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An Improved Transverse Shear Deformation Theory for Laminated Anisotropic Plates03615_1982003615
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8/13/2019 An Improved Transverse Shear Deformation Theory for Laminated Anisotropic Plates03615_1982003615
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
8/13/2019 An Improved Transverse Shear Deformation Theory for Laminated Anisotropic Plates03615_1982003615
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
8/13/2019 An Improved Transverse Shear Deformation Theory for Laminated Anisotropic Plates03615_1982003615
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 )
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8/13/2019 An Improved Transverse Shear Deformation Theory for Laminated Anisotropic Plates03615_1982003615
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
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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
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8/13/2019 An Improved Transverse Shear Deformation Theory for Laminated Anisotropic Plates03615_1982003615
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.
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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
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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
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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).
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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
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8/13/2019 An Improved Transverse Shear Deformation Theory for Laminated Anisotropic Plates03615_1982003615
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 .
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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 .
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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 .
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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.
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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