Critical_review_fiber Reinforced Composites and Sandwiches

8/12/2019 Critical_review_fiber Reinforced Composites and Sandwiches

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Com posite Structures 2 3 1 9 9 3 ) 2 9 3 - 3 1 2

A c r it ic a l r e v ie w a n d s o m e r e s u l t s o f r e c e n t lyd e v e l o p e d r e fin e d t h e o r i e s o f f ib e r - re in f o r ce dla m in a te d c o m p o s i t e s a n d s a n d w i c h e s

Mal l i k ar j u n aMe chan ical Engineering Department University of Toronto Toronto Cana da - M 5S 1A 4

T . Kan tCivil Engineering Department India n Institute o f Technology Bom bay India - 400 076

A c r i ti c a l re v i e w o f l i t e ra t u r e p e r t i n e n t t o t h e s u b j e c t m a t t e r o f t h is p a p e r w a sc a r d e d o u t u n d e r t h e f o l l o win g two b r o a d h e a d i n g s: f r e e v ib r a t i o n a n d t r a n s i e n td y n a m i c s . Ea c h o f t h e s e g r o u p s d e s c r i b e s th e v a r i o u s t h e o r e t i c a l d e v e l o p m e n t si n f i b e r r e i n f o r c e d l a m i n a t e d c o m p o s i t e a n d s a n d wi c h p l a t e s . Th e t h e o r e t i c a ld e v e l o p m e n t s a r e f u r t h e r c la s s if i ed a c c o r d i n g t o t h e r e f i n e m e n t / a c c u r a c y o f t h et h e o r i e s d e v e l o p e d , s u c h a s t h e c l a s s ic a l t h e o r y , t h e f i r s t -o r d e r s h e a r d e f o r m a -t i o n t h e o r y , a n d t h e t h r e e - d i m e n s i o n a l e l a s t i c i t y / h i g h e r - o r d e r s h e a r d e f o r m a -t i o n t h e o r i e s . Th e p r e s e n t l i t e r a t u r e r e v i e w i s l i m i t e d t o l i n e a r f r e e v i b r a t i o na n d t r a n s i e n t d y n a m i c a n a l y se s , a n d g e o m e t r i c n o n l i n e a r t r a n s ie n t r e s p o n s e o fm u l t i l a y e r s a n d wi c h / f i b e r - r e i n f o r c e d c o m p o s i t e p l a te s . A c o m p a r a t i v e s t u d y o fr e c e n t l y d e v e l o p e d r e f i n e d t h e o r i e s i n c o n j u n c t i o n w i t h t h e C i s o p a r a m e t r i cf i n i t e d e m e n t f o r m u l a t i o n h a s b e e n m a d e a n d t h e c o n c l u s i o n s w e r e d r a w nb a s e d o n t h e l i t e r a t u r e r e v i e w a n d t h e r e f i n e d t h e o r i e s r e s u l t s . I n o r d e r t oc o m p a r e t h e p r e s e n t r e s u l t s wi t h t h e a v a i l a b l e r e s u l t s a n d t o p r o v i d e a n e a s ym e a n s f o r f u t u r e c o m p a r i s o n s b y o t h e r i n v e s t i g a to r s , th e n u m e r i c a l r e s u lt s a r ep r e s e n t e d i n t a b u l a r fo r m .

I N T R O D U C T I O NE n g i n e e r s h a v e a w i d e s c o p e i n c o m p o s i t e s tr u c -tu ra l des ign because o f the var i e ty o f cons t i tuen tm a t e r i a ls w h i c h c a n b e e m p l o y e d a n d t h e n u m e r -o u s o p t i o n s i n f ib e r o r i e n t a ti o n s a n d t h e l a m i n a sa r r a n g e m e n t s . M o s t s t r u c tu r e s , w h e t h e r t h e y a r eused on l and , s ea o r in the a i r a r e sub jec ted tod y n a m i c l o a d s d u r i n g t h e i r o p e r a t i o n . T h u s ,t h e o r i e s w h i c h c a n p r e d i c t t h e c o m p l e t e b e h a v i o rb e c o m e n e c e s s a r y fo r b e t t e r u n d e r s t a n d i n g o f t h ec o m p l e x f a i l u r e m e c h a n i s m a n d s t r e n g th o f m u l t i-l ayer comp os i t e s t ruc tu res .

F o r m a t h e m a t i c a l m o d e l i n g p u r p o s e s , t h e i n d i-v i d u a l l a y e r ( l a m i n a ) i s c o n s i d e r e d t o b e h o m o -g e n e o u s a n d o r t h o t r o p i c w h i l e t h e l a m i n a t e i sh e t e r o g e n e o u s t h r o u g h t h e t h i c k n e s s a n d g e n e r -a l ly an i so t rop ic . T he grea te r d i f f e rences in e las t icp r o p e r t i e s b e t w e e n f i b e r f i l a m e n t s a n d m a t r i xma ter i a l s l ead to a h igh r a t io o f in -p lane Youn g sm o d u l u s t o t r a n s v e r s e s h e a r m o d u l u s f o r m o s t o ft h e c o m p o s i t e l a m i n a t e s d e v e l o p e d t o d a t e . T h i s 293

makes the c l as s i ca l l amina t ion theory , whichneg lec t s the e f f ec t o f ou t -o f -p lane s t r a ins , i nade-q u a t e f o r t h e a n a l y s i s o f m u l t i l a y e r c o m p o s i t ep la t es . Thus , in o rder to have a r e l i ab le ana lys i sand s a fe des igns , more accura te theor i es whichinc lude the e f f ec t s o f t r ansver se shear deforma-t ion becom e neces sary . As a r esu l t, a cons ide rab leam oun t o f wo rk in inves t iga t ing the e f f ec t s o ft r a n s v e rs e s h e a r d e f o r m a t i o n a n d r o t a t o r y i n e r ti ahas been conduc ted l ead ing to the so-ca l l ed f i r s t -o r d e r s h e a r d e f o r m a t i o n t h e o ri e s . T h e f i r s t -o r d e rt h e o r ie s a s s u m e a c o n s t a n t s h e a r r o t a t i o n t h r o u g hthe p la t e th i cknes s and thus r equ i r e the use o f ashear cor rec t ion coef f i c i en t whose accura te p re -d i c t i o n f o r a n i s o t r o p i c l a m i n a t e s i s c u m b e r s o m ea n d p r o b l e m d e p e n d e n t . I t i s c l e a r t h a t t h e s etheor i es do no t inc lude the e f f ec t s o f c ros s - sec-t iona l warp ing which i s def in i t e ly es sen t i a l fo rt h i c k s a n d w i c h p l a t e s w h i c h a r e g e n e r a l l y c o m -p o s e d o f a m i d d l e w e a k c o r e s a n d w i c h e d b e t w e e ns t i f f facings . Fur ther , th e ef fects of t ran sve rsenormal s t r es s / s t r a in which a re neg lec ted in f i r s t -

Com posite Structures 0 2 6 3 - 8 2 2 3 / 9 3 / S0 6 . 0 0 1 9 9 3 E l s e v i e r Sc i e n c e Pu b l i s h e r s L t d , En g l a n d . P r i n t e d in Gr e a t Br it a i n


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Refin ed theories of iber reinforced lam inated com posites and sandwiches 295ized formula t ions appl icab le to mul t i layer sand-w ic h p l a t e s h a v e b e e n r e p o r t e d b y K h a tu a a n dC h e u n g2 u s in g n o n -c o n fo rm in g r e c t a n g u la re l e m e n ts . T h e c o n d i t i o n o f c o m m o n sh e a r a n g lefo r a l l c o re s h a s n o t b e e n a s su m e d in t h e fo rm u la -t ion.L e e a n d C h a n g 2 d e r iv e d a n a ly t ic a l f r e q u e n c ye q u a t io n s fo r p l a n e w a v e s f ro m th e th r e e -d im e n -s iona l equa t ions of e las t ic i ty . They computeddispers ion curves for two ve ry d i f fe ren t sandwichp la te s a n d d i s cu s se d th e c o r r e sp o n d in g d i f f e r e n c ein t h e p h y s i c a l b e h a v io r . T h e i r t r e a tm e n t c o u ld b every usefu l in tes t ing num er ica l ly the accuracy o fa p p ro x im a te t h e o ri e s . N g a n d L a m 22 p re se n te dd i sp l a c e m e n t -b a se d f in i t e e l e m e n t fo rm u la t io n sus ing a para l le logrammic e lement hav ing f ived e g re e s o f f r e e d o m p e r n o d e fo r s t a t i c a n ddynamic ana lys is o f skew sandwich p la tes . Thefo rm u la t io n s a r e v a l id fo r a s a n d w ic h p l a t e m a d eu p o f i so t ro p ic f a c e s o f e q u a l t h i c k n e s s a n d a no r th o t ro p ic c o re . S a y i r a n d K o l l e r23 discussed thep h y s i c a l b e h a v io r o f b e n d in g w a v e s i n s a n d w ic hpla tes in which the fac ings a re th in , s t i ff and heavya s c o m p a re d w i th t h e c o re . B y m e a n s o f a sy m p -to t ic expans ions of the bas ic equa t ions of l ineare las t ic i ty , i t was shown tha t d i f fe ren t ' phys ica lm e c h a n i sm s ' p r e d o m in a te i n d i f f e r e n t f r e q u e n c yranges. Ravi l le and Uen g 24 exper im enta l ly de te r -m in e d th e n a tu ra l f r e q u e n c ie s o f v ib ra t io n o f asandwich p la te .Fiber re inforced com po si te platesT h e a v a i la b l e l i t e ra tu re r e l a t e d to f r e e v ib ra t io nana lys is o f f iber re in forced composi te p la tes i sc lass i f ied in to th ree groups . The f i r s t g roupdescr ibes the c lass ica l ( th in p la te ) theory whichneglec ts the e f fec ts o f shear s t ra ins , norm al s t ra ina n d n o rm a l s t r es s i n t h e t r a n sv e r se d i r ec t io n . T h i si s c o m m o n ly k n o w n a s t h e c l a s s i c a l l a m in a t io ntheo ry (CLT) and i t i s an ex tens ion of the c lass ica lp l a t e t h e o ry25-28 to lam ina ted p lates. T he l imita-t i o n s o f C L T h a v e l e d to t h e d e v e lo p m e n t o f t h ef i r s t -o rd e r sh e a r d e fo rm a t io n th e o ry (F O S T )w h ic h r e q u i r e s t h e u se o f sh e a r c o r r e c t io n c o e ff i -c ien ts and the l i te ra ture re la ted to th is i s descr ibedin th e s e c o n d g ro u p . T h e th i rd g ro u p d e sc r ib e sth e r e f in e d th e o r i e s w h ic h a r e e i t h e r b a se d o n th eth re e -d im e n s io n a l (3 D ) a p p ro a c h o r t h e tw o -d im e n s io n a l (2 D ) a p p ro a c h w i th h ig h e r -o rd e rd i sp l a c e m e n t m o d e l s g ivin g p a ra b o l i c v a r i a t i o n o ft ransverse shear s t ra ins th rough the p la te th ick-n e s s a n d th u s r e q u i t i n g n o sh e a r c o r r e c t io n c o e f f i-cients . Be rt and Francis , 29 and Be rt 3-33 hav e

presen ted a de ta i led rev iew of the l i te ra turere la ted to the s t ruc tura l mechanics ( s ta t ics andd y n a m ic s ) a sp e c ts o f c o m p o s i t e b e a m s , p l a t e s a n dshe l l s . A rev iew on f in i te e lement model ing ofp la tes i s g iven in Refs 34 and 35 .C l a s s i c a l l a m i n a t i o n t h e o r yThe c lass ica l lamina te theory ignores the th reet r a n sv e r se s t r a in c o m p o n e n t s a n d th e t r a n sv e r sen o rm a l s tr e ss c o m p o n e n t s a n d m o d e l s t h e l a m in -a te as a two-d imens iona l equ iva len t s ing le layer .T h i s s im p le th e o ry c a n p ro v id e r e a so n a b ly a c cu r -a te p red ic t ion on ly for re la t ive ly th in p la tes .Various texts 36-44 hav e desc ribed this t heo ry an di ts a p p l i c a t io n to t h e a n a ly s i s o f l a m in a te d c o m p o -s i te s t ruc tures . Lam ina te d p la te theor ies based onth e K i r c h h o f f h y p o th e se s h a v e b e e n d e v e lo p e d b yReissner and S tavsky ,45 Dong e t a l 4 6 a n dStavsky,47 and these deve lopments a re summar-ized in Ref . 40 . These w orks a re based o n a l inearlongi tud ina l d isp lacement d is t r ibu t ion across thee n t i r e l a m in a te w i th sh e a r d e fo rm a t io n n e g le c te d .F o r b e n d in g , b u c k l in g a n d f r e e v ib ra t io n a lana lyses o f a c lamped an iso t rop ic p la te a modi-f ied Four ie r se r ies me thod was used by W hi tney .48T sa y a n d R e d d y 49 presented a mixed f ini te ele-m e n t fo rm u la t io n b a se d o n R e i s sn e r' s v a r i a t io n a lpr inc ip le us ing a rec tangu lar e lement fo r bending ,s tab i l i ty and f ree v ibra t ion ana lyses o f i so t rop ica n d o r th o t ro p ic t h in p l a t e s . I t w a s sh o w n b yL a u r a et al .5.5~ t h a t t h e p o ly n o m ia l c o o rd in a t efu n c t io n s u se d to o b ta in t h e fu n d a m e n ta l f r e -quency of t ransverse v ibra t ion of th in , e las t icp l a te s b y m a k in g u se o f t h e R a y le ig h -R i t z m e th o dyie ld be t te r accuracy . Don g and Lop ez 52 de te r -m in e d th e n a tu ra l f r e q u en c ie s a n d m o d e sh a p e s o fa c lamped c i rcu la r p la te wi th rec t i l inear o r tho-t ro p y b y a m o d i f i e d a p p l i c a t io n o f t h e i n t e r io rco l loca t io n meth od . A leas t -squares e r ror f i t wasthen used to genera te the govern ing e igenvaluep ro b le m . T h i s m e th o d w a s s im p le t o im p le m e n ta n d h a d a n a d v a n ta g e o v e r t h e R a y le ig h -R i t z a n dG a le rk in m e th o d s w h e re e n e rg y o r e r ro r i n te g ra lsh a d to b e e v a lu a te d a n a ly t i c a lly b e fo re n u m e r i c a la n a ly s is . H o w e v e r , c o l lo c a t io n d o e s n o t p ro v id ea n u p p e r b o u n d a s i n R a y le ig h -R i t z . I y e n g a rU m a r e t i y a53 made an a t tempt to ob ta in the f reev ib ra t io n r e sp o n se o f h y b r id , l a m in a te d r e c t a n g u -l a r a n d sk e w p lat e s. T h e G a le rk in t e c h n iq u e w a se m p lo y e d to o b ta in a n a p p ro x im a te so lu t io n o fthe govern ing d i f fe ren t ia l equa t ions . Opt imald e s ig n s o f l a m in a te d p l a t e s h a v e b e e n g iv e n inseveral s tudies 54, 55 of wh ich only a few are refer-enced here , w i th respec t to na tu ra l f requenc ies in


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296 Mallikar]una T. Ka ntwhich the e f f ec t o f shear deformat ion was neg-lec ted . The neg lec t o f shear deformat ion l eads todes igns tha t a r e on ly subopt imal . Indeed , theva lues o f op t imum f iber o r i en ta t ions and l ayerth icknes ses depend on the s ide- to - th icknes s r a t ioa n d c o n s e q u e n t l y F O S T a n d C P T y i e ld d i ff e r e n to p t i m u m p o i n ts .Firs t-order shear deform at ion theor iesTh e c las s i ca l p l a t e theory w hich ignores the t r ans -v e r se s h e a r a n d t r a n s v e r se n o r m a l d e f o r m a t i o n s i si n a d e q u a t e f o r t h e a n a l y s i s o f m o d e r a t e l y t h i c kp la tes . Fur ther , t he shear deformat ion e f f ec t s a r em o r e p r o n o u n c e d i n f i b e r -r e i n f o rc e d c o m p o s i t el amina tes in compar i son wi th the i so t rop ic p la t esdue to the h igh r a t io o f in -p lane m odulu s to t r ans -ver se shear m odulus o f f iber r e in forced m ater i a ls .T h e s e h i g h r a t i o s m a k e c l a s s i c a l l a m i n a t i o ntheory inadequate fo r the ana lys i s o f f iber - r e in -f o r c e d c o m p o s i t e p l a t e s . A n a d e q u a t e t h e o r ymus t account fo r t r ansver se shear e f f ec t s . Thed e v e l o p m e n t o f F O S T b e g a n w i t h t h e w o r k o fRei s sner 56 and M indl in ~ fo r i so t rop ic p la t es . Th ea p p r o a c h w a s e x t e n d e d t o e m b r a c e l a m i n a t e dcompos i t e p la t es by Yang , Nor r i s and S tavsky( Y N S y a n d W h i t n e y a n d P a g a n o 58,59 f o rdyna mic ana lys i s. For b end ing , s t ab i l ity and v ibra-t ion ana lys i s o f spec ia l ly o r tho t rop ic o r t r ans -ver se ly i so t rop ic p la t es, Am ba r t sum ya n 6 hass y s t em a t i c a l ly p r e s e n t e d t h e g o v e r n i n g e q u a t i o n swh ich inco rpora te the t r ansver se she ar e ff ec ts . O ns imi la r l ines , Vinson Cho u 61 p u b l i s h e d a b o o kwhich inc ludes compos i t e p la t e as wel l as she l ls t r u c t u r e s . W h i t n e y62 noted that the values ofshear cor rec t ion coef f i c i en t s fo r o r tho t rop icl a m i n a te s d e p e n d e d o n t h e d e t a il s o f t h e l a m i n a t econs t ruc t ion . Do ng and Ne l son 63 s tud ied thev i b r a t i o n s o f a l a m i n a t e d p l a t e c o m p o s e d o f a na r b i t r a r y n u m b e r o f b o n d e d e l a s t i c , o r t h o t r o p i clayer s . The ana lys i s was car r i ed ou t wi th in thef r ame wo rk of l inear e l as t ic i ty fo r p lane- s t r a inbehav ior .

S u n a n d Whitney64 i nves t iga ted the e f f ec t o fh e t e r o g e n e o u s s h e a r d e f o r m a t i o n o v e r t h e t h i c k -n e s s o f p l a t e o n t h e d y n a m i c b e h a v i o r o f l a m i n -a ted p la t es . Three s e t s o f govern ing equa t ionsw e r e d e r i v e d a c c o r d i n g t o d i f f e r e n t a s s u m p t i o n so n t h e l o c a l t r a n s v e r se s h e a r d e f o r m a t i o n a n d t h ein te r f ace condi t ions . For t i e r Rosse t tos 65ana lysed f r ee v ib ra t ion o f r ec tangular p la t es o fu n s y m m e t r i c c r o s s - p l y c o n s t r u c t i o n w h i l e S i n h aa n d R a t h c o n s i d e r e d b o t h v i b r a t i o n a n d b u c k -l ing fo r the s ame type o f p la tes . Us ing a th ick f in it es t r ip approach , Hi nt on 67 p r e s e n t e d a n o t e o n t h e

f r ee v ib ra t ion o f l amina ted p la t es inc lud ing t r ans -ver se shear e f f ec t s and ro ta ry iner t i a . Ber t andCh en 68 have g iven a c losed fo rm so lu t ion us ing aYNS theory fo r the f r ee v ib ra t ion o f s imply sup-por ted r ec tangular p la t es o f an t i symmet r i c ang le-p ly l amina tes . Th e e f f ec t o f de le t ing ro ta ry iner t i aand in -p lane iner t ia , s ing ly and in comb ina t ionwere a l so inves t iga ted . Cra ig and Da we 69 7s tud ied the f l exura l v ib ra t ion o f r ec tangular l am-ina ted p la t es us ing FOST. Two numer ica l t ech-n iques were employed in the s tudy , v iz . t heR a y l e i g h - R i t z m e t h o d a n d t h e f i n i t e - s t r i pmethod , and in bo th , the t r i a l d i sp lacement func-t io n s m a k e u s e o f t h e n o r m a l m o d e s o f v i b r a t io no f T i m o s h e n k o B e a m s . C h e n R a r n k u m a r 7~formu la ted a the ory fo r the ana lys i s o f c l ampedor tho t rop ic p la t es by us ing a Lagrang ian mul t i -p l i e r t echn ique fo r the so lu t ion o f the s t a t i c ande igenvalue p rob lem.

W h i l e c o n s i d e r a b l e e f f o rt h a s b e e n e x p a n d e d i nthe f in i t e e l ement v ib ra t ion ana lys i s o f i so t rop icp la tes , on ly l imi ted inves t iga t ions o f l amina tedan i so t rop ic p la t es can be found in the l i t e r a tu re .H i n t o n e t a l . 7 2 7 3 out l ined a par t i cu la r lumpingp r o c e s s t o s h o w t h a t g o o d a c c u r a c y c a n b eo b t a i n e d i n a l i n e a r a n d n o n - l i n e a r d y n a m i cprob lem us ing i soparamet r i c parabo l i c e l ement s .T h e p r o c e d u r e o f l u m p i n g r e c o m m e n d e d i n v i ewof the in f in i t e pos s ib i l i t i es o f f e red i s to computethe d iagonal t e rms of the cons i s t en t mass mat r ixand then s ca le these t e rms so as to p reserve theto ta l mass o f the e l ement . Reddy T u s e d t h e Y N Stheory fo r f r ee v ib ra t ion o f an t i symmet r i c ang le-p ly l amina ted p la t es wi th a f in it e e l em ent fo rmu la-tion.Ref ined theor iesT h e f i r s t - o r d e r s h e a r d e f o r m a t i o n t h e o r y w h i c hignores the e f f ec t s o f c ros s - sec t iona l warp ingleads to an unrea l i s t i c ( cons tan t ) var i a t ion o f thet r ansver se shear s t r es ses th rough the l amina teth icknes s . Development o f r e f ined 2D theor ies ,w h i c h i n c o r p o r a t e h i g h e r - o r d e r m o d e s o f t r a n s -v e r se c r o s s - s e c ti o n a l d e f o r m a t i o n a n d a c c o u n t f o r2 D / 3 D s ta t e o f s t r e ss / s tr a i n h a s b e e n a t t e m p t e d i nrecen t year s . These theor i es dep ic t a r ea l i s t i cparabo l i c var i a t ion o f t r ansver se shear s t r es sest h r o u g h t h e l a m i n a t e t h i c k n e s s a n d d o n o t r e q u i r ethe use o f as sumed shear cor rec t ion coef f i c i en t sas in the case o f f i r s t -o rder Rei s sner /Mindl intheory.

Sr in ivas and Ra o 75,76 der ived the gove rn inge q u a t i o n s f o r t h e b e n d i n g , f r e e v i b r a t i o n a n db u c k l i n g a n a l y s e s o f s im p l y s u p p o r t e d t h i c k is o -


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Refined theories of iber reinforced laminated composites a nd sandwiches 297t ro p ic a n d o r th o t ro p ic r e c t a n g u la r l a m in a t e s u sin gthree-d imens iona l theory of e las t ic i ty . So lu t ionso f t h e 3 D e l a st i ci t y t h e o ry fo r f r e e v ib ra t io n o fm u l t i l a y e re d c o m p o s i t e p l a t e s w e re o b ta in e d b yN o o r 7 7 USing a h igh er-ord er f in i te d i f fe rencescheme. Such a sch eme was s how n to g ive h igh lyaccura te resu l t s fo r the respons e charac te r i s t ics o fthe p la te . Re issner78 formula ted a theory forf lexura l respo nse o f i so t rop ic p la tes by assum ing ac u b ic v a r i a t i o n o f i n -p l a n e d i sp l a c e m e n t s a n dp a ra b o l i c v a r i a t i o n o f t r a n sv e r se d i sp l a c e m e n tacross the p la te th ickness . Lo e t a / . 79 80 p re se n te da th e o ry fo r h o m o g e n e o u s i so t ro p ic79 an d lamin -a t e d c o m p o s i t e8 p la tes wh ich is o f the sam e o rderof app roxim at ion as tha t o f Re issner , 78 bu tinc ludes the te rms cont r ibu t ing to the in -p lanem o d e s o f d e fo rm a t io n . T h u s , t h e t h e o ry a c c o u n t sfo r t h e p a ra b o l i c v a r i a t i o n o f t r a n sv e r se sh e a rs t resses , t ransverse normal s t ra in /s t ress and ac u b ic v a r i a t i o n o f t h e i n -p l a n e d i sp l a c e m e n t sacross the p la te th ickness . The pr inc ip le o fs t a t i o n a ry p o te n t i a l e n e rg y h a s b e e n u se d toder ive the govern ing d i f fe ren t ia l equa t ions .B a se d o n th e a s su m e d d i sp l a c e m e n t f i e ld o fReissner, T b o th L e v in so n81 and M urth y82 used thee q u i l i b r iu m e q u a t io n s o f t h e F O S T w h ic h a r ev a r i a t i o n a lly i n c o n s i s t e n t fo r t h e h ig h e r -o rd e rd i sp l a c e m e n t f i e ld u se d b y th e m w i th t h o sed e r iv e d f ro m th e p r in c ip l e o f v i r t u a l d i sp l a c e -ments . Th is fac t was no te d by R edd y 83 and hep re se n te d a c o n s i s t e n t d e r iv a t io n o f t h e a s so c i a t e de q u i l i b r iu m e q u a t io n s . T h e d i sp l a c e m e n t m o d e lu se d b y M u r th y , L e v in so n a n d R e d d y i s th e s a m ea n d i t c o n ta in s t h e s a m e n u m b e r o f d e p e n d e n tv a r i a b le s a s i n t h e F O S T a n d th e t h e o ry im p l i ci t lysa t i s f ies the f ree t ransverse shear condi t ions onto p a n d b o t to m su r fa c e s o f th e p l a t e. C lo se d - fo rmso lu t io n s w e re p re se n te d fo r s im p ly su p p o r t e dsy m m e t r i c c ro s s -p ly l a m in a te s . Mu r th y 84 real isedthe inab i l i ty o f ea r l ie r 81' 83 hig her -or der theo ries toeva lua te the t ransverse shear s t ra ins a t po in ts inth e p l a t e w h e re d i sp l a c e m e n t s a r e c o n s t r a in e d tobe zero , such as those on f ixed edges . To over-c o m e th i s l im i t a t i o n , a n a d d i t i o n a l p a r t i a l sh e a rd e f l e c t io n v a r i a b l e w a s in t ro d u c e d . T h u s , b a se don four bas ic d isp lacement var iab les ( two par t ia lt r a n sv e r se d e f l e c t io n s a n d tw o in -p l a n e d i sp la c e -m e n t s ) , t h e g o v e rn in g e q u a t io n s h a v e b e e nd e r iv e d u s in g a v a r i a t i o n a l p r in c ip l e a n d a r ep re se n te d in t h e fo rm o f fo u r s im u l t a n e o u s p a r t ia ld i f f e r e n t i a l e q u a t io n s . T h e f r e e t r a n sv e r se sh e a rc o n d i t i o n s o n th e b o u n d in g p l a n e o f t h e p l a t e a r en o t s a t i s f i e d d u e to t h e i n t ro d u c t io n o f p a r t i a lsh e a r d e f l e c t io n in t h e fo rm u la t io n .

K a n t 85 a d o p te d th e s e g m e n ta t io n m e th o d a n dder ived the govern ing equa t ions for l inear e las ticana lys is o f homogeneous i so t rop ic p la tes . La te r ,K a n t e t a l 8 6 p re se n te d a d i sp l a c e m e n t -b a se df in i te e lement fo rmula t ion us ing the d isp lace-me nt mo del o f K ant . 85 Pand ya and Ka nt 87-9i n v e s tig a te d th e b e h a v io r o f a n i so t ro p ic l a m in a t e dc o m p o s i t e p l a t e s b a se d o n v a r io u s a s su m e d d i s -p lace me nt f ie lds wi th s imp le i soparam etr ic f 'mi tee l e m e n t fo rm u la t io n s . T h e se st ud ie s 78-8,82-9were conf ine d to s ta t ic ana lys is .W hi tney a nd Sun 91 ex tend ed the theor ies o fY NS 57 a n d W h i tn e y a n d P a g a n o 59 to i n c lu d e th ef i r s t symmetr ic th ickness shear and th icknesss t r et c h m o d e s b y in c lu d in g h ig h e r -o rd e r t e rm s inth e d i sp l a c e m e n t e x p a n s io n a b o u t t h e m id -p l a n eo f t h e l a m in a t e i n a m a n n e r s im i l a r t o t h a t o fMin d l in Me d ic k 9e fo r h o m o g e n e o u s i so t ro p icp l at e s. B h im a ra d d i a n d Ste ven s 93 h a v e g iv e n so m eresu l t s fo r f ree v ibra t ion of o r tho t rop ic p la tes byu s in g a h ig h e r -o rd e r t h e o ry w i th c lo se d fo rmsolu t ion . They cons idered a to ta l o f f iveunkno wns , which a re the midd le sur face d isp lace-m e n t q u a n ti t ie s . T h e y m a in t a in th e h ig h e r -o rd e r( c ub ic) p o ly n o m ia l fo rm fo r i n -p l a n e d i sp l a c e m e n texpress ions and a t the same t ime the more rea l -i s t ic parabol ic var ia t ion for t ransverse shearstrains is achieve d.

T o d e t e rm in e th e n a tu ra l f r e q u e n c ie s a n db u c k l in g lo a d s o f o r th o t ro p ic l a m in a t e d p l a t e s ,R e d d y a n d P h a n 94 a n d P u tc h a a n d R e d d y 95 p re -sen ted a c losed form so lu t ion and mixed f in i tee lement fo rmula t ion , respec t ive ly , wi th theassumed d isp lacement f ie ld used ear l ie r in Ref .8 3 . O w e n a n d Li 96 97 presen ted a loca l f in i tee l e m e n t m o d e l b a s e d o n a n a p p r o x i m a t e t h e o r yfo r t h i c k a n i so t ro p ic l a m in a t e d p l a te s . T h e th r e e -d im e n s io n a l p ro b le m w a s r e d u c e d to a tw o -d im e n s io n a l o n e b y a s su m in g p i e c e w ise l i n e a rvar ia t ion of the in -p lane d isp lacemen ts u and v ,a n d a c o n s t a n t v a lu e o f t h e l a t e ra l d i sp la c e m e n t wacross the th ickness . A subs t ruc tur ing techniquewas used in the bending , v ibra t ion and buckl ingana lys is . A 3D e igh t -nod e hybr id s t ress f in itee l e m e n t w a s d e v e lo p e d fo r t h e f r e e v ib ra t io nana lyses o f lamin a ted p la tes b y Sun and L iou . 98T h i s h y b r id s t r es s m o d e l w a s b a se d o n th e m o d i -f i ed c o m p le m e n ta ry e n e rg y p r in c ip le a n d a l l t h r e ed i sp l a c e m e n t c o m p o n e n t s w e re a s su m e d to v a ryl inear ly th rou gh the th ickness o f each lam ina .Recent ly , M al l ik ar ju na and Ka nt 99-12 em pha-s ized on es tab l i sh ing the c red ib i l i ty o f h igher-o rd e r t h e o r i e s w i th d if f e r e n t d i sp l a c e m e n t m o d e l sfo r f r e e v ib ra t io n a n a ly s i s o f a n i so t ro p ic l a m in -


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298 M a l l i k a r ju n a T . K a n ta t e d c o m p o s i t e a n d s a n d w i c h p l a t e s . A s i m p l ei s o p a r a m e t r ic C O f i n it e e l e m e n t f o r m u l a t i o n w a spresen ted . The spec ia l mass mat r ix d iagona l i za -t i o n s c h e m e w a s a d o p t e d w h i c h c o n s e r v e s t h eto ta l mass o f the e l ement and inc luded the e f f ec t so f m a s s i n e r t i a t e r m s c o r r e s p o n d i n g t o a l l t h ed e g r e e s o f f r e e d o m .

T R A N S I E N T D Y N A M I C STh e ana lys t usua l ly has a t h i s d i sposa l a var i e ty o fm a t h e m a t i c a l m o d e l s o f v a r y i n g a n a l y t i c a l c o m -plex i ty and phys ica l f ide li ty . I f cons idera t ion s a reres t r i c t ed to suf f i c i en t ly smal l e l as ti c defo rma-t ions , the r esu l t ing theor i es can be cons idered tob e m o r e o r l e ss s a ti s fa c t o r y a p p r o x i m a t i o n s o f 3 Dclas s ica l e l as t i c i ty theory . Th e l a t t e r r epresen t s theu n d i s p u t e d s t a n d a r d o f a c c u r a c y w i t h i n th e l i m i ta -t ions o f e l as ti c ac t ion an d sm al l deform at ions . Th es t a t i c b e h a v i o r o f c o m p o s i t e l a m i n a t e s h a s b e e nr e a s o n a b l y w e l l e s t a b l is h e d a n a l y t ic a l l y a n dexper imenta l ly . In the case o f dynamic load ing ,however , the s t a t e o f the a r t i s in a deve lop ingstage.The l inear e l as t i c t r ans ien t r esponse o f i so -t rop ic p la t es has been inves t iga ted by s evera lresea rchers . Re ism an n an d his c ol league s ~3-15ana lysed a s imple suppor ted , r ec tangula r , i so -t rop ic p la t e sub jec ted to a sudden ly app l i edu n i f o r m l y d i s t r i b u t e d l o a d o v e r a s q u a r e a r e a o fthe p la t e . Exac t so lu t ion was ob ta ined us ing(class ical ) 3D elas t ici ty theory , and c lass icali m p r o v e d t h e o r ie s . T h e f i n it e d if f e r e n c e m e t h o d ,wide ly used in the so lu t ion o f the equa t ions o fm o t i o n w h i c h g o v e r n t h e t r a n s i e n t r e s p o n s e o fs t ruc tu res such as p la t es and she l l s , can becomeuns tab le un les s the r a t io o f the t ime mesh to thespace mesh s a t i s f i es a ce r t a in condi t ion . Thecondi t ion o f s t ab i l i ty o f the f in i t e d i f f e rencee q u a t i o n f o r t h e t r a n s i e n t r e s p o n s e o f a t h i n f l a tp l a t e a n d m o d e r a t e l y t h i c k p l a t e h a s b e e n g i v e nby Le ech , 16 an d Tsui an d Ton g, ~7 respect ively.Ro ck and H in to n ~8 prese n ted t r ans ien t f in i tee lem ent ana lys i s o f th i ck and th in i so t rop ic p la t es .T h e e l e m e n t is b a s e d o n t h e R e i s s n e r - M i n d l i n( R - M ) t h i c k p l a t e t h e o r y f o r h o m o g e n e o u s , i s o -t rop ic p la t es . Exce l l en t agreement o f the f in i t ee l e m e n t s o l u t i o n s w i t h t h e a n a l y t i c a l s o l u t i o n o fRe i sm ann Lee 13 was ob ta ined . H in to n 19a d o p t e d t h e F O S T f o r c i r c u l a r p l a t e b e n d i n gp r o b l e m s b y u s i n g a x i s y m m e t r i c p a r a b o l i c i s o -p a r a m e t r i c e l e m e n t s a n d a n e x p l i c i t t i m e m a r c h -

i n g s c h e m e w i t h a s p e c i a l m a s s l u m p i n gp r o c e d u r e . A u n i f o r m r e d u c e d i n t e g r a ti o n t e c h -n i q u e w a s u s e d . T h e s e pa pe rs 13 19 dea l t on lywi th l inear t rans ien t r espon se o f i so t rop ic p la t es .T h e s o l u t i o n o f l i n e a r a n d n o n l i n e a r d y n a m i ct r a n s ie n t p l a t e b e n d i n g p r o b l e m s w a s c o n s i d e r e db y H i n t o n e t a l . ~ and th ree s i tua t ions weree x a m i n e d : s m a ll d e f o r m a t i o n a n d l a rg e d e f o r m a -t ion e l as t i c r esponse , and smal l deformat ione las to -p las t i c r esponse employ ing the y ie ld c r i -t e r i a o f Von M ises and Tresca . An es t imate o f thecr i t i ca l t ime s t ep l eng th fo r the t r ans ien t so lu t ionof R -M p la tes , g iven by Tsu i an d Tong 17 wasu s e d w i t h m i n o r m o d i f i c a t i o n i n R e f s 1 0 9 a n d1 1 0 . S h a n t a r a m e t a l . 111 em ploy ed the F E M in thep r e d i c t i o n o f t h e t ra n s i e n t r e s p o n s e o f 2 D a n d 3 Dsol ids exh ib i t ing geom et r i c ( l a rge deformat ion)and mater i a l ( e l as to -p las t i c ) non l inear i t i es . P icaand H in to n t 12.113 presen ted a un i f i ed app roachfor the s t a t i c and t r ans ien t dynamic l inear andg e o m e t r i c a l l y n o n l i n e a r a n a l y s i s o f R - M p l a t e sinc lud ing in i t i a l imper fec t ions . A f in i t e e l ementi d e a l i z a t i o n w a s a d o p t e d a n d t h e q u a d r a t i cL a g r a n g i a n e l e m e n t s w e r e u s e d t o g e t h e r w i t hse lec t ive in tegra t ion . Ak ay ~~4 analysed large de-f l ec t ion t r ans ien t r esponse o f i so t rop ic p la t esu s i n g a f o u r - n o d e i s o p a r a m e t r i c m i x e d q u a d r i -l a te r a l e l e m e n t . D y n a m i c V o n K a r m a n p l a t e e q u a -t ions a re modi f i ed to inc lude the e f f ec t o ft r ansver se shear deformat ions as in Re i s sner p la t et h e o r y . F i n i t e e l e m e n t e q u a t i o n s o f m o t i o n a r eo b t a i n e d v i a a m i x e d G a l e r k i n a p p r o a c h w i t ht h r e e m o m e n t a n d t h r e e d i s p l a c e m e n t c o m p o -nen t s as dependent var i ab les . Al l o f these s tud ieswere conf ined to homogeneous , i so t rop ic p la t es .M oon ~ 5, ~6 inves t iga ted the r espon se o f in -f in i t e l amina ted p la t es sub jec ted to t r ansver seimpac t loads a t the cen te r o f the p la t e . F ive par -t i a l -d i f f e ren t ia l equa t ions o f mot ion wi th a math e-m a t i c a l m o d e l b a s e d o n t h e w o r k o f M i n d l i n a n dco-wo rker s were ob ta in ed 115 fo r o r tho t ro p ics y m m e t r y i n w h i c h t h e i n - p l a n e a n d f l e x u r a lm o t i o n w e r e d e s c r i b e d . T h e t w o - d i m e n s i o n a lve loc i ty and wave sur faces and the p r inc ipa lv i b r a t o r y d i r e c t i o n o f p a r t i c l e m o t i o n f o r e a c hwave norm al were p resen ted . Th e ana lys i s 116 wasb a s e d o n t h e u s e o f a L a p l a c e t r a n s f o r m o n t i m ea n d a 2 D F o u r i e r t r a n s f o r m o n t h e s p a c e v a r i -ab les . Th e so lu t ion perm i t s the ana ly t i ca l inver -s i o n o f t h e L a p l a c e t r a n s f o r m w h i l e acomputa t iona l too l ca l l ed the Fas t Four ie r T rans -f o r m w a s u s e d t o n u m e r i c a l l y i n v e r t t h e F o u r i e rt r ans form so lu t ion .


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R e f i n e d t h e o r i e s o f i b e r - r e i n fo r c e d l a m i n a t e d c o m p o s i t e s a n d s a n d w i c h e s 299C h o w 117 e m p l o y e d t h e L a p l a c e t r a n s f o r m t e ch -n i q u e t o s t u d y t h e d y n a m i c r e s p o n s e o f o r t h o -t r o p i c l a m i n a t e d p l a t e s . T h e d y n a m i c e q u a t i o n sw e r e d e r i v e d f r o m t h e c o n c e p t s o f T i m o s h e n k o ' sb e a m t h e o r y t o i n c l u d e t h e e f f e c t s o f t r a n s v e r s es h e a r a n d r o t a r y i n e r ti a . T h e i n f lu e n c e o f i n t e rn a lf r i c t i o n r e l a t e d t o t h e d a m p i n g o n t h e r e s p o n s e o ft h e p l a t e w a s a l s o c o n s i d e r e d . W a n g e t a l . ~ 8a p p l i e d t h e m e t h o d o f c h a r a c t e r i s t i c s t o u n s y m -m e t r i c a l o r t h o t r o p i c l a m i n a t e d p l a t e s . I n a s e r i e sof pa pe rs , Su n an d h is co l leagu es 119-122 use d the

c l a ss i ca l m e t h o d o f s e p a r a t i o n o f v a r i a b l e s c o m -b i n e d w i t h t h e M i n d l i n - G o o d m a n 123 p r o c e d u r ef o r tr e a ti n g t i m e - d e p e n d e n t b o u n d a r y c o n d i ti o n sa n d / o r d y n a m i c e x t e r n a l l o a d i n g s . Yu 124 s ubs e -q u e n t l y a p p l i e d i t t o s a n d w i c h p l a t e s . H o w e v e r ,thes e pap e r s 115-124 we re con f in ed to p la t e s un de rcy l ind r i ca l bend ing .F o r t w o d i f f e r e n t l a m i n a t i o n s c h e m e s , u n d e ra p p r o p r i a t e b o u n d a r y c o n d i t i o n s a n d s i n u s o i d a ld i s t r i b u t i o n o f t h e l o a d , t h e e x a c t f o r m o f t h es p a t i a l v a r i a t i o n o f t h e s o l u t i o n w a s o b t a i n e d b yR e d d y 125 a n d t h e p r o b l e m w a s r e d u c e d t o t h es o l u t io n o f a s y s t e m o f o r d i n a r y d i f fe r e n t i a l e q u a -t ions in t ime . R ed dy 126 has a l s o p r es en ted thel i n e a r tr a n s i e n t r e s p o n s e o f c o m p o s i t e p l a t e sus ing f in i te e le m ents . In b oth the pa pe rs , 125,126 thet h e o r y u s e d w a s a g e ne r a li z a ti o n o f t h e R - M t h ic kp l a t e t h e o r y f o r h o m o g e n e o u s , i s o t r o p i c p l a t e s t oa r b i t r a r i l y l a m i n a t e d a n i s o t r o p i c p l a t e s a n di n c l u d e d s h e a r d e f o r m a t i o n a n d r o t a r y i n e r t iaef fects .A g e n e r a l i z a t i o n o f t h e V o n K a r m a n ~27 n o n -l i n e a r p l a t e t h e o r y f o r i s o t r o p i c p l a t e s t o i n c l u d et h e e f f e c ts o f tr a n s v e r s e s h e a r a n d r o t a r y i n e r t i a i nt h e t h e o r y o f o rt h o t r o p i c p l a te s is d u e t o M e d w a -dow s k i 128 and tha t f o r an i s o t ro p ic p la t e s i s due toE b c i o g l u . 129 F o r c e d m o t i o n s o f l a m i n a t e d c o m p o -s i te p la t e s a r e inves t iga ted by R e dd y ~3 u s ing af i n i t e e l e m e n t t h a t a c c o u n t s f o r t h e t r a n s v e r s es hea r s t r a in s , r o t a ry ine r t i a , and l a rge ro ta t ions ( int h e V o n K a r m a n s e ns e) . C h e n S u n TM u s e d t h eF E M f o r n o n l in e a r t r a n s v e r s e r e s p o n s e o f la m i n -a t e d c o m p o s i t e p l a t e s u n d e r i n i t i a l d e f o r m a t i o na n d i n i ti a l s t re s s a c c o r d i n g t o t h e R - M p l a t et h e o r y a n d V o n K a r m a n l a r g e d e f o r m a t i o na s s u m p t i o n s . K a n t M a l l i k a r j u n a 132 p r e s e n t e dt h e l in e a r t r a n s i e n t r e s p o n s e o f c o m p o s i t e - s a n d -w i c h p l a t e s u si n g a F O S T w i t h 4 - , 8 - , a n d 9 - n o d e di s o p a r a m e t r i c q u a d r i l a t e r a l e l e m e n t s . A l l o f t h eab ov e stud ies 13 132 w e r e b a s e d o n e i t h e r t h ec la s s i ca l 3D e la s t i c i ty theo ry o r the c l a s s i ca l( K i r c h h o f f ) p l a t e t h e o r y o r t h e f i r s t - o r d e r s h e a rd e f o r m a t i o n ( R e i s s n e r- M i n d l i n ) t h e o ry . R e c e n t ly

M a l l i k a r j u n a a n d K a n t u s e d t h e h i g h e r - o r d e rd i s p l a c e m e n t m o d e l s ( s e e b e l o w e q n s ( 1 ) - ( 5 )) w i t hs i m p l e C fi n it e e l e m e n t f o r m u l a t i o n f o r f r e ev ib r a t ion 99-~ 2 and t r an s ien t dynamic99 133-143ana ly s es , and in ob ta in ing s o lu t ions to gene ra ll a m i n a t e d f i b e r - r e i n f o r c e d c o m p o s i t e a n d s a n d -w i c h p l a t e p r o b l e m s .

H I G H E R - O R D E R S H E A R D E F O R M A T I O NT H E O R I E SRefin ed theories 99 f o r f r e e v i b r a ti o n a n d t r a n s i e n td y n a m i c a n a ly s i s o f a n i s o t r o p i c c o m p o s i t e a n ds a n d w i c h l a m i n a t e s h a v e b e e n d e v e l o p e d , s e p a r -a t e l y f o r s y m m e t r i c a n d u n s y m m e t r i c l a m i n a t i o ns c h e m e s , u s i n g t h e f o l l o w i n g d i s p l a c e m e n t f i e l d sb a s e d o n t h e T a y l o r ' s s e ri e s e x p a n s i o n :S y m m e t r i c l a m i n a t e( a) H i g h e r - o r d e r s h e a r d e f o r m a b l e( H O S T 5 ) , 5 d . o . f . / n o d e

u x , y , z , t ) = Z O x X , y , t ) + z 3 0 * ( x , y , t )v x , y , z , t ) = Z O y X , y , t ) + z 3 0 * x , y , t ) (1)w x , y , z , t ) = W o X , y , t )

( b ) H i g h e r - o r d e r s h e a r d e f o r m a b l e( H O S T 6 ) , 6 d . o . f . / n o d eu x , y , z , t ) = z Ox x , y , t ) + z 3 O* x , y , t )v x , y , z , t ) = Z O y X , y , t ) + z 3 0 ~ x , y , t ) (2)w x , y , z , t) = W o X , y , t ) + z 2 ~ x , y , t )

U n s y m m e t r i c l a m i n a t e( a) H i g h e r - o r d e r s h e a r d e f o r m a b l e( H O S T 7 ) , 7 d . o . f . / n o d eu x , y , z , t ) = U o X , y , t ) + z O x x , y , t )

+ z 30*x x , y , t )v x , y , z , t ) = v 0 x , y , t) + z O y x , y , t)

+ z3 0 ~ x , y , t) (3)w x , y , z , t ) - - W o X , y , t )

( b ) H i g h e r - o r d e r s h e a r d e f o r m a b l e( H O S T 9 ) , 9 d . o . f . / n o d eu x , y , z , t ) = U o X , y ,

+ Z 2 u~v x , y , z , t ) -- V o X , y ,

+ Z 2 [ ~w x , y , z , t ) = W o X , y ,

t ) + z O x X , y , t )x , y , t ) + z 3 0 ? x , y , t )

t ) + Z O y X , y , t )X , y , t ) + Z 3 { g y X , y , t)t )

t h e o r y

t h e o r y

t h e o r y

t h e o r y

4 )


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300 M a l l ik a r j u n a , T . K a n t( c ) H ig h e r -o rd e r sh e a r d e fo rm a b le t h e o ry(HO ST 11 ), 11 d.o . f . /node

u ( x , y , z , t ) = u o ( x , y , t ) + z O x ( X , y , t )+ z 2 u~ (x, y, t) -t-z3 0 x * X , y, t)

v ( x , y , Z , t ) = v 0 (x , y , t ) z O r (X , y , t )+ Z 2 V ~ ( X , y , t ) + z a O ~ ( x , y , t

w x , y , z , t ) = W o x , y , t ) + Z O z x , y , t )+ z 2 W * oo x , y , t )

5 )

whe re t is the t ime , u , v and w def ine the d isp lace-ments o f any gener ic po in t (x , y , z ) in the p la tespace, u 0, v0 and w0 d e n o te t h e d i sp l a c e m e n t s o f agener ic po in t (x, y ) on the m idplane , O x and 0y a reth e ro t a t i o n s o f n o rm a l s t o m id p la n e a b o u t t h e yand x axes , respec tive ly . Th e param eters u~, v~ ,w * , 0 , O ~ a n d 0 z a re h ig h e r -o rd e r t e rm s in t h eTaylo r s se r ies expa ns ion and a re a lso def ined a tt h e m id - su r fa c e . T h e d e v e lo p m e n t o f t h e r e f in e dtheor ies and the i soparam etr ic C f in i te e lem entform ula t io n can be seen in Ref . 99 .

N U M E R I C A L R E S U L T S A N D D I S C U S S I O NT o d e m o n s t r a t e t h e v e r sa ti l it y o f t h e r e f in e dth e o r i e s d e v e lo p e d , v a r io u s n u m e r i c a l e x a m p le sd ra w n f ro m th e l i t e r a tu re a r e d e sc r ib e d , e v a lu at e dand d iscussed . The f in i te e lement so lu t ion tech-n iq u e a d o p te d h e re h a s a w id e r a n g e o f a p p l i c -ab i l i ty fo r lamina tes wi th a rb i t ra ry geometry ,lo a d in g a n d b o u n d a ry c o n d i t i o n s . T h e c o m p u te rp ro g ra m s h a v e b e e n d e v e lo p e d s e p a ra t e ly t op re d ic t t h e f r e e v ib ra t io n a n d t r a n s i e n t re sp o n seo f sy m m e t r i c a n d u n sy m m e t r i c l a m in a te s . I na d d i t i o n to t h e r e f in e d th e o r i e s , p ro g ra m s w e red e v e lo p e d fo r t h e f i r s t -o rd e r sh e a r d e fo rm a t io nth e o ry w i th t h r e e d e g re e s o f f r e e d o m (F O S T 3 ) i . e .w , Ox, Oy fo r sy m m e t r i c l a m in a te s a n d w i th f i v edegrees o f f reedo m (FOS TS) i .e . u , v , w , O x , O y fo runsymmetr ic lamina tes . I t i s wel l known tha t thesh e a r c o r r e c t io n c o e f f i c i e n t s d e p e n d o n th el a m in a t io n s c h e m e a n d th e l a m in a m a te r i a l p ro -per t ies . But due to lack of wel l accep ted coeff i -c ien ts fo r f in i te p la tes , the t ransverse sh ear en ergyte rm in F O S T i s c o r r e c t e d u s in g a m u l t i p l ie r 5 /6for a l l the mater ia ls excep t fo r the core o f a sand-w ic h p l a t e w h e re a c o e f f i c i e n t o f u n i ty h a s b e e nused . The resu l t s o f the presen t re f ined theor iesh a v e b e e n c o m p a r e d w i t h t h e p r e s e n t F O S Tw h e re v e r so lu t io n s b y o th e r m e th o d s a r e n o tava i lab le in the l i te ra ture .

F o r t h e 9 -n o d e d L a g ra n g ia n q u a d r i l a t e r a li so p a ra m e t r ic e l e m e n t u se d th ro u g h o u t h e re , th ese lec t ive numer ica l in tegra t ion scheme, based onthe G auss-qu adra ture ru les , v iz . 3 x 3 for m em-b ra n e , f l e x u re a n d c o u p l in g b e tw e e n m e m b ra n eand f lexure te rms , and 2 x 2 for shear te rm s in thee n e rg y e x p re s sio n i s e m p lo y e d in t h e e v a lu a t io nof the e lement s t i f fness p roper ty . The e lementmass matr ix i s eva lua ted us ing a 3 x 3 Gaussquadra ture ru le . An expl ic i t cen t ra l d i f fe rencet e c h n iq u e a n d su b sp a c e i t e r a t i o n s c h e m e fo r t h eso lu t ion of t rans ien t dynamics and f ree v ibra t ion ,respec t ive ly , a re employed wi th a spec ia l massmatr ix d iagona l iza t ion schem e appl icab le toquadr i la te ra l i soparam etr ic C f in i te e lements . Aconverg ence s tudy was car r ied o u t wi th a v iew toge t t ing reasonably convergent re l iab le so lu t ionsw i th a n o p t im u m n u m b e r o f e l e m e n t s. A 2 x 2mesh (4 e lements ) in a quar te r p la te and a 4x 4mesh (16 e lements ) in a fu l l p la te d isc re t iza t ionwere seen to g ive genera l ly converged d isp lace-ments , s t resses and s t ress - resu l tan ts , and there-fore un less o therwise spec i f ied , these d is -c re t iza t ions were adopted in the presen t work .A f t e r h a v in g e s t a b l i sh e d a n o p t im u m sp a c e d i s -c re t iza t ion , the assoc ia ted c r i t ica l t ime s tep wasobta ined for the t rans ien t dynamic ana lyses . Aquar te r p la te i s used for i so t rop ic , 0 - - o r tho -t rop ic and c ross-p ly (0 /90 / . . . ) lamina tes , whi lea fu l l p la te geom etr ic mod el i s used for ang le-p lylamina tes . Fur ther , a fu l l p la te d isc re te model i sinvar iab ly used in a f ree v ibra t ion ana lys is fo ro b ta in in g h ig h e r f r e q u e n c ie s a n d m o d e s . I n t r a n -s ien t response ana lyses , ze ro in i t ia l cond i t ions ondisp lacements and the i r t ime der iva t ives wereassumed for a l l the cases . Al l the computa t ionswere car r ied ou t in s ing le p rec is ion on a CDCC Y B E R 1 8 0 /8 4 0 c o m p u te r a t t h e In d ia n In st i-t u t e o f T e c h n o lo g y , B o m b a y , In d ia . T h e b o u n d a ryc o n d i t i o n s c o r r e sp o n d in g to d i f f e r e n t t y p e s o fe d g e s m o s t c o m m o n ly o c c u r r in g in p ra c t i c e ,namely , the s imple suppor ted , the c lamped sup-p o r t a n d sy m m e t ry c o n d i t i o n s a lo n g th e e d g e a r el i s ted in Table 1 . The mater ia l charac te r i s t ics o fthe ind iv idua l layers a re g iven in Table 2 fo r d i f -fe ren t se ts o f da ta .E x a m p l e 1. S imply suppor ted square p la tes o fm u l t i l a y e re d sy m m e t r i c c ro s s -p ly w e re a n a ly se dw i th t h e m a te r i a l p ro p e r t i e s t y p i c a l o f h ig h f ib ro u sc o m p o s i t e s (D A T A - l ) g iv en in T a b le 2 . T h e r a t i oo f E 1 / E 2 w s v a r i e d b e tw e e n 3 a n d 4 0 , a n d th en u m b e r o f l a y e rs b e tw e e n 3 a n d 9 . B e c a u se o f t h eex is tence of b iax ia l symmetry in the c ross-p lylamina tes , on ly a quadran t o f the lamina te i s


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R e f i n e d t h e o ri e s o f i b e r - r e in f o r c e d l a m i n a t e d c o m p o s i t e s a n d s a n d w i c h e sTable 1 . Deta i l s o f bound ary cond i t ions for laminated p lates

3 0 1

E d g e T h e o r y S i m p l y - su p p o r t e d C l a m p e d S y m m e t r y l in e ~S S 1 S S 2

FO ST 5 v o - - w o = 0y = 0 Uo = wo- - 0y = 0 uo = Vo = W o- 0 uo = 0~ = 0O~ = O y =H O S T 7 v o = w o = 0 Uo = Wo = 0 u o = v o = w o - - 0 u o = 0

H O S T 9 v o - v ~ = w o = 0 U o = u ~ = w o = 0 u o = v o = u~ = v~ = 0 u o = u~ = 00 O y = O y = O O y = Oy * 0 O x = O v = --0~=0 0x=~x=0II W o = 0H O S T 11 v o = v~ = 0 uo = u~ = 0 Uo = Vo = u* = v~ = 0 Uo = u~ = 0w0= =0 wo= =0 0 =0

O y ~ - O ~ , = 0 z = O O y = O~y ~ - O z = O W O = ~o0 = O z = OF O S T 5 u0 = wo = 0~ = 0 vo = wo = Ox = 0 Uo = vo = Wo = 0 Vo = Oy = 00 ~ = 0 ~ = 0H O S T 7 u o - - wo = 0 vo = wo = 0 Uo = vi i = wo = 0 vo = 00 =o*=o= H O S T 9 Uo = u* = Wo = 0 Vo = v~ = Wo = 0 Uo = v(~ = u~ - - v~ = 0 v o = v~ = 0

8 0 x ~ O* = 0 0 x = O* = 0 Ox = Oy = O~x = ~ y = 0 Oy = ~yy = 0II W o = 0

H O S T l l u o = u ~ = 0 V o = V ~ = 0 u o = v o = u ~ = v ~ = 0 v o = v ~ = 0W 0 = W~o = 0 W 0 = W ~ = 0 0 x = Oy = O* = ~. , = 0 Oy = ~yy = 0ox= Ox*=o = 0 O x * = O Wo= =o =

B o u n d a r y c o n d i t i o n s a l o n g t h e s y m m e t r y l in e s a r e u s e d o n l y f o r q u a r t e r p l a t e a n a l y s e s . S S 1 is u s e d f o r b o t h q u a r t e r a n d f u l lp l a t e a n a l y s e s , b u t S S 2 i s u s e d o n l y f o r f u l l p l a t e a n a l y s is .

Table 2 . Mater ia l propert iesD A T A M a t e r i a l p r o p e r t i e s

T y p i c a l h i g h - m o d u l u s g r a p h i t e / e p o x y ( d i m e n s i o n le s s p r o p e r t i e s )E t / E 2 = O pe n; E 2 = E 3 = 1 ; G I 2 = Gl3 = 0 6 E 2; G23 = 0 5 E2; 1/12 1 2 3 ~ 1 3 = 0 25 ; p = 1 0F o r f a c e sh e e ts , th e a s s u m e d p l y d a t a b a s e d o n H e r c u l e s A S 1 / 3 5 0 1 - 6 g r a p h i t e / e p o x y p r e p r e g s y s t e mE~ = 13 08 x 106 N/cm2 ; E2 = E 3 = 1 06 x 106 N/cm2; Gj2 = GI3 = 0 6 106 N/cm2 ; G23 = 0 3 9 x 106 N/cm 2;vj2 = v l3 = 0 28; v23 = 0 3 4 ; p = 1 5 8 x 1 0 - 6 N - s e c 2 / c m 4C o r e m a t e r i a l is o f U . S. C o m m e r c i a l a l u m i n i u m h o n e y c o m b ( 1 / 4 i n c h c e l l s iz e , 0 . 0 0 3 i n c h f o il )6 2 3 G y = 1 7 7 2 x 1 0 4 N / c m 2 ; G l 3 = Gxz = 5 - 2 0 6 x 1 0 4 N / c m 2 ; E 3 = z = 3 0 1 3 x 1 05 N / c m 2 ;p = 0 1 0 0 9 x 1 0 - 5 N _ s e c 2 / c m 4F o r f a c e s h e e t s ( t y p i c a l g r a p h i t e / e p o x y )E l = 0 . 1 2 x 1 0 8 N / c m 2 ; E 2 = E 3 = 0 79 x 106 N/cm 2; Gj2 = G23 = G j3 = 0 .55 x 106 N/cm 2;vl2 = v23 = v l3 = 0 .3 ; p = 1 .58 x 10 - 5 N_ sec2 /cm 4F o r c o r e m a t e r i a l ( U . S . C o m m e r c i a l a l u m i n i u m h o n e y c o m b , 1 / 4 i n c h c e l l s i z e, 0 . 0 0 7 i n c h fo i l)G 2 3 G y z = 0 ' 7 0 3 4 x 1 0 4 N / c m 2 ; G l 3 = Gxz 0 ' 1 4 0 7 x 1 05 N / c m 2 ; p = 0 . 3 4 1 5 x 1 0 - 6 N - s e c 2 / c m 4

m o d e l l e d f o r f i n d i n g f u n d a m e n t a l f r e q u e n c i e s .T h e e f f e c t s o f t h e n u m b e r o f l a y e r s a n d d e g r e e o fo r t h o t r o p y o f t h e i n d i v i d u a l l a y e r s o n t h e d i m e n -s i o n l e s s f u n d a m e n t a l f r e q u e n c y a r e p r e s e n t e d i nT a b l e 3 .

T h e r e s u l t s a r e c o m p a r e d w i t h N o o r ' s s o l u -t i o n 77 o f t h e 3 D e l a s t i c i t y t h e o r y u s i n g h i g h e r -o r d e r f i n i t e d i f f e r e n c e s c h e m e s . V e r y g o o da g r e e m e n t s a r e o b s e r v e d b e t w e e n t h e p r e s e n tr e f i n e d t h e o r i e s a n d 3 D e l a s t i c i t y t h e o r y . T h er e s u l t s o b t a i n e d u s i n g c l a s s i c a l l a m i n a t i o nt h e o r y , 97 h y b r i d s t r e s s f i n i t e d e m e n t m e t h o d 98

a n d a l o c a l f i n i t e e l e m e n t m o d e l b a s e d o n ar e f i n e d a p p r o x i m a t e t h e o r y 97 a r e a l s o i n c l u d e d f o rc o m p a r i s o n . T h e C P T o v e r e s t im a t e s t h e f u n d a -m e n t a l f r e q u e n c i e s , e s p e c i a l l y w h e n t h e d e g r e e o fa n i s o t r o p y i s g r e a t e r . T h e f u n d a m e n t a l f r e q u e n -c i e s i n c r e a s e w i t h t h e i n c r e a s e i n d e g r e e o f o r t h o -t r o p y a n d a l s o t h e i n c r e a s e i n n u m b e r o f l a y e r s .E x a m p l e 2 . T o s h o w t h e e f f e c t s o f t r a n s v e r s es h e a r r i g i d i t i e s o f s t i f f l a y e r s a n d l e n g t h / t h i c k n e s sr a t i o o n t h e n a t u r a l f r e q u e n c i e s , a s e v e n - l a y e r

0 / 4 5 / 9 0 / c o r e / 9 0 / 4 5 / 0 ) s q u a r e s y m m e t r i cc o m p o s i t e - s a n d w i c h p l a t e i s a n a l y s e d w i t h d i f f e r -


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302 M a l l i k a r ]u n a T . K a n te n t b o u n d a ry c o n d i t i o n s : s im p ly - su p p o r t e d S S2 )and c lamped . Th e mater ia l p rope r t ies DATA-2)g iven in Table 2 a re used .The resu l t s ob ta ined us ing the presen t re f inedth e o r i e s a n d th e p re se n t F O S T a re p re se n te d inT a b les 4 a n d 5 fo r s im p ly - su p p o r t e d a n d c l a m p e dboundary condi t ions , respec t ive ly . I t i s seen tha tfo r a m o d e ra t e ly t h i c k p l a t e a/h=10) with thet r a n sv e r se sh e a r m o d u l i i G23 a n d GI3 ) of st ifflayers inc luded , the d i f fe rence be tween the pre-d ic t ions of na tura l f requenc ies us ing theor iesH O S T a n d F O S T in c rea se s w i th i n c re a sin g m o d e

n u m b e r s . T h e F O S T e s tim a te s h ig h e r f re q u e n c ie s .This i s due to s impl i fy ing assump t ions mad e inFOST. I t is conc lu ded tha t the e f fec t o f t ransversesh e a r m o d u l i i o f s ti ff l a y e rs i s m o re p ro n o u n c e d inth icker lamina tes low a/h ra t io ) than for th inlamina tes h igh a/h ra t io ) . Th e e f fec t o f boun darycondi t ion can be seen f rom the Tables 4 and 5 . Th efrequenc ies fo r c lamped lamina te a re a lwaysh ig h e r t h a n th a t o f s im p ly - su p p o r t e d l a m in a te .Example 3 . S im p ly su p p o r t e d sq u a re o r th o t ro p iclamina tes hav ing skew-symmetr ic lamina t ionswi th respec t to the middle p lane a re cons idered .

Table 3 . Ef fec t o f number of layers and degree o f or thotropy of indiv idual layers on the fundamenta l frequency of s imply-s u p p o r t e d s q u a r e m u l t i l a ye r e d s y m m e t r ic c o m p o s i t e p l a t e s(D A T A - 1 w i t h v a r yi n g E l ~ E 2 , 0 / 9 0 / 0 / . . . / 0 , 2 2 m e s h , q u a r t e r p l a t e, a / h = 5 , 6~ = w p h 2 / E 2 ) 1 /2 1 0 )No. of Source E t / E 2layers 3 10 20 30 40

Noor 2.6474 3.2841 3'8341 4.1089 4.3006HOST5 2.6260(-0.8) 3.2672(-0 5) 3 7801(- 1. 4) 4'03 00(- 1- 9) 4.1998 (- 2.3)HOST6 2.6126(-1.3) 3.2528(-0-9) 3 7253(-2 5) 3'9884(-2.9) 4.1521(-3.4)3 FOST3 2.6124( -1.3) 3 2519(-0-9) 3 7221(-2.6) 3 9721(-3.3) 4-1501(-3.5)Owen & Li 2.6948(+1.8) 3.3917(+3.2) 3 8979(+1.9) 4 1941(+2.1) 4.3951(+2.2)Sun & Liou 2.6524(+0.2) 3 3364(+1'6) 3-8289(-0.1) 4.1142(+0.1) 4.3062(+0.1)CPT 2.9198(+10) 4.1264(+25) 5 4043(+41) 6-4336(+56) 7.3196(+70)Noor 2.6587 3.4089 3 9792 4'3140 4.5374HOST5 2 6389(-0 7) 3'3766(-0 9) 3 9337(- 1' 1) 4'26 22(- 1. 2) 4.4831 (- 1.1)HOST6 2.6255(- 1. 2) 3.3621(- 1. 3) 3 9192(- 1. 5) 4.2482(- 1. 5) 4.4695(- 1.5)5 FOST3 2'6255(- 1. 2) 3'3622(- 1. 3) 3 9190(- 1. 5) 4 2456(- 1. 6) 4'4628(- 1.6)Owen & Li 2.6988( + 1. 5) 3. 45 34 ( 1 .3) 4.0297( + 1. 3) 4.3704( + 1. 3) 4.5992( + 1.4)Sun & Liou 2.6608(0.08) 3. 41 03 (0 .0 4) 3- 98 03 (0 .0 3) 4- 31 49 (0 .0 2) 4.5380(0.01)CPT 2.9198(+9-8) 4.1264(+21) 5 4043(+36) 6-4336(+51) 7.3196(+61)Noor 2.6640 3.4432 4-0547 4.4210 4'6679HOST5 2'6433(-0 7) 3.4184(-0.7) 4 0259(-0 7) 4'3904(-0.7) 4.6367(- 0'6)9 HOST6 2'6298(- 1 3) 3.4035(- 1. 1) 4 0107(- 1 1) 4'3755(- 1. 0) 4.6222(-0 9)FOST3 2.6297(- 1. 3) 3'4035(- 1. 1) 4 0107(- 1. 1) 4.3756(- 1. 0) 4.6225(-0 9)Owen & Li 2.6971(+1.2) 3.4708(+0.8) 4.0746(+0.5) 4.4360(+0.5) 4.6803(+0.3)CPT 2.9198(+9.6) 4.1264(+20) 5-4043(+33) 6.4336(+45) 7.3196(+57)

Values in brackets give percentage errors with respect to 3D-elasticity solution.

Table 4 . Ef fec t o f shear r ig id i ty o f s t i f f layers and length- to- th ickness rat io on the natura l frequenc ies w / 2 x c y c l e s / s e c ) o fseven- layer 045 0/ 90 co r e/9 045 0 s i m p l y -s u p p o r t ed s q u a r e s y m m e t r ic c o m p o s i t e - s a n d w i c h p l a t es(D A T A - 2 , 4 4 m e s h , f u ll p l a t e , a = b = 1 0 0 c m , h 0 - - h 4 s = h 9 0 = 0 0 5 h , h . . . . = 0 7 h )

ModalNo. Considering G 3and Gj 3 of stiff layers Neglec ting G 3and Gl3 of stiff layersa / h = 10 a / h = 100 a / h = 10 a / h = 100HOST5 HOST6 FOST3 HOST5 HOST6 FOST3 HOST5 HOST6 FOST3 HOST5 HOST6 FOST3

1 473 473 593 70 70 70 333 334 356 69 69 692 775 774 1203 166 166 168 517 518 524 161 161 1613 1004 1003 1331 194 194 196 616 617 707 188 188 1894 1096 1097 1363 267 268 271 718 720 713 256 257 2575 1173 1173 1719 344 345 358 729 731 802 322 323 3216 1320 1321 2005 400 400 407 862 865 820 377 375 3837 1376 1376 2172 408 409 421 870 872 827 382 383 3858 1436 1476 2180 478 479 492 884 885 907 442 444 450


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Refined theories of iber-reinforced lam inated composites and sandwiches 303T a b l e 5 . E f f e c t o f s h e a r r i g id i t y o f s t if f l a y e r s a n d l e n g t h - t o - t h i c k n e s s r a t i o o n t h e n a t u r a l f r e q u e n c i e s (m / 2 : ~ y c l e s / s e c ) o fs e v e n - l a y e r (0 /4 5 /90 /cor e/900/4 50 ) l a m p e d s q u a r e s y m m e t r i c c o m p o s i t e - sa n d w i c h p l a te s

( D A T A - 2 , 4 x 4 m e s h , f u ll p l a te , a - b = 1 0 0 c m , h 0 = h 4 5 = h g 0 = 0 0 5 h , h c o ce = 0 7 h )ModalNo. Considering G 3 and G~3 of stiff layersa /h = 1 0 a /h = 100

HOST5 HOST6 FOST3 HOST5 HOST6 FOST3

Neglecting G 3and G~3 of stiff layersa / h - - 10 a /h = 100

HOST5 HOST6 FOST3 HOST5 HOST6 FOST31 605 606 866 133 134 135 384 385 407 128 128 1282 854 856 1 399 250 251 259 545 547 546 232 233 2323 1 080 1 081 1 512 290 291 297 661 663 717 270 271 2754 1 173 1 176 1 877 373 375 388 740 741 728 342 344 3465 1 244 1 245 2 128 475 477 514 764 766 815 415 416 4106 1 377 1 383 2 180 550 552 584 875 876 820 483 485 4847 1450 1455 2285 564 566 587 893 896 831 499 501 5198 1 549 1 556 2408 644 647 683 940 938 916 559 560 558

T h e f i b e r o r i e n t a t i o n s o f t h e d i f f e r e n t l a m i n a ea l t e rna te be tw een 0 and 90 wi th r espe c t to thex-ax i s. The mater i a l ch arac te r i s t i cs o f the ind iv id-u a l l a y e r s a r e t a k e n t o b e t h o s e o f h i g h f i b r o u scompos i t es (DATA- l ) which a re g iven in Tab le 2 .T w o p a r a m e t e r s w e r e v a r i e d , n a m e l y t h e d e g r e eo f o r t h o t r o p y o f t h e i n d i v i d u a l l a y e r sEI /Ez ) , and the l eng th- to - th icknes s r a t io o f thel a m i n a t e . T h e r a t i o E l ~ E 2 w a s v a r i e d b e t w e e n 3a n d 4 0 , a n d n u m b e r o f l a y e r s v a r i e d b e t w e e n 2and 10 .

In Tab le 6 , the fundamenta l f r equenc ies ,ob ta ined by the p resen t theor i es , 3D e las t i c i tytheory 77 us ing h ighe r -order f in i te d i f f e renceschemes , a 3D e igh t -node hybr id s t r es s f in i t ee lem ent so lu t ion , 98 a loca l f in i t e e l em ent mo delbased on a r e f ined a pprox ima te theory , 97 a mixedF E M b a s e d o n a h i g h e r - o r d e r t h e o r y 95 a n d aclass ical thin plate the ory , 97 a re presen ted . I t i sfound tha t fo r skew-symmet r i c l amina tes , as thenum ber o f l ayer s increases f rom 2 to 4 , the accur -a c y o f t h e C P T s h a r p l y d e t e r i o r a t e s . F u r t h e ri n c r e a s e o f t h e n u m b e r o f la y e r s d o e s n o t h a v e as ign i f i can t e f f ec t on the accuracy . On the o therhand , fo r symm et r i ca l ly l amin a ted p la t es ( seeE x a m p l e 1 ), t h e e r r o r d e c r e a s e s a s t h e n u m b e r o fl a y e r s i n c r ea s e s . T h e e r r o r i n t h e C P T p r e d i c t io n si s main ly a t t r ibu ted to the neg lec t o f shear defor -m a t i o n . W h e n t h e r e s u l t s o f p r e s e n t t h e o r i e s a r ecom pare d wi th th e 3D e las t i c i ty so lu t ion , 77 theagreem ent i s s een to be exce l l en t. Th e e r ro r in thep r ed i ct io n s o f H O S T 7 , H O S T 9 a n d H O S T 1 1 di dno t exceed 2 .59%, 1 3% and 1 .63% respec t ive ly ,e v e n f o r t h e c a s e o f a h i g h l y o r t h o t r o p i c t h i c kl a m i n a t e w i t h E l ~ E 2 = 4 0 . T h e c o r r e s p o n d i n ge r r o r e s t im a t e f o r t h e p r e s e n t F O S T 5 a n d C F S o fa h i g h e r o r d e r th eo ry 95 i s s e e n t o b e 5 . 1 % a n d

6 .12% respec t ive ly , whereas fo r smal l degrees o fo r t h o t r o p y E I / E 2 = 3-10) , e r ro r i s a lmo s t neg lig -ib le . F rom Table 6 , it is conc luded tha t the r esu l t sr eaf f i rm the f ac t tha t the e f f ec t o f coupl ingb e t w e e n b e n d i n g a n d s t r e t c h i n g a n d o r t h o t r o p yc a n n o t b e i g n o r e d e v e n a t lo w m o d u l u s r a ti o . T h ef u n d a m e n t a l f r e q u e n c y i n c r e a s e s w i t h t h ei n c r e a s e i n d e g r e e o f o r t h o t r o p y a n d / o r i n c r e a s ei n n u m b e r o f l a y e rs .Exampl e 4 . A n e i g h t -l a y e r ( 0 / 4 5 / 9 0 / c o r e / 9 0 /450 /300/0 ) unsym me t r i c squa re com pos i t e - s and-wich p la t e i s ana lysed fo r two d i f f e ren t boundarycondi t ions : s imply- suppor ted (SS2) and c lamped .Th e e las ti c mater i a l p roper t i es (DATA-2) g iven inTable 2 a re used . The na tura l f r equenc iesob ta ined us ing the p resen t r e f ined theor i es( H O S T ) a n d t h e p r e s e n t F O S T a r e p r e s e n t e d i nT a b l e s 7 a n d 8 f o r s im p l y - s u p p o r t e d a n d c l a m p e dboundary condi t ions , r espec t ive ly . A compar i sonof the e f f ec t s o f t r ansver se shear r ig id i ty o f s t i f fl a y e r s a n d l e n g t h / t h i c k n e s s r a t i o o n t h e n a t u r a lf r equenc ies o f unsym me t r i c l amina tes i s made .I t i s s een f rom Tables 7 an d 8 tha t the e f f ec t o ft r a n s v e r s e s h e a r m o d u l i i G23 a n d GI3) of stiffl a y e r s i s m o r e p r o n o u n c e d i n t h i c k e r l a m i n a t e s( low a / h r a t io ) than fo r th in l amina tes (h igh a / hr a tio ). For a m ode ra te ly th ick p la t e , t he d i f f e rencei n t h e p r e d ic t io n s o f F O S T 5 w i t h H O S T 9 a n dH O S T l l is m o r e t h a n w it h H O S T 7 . T h i s di s-c r e p a n c y i s d u e t o s i m p l i fy i n g a s s u m p t i o n s m a d ei n F O S T , w h e r e a s t h e p r e s e n t r e f i n e d t h e o r i e srepresen t the r ea l i s t i c c ros s - sec t iona l deforma-t i o n . I n H O S T 7 t h e h i g h e r - o r d e r i n - p l a n edegree s -of - f r eedom (u~ ', v~) a re neg lec ted , theef fec t o f which i s s een in Tab les 7 and 8 . Thed i f f e rence in the r esu l t s be tween r e f ined theor i esa n d F O S T i n c r e a s e s w i t h i n c r e a s i n g m o d e


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3 0 4 Mallikar]una T. K a n tTable 6. Effect of numb er of layers and degree of orthotropy of individual layers on the fundam ental frequency of simply-supported (S S 1) square mutli layered skew-symm etric com pos ite plates(DAT A-I w ith varying E l ~ E 2 , 0 / 9 0 / . . . / 9 0 , 2 2 m es h , q uar ter p la t e, a / h = 5 , ~ = m p h 2 / E 2 ) I / 2 1 0 )No. of Source El~E2layers

3 10 20 30 40N o o rH O S T 1 1H O S T 9H O S T 72 F O S T 5Put cha & R eddyO w e n & LiS u n & L i o uC P TN o o rH O S T 1 1H O S T 9H O S T 74 F O S T 5Put cha & R eddyO w e n & LiS u n & L i o uC P TN o o rH O S T 1 1H O S T 9H O S T 76 F O S T 5Put cha & R eddyO w e n & LiSun & Li ouC P TN o o rH O S T 1 1H O S T 9H O S T 71 0 F O S T 5Put cha & R eddyO w e n & LiSun & Li ouC P T

2 .50312.4782 - 0 992-4909 - 0 482 4909 - 0 482.4829 - 0 802.4868 - 0 652 '5601 + 2 272 .5148 + 0 472 . 7 0 8 2 + 8 ' 1 92 61822 5997 ( - 0 702 603 7( - 0 552 605 5 ( - 0 482 6012 ( - 0 652-6003( - 0 682 6691 ( + 1 942 6 2 1 9 ( + 0 1 42 8676 ( + 9 522 .64402 . 6 1 9 4 ( - 0 9 3 )2 6 2 4 3 ( - 0 7 4 )2 6 2 7 5 ( - 0 6 2 )2 6 2 2 2 ( - 0 . 8 2 )2 . 6 2 2 3 - 0 . 8 2 )2.6839( + 1 50)2 6458( + 0 06)2 . 8 9 6 6 ( + 9 5 5 )2 .65832 6 331 ( - 0-94)2 . 6 3 8 5 ( - 0 - 7 4 )2 . 6 3 8 9 ( - 0 - 7 2 )2 . 6 3 2 9 ( - 0 9 6 )2 .6 3 3 7 - 0 .9 2 )2.691 6( + 1 .25)2 . 9 1 1 5 ( + 9 . 5 2 )

2 79382 . 7 7 6 4 ( - 0 6 2 )2 7 9 0 5 ( - 0 . 1 1 )2 .798 1( + 0 15)2 . 7 7 5 1 ( - 0 . 6 7 )2 .7955 ( + 0 06)2 - 8 7 1 2 ( + 2 7 7 )2.8030 ( + 0-33)3 0968( + 10 8)3 .25783 '2486( - 0 .283 ' 2 6 2 1 ( + 0 . 1 33 2870( + 0 89)3 2889 ( + 0 95)3 2782( + 0 62)3 3250( + 2.06)3 2 6 2 1 ( + 0 . 1 3 )3.887 7( + 19.3)3 36573 '342 3 ( - 0 69)3 . 3 5 4 5 ( - 0 ' 3 3 )3 . 3 7 1 2 ( + 0 . 1 6 )3.3664 ( + 0 02)3 3 6 2 1 ( - 0 1 1 )3 4085( + 1 27)3 3666( + 0 .02)4 .0215(+ 19 .5 )3 .42503 . 3 9 8 9 ( - 0 . 7 6 )3 . 4 0 8 3 ( - 0 . 4 8 )3 4 1 4 2 ( - 0 - 3 1 )3 . 4 0 4 3 ( - 0 - 6 0 )3.405 0( - 0 58)3 4527( + 0 .80)4 .088 8( + 19.4)

3 06983-0737( + 0 .12)3 0702 (+ 0 01 )3 1252 + 1.80)3 0998 + 0 98)3 1284 + 1.91)3 1558 + 2 .80 )3 0768 +0 23)3 5422 + 15.3)3 76223 7801, + 0 47)3 '7835 , + 0 56 )3'8014~ + 1.04)3 8741~ +2.97)3 8506~ + 2'35)3 8454~ + 2 '21 )3 7675q +0 14)4 99071 + 32 6)3 93593 92 49( - 0-27)3 9373( + 0 03)3 9784 ( + 1.07)3 '975 6 ( + 1 .00)3 '9672( + 0 .79 )3 '9758( + 1 .01)3 '9359( + 0 00)5 '22 34( + 32.7)4 03374 0069 - 0 66)4 0 22 1 - 0 ' 2 8 )4 0377 + 0 09)4 0239 - 0 .24)4 0 2 70 - 0 . 1 6 )4 .052 6 + 0 .47)5 3397( + 32.4)

3 .27053 3003~ +0 913 2979~ +0 833.3414~ + 2'163.3771q +3.263.4020d + 4.023 36101 + 2 .763.2763q +0 183 93351 + 20 .24 06604.1041 ( + 0 934.0923( + 0 644.1247( + 1 .444 .2462 ( + 4 .434 2139( + 3 644 1 6 1 2 ( + 2 . 3 44 . 0 7 1 9 ( + 0 1 45 '8900( + 44-84 27834 2766 - 0-044-2890 + 0-254.352 6 + 1 .734.351 2 + 1 704-3419 + 1 484-3233 + 1 .054 . 2 7 7 5 - 0 . 0 26 . 1 9 6 3 + 4 4 . 84 .40114 . 3 7 8 0 ( - 0 - 5 2 )4 3 9 2 9 ( - 0 ' 1 8 )4-4178 ( + 0 37)4 - 4 0 0 3 ( - 0 . 0 2 )4 . 4 0 7 9 ( + 0 . 1 5 )4 .4140( + 0 .29 )6 .3489( + 44.2)

3 42503.4810 ( + 1 .63)3 .4698 ( + 1 .30)3 5 1 3 8 ( + 2 . 5 9 )3 '5995( + 5 10)3 . 6 3 4 8 ( + 6 . 1 2 )3 ' 5 1 8 5 ( + 2 ' 7 3 )3 . 4 3 0 1 ( + 0 . 1 5 )4.2884( + 25.2)4 27194 3240 + 1.21)4 . 3 0 6 9 + 0 ' 8 1 )4 .3786 + 2 .49)4 5062 + 5 48)4 .4686 + 4 60)4 3763 + 2 .44)4 .2780 + 0 14 )6 6690 + 56 1 )4 50914 5 1 4 1 ( + 0 1 1 )4 5 2 6 2 ( + 0 . 3 7 )4.6090( + 2 .21)4 6083( + 2 .19)4 .6005( + 2 .02 )4 5558( + 1.03)4 . 5 0 7 7 ( - 0 0 1)7 '0359( + 56 0)4 . 6 4 9 84 . 6 2 9 5 ( - 0 ' 4 3 )4 . 6 4 4 1 ( - 0 . 1 2 )4 6 7 7 1 ( + 0 . 5 8 )4 . 6 5 5 4 ( + 0 . 1 2 )4.6692( + 0 .41)4 . 6 5 9 0 ( + 0 . 1 9 )7.2184( + 55.2)

Values in brackets give percentage errors w ith respect to the 3D -elasticity solution.

n u m b e r s . T h e e f f e c t o f b o u n d a r y c o n d i t i o n s o nt h e fr e q u e n c i e s c a n b e s e e n i n T a b l es 7 a n d 8 . T h ef r e q u e n c i e s o f c l a m p e d l a m i n a te s a r e h i g h e r th a nth ose o f s i m p l y - su p p or te d l am i n ate s , wh i c h i sob v i ou s .Example 5 . A n a n i s o t ro p i c la m i n a t e d c o m p o s i t e -s a n d w i c h p l a t e 0304560core/6045300 ) c l am p e d on a l l th e fou r s i d e s i s a n a l yse d fo rsu d d e n l y ap p l i e d u n i for m l y d i s tr i b u te d p u l sel oad i n g . Th e l e n gth - to - th i c kn e s s r a t i o a / h = 1 0( m od e r ate l y th i c k p l a te ) an d a/h = 5 0 ( r e a so n a b l yth i n p la te ) ar e c on s i d e r e d . A fu l l p l a te is d i s c r e t -

i z e d w i t h 4 x 4 m e s h . T h e e l a s t i c m a t e r i a l p r o p e r -t ie s ( D A T A - 3 ) g i v e n i n T a b l e 2 a r e u s e d . Ac om p ar i son o f th e r e su l t s ob ta i n e d b y th e r e f i n e dt h e o r i e s ( H O S T ) w i t h t h o s e o f F O S T r e s u l t s i sm a d e i n T a b l e s 9 - 1 2 . T h e s t a ti c r e su l ts o f F O S T 3a n d H O S T 5 a r e a ls o i n c l u d e d i n t h e s e t a b le s . T h em ai n p u r p ose o f tab u l a t i n g th e se r e su l t s i s top r o v i d e a n e a s y m e a n s f o r f u t u r e c o m p a r i s o n b yoth e r i n ve s t iga tor s .Th e var i a t i on o f i n -p l an e d i sp l ac e m e n ts ( u an dv ) a n d c e n t e r t r a n s v er s e d e f l e c t io n ( w 0 ) w i t hr e sp e c t to t i m e for a/h- - -10 a n d 5 0 i s s h o w n i n


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T a b l e 7 . E f f e c t o f s h e a r r i g i d i t y o f s ti f f l a y e r s a n d l e n g t h - t o - t h i c k n e s s r a t i o o n t h e n a tu r a l f r e q u e n c i e s w /2 : ~ c y c l e s / s e c ) o f e i g h t - la y e r 04590c0r e/90 0/4 530 0/0 o) i m p l y -s u p p o r t e d ( S S 2 ) s q u a r e u n s y m m e t r i c c o m p o s i t e - s a n d w i c h p l a t e s

( D A T A - 2 , 4 x 4 m e s h , f u ll p la t e , a = b = 1 0 0 c m , t h i c k n e s s o f e a c h t o p s t i f fl a y e r = 0 - 0 2 5 h , t h i c k n e s s o f e a c h b o t t o m s t i f f l a ye r = 0 - 0 8 1 2 5 h , t h i c k n e s s o f c o r e = 0 -6 h )Mo d a l C o n s id e r in g G 2 3 a n d G ~ ~ o f s t i f f l a y e r s N e g le c t in g G 2 3 a n d G ~ 3 o f s t i f f l a y e r sN o .

a h= 10 a h= 1 0 0 a h= 10 a h= 1 0 0H O S T l l H O S T 9 H O S T 7 FO S T 5 H O S T l l H O S T 9 H O S T 7 FO S T 5 H O S T l l H O S T 9 H O S T 7 F O S T 5 H O S T l l H O S T 9 H O S T 7 F O S T 5

1 4 6 4 4 6 4 4 8 5 5 1 6 5 9 5 9 5 9 5 9 2 8 1 2 8 0 3 0 5 2 9 7 5 7 5 7 5 8 5 8 ~ .2 8 5 3 8 5 3 9 2 6 1 0 1 3 1 2 7 1 2 7 1 2 7 1 2 7 4 3 1 4 3 0 4 5 2 4 3 0 1 2 0 1 2 0 1 2 3 1 2 33 9 4 3 9 3 4 1 0 6 3 1 1 5 4 1 5 4 1 5 4 1 5 4 1 5 4 5 3 0 5 2 8 5 8 0 5 7 9 1 4 2 1 4 1 1 5 0 1 5 0 ~ -4 9 5 6 9 4 1 1 3 5 5 1 5 0 1 2 1 1 2 1 0 2 1 0 2 1 1 5 8 2 5 8 1 6 1 9 5 8 2 1 9 2 1 9 1 2 0 2 2 0 15 1 0 0 2 1 0 0 0 1 5 3 1 1 7 7 3 2 6 4 2 6 3 2 6 5 2 6 5 6 0 3 6 0 2 6 7 3 6 5 6 2 3 6 2 3 5 2 4 6 2 4 3 , ~6 1 2 0 1 1 1 8 8 1 7 4 7 1 9 9 3 3 2 1 3 2 0 3 2 1 3 2 2 6 2 8 6 2 4 7 3 1 6 7 3 2 7 9 2 7 8 2 9 9 2 9 7 ~7 1 2 2 6 1 2 2 4 1 7 8 1 2 0 4 2 3 2 6 3 2 5 3 2 6 3 2 7 6 3 8 6 3 6 7 3 7 6 7 8 2 8 2 2 8 0 3 0 9 3 0 98 1 2 4 5 1 2 4 6 1 7 9 1 2 1 7 3 3 8 7 3 8 6 3 8 7 3 8 9 6 6 5 6 5 9 7 8 0 7 4 4 3 2 7 3 2 5 3 5 9 3 5 7 ~ .

T a b l e 8 . E f f e c t o f s h e a r r i g i d i t y o f s t i f f l a y e r s a n d l e n g t h - t o - t h i c k n e s s r a t i o o n t h e n a t u r a l f r e q u e n c i e s w / 2 ~ c y c l e s / s e c ) o f e i g h t - la y e r 045 90c0 r e/ 90 4530 0 c l a m p e d s q u a r e u n s y m m e t r i c c o m p o s i t e - s a n d w i c h p l a t e s( D A T A - 2 , 4 x 4 m e s h , f u ll p la t e , a = b = 1 0 0 c m , t h i c k n e s s o f e a c h t o p s t if f l a y e r = 0 0 2 5 h , t h i c k n e s s o f e a c h b o t t o m s t i f f l a y er = 0 0 8 1 2 5 h , t h i c k n e s s o f c o r e = 0 6 h )

~o

M o d a l C o n s i d e r i n g G 2 3 a n d G ~ 3 o f s t i ff l a y e r s N e g l e c t i n g G 23 a n d G j 3 o f s t i ff l a y e r sN o .a h= 10 a h= 1 0 0 a h= 1 0 a h= 1 0 0

H O S T l l H O S T 9 H O S T 7 F O S T 5 H O S T l l H O S T 9 H O S T 7 F O S T 5 H O S T l l H O S T 9 H O S T 7 F O S T 5 H O S T l l H O S T 9 H O S T 7 F O S T 5

ga.

1 6 4 1 6 3 9 6 8 6 7 5 4 1 0 3 1 0 2 1 0 2 1 0 2 32 1 3 1 9 3 4 1 3 3 2 9 4 9 3 9 8 9 82 9 9 5 9 8 8 1 0 9 3 1 2 4 4 1 9 2 1 9 1 1 9 2 1 9 2 4 5 6 4 5 5 4 7 0 4 4 6 1 6 8 1 6 7 1 7 7 1 7 63 9 9 7 9 9 4 1 2 3 8 1 3 8 2 23 1 2 3 0 2 3 1 2 3 1 5 8 0 5 7 8 6 0 7 5 8 6 1 9 4 1 9 3 2 1 6 2 1 64 1 0 5 3 1 0 4 3 1 5 0 8 1 7 0 6 2 9 5 2 9 3 2 9 5 2 9 6 5 9 7 5 9 6 6 2 8 5 9 5 2 4 5 2 4 4 2 6 9 2 6 85 1 1 6 1 1 1 6 0 1 6 6 4 1 9 6 1 3 7 4 3 7 1 3 7 5 3 7 8 6 2 1 6 2 0 6 9 1 6 6 6 3 0 2 3 0 0 3 2 0 3 1 46 1 3 8 5 1 3 9 6 1 8 2 5 2 1 5 0 4 4 0 4 3 7 4 40 4 4 4 6 4 1 6 3 8 7 3 5 6 7 4 3 4 6 3 4 4 38 0 3 7 47 1 3 9 9 1 4 1 0 1 9 1 6 2 1 7 3 4 5 9 4 5 6 4 5 9 4 6 2 6 7 3 6 7 1 7 3 7 6 8 0 3 7 5 3 4 5 4 1 1 4 1 18 1 4 2 9 1 4 3 2 1 9 2 1 2 2 2 2 5 2 5 5 2 2 5 2 6 5 3 1 6 7 8 6 7 3 7 9 2 7 5 0 3 9 6 3 9 4 4 4 5 4 3 2


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3 0 6 M a l l i k a r] u n a T . K a n tTables 9 and 10 respectively. The in-plane dis-placements in refined theories (HOST) are about50 larger compared to those of FOST formoderately thick laminate a/h = 10), and for thinlaminate a/h=50) it is about 12 larger,whereas the transverse displacements in HOSTsare about 70 and 24 larger compared to thatin FOST for a/h = 10 and 50 respectively.

The variation of center normal stresses andcorner in-plane shear stress with time for a h = 1 0and 50 is given in Tables 11 and 12 respectively.The maximum central normal stress obtained byHOSTs is about 40 larger compared to that ofFOST for a/h=lO, and it is about 12 fora/h = 50. The in-plane shear stress obtained byHOSTs is about 50 larger compared to that of

T a b l e 9 . C o m p a r i s o n o f i n - p l a n e d i s p l a c e m e n t s (U t p X 1 0 4 c m a t x = 2 1 8 7 5 c m a n d y = 1 2 5 c m , v t P x 1 0 4 c m a t x = 1 2 ' 5 c ma n d y = 2 1 . 8 7 5 c m ) a n d c e n t e r t r a n s v e rs e d i s p l a c e m e n t s ( w 0 x 1 0 3 c m ) o f a c l a m p e d , n i n e - la y e r 0304560 co r e / 6 045

3 0 / 0 c o m p o s i t e - s a n d w i c h p l a t e u n d e r s u d d e n l y a p p l i e d u n i fo r m l y d i st r i b u te d p u l s e l o a d i n g( D A T A - 3 , 4 x 4 m e s h , f u l l p l a t e, a = b = 2 5 c m , a / h = 1 0 , A t = 1 - 0 k~s e c , q o = 1 N / c m 2, h o = h 3 o = h 45 = h 6 0 = 0 . 0 6 2 5 c m ,

h . . . . = 2 0 c m )T i m e( p s e c ) I n - p l a n e d i s p l a c e m e n t , u x 1 4 I n - p l a n e d i s p l a c e m e n t , v x 1 4 T r a n s v e r s e d i s p l a c e m e n t , w o 103F O S T 3 H O S T 5 H O S T 6 F O S T 3 H O S T 5 H O S T 6 F O S T 3 H O S T 5 H O S T 64 0 0 0 6 2 6 0 0 5 0 0 0 0 4 8 4 0 ' 1 0 9 4 0 ' 0 9 3 7 0 0 8 8 7 0 0 9 3 4 0 ' 1 0 4 6 0 1 0 3 78 0 0 2 4 5 1 0 1 5 8 7 0 ' 1 5 6 4 0 ' 3 9 9 6 0 ' 3 1 4 5 0 3 1 1 0 0 4 2 7 8 0 ' 4 1 5 8 0 4 1 5 21 2 0 0 4 5 0 2 0 3 2 2 9 0 ' 3 2 0 9 0 6 2 8 6 0 ' 6 3 5 2 0 ' 6 2 1 8 0 7 9 7 0 0 ' 9 3 9 0 0 9 3 8 01 6 0 0 ' 4 7 9 1 0 5 0 9 0 0 5 0 8 3 0 6 8 5 0 0 9 8 2 1 0 9 6 7 6 0 8 1 0 4 1 6 4 1 0 1 6 3 6 32 0 0 0 3 3 3 6 0 7 0 1 1 0 ' 6 9 5 1 0 5 3 9 1 1 ' 2 3 7 6 1 2 2 0 2 0 6 0 2 2 2 ' 3 6 9 1 2 3 6 0 62 4 0 0 1 5 3 4 0 ' 8 3 7 2 0 ' 8 2 8 3 0 ' 1 9 7 5 1 ' 3 6 9 5 1 - 34 5 6 0 2 6 5 8 2 ' 8 9 3 6 2 ' 8 8 5 72 8 0 0 ' 0 0 6 9 0 9 0 7 4 0 ' 8 9 4 0 0 ' 0 2 7 0 1 ' 3 6 1 6 1 ' 3 3 9 3 - 0 0 2 4 7 3 ' 0 2 4 3 3 ' 0 2 0 53 2 0 0 0 0 3 8 0 8 5 1 9 0 ' 8 4 5 2 0 ' 0 3 7 2 1 2 7 6 6 1 2 5 6 4 0 - 0 2 5 3 2 ' 7 6 2 8 2 ' 7 5 0 83 6 0 0 1 6 2 7 0 6 9 1 6 0 ' 6 8 5 9 0 ' 2 4 1 6 1 ' 1 5 2 4 1 1 3 7 0 0 2 7 2 7 2 ' 2 2 7 0 2 2 0 5 04 0 0 0 ' 3 7 6 1 0 4 7 1 9 0 '4 6 2 4 0 ' 5 5 0 8 0 ' 9 4 8 3 0 9 2 4 6 0 6 4 7 9 1 ' 5 6 6 5 1 ' 5 5 1 54 4 0 0 ' 4 8 5 8 0 ' 2 7 1 6 0 ' 2 6 3 8 0 ' 7 1 1 7 0 ' 6 3 3 2 0 6 1 1 9 0 8 5 6 3 0 ' 9 0 7 4 0 ' 9 0 7 04 8 0 0 4 1 1 9 0 1 1 8 7 0 1 1 4 4 0 5 9 5 3 0 3 1 8 4 0 3 0 9 4 0 ' 7 0 5 8 0 3 5 4 1 0 3 5 2 65 2 0 0 2 3 7 1 0 0 3 3 5 0 0 3 8 2 0 3 6 0 8 0 0 4 5 6 0 0 3 9 0 0 4 2 2 5 - 0 0 0 0 4 - 0 ' 0 0 8 95 6 0 0 ' 0 5 5 5 0 0 2 40 0 0 2 63 0 1 0 92 - 0 0 2 5 9 - 0 0 3 5 0 0 0 8 74 - 0 ' 0 8 8 6 7 - 0 ' 0 9 2 46 0 0 - 0 ' 0 1 3 8 0 ' 0 7 2 6 0 0 6 86 - 0 0 3 2 8 0 0 8 00 0 0 9 44 - 0 ' 0 5 1 1 0 ' 1 2 7 4 0 ' 1 3 2 3

6 4 0 0 ' 0 6 6 4 0 ' 1 7 7 1 0 - 1 7 9 0 0 1 5 8 9 0 ' 3 8 6 7 0 3 7 9 9 0 1 5 0 8 0 5 9 1 5 0 - 6 0 3 86 8 0 0 ' 2 9 8 9 0 3 4 9 1 0 3 5 0 2 0 4 2 2 7 0 7 2 6 9 0 7 0 6 4 0 ' 4 8 5 2 1 ' 1 8 4 5 1 ' 1 8 4 7S t a ti c 0 . 2 3 1 9 0 . 4 2 9 3 - - 0 . 3 4 7 0 0 . 7 4 6 6 - - 0 . 4 0 0 6 1 . 3 8 89 - -

T a b l e 1 0 . C o m p a r i s o n o f i n - p l a n e d i s p l a c e m e n t s ( f / top X 1 0 4 c m a t x = 2 1 8 7 5 c m a n d y = 1 2 ' 5 c m , v t p x 1 0 4 c m a t x = 1 2 . 5 c ma n d y = 2 1 . 8 7 5 c m ) a n d c e n t e r t r a n s v e r s e d i s p l a c e m e n t s ( w 0 x 1 0 3 c m ) o f a c l a m p e d , n i n e - la y e r 03045 60 co r e / 6 045

3 0 0 / 0 c o m p o s i t e - s a n d w i c h p l a te u n d e r s u d d e n l y a p p l i e d u n i fo r m l y d i s tr i b u t ed p u l s e l o a d i n g( D A T A - 3 , 4 x 4 m e s h , f u l l p l a t e, a = b = 2 5 c m , a / h = 5 0 , A t = 0 . 5 k z s e c , q 0 = 1 N / c m 2 , h 0 = h 3 0 = h 4 5 = h 6 0 = 0 0 1 2 5 c m ,

h . . . . = 0 ' 4 c m )T i m e( p s e c )

I n - p l a n e d i s p l a c e m e n t , u 1 4 I n - p l a n e d i s p l a c e m e n t , v x 1 04 T r a n s v e r s e d i s p l a c e m e n t , wo 103F O S T 3 H O S T 5 H O S T 6 F O S T 3 H O S T 5 H O S T 6 F O S T 3 H O S T 5 H O S T 6

8 0 0 9 2 6 6 0 7 5 0 9 0 7 4 6 9 1 1 2 6 7 0 8 7 5 9 0 8 6 7 3 1 5 3 1 4 2 1 3 9 2 2 1 3 7 91 6 0 2 7 9 5 5 2 4 3 6 8 2 4 2 5 5 4 1 1 4 5 3 2 8 7 1 3 2 6 1 2 9 2 5 3 2 8 0 6 5 5 8 0 5 8 62 4 0 5 4 6 5 6 4 ' 6 5 0 2 4 6 3 0 9 6 4 8 9 6 6 5 7 5 3 6 5 2 5 0 2 2 ' 3 9 7 2 1 4 0 7 2 1 4 0 63 2 0 8 6 6 4 4 7 ' 7 3 2 3 7 6 9 9 3 8 ' 3 4 3 5 8 9 3 9 4 8 8 6 2 4 3 4 6 4 6 3 7 5 6 6 3 7 5 4 74 0 0 1 0 1 3 2 1 0 ' 7 0 5 1 0 6 4 7 1 0 9 3 2 1 0 5 7 7 1 0 4 8 7 4 3 6 1 7 5 0 3 3 9 5 0 2 5 64 8 0 1 0 9 0 6 1 2 ' 5 3 4 1 2 4 5 5 1 2 2 3 3 1 2 2 9 6 1 2 - 1 9 3 4 6 1 8 3 5 7 ' 9 6 7 5 7 8 1 35 6 0 1 0 9 9 0 1 2 ' 3 0 0 1 2 ' 1 9 6 1 1 ' 2 0 3 1 3 7 3 3 1 3 6 0 6 4 2 4 3 3 6 0 ' 5 4 7 6 0 - 2 9 26 4 0 8 4 2 3 4 1 1 1 8 4 1 1 ' 0 7 7 9 3 3 7 3 1 3 3 1 4 1 3 1 5 9 3 5 1 6 1 5 4 8 2 5 5 4 5 0 67 2 0 5 3 9 6 6 9 9 8 6 7 9 8 8 7 0 6 2 3 2 7 1 0 7 8 2 1 0 6 1 9 2 3 5 4 2 4 4 ' 7 9 1 4 4 4 4 98 0 0 3 2 7 9 0 7 4 2 9 2 7 ' 3 1 6 4 3 ' 1 4 2 1 7 - 5 2 4 5 7 3 8 4 1 1 1 1 4 3 3 2 ' 9 5 6 3 2 5 8 28 8 0 0 9 1 2 9 4 0 2 2 2 3 ' 9 3 2 3 1 ' 6 7 2 4 4 6 6 6 2 4 5 6 7 8 2 6 1 0 0 2 0 ' 3 6 4 2 0 0 2 39 6 0 0 1 0 1 3 1 6 9 7 6 1 6 3 8 8 1 1 5 1 9 2 4 0 8 6 2 3 4 7 3 - 2 5 5 9 6 8 2 1 6 9 7 9 3 7 41 0 4 0 0 7 3 8 2 0 4 5 9 4 0 - 4 2 5 1 1 1 8 77 0 8 2 9 3 0 7 9 3 8 0 9 2 0 9 - 1 ' 7 1 5 5 - 1 8 6 5 31 1 2 0 2 4 9 23 - 0 - 0 7 2 6 - 0 - 0 4 9 8 2 6 2 83 0 6 9 60 0 . 69 4 1 1 1 3 25 - 2 ' 4 5 6 3 - 2 3 6 8 61 2 0 0 5 6 6 3 4 0 - 9 3 1 8 0 9 6 9 4 5 7 0 8 7 1 9 6 2 8 1 9 8 5 6 2 1 0 6 5 3 ' 5 3 0 3 3 7 4 0 5S t a t i c 5 . 6 2 7 5 6 - 2 5 4 6 - - 6 . 2 4 1 7 7 . 0 0 8 1 - - 2 2 . 4 6 4 2 8 . 9 3 6 - -


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Refined theories of iber reinforced lamina ted comp osites and san dwiches 307T a b l e 1 1 . C o m p a r i s o n o f c e n t e r n o r m a l s t r e s s e s ( a top a n d a ~ o p i n N / c m 2 ) a n d i n - p l a n e s h e a r s t r e s s e s ( l r ~ i n N / c m top o f ac l a m p e d , n i n e - l a y e r (030 / 4560 / cor e/60 * /4530 /0 ) c o m p o s i t e - s a n d w i c h p l a t e u n d e r s u d d e n l y a p p l i e d u n i f o r m l y d i s t r i -b u t e d p u l s e l o a d i n g( D A T A - 3 , 4 x 4 m e s h , f u l l p l a t e , a = b - - 2 5 c m , a / h = 1 0 , A t = 1 . 0 / z s e c , q 0 = 1 N / c m 2 , h 0 = h 30 = h 45 = h 6 0 = 0 0 6 2 5 c m ,h . . . . = 2 0 cm)Time Bending stress a~ Bending stress oy Shear stress r~(~sec) FOST3 HOST5 HOST6 FOST3 HOST5 HOST6 FOST3 HOST5 HOST6

40 1 9236 4-0605 5 3069 - 0-0 47 6 0 6385 1 6677 - 0 535 8 -0 80 14 -0 783 480 65 836 21 526 22 775 5 3596 2 2350 3 6950 -1 395 8 -2 36 40 -2 361 1120 148 55 54 841 57 173 17 108 4 6381 5 2782 -2 21 42 -3 56 61 -3 53 36160 140 81 130 07 133 36 16 103 11 339 12 686 -2 -6 34 7 -4 -00 55 -3 921 7200 104 78 207 64 207 59 8 8519 19 057 21-014 -1 777 7 -4 10 59 -4 03 60240 36 692 249 67 249 49 5 1846 27 238 28-899 -0 792 8 -4 365 5 -4 348 9280 -16 263 245 07 248'31 -2 154 8 29 358 32-290 -0 3114 -4 996 0 -4 9997320 7 4574 212 49 218 32 -1 40 23 26 353 28 508 -0 086 6 -5 21 57 -5- 171 1360 20 940 177 86 180 34 3 1356 18 282 19 578 -0 91 15 -4 57 11 -4 49 27400 122 06 132 90 130 79 12 922 10 384 11 696 -2 07 24 -3 45 61 -3 40 92440 164 21 80 666 81 140 16 751 4 7036 5 1654 -2 471 8 -2 55 17 -2- 54 60480 102 83 20 261 23'308 12-365 1 2816 1 5971 -2-2 05 4 -1 872 7 - 1 89 07520 77 434 -19 594 -21 286 7 3379 -1 0825 0 1142 -1 2958 -1 3733 -1 3162560 17 677 -27 577 -28 916 0 7743 -0 8900 -0 8138 -0 4365 -0 8591 -0 8063600 -37 018 3 5468 4 4951 -2 399 4 1 1277 1 3388 -0 0747 -0 8079 -0 7667640 22 641 50-359 54 425 -0 063 1 3 9069 5 9228 -0 44 40 -1 685 5 -1 690 1680 88 304 104 86 106 04 9 3167 8 7733 9 6446 -1 673 5 -3 '2 16 7 -3 204 5Static 65.58 109.60 -- 6.769 11.24 -- -1. 28 6 -3 .1 30 --

T a b l e 1 2 . C o m p a r i s o n o f c e n t e r n o r m a l s t r e s s e s ( o top a n d a ~ p i n N / c m 2 ) a n d i n - p l a n e s h e a r s t r e s s e s ( t' t_ p i n N / c m top o f ac l a m p e d , n i n e - l a y e r ( 0304560co r e/ 6045300 c o m p o s i t e - s a n d w i c h p l a t e u n d e r s u d d e n l y a p p l i e ~ u n i f o r m l y d i s tr i -b u t e d p u l s e l o a d i n g( D A T A - 3 , 4 x 4 m e s h , f ul l p l at e , a = b = 2 5 c m , a / h = 5 0 , A t = 0 . 5 / ~ se c , q 0 = l N / c m 2, h o = h 3 o = h 4 5 = h 6 o =O . 0 1 2 5 c m ,h . . . - - 0 . 4 cm)Tim e Bending stress trx Bend ing stress oy Shear stress Ly(psec)

FOST3 HOST5 HOST6 FOST3 HOST5 HOST6 FOST3 HOST5 HOST680 - 156-06 - 29 528 - 29-026 - 13 868 1'255 5 2 0682 - 9-7087 - 8 0415 - 7'97 91160 486 77 111 16 111 26 1 4424 -7 9831 -7-9 845 -16 469 -19 005 -18 836240 1 796 5 1 276 2 1 276 9 112-07 49 284 49 585 - 21-244 - 25 492 - 25 234

320 2663 6 2456 8 2460 2 249 34 181 59 184 76 -3 9 140 -3 4 80 7 -34 452400 3 786 8 3 297 5 3 295-8 297 86 298 48 300-85 - 49-506 - 48 820 - 48 333480 3 816-3 3 775-0 3 769-0 276 69 351 66 355-00 - 46-901 - 58 642 - 58 036560 2 991 7 4 268 1 4 256 0 257-97 333 42 335 21 - 44-679 - 60' 287 - 59-556640 2 827 3 3 777 5 3758 1 215 91 257 96 258-65 - 35-255 - 54 727 - 53-920720 2080-4 2554.2 2531 1 147 65 210 76 212 38 -3 1- 04 7 -4 4 60 0 -43 842800 654 22 1 851 3 1 830 9 67-442 169 00 167 20 - 18 812 - 32 648 - 32 039800 51 075 1 520-0 1 503 2 - 19 063 109-40 110 39 - 1'72 87 - 22 319 - 21-890960 - 54 437 619 88 599 06 - 84 155 40 529 39-609 -4 28 19 - 15 680 - 15 3561040 - 186 29 - 511 43 - 524 92 - 28 272 - 48 132 - 50 059 - 10 761 - 9 8985 - 9 6544

1 120 1016 9 -511 55 -508 48 90 696 -66 862 -66 040 - 11 740 -3 0369 -2 93551200 1390-8 -87 275 -70 608 119 60 -35-85 8 -36-0 30 -25 815 -1 9424 -2 0707Static 1 751.0 180 7.0 -- 128.6 139-7 -- - 26-43 - 31.46 --

F O S T f o r a / h = l O and i t i s abo ut 18 fora /h = 50.A s e x p e c t ed , t h e m a x im u m d i sp l a c e m e n t fo r acons tan t fo rce appl ied suddenly i s twice the d is -p l a c e m e n t c a u se d b y th e s a m e fo rc e a p p l i e ds ta t is t ica l ly (s lowly). Un l ike in i so t rop ic p la tes , thein t e rn a l fo r c e s a n d

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