An Improved Transverse Shear Deformation Theory for Laminated Anisotropic Plates03615_1982003615

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NASA

TP

1903

c . 1

NASA Technical Paper 1903

An Improved Transverse Shear

Deformation Theory for

Laminated Anisotropic Plates

M . V.V.Murthy

NOVEMBER 1981

NASA I



TECH LIBRARY KAFB, NM

I llllllllllI1I1NASA Technical Paper 1903

An Improved Transverse Shear

Deformation Theory for

Laminated Anisotropic Plates

M. V. V. Murthy

Langley Research Cm ter

Hampton Virginia

NASANational Aeronautics

and Space Adm inistration

Scientif ic and Technical

Information Branch

1981



SUMMARY

An im pro ve d t r a n s v e r s e s h e a r d e fo r m at io n t h e o r y f o r l a m i na te d a n i s o t r o p i c

p l a t e s under bending i s p r e s e n t e d . The t h e o r y e l i m i n a t e s t h e ne ed f o r a n a r b i

t r a r i l y ch os en s h e a r c o r r e c t i o n f a c t o r . F or a g e n e r a l l a m in a t e w i t h c o up l ed

b en di ng an d s t r e t c h i n g , t h e c o n s t i t u t i v e e q u a t io n s c on ne ct in g s t r e s s r e s u l t a n t s

w i th a ve r a ge d i sp l a c e m e n t s a nd r o t a t ions a re der ived . S i m pl i f i ed forms of

t h e s e r e l a t i o n s a r e a l s o o b ta in e d f o r t h e s p e c i a l c a se of a symmetr ic laminate

wi th uncoupled be nding . The gover n ing e qua t ion f o r t h i s sp e c i a l case i s

o b t a i n e d as a s i x t h - o r d e r e q u a t i o n f o r t h e no rm al d i sp l ac e m en t r e q u i r i n g pre-

s c r i p t i o n of t h e t h r e e p h y s i c a l l y n a t u r a l b ou ndary c o n d i t i o n s a l o n g ea ch e dg e.

F o r t h e l i m i t i n g c a s e of i s o t r o p y , t h e p r e s e n t t h e o r y r e d u c e s t o an im pro ved

v e r s i o n of M i n d l i n ' s t h e o r y .

N u m e r i c a l r e s u l t s a r e ob ta i ne d f rom th e p r e se n t t he o r y f o r a n exam ple of a

la m inate d p l a t e under c y l i nd r i c a l bend ing . C om pa rison w i th r e s u l t s from e xa c t

t h r ee - d im e n s i on a l a n a l y s i s shows t h a t t h e p r e s e n t t h e o r y i s m ore a c c u r a t e t ha n

o t h e r t h e o r i e s of e q u i v a l e n t o r d e r.

I N T R O D U C T I O N

C l a s s i c a l b en di ng t h e o r y p ro du ce s e r r o r s when t h e r a t i o of t h e e l a s t i c

modulus t o sh ea r modulus becomes lar ge . Advanced composites l i k e gr ap hi te -

epoxy a nd bo r on - e poq have r a t i o s of a bou t 25 a nd 45, r e s pe c t i ve ly , i n c on t r a s t

t o 2.6 which i s t y p i c a l of i s o t r o p i c m a t e r i a l s . T hes e h ig h r a t i o s r en d er c la s

s i c a l p l a t e t h e o r y i n a c c u r a t e f o r a n a l y s i s of c om po site p l a t e s . S t r u c t u r e s

f rom a dvance d c om pos it es a r e c ons t r uc t e d i n l a ye r e d f or m wi th e a c h l a y e r be ingo r t h o t r o p i c . C o ns eq u en tl y, f o r a n a l y s i s of c o mp os it e p l a t e s , a s a t i s f a c t o r y

t r a n s v e r s e s h e a r d e fo rm a ti on t h e o r y f o r l am i na te d a n i s o t r o p i c p l a t e s i s needed.

C u r r en t s h e a r d e f o rm a t io n t h e o r i e s f o r l a mi n at e d a n i s o t r o p i c p l a t e s h av e

one drawback o r an ot he r . The m o r e r ig o ro u s t h e o r i e s ( r e f s . 1 and 2 ) are u -

bersome because of t h e i r h igh ord e r and inconven ient boundary con di t ion s . Most

o f t h e s i m p l e r , s i x t h -o r d e r t h e o r i e s ( r e f s . 3 t o 5 ) r e q u i r e an a r b i t r a r y co r

r e c t i o n t o t h e t r a n s v e r s e s h e ar s t i f f n e s s m a tr ix . C oh en 's s i x t h - o r d e r t h e o r y

( r e f . 6 ) d oe s n o t r e q u i r e a c o r r e c t i o n f a c t o r b u t , l i k e a l l o t h e r s i x t h - o r d e r

t h e o r i e s ( r e f s . 3 t o 5 1, h i s t h eo r y i s no b e t t e r t ha n c l a s s i c a l p l a t e t he o ry i n

p r e d i c t i n g s t r e s s e s .

H e re in , a s i x t h - o r d e r b e nd in g t h e o r y f o r la m i n a te d a n i s o t r o p i c p l a t e s i sd ev el op ed t h a t r e q u i r e s j u s t t h e t h r e e n a t u r a l b ou nd ary c o n d i t i o n s i n u nc ou pl ed

be nd ing p rob le m s. The se th r e e c on d i t ion s a r e t h e no rm al moment, t h e tw i s t i ng

moment, a nd th e t r a n sv e r se she a r f o r c e . A l t e r n a t i ve ly , a ny one o f t h e s e c ond i

t i o n s c o ul d b e re p l a c e d by p r e s c r i p t i o n o f t h e c o r re s p o n di n g d is p la c em e n t

degree of freedom, such as r o t a t i on o r no rm al di sp la c e m e n t. The the o r y does



n o t r e q u i r e an a r b i t r a r y s h e a r c o r r e c t i o n f a c t o r . U n li ke o t h e r t h e o r i e s of

e q u i v a l e n t o r d e r ( r e f s . 3 t o 61, t h e p r e s e n t t h e o r y g i v e s an a c c u r a t e p r e d i c

t i o n of stresses.

T h e p r e s e n t t h e o r y uses a d i sp l ac e m en t a p pr oa ch l i k e t h a t i n M i n d l in ' s

t h e o r y ( r e f . 7 ) and i t s s t r a i g h t f o r w a r d e x t e n s i o n s t o l a m i n a t e d a n i s o t r o p i cp l a t e s ( r e f s . 3 t o 5 ) . However, i n c on t r a s t t o e a r l i e r t h e o r i e s , a sp ec ia l

d i s p l a c e m e n t f i e l d i s used . The d i sp la c e m e n t f i e l d i s chosen so t h a t t h e

t r a n s v e r s e s h e a r stress va n i she s on th e p l a t e s u r f a c e s . T h i s i m p o r ta n t m o d if i

c a t i o n e l i m i n a te s t h e need f o r u s i n g an a r b i t r a r y s h e a r c o r r e c t i o n f a c t o r ,

c h a r a c t e r i s t i c of o t h e r t h e o r i e s ( r e f s . 3 t o 5).

The theory i s f i r s t de veloped f o r a ge ne r a l unsym m et ri c l a m ina te . S im p l i

f i e d r e l a t i o n s a r e t he n d e r iv e d f o r a symmetr ic l am ina te i n which bending and

s t r e t c h i n g a r e u nco up led . The r e s u l t s a r e f u r t h e r s p e c i a l i z e d f o r a s ym m etric

c ross-p ly lamina te , which i s a c a s e of c l a s s i c a l o r t h o t r o p y . F or t h e l i m i t i n g

c a s e of i s o t r o p y , t h e p r e s e n t t h e o r y r e d u ce s t o a n im pro ved v e r s i o n of

M i n d l i n ' s t h e o r y ( r e f . 7).

C y l in d r i c a l be nding of a t h r e e - l a y e r e d l a m i n a t e i s c ons ide r e d , as a nwner

i c a l e xa mp le , t o c om pare r e s u l t s from t h e p r e s e n t t h e o r y w i th t h o s e f ro m t h e

e x a c t s o l u t i o n and o t h e r t h e o r i e s .

SYMBOLS

Aid i f f e r e n t i a l b en din g and t w i s t i n g r i g i d i t i e s f o r la m in at ed

ja n i s o t r o p i c p l a t e ( i , = 1, 2 , 3 )

i je le m en ts of s t i f f n e s s m a t r ix f o r i n d i v i d u a l l am in a c o n n e c t i n g

s t r e s s e s a nd s t r a i n s ( i = 1, 2 , 3 , . . ., 6 )

- i j

e le m en ts o f r e d uc ed s t i f f n e s s m a t r i x f o r i n d i v i d u a l l am i na

d e f i n in g c o n s t i t u t i v e r e l a t i o n s of p l a ne s t r e s s t yp e

( i= 1, 2, 3 )

- ( n ) i j

3Eh

D b en din g r i g i d i t y of i s o t r o p i c p l a t e ,12 ( 1 - v 2 j

Di b en din g and t w i s t i n g r i g i d i t i e s f o r la m in at ed a n i s o t r o p i c p l a t e

( i , j = 1, 2 , 3 )

E Young 's modulus of i s o t r o pi c p l a t e

E R ' E te l a s t i c mo du li o f i n d i v i d u a l l am in a i n l o n g i t u d i n a l an d

t r a n s v e r s e d i re c t i o n s , r e s p e c t i v e l y

G s h e a r modulus o f i s o t r o p i c p l a t e

2



GA t Gtt

sh ea r moduli of in d i v i du al lamina

h l a m i n a t e t h i c k n e s s

k s he a r co r r e c t i on f a c t o r i n Whitney and Pagano 's t heo ry

K1,KZ t r a n s v e r s e s h ea r s t i f f n e s s c o e f f i c i e n t s f o r l a m in at ed a n i s o t r o p i cp l a t e

Mx,My,Mw bending and tw is t i n g moments per u n i t l eng t h o f l ami na t e e lement

N x , N y , N w membrane forces per u n i t l eng t h of l ami na t e el emen t

t r a n s v e r s e s h e a r f o rc e s per un i t l eng t h of l ami na t e el emen t

d i s t r i b u t e d no rm al l o a d per u n i t area of l ami na t e s u r f a ce ,

p o s i t i v e i n d i r e c t i o n of i n c r e a s i n g z

span l en g th i n numer ica l example

d i s p lacemen t s i n x-, y -, and z - d i r ec t i o ns , r e s p ec t i v e l y

average va lues of d i sp lacement components over th i ck ne ss

C a r t e s i a n c o o r d i n a t e s w it h z - a xi s o r i e n t e d i n t h i c kn e s s w i se

d i r e c t i o n and meas ured f rom mi dd le p l ane o f l ami na t e ( f i g . 1)

a v e ra g e v a l u e s of r o t a t i o n s of l i n e no rm al t o m id dle s u r f a c e ov e r

t h i c k n e s s

A 1 , A 2 , r A 1l i n e a r d i f f e r e n t i a l o p er a t or s

1 s t r a i n com pone nts i n C a r t e s i a n c o o r d i n a t e s

Yxy Yyz Y x z

3



h a n g l e of f i b e r o r i e n t a t i o n

V P o i s s o n 's r a t i o o f i s o tr o p i c p l a t e

VA t Vtt

P o i s s o n ' s r a t i o s of in d i v i du a l la mina

ox, oy, Uz

stress c om pone nts i n C a r t e s i a n c o o r d ina t e s

Tx y G y z % Z

X t r a n s v e r s e s h e a r f u n c t io n i n R e i s s n e r 's t h e o r y f o r i s o t r o p i c p l a t e s

under bending

S u b s c r i p t s :

R , t l o n g i t u d i n a l an d t r a n s v e r s e d i r e c t i o n s of u n i d i r e c t i o n a l l am in a,

r e s p e c t i v e l y

max maximum v a lu e

S , B s t r e t c h in g a nd be nd ing componen ts , r e s pe c t i ve ly

SS,BB a pp l i e d t o matrices, r e f e r t o s t r e t c h i n g a nd b e nd in g p a r t s ,

r e s p e c t i v e l y

SB,BS a p pl ie d t o matrices, r e f e r t o s t r e t c h i n g - b e n d i n g c o u p l i n g

S u p e r s c r i p t :

T t r a n s p o s e of m a t r i x

M a t r i x n o t a t i o n :

c 3 m a t r i x

column matr ix

L J row m a t r i x

Col n u l l m a t r ix

DEVELOPMENT OF THE THEORY

The p r e se n t t he o r y i s developed by u s i n g a displacem ent approach. The

inp la ne d i sp l a c e m e n t s u and v are approximated by c u b i c p o ly n o m ia ls i n

z. The out-of-plane displacem ent w i s assumed t o be c ons t a n t w i th respec t

t o z. By r e q u i r i n g t h a t s h e a r s t resses v a ni sh o n t h e s u r f a c e , c o e f f i c i e n t s of

4



t h e p o ly n o m ia ls a re r e l a t e d . By u s i n g t h e s e r e l a t i o n s , t h e d is pl ac em e nt f i e l d

is c om plet e ly de f ine d i n t e r m s o f z, average d isp lacements U, V, and W,

a nd a ve ra ge r o t a t i o n s o f a l i n e n orm al t o t h e m i dd le s u r f a c e $ and By.

The n the stress d i s t r i b u t i o n th ro ug h t h e t h i c k n es s i s de te rmined f rom the

c o n s t i t u t i v e r e l a t i o n s of i n d i v i d u a l l a y e r s . N e x t t h e l am in at e c o n s t i t u t i v e

e qua t ions , which r e l a t e t h e f o r c e s a nd m oments t o d i sp l a c em e n t s U , V, and

W a n d r o t a t i o n s @ and gY, axe ob ta i ne d f rom st resses by i n t e g r a t i o n

t hr o ug h t h e t h i c k n e s s .

F i n a l l y , f o r t h e case of a symmetric lamina te , th e moments and t r an sv er se

s h e a r f o r c e s f r o m t h e p l a t e c o n s t i t u t i v e e q u a ti o n s a re s u b s t i t u t e d i n t o t h e

t h r e e p l a t e e q u i l i b r i u m e q u a t io n s . By e l i m i n a t i n g a l l q u a n t i t i e s e x c e p t W, a

s i n g l e g ov er ni ng e q u a t i o n i n terms of W i s obta ined .

Unsymmetric Laminate

The genera l case of an unsymmetr ic lam ina te wi th s t re t ch i ng -be nd ing cou

p l i n g i s c on si de re d f i r s t . The c o n s t i t u t i v e r e l a t i o n s f o r any i n d i v i d u a ll a m ina a r e

( 1

13 23 33

-

whe r e the c oo r d ina t e sys t e m i s shown i n f i g u r e 1. The oZ i s assumed t o be

small i n comparison wi th o t he r normal stresses and i s negl ected . Whitney and

P a ga no ( r e f . 4 ) have shown t h a t , w i th t h i s as sum pt ion , e qua t ion ( 1 ) reduces t o

t h e f o l l o w i n g c o n t r a c t e d fo rm :

- c 1 1 c 1 2

- c 1 2 c 22 (3 - 23

5



where

i 4ci j c i j -c c

j 4

c44

The s t r a i n s i n e q u a t io n ( 3 ) a re de te r m ine d f r om a n a s sum e d d i s t r i bu t ion of d i s

p l a c em e n t s t h r ough th e th i c k ne s s . The d i sp l a c e m e n t s are approximated as

The polynomia l r e pr es en ta t i o n of u and v as shown i n e qu a t ion s 5)

and ( 6 ) i s a n im p o r t a nt d e p a r t u r e fro m M i n d l i n ’ s t h e o r y ( r e f . 7 ) and i t s exten

s i o n s t o l am i na te d p l a t e s ( r e f s . 3 t o 51, which assume u and v as l i n e a r

f u n c t i o n s of z . The assumpt ion of a h ig he r ord e r polynomia l i n z f o r u

a nd v i n p l a t e b en di ng t h e o r i e s i s not new. Lo an d o t h e r s ( r e f . 2 ) have

p r e v i o u s l y u se d a c ub ic po lynom ial f o r u a nd v as i n e q u a ti o n s 5 )

and 6). H ow ever, i n c o n t r a s t t o t h e i r t h e o r y , u i, vi ( i= 0 , 1 , 2 , 3 ) i n

e q u a t i o n s 5 ) and ( 6 ) a r e no t i ndepe nde n t. I n s t e a d , t he y a re chosen so t h a t

t h e c o n d i t i o n zXz ‘tYz = 0 a t z = +h/2 i s s a t i s f i e d . When t h e s e f o u r

c o n d i t i o n s a r e s a t i s f i e d , t h e f i n a l number o f i n de p en d e nt unknowns i s f i v e .

T hu s, t h e o r d e r of t h e t h e o r y i s t h e s a m e as t h a t of e a r l i e r l o w - o r d e r t h e o r i e s

( r e f s . 3 t o 5 ) . I n a d d i t i o n , t h e p h y s i c a l c o n d i t i o n s T= zyz = 0 a t

z = f h/ 2 a r e a l s o s a t i s f i e d . It w a s n o t p o s s i b le t o s a t i s f y t h e s e c o nd i t io n s

i n t h e e a r l i e r th e o r i es ( r e f s . 3 t o 5).

B ec au se t h e s h e a r s t resses T~~ and T a r e z e r o on t h e tw o l a m i n a t eYZ

s u r f a c e s , e q u a t i o n 2 ) i n d i c a t e s t h a t t h e s h e a r s t r a i n s a l s o must v a ni sh .

C onseque nt ly , t h e she a r s t r a i n s de te r m ine d from e qua t ion s 5) o 7 )must equal

z e r o a s

6



E q u a t i o n s 8) and (9) l ea d t o

Equa t ions ( 8 ) t o ( 10 ) i n d i c a t e t h a t t h e r e a r e i n a l l o n ly f i v e i nd ep en de ntunknowns uo , u l , vo , v l , and W; h ow ev er, t h e f i r s t f o u r l a c k d i r e c t p hy s

i c a l i n t e r p r e t a t i o n a nd are i nc onve n ie n t from th e po in t o f v iew of p r e s c r i b in g

bounda ry c ond i t ions . The r e f o r e uo: u l , vo , a nd v1 a r e r e p l a c e d by f o u r

o t he r qu a n t i t i e s w i th ph ys i c a l m ea ning; na me ly , a ver age inp la n e d i sp l a ce m e n t s

U and V a nd a ve ra ge r o t a t i on s of a l i n e n orm al t o t h e m i dd le s u r f a c e ,

which are d e f i n e d as

Averaging he re means avera ging through t h e th ic kn ess . Equa t ion s ( 11

a nd ( 12 ) are ob ta ine d f r om a l e a s t s q u a re a p pr o xi m at io n . D e t a i l s o f d e r i v a t i o n

o f e q u a t i o n s ( 1 1 ) a nd ( 12 ) are g iven i n a ppe nd ix A.

BY u s i n g e q u at i on ( l o ) , d e f i n i t i o n s ( 1 1 ) a nd ( 1 2 ) y i e l d

u = uo

v = vo

23h

px = u1 +-u3

( 1 5 )20

( 1 6 )

7



Solving equations (81 , (91 , (151 , and ( 1 6 ) leads to

u = - $ + - 1 aw1 4 x 4 a x

v = - $ + - 1 aw1 4 Y 4 a y

v3

= - - p53h2 +E)

( 1 7 )

( 1 8 )

( 1 9 )

From equations (31 , 51, (61, and (101 , the strain-displacement relations,

and the definitions of force and moment stress resultants, there follows

4 = k N d'= (3:9) (9 x 1 )

( 2 1

( 3 x 1 1

where

I = Ess LSJ

( 3 x 3 ) ( 3 x 6 )

Lss =

8



( 3 x 3 ) ( 3 x 3 )

( 3 x 3 ) ( 3 x 3 )

-(n) -(n) -(n)c 1 1 ‘12 ‘1 3

- ( n ) - ( n ) - ( n ) ( 2 9 a )

‘12 ‘22 ‘23

- ( n ) - ( n ) - ( n )

‘13 ‘23 ‘33

6

=(;:)

9



The m a t ri x c o n s t i t u t i v e e q u a t i o n s ( 2 1 ) an d ( 2 2 ) are a r r a nge d so t h a t

matrices c o rr e sp o n d in g t o s t r e t c h i n g , b en di ng , a nd s t r e t ch i n g - b e n d i n g c o u p l i n g

a pp ea r s e p a r a t e l y i n t h e t o t a l m a tr ix . These matrices a re denoted by t h e sub

s c r i p t s SS, BB, and SB o r BS, r e s p e c t i v e l y . A l s o , e le m e n t s of t h e column

m a t r i x (0 a r e a r r a n g e d so t h a t t h e v a r i a b l e s s i g n i f y i n g s t r e t c h i n g a nd be nd in g

are grouped se pa ra t e ly . These groups a re denoted by t h e s u b s c r i p t s S and B,

r e s p e c t i v e l y . A lt ho ug h t h e p r e s e n t f o r m u l at i o n i s f o r a ge ne r a l l a m ina te w i th

c oup le d s t r e t c h i ng a nd bend ing , t h e a r ra nge m e nt o f matrices al lows e a sy extrac-

t i o n o f matrices f o r sym me tr ic l am ina te r e l a t i on s . The sym m et ri c l a m ina te c a se

i s d i sc u s se d i n a s u b s eq u e n t s e c t i o n .

By combining equa t ions (131 , (141 , (171 , (181 , (191 , and (201 , t h e column

m a t r ix $ i s e xpr e s se d as a f u n c t i o n of t h e d e r i v a t i v e s of U , V, W ,

and y a s

= H 3,( 9 x 1 ) ( 9 x 1 1 ) ( 1 1 x 1 )

where

4 =6)

H =

HSS = [ ;

BX

(34 1

(35 1

( 3 6 )

(37 1

( 3 8 )

(391

10



-5/4 0

0 5/4

0 0

-5/3h 2 o

0 -5/3h2

0 0 -

where is a null matrix.

N = I H

-0 0 1/4 0 0

0 0 0 1/4 0

5/4 5/4 0 0 1/2

0 0 -5/3h2 o 0

0 0 0 -5/3h2 0

-5/3h2 -5/3h 2 0 0 -10/3h2-

J,(3x1) (3x9) (9x11) (11x1)

M = J H J,

(3x1) (3x9) (9x11) (11x1)

BY using the strain-displacement relations and equations (51, ( 61 , (71,(io), (17), (18), (19), and (20), equation (2) yields on integration through

the thickness

where

1 1



-c55

-56

3c552 2 ‘56 3 c 5 6 2 7

dz ( 4 5 )

3c562 2 ‘66 3c6622_]

= 5 j dz

‘66

E q ua ti on s ( 4 1 ) t o ( 4 3 ) are t h e c o n s t i t u t i v e e q ua ti on s f o r a g e n e r a l ,

uns ymmetr ic l ami na t e w i t h s t r e t ch i ng -b end i ng coup l i ng . These equa t i ons de f i ne

t h e fo rc e and moment stress r e s u l t a n t s i n terms of ave rage i np l ane d i s p l ace

ments U and V, n o r m a l d e f l e c t i o n W, and average ro t a t i on s of a l i ne no rma lt o t h e m iddle s u r f a c e p x and By. The governing equ at i on s can be de riv ed by

s u b s t i t u t i n g from t h e s e c o n s t i t u t i v e e q u a ti o ns i n t o t h e e q u i li b r iu m e q u at io n s.

Because they are length y and cumbersome, the y a r e not shown he re in . De rjv at i on

o f t h e gove rn i ng equa t i o n i s c a r r i e d o ut h e r e on ly f o r t h e symmetric l aminate

c a s e , i n w hich t h e r e i s no coupl ing between s t re tc h in g and bending .

Symm etric Lam inate Under B ending

For a s ymmetr ic l ami na te , t h e ma t r i c e s LSB and LBs which couple

s t r e t c h i ng and bend ing become nu l l matrices. C o ns eq ue nt ly , t h e c o n s t i t u t i v e

e q u a t i o n ( 4 2 ) f o r moments s i m p l i f i e s t o

M = LBB

HBB ‘B

(47 1

( 3 x 1 ) (3x6)(6x7) (7x1)

,where the‘ vari ou s matrices appea r i ng on t h e r i gh t-hand s i d e of equa t i on ( 4 7 )are def ined by e q u a t i o n s (281, (291, (371, and ( 4 0 ) . The c o n s t i t u t i v e e qua

t i o n ( 4 3 ) f o r tr a n s v e r s e s h e a r f o r c e s i s n o t s i m p l i f i e d . F u r t h e r t r e a t m e n t of

symmet r ic l aminates here in i s c on fi ne d t o d e r i v a t i o n of r e l a t i o n s p e r t a i n i n g t o

b en di ng . The s t r e t c h i n g r e l a t i o n s become i d e n t i c a l w i t h t h o s e fro m c l a s s i c a l

l ami na t ion t h eo ry and need no s ep a r a t e t r ea t me n t .

The moment and fo rc e equ i l ib r ium eq uat ions f o r th e l a mi nate under bending

a r e

12



On s u b s t i t u t i o n fro m t h e c o n s t i t u t i v e e q u a t i o n s ( 4 3 ) and (471 , e q u a t i o n s

( 4 8 ) t o (50) y ie ld t h r e e gove r n ing e qua t ion s f o r t h r e e unknowns: nam ely, W,

BX and By. These c an be r educ e d t o a s i n g l e e q u at io n i n terms of W by

e l i m i n a t i n g B X and f l y , as f o l l o w s :

Equa t ion ( 4 3 ) can be r e w r i t t e n as

where

[PI = [-e -1 r) 5-11

The [ E ] and [ r ) ] i n e q ua t io n (52) a re 2 x 2 submatrices obta ined by

p a r t i t i o n i n g t h e 2 x 4 m a t r i x [R] of e qua t ion ( 4 4 ) a s

= r 5I2x2) I (2x2)

(53

A column matr ix Q i s now defined as

From equation (511, a r e l a t i on sh ip betwe en th e c olumn matrices and

6 i s found as

Js = r P

(7x1I(7x7) (7x1)

(55 1

13



where

p 1 1 0 p12 '13 0 '14 0

0 p22 p21 0 ' 3 0 '24

0 p12 p11 0 '13 0 '14

r = p2 1 0 p22 '2 3 0 '24 0

1 0 0 0 0 0 0

0 1 0 0 0 0 0

0 0 1 0 0 0 0 -

The elements Pij i n t h e f o r e g o i n g e q u at io n are t h e e lemen t s o f t h e [PImat r i x ob t a i ne d f rom equa t i on (52 ) . By u s i n g equa t i on (551, e q u a t i o n ( 4 7 ) i s

r ed uc ed t o

~ = 5-2 e(3x1) (3x7) (7x1)

where

e = LBB

HBB

r ( 5 7 )

(3x6 ) (6x7 ) (7x7 )

By us i ng equat ion (5 6 ) , th e moment equ i l ib r ium e qua t ion s (48 ) and (4 9)

become

-A4Qx + A5Qy = A6W ( 5 9 )

Here, A1 t o A6 are l i n e a r d i f f e r e n t i a l o p e r a t o r s, t h e c o e f f i c i e n t s of which

a r e def i ned i n te rms o f e l emen ts of t h e ma t r i x 8 . Equat ions (58 ) and (59 )

y i e l d

14



A7Qx = A3A5 + A2A6)W

A7Qy = A1A6 + A3A4)W

where

A7 = Al A5 - A2A4) ( 621

El i mnat i on of Qx and Qy f r omequat i ons ( 501, ( 601, and ( 61) resul t s

i n

4 6a w + a w

i =O l ,2, . . .

a(4- i ) i

ax

(4- i )

ay i =O l , 2, . . .

a(6- i ) i

ax

(6- i )

aY

i

where t he coef f i ci ent s a and b are der i ved i n appendi x B i n t erms of el ement s of t he matr i x 8 The convent i on f ol l owed f or t he subscr i pt s on a

and b i n equat i on ( 63) i s t hat t he t wo subscr i pt s ( t he f i r st one bei ng i n

parent heses) are not t o be mul t i pl i ed w th each ot her but wr i t t en adj acent t o

each other. For exampl e, a(6- i ) if or i = 1 i s aS1.

Equat i on ( 63) i s a si xth- order governi ng equat i on f or t he normal di spl acement. I t permt s the t hree nat ural physi cal boundary condi t i ons t o be pre

scr i bed over each boundary as i n Rei ssner s ( ref . 8 ) or Mndl i n s (r ef . 7

t heory f or i sot ropi c pl ates. Once W i s det ermned f romequat i on ( 63) and t heprescr i bed boundary condi t i ons, al l t he ot her physi cal quant i t i es can be det er

mned i n t erms of W To t hi s end, t he t ransver se shear f or ces are f oundt hrough equati ons ( 58 ) and ( 591, moment s t hrough t he matr i x equat i on ( 561, and

rot at i ons through equat i on (511.

15



Symmetric Cros s-Ply Lamin ate Under Bending

The c o ns t i t u t i ve e qua t io ns f o r a n unsymm et ri c l a m ina te a nd a general sym

m e t r i c lamina te w e r e d e r i v e d i n m a t r i x fo rm so t h a t t h e c o e f f i c i e n t s a pp e ar in g

i n them could be c a l c u l a t e d r o u t i n e l y w i t h m a t r i x a l g e b r a . The s pec i a l case of

asymmetric

cross p l y i s now considered.symmetric

c r os s - p ly l a m ina te i sd e f i n e d as one which i s comprised of only O o and 90° l a ye r s s t a c k e d symm et ri

c a l l y w it h respect t o t h e m id dle su r fa c e. I n t h i s case, t h e r e i s no c oup l ing

b etw een b e n di ng an d t w i s t i n g , an d t h e m a t r i x r e l a t i o n s d e r i v e d p r e v i o u s l y f o r a

g e n e r a l symmetric lamina te reduce t o s i m p l e formulas. These formulas are now

d e r i v e d .

F o r convenience , the x- and y-axes ( f ig . 1 ) a re c ho se n t o be o r i e n t e d

a long th e p r in c i pa l axe s o f t h e l a m ina te . Thus

-C 1 3 = c23 c34 = 0

a nd c onse que n tly some e l em e n ts o f t h e r e duc ed s t i f f n e s s m a t r ix a l so va n i sh ;

t h u s

By u s i n g e q u a t i o n s ( 6 5 ) and ( 6 6 1 , e q u a t i o n s ( 4 7 ) and ( 4 3 ) r ed uc e t o

aP a w- a P X - 2

Mx D 1 l ax D12 ay 2 A12 2ax a Y

aP 2 a w a w P X - -

My = D12 ax D22 ay A12 2 A22 2ax 3Y

ayxy = D33i +22A33

a2w

Q x = K I ( @x + -E)

16



QY = K ( B +E)

where

D =

1 - ( 2 ) 5 - ( 4 )( 7 31- -

A i j 4 ' i j 3h2C

i j

H e r e D i j can be i n t e r p r e t e d a s a n i s o t r o p i c p l a t e b en din g r i g i d i t i e s . A s

shown subseq uent ly , the y become equa l t o th e i s o t r o p i c v a lu e s f o r t h e l i m i t i n g

c a s e of i s o t r o p y . The c o e f f i c i e n t s A i . v a ni sh f o r t h e i s o t r o p i c c as e an d

m ig ht t h e r e f o r e b e r e f e r r e d t o as a n i s o g r o p i c d i f f e r e n t i a l p l a t e b en di ng

r i g i d i t i e s .

S u b s t i t u t i n g fro m e q u at i o n s ( 7 0 ) an d ( 7 1 ) i n t o e q u a t i o n s ( 6 7 ) t o ( 6 9 ) an d

u s i n g e q u a t io n ( 5 0 ) y i e l d s

MX

= - C l l2 c12

2D12 aQx D12( 2 ) a

2w - ( 2 ) fi (?---)---

qax K 2ax a Y

2 2D12

M = -C- ( 2 ) a w - ( 2 ) a w 2 2 _ 2 Y

Y l 2 ax2 c22

a Y 2(2 z ):: K1

17



--

Equat i ons ( 67) to 69) and ( 76) t o ( 78) are two al t ernat e forms of con st i t ut i ve equat i ons for moments f or a symmet r i c cross- pl y l amnate. Equat i ons

70) and ( 71) are t he const i t ut i ve equat i ons f or t ransverse shear f orces.

By usi ng equati ons 501, (76), (77) , and ( 781, the moment equi l i br i um equati ons 48) and ( 49) can be wr i t t en as

A Q = -AloW - c1 ( 79)x8 x

AgQy = -A11

W - c2 aY

where A8 t o A l l are l i near di f f erent i al operat ors def i ned by

3 3 3 2A l 0 = e

1a ( )/ax + e2 a ( )/ax ay

1 Dl 1dl T ( Dl 2 + D33)

2 K1

18



d2 = -D33/K1

d3 = -D33

/K2

1 D22d4 - D

12+ D33) - 1 K

2

- 2 )e = C l l

1

- 2 )e3

= C 2 2

1c1 = y D 1 2 + D33

2

1c2=-(D

12 D33 1

Elimination of Qx and Q from equations (501, 7 9 1 ,

in the governing equation for as

8 6 )

8 7

8 8 )

(931

and 8 0 ) results

B y substituting for the differential operators, equation 9 4 ) becomes

19



4 4- a w+ e3- a

a6wa w + ~ e 2 a

4]: + %el --g[e2(d1 d4el] ax4ay2l

ax4

ax ay a Y ax

2 2

= q + ( d l + d3 - C 1 ) % + + d4 - c,) asax aY

4

+ d 2 ( d 4 - c,)as (9 5 1

3Y

E q u at io n ( 9 5 ) h a s no d e r i v a t i v e i n v o l v i n g odd number of d i f f e r e n t i a t i o n s

wi th respect t o x o r y. I n c o n t r a s t , t h e s e d e r i v a t i v e s a pp ea r i n t h e

g o v er n in g e q u a t i o n ( 6 3 ) f o r a g e n e r a l symmetric lamina te .

Reduction t o I s o t r o p i c P l a t e R e l a t i o n s

For t h e s p e c i a l c a s e of i s o t r o p y

( 9 6 )

- 2C12 = vE/(1 - v (97 1

3Eh- - - (2 ) --(2) = P l a t e b en din g r i g i d i t y , D =

D1l - D22 = c l l c221 2 ( 1 - v

21

- ( 2 )= v D

D12 c12

20



K 1 = K2 = Gh ( 1 0 2 )6

By u s i n g e q u a t io n s ( 9 6 ) t o (1021 , t h e c o n s t i t u t i v e e q ua ti on s ( 67 ) t o ( 7 1 )a nd ( 7 6 ) t o ( 7 8 ) re d uc e t o a form s i m i l a r t o R e i ss n e r' s ( r e f . 8 ) . The only

d i sc r e pa nc y invo lve s terms of q. F o r t h e homogeneous case q = 0, t h e t w o

t h e o r i e s are i d e n t i c a l . The d i s cr e p an c y f o r q 0 v a n is h e s i f t h e c o n tr i bu

t i o n f r o mbz

t o s t r a i n e nerg y i s n e g l e c t e d i n R e i s s n e r ' s e ne rg y f o r m u l a t i o n

( r e f . 8 ) . As t h i s c o n t r ib u t io n is r e l a t i v e l y s m a l l , t h e d i sc r e pa nc y be twee n

t h e tw o t h e o r i e s c an be c o n s i d e r e d t o be a h i g h e r o r d e r e f f e c t an d n e g l i g i b l e .

The d i sc r e pa nc y i n terms i n v o l v i n g q i s a consequence of t h e assumption

of c o n s t a n t w t hr ou gh t h i c k n e s s i n t h e p r es e n t t h eo r y . R e i s s n e r ' s i s o t r o p i c

p l a t e t h e o r y i s b as ed o n e xa c t s a t i s f a c t i o n of t h e e q u i l i b r i u m e q u at i on i n t h e

z - d i r e c t i o n f o r a l l z which i m p l i e s v a r i a t i o n o f w wi th z .

DISCUSSION

A s i x t h - o r d e r g o v er n in g e q u a t i o n f o r W i s ob ta ine d he r e f o r a sym m et ri c

lamina te . This i s i n c o n t r a s t t o t h e R ei s sn e r ( r e f . 8 ) and M in dlin ( r e f . 7 )

t h e o r i e s f o r an i s o t r o p i c p l a t e which g i ve a f o u rt h - o r d er e q u a t i o n f o r W

t o g e t h e r w i t h an a u x i l i a r y e q u a t i o n of s ec on d o r d e r f o r a t r a n s v e r s e s h e a r

f u n c t i o n x H ow ever, t h e t o t a l o r d e r i s t h e s a m e i n b ot h cases, t h e r e b y

r e q u i r i n g p r e s c r i p t i o n of t h e same number of boundary co nd it i on s.

A c l o s e in s p e c t i o n r e v e a l s t h a t , f o r t h e l i m i t i n g c a se of i s o t r o p y , t h e

p r e s e n t t h e o r y a l s o l e a d s t o lo wer o r d e r e q u at i on s f o r W and x l i k e t h e

R e i s s n e r ( r e f . 8 ) a nd Mind lin ( r e f . 7 ) t h e o r i e s . T h i s h ap pe ns b ec au s e, f o r

i s o t ro p y , t h e d i f f e r e n t i a l o p e ra t or s A8 and Ag i n e q u a t i o n s ( 7 9 ) a nd ( 8 0 )

become i d e n t i c a l . T hu s, f ew er d i f f e r e n t i a t i o n s w ou ld b e r e q u i r e d t o e l i m i

n a t e Qx and Qy f r om e qua t ions 501, ( 791 , a nd ( 8 0 ) . ( S e e de r iv a t i on o f

eq. (941.1

The Qx and Qy f o r a l a m ina te d p l a t e a re de ter m ine d com ple te ly i n terms

of W as p a r t i c u l a r i n t e g r a l s of equ a t io ns ( 79 ) and (80 ) . Complementa ry so lu

t i o n s of t h e s e e q u at i o ns a re n o t a d m i ss i bl e a s t h e y v i o l a t e t h e e q u il i br i um

e q u a t i o n 50). R eca l l t h a t , i n c o n t r a s t , com plem entary s o l u t i o n s a r e u se d i n

i s o t r o p i c p l a t e s ( r e f s . 7 an d 8 ) be ca use comple me ntary so lu t i on s i n t h e i so

t r o p i c c a s e c an b e c ho se n t o s a t i s f y t h e e q u i l ib r i u m e q u a ti o n (50).

S h ea r C o r r e c t i o n F a c t o r

The term s h ea r c o r r e c t i o n f a c t o r i n t r a n s v e r s e s h e a r d ef or m at io n

t h e o r i e s i s usua l ly meant t o r e f e r t o a n a r b i t r a r y c o r re c t io n a p p li ed t o t h e

s h e a r s t i f f n e s s p r e v i o u s l y d et er mi ne d. I n t h e p r e s e n t t h e o r y , t h e r e i s no

s h e a r co r r e c t i o n f a c t o r i n t h i s s en s e of t h e t e r m b ec au se t h e t r a n s v e r s e s h e a r

s t i f f n e s s i s e x p l i c i t l y d e te rm i ne d an d no c o r r e c t i o n i s ne c e s sa r y . Th i s

21



becomes c l ea r by exa mining th e l im i t in g case of i so t r o py i n d e t a i l . From equa

t i o n s (701, ( 711 , and ( 102 l t r a n s v e r s e s h e ar f o r c e s f o r an i s o t r o p i c p l a t e are

given by

where G i s t h e i s o t r op ic she a r m odulus . Equa t ions ( 103 ) a nd ( 104 ) a g r e e w i th

t h e r e s u l t s f rom R e i s s n er ' s t h e o ry f o r i s o t r o p i c p l a t e s ( r e f . 8 ) .

T h e q u a n t i t i e s Px x and By +-

r e p r e s e n t a ve ra ge s h e a r s t r a i n sa Y

t h ro u g h t h e t h i c k n e s s . T hus, t h e f a c t o r 5/6 ( 0 .8 3 3 ) i n e q u a t i o n s ( 1 0 3 )

and ( 1 0 4 ) c an be l oo ke d upon a s a c o r r e c t i o n t o b e a p p l i e d t o t h e t r a n s v e r s e

s h e a r s t i f f n e s s t o ac co un t f o r v a r i a t i o n of s h e a r s t ress t h ro u g h t h e t h i c k n e s s .

I n M i n d l i n ' s th e o r y ( r e f . 71, t h e a ss um pt io n of c o n s t a n t s h e a r s t r a i n l e d t o a

fa c t o r of un i ty in s t ea d of 5/6 (0 .833) i n equ a t io ns (103) and (10 4) . However,

M i nd li n r e pl a ce d t h i s f a c t o r of u n i t y by a n a r b i t r a r y f a c t o r , t h e v a l u e o f

which was adjusted so t h a t r e s u l t s f rom t h e t h e o r y a g r ee d w it h t h e e x a c t s o l u

t i o n f o r a chosen example. H e a r r i v e d a t a v al u e of 7t2/12 (0. 82 2) f o r t h i s

f a c t o r by co ns id e r i ng one example . By co ns id e r i ng a second example, he

obt a in ed anothe r va lu e which depends on Po iss on 's r a t i o v a nd va r i e s f r om

0.76 t o 0 .91 as v v a r i e s f r o m 0 t o 0.5. The f i r s t va l ue 7t2/12 ( 0 . 8 2 2 ) i s

c l os e t o 5 /6 ( 0 .833 ) ob ta ine d from R e i s sn e r ' s t h e o r y . However, t h e m anner i nw hich t h e s h e a r c o r r e c t i o n f a c t o r i s de r ive d in Mind l in ' s a ppr oac h i s a r b i t r a r y

b e ca u se t h e v a l u e a r r i v e d a t depends on t h e example chosen. The p re se n t th eo ry

i s a l so ba se d on a d i sp l ac e m en t f o r m u l a t i o n s i m i l a r t o M i n d l i n ' s b ut t h e s h e a r

c o r r e c t i o n i s d e r iv e d i n a l o g ic a l way. Thus, t h e p r e se n t t he o r y c an be looked

upon a s an improvement of Mindl in ' s the ory f o r i s o t r o p ic p la t e s .

S t r a i g h t f o r w a r d e x t e n s i o n s o f M i n d l in ' s t h e o r y t o l a m in a t ed c o m po si te

p l a t e s ( r e f s . 3 t o 5 ) have th e same de g re e of a r b i t r a r i n e s s as M i n d l i n ' s t h e o r y

i t s e l f . F o r i n s t a n c e , Whitney a nd P agano ( r e f . 4 ) a r r i v e d a t d i f f e r e n t s h e a r

c o r r e c t i o n f a c t o r s f o r t w o- la ye re d a nd t h r e e - l a y e r e d p l a t e s . The suggested

p r oc e dur e i n t h e i r method a ppe a r s t o be t o d e r i v e t h e s h e a r c o r r e c ti o n f a c t o r

f o r e a c h s e t of lamina t ion pa ramete r s by c o n s i d e r i n g t h e known e x a c t s o l u t i o n

f o r a c e r t a i n p ro ble m. B ut w i t h t h e p r e s e n t t h e o r y , n o s h e a r c o r r e c t i o n f a c t o ri s n e ce ss a ry a nd t h e t r a n s v e r s e s h e a r s t i f f n e s s i s o b t a i n e d as a f u n c t i o n o f

e l a s t i c c o n s t a n t s a nd t h e s t a c k i n g s eq u en ce w i t h o ut c o n s i d e r i n g t h e e x a c t s o l u

t i o n f o r a sp e c i f i c p roblem . The r e q u i r e d e xp r e s s ion s a re given by equa

t i o n s ( 7 4 ) a nd 75).

22



I n t e r l a m i n a r S h ea r S t r e s s e s

I n t e r l a m i n a r s h e a r i s o ne of t h e s o u r c e s o f f a i l u r e i n l a m in a t ed p l a t e s .

T h e r e f o re , c a l c u l a t i o n of i n t e r l a m i n a r s h e a r s t r e s s e s i s impor tan t i n any bend

i n g p ro blem . W ith t h e u s e o f t h e p r e s e n t t h e o r y , t h e s e s t resses can be calcu

l a t e d a s f o l lows : A problem i s c o ns i de re d s o l v e d i f Bx fly, and W a r e

de te rmined as fu nc t i on s of x and y . The d isp lacem ents can be de te rmined a sf u nc t io ns o f x , y , a nd z by t h e u s e o f e q u a t i o n s ( 10 1, ( 1 7 ) t o (201 , and

5) o ( 7 ) . Thus , ax, ay, and zXy

a r e a l s o de te r m ine d a s f unc t ions o f

x, y , and z fro m t h e c o n s t i t u t i v e r e l a t i o n s of i n d i v i d u a l lam in ae. The

i n t e r l a m i n a r s h e a r s t r e s ses are t he n de te rm ined f rom th e f o l low ing e qu i l i b r ium

e q u a t i o n s:

a x-- / ” (%+$) dz

yz -h/2

T h i s method g i v e s s i n g l e - v a l u e d i n t e r l a m i n a r s h e a r s t r e s s e s a t t h e i n t e r

f a c e s . A n a l t e r n a t e m ethod would be t o c a l c u l a t e zxz and zY Z

d i r e c t l y from

t h e d i sp l ac e m e n t s g ive n by e qua t io ns 5 ) t o ( 7 ) . However, t h i s m ethod g ive s

tw o v al u e s f o r t h e i n t e r l a m i n a r s h e a r st ress a t e ac h in t e r f a c e de pe nd ing upon

t h e lamina chosen .

NUMERICAL EXAMPLE

A numerical example i s u se d t o c om pare t h e p r e s e n t t h e o r y w i t h t h e e x i s t

i n g th eo r i es . The example chosen i s t h a t of c y l i n d r i c a l b en di ng of a t h r e e -

lay e re d , symmetric c ross-p ly lamina te Oo/900/Oo) of high modulus gra ph ite -

e poxy . The l a ye r s a r e a l l t a ke n t o be of e q u al th i c k n e s s wi th f i b e r s i n t h e

o u t e r l a y e r s o r i e n t e d i n t h e d i r e c t i o n of b en din g ( f i g . 2 ) . The la ye r proper

t i e s used a r e

= 25E R / E t

GRt/Et = 0.5

G t t / E t = 0.2

vRt = vtt = 0.25

23



The p l a t e i s c o n s id e r e d t o be s e m i - i n f i n i t e w i t h a f i n i t e s pa n s i n t h e

x - d i r e c t i o n ( f i g . 2 ) and s ub j ec t ed t o a s i n u s o i d a l , d i s t r i b u t e d no rm al l o a d o f

i n t e n s i t y q = qmax s i n p. The boundary co nd i t i ons cons i de red a r e

A t x = 0 and s ,

W = f = M x = O (105)Y

F o r t h e c a s e of c y l i n d r i c a l be nd in g, a l l d e r i v a t i v e s w i t h r e s p e c t t o y

are zero. Thus equa t ion (95) y i e l d s

7tXw = w

m a Xs i n - ( 1 0 6 )

S

where

The t r a n s v e r s e s h e a r f o r c e Q x i s ob t a i ne d from equ a t i on (79) a s

7tX

Qx - Qx,maxcos - ( 1 0 8 )

S

where

Equat ion (70) y i e l d s t h e r o t a t i o n a sBX

7tX

px = Bx,maxcos - 110 )

s

24



where

1

px, max K1 Qx,max- (n/s)Wmx (111)

F r o m e q u a t i o n s ( 8 0 ) and (71 ) and th e boundary co nd i t i on onBY

Q E O $ : oY Y

S t r e s s e s i n t h e d i f f e r e n t lam in ae are now determined as f o l lows . I n pu r e

be nd ing , t he terms u o a nd v o i n e q u a t i o n s 5) and ( 6 ) van ish. Conse

qu e n t ly , t h e inp la n e d i sp l a c em e n t s u a nd v a r e de f ine d , i n v ie w o f e qua

t ion (101 , by

3u = u z + u z ( 1 1 3 )

1 3

3v = v z + v z ( 1 1 4 )

1 3

The u1 and u3 a r e de te rmined from equa t io ns (17 ) and ( 1 8 ) . It f o l lows

f rom equa t ions (191 , 201, (1061, (10 7) , and (112) t h a t v1 and v3 a re ze ro .

The i np l a ne d i sp l a c e m e n t s a r e th us determined. Bending s t ress d i s t r i b u t i o n

t h ro u gh t h e t h i c k n e s s i s naw de te r mined f rom th e c o n s t i t u t i v e r e l a t i on s of

d i f f e r e n t l a m i n a e .

E x ac t s o l u t i o n f o r t h i s p ro ble m b as ed on t h re e -d i m e ns i o n al e l a s t i c i t ya n a l y s i s w a s given by P a ga no ( r e f . 9 ) . Fi gu re s 3 and 4 compare r e s u l t s from

t h e p r e s e n t t h e o r y w i t h t h o s e f ro m t h e e x a c t s o l u t i o n an d t h e t h e o r y o f W h itn ey

and Pagano ( r e f . 4 1 . The t h e o r i e s p r e s e n t e d i n r e f e r e n c e s 3 t o 5 are a l l a

s i m p l i f i e d s e t r e q u i r i n g an a r b i t r a r y s h ea r c o r re c t io n f a c to r . I n t h e r e s u l t s

p r e s e nt e d i n t h e s e r e f e r e n ce s , t h e v al ue o f t h e s h e a r c o r r e c t i o n f a c t o r i s

a d j u s t e d so t h a t t h e r e s u l t s c o m e close t o t h e e x a c t s o l u t i o n . T h e re fo r e a

compar ison of r e s u l t s from a l l t h e s e t h e o r i e s c ou ld be m i sl ea d in g . F or t h i s

purpose , on ly Whitney and Pagano 's the ory ( r e f . 4 ) i s chosen f o r comparison and

i s t r e a t e d as b e i n g r e p r e s e n t a t i v e of t h e s i m p l i f i e d t h e o r i e s ( r e f s . 3 t o 5 ) .

Figure 3 shows a p l o t of t h e maximum d e f l ec t i o nWmax

a t t h e c e n t e r o f

t h e s p a n as a f un c t io n o f t h e spa n -to -th i ckne ss r a t i o . The p r e se n t t he o r y i s

c lo se r t o t h e e x a c t s o l u t i o n t h a n W hitney a n d P ag an o 's t h e o r y ( r e f . 4 ) wi th ashe a r c o r r e c t i on f a c t o r k o f un i ty . On th e o t he r hand , W hitney a nd P a ga no' s

t h e o r y shows b e t t e r c o r r e l a t i o n i f k is shown a s 2/3. However, t h e f a c t o r

2/3 w a s a r r i v e d a t by Whitney and Pagano by a t r i a l - a n d - e r r o r p r o c e d u r e . There

a r e two d isadvantag es i n Whitney and Pagano's method. F i r s t , t h e t r i a l - an d

er ro r pr oc e dur e i s t o be r e p e a te d i f t h e l am i na ti o n parameters a r e d i f f e r e n t .

25



There i s no s i n g le va lue of k which ho l ds good f o r a l l l a m i n a t i o n parameters.

(Fo r example, Whitney and Pagano a r r i v e d a t ano the r va lue fo r k , namely , 5 / 6 ,

f o r a t wo -l ay er ed p l a t e . ) S ec on dly , t h e s h e a r c o r r e c t i o n f a c t o r a r r i v e d a t by

t h i s p ro ce du re i s problem-dependent and it i s n o t s u r e w h et he r t h e v a l u e f o r

k so d e r i v e d i s v a l i d f o r a p ro ble m o t h e r t h a n c y l i n d r i c a l b en di ng . R e c a l l

t h a t M in dli n ( r e f . 7 ) a r r i v e d a t d i f f e r e n t v a l u e s f o r k by c o n s i d e r i n g d i f f e re n t p robl ems .

F i g u r e 4 p r e s e n t s t h e b e n d i n g stress ax d i s t r i b u t i o n th ro ug h t h e t h i ck

n e ss f o r a l aminate wi th a span- to- th ickness r a t i o s/h of 4. The exact solu

t i o n of Pagano ( r e f . 9 ) and r e s u l t s from Whitney and Pagano' s the or y and th e

p r e s e n t t h e o r y a r e p r e s e n te d i n t h i s f i g u r e . R e s u l t s from t h e c l a s s i c a l l a m i

n a t e d p l a t e t h e o r y a r e a l s o i nc luded fo r compar is on . The exac t s o l u t i o n and

t h e p r e s e n t t h e o r y show c o n s i d e r a b l e d e v i a t i o n f ro m t h e c l a s s i c a l t h e o r y .

F u rt he rm o re , t h e e x a c t s o l u t i o n a nd t h e p r e s e n t t h e o r y are i n good agreement

f o r m ost of t h e t h i c k n e s s . N ote t h a t i n t h e m id dle l a y e r , t h e 90° l ami na , t he

stresses ;care extremely smal l and a l l t h e o r i e s p r e d i c t n e a r z e r o va l u e s .

F i g u r e 4 shows t h a t Whitney and Pagano' s the or y ( r e f . 4 ) y i e l d s t h e s a m es t ress d i s t r i b u t i o n a s t h e c l a s s i c a l l am in ate d p l a t e t h e o r y i r r e s p e c t i v e of t h e

v a l u e of t h e s h e a r c o r r e c t i o n f a c t o r u se d. T h i s i s t r u e of a l l c u r r e n t s i x t h -

o r de r t h e o r i e s f o r la min ate d a n i s o t r o p i c p l a t e s ( r e f s . 3 t o 6 ) . The f a c t t h a t

t h e s e t h e o r i e s p r e d i c t s t r e s s e s no d i f f e r e n t from t h e c l a s s i c a l t h e o r y i s a

s eve re l i m i t a t i o n . Even Cohen 's t heo ry ( r e f . 61, w h i c h p r e d i c t s W with good

a c cu r ac y w i th o u t a s h e a r c o r r e c t i o n f a c t o r , h a s t h i s draw bac k. A c a r e f u l

i n s p e c t i o n r e v e a l s t h a t t h i s drawback i n c u r r e n t s i x t h - o r d e r b en di ng t h e o r i e s

i s a consequence of t h e assumpt ion of l i n e a r i t y of sX, cy, and yxy w i t h

r e sp e c t t o z . The p re s en t theo ry a l l ows fo r a m o r e g e n e r a l v a r i a t i o n of t h e s e

s t r a i n s w it h respect t o z and does not have t h i s drawback.

CONCLUDING REMARKS

A s h e a r d e f o rm a ti on t h e o r y f o r l am i na te d a n i s o t r o p i c p la t e s i s devel

oped. In th e case of uncoupled bending , t h e p r es en t theo ry i s one of s i x t h

o r d e r an d r e q u i r e s j u s t t h r e e n a t u r a l b ou nd ar y c o n d i t i o n s . Most of t h e c u r r e n t

s i x t h - o r d e r t h e o r i e s r e q u i r e an a r b i t r a r y s h e a r c o r r e c t i o n f a c t o r a nd a l l of

them have t h e drawback t ha t t h e i r stress p r e d i c t i o n i s h i gh l y i nac cu r a t e . The

p re s e n t t heo ry does no t have e i t h e r of t he s e d rawbacks ,

The p re s en t t heo ry i s no t p re s en t ed as an improvement over th e cu rr en t

h i g h e r o r d e r t h e o r i e s . S u r e l y , th e y s h o ul d be m o r e a c c u r a t e b ut t h e y r e q u i r e

p r es c r i p t i o n of i nconven ien t boundary cond i t i on s . From t h e eng i n ee r i ng po i n t

of view, it i s d i f f i c u l t t o p r e sc r i be a n yt hi ng o th e r t ha n t h e t h r e e n a t u r a l

boundary c on di t io ns i nv olv ing moments and fo rc e s or t h e c o r r e s p o n d i n g r o t a t i o n s

and displaceme nts . Consequent ly , a s i x t h -o r de r bend ing t heo ry which r equ i r e s

t h e t h r ee boundary cond i t i ons i s d e s i r a b l e . T h i s paper w a s aimed a t develop ing

t h e b e s t p o s s i b le t h eo r y r e s t r i c t i n g t h e o r d e r of t h e t h e o ry t o s i x .

The theory i s devel oped f o r t h r ee cas es . F i r s t , t h e f o r m u l a t i o n i s

ca r r i e d ou t f o r an unsymmetr ic l ami na t e . Here, t h e c o n s t i t u t i v e e q u a t i o ns con

ne ct in g moments and fo rc es t o ave rage d i s p l acemen t s and ro t a t i ons are der ived .

26



S eco nd , s i m p l i f i e d fo rm s of t h e s e c o n s t i t u t i v e eq u a t i o n s a re d e r iv e d f o r a

symmetr ic lamina te . In both th es e cases, t h e v a r io u s r e l a t i o n s a re d e ri v ed i n

m a t r i x fo rm t o a v o i d t e d i o u s a l g e b r a a nd a l s o t o f a c i l i t a t e r o u t i n e c om pu tation

on a computer. L a s t , f u r t h e r s i m p l i f i e d r e l a t i o n s a r e d e r iv e d f o r a symmetric

c r os s p ly wh ic h i s a c a s e of c l a s s i c a l o r th o t ro p y . I n t h i s case, it w a s found

t h a t m a t r ix f o r m u l a t i o n co u ld be d i sp e ns e d w i th an d t h e v a r i o u s r e l a t i o n s a r e

o b t a i n e d i n t h e fo rm of s imple formulas .

For t h e l i m i t i n g case of i s o t r o p y , t h e p r e s e n t t h e o r y re d uc e s t o an

improve d ve r s io n o f Mind l in ' s t he o r y . The she a r c o r r e c t i on f a c t o r o f 5 /6 i n

t h i s c as e i s o b t a i n e d from t h e t h e o r y r a t h e r t h a n w i t h a spe c i f i c e xa m ple as i n

Mindl in ' s approach .

Of p a r t i c u l a r i n t e r e s t s h o u ld be t h e method of c om p ut in g t h e i n t e r l a m i n a r

shea r stress s u gg e st ed i n t h i s paper. The use of e qu i l i b r iu m e qua t ions l e a d s

t o s i n g le - v al u e d s h e a r s t resses u n l i k e s t r a i g h t f o r w a r d co m pu ta ti on fro m

disp lacements .

The a cc u r ac y o f t h e p r e se n t t he o r y i s demonst ra ted by c o n s i d e r i n g a

numer ica l example of c y l i n d r i c a l bending of a t h r e e - l a y e r e d p l a t e .

Langley Research Center

Na t io na l Aeronaut ic s and Space Adm inis t ra t ion

Hampton, VA 23665

June 30, 1981

27



APPENDIX A

D E F I N I T I O N O F AVERAGE VALUES O F DISPLACEMENTS AND ROTATIONS

L e t t h e d i sp l a c e m e n t s u a nd v be a r b i t r a r y f u n c t i o n s of z . L e t Uand V r e p r e s e n t a v e r a g e values of d isp lacem ents and f i x and By r e p r e s e n t

a ve r a ge va lu e s o f r o t a t i on s . Aver ag ing he r e m eans a ve r a g ing th r ough th e

t h i c k n e s s . E s s e n t i a l l y , t h e defo rm ed s h ap e o f a l i n e normal t o t h e m id dl e

s u r f a c e i s sough t t o be a pp rox im a ted by a s t r a i g h t l i n e s o t h a t i t s d e v i a t i o n

f rom the t rue de formed shape i s minimum. To t h i s e nd , t he . d i sp l a c e m e n t s u

a nd v a r e e xp r e s se d as U + Bxz and V + Byz, re sp ec t i ve ly . Then, th e devi

a t i o n f rom th e t r u e de fo rm ed shape i s r e p r e s e n t e d by t h e f o l l o w i n g i n t e g r a l s o f

s q u a r e s o f e r r o r s i n d e t er m i n in g u a nd v :

h/2

( u - u - Bx Z l 2 dzEU =

The be s t l e a s t squ a r e a pp r ox im a t ion i s t h a t w h i c h s a t i s f i e s t h e c o n d i t i o n s

Equa t ions A I ) y i e l d t h e d e f i n i t i o n s g iv en by e q u a t i o n s ( 1 1 ) an d ( 1 2 ) . The

d e f i n i t i o n s g i v en by e q u a t i o n s ( 1 1 ) a nd ( 1 2 ) are i d e n t i c a l w i th t h o s e o b t a in e d

by Timoshenko and Woinowsky-Krieger ( r e f . 10) from co n si d er at io n s of work done

by t h e f o r c e s and m om ents. However, t h e de r iv a t i on i n r e f e r e nc e 10 i s v a l i d

on ly i f t h e st ress v a r i e s l i n e a r l y t h ro ug h t h e t h i c k n e s s . The stress v a r i a t i o n

f o r a l a m ina te d p l a t e can, a t t h e b e s t , be only piecewise l i n e a r . I n s uch a

case , it would be necessa ry t o resor t t o a m a th em a ti ca l d e f i n i t i o n a s g i ve n

h e r e i n .

28



APPENDIX B

DERIVATION OF COEFFICIENTS IN THE GOVERNING EQUATION

FOR A SYMMETRIC LAMINATE

E l i m i n a t i o n of Qx and Q f rom equat ions S O ) , (601, and ( 6 1 ) r e s u l t s

i n -

By s u b s t i t u t i n g f o r t h e d i f f e r e n t i a l o p e r at o rs A 1 t o A7 i n terms of ele-

ment s of t h e ma t r i x [ e ] , t h e f o ll o w in g e x p re s si o ns f o r c o e f f i c i e n t s i n t h e

gove rn i ng equa t i on ( 6 3 ) are obta ined :

a60 = 16 31 - 11 36

a33 = e 1 1 e 2 5 + el6eZ2 + e 3 4 e 1 2 + e 3 3 ) + e37 ( e21 + e33)

( 13 + e31 ) ( e z 4 + e 3 5 ) + e l 7 + e 3 6 ) e 2 3 + e 3 2 - e31 e35 - '32 36

- ,4( 23 + 8 3 2 ) - e2, e13 + e 3 1 ) - e l 2 + e33) (e26 + e37)

- ( e l 5 + e34) e21 + e 3 3 )

29



a06

= e25 32 - '22 '35

a40

= e l l

a31

= €Il3 + 2831

a22 = e12 + 821 + 2833

a1 3

= 823 + 2832

a = 82204

-b20 - '14 '36

-b l l - '15 '26 '34 '37

-b02 '27 '35

b40 = '16'34 - '14'36

3



b22 = 16 25 34 37 (' 17 + e36 (e24 + e35

- 14 27 - e,, ,, - e , s + e34)(e26 + e3 ,

b04 = 25 37 - 27 35

31



REFERENCES

1. Reissner , E.: Note on th e E f fe c t of Transverse Shear Deformat ion i n

Laminated An iso tro pi c P la te s . Comput. Methods A p p l . Mech. & Eng.,

vol . 20, no. 2, Nov. 1979, pp. 203-209.

2 . Lo, K. H.; C h r i s t e n s e n , R. M.; and Wu, E. M.: A High-Order Theory of P la t e

Deformat ion . Pa r t 2: Lam inated P l a t e s . Tra ns. ASME, Ser. E: J. A p p l .

Mech., vol. 44, no. 4, D e c . 1977, pp. 669-676.

3. Yang, P. Constance; N o r r i s , Char les H.; and Stav sky , Yehuda: E l a s t i c Wave

P r opa ga t ion i n He te rogeneous P la t e s . I n t . J. S o l i d s & S t r u c t . , v o l . 2,

no. 4, Oct. 1966, pp. 665-684.

4 . Whitney, J. M.; and Pagano, N. J. : Shear Deformation i n Heterogeneous

A n i s o t r o p i c P l a t e s . J. Ap p l . Mech., vol. 37, no. 4, D e c . 1970,

pp. 1031-1036.

5. Chou, Pei C h i ; and Car leone , Joseph: Transverse Shear i n Lamina ted P l a t eTheories . AIAA J., v o l . 1 1 , no. 9 , Sept. 1973, pp. 1333-1336.

6 . Cohen, Gerald A . : Transverse Shear S t i f f n es s of Lamina ted Aniso t rop ic

S h e l l s . Comput. Methods Appl. Mech. & Eng., vol . 13, no. 2, Feb. 1978,

pp. 205-220.

7 . Mindlin, R. D. : I n f l u e n c e of Rota tory I n e r t i a and Shear on F lexura l

M otio ns of I s o t r o p i c , E l a s t i c P l a t e s . J. Ap p l . Mech., vol. 18, no. 1 ,

Mar. 1951, pp. 31-38.

8 . Reissner , E r i c : The Ef fe ct of Tran svers e Shear Deformation on t h e Bending

o f E l a s t i c P l a t e s . J. Appl. Mech., vol. 12, no. 2, June 1945,

pp. A69-A77.

9 . Pagano, N. J.: Exa ct S o lu t ion s f o r C om posite La m inates i n C y l in d r i c a l

Bending. J. Compos. Mater., v o l . 3, J u l y 1969, pp. 398-411.

10. Timoshenko, S . ; and Woinowsky-Krieger, S.: The or y o f P l a t e s a nd S h e l l s ,

Second ed. M c G r a w - H i l l book Co., In c., 1959, pp. 165-171.

32



QY

Z

Figure 1 . - System of coordinates and stress resultants.

33



sin nX

Figure 2.- Cylindrical bending of three-layered symmetric cross ply.

34



2 . 0

1 . 8

1 . 6xrcI

E

m X1= mE

1 . 4W

0

1 . 2

1 . 0E3

.-EXro . 8E

.6

. 4

. 2

0

Exact so lu ti on (ref. 9

ey and Pagano (ref. 4)

Classical laminated plate theory

1 . ..

0 1 5 20 25

Thickness parameter, slh

Figure 3.- Maximum deflection as function of thickness parameter.

35



20

15

10

5

0

5

/Exact solution (ref. 9

Classical laminated plate theory

as well as

Whi tn ey and Pagano's th eo ryw i t h k =1 and k = 213 (ref. 4)

I

0 .1 . 5

I / z'h

j-90° layer 0 layer

,/[surfaceMiddle

F i g u r e 4. - D i s t r i b u t i o n of b e n di ng s t ress t h r o u g h t h i c k n e s s .

36



1. Report No. 2. Government Accession NO.

NASA TP-1903

7. Author(s1

M. V. V. Murthy

..-. -

9. Performing Organization Name and Address

NASA Langley Re search C en ter

Hampton, VA 23665

.~~

12. Sponsoring Agency Name and Address

Na tiona l Aerona utics and Space Adm inistration

Washington, DC 20546

5. Supplementary Notes

3. Recipient's Catalog No.

5. Repor t Date

November 1981

6. Performing Organization Code

505-33-33-05.

8. Performing Organization Report No.

L- 14533

10. Work Unit No.

11. Contract or Grant No.

13. Type of R e p o r t and Period Covered

Technical Paper

14. Sponsoring Agency Code

M. V. V. Murthy: NRC-NASA Resid ent Research As soc iat e. S c i e n t i s t (on lea ve from )Na tiona l Aero nautical Labor atory, Bangalore-560017, In di a.

. . .. . ... -

6. Abstract

An improved t rans vers e shear deformation theory fo r l aminated an i so t rop ic p la te s

under bending i s presen ted . The theory e l iminates the need for an a rb i t ra r i ly chose]

shea r co rre ct io n fac tor . For a general laminate wi th coupled bending and s t re tc h in g

the cons t i tu t ive equat ions connect ing s t ress resul tants wi th average displacements

and ro ta t ion s a re der ived . Simp l i f i ed forms of these r e l a t io ns are a l s o ob ta ined fo i

the spec ia l case of a symmetric lam inate with uncoupled bending. The governing

e q ua t io n f o r t h i s s p e c i a l case i s obtained as a s ixth-ord er equat ion fo r th e normal

d isp lacement req u i r ing pres cr ip t ion of the th ree phy s ica l ly na tur a l boundarycondi t ions along each edge. For th e l im it i ng case of iso trop y, t he pres ent theory

reduces t o an improved vers ion of Mindl in 's theory. Numerical re su l t s ar e obtained

from th e pre sen t theory f o r an example of a l aminated p la te under c y l in dr i ca l

bending. Comparison with r e s u lt s from exact three-dimen sional an al ys is shows t h a t

t h e p re s en t t heo ry i s more accura te than othe r th eo rie s of equiva lent order .

7. Key Words (Suggested by Author(s1) 18. Distribution Statement

Shear deformation Unclass i f i ed - Unlimited

Laminated plates

An i s o t rop i c p l a t e s

Pla te bending theory

Higher o rder p la te theory Subject Category

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An Improved Transverse Shear Deformation Theory for Laminated Anisotropic Plates03615_1982003615

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