and r2_

..._t._ _2

_i,*3p_ 3+/_

7 r4|

(7)

J

where / /

(i

-_,)

_l=

pW 2a 2b 2 8

[((_ +.)b_(1+.) 1 + p) b2 + (1 p) aa_] 2

(8)

and

(3 +,)pJ h= for

values

(9)

S of r greater

than

a.

[ f

FIGURE

G1.2.

I-I.

CONFIGURATION

OF DISK

WITH

FULLY

CLAMPED

HUB

Section

Gi. 2

i May Page For

a solid

field,

the radial

given

in Paragraph

with

the

i.2.2

with a uniform

and tangential i .3.2

modifications

(Disks

For

circular

thickness

stresses

(Ref.

rotating

with a Hole

at the

using

using

Center

4 temperature

the procedure -- Variable

Thickness)

i .2.2.

at a constant

temperature

3) are calculated

and a varying

calculated

in Paragraph

disk

or a varying

are

with a Hole

described THICKNESS.

a solid

thickness

stresses

VARIABLE

varying

(Disks

disk

i97i

field,

angular

the radial

the procedure

given

at the

Center

-- Variable

Thickness)

point

i should

be chosen

at 5 percent

velocity

with a

and tangential in Paragraph

i.3.2

with the following

modifications:

to the i.2.3.

i,

Station

2.

The initial

3.

The stresses

stresses

shown

point

radial

in the following

Problem

of the

disk

FOR ROTATING

(b).

be i .0. (r = 0) are

assumed

equal

SOLID

CIRCULAR

DISKS.

i.

and tangential sketch.

33 should

radius

i.

PROBLEMS

Example

Find the

in column

at the center

at station

EXAMPLE I.

value

of the rim

stresses

for the

solid

circular

disk

I

Section

Gi.

i May, Page

1971 5

Q_)

16 000 rpm

MATERIAL ROOM

6AI-4V

Solution:

E =

i6.0 x 106 Ib/in. 2

p = 0. i 6 Ib/in.3.

/_ =

0.3i3

g =

32.2 ft/sec 2= 386.4

in./sec 2

u) =

16 000 rpm

r/sec

= 266.7

.

= i675.52/sec

TITANIUM

TEMPERATURE

.

2

Section

Gt. 2

1 May, 1971 Page 6

From

equation

(4)

v = bw = t5 × t675.52

= 25 132.8

ln./sec

and v 2 = 6.3166

× 108 in./sec

2.

Equation (1) becomes

e

r

=

0. t6 x 6.3i66 × 10 s • x 3.3t3 8x3.864 × 102

= t08

/h-

3t0

k

The following sketch

lb/in.'

225/

depicts

x( lb sin'2 t - _552) -In. sec sec2 2 in.

cr . r

1(_.31

v

6.0

I0.0 RADIUt

Equation (2) _0 =

The following

becomes 108

310

sketch

r2 (1 - 3-_)lb/ln.

depicts

e 0.

16.0 r fin.)

Section

G1.2

I May,

1971

Page

7

100.31 F"

64.16e

o

I

t

5.0 RACHUS

II. Find in the

Example the

following

Problem

2.

and

tangential

radial

0=

10.0

stresses

15.0

r (in.)

for

the

solid

circular

sketch.

MATERIAL ROOM

el------

16 in.

/

6AI-4V

TITANIUM

TEMPERATURE

disk

shown

Section

Gi. 2

1 May,

1971

Page

8

Solution: E = i6.0xl0

elb/in.2

p = 0.i61b/ln.

$

= 0.3i3

.

.

.

g = 386.4

in./sec

2 .

o_ = 16 000 rpm = t.675.52/sec. The idealization for the finite-dlfference-type analysis is given in the following sketch.

1076.62/m

2

1

1_

_;

4

s

0.76 6.376---.-D 10.0 12.6

•

NOTE:

16.0

DIMENSIONS

ARE

7-

IN INCHEE.

The computations for the finite-difference-typeanalysis are given in Table Gi .2.3-i.

J

OF i:_O_

QUALirf

Section

G1.2

1 May,

1971

Page

g_

F O

-_i4-

,,'.'4

r_

o _n

O 7.

c_

=

:

:

r_

_3

o u]

"4

T

_o

,...<

dddd

<

T

¢9 w

_d :" =" d d

_9 (D

I

_c

d

d

_

d

s

:_

:

-

•

*,..4

o ,"3 -s ::

I

-

¢,q

:'

-

::

d d

.Q

+

d

d

d

=: d

%

-" d

d

d

9

Sect ion Gi. 2 i May, 1971 Page i0

REFERENCES Timoshenko,

Ii

Company, o

S. : Inc.,

Eversman,

Strength

New York,

W.; and Dodson,

of Materials, i956,

Manson, NACA

S. S. Report

: Determination 87i,

Cleveland,

If.

D.

Van Nostrand

and

p. 2i8.

Jr., R. O.

Clamped Spinning Circular Disk. 1969, pp. 2010- 2012.

a

Part

AIAA

of Elastic Ohio,

Feb.

: Free Vibration of a Centrally Journal, vol. 7, no. i0, Oct.

Stresses 27,

in Gas-Turbine

Disks.

i947.

'\

TABLE OFCONTENTS Pa ge Gi.3

DISKS WITH

A HOLE

AT

1"HE CENTER

li

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

CONSTANT

(UNIFORM)

1.3.2

VARIABLE

THICKNESS

1.3.3

EXAMPLE PROBLEMS DISKS WITH CENTER

FOR ROTATING HOLES. ...............

I.

Example

Problem

i ....................

16

II.

Example

Problem

2 ....................

2O

REFERENCES

THICKNESS

...........

li

1.3.1

13

.................... CIRCULAR

16

24

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

G1.3-iii

DEFINITIONOF SYMBOLS Definition

Symbol a

Inner

surface

radius

b

Rim

E

Modulus

g

Gravity constant

h (r)

Disk thickness at location r.

n

Node

r

Distance from axis of revolution

T

Temperature

C_

Coefficient

radius of elasticity

(station) index

(° F) of thermal

expansion

Poisson's ratio P

Disk material density

ff r

Stress in radial direction; positive denotes tension Stress in tangential (hoop) direction; positive denotes tension Constant angular velocity (rad/sec)

Gi. 3-iv

Section

Gi.

3

i May, i971 Page 11

Gi.3

DISKS WITH

f

In this with

circular

axis

which

analysis for

THE

at the center

are

to the

disk.

Because

hardware

are

quite

Since

of failure

will

be discussed

at this

and

stress,

stresses

for

involved,

considered.

coordinates,

constant

rotate

methods

are

in Figure

assume

the

circular

disks

thickness

geometry,

for

about

an

a final

only

or linearly the

disks

methods varying

methods

are

time.

a rotating

circular

disk

GI.3-i. (UNIFORM)

circular

disks

with

stresses

r

The

the disk

velocity

tangential

presented.

rotating

which

CONSTANT For

of analyzing

analysis

no modes

shown

CENTER.

methods

of turbomarchinery-type

The

angular

of the

is perpendicular

across

i.3.i

some

cutouts

preliminary,

are

AT

section

preliminary

stress,

A HOLE

with

uniform (Ref.

THICKNESS. a center thickness

hole

that

rotate

and temperature

at a constant fields,

the

radial

and

1) are

8g

(1)

and

_0 = (P_V2)(3 8g

+_)(1

+y2-

(t3 +3u)x2 + ]a

+ x_- )

(2)

whe re

2/

a

b

'

(3)

Section GI.

3

I May, 1971 Page 12

FIGURE

GI. 3-I.

CONFIGURATION WITH A CENTER

OF A CIRCULAR HOLE

DISK

r b

'

(4)

boo

.

(5)

and v= The maximum

stresses occur at r = _aab and are

(_) r

=

(or 2) (3+/z)

(I-7}

2

(6)

8g

max

and

(_e)

max

=

(ov_

(3+/_) 4g

Ii+(f-u)'v21 3 +/_

"

(7)

Section

GI. 3

i May. Page

The

stresses

may

be determined

by the computer

pr(_gram

1971 13

documented

in

r /

a NASA

technical For

the

radial

Paragraph 1.3.2

note.

a disk and

with

1 a uniform

tangential

stresses

thickness are

and

calculated

a varying using

the

temperature procedure

field, given

in

1.3.2. VARIABLE

THICKNESS.

The stresses in a circular disk with a center hole and a variable thickness or a variable temperature difference method

and material properties.

stations along the radius. on the outer

(rim)

Intermediate percent thickness,

of the

The

are easily executed in a tabular format.

An idealizationof the disk is made

lies

using a finite-

(Ref. 2) . This method considers the point-to-point

variation in thickness, temperature, computations

field may be determined

rim

by selecting

Station i lies on the inner surface, and station N

surface. stations

diameter

temperature,

(Fig. GI.3.2-I)

should from

or material

be located the

inside property

at distances boundary variations.

and

of 1, 2, at locations The

radius

3, and

5

of at each

Byron Foster and Jerrell Thomas: Automated Shell Theory for Rotating Structures (ASTROS). NASA TN-D-, Marshall Space Flight Center, to be published.

Section

Gi. 3

i May, i97i Page i4

1

FIGURE

3

..-

N

Gi.3.2-i. IDEALIZATION OF DISK FINITE-DIFFERENCE ANALYSIS

station is entered in column thickness is entered.

2

i (Table Gi.3.2-i).

In column

FOR

2 the idealized

If a sharp discontinuity in thickness occurs, such as an

abrupt flange, the thickness should be faired in the disk contour, and the faired disk used in determining

the thickness.

The mass

density (corrected

if in a faired section) multiplied by the square of the rotational speed is entered in column columns

3.

1_oisson's ratio and the modulus

4 and 5, respectively.

of elasticityare entered in

The coefficientof thermal expansion, which

must be an average value applicable to the range between actually existing and the temperatures entered in column temperature

6.

the temperature

at which there is no thermal stress, is

The difference between the actual temperature

at which there is no thermal stress is entered in column

The manipulations required in columns

8 through 34 are shown

and the 7. in the

_.,i:__¸"

.....

Section

GI. 3

i May, i97i Page i5 q

"7 +

T

×

y

O

/ i _2

t_

< (D _J

_J

! aJ

"I

o_

e

I o,1

¢9

%

Section

G1.3

i May, Page

respective

columns.

column

33 having

of 1.000.

an initial

Columns

an initial-value

the total

37,

cre,

In using

Ibid.

total

_r'

calculated

at each

34 having

with

an initial

simultaneously,

the blade

centrifugal

force

station

station

at the

loading

value

but each

at the

roots

is calculated

is calculated method,

of station

has

rim

of the blades

and by

points

program

in column

the accuracy

increases. already

FOR ROTATING

in column

The

39.

of the

results

The stresses cited.

38.

may

also

_

CIRCULAR

DISKS

HOLES

Problem

radial

denotes

r,b

at each

PROBLEMS

Example

Find the

_

by the computer

WITH CENTER

2.

also

and column

simultaneously,

area.

number

EXAMPLE

I.

of 0.000

the finite-difference

as the

be determined i.3.3

the

stress,

stress,

increases

the term

peripheral

The radial tangential

value

calculated

of 0.000.

by dividing rim

33 and 34 are

35 and 36 are

In column is obtained

Columns

1971 i6

i.

and tangential

stresses

for

the following

circular

disk:

Section

Gi.

t May,

i971

Page

MATERIAL ROOM

6AI-4V TITANIUM TEMPERATURE

0.26

in.

Solution:

E =

i6.0×

p = 0.i6

106 psi, Ib/in.

3,

= 0.313, g =

386.4

in./sec

w = 16 000 rpm

T -

2, = 1675.52/sec,

2 15 - 0.t333,

v = 15 × i675.52 in./sec = 25 132.8 in./see. and v2= 6.3167 × 108 in2/sec2

.

17

3

Section

GI. 3

I May, Page

1971 18

Equation (I) becomes ( 0.16x _y

6.3167x

10s)(3.0

r2 4. 005 + 0.313) (I + O. 1333 z - 225 r-T--/ 3091.2

=

1.0832x

The following

10 s (1.0178-0,00444r2-_

sketch

depicts

5.0

7.5 RADIUS

Equation

a 0=

(2)

t.0832

12.5

15.0

r (in.)

I + 0.939_ (3 + 0.313) 225

x 105. /l +0.13332-

x 105 _1.0t78-0.00444

sketch

psi.

4. 005 I

following

10.0

becomes

/

= 1.0832

psi.

"Q_)

a . r

2.6

The

psi

=

r

depicts

a0 .

r2

psi.

Section

G1.3

1 May,

1971

Page

216.64

162.48

108.32

54.16

2.5

5.0

7.5 RADIUS

The

maximum

stress

r = 5.4772 The

stresses

at that

occurs

at

in. position

are

= 1.0832×

0.7511

= 0.8136x

105

= 1.0832x

1.0733

× l0 s

r

psi

psi

and

= i.1626xi05

× 105 psi

psi.

10.0 r (in.)

12.5

15.0

i9

Section Gi. i May, Page

If. Example

Problem

3

197i 20

2.

Find the radial and tangential stresses for the following

la_w

0.25

Solution: E=

i6.0 x i06 psi.

p =

O.i61b/in.S

/_ =

0.313.

g =

386.4 in./sec z.

co=

16 000 rprn = 1675.52/sec.

in.

circular disk.

Section

Gl. 3

1 May, 1971 Page 2i

The idealization

for

the

finite-difference-type

analysis

is given

in the

following sketch.

123

6

['

8

.....

9

b

'

I

lip _--2.3--_

ii

•-- 2.45_ el.---." :

I

2.75 5.0 5.5 10.0

i-" _-" NOTE:

The Table

computations

G1,3.3-i.

12.5 15.0 DiMENSiONS

for the

ARE iN iNCHES

finite-difference-type

analysis

are

given

in

Section

Gi.3

i May,

1971

Page

N q=4

oooo,,°o°

÷

×

x

t0=4 C) nn

v

.o ,jJ

N

A

×

.=

I 0

t_ X

_

°l=d v

I •Jq

I I;I

× •

o

•

•

,

•

o

•

•

o

•

,

•

o

•

o

•

o

,==. o•ooo•ooo°

•

•

oo•°o,o,oo

22

Section

G1.3

t May, 1971 Page 23

×

_

b

T

t

_ _

_

_

_

_

_

_

_

"_

_

_

_

_

_

_

_

'_

_

_

_

_

_

,_

•

.

_

"7

x

+

×

"7

• _

.

×

"7

(o Z © co

×

x

___

I

× m

"

_

E

_

_

_

E

_

E

£

4

4

E

_

_

_

_

_

_

_

_

_

_

_,

_

_

_

=

=

"7 m

,,P-I

rO

×

+

×

_,

¢)

,--,,4

.D ×

×

x

m

×

' _

_

_

_

N

N

_

N

_

N

N

_

N

_

N

_

_

_4_j'_

_

_

_

_

_

_

_..._

N

i ÷

_

,

Section

G1.3

I May,

i97i

Page

24

REFERENCES I.

Timoshenko, S. : Strength of Materials, Company, Inc., New York, 1956.

l_art II. D.

2.

Manson,

Stresses

NACA

S. S. : Determination Report

871,

Cleveland,

of Elastic Ohio,

Feb.

27,

Van Nostrand

in Gas-Turbine

Disks.

1947.

v

SECTION HI STATISTICAL METHOD S

j

_a

TABI,E

OF

CONTENTS

Page Hi

STA TISTICA L ME THODS i.t

INTRODUCTION

1.2

METHODS

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

1 1

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

f FOR

MEASURING

PERFORMANCE

O F A MA'rEI',IA

L ............................

1.2.1

Probability Curve

Normal

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

3

r

Properties .................... Estimate of Average Performance ...................

4

i.2.i.5

Example Estimate Confidence

4 5 5

i.2.1.6

Example

Problem

2

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

5

1.2.i.7 1.2.1.8

Example Estimating

Problem 3 Variability

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

6

i.2.1.9

Example Number

i.2.i.iO

Required Tolerance

i.2.i.3 i.2.1.4

1.2.2

Log-Normal 1.2.2.1 i.2.2.2 1.2.2.3

i.3

Problem i .............. of Standard Deviation Interval Estimate

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

Problem ................ of Measurements

8

...................... Limits ................

11 13

Probability Properties Estimate

3

Curve

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

16

.................... of Average Performance

i6 17

1.2.2.4 1.2.2.5

Example Estimate Interval

1.2.2.6

Example

Problem

5 ...............

21

1.2.2.7

Example

Problem

6 ...............

2i

REFERENCES

Problem 4 .............. of Standard Deviation Estimates ................

.....

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

PAGE BLANK

17 17 i8

23

Hi-ill

Jalq_CE_lltG

.......

NOT

Ffl.MED

LIST OF

ILLUSTRATIONS Ti tle

Figure Hi-i.

HISTOGRAM

Hi -2.

DISTRIBU TION FUNC "FAIRED"HISTOGRAM

Hi-3.

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

NORMAL

Hi-5.

GUIDES

FOR

NUMBER

OF

REQUIRED DEVIATE)N TRUE

AND

Hi-

HI-8.

Hi-9.

.

2

3

PERFORMANCE

NORMAL

DISTRIBUTION

DEGREES

OF

...........

3

FREEDOM

TO ESTIMATE THE STANDARD WITHIN P PERCENT OF ITS

VALUE

COEFFICIENT Hi--6.

2

DISTRIBUTION

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

PARAMETERS

Hi-4.

TION FRO M ......................

(GAUSSIAN)

CURVE

P_go

FACTOR FOR POINT RANGE

WITH y

CONFIDENCE .........................

DETERMINING DATA ...........................

13

19

_

PROBABILITY

FACTOR

kpT

PROBABILITY

FACTOR

kp,/ FOR

"y : 0.90 ......

20

PROBABILITY

FACTOR

kpy

7

20

Hi-iv

FOR

FOR

T

= 0.95 ......

- 0.50 ......

i9

LIST Table

Ultimate

Hi -2.

Percentiles

HI-3.

Factors Limits Factors for

Hi -5.

TABLES

Ti fie

Hi-1.

H1-4.

OF

Strength of the

"t"

"A"

for

Life

One-sided Distributions of Product

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

Distribution

for Computing Two-sided for a .............................

Normal

Fatigue

of Product

Page

Tolerance

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

Hi-v

7

Confidence 9 Limits

..................... "B"

4

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

14 t7

LIBT OF

SYMBOLS

COMMONLY

.Symbol

USED

IN STATISTICS

Definition

c (x)

Coefficient of variation

=

C.D.F.

Cumulative

function

Coy

Covariance

distribution

ill

M(x)

[2l [2]

n

C r

(n

-

[21

r) _. r:

D.F.

Degrees

df

Distribution

Exp X

e

_fF

F r_Uo

of freedom

[2J

function/degrees

of freedom

K

.

f

I!

[21 or variance

(N-l)

Degrees

sampled; k

kp

ratio

[21

of freedom;

fraction

of parts

frequency

[2]

A standardized

variable

expressing

about

In terms

of

the mean

A constant

for

a specified

correlates

population

sometimes

called

the dispersion

[3]

o probability

density

level

with standard

a standardized

variable,

which deviation; applies

to normal distributions Composite variable value

probability relating

of the

P,

variable,

[3] factor; 7,

and

applied

Hi -vt

a standardized n

to some

to normal

limiting

distributions

[3]

LIST OF

SYMBOLS

COMMONLY

USED

IN STATISTICS

(Continued)

f Defin[ lion The mean

of a stochastic

M.G,F.

Moment

Me

Mode

m

Population

N

Sample

size

failure

[ 3]

N

generating

variable

[ll

x

[2]

function

[21 [ 4]

mean [ 2] ; number

An arbitrary

lifetime

of loading

to which

cycles

fatigue

to

test

data

@

are

Same

N1

relationship

Lower

Npy

limiting

L3]

to N as x l has to x

life

can be expected

lq

[3]

to be extrapolated

above

to fall

which

any test

value

with a probability

of P [31

and a confidence

of

,/

Arithmetic

mean

of test

Arithmetic

mean

lifo

an infinite

number

sample

[31

lives

g I

N

that

could

be expected

if

U

,

N4

Number four

_4

of samples

of loading

specimens

Arithmetic mean

cycles

could

at which

be tested weakest

[3]

of

fails

[3l

of least-of-four test sample [3l

lives

Hi -vii

LIST OF SYMBOLSCOMMONLYUSEDIN STATISTICS (Cofitinued) Definition

Symbol n

Number

of independent single test specimens

n t.

Factorial n = Number

n4

(;)

C (:)

i x 2 ×

... x (n-I)xn

of least-of-four test points

=

[3] [21 [3]

[2]

n'. 1.r'. (n-r)

Probability; the percent of a group of specimens

P

P (x) P4

expected to fallwithin a certain range

[3]

Probability of an event x occurring

[2]

Probability of four consecutive test values exceeding some

specified limiting value

[2]

R. M.S.,

Root mean

r

Correlation coefficient

S

Loading stress level at which failure occurs at some

square

number

[3]

[2]

of cycles (N) in fatigue work

[3]

A value of stress on the estimated average

S a

S-N curve corresponding

to some

arbitrary test

[3l

point life,N. l SD

Standard deviation of a sample

[2]

A value of stress on the estimated average

S e

S-N curve ccrresponding to the lifetime (N e) to which fatigue data are to be extrapolated

Hi-viii

[31

I,[S'I'()F

SYMB()[,S

COMMONLY

USEI)

{Continued)

1) c/'i n t tl on

S3:ml)ol

An

S.

IN STATISTICS

at'l)itral'y

value

of stt'ess

occul'J'ing

in a

1

l)a.t'tictflat

• pt'oblen_

at

lifetime

[:;1

N. l

A derived

S.

stress

value

at

lifetime

(N) (2

112

to

COl't'esl)oll([ing

S. at

lifetime

an

ol)sct'vcd

stress

va.luo

[:_]

N.

1

1

of the

stan(lat'd

[2]

Stan(lavd

(2vvov

Va_'iance

()f :t samp[(2

tel

Stu(lent's

"t"

statistic

[2]

U

Stanclar(li

ze(I

v:u'iable

V

Coeftici(2nt

of variation

[2]

W

Often

for

[2]

X

A

£

Arithmetic

S

deviation

S

8 2

x -

[1]

o

used

random

numl)er Any

X°

l"ange

statistical

mean

of x-values

of values

arbitrary

[a]

variable

vcgav(lless

of the

involved

value

o[ x oc,curl'ing

in a sl)ecifi(2d

1

17roble

x

P

Xpy

[n]

m

A variable

expressed

A limiting

value

as

a function

of x dependent

Hl-ix

on

of P and

p,

Y ,

n,

[.-,]

o-

and

_:

[ ._,1

LIST

OF SYMBOLS

COMMONLY

Defini

Symbol X

USED

IN STA TIS'FlCS

(Continued)

tion

Arithmetic

mean

of values

from

Arithmetic

mean

that could

a limited

sample

size

13]

S

be expected

from

an

U

infinite

number

Frequency

Y

of occurrence

intervals

frequency;

percentage

of test

Ot

t_ risk,

Type

ot

Limiting given

ot

1,

points

of total

in a given

variable

131

increment

tin1

of standardized

probability

mean

in given

I error

value

Limiting

points

131

_Ax!

number

Pi

of test

of the variable

Percentage

,f

[31

of specimens

variable

and an unknown

value

of standardized

for a given

confidence

for a

131

distribution

variation

of the

and an unknown

13l

distribution B risk, 1,

Type

Confidence; values [3];

II error;

the

falling

Population

percentage

within

associated

Standard

a given

with

equals

deviation

12]

I1 - a)

of sample range

mean

of the universe

the tolerance

limit

tables

mean

[2]

[21

mean

Mean of a stochastic ¢y

also

variable of variable

Hi-x

x

=

[11

M{x}

x about

,_

[31

LIST

OF

SYMBOLS

COMMONLY Defini

Symbol Standard

o

USED

deviation

IN STATISTICS

(Concluded)

tion

of a limited

number

of

c

derived

stress

points

about

S e

_4

Standard

error

of least-of-four

u

Standard

error

of the

standard

deviation

[31

mean

standard

(N4)

deviation

[3]

[2] ;

S

sample

size

Unbiased

Lr

computed

from

a limited

deviation

of the

universe;

[3]

standard

U

corresponds Standard

O

to an infinite

sample

deviation

of universe

deviation

of x , about

size

of least-of-

U 4

four

points

Standard

(3_

S

X

called

(5 4

the

Standard failure

standard deviation

errol

Variance

)(2

Chi-square

U

of the

of least-of-four

mean test

[31

points

o2

x ; sometimes

of a population

tel [2J

Hi-xi

LIST

OF DEFINITIONS

COMMON

IN

t'SI':I)

IN: STATISTI('S

I)cfinition

Alternative IIypothcsis

Possible

true

sis

statistically

I×,ing

sample

size,

rejecting

the

to the

tested. greater

The

the when

hyl)othe-

larger

the

I)ossibility

of

an alternate

answe_

[2J.

AOQL (Average Outgoing

UPl)er

Quality Limit)

CXl)ect(:d

limit

on outgoing

in the long

lots

are

subjected

with

all

defective

replaced

N times

and

quality

run,

th:tt

when

all

to 100 percent

by good

If in a number

of its

answer

a hypothesis

is true

A Posteriori Probability

alternate

articles

removed

articles

[ 2].

of trials failed

occurring

an event

M times,

the

in the next

may

bc

rejected insl)ection, and

has

occurred

probability

trial

N

is

M

+ N

[2]. A Priori Probability

Let

N bc the

exclusive,

and

under

a given

cases

are

mathematical,

Hi -xii

number

of exhaustive,

equally set

known

likely

cases

of conditions. as the

or a priori

event

mutually of an event If M of these

A,

then

probability

the of event

LIST OF

DEFINITIONS

COMMONLY

USED

IN STATISTICS

f

(Continued)

Definitions A occurring is M/N AQL

(Acceptable Quality

Level

under

the

given

set

of conditions

[2].

Percentage lot that

of defective a sampling

the usual

case)

items

plan

will

in an inspection accept

an associated

_

with Risk

(in of 0.05

[2]. Biased

Sample

If some

individuals in the Universe are more

likely to be chosen than others, the sample is said to be biased. Class Interval

When the

the

number

range

of the

limited

Confidence

Intervals

number

The

segments

cells

[ 2] .

This

provides

estimate interval

data

are

is the

proportion

lmown

a method

associated coefficient.

can

true with The

is large,

be broken

of segments

is to the

dence

Hl-xiii

of observations

into

of equal as class

of stating value

[ 2].

a prescribed confidence

of samples

of size

a

length.

intervals

how close It is the conficoefficient n for

or

an

LIST OF

DEFINITIONS

COMMONLY

USED

IN STATISTICS

(Co_Unued)

Definition which

intervals

metht)ds Continuous Distribution

Cumulative

Distribution

may

computed be expected

One in whtch

to bracket

tim only limit

measured

is the

apparatus

[ 2].

Indicates

by its

the

by the prescribed

Univeroe

oensltivity

to si_._

a value

[ 4].

tatervsls

of the measuring

magnitude

the

(or sample)

proportion

of

to the Left of that.

point [Zl. Curve Fitting

Method imate

utilizing

computed

parameters

as spprox*

statistics

for theoretical

distributions

[21. Degree

of Freedom

Number

of free

independent entering sample are Discrete Distribution

variables

in the

of random

into a statistic. of size

N°!

N,

degrees

If a random

from

values,

distribution

[ 2].

In the case a UniverN,

of freedom

variable

of possible

H i-xiv

sense

(unrestricted

and sampling) of a there

[ 2].

has only

a finite

amber

then it will

form

a discretB

LIST OF DEFINITIONS

COMMONLY

USED IN STATISTICS

f-

(Continued)

Definition Double

Sampling

Involves

the

disposition test

possibility

of an inspection

sample

is taken.

on the basis

,:

of the

are

very

good

are

very

poor.

sample good

of putting

are

lot until

A lot will first

or will If the

sample

results

if the

results

if the

from

the

nature

a second

a second

be accepted

be rejected

of a borderline

and poor},

off the

results first

(between

sample

must

be

taken. On the first

basis

and

second

accepted Error

In statistics

then

false, Kind Distribution

the

to reject then

of the

samples,

are

null

the

combined

lot is either

[2].

there

we make

we fail

results

or rejected

we reject

"F"

of the

two types

hypothesis

an Error

of error.

when

of the

we make

an Error

it is true,

First

a null hypothesis

If

Kind. when

of the

it is

Second

[ 4].

Sampling

H l -xv

distribution

of the

variance

If

[ 2].

LIST

OF DEFINITIDNB

COMMONLY

USED

IN STATISTICS

(Continued) \

Definition Frequency

Table

Histogram

_

Tabulation

of the number

occur

in each

Block

representation

dispersion Hypothesis

'

Based

Inference

through on the

cations Square

"Analysis

Squares

mathematical

design arise

values

of several Method

of a quantity

upon

for description, decisions because

[ 2]. of

allowing

that

of the

compli-

technique

the principle

squares

Ht-xvi

of variation

[ 2].

the sum

(from

the

of variability

or observations

which

prediction,

despite

ordering

measurements

observations

it may be

framework

that can be deduced

of the

the

statistical

in an experiment,

sources based

to show

a way that

of probability,

of Variance"

observed

Least

theory

which

[ 2].

[ 2]. in such

a technique

and rational

of a histogram

statistics.

is that

supplies

that

of data arranged

formulated

inference

Latin

interval

of the data

Statement refuted

class

of observations

control

the best

from

value

a set of

is that for which deviations

it) is a minimum

of the [ 2].

LIST

OF I)EFINITIONS

COMMONLY

USED

IN STATISTICS

(Continued)

r

Definition Lot

LTPD

(Lot

Percent

Tolerance

Defective)

Group

of manufactured

alike,

such

Usually there

as

refers

1 day's to the

is a small

It is usually

articles production incoming

chance

taken

which

with

that

are

essentially

[2]. quality,

a lot will

a consumer's

above

which

be accepted. l"isk

of

= 0.1012]. The arithmetic average of a gl'ouv of observations

Mean

[2].

It is the location parameter

o[ ,tnormal

distribution locating the "centel"of _1'avity"of the distribution [4]. Mean

Deviation

Arithmetic mean

of the absolute distance of each

observation from tilemean Median

[2].

Middle value of a group of obscl'vations. In the case of an even numbered

set of obsel'vations, it is tile

aver_tge of the middle pair [2]. Midrange

Arithmetic average of the extreme

values of a set

of observations [2]. Mode

Most frequent value of a set of obserwltions [2].

Hl-xvii

LIST

OF DFFINITIONS

COMMONIN

USI,:D IN STATISTICS

(Continued)

Definition Moments '

In statistics,

moments

in mechanics

in several

ix_dies

are

some

completely

expected

value

The

moment

first

probability

above

the

Nonparametrte

Statistics

Statistical

Curve

(GaussianJ

random

the center

is also

known

as the

Bell-shaped

curve

developed

mean [ 2].

Hi-xviii

variable.

of gravity moment moment

to test

of normality,

than

that

of

from

the

and

hypotheses or any

of continuity

Gaussian

It is a two parameter the

to the

[2].

the assumption

[ 2].

description

of the

second

distribution

The

is equivalent

The

other

are

moments.

mass.

assumption,

requiring

origin

is also

techniques

distribution.

distributions

mean)

and variance

without

Normal

mean

as some by their

by their

the

(the

of the

inertia

probability

about

Just

to moments

characterized

characterized

moment

analogous

ways.

completely

moments,

first

arc

the variance

other

of a

probability distribution for

its

LIST OF

DEFINITIONS

COMMONLY

USED

IN STATISTICS

(Continued)

Definition Normal

Probability Paper

Special graph paper which reduces the cumtflative normal The log-normal same

curve to a straight line.

probability paper does the

with log values as the normal

paper

does with linear values [2]. OC Curves

(Operating

Characteristics Population

Give the chance of accepting a hypothesis [2].

Cmwes) Any set of individuals (or objects) having some

common

observable characteristic. It

is synonymous Proof

with Universe

Differs from mathematical not fallwithin the framework

[2].

proof as it does mathematics,

but results from experimentation, with an accompanying Random

Digits

probability statement [2].

Digits picked in such a way that each digit has an equal chance of occurring at any time [2].

Randomization

Assignment test program

of a sequence of operations to a by the use of some

such as random

tables, to avoid bias in the

test results [2].

Hi -xix

technique,

LIST

OF DEFINITIONS

COMMONLY

USEI)

IN STATISTICS

(Continued)

Definition Random

Sample

Picked the

in such

population

sclcctioa

values

Root

Mean

Square

have

all

members

an equal

difference of a set

Square

that

of

chance

of

[ 2].

Absolut_

Range

a way

root

squares

Ix_tween

the

of observations

of the average

of a set

extreme [2].

of the

sum

of observations

of the

[ 2].

X. •

R MS

:

Set of observations

Sample

l

chosen

from

a population

[2]. Sampling

Distribution

Distribution samples

of a specific

Universe Sequential

Significance

Samplings

Level

of a statistic

in the

size

set

from

of all

a given

[2].

Acceptance

plans

permitting

from

three,

of samples

[ 2].

to an unlimited

number

This

our

reluctance

to give

the null

hypothesis

and

or

expresses "reject"

by the

Hi-xx

magnitude

of the

_

Risk.

up

up

is given The

LIST OF

DEFINITIONS

COMMONLY

USED

IN STATISTICS

(Continued)

Definition

f_

Single

Samphng

smaller

the

we are

willing

When

of only

lot one

described Standard Deviation

Positive

Statistic

Estimation

sample,

made

the

from

probability

function even

Tolerance

Limits population

a specified

that

computed

[ 2]. which

may

the

have

a

or

stochastic

frequency

function

shall

of the

a certain

probability o[ a parameter

corrected

l or sample

equivalent

to the

total

mean

[ 2].

portion

lie within

Estimate

H i-xxi

[ 2].

[2].

distribution such

is

[ 2].

variance

is a chance

though

evidence plan

plan

[ 4].

of an

parameter,

a sample any variable

Sampling

acceptance

of the

less

hypothesis

on the

sampling

root

the

disposition

of a population

is not knmvn

Unbiased Estimate

is always

In general,

"t"Distribution

null

as to the

square

variable,

Limits

the

as a single

entirely Variable

of significance

to reject

the decision

inspection

Stochastic

magnitude

of the

or above

them

with

[ 2]. which size

effects

population

has

been and

is

parameter

[ 3].

lIST OF DEFINITIONS

COMMONLY

USED

IN STATISTICS

(Concluded)

Definition Comprised

Universe

Variables

Sampling

of any set of individuals having

some

common

observable characteristic [2].

When

a record is made

of an actual measured

quality characteristic, such as a dimension expressed known Variance

Sum

in thousandths of an inch, it is

as variables sampling [2].

of the squares of the deviations from the

mean

divided by the number

of observations

less than one [2]. a Error

Risk of rejecting a true hypothesis. also known

as an

This is

a Risk, Consumer's

Risk,

or Type I [2]. /_ Error

Risk of accepting a false hypothesis. is also known as

#3 risk, Prodttcerts

Risk, or Type II Error (1 -a)

.

HI -xxii

[2]. Equals

This

f

f-

Ill

STA TIST] CA L ME TItODS

1.1

1NTI/OI)UC "One

description and

the

\Vhile a set

of the

main

observation

characteristic

characteristic

fetttures

Various developed,

one

set

let

of values

number known

of test as the

vertical

through

the

shape

curves

(Normal,

but only two

characterized

I May

J 972

the

features

which

can

be reproduced;

statistics

attempt

of describing

test

axis

be divided in each If the

various

may

ling-Normal,

)¢_, "t",

Log-Normal)

laid

process,

that

produces

it is these

number

tI1-2). Weibuli)

Consider

a count

of points are

o ltl-1).

(Fig.

h_tve been a

per

now laid From

the

by fairing

a smooth

Several

frequency

have

be discussed

order

of a graph.

and

frequencies

will

shown

off in ascending

intervals

be obtained (Fig.

features

the abscissa

The

is produced

curve

and

to form

interval.

and

of a histogram.

as being

into equal

[ 1].

to describe.

the use

values

phenomena

has

of some

characteristic

includes

observed

experience

repetition

that

a mathematical

by a few numbers"

be reproduced,

of the histogram

(Normal

that

the

a histogTam

distribution

a maturer

from

frequency.

a h'equency

is to give

resulting

a horizontal

points

scale,

are

of recorded

along

range

in such

method

of magnitude

of statistics

cannot

methods

and

hypothetical

the

data

of obscrwltions,

certain

objectives

of observation

a single

II1

TION

of observed method

Section

been here.

Now

m_tde

of the

interval eli

is

on a

histogram line distribution

est:dflished,

Section 1 May

I11 1 !)7"2

I k l go: 2

).

W tat D O tu tr L

MEASURED

I"IGUI{E

VALUE

Ill-1.

MEASURED

FIGUI{E

11l-2

The

necessary has

been

and

fracture

I)ISTItlI_UTI()N

statistical

for found

that

[ 5]

data

discussed

material,

material

obey

VALUE

I,'UNCTI()N

melhod_

evaluating

IIISTO(;IUkM.

data

log-normal

I.'I{()M will

fatigue, obey

normal

be

"I:AII_.FI)" those

methods

fracture

mechanics

data.

dJ:_tribution

and

fatigue

and

distributions.

limited

to

II1ST()GRAM

that

It [ 31

Suction

H1

1 May __

I>:tge 1.2

,_II':TII()I)S

1.2.1

Normal

1.2.1.1

Prol)ovties

The It is defines one.

by

The

or

tho

sln'cad

the

relation

significance

curve h)czttcs

of the

curve.

The

of the

mean

is

()I," ,k _IA'I'I,]IIL\I,

(p)

shown

in

,

is a t\vo-lmr:tmctcv the

curve

curxc

anti

the

at'ca

under

the

cttJ'vc

the

stan(t',tL'd

Fig.

(l,'i_,.


\vhich

is :t[\v:tyb


(tr)

I11-32.

equal

, ;in(I

to

the

111-4.

OF

o

o

-3o FIGURE

(;aussi:m, \\hich

(o:)

t_I.:III.'()IIMANCI.;

Curve

mean

level

VARIATION

_II.:A,S['I_IN(;

I>rol)ability

nornutl,

(IoI'incd

FOIl

1!)72 ;I

-2o lIl-:l.

-o

NOIIMAI_

(GAUSSIAN)

I

I

o

20

30

I)ISTI{II'I;TI()N

CUIIVI.].

It VARIATION

OF

_

POPU LAT ION

I_ _

STANDARD

_
ERROR

OF THE MEAN

_

._"_'_._'x._._"_'__'_'_.

/

V ARIATION

OF THE

EVIATION -3o

-20

-o

I

t

I

2.5 PERCENT

OF AREA

FOR

Itl-4.

20

I

I

7

4 2.5I PERCENT

OF AREA

o_,, 0.05 LEVEL

FIGURE

I

o

OF SIGNIFICANCE

PAIC\3[ETI.:I{S FOIl

NOII_L\

a-

AND

0.05

t)I.:III.'C)IIMANCI.:

l. I)ISTI'dI'UTION.

GUII)I.:%

FOR

a = 0.05

Section

II1

1 May

1972

Page 1.2.1.2

Estimate The

lation

most

mean,

of Average

common

and

m , is simply

Performance

ordinarily

the

tile arithmetic

Imst

single

mean

of the

estimate

of the 1)o;,:- -

measurements.

n

m _ x

1.2.1.3

= nl

E_mmple Determine

havhlg

i=_l

the

test

"

Problem

the

mean

results Table

xi

1

value

in Table HI-1.

Test

of the ultimate

H 1-1

Ultimate

Specimen

streng'th

of product

[ 3]. Strength

of Product

Ultimate x.

"A"

Strength (lb)

t

1

578

2

572

3 4

570

5

568 572

6 7

570 570

8

572

9

596

10

584

n =

1 n

n

V

X°

_"

l

4

10

1 .---: (5752)

_ xi

= 575.2

=

lb

5752

2

= t'tle an value

"A"

Section

HI

I May

1972

Page 1.2.1.4

Estimate The

root

of the

estimate best

seldom

that

it is fairly

confident

close,

will

of the

level,

our

contain

the

true

level.

Confidence correspond

1.2.1.6 Using percent

the

time,

long

same

levels to

y

as the

100

and

the

in Table

Hi-i,

for the

the

intervals

included

at the used

percent

100

are

average

confidence intervals.

of our

(1 - a)

we are

the true

confidence

a)

average

We do hope

which

at a 95 percent

(1 -

sample

average.

an interval

95 percent

commonly

2

interval

to state

If our

operating

Problem

confidence

like

be operating

Example data

lot or population

be called

a = 0.01

[ 4].

a lot or a population,

lot mean.

we are

the square

(2)

run we expect

value,

as

1)

from

would

taken

Estimate

we would

intervals

if in the

-

Interval

the

is usually

xi

and we would

bracket

95 percent and

-

l

a sample

be exactly

deviation of variance

n(n

we take

will

standard

x.

Confidence When

Deviation

estimate

i

=

1.2.1.5

which

of the

unbiased

a _ s

general,

of Standard

5

intervals

percent

In to

confidence

99 percent

and

95 percent

two-sided

100 (1 - c_)

c_ = 0.05.

what true

is the

mean

m

of the

total

population

[ 4] ?

Section H i i May 1972 Page 6 Choose

desired

confidence

Let

level,

i

0.95 =

Compute Compute Look

X =

X s from Eq. (2)

up

t =

t

lb

8.24

S

forn-

ot

575.2

0.05

t = t for O. 975

1

9 deg

of freedom

2 degrees

of freedom

in Table

= 2.262

HI-2

S

Compute

Xu =

x

X + t

= 575.2

u

=

+

xL =

X -

2.262 x L --

t _nn

575.2

=

Conclude:

The

interval dence

for that

interval

x L to

the population the population

t. 2. t. 7 Using

mean; mean

i.e.,

3

the

in Table

Hi-t,

confidence

Choose

desired

interval confidence

for the level,

569.3 Ib

we may assert 569.3

what true

percent

with

is a one-sided

1

mean -

_

100 [ 4] ?

= 0.99 = 0.0t

Compute

X S

= 575.7 s

= 8.24

lb

confidence

95 percent

lb and 581.1

population Let

(8.24)

-

i00 (I - a)

is between

Problem

percent

isa

xu

Example data

JT6

581.1 Ib

S

Compute

2. 262 (8.24)

confi-

lb.

(1

-

_)

Section

111

1 May

1972

OF POOR QUi,LiCV r

Page Table df

to.7o

II1-2. to. 80

Percentiles

of tile

tO. 90

"t"

7

Distributiol_

to. 95

975

to.99

tO.99.5

:'.

O. 325

[).727

0.289 0.277

1. 376

3. 078

6.314

0.617

1.061

1.88(;

2.920

0.584

0.978

1.638

2. 353

0.271

0. 569

0.941

l. 533

0.267

0.559

0.920

1.476

6

0.265

0. 553

.906

7

0.263

0.549

.8!)6

12.706

31.821

63.657

4 .303

6.965

9.925

3.182

4.541

5.841

2. 132

2. 776

3.747

4.604

2.015

2.571

3. 365

4.032

1. 440

l .943

2.447

3.t43

3. 707

1.415

1. 895

2. 365

2.998

3.499

0.262

0.546

• a89

1.397

1. ,q60

2. 306

2. 896

3. :355

9

0.261

0.5,1:3

._3

l .:;_3

1 .,_3:1

2. 262

2._21

10

0.260

0.542

.s79

l. ;372

1.812

2.228

2. 764

3.250 3. 169

11

O. 260

0.5t0

. S76

1.363

1.79(;

2.201

2.718

3. 106

0.259

0.539

.873

1.356

1. 782

2. 179

2.681

3.055

0.259

0.538

.8 70

1. 350

1.771

2.

2.650

3.012

0.258

0,537

.868

1. 345

1.761

2.145

2.624

2. 977

0.258

0. 536

.866

1.3t

1. 753

2. 131

2.602

2.947

6O 1 oo

l

160

0.258

0. 535

0.865

1. 337

1. 746

2. 120

2. 583

2.921

0.257

0.534

0. 863

1. 333

1. 740

2. 110

2. 567

2. 898

0.257

0.5:34

0.862

1. 330

1.734

2.101

2.552

2 .S78

0.257

O. 53:3

0.861

1.328

1. 729

2. 093

2. 539

2. U61

0.257

O. 5:33

0. 860

1.325

1. 725

2.0

2.528

2.845

0.257

0.532

0. 859

1. 323

1.721

2 .O80

2.518

2.831

0.256

0.532

O. 858

i .321

1.717

2.0

2. 508

2.819

0.256

0.532

O. 858

1.319

1.714

2. O69

2.50O

2.807

O. 256

0.531

O. 857

1. 318

1.711

2.064

2.492

2.797

0.256

0.5:31

O. 856

1. 316

1. 708

2. O60

2.485

2.787

0.256

0.531

0.856

1.315

1. 706

2.056

2.479

2. 779

0. 256

0.5:31

0.855

1.314

1. 703

2.052

2.473

2.771

0.256

0. 530

0.855

1.313

i .701

2.048

2.467

2. 763

0.256

0.530

0.854

1.311

1.699

2.045

2.462

2. 756

0. 256

0.530

O.854

1.310

1.697

2.042

2.457

2.750

0.255

0.529

0.851

1.303

1.6_4

2.021

2.423

2.704

0.254

0.527

0.848

1.296

1.671

2.000

2. 390

0.254

0.526

0.845

1.289

1.658

1.980

2.358

0.253

0.524

0.842

1.282

1.645

1.690

2. 326

$6

7,t

2.660

}

2.617

1

Section

Hi

I May 1972 Page 8 Look up

t

= tI

deg of freedom

.

a

for

in Table

n

-

for

_.99

I

9 deg of freedom

= 2.82!

Hi-2

!

Compute ._

xL 'i_

= 2

-

xL

t 4n

=

575.7

-

= 568.4

lb

"

rcompute

u

= x

Conclude:

We are

is greater

than

population

mean

+ t

i00

(!

-

a)

percent

confident

that the

population

mean

!

i. 2. I. 8

xL

; i.e.,

we may

is greater

Estimating

We have

assert

than 568.4 Variability

estimated

the

with

99 percent

confidence

that

the

lb. Example

standard

Problem

deviation;

and,

in a manner

similar k

to determining confidence

the confidence interval

the data in Table of the standard Choose I

-

for Hi-l,

the deviation determine

deviation

the desired

interval

for the which

mean,

we may

is termed

an interval

which

determine

the variability. brackets

the true

a Using value

[ 4].

confidence

level,

Let

1

-

eL = 0.95 0.05

at I

Compute _Lookup

s

s B u and

B L for

n -

1

= 8.24

for

9 deg

of freedom \

deg

of freedom

in Table

HI-3

BL = B

0.6657

= t.746 U

O_

PO0_

(_U,kL(,'/

Section 1 May Page

HI 1972

Table

H 1-3.

Factors

Degrees of Freedom

a-

df

BU

for

Computing

Two-sided

0.05

_

BL

BU

Confidence

Limits

0.0t

_-

BL

9

for

0.00i

BU

17.79

0.3576

86.31

0. 2969

4.

H59

0.45_1

10.70

0.3879

33.29

0.3291

3

3.

i83

0.5178

5.449

0.4453

11.65

0.3_24

4

2.567

0 •5590

3. X!}2

0.48_5

6.938

0.4218

5

2.248

0,5899

3.

0.51X2

5.085

11.4fi29

6

2,052

0.6143

2.764

i).5437

4.128

0.47_4

7

1.91S

0.6344

2.49H

0.5650

3.551

o. 50O[)

8

I.X20

0.6513

2.311

0.5830

3.167

6.51_

9

I. 746

0.6657

2.173

0.5987

2.804

6.5:14_

io

I, 686

0.6784

2.0_5

0.6125

2.680

6.5492

tl

1.638

0.6896

1.98O

0.6248

12

1,598

0.6995

I, 9O0

0.6358

2.530 2.402

6.562| 0.5738

13

0.7084

1.851

0.6458

2,298

0.5845

14

1.5!; 4 I •534

0.7166

1.8[11

0.6540

2.2|0

O. 5042

15

t. 5([9

0.7240

1• 77, 8

0.6632

2.136

0.6032 0.6116

175

844.4

BL

1 2

II. 2481_

10

t. 486

0.7308

1.721

0.6710

2.073

17

1.466

0.7372

1.6X8

0.6781

2,017

18

_.448

0.7430

1.115½

0.6848

1.968

0.6266

19

1 • 432

0.74A4

I .li32

0,6909

1.925

0,6333

20

1.4t7

0.7535

I • 609

6.(1968

1.8_6

_1.6307

21

I. 404

0.7582

1. 587

0.7022

t.851

0.6437

22

1,391

0.7627

I. 56fl

0.7674

1.820

0.6514

23

I. 3 _0

0.7660

l. 550

0.7122

1.791

0.8566

24

1,370

0.7709

I. 533

0.7169

1.765

0.6619

0.7212

1.741

1'.8668

25

1. 360

0.7747

1.518

26

t.351

0.7703

!. 504

O.7253

1.719

0.6713

27

1.343

0.7817

1.491

0.7293

1.89_

(1.675_

28

1. 335

0.7849

1.479

0.7331

1,670

0. 6800

29

1.327

0,7880

1.467

0.7_17

1.861

0.6841

0.7401

1.645

30

1.321

0.7909

1.457

31

1.314

0.7937

1.447

0.7434

1.629

32

t. 308

0.79_4

1.437

0.7467

1.615

0.6953 0.69_7

0.8917

33

1.302

0.7990

1,428

0.7497

1.601

34

1 • 296

0.8015

1,420

0.7526

1.5X8

o. 7020

35

t,29t

0.8039

1.412

0.7554

1.576

0.7652

36

I • 286

0.8P_62

1. 404

0.7582

1.564

o.

37

1.281

0.8085

1. 397

0.760X

1.553

0 •7113

38

i • 277

0.8106

1 • 300

0.7633

1.543

0.7141

39

I • 272

0.q126

1 • 383

0.7658

1.533

O. 7169

40

i • 268

0.8146

|..177

0.7681

1.523

0.7107

1.515

0.7223

7083

41

1.2[;4

0.8166

1,371

0.7705

42

I • 260

0.8{84

|.3C;3

0.7727

1.506

O. 7248

43

1.257

0.8202

1.360

0.774_

1.498

0.7273

44

1,253

0.8220

1. 355

0.7769

1.490

0.7207

45

1.249

0.8237

1.349

0.7789

1.482

O. 7320

0.7809

1.475

0.7342 0 • "_364

46

I. 246

0.8253

1.345

47

l. 243

0.8269

I • 340

0.7_28

1.468

48

1 • 240

0.82_5

1. 335

0.7X47

1.462

0.7386

49

I. 237

0._300

1.331

O. 7064

1.455

O. 7407

1.449

0.7427

50

1.234

0.8314

1. 327

0.7882

OR|GiNAL

P_,,C£ _;

OF POOR QU ..iTY

Section

H1

1 May 1972 Page Table

DII_

a

=

H1-3

0.05

•

-

10

(Continued)

0.01

a

-

0.911

of Freedom _d

BU

II L

BU

BL

BU

51

i.532

0 • 8320

1.323

0 • 70041

1.443

0.7441

BL

53

i.228

0.8342

1.319

0.7910

i.437

O. 7491

53

i.2N

0.838

1.315

O. 7922

1.482

0.74

54

i.224

0.2370

1.31i

0 • 7940

1,4Zg

0.7503

03

50

i.221

0.0383

1. 308

0 • 7984

1,421

0.7621

M

i.219

0.8308

1.304

0 • 7279

1.410

O. 7991

37

1.2i7

0,8400

i,30i

0 • 7914

1.41i

0.75M

M

1.2i4

O. 8420

1. 290

0 • 8000

i.491

0.7579

50

i.Zi5

0.0431

1 • 295

0 • 8022

i.402

O. 7591

0.

|091

1.307

0 • 7008

00

iotiO

0.8442

1 • 292

01

1.208

0.8454

t.289

O. 8060

1.388

0.7821

0|

i.JOe

0.8445

i,388

0.I_82

1,088

0.768

113

1.204

0.84?5

1.282

0 • 8076

i.206

0.7081

O. 80N

114

1.202

0.0486

i.238

1.500

0.8438

i.2?7

0.810i

1.311 1.377

O. _91

85 64

i.191

0.0506

i.279

0.8112

1.374

0.7694

01'

1.i57

0.8518

i.272

0.8120

1.370

0 • 7708

08

I.i00

0.8525

i.270

0.8137

1.344

O. 7722

80

1.194

0,0535

i,918

0.0148

1.363

O. 7738

I0

I • i02

0.8544

1. 208

0.8159

1.910

0 • 7740

71

1.191

0.85U

1.242

0.21_0

i.291

0.7711

79

I.tm

0.882

i.20i

O.SiOi

! .3U

O. 7774

73

1.i87

0.8571

t.250

0.819i

1.280

O. 7717

74

i.i88

0.0880

i.257

0.8202

1.347

0.77110

79

i,184 i.i88

0 • 8588 0.0588

i.258 1.253

0.0212 0,8222

1.244 1,341

0.7911

?6 17

1.183

0.8804

1.25i

0.2222

1.330

O. 7114

78

1.iOi

0.8812

1.242

0.8242

1.338

O. 7840

79

1.179

0 • 8620

1.247

0.0252

1.330

0.798

8O

1. i 78

0.3827

i.245

O.Sml

1.230

0 • 738|

81

i,i78

0 • 0838

i .|40

0 • 0270

1.320

O. 7171

82

i.179

0 • 8842

i.241

0.2279

!.326

0.7138

83

i.i74

0.9150

i .239

0.8288

1.220

o. 7991

84

I.i74

0.8857

1.238

0.8297

1.320

0.7109

05

L.i73

0.6664

1.28

0.8388

1.318

0.7938

38

i.1?i

0.6471

1.235

0.8314

1.318

0.7930

57

1.170

0.86?8

1.232

0.8212

1.2i$

0.7991

91

1,108

0 • 8884

1.221

0.0301

1.211

0.7940

N

I.i07

0.5338 0.0:144

0.7169

I. i00

1.230 1.228

1,291

O0

O.MS! 0 • 6697

1.307

0,7942

111

1.105

0.8704

1.227

0.0354

1.305

0.79?7

N

1.154

0.87i0

1.228

0.8382

I. 302

0.7917

03

|,100

0.8710

1.224

0.0370

1.301

0.71111

1.152

q.8_2

1.222

0.2377

i.291

0.11004

N

1,101

0.8721

1.221

0.8385

t .207

0.8813

M

1.180

0.8734

1.210

0.8392

1.215

0.8822

07

1.159

0.0741

1,210

0.$391

i.293

0.8021

91 M

i.150

0.8744

1.217

0.8406

1.201

0.0029

I,i85

0.8762

1.210

0.8413

1.289

0.8847

0,8757

1.214

0.8420

1.288

0.3850

i00

1.i57

O. 79e0

0. 791"_

Section HI I May Page Compute

sL

= BLS

s

sL

= Bs U

s

U

U

=

(8.24)

=

5.48

=

(8.24)

1972 II

(0.6657)

(1.746)

= 14.38

Conclude:

The

confidence

interval

confidence

that

1.2.

i. 9

interval

estimate

Nmnber

in order accuracy.

assume

a

the

mean

Choose

is a two-sided

5.48

and

we may

assert

(1 - a)

with

percent

95 percent

14.38. Required

we may

need

a parameter

If an estimate ascertain

100

s the

or

to lmow

of some

how many

distribution

cr is available

required

sample

measurements with

or if we are size

n

prewilling

for determining

[4].

and

estimate

Su

(r ; i.e.,

experiments,

(y ; we may

d,

error,

to

of Measurements

to determine

scribed

sL for

(r is between

In planning to take

from

of

the allowable

margin

c_ , the

risk

m

be off by

will

that

of

Let

=

c_ =

our d

d

0.2 0.05

or

more Look

up

t =

t

for df deg (X

of

t :

t0.975

=

2. 262

2 freedom

in Table

tti-2

for 9 deg of freedom

to

Sc_'tion

II 1

1 .May 1972 Page

Compute

n

t2 s 2

(2.262):'(s.24J_

d2

((1.2) 2

-

12

s7()[) Conciu(le:

We may

sample

of size

n

interval

._ -

0.2

=

8700

to

A similar tion.

conclude

_

cient

deviation

equal

Specify

P , the

deviation

from

in Figure horizontal for the the

scale,

scale

percentage

of its

when

computing

the

be required

+

i

that

l'tllldolli

the

value,

with

st:mdard

(levia-

to estimate confidence

find and

P use _/ .

the

Let

P

the coeffi-

= 20 percent

coefficient

Let

-y = 0.95

on the

For

y

:

I).95,

the

curve

df

=

46

Read

on

required

P

degree

of freedom n=df

O[" ;I

lot mean.

would tt'ue

confidence

._

value

confidence

appropriate

vertical

a sample

the

mean

standard

its true

Hi-5,

95 l)el'c'ent

include

be used

20 percent

allowable

the

may

ha\e

the

[ 4] ?

of the estimated

y,

will

how large

deviation

Choose

+ 0.2

within

to 0.95

if we now ('Omlmtc

, we may

procedure

As an e,'m_ple,

standard

that

n

:

46

+

1 = 47

= 20 percent

-x

OF POOR

'_ _ ' "'" Section

1000 8OO

It 1

1 May

1972

Page _00

%

500 400

x • \

300

\

X

\

\

\

\

"

\\

\\ \

=,2oo

\

\ ..a

..

80

w ,,, ¢¢ 0 ,-

60 50 40

i 3

o

_ .o .o_ \

....

\

\

\

30

k

x

\

\

\\

",

% \

\

\\

\ \\

lo

......

\ \

\

8

\

\

,,

5 5

8

8

10

20

30

40

50

P PERCENT FIGUILI,:

111-5.

ESTIMa\TE

NUMI_I_It

Till'] TI/UE

1.2.1.10

in

VAI,UL"

Tolerance

Sometimes

a lot

or

\\'e

than

to

a prescribedl)J'opovtion

H1-4

\V1TI!

limits

II_()l'C

\re

will

one-sided

[4].

As

()l"

I'I{I,',I"_D()M

\\'ITIIIN

('ONI"IDI"NCI',

l)(,(\_eon,

lie

(one-side(I

lower

limit.

anexa.ml)le,

inh,

:tro

o[

,\

in

lll"QUII_.I,'D

I ) PEIICI.:NT

COF.

I,'FICII.:NT

In

this

al)l)VOl),iatc

the

TO OF

ITS

3/ .

range

Statistical

which

wo

of

case,

c,.)nfident!v

in

l'or

example

,_

N.

values

cxpocL

a population.

x L

of

tolerance

Thus,

:It I(,:t,_t a proportion

values

(Iztt:t

approximate

v,llue.

items

:/I)o\e which

consi(lei

the

())'l)('l<,\v

in(lixi(l_utl

limit).

The

reste(I

i1._ :/vc,.v:t;,;e.

:,l>)ve.

be able to give :l v:lluc

population

the

al'e

population

furnish

might

I)EGI_I,_I,_S I)I.]VtATI()N

Limits

limits

find

()1"

S'I'ANi)Alll)

_

are

lwoblem

-

P

we

of the

Ks

will

be

given

in

Table

1 and

find

a

Section i May Page Table

Hi i972 i4

Hi-4. Factors for One-sided Tolerance Limits for Normal Distributions

Factors K such that the probability Is T that at least a proportion the distribution will be less than _ + Ks (or greater than _ - Ks), i_ and s are estimates of the mean and the standard deviation

computed

-

from

a sample

size

0.75

of

P of where

n .

_ =

0.90

0.75

0.90

0.95

0.99

0.999

0.75

0.90

0.95

0.99

0.999

9 4

1.464 1.250

2.501 2.134

3.152 2.600

4.356 3.726

5.805 4.910

2.002 i.972

4.258 3.i07

5.310 3,957

7.340 5.437

9.051 7.128

5

1.152

1.961

2.463

3.421

4.507

1.698

2.742

3.400

4.666

0.112

0 7

1.087 1.043

1.000 1.791

2.336 2.250

3.243 3.126

4.273 4.1i8

1.540 1,435

2.494 2.333

3.091 2.894

4.242 3.972

6.556 5.201

S 9

1.010 0.984

1.740 t.702

2.190 2. 141

3.042 2.977

4.008 3.924

1.360 i.302

2.219 2.133

2.755 2.649

3.783 3.64i

4.955 4.772

|0

0.964

1.67/

2.|03

2.92?

3.858

1.257

2.065

2.508

3.582

4.629

I|

0.947

1.846

2.073

2.885

3.804

i.219

2.012

2.503

3.444

4.515

13 13

0.033 0.919

/.624 |.006

2.048 2.020

2.851 2.822

3.700 3.722

1.188 1.162

1.960 /.928

2.448 2.403

3.371 3.310

4.420 4.341

14

0.909

t.891

2.007

2.796

3.690

/.139

t.895

2.363

3.257

4.274

15

0.899

1.577

1.991

2.776

3.661

i.il9

1.866

2.329

3.2i2

4.215

t0

0.891

|.566

i.977

2.756

3.637

l.iOl

1.842

2.299

3.172

4.164

17 i8

0.883 0.876

1.554 1.644

1.964 t.051

2.739 2.723

3.615 3.596

1.085 i.07t

1.820 t.800

2.272 2.249

3.136 3.t06

4.118 4.078

19 20

0.870 0.865

1.536 /.528

i.942 1.933

2.7/0 2.697

3.577 3.56t

1.058 1.046

1.78| i.765

2.228 2.208

3.078 3.052

4.04t 4.009

21

0.859

1.520

1.923

2.686

3.545

1.035

t.750

2,190

3.028

3.979

22

0.854

1.514

1.916

2.075

3.532

1.025

1.736

2./74

3.007

3.952

23 24

0.849 0.848

t.508 1.502

1.907 1.901

2.665 2.656

3.520 3.509

t.016 1.007

1.724 1.712

2.t59 2.145

2.987 2.969

3.927 3.904

25

0.842

1.496

1.895

2.647

3.497

0.999

1.702

2.132

2.952

3.882

30

0.825

1.475

1.869

2.613

3.454

0.966

i.657

2,080

2.884

3.794

35

0.812

1.458

1.849

2.588

3.42/

0.942

1.023

2,041

2.833

3.730

40

0.803

t.445

t.834

2.568

3.395

0.923

1.598

2.0i0

2.793

3.079

45 50

0.795 0.788

1.435 1.426

1.82/ 1.8ii

2.552 2.538

3.375 3.358

0.908 0.894

i.577 1.560

i.956 1.965

2.762 2.735

3.638 3.604

Section H I i May 1972 Page 15 Table

7 =

H 1 .-4.

(Continued)

7 =

0.95

0.75

0.90

0.95

0.99

3 4

3.804 2.619

6.158 4.163

7.655 5.145

5

2.149

3.407

4.202

5.741

7.501

6 7

1.895 1.732

3.006 2.755

3.707 3.399

5.062 4.64t

6.612 6.061

8

t0.552 7.042

0.999

13.857 9.215

0.75

0.99

0.90

0.95

0.99

0.999

2.849 2.490

4.408 3.856

5.409 4.730

7.334 6.411

9.540 8.348

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

1.617

2.582

3. t88

4.353

5.686

2.252

3.496

4.287

5.8tt

7.566

9 10

t.532 1.465

2.454 2.355

3.031 2.911

4.t43 3.981

5.4t4 5.203

2.085 1.954

3.242 3.048

3.97t 3.739

5.389 5.075

7.014 6.603

II

i.411

2,275

2.815

3.852

5.036

1.854

2.897

3.557

4.828

6.284

t2 t3

1.366 1.329

2.2i0 2.155

2.736 2.670

3.747 3.659

4.900 4.787

t.77t 1.702

2.773 2.677

3.410 3.290

4.633 4.472

6.032 5.826

t4 15

1.296 1.268

2.108 2.068

2,6t4 2.566

3.585 3.520

4.690 4.607

i.645 1.596

2.592 2.521

3.189 3.102

4.336 4.224

5.651 5.507

16 17

i.242 1.220

2.032 2.00i

2.523 2.486

3.483 3.415

4.534 4.471

1.553 1.514

2.458 2.405

3.028 2.962

4.124 4.038

5.374 5.268

18

1.200

1.974

2.453

3.370

4.4i5

t.48i

2.357

2.906

3.961

5.167

19 20

1.183 1.167

1.949 1.926

2.423 2.396

3.33i 3.295

4.364 4.319

1.450 1.424

2.315 2.275

2.855 2.807

3.893 3.832

5.078 5.003

21

1.152

t.905

2.371

3.262

4.276

1.397

2.241

2.768

3.776

4.932

22 23

1.t38 1.126

t.887 1.869

2.350 2.329

3.233 3.206

4.238 4.204

1.376 1.355

2.208 2.179

2.729 2.693

3.727 3.680

4.866 4.806

24 25

1.114 1.103

1.853 1.838

2.309 2.292

3.181 3.158

4.171 4.143

1.336 1.319

2.154 2.129

2.663 2.632

3.638 3.601

4.755 4.705

30

t.059

t.778

2.220

3.064

4.022

t.249

2.029

2.516

3.446

4.508

35

1.025

1.732

2.166

2.994

3.934

1.195

1.957

2.431

3.334

4.364

40 45

0.999 0.978

1.697 1.669

2.126 2.092

2.941 2.897

3.866 3.811

i.154 1.122

1.902 1.857

2.365 2.313

3.250 3.181

4.255 4.168

50

0.961

1.646

2.065

2.863

3.766

1.096

1.821

2.296

3.124

4.096

Section

Hl

1 May 1972 Page 16 single

value

above

which

percent

of the population

Choose

P

the

we may will

the proportion,

confidence

predict

with

and

Let

_/

s

K

in Table

appropriate Compute

n,

H1-4

-f,

xL = £

same

and -

procedure

the primary

Fbry)

that

99

0.99

= 0.90 lb

= 8.24 i0,P

= 0.99

,'y

:

0.90

K = 3. 532

P

xL

in Ref.

properties

= 546.6

lb

Thus

we are

90 percent

that

99 percent

material

ultimate

strengths

for

product

"A" will

be above

546.6

lb.

[ 6] to determine

(Ftu,

of the

confident

Fty,

the

Fcy,

"A"

and

Fsu,

"B"

values

Fbr u,

and

•

1.2.2.1

Properties A log-normal

curve

T

n=

Ks

strength

LoG-Normal

The

for

is used

1.2.2

from

=

coefficient

s

for

P

= 575.7

Look up

confidence

lie.

Compute

This

90 percent

the

use

log-normal and

has

Probability

distribution

of the logarithm distribution the

properties

Curve

is the of a variable

curve

is bell

of a normal

frequency

distribution

rather shaped

than like

distribution.

the the

curve

variable normal

resulting itself.

probability

J

Section

HI

1 May 1972 Page 17

f

1.2.2.2

Estimate Similar

of the ments

to the

population

Performance

normally

mean,

distributed

m , is simply

variables,

the

the best

arithmetic

mean

single

of the

estimate

measure-

[ 3].

Iogm

1.2.2.3

1 = -n

_IogR

Example Using

Table

of Average

the

H1-5,

fatigue

for

mean

HI-5.

Specimen

stress

of the

parent

universe

Fatigue

Life

of Product

level

(50 ksi)

'_B"

Log N.

1

1

13000

4.1139

2 3

13100 24000

4.1173 4.3802

4 5

28000 40000

4.4472 4.6021

=

5

: :

= antilog

_logN.

4. 3321

similar

estimate

of the standard

variance

[3l

•

1

= 21. 6607

1

,log Ni : : (21.6607;

Estimate Again,

given

[ 3].

N. (Cycles)

1

1.2.2.4

a constant

1

log N

N

(3)

4

lives the

Table

n

_ Iogx. i=l

Problem

estimate

Test

n

= 4.3321

= Mean

log value

= 21485cycles

of Standard to normally deviation

Deviation distributed is thc

square

variables, root

of the

the best estimate

unbiased of

in

Section H1 i May 1972 Page

logx S_

a_

i

2

Interval Confidence

The range mean

intervals

with

of 50 percent.

logx

i

-

1)

(4)

Estimates

of a percentage

coinciding

_ i=l

n (n 1.2.2.5

_

=

i8

may

be determined

of data points

the

population

The range

may

mean,

is computed

for log-normal be computed

which

occurs

distributions.

for the sample with a confidence

as

m

logx

where

kp

interval)

= logx

+ kps

is defined

in Fig.

may be computed

log x L = o_

where size,

kpT

is defined

the percentage

confidence

value.

,

H1-6.

(5)

A lower

limit

(one-sided

confidence

as

-

kpTs

in Figs. of data

,

Hi-7

points

(6)

through occurring

HI-9 above

as a function the lower

of the sample limit,

and

the

Section t May Page

H1 1972 19

0.60 _

w

0.40

n, L E w

o ..<. 0.2o K

¢( < w

0.1o

0

1

2

3

4

5

h, FIGURE

H 1-6.

FACTOR

FOR

DETERMINING

/

DATA

POINT

RANGE,

CENT

--...

P " 99 PERCENT I "-'-----P " 90PERCENT

_ _--.-__

6

P -

7

10

,

50 PERCENT

20

30

5Q

NUMBER OF SAMPLES

FIGURE

ttl-7.

PROBABILITY

FACTOR

kpT

FOR

7

= 0.95.

100

Sc,(.ti_m

]!1

1 May

1.),_

14

12

m

P " NPIERCENT

"--------.----.

P - IIOPIERCIENT I

I P - 60 PERCENT

0 3

S

7

10

IO

NUMIEIt

l,'l(;I.;Itl,:lll-x.

30

60

100

OF SAMPLES

I'I{()]LA.IHI.I'I'Y

I:AC'I'()]|

k

().!)().

l.'()l{_, 1)-_,

4.0

P - 99.9

3.O

PERCENT

.,,_ 2.0 a

P-95PERCENT

P-90PERCENT

1.0 P-80PERCENT

,i

ii

"'

"'

I

"

4

FIGURE

"

"

I II NUMBER

111-9.

"

I

'

12 OF SAMPLES

IHtOI3ABILITY

l,'AC'r(.)ll

I 16

ki) Y

"

"

"

'

1 P'SOPERCENT 20

FOI{

y

=

0.50.

Section

H1

1 May Page 1.2.2.6

Example Using

percent

Problem

the data

of all points logN s

From

=

0. 213

Figure

H1-6,

max

log Nmi n

Conclude:

interval confidence

kp

=

the

range

within

which

68

1.0

= 4.3321

*

0.213

= ,;.545

N

= 4. 3321

-

0.21;I

= ,:. 11.9

Nmi n

to

Xu

interval

xL

P

percent

of the

the

determine

[ 3].

The

for

H1-5,

4. 3321

=

logN

5

in Table fall

1972 21

68 percent

is :l

pol)u]:',ti,)n;

of the

100 (1

i.e.,

i)opulation

-

lnax

fall

35100cycles

=

13700

a)percent

we may

will

=

confidence

assert

within

cycles

with

13700

50 percent

< N < 35100

cycles. 1.2.2.7

Example Using

of all

the

points

Choose

data

in the

desired

Problem

6

in Table

total

II1-5,

population

confidence

will

determine

the

lie with

above

a confidence

I__t 1

level

life

-

o_ = 0.95 c_ = 0.05

Choose which Compute

P should

percent exceed

log x

of data

points

the lower

Let

P

=

90 percent

limit. logN

= 4.3321

which

90 percent

of 95 percent.

Section i May Page Compute

Find

s

s

rk_y in Figure

= 90 percent,

n = 5

Compute

logN

L =

LogN

-

logN

kpys

kpy

We are

100

(1

-

a)

percent

y

=

95 percent,

= 3.35

L = 3.61855

N L = 4155 Conclude:

1972 22

= 0.213

for P

HI-7

HI

confident

that

cycles P

percent

of the

?

data points

dence

that

are

greater

90 percent

than

xL

of the lives

; i. e.,

are

we may

greater

assert

than

4155

with

cycles.

95 percent

confi-

Section

H1

1 May 1972 Page 1.3 HI-I.

REFERENCES Hald, John

A. : Statistical Wiley,

Theory

Structures

Manual.

Convair,

HI-3.

Structural

Methods

Handbook.

H1-4.

H1-5.

no.

M. G. :

Standards

Handbook

A.

Airframe

H1-6,

00.80.39,

Natrella,

Liu,

91,

Fracture

AFFDL

TR

70-144,

Texas.

Aircraft

Statistics,

August

1,

Corporation,

and

GOVERNMENT

Bureau

of

1963.

of the

of Aircraft December

National

in Fracture

Proceedings

of Defense, ¢t U. S.

Worth,

Temco

Variation

Materials. and

Department

Application.

1956.

F. : Statistical

Materials

Fort

Experimental

Fatigue

Metallic

Engineering

1952.

HI-2.

Report

with

Touglmess

Air

Structures

Force and

Data

of

Conference

on

Materials,

1969.

Elements

for Aerospace

MIL-HDBK-5A, PRINTING

OFFICE:

February

Vehicle 8,

1972-74,5-386/Region

Structures. 1966. No.

4

23

!