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1/234

NASA

CONTRACTOR

REPORT

NASA CR-620

- sa :

1 9 9 6 0 4 1 0 0 7 9

ANALYSES F OMPOSITE TRUCTURES

by

Stephen

W.

Tsai,

Donald F ,

Adams,

and Douglas

R. Doner

K

Prepared

by

PHILCO

ORPORATION

Newport Beach,

Calif.

for

Western

Operations

Office

"Ŝ ved

ot P^lic

dea

I&trib-aüon Unlisted

NATIONAL ERONAUTICS ND

PACE

DMINISTRATION

• WASHINGTON,

.

.

•

NOVEMBER

m

•„


2/234

D I S C L A I M E R

N O T I C E

THIS

OCUMENT

S EST

QUALITY AVAILABLE. HE

COPY

FURNISHED

TO

DTIC CONTAINED

A IGNIFICANT

UMBER

F

PAGES

HICH

O

OT

REPRODUCE LEGIBLY.


3/234

NASA CR-620

ANALYSES

OF COMPOSITE

STRUCTURES

By Stephen

W .

Tsai,

Donald

F. Adams, and

Douglas

R. Doner

Distribution

of

this report

is provided

in

the interest

of

information

exchange. esponsibility

for

the

contents

resides n

the

uthor or rganization

that

prepared

it.

Prepared under

Contract

No.

NAS

7-215

by

PHILCO CORPORATION

Newport

Beach,

Calif.

for

Western

Operations Office

NATIONAL

AERONAUTICS

AND SPACE ADMINISTRATION


4/234

F O R E W O R D

This s an annual report of the

work

done under

National

Aeronautics

and

Space

Administration Contract NAS

7-215,

entitled

"Structural Behavior

of

Composite

Materials,

"

for

he

period

January

1965

o

January

1966.

The

program

s monitored

by

Mr. Norman

J.

Mayer,

Chief,

Advanced

Struc-

tures and Materials Application,

Office

of

Advanced

Research

and

Technology.

The authors wish

o acknowledge he

ontributions of their consul-

tants

Dr. G . S. Springer of the Massachusetts

nstitute of Technology,

Dr.

A.

B. Schultz of the University

of llinois,

and

Dr.

H .

B.

Wilson,

Jr.

of the University of

Alabama.

The assistance of Mr. R. L. Thomas and

Mrs.

V. A.

Tischler

of

Aeronutronic

s

also

gratefully

acknowledged.

Particular recognition s given o Dr. Wilson for his

work

n estab-

lishing

he

undamental concepts

upon

which

he

periodic

nclusion problems

of Sections and 4

are based.


5/234

AB STRACT

r

[The

tiffness nd

trength analyses f

composite materials

pre-

viously

presented

have

been eviewed

and

extended

o

cross-ply

and

helical-wound

cylinders,

as

well

s

lat

laminates ̂

Consideration

has een

given

o

he

omposite ehavior fter nitial yielding,

including he

nfluence

of filament crossovers n helical-wound cylinders. In doing o,

a

modified

"netting analysis" has

een

used

n conjunction

with

he

ontinuum

analysis

to predict both initial

yielding

and

post-yielding ehavior.

Cylinders

were

assumed o

e

ubjected

o various

oading

ondi-

tions,

including

axial

tension

and compression,

torsion,

and

internal

pres-

sure. Theoretical esults were hen compared

with

experimental ata

obtained

using

glass-epoxy

composites.

Investigations

have

also

een

made f

he elative ontributions

f

the constituent material

properties o

he

gross ehavior f

a unidirectional

fiber-reinforced

composite

when

ubjected

o

various

oading

onditions.

Theoretical

values btained

for he

prediction

of

he

tiffness

nd

trength

of he omposite

s

unction

of constituent

properties

have

been compared

with

experimental

data

obtained

using both

glass-epoxy

and

boron-epoxy

systems.

Complete digital omputer

programs,

developed

n

conjunction

with

the

trength analyses f

flat

laminates

nd

laminated

omposite cylinders,

and

he

nvestigation

of tress distributions

n

he

fibers nd

matrix

of

a

composite

ubjected

o

either

ongitudinal

hear

or

ransverse

normal

oading,

are presented

n

Appendices

A,

B,

and

C.


6/234

CONTENTS

SECTION AGE

1

NTRODUCTION

2

TRENGTH ANALYSIS

Anisotropie

Yield

Condition

Strength of

Laminated

Composites

1

Cross-Ply Composites 3

Helical-Wound ubes

9

3

ONGITUDINAL

SHEAR

OADING

Introduction 9

Description of

Problem 0

Method

of

Analysis

2

Solution

echnique 6

Presentation

of

Results

7

4

RANSVERSE

NORMAL

LOADING

Introduction

3

Method

of

Analysis

6

Discussion

of Results 1

5

ONCLUSIONS 7

Stiffness Ratios

8

Fiber

Volume 0

Fiber

Cross ection 2

Filament Crossovers 3

Future

Research 4

Vll


7/234

CONTENTS

Continued)

SECTION AGE

REFERENCES 97

APPENDIX

A 99

APPENDIX

B

25

APPENDIX C

65

Vlll


8/234

ILLUSTRATIONS

FIGURE

PAGE

1

omparative

Yield

Surfaces

2 ield Surfaces

or Glass-Epoxy

Composites

3 niaxial Properties f Glass-Epoxy Composites

.

2

4

etting Analysis Notation

1

5

lass-Epoxy

Cross-Ply Composites

ubjected

o

Uniaxial

Loads 6

6

ross-Ply

Pressure

Vessels 7

7 lass-Epoxy Cross-Ply

Pressure Vessels,

m

0.4 4

8

lass-Epoxy

Cross-Ply Pressure Vessels,

m 1.0 5

9

lass-Epoxy

Cross-Ply

Pressure

Vessels,

m

.

6

10

ypical

Pressure Vessel

Failures 8

11

elical-Wound ubes, Glass-Epoxy

0

12 niaxial ension

Test

1

13 niaxial Compression Test

2

14 orsion Test

3

15

niaxial

ension

est, Glass-Epoxy

Helical-

Wound

ubes

5


9/234

ILLUSTRATIONS

(Cont inued)

FIGURE

16

niaxial

Compression

Test,

Glass-Epoxy

Helical-Wound

ubes

6

PAGE

17

ure Torsion Test,

Glass-Epoxy Helical-

Wound ubes

7

18

nternal Pressure Test,

Glass-Epoxy Helical-

Wound ubes

8

19

elical-Wound

ubes

After Failure

2

20

niaxial

ension

Test

of

a

3-Inch

Diameter Glass-

Epoxy Helical-Wound ube 3

21

niaxial

ension

Test

of

a

1-1/2

nch

Diameter

Glass-Epoxy

Helical-Wound ube 4

22

orsion

Test

of

a

1-1/2 nch Diameter

Glass-

Epoxy Helical-Wound ube

6

23 nternal Pressure Test

of

a 1-1/2 nch

Diameter

Glass-Epoxy Helical-Wound

ube

7

24 omposite Containing

Rectangular

Array

of

Filaments

mbedded

n n

Elastic

Matrix

1

25 irst Quadrant

of

he

Fundamental

Region

Longitudinal

Shear Loading

2

26

hear

Modulus

G)

nd

Stress

Concentration

Factor SCF)

or

Glass-Epoxy

Composites

Subjected

o n

Applied

Shear Stress f 8

27

omposite Shear

Modulus

or

Circular Fibers n a

Square

acking

Array

9

28

omposite Shear Modulus

or

Boron

Fibers

s

Function

of

Matrix Shear

Modulus

nd

Fiber

Volume 1

29

omposite Containing a Rectangular

Array

of

Filaments

mbedded

n n

Elastic

Matrix

and

Subjected

o

Uniform

Transverse

Normal

Stress

Components

t

nfinity

4


10/234

I LLUSTRAT IONS

Continued)

_

FIGURE

AGE

30

irst

Quadrant

of

he

Fundamental

Region

77

3 1

ethod

of

Combining

Problems

1,

2,

and o

Obtain

Desired

Solution 82

32

omposite

Transverse Stiffness

or

Circular

Fibers

in a Square

Array

84

33

omposite

Transverse Stiffness or

Boron

Fibers

as

Function

of

Matrix Shear

Modulus

nd

Fiber

Volume

85

B-l irst

Quadrant of

the

Fundamental

Region

Showing

Typical

Grid Lines

nd

Notation Used

126

B-2

ode

dentification Numbering System

128

C-l

irst Quadrant of

he

undamental

Region

Showing

Typical

Grid Lines nd Notation

Used

166

C-2 ode Identification

Numbering System

168

XI


11/234

NOMENCLATURE

B..

_

B

B ' . ' ~ .

=

B

B..

=

B

A..

=

A

In-plane tiffness matrix,

lb/in.

A.'. = A' =

Intermediate in-plane

matrix, in./lb

ij

A.'.

=

A' - In-plane ompliance

matrix,

in./lb

a ength

of

he

upper nd ower boundaries

f

he

first

quadrant

of

he

fundamental

region urrounding

one

inclusion,

in.

=

Stiffness

oupling matrix,

lb

=

ntermediate oupling

matrix,

in.

=

ompliance oupling

matrix, 1/lb

b

ength

of he left and right boundaries f

he

first

quadrant

of

he

fundamental

region urrounding

one nclusion,

in.

C..

=

nisotropie

tiffness

matrix,

psi

ij

D.. = D

=

lexural

tiffness

matrix,

lb-in.

D..

=

D

ntermediate

flexural matrix,

lb-in.

D . =

D

1

=

lexural compliance

matrix, 1/lb-in.

E

odulus

f

elasticity, psi

E, = omposite axial

tiffness,

psi

E

?

- =

omposite

transverse tiffness,

psi

Xll


12/234

G hear

modulus,

psi

H 7.

' =

Intermediate

coupling,

matrix,

in.

h

otal

thickness,

in.

M.

=

Distributed bending

and

wisting) moments,

lb

T

M = Thermal moments,

lb

l

_

M.

=

Effective moment = M.

+

M

l

m

os

or

cross-ply

ratio

total

thickness

f

odd

layers

over

that

of

even

layers)

N.

=

Stress esultant,

lb/in.

l

N. =

Thermal

tress

esultant,

lb/in.

l

_

N.

= Effective tress

esultant

= N. + N.

l

N,

tress

n

he

direction

of

he

fibers

per

nch

of

thickness,

f

b/in.

n

in

9,

or

total

number

f

layers

P

nternal

pressure, psi

R

adius,

in.

r

atio

of

normal trengths

=

X/Y

S hear trength

of

unidirectional composite,

psi

s hear

trength ratio = X/S,

or tandard

deviation

of

fiber

trength

SCF

tress

oncentration factor

T

emperature,

degree

F

u,

v,

w

isplacement

components,

in.

v,

ercent

fiber ontent by volume

X

xial

tensile

trength

of

unidirectional

omposite,

psi

xiv


13/234

X' xial

compressive

trength of

unidirectional

composite,

psi

Y

ransverse

tensile

trength

of

unidirectional

composite, psi

Y'

ransverse

compressive

trength of

unidirectional

composite,

psi

z

istance

as

measured

from

he

middle

urface,

in.

C i.

hermal xpansion coefficient,

in./in./degree

ß atrix

effectiveness n "shear

transfer"

C.

train

component,

in./in.

e.

n-plane

train

component,

in./in.

l

Fiber orientation

or lamination angle,

degree

Jt .

Curvature,

1/in.

V Poisson's

atio

a. Stress

omponent, psi

C T - p . = Fiber bundle

trength,

psi

O

Average

deviation

of

he

fiber trength

T..

= Shear tress, psi

SUBSCRIPTS

f

fiber

m

matrix

i, j, k =

1,

2,

...

6

or

x,

y,

n 3-dimensional

pace,

or

1,

2,

6

or

x,

y,

n

2-dimensional

pace


14/234

SUPERSCRIPTS

k

kth

layer

of

a

laminated

composite

-1

Inverse

matrix

H

Hoop

ayers odd

ayers)

of

a cross-ply cylinder or

pressure

vessel

L Longitudinal

ayers

even ayers) of a

cross-ply

cylinder or

pressure

vessel


15/234

SECTION

INTRODUCTION

This s

continuing

attempt

o evelop

a ational approach o

he

design

and

utilization

of composite materials n tructural applications.

1

2*

Previous

efforts '

were concerned

with he establishment

of he

ndepend-

ent

elastic

moduli

and

trength

parameters from

he macroscopic viewpoint.

The current

effort

s

oncerned

with

he

evelopment

of

guidelines

for he esign

of composite tructures.

The

determination

of

he deforma-

tion

and

load-carrying apacity of filamentary tructures

s

utlined.

Helical-wound

ubes

ubjected o various oading onditions re xamined n

detail.

The

ehavior

of

his tructural element

s expressed

n

terms f

various amination

parameters ncluding he

helical wrap

angle, number

of

layers,

etc.

and material

parameters such

as

he

properties f he

on-

stituent materials, the cross-sectional

hape

f

he

filaments,

etc. The

present

theory

of

design

of

composite

materials an

be pplied o

he nal-

ysis

nd

design

of

filamentary

tructures.

The

weak

ink

n

a fiber-reinforced composite, as xhibited

by he

initial

yielding,

is losely

associated with

he ow

trength

levels ttainable

in

a

direction

transverse o

he fibers nd

n

hear.

For this

eason,

the

transverse

and hear

properties f

a

unidirectional omposite re

analyzed,

the esults

providing

information

needed

n

mproving omposite materials.

References

re

listed

at

he

nd of

this eport.


16/234

The present heory of

design

of composite

materials

s only prelimi-

nary.

A

number

of

refinements

and

appropriate

experimental

verification

remain to

be

explored.

In

particular,

inelastic

behavior

both

on

the

macro-

scopic and

microscopic evels

and

the

effect

of

filament

crossovers

are

wo

problems

hat

deserve

mmediate

attention. It s hoped that

as

he

heory s

improved, the extent

of

empiricism

can

be

substantially

reduced

n

he de-

sign

and

utilization of

composite

materials.


17/234

SECTION

STRENGTH

ANALYSIS

Anisotropie

Yield Condit ion

The anisotropic yield

condition,

as

reported in Reference Z, is

derived

from

a generalization

of

he

von

Mises yield condition for so-

3

tropic materials, It

s assumed

hat

he yield

condition

is a

quadratic

function

of

he

tress

components

Zfdry) = F(o

y

of

+

G (a

z

aj + (a

x

a/

(1 )

+

2LT

2

+

ZM

2

+

ZN

2

=1

yz x

y

where

F,

G, H, L,

M,

N are

material

coefficients

characteristic

of

he

state of anisotropy, and x,

y,

z,

are he axes of he assumed orthotropic

material

symmetry.

Equation

1)

reduces

o

he

von

Mises

condition

if

F =

G

=

H

=

l/6k

2

M

= N

=

1/Zk

2

where k

is a

material

parameter

governing

he

yielding

of

sotropic

materials.


18/234

Since he

composite material

of

present nterest

s

n

a

orm

of rela-

tively hin

plates,

a

state

of

plane

stress s assumed. Equation

1)

can be

reduced o:

\x)

X

Y

Y

= 1

(2)

The

validity of his yield

condition

has

been

demonstrated in Reference Z ,

using unidirectional glass-epoxy composites ubjected

o

ensile

oads.

For

he strength analysis of

a

filamentary

structure

ubjected

o

combined

oading,

compressive

properties

must

be

known.

Analogous o

the

ensile

trengths

X

and

Y ,

the

compressive

trengths

X

and

Y'

are

determined from 0- and 90-degree

pecimens ubjected

o

uniaxial com-

pressive

oads,

respectively.

Shear has no

directional property,

hence,

S = S'.

It s assumed

hat he anisotropic

yield

condition

remains applicable

for materials with properties different

n ension

and compression. It

s

only necessary

o

use

he

principal

strengths

compatible with he

prevailing

stress

components,

i.e.,

tensile

trength

for

positive

normal stress and

compressive

trength

for negative

normal

stress.

This method of

aking

into account different ensile and compressive properties ollows hose

used

4

5

previously

by other nvestigators. '

Equation Z) can now be written m

four

forms

corresponding

o

he our

quadrants of

he

O^

stress

pace.

The quadrant

descriptions

are as ollows:

Axial

T

ransverse

Strength

Quadrant

1

C T

x

a

positive

Strength

St

rength

Rat

io

positive

X

Y

r

l

=

X/Y

Z

negative

positive

X'

Y

r

z

X'/Y

3

negative

negative

X'

Y'

r

3 =

X'/Y"

4

positive

negative

X

Y'

r

4

=

X/Y'


19/234

In

erms

of hese definitions, the ield condition

given by Equation Z)

becomes, in he order of he corresponding quadrant:

(̂ ) T;

# ff

(?f)

2

(F)

•

5

>

ft)

2

t f f

(

£)' (

#

'

The

igns

or

he

principal strengths are

always

positive;

hose

or he

stress

components

are positive or negative, corresponding

o

he

appro-

priate

quadrant n

he

stress pace. Diagrammatically,

the yield

surface

can

be

represented

in

dimensionless

orm

as

hown

in Figure 1.

For

unidirectional glass-epoxy composites

v£

=

70%),

r =

X/Y

=

150/4

=

37.5

r

2

= X'/Y = 150/4 = 37.5

r

3

=

X'/Y'

= 150/20 = 7.5

r

4

=

X/Y'

=

150/20

=

7.5

This

s

represented

by he solid curves n

Figure 2.


20/234

ANISO T RO PIC

YIELD

S U R F A C E S

X

_

=

1

(VON MISES)

=

2

Figure h Comparat ive

Yield

Surfaces


21/234

Figure

2. Yield Surfaces or

Glass-Epoxy

Composites


22/234

The yield

conditions

of Equations

2 )

hrough

6) apply

o

an ortho-

tropic

material

n he directions of ts

material

symmetry axes. For uni-

directional composites,

the

ymmetry axes

are parallel and perpendicular

to

he fibers. If

he ibers are oriented other

han

0- or

90-degrees with

respect

o

he

externally

applied

oad,

the

applied

stress

components

r,

i =1,2, 6, must

be ransformed o

he

symmetry axes, i

x,

y,

s, before

the

yield condition can be applied.

The

usual

ransformation

equation

for

stress

components,

in

matrix form,

is

m

mn

n 2mn

2 2

-mn

n

-n

(7 )

For

uniaxial ension,

0 " ,

=

positive, 0

?

=

O,

(8)

From Equation

7),

2

x

s

(9 )

Substituting hese values nto he appropriate yield

condition,

Equation 3),

one obtains:

4

m

+

(

2

2

2 24

/v/

.2

s,

- 1 m

n

+ r

,

-

(X/O

(10)

which

is

dentical

with

Equation

9)

of

Reference

2,

where

Sl

s

=

X/S,

r

l

= r

=

X/Y


23/234

In he

ame manner, for uniaxial compression, the appropriate

yield

condi-

tion equation

is

4

m

+

(s

Z

-

l]

m

2

n

+

r

2

n

4

=

(X'/o^)

2

11)

where

3

= s = X'/S, r

3

= r =

X'/Y'

For

pure

shear,

the yield

condition

corresponding

o

he

econd

or

fourth

quadrant will

be needed. This can easily

be

derived

by aking O, as

the

only

nonzero

stress

component. If

r,

nd

r

. are

different,

which

is

usually he case, the

hear

strength

of a

unidirectional

composite will

have

different

values

depending

on

he direction

of he applied shear, i.e. posi-

tive

or

negative

shear.

In summary,

the

nitial

yielding of

a

unidirectional

composite, when

subjected

o

a

complex

state of stress, is

governed

by

one

of

our

possible

yield

conditions.

The appropriate

condition o

be used is determined

by

he

signs of

he normal

stress

components. If he ensile and compressive

strengths

are equal, the our conditions reduce

o

one equation; such is he

case

n

Equation

4) of

Reference

.

Compressive

Properties

In

a

previous tudy,

he principal strengths

were

imited

o

ensile

loading only.

owever,

n

he

trength analysis of

a

structure

ubjected

o

combined

oading,

he

compressive properties of unidirectional composites

must also be known.

Compressive

elastic

moduli have been found

o

be approximately

he

same

as

ensile

moduli or

glass

-epoxy

composites and

boron-epoxy

composites.

Compressive

axial

and ransverse

strengths,

X

1

and Y',


24/234

respectively,

can be

determined

by he

compressive oading of 0- and

90-degree

specimens.

Compression

ests

are

known

o

be difficult

o

perform.

Test results

often

are affected

by he

geometric

configuration

of

he

pecimen. Competing modes

of failure,

i.e.,

buckling and

strength,

are

operative.

As an indication

of he difficulty of direct

measurement

of he com-

pressive axial

strength,

X,

he

numerical

value

of

X or glass-epoxy

composites has

been reported

as

anywhere within a

range

of

from

00

o

Z 50

ksi,

depending upon

he

est

method used.

In flexural ests of 0-degree

specimens, which include

a

hoop-wound ring pin-loaded at diametrically

opposite points,

most

failures are

of

he

ensile

ype. It

appears reason-

able

o

assume

hat

he

compressive strength

is

at

east equal o,

if

not

higher

han,

the

ensile

strength.

In

he

present

work,

a

value

of

50

ksi

is

assumed

for

both

he ensile and compressive strengths of he glass -

epoxy

composite.

This value s undoubtedly conservative.

The compressive

ransverse

strength Y' s

comparatively

simple

o

determine because of ts ow

numerical

value. For glass-epoxy composites,

with v

f

=

70

percent,

the

value

of Y'

s

between 16

and

24

ksi. The

ower

values

were

obtained

using

pecimens having rectangular cross

ections;

the higher values,

circumferentially wound

ubes

with over-wound (rein-

forced)

ends.

No

gross

buckling

of

he

specimens

was

observed.

Using

he

experimentally determined

principal

strengths,

X =

15 0

ksi

2 0 ksi

-

6 ksi

10


25/234

from

which,

r

3

=

X'/Y'

=

150/ZO

=

7.5

s

3

=

X'/S

= 150/6 =

2 5

one

can

determine,

using

Equation

11),

the

uniaxial

compressive

trength

0\ s a function of fiber orientation. The

resulting curve,

together with

experimental

data,

is

hown in Figure

. The corresponding uniaxial stiff-

ness and

ensile

trength are also shown. The ensile and compressive

stiffnesses are practically

dentical when

he

train is

mall,

i.e. in he

order of 0.

percent.

Strength of

Laminated

Composites

For

he

ake of

completeness,

the

trength

analysis

of laminated

composites described in

Reference s

ummarized

here.

Essentially,

the

strength

of

materials approach is used, whereby

he normals

o

he middle

surface

remain

undeformed

during he

tretching and bending of he compos-

ite plate. The otal strain

at any

point

n he

plate

s

defined

as

€ .

=

e°

zx.

12)

It

s further assumed hat each

constituent

ayer

of

he aminated composite

is

mechanically

and hermally

anisotropic,

i.e.,

o

{

=

y

e.

a.T) 13)

where , =1,2, and 6.

11


26/234

10

Q.

o

CO

CO

LL

LL .

X


27/234

Equation

(13),

when integ

ra

ted across

the thickness

of

he

la

minated

compo

site, becomes:

N,

=

N.

T

+ :

= A..

o

+

B..

x

i

(14)

l

l

i

ij

J

ij

J

_

o

M. = M. + M. = B.. e. D..

.

i

j J

j

J

(15)

where

r

h/2

(N . .

.)

= f a

(1 ,

-̂h/2

z)

dz

(16)

(N ;

r

,

T

)

f

h/2

C.a.T (1,

z)

dz

h/2

J

(17)

(A..,

B..,

D..)

= f

ij ij ij /_

h/2

C.

(1,

z,

z

2

)

dz

h/2

(18)

Equations 14) and 15)

are he

basic

constitutive

equations

or

a

aminated

anisotropic composite,

taking

nto

account

equivalent hermal loadings.

The

stress at

any

location across he

hickness

of

he

composite

can

2

be

expressed

in

he

ollowing manner.

Having established hat

N

A

|

=

—

-4-

M

B

D

(19)

13


28/234

then,

by

matrix inversion,

M

A* ' B*

_

i

H*

|

D*

I

N

(20)

r < n

H'

D'

N

M

(21)

where

A

=

A

A B̂

H

= BA

D = D

BA

B

(22)

A ' = A" B"D"~

H

B'

= H' =

B

D

*-l

D' =

D

*-l

Substituting

Equation

21)

nto

12)

e.

= (A .

+

zB.'.)

N.

+

B.

.

+

D .)

M.

i

j

j

J

j

j

J

(23)

14


29/234

From

Equation 13) ,

the

tress

components or he

kth

layer

are:

a

(k)

=C

<

k)

(. -o/

k)

T ) 24)

cf

k

>

(A .

+

zB'

)

N,

+

(B .

+

zD . M. -

a

(k)

T

jk

k' k

k

k'

k

This s he most general expression for stresses as unctions of stress

resultants, bending moments, and emperature. The

same

material

coeffi-

cients

A',

B',

and

D',

as reported

n

Reference 2, can

be used

for

he thermal

stress

analysis.

This

imple

ink

between

he

sothermal

and

nonisothermal

analyses s achieved

by reating hermal

effects

as equivalent mechanical

T

loads,

e.g.

N.

and

M.

n

Equation

17).

Determining

he

evel

of

external

load

N.

and/or

bending

moment M.- that

will

nitiate

failure

n

one or

several

of he constituent ayers

s

not

a

straightforward

calculation. This s due o

the

act hat he

stress

components . i =1,2,6) computed from Equation

(24)

must

be ransformed into he

x-y

coordinates i

=

x,

y,

s),

which

repre-

sent he

material

symmetry

axes, before

he

signs

of

he

stresses

r

and

a

whether

positive

or

negative,

can

be

determined.

Only

after

he

igns

of a

and are known, can

he

proper

yield

condition

be

selected.

The

actual

numerical

method by

which he maximum

allowable

oadings

N.

and/or M.)

are determined is outlined in detail in Appendix

A.

A

cylindrical shell

s one of he

basic

structural shapes. When a

shell

s

ubjected o

homogeneous oading,

e.g.

uniaxial

ension

or

com-

pression, internal or external hydrostatic pressure,

or pure

shear,

the

shell

maintains

ts

shape. There s

no

change n curvature n

either

he circum-

ferential

or he ongitudinal direction.

Because

of his

geometric

constraint

imposed

on cylindrical shells

under

homogeneous

oadings,

the

nduced

stress

distribution

can be

represented by

simpler

relations

han

hose just

outlined.

By

assuming

no

change n curvature

this

can

be

represented by letting

K

=

0),

the

otal

strain

is

now

equal

o

he

n-plane

strain.

This

s

obtained

directly

from Equation

12)

by letting x = . Strain is

herefore

homogeneous

across

he hickness

of

he

hell,

i.

e.

independent

of

z.

15


30/234

c r<

k

>

c<

k

>

A . N,

- v <

k

>T

l

ij

jk

N

k

J

For

cylindrical

hells,

the

tress omponents or

ach

layer are also

constant,

as

iven

by

Equation

13).

Using

quation

20), one

an

immediately

determine

he

in-plane, i.e. total train

caused by N-,

e

°

=

A

iĵ j

25)

The

stress

components

are:

(26)

Being ndependent of

z,

this

equation is considerably simpler han

Equation

24).

The trength

analysis

of cylindrical shells ubjected

o

a

ew fre-

quently

occurring

oading

conditions

has

also

been

programmed. The entire

program

s outlined

n detail

n

Appendix A.

Post-

Yielding Behavior

For most

fiber-reinforced

composites presently available,

initial

yielding s often

dictated

by he

values of

he

ransverse and shear

strengths,

which

are

significantly ower han

he

axial

strength.

The

nitial yielding

introduces failures

parallel

o

he

fibers.

These

failures are

audible

during

the oading and

become

visible

soon

after he

heoretically

predicted

yield

stress s attained.

The post-yielding

behavior

of

cross-ply

composites has been investi-

gated previously.

For

a

cross-ply

composite

ubjected o

a

uniaxial ensile

load in he direction of

he fibers of one of he

onstituent layers, additional

load

can be

upported after

nitial yielding

until

ultimate

fiber

failure

s

induced.

Thus,

initial

yielding

does

not

necessarily

determine

he

oad-

carrying capacity of

a

aminated composite. After one or more ayers have

yielded,

the

ayers of he aminated composite which are

till

ntact must be

16


31/234

investigated

o

ascertain whether or not hey can support

he

prevailing

externally

applied

oad.

However,

in

he

case

of

an

angle-ply

composite

under

uniaxial

ension,

the

still

ntact

ayers

cannot

carry

he existing oad

after

nitial yielding.

For

2

this

reason,

there

s

no post-yielding

oad-carrying capability.

Thus, under

uniaxial

ension

applied along

one of

he material

symmetry

axes

of

he

com-

posite,

ross-ply

composites

can

carry additional

oad

after he nitial

yield-

ing

but

angle-ply

composites

cannot.

A general

heory

or he analysis of

he post-yielding

behavior

of

a

laminated composite s difficult

o

formulate because he material

s

trans-

formed from

a

continuum

o

a

"discontinuum"

on

he microscopic

cale. A

theory will

be

proposed

n his report, using some of he assumptions of

he

conventional

netting

analysis.

It

s assumed

hat, after

nitial

yielding,*

he

unidirectional ayers of

a

composite

can carry ensile

oad

only along he

fiber axis.

To

maintain static

equilibrium,

load ransverse

o

he

fibers

and distortional load must be

carried

by other nternal agencies of he

composite. Such agencies

may be

derived

from

filament

crossovers n

he

case of a helical-wound structure,

or rom

some end constraint

ypical

of

shell-type

tructures, e.g., at he shell-and-head junction.

An

internal

agency

is

necessary

for

he

ransfer

of

the

externally

applied

oads

o

axial

loads

along

he

unidirectional fibers. Before

nitial

yielding,

this

nternal

agency

s

achieved by

he

binding matrix.

The

entire

composite

s

a

continuum.

After

nitial yielding,

failure n

he

matrix and/or

at he

fiber-matrix

interface

s

ntroduced.

Fibers

are

apparently

still ntact.

In

he

case of

angle-ply

composites

under uniaxial oading,

no nternal

agency

A

composite,

after

nitial

yielding

occurs,

is

referred

to

as

a

"degraded'

composite n

Reference

2.

17


32/234

is operative after he nitial failure. Complete failure of the

composite

occurs

immediately

after nitial yielding.

However,

in he case of

cross-ply

composites, n

internal

agency

is

not

needed for ransferring

he

external

load.

Since

some

of

he

ilaments

are

aligned

parallel

o

he

applied

oad,

they

can continue

o

carry

oad until filament

failure

s

eached.

Filament-wound

structures

often

acquire

filament

crossovers during

winding

with a

helical

pattern.

This ype

of

composite

may be

represented

by

an angle-ply with filament crossovers.

The

geometric distribution

and

he

frequency

of

occurrence of filament crossovers or

a

given helical-wound ube

depend

on

he

helical

angle, the

width

of

he roving,

the

diameter

of he

ube,

and

other

process

parameters,

which

may

include

he characteristics of he

winding

machine. In he present nvestigation, it s

assumed

hat

he

effect

of

filament crossovers

ntroduces

wo

factors:

(1 )

As

an internal

agency,

filament

crossovers

provide

additional oad-carrying capacity o helical-wound

composites.

This

trengthening of

angle-ply

composites s exhibited

by higher effective

ransverse

and shear strengths, designated as

Y

and

S,

respectively.

(Z )

In contradiction o he

strengthening

effect

above, ilament

crossovers

will

be

sources

of

stress

concentrations,

since

filaments

can be subjected

o

direct abrasion

among

themselves. Therefore,

crossovers

will

end

o reduce

the axial strength X

of

he

constituent ayers.

Because

of

he existence of filament crossovers, it

may

be necessary

to

reat

helical-wound composites

differently

han

angle-ply

composites.

It

may

be

possible for

helical-wound

composites

o carry a

higher

oad

because

of

he

nternal

agency

generated

by he crossovers. The

ultimate oad

hat

the composite

can

carry

will

be

governed

by

either

he breakdown of

he

internal

agency

which

is

needed

o

ransfer

external oads or

ilament

failure.

18


33/234

In

conclusion, the post-yielding behavior

of

laminated composites s

dictated by he ability

of

he

filaments which are

till

intact

o ustain

con-

tinued oading. This

is

ccomplished n

cross-ply

composites when subjected

to

uniaxial

tension

or

internal

pressure, or example,

by having filaments

aligned

parallel o he

pplied oad.

The

post-yielding

capability

can

also

be

achieved by means

f

an

internal

agency n he omposite, an example

of

which

is

due

o he

filament

crossovers

which

exist

n

woven

fabric

and

helical-wound tructures.

Angle-ply

composites

under

uniaxial oad do

not

have

post-yielding apability because

fibers

re

not

aligned

n he direction

of applied

loads,

nor

is

here

n

internal

agency

for

oad

transfer.

Assuming

that an

internal

agency is vailable

n

a composite uch

hat

he externally

applied

oad,

N.,

i

= 1,

2 , 6,

can

be

transferred

o

an

axial oad,

N,,

in

he

unidirectional layers,

one

an

derive

he elation between he xial tress;

N

f

of

a

unidirectional constituent layer

and

N .

s

ollows.

As

hown

n

Figure a, the equilibrium

of forces etween

he exter-

nally

applied oad,

N^

and

he nduced

oad, N in

he direction

of

he fibers

must

atisfy

he

elation:

N

f

cos ,

A cos

(27)

or

N

f

=

N

1

/c

os

2

a = N

x

/

m

2

28)

In

order

o maintain

equilibrium

n he

2-direction, an

internal

force,

N

?1

,

must be:

N

21

=

N

f

sin

«

A

sin

a

(29)

19


34/234

or

N

2_

A

T

2

= - N

f

in a = -

n

N

f

= -

n

N^m

(30)

Similarly,

in

Figure

b,

the

equilibrium

of

forces between he

externally

applied

oad, N

2

, and he nduced oad,

N

f

, results n

he

condition:

N

£

= N

2

/

n

2

3

1

)

N

1

2

= m

2

N

f

= m

2

N

2

/n

2

32)

In

he case

f

an

externally

applied

hear

force, N

&

,

the equilibrium

condition,

as

hown n Figure c

must

atisfy:

Ü 6

5in

a

N

6

cos

a

_

+

6̂_

33)

A

—A

cos

A

in

a

Amn

N

f

=

+

N

6

/

mn

3

4

)

The internally nduced oad, N ,̂ in this

ase is

ero because

Ü66

N

6

OS

a

_

N

6

Sin

a

=

0

3

5)

A

cos

in

a

Equations

28),

(31),

and

34) how

he

ontribution

of each

externally

applied

oad,

N

p

N

2>

and

N

6>

to

he

xial tress

long

he unidirectional

layer

with an

orientation

of

a

degrees rom

he

1-axis. The

total

axial

stress s,

by

superposition:

i

. n

m

20


35/234

4a

.

2

4c

A

cos

a

4b

Figure

4.

Netting Analysis Notation

21


36/234

This

equation gives he maximum

oad-carrying capacity of

each

uni-

directional

constituent ayer

of

a

aminated composite.

The

ultimate

oad

s

governed

by

he

axial strength,

X, of

each

unidirectional ayer. It s, of

course,

assumed

hat

some

nternal

agency

of

he

aminated

composite,

by

virtue

of

the ilament crossovers, s

capable

of supporting he nternal

forces

N,^

an<

^

N

?

t

east

up o

he axial strength of

he constituent

ayers.

The

validity of his analysis s imited

o

he capability

of

he nternal

agency

o

ransfer

he oad. In

particular,

the ilament crossovers

n

helical-

wound ubes

will be examined as a specific nternal agency. As

tated

pre-

viously, the

effect

of

crossovers

may

be characterized by

effective

ransverse

and shear

strengths, Y and

S,

higher

han

hose of unidirectional composites,

and

by

a reduction

in

he

effective

axial strength

X,

possibly caused

by

he

abrasive

action between

filaments at crossover points. Presently,

the

exact

change n magnitude of

hese

effective

trengths

must be

determined experi-

mentally. Future

nvestigations may

provide

a

basis

or

he

heoretical

pre-

diction

of

hese

values.

In

he

next wo

sections,

detailed

procedures

or

he determination

of

the

oad-carrying

capacity

of cross-ply

and

helical-wound

ubes will

be

out-

lined.

The heoretical

results

will

be compared with

experimental

data,

using E

glass

and epoxy

as

he

constituent

materials.

22


37/234

Cross-Ply

Composi tes

In this paragraph, the deformation and ultimate

trength of

cross-ply composites are discussed.

Theoretical predictions, using the

strength analysis program outlined

in

Appendix A, are made. A sample

problem s presented in detail and numerical results are abulated.

The

theoretical results are

hen

compared with experimental

data.

A cross-ply composite consists of

two

systems of unidirectional

constituent

ayers

with adjacent ayers oriented

orthogonal

o each other.

There

are

wo

amination

parameters:

(1)

the

otal

number

of

layers,

n,

each

layer

may

consist of

one

or

more

unidirectional

plies of

roving, all

of which

must have he same iber orientation),

and

2 ) the cross-ply

atio,

m, which

is

defined

as

he

atio of

the

otal hickness of all the ayers oriented in one

direction

o

he otal

hickness of the

ayers

n

he orthogonal

direction.

For

laminated beams

nd plates, as

reported

in

References

1 and ,

the cross-ply

ratio s

omputed

using

he

ayers

with

degree

orientation,

as

measured

from he

eference

coordinate

ystem, as

he

first

ystem of layers. In

the

case of

cylindrical

pressure vessels, which will be discussed

in

his para-

graph,

the

cross-ply

ratio

s

defined

on

the

basis

of

he outermost

ayer

as

being

n

the

first

system of ayers.

If

the outermost ayer s

a

hoop winding,

which

is usually he case,

then

he

cross-ply ratio s he ratio of

the thick-

ness of all he hoop windings o hat of the ongitudinal windings.

The

deformation

and

ltimate

strength

of

cross-ply

specimens

12

7

subjected

to

uniaxial

ension has been

reported

previously. ' ' However,

a

computational error n he calculation of the

tress at

nitial yielding the

knee) has

been

discovered.

The

corrected theoretical result is as ollows:

Cross-ply

Ratio,

m

nitial

Yielding,

Nj/h,

ksi

0.25 .9

1.00 3.7

2.50

7.6

4.00

9.1

23


38/234

These esults

ave

been omputed

using

he ollowing material

properties, which

are

he same

s

hose eported previously:

c

n

=

c

2

2

2

=

-

9

x

1Q6

s

i

C(

12

=

C

[

Z

=

°-

6

x

0

6

P

si

C

2

2

=

n

=

-

66

x

1Q6

psi

clV

=

C

Z

= 1 . 2 5

x 0

6

ps i

66 6

ü

< ' i? 4?

o^

1

) =aW

=

3.5

x 0"

6

in/in/°F

C C ^ =

a[

Z)

= 11.4

x 0"

6

in.

/in.

/°F

4

1

)

=

<

2

>

=

0

6

T

=

-200°F

(lamination temperature)

n =

3 number

of

layers)

(37)

24


39/234

In addition, the ollowing

trength

data

are used:

X

= X' =

15 0

ksi

Y

=

4

ksi

(38)

Y'

= Z0 ksi

S =

ksi

These

material

properties

re

equired nputs

n

he

strength

analysis

program

outlined n

Appendix A. The

corrected theoretical

results

show

better greement

with

he experimental

esults,

as an

be

een

n

Figure

(which

s

Figure

of

Reference

2

and

Figure

of

Reference

with

he

cor-

rected

initial yielding

curve

hown). The procedure

for he

determination

of

the ost

yielding tiffness nd he

ultimate oad is lso utlined

n

these

references.

Essentially,

post-yield oad carrying

capability

s

possible

for

cross-ply composites because he filaments n he direction

of

he

pplied

uniaxial

oad

can

carry he prevailing

oad.

No

internal

agency

for

oad

transfer s

required

n

this

ase.

The

ultimate oad s btained

when he

axial

trength of

he

unidirectional ayer

s

eached, i.

e.

when

X = 150

ksi.

It

is mportant

o ecognize

hat

he

value

f

he

xial

trength

X

s

experimentally determined. It

s ot alculated from

he

fiber trength

using

the

ule-of-mixtures quation,

from which,

for E glass,

the omputed

axial

strength

would

be

400

/3

=

266

ksi

filament

trength times percent

fila-

ment

volume).

Cross-ply

pressure

vessels

will

now

be

xamined.

A

typical

vessel

is

hown n Figure . The middle hird of

he

vessel is he

est

ection,

the

nds

eing

built

up

from special

aluminum

fittings. The basic esign of

the vessel

was

eveloped

at

Aeronutronic

nder

another

esearch

program.

The ongitudinal

layers

were

aid

up by hand

and

the

hoop

layers

wound

by

machine.

The

ovings sed were 0-end

E glass preimpregnated with

epoxy

25


40/234

" S . 8

O

r-H

^

A

I/)

o

L U

U.

LL.

R

4


41/234

Figure . Cross Ply Pressure Vessels

27


42/234

resin. Two-element

train

ages were bonded o ach pressure vessel with

the

elements

oriented n

he

oop and

ongitudinal

directions.

Internal

pres-

surization was chieved

using hydraulic

il

and

a pumping arrangement

specifically

esigned

for

testing

pressure

vessels.

Internal

pressure

nd

strains

were

ecorded by a

multi-channel

continuous

ecorder.

Using

he

material

properties

isted

n

quations

(37)

nd

38)

n

he

program

outlined

in

Appendix

A,

the

esults

iven

n

able

were

btained

for cross-ply

ratios

f

0.4,

1.

0

nd

4.

0.

*

TABLE

I

CROSS-PLY PRESSURE

VESSELS -INTERNAL

PRESSURE

Cross-ply

Ratio

m)

0.4

1.0

4.0

A

11

0.

158

0.

19 1

0.

273

A

12

(10

-

6

in/lb)-

-0.025

-0.024

-0.026

A

22

0.

244

0.

191

0.

147

N

2

/h

(hoop tress

at

initial

yielding)

9.

3

ksi

12.

8

ksi

14.6

ksi

Yielding

Location

Long.

Long.

Hoop

*The

numerical

values

f

he

A*

matrix

are

lso

given

on

pp

65,

67,

and

69

of

Reference

with he xes

1

nd interchanged.

This

hange

s

necessary

because f he

differences

n

he

efinitions

f he cross-ply atio ited

earlier

n

this

section.

28


43/234

Using a eference oordinate

ystem with he 1-axis n

he

longitudinal direction nd he

2-axis n he oop direction,

strains

long

these xes

an be omputed using

Equation 25):

Longitudinal

Strain =

e

.

=

A.

N +

A.

2

N

- i

A

ti

+

I

2

N

2

Hoop

Strain

=

^

=

A. N. +

A^

z

N

2

=

(i

A

tz

+

A

22

N

2

(39)

(40)

where N, =

N

?

= PR s ssumed

and

P

= internal

pressure,

R

= radius.

Strain

after initial yielding

s

btained

by he usual

neeting

analysis,

which

assumes hat

each

unidirectional layer etains

nly its xial tiffness,

E..,

the

ransverse tiffness nd shear

modulus

being zero. The esulting

relations,

as

shown

n

Equation 9-5)

f

Reference 1,

are:

11

+

m

=

J_Z-1±}

41)

PR

E

ll

h

°

+

m

PR

2

=

x

m

42)

where h represents he

total

wall

thickness

f

he

pressure vessel.

Taking

E..

s 7.8x

0

psi, which s

epresentative

f an

E lass

-

epoxy

omposite

with a

fiber

volume

f

approximately

5

percent,

the

longitudinal and hoop

trains,

before

and

after initial yielding

the knee),

are

obtained

from

Equations

(39)

hrough

42).

These

re

iven

n

able

I.

2 9


44/234

TABLE

I

LONGITUDINAL

AND

HOOP STRAINS OF

CROSS-PLY

VESSELS

Cross

-ply

Before

Yielding

After Yielding

Ratio

(m)

E

ll

h

e°

PR

1

E

H

e

„

PR 2

E

n

^

PR

1

E

n

?0

PR

2

0.4

1.0

4.0

0.42

0.55

0.86

1.81

1.40

1.05

0.

70

1.00

2 .

50

3. 50

2.00

1.

25

The burst pressure

of

he cross-ply

vessels

may be predicted as

follows:

First,

the axial tress

n

he

unidirectional omposite t

he

initial

yielding

must

be

determined.

Assuming that

he

outermost layer

of all

vessels

s n he hoop direction along he

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