Thermal buckling analysis of skew bre-reinforced composite and sandwich plates using shear deformable nite element models

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1 Composit Structurs 49 () ±85 Thrmal buckling analsis of skw br-rinforcd composit and sandwich plats using shar dformabl nit lmnt modls T. Kant *, C.S. Babu Dpartmnt of Ciil Enginring, Indian Institut of Tchnolog Bomba, Powai, Mumbai 4, India Abstract Th papr considrs th lastic buckling of skw br-rinforcd composit and sandwich plats subjctd to thrmal loads. To th bst of th authors' knowldg, thr is no papr in th opn litratur on this subjct and th prsnt papr attmpts to ll this gap. Two shar dformabl nit lmnt modls, on basd on rst-ordr shar dformation thor and th othr basd on highrordr shar dformation thor, ar mplod to obtain thrmal buckling solutions. Etnsi numrical rsults ar prsntd for both thin and thick laminatd composit plats with arious skw angls, lamination paramtrs and boundar conditions. A fw rsults for skw sandwichs ar obtaind for arious gomtric paramtrs and skw angls. Rsults prsntd, not aailabl so far, could b usful to dsignrs and rsarchrs who ma us thm as bnchmark alus to alidat thir numrical tchniqus and softwar for similar problms. Ó Elsir Scinc Ltd. All rights rsrd. Kwords: Buckling Highr-ordr thor Shar dformation Skw plat Skw laminats Skw sandwichs Thrmal buckling 1. Introduction Fibr-rinforcd composit matrials du to thir high spci c strngth and sti nss ar bcoming incrasingl usd in man nginring applications, spciall in th arospac and ship building industris. It is wll known that skw or obliqu plats mad of ths matrials ar important structural componnts of ship hulls and swpt wings of aroplans. Buckling is on of th primar mods of failur of ths lmnts whn th ar subjctd to mmbran strsss causd b ithr thrmal loads or mchanical loads or a combination of ths loads. Thus th buckling strngth of skw composit laminats is on of th factors gorning thir dsign and its accurat dtrmination is of intrst to dsignrs. Considrabl rsarch [1±4] has bn publishd instigating th buckling rspons of skw composit laminats. In ths instigations, numrical mthods such as th nit lmnt mthod, th Raligh±Ritz mthod, tc. ar usd as act solutions cannot b obtaind du to th us of a non-orthogonal coordinat sstm in th driation of th gorning di rntial quations. Rdd and Palaninathan [1] ha mplod a * Corrsponding author. addrsss: tkant@ciil.iitb.rnt.in (T. Kant), rssb@ciil.iitb.rnt.in (C.S. Babu). triangular nit lmnt basd on classical laminatd plat thor. Jaunk t al. [] and Wang [] ha usd th Raligh±Ritz mthod incorporating rst-ordr shar dformation cts. Vr rcntl, Babu and Kant [4] ha prsntd two C shar dformabl nit lmnt formulations for th buckling analsis of skw laminatd composit and sandwich panls. A 1-nod bi-cubic Lagrang lmnt is usd in th formulations. In all ths instigations, skw laminats subjctd to onl mchanical loads ar considrd. Surprisingl, in th cas of skw composit laminatd and sandwich plats subjctd to thrmal loads, thr ar irtuall no paprs in th opn litratur although thr ar a fw studis [5±] on thrmal buckling of isotropic skw plats. Th objcti of this work is to ll this gap b prsnting a stud of th thrmal buckling of skw laminatd composit and sandwich plats. Th two shar dformabl nit lmnt modls prsntd b th authors [4] priousl for mchanical buckling analss ar mplod hr for thrmal buckling analss. On of ths modls is basd on Rissnr±Mindlin rst-ordr thor and th othr is basd on a highr-ordr thor dlopd b Kant and his fllow rsarchrs [8±1]. Th accurac of th modls is ri d against th litratur alus for isotropic skw plats. Nw rsults ar prsntd for skw laminatd composit and sandwich plats using th standard matrial proprtis aailabl in th litratur. -8//$ - s front mattr Ó Elsir Scinc Ltd. All rights rsrd. PII: S - 8 ( 9 9 ) 1-

2 8 T. Kant, C.S. Babu / Composit Structurs 49 () ±85. Thortical formulation Th two shar dformation thoris considrd for instigation in th prsnt work ar basd on th assumption of th displacmnt lds in th following form: (a) First-ordr shar dformation thor (FSDT), 5 dgrs of frdom/nod u z ˆu zh z ˆ zh w z ˆw : 1 (b) Highr-ordr shar dformation thor (HSDT), 9 dgrs of frdom/nod u z ˆu zh z u z h z ˆ zh z z h w z ˆw whr u, and w d n th displacmnts of an gnric point (,, z) in th plat spac, u, and w dnot th displacmnts (Fig. 1) of a point on th middl plan, h and h ar th rotations of normal to middlplan about - and -as, rspctil. Th paramtrs u,, h and h ar highr-ordr trms in th Talor sris pansion and ar also d nd at th mid-surfac. Nglcting th transrs strss r z and strain z, th Duhaml±Numann form of Hook's law for th L th lamina in th laminat co-ordinats z is writtn as 8 r >< r s >: s z s z 9 Q 11 Q 1 Q 1 >= Q 1 Q Q ˆ Q 1 Q Q 4 Q > 44 Q 45 Q 45 Q 55 or in short form r ˆ Q t in which 8 >< 5 >: Fig. 1. Th gomtr of skw laminat. a DT a DT c a DT c z c z 9 >= > 4 T r ˆ r r s s z s z T ˆ c c z c z T t ˆ a a a DT 5 ar rspctil th strss, th total strain and th thrmal strain ctors. Th Q ij 's ar th plan strss rducd sti nss co cints. Th transformation of th strsss/ strains btwn th lamina and laminat co-ordinat sstms follows th usual strss tnsor transformation rul. DT is tmpratur ris. Th a, a and a trms ar d nd as follows: a ˆ a 1 cos h a sin h a ˆ a 1 sin h a cos h a ˆ a 1 a sinh cosh whr a 1 and a ar th thrmal pansion co cints of th lamina along th longitudinal and transrs dirctions of th brs, rspctil. It will b notd that th thortical formulation gin in th rst of th papr is basd on HSDT and th formulation corrsponding to FSDT can b obtaind from that of HSDT b truncating th trms corrsponding to th highr-ordr displacmnt dgrs of frdom. Substituting Eq. () into Grn's strain tnsor [11], with z ˆ, th gnralizd strain ctor componnts ar obtaind as: ˆ z z z z4 ˆ z z z z4 z 5 z 5 z z c ˆ z z z z4 z5 z c z ˆ / zw z / z w z4 / c z ˆ / zw z / z w z4 / z 5 w z 5 w : Not that th gnralizd strain componnts ar prssd in trms of mid-surfac strain componnts dnotd b th ctor,. Each of th componnts of th strain ctor, has both linar as wll as non-linar parts L which ar prssd in trms of mid-plan displacmnt componnts (s Appndi A). In th absnc of trnal loads, th total potntial nrg of th laminat for th thrmal buckling problm is prssd as P ˆ 1 Z Z f t g T rd T L r d 8 whr and L ar rspctil th linar and non-linar parts of th gnralizd strain ctor,. Th scond prssion in th abo quation is th work don b initial (or prbuckling) strsss. Substituting for, L and r in Eq. (8) and prforming an plicit intgration through th laminat thicknss, th potntial nrg of th laminat is rwrittn as

3 T. Kant, C.S. Babu / Composit Structurs 49 () ±85 9 P ˆ 1 Z Z Z T D da TrdA L Tr r ˆ D r t da t 15 A A A whr f Q ij a i 9 D M D C in which f is a function of Q ij and a i, and th nw D ˆ 4 D T C D B 5 1 ctors in concis form ar prssd as D S r ˆfN T M T Q T g T and r t ˆfN T t M T t O T t gt 1 in which D M, D B, D C and D S ar th mmbran, ural, mmbran± ural coupling and shar rigidit matrics, whr rspctil. Th componnts of th rigidit matri ar n o T N ˆ N N N N N N N N N N N N gin in Appndi A. n o Following th standard procdur of th nit lmnt formulation and th transformation of th sti nss T M ˆ M M M M M M M M M n o T matrics and th load ctor from global as to Q ˆ Q Q S S Q Q S S Q Q S S local as 1 1 (Fig. 1) as plaind in Rf. [4], th n o T N t ˆ N t N t N t N t N t N t N t N t N t N quilibrium and stabilit conditions ar obtaind as t N t N t n o T K d ˆ R 1 M t ˆ M t M t M t M t M t M t M t M t M t K O t ˆ f g T kk g Šd d ˆ 18 : whr K, K g and R ar th linar sti nss matri, th gomtric sti nss matri and th thrmal load ctor, rspctil. d is th nodal displacmnt ctor of th plat. Th componnts of strss rsultant and strss coupl ctors in Eq. (1) ar d nd as follows: N N N Z 1 N N N 4 N N N 5 ˆ XNL ZL 1 z 4 Z L z 4 5 r r s dz z N N N M M M 4 M M M 5 ˆ XNL 4 M Q S Q S Q S M Q S Q S Q S 5 M ˆ XNL Z ZL 1 Z L 4 Z ZL 1 Z L 1 z z z z 4 z 5 z 4 z 5 r r s dz z s z s z dz 1 5 Q ij H N t ˆ XNL Q ij H < a = 4 Q ij H 5 5 a : DT a Q ij H and 8 M t ˆ XNL Q ij H 4 Q ij H 4 5 a 9 < = a DT 14 : Q ij H a whr i j ˆ 1 and H i ˆ zi L 1 zi L i Aftr intgration, th strss rsultants in Eqs. (11)± (1) ar writtn in matri form as.1. Thrmal buckling analsis Th calculation of th critical buckling tmpratur consists of two stags. For a spci d ris in tmpratur DT, a linar static analsis (1) is carrid out to dtrmin th thrmal strss rsultants. Ths strss rsultants ar thn usd to comput th gomtric sti nss matri which is subsquntl usd in Eq. (18) to dtrmin th smallst ignalu, k and th associatd mod shap, d d. Th critical tmpratur, T cr of th plat is calculatd using T cr ˆ k DT : 19. Numrical rsults and discussion Computr programs ha bn dlopd, basd on th forgoing nit lmnt modls, to sol a numbr of thrmal buckling ampls of skw composit laminatd and sandwich plats. Th programs can handl panls subjctd to non-uniform tmpratur ris or th surfac and through th thicknss. Howr, in all th ampls considrd hr, th tmpratur ris is assumd to b uniform. In gnral a skw msh of 1-nod lmnts has bn usd in th computations cpt for th conrgnc stud prsntd on isotropic plat. Th slcti intgration schm, naml 4 4 Gauss±Lgndr for th mmbran, ur, mmbran± ur and for th shar nrg contributions was usd for thin plats sid to thicknss ratio a=h > and a full 4 4 intgration schm is usd for thick

4 8 T. Kant, C.S. Babu / Composit Structurs 49 () ±85 Tabl 1 Critical tmpratur paramtr, k T for clampd isotropic skw plats W Msh Rf. [] HSDT FSDT FI SI FI SI ) ) ) laminats. All of th laminats considrd wr assumd to ha an aspct ratio of a=b ˆ 1, though th gnral cas a ˆ b ma also b studid without an di cult. In all of th FSDT modl computations, a shar corrction factor of 5= was usd. Du to th lack of comparati rsults for skw composit laminatd plats, th accurac of th prsnt formulations was aluatd onl with rspct to rsults for isotropic plats. Subsquntl, som nw rsults ar prsntd for laminatd composit and sandwich skw plats..1. Isotropic skw plats Critical buckling tmpratur alus of clampd isotropic skw plats ar gin b Prabhu and Durasula []. Th usd classical plat thor in conjunction with th Ritz mthod. To obtain th classical plat thor solution, a thin plat with a=h ˆ 1 is analsd hr. Th critical tmpratur alus ar aluatd in nondimnsional form, prssd as k T ˆ Eab ht cr =p D whr T cr is th critical tmpratur, a th thrmal pansion co cint, E th Young's Modulus and D ˆ Eh =1 1 m. Poisson's ratio, m is.. Th rsults obtaind with full intgration (FI) and slcti intgration (SI) schms ar gin in Tabl 1 for four skw angls, W ˆ 15 and 45. For W ˆ 45, th rsults obtaind with succssi r nd mshs ar also prsntd. Th rsults of th prsnt modls obtaind with th SI schm ar almost idntical with th rsults gin in [] and th di rnc btwn th two intgration schms is ngligibl. Thus, th rsults dmonstrat that th 1-nod lmnt is lss suscptibl to shar locking n in a distortd msh. It is notd that th lmnt hibits monotonic conrgnc and a skw msh is considrd adquat for furthr analsis... Composit skw laminats Cross-pl and angl-pl laminats ar considrd. Th lamination schms usd ar: (i) smmtric crosspl ( /9 /9 / ) and (ii) anti-smmtric angl-pl (45 / )45 /...) with th numbr of lars, NL ˆ 4 and 1. Both thin and thick skw laminats with four di rnt dg conditions ar analsd. Th dg conditions considrd ar: (i) all dgs simpl supportd (SSSS), (ii) straight dgs clampd and skwd dgs simpl supportd (CSCS), (iii) straight dgs simpl supportd and skwd dgs clampd (SCSC) and (i) all dgs clampd (CCCC). Th simpl supportd (SS) and clampd boundar conditions usd hr ar gin in Rf. [4]. Th skw angl, W is arid from to 45. Th matrial charactristics [1] of indiidual lamina usd hr ar: Tabl Critical tmpratur paramtr k T 1 of smmtric cross-pl =9 S Š skw laminat with arious dg conditions a=h ˆ 1 mshš W Thor Edg conditions SSSS CSCS SCSC CCCC HSDT FSDT HSDT FSDT HSDT FSDT HSDT FSDT

5 T. Kant, C.S. Babu / Composit Structurs 49 () ±85 81 Tabl Critical tmpratur paramtr k T 1 of anti-smmtric angl-pl (45 /)45 /...) skw laminats with arious dg conditions a=h ˆ 1 mshš NL W Thor Edg conditions SSSS CSCS SCSC CCCC 4 HSDT FSDT HSDT FSDT HSDT FSDT HSDT FSDT HSDT FSDT HSDT FSDT HSDT FSDT HSDT FSDT Tabl 4 Critical tmpratur paramtr k T of smmtric cross-pl =9 S Š skw laminat with arious dg conditions a=h ˆ 1 mshš W Thor Edg conditions SSSS CSCS SCSC CCCC HSDT FSDT HSDT FSDT HSDT FSDT HSDT FSDT Tabl 5 Critical tmpratur paramtr k T of anti-smmtric angl-pl (45 /)45 /...) skw laminats with arious dg conditions a=h ˆ 1 mshš NL W Thor Edg conditions SSSS CSCS SCSC CCCC 4 HSDT FSDT HSDT FSDT HSDT FSDT HSDT FSDT HSDT FSDT HSDT FSDT HSDT FSDT HSDT FSDT

6 8 T. Kant, C.S. Babu / Composit Structurs 49 () ±85 E 1 =E ˆ 15 E ˆ E G 1 =E ˆ G 1 =E ˆ :5 G =E ˆ :5 m 1 ˆ m 1 ˆ : m ˆ :49 a 1 =a ˆ :15 a =a ˆ a =a ˆ 1: whr a is a normalisation factor for th co cint of thrmal pansion. Tabls and show th critical tmpratur paramtr, k T ˆ a T cr for smmtric cross-pl ( /9 /9 / ) and anti-smmtric angl-pl 45 = 45 =... thin a=h ˆ 1 skw laminats, rspctil. As pctd, th rsults of th rst-ordr and highr-ordr thoris show good agrmnt for all th dg conditions and for all th skw angls considrd. Tabls 4 and 5 show th critical tmpratur paramtr for thick a=h ˆ 1 smmtric cross-pl ( /9 / 9 / ) and anti-smmtric angl-pl 45 = 45 =... laminats, rspctil. It is clar that for both crossand angl-pl laminats, th rst-ordr thor or-stimats th critical tmpratur and th di rnc btwn th two thoris incrass with incrasing skw angl. Th maimum di rnc btwn th two thoris occurs with CCCC laminats whn W ˆ 45. Th diffrnc is about 4.% in th cross-pl laminat and in th angl-pl laminat with NL ˆ 4, th di rnc is about.%. Howr, for th tn lard angl-pl laminat, th rsults of both thoris ar narl th sam with a maimum di rnc of about 1%. In gnral, th critical tmpratur of both th thin and thick laminats incrass with skw angl. But th incras is mor pronouncd in thin laminats. Figs. (a) and (b) show th ct of width-to-thicknss ratio on th critical tmpratur of four-lard anti-smmtric angl-pl (45 /)45 /45 /)45 ) skw laminat for SSSS and CCCC support conditions, rspctil. Th rsults obtaind with HSDT analsis ar usd hr. Th critical tmpratur paramtr dcrass, i.., th ct of transrs shar dformation incrass, with incras in laminat thicknss. Th ct of transrs shar dformation is sn to incras with incras in th skw angl for both simpl supportd and clampd laminats, but th incras is mor signi cant in clampd skw laminats. Th ct of skw angl on critical tmpratur dcrass with incras in thicknss and for thick laminats with a=h ˆ 5, th skw angl has ngligibl in unc on critical tmpratur... Skw sandwichs Smmtric skw sandwich panls with cross-pl composit fac shts and a honcomb cor ar considrd hr. Th stacking squnc of th panl is [( / 9 ) 5 /cor/(9 / ) 5 ]. Th matrial charactristics [1] of th fac shts and cor ar: Fac shts E 1 =E ˆ 19 E ˆ E G 1 =E ˆ :5 G =E ˆ :8 m 1 ˆ m 1 ˆ : m ˆ :49 a 1 =a ˆ :1 a =a ˆ a =a ˆ 1:: Cor E 1 =E f ˆ : 1 5 E =E f ˆ :9 1 5 E =E f ˆ :4 G 1 =E f ˆ :4 1 G 1 =E f ˆ :9 1 G =E f ˆ : 1 m 1 ˆ :99 m 1 ˆ m ˆ 1 5 a 1 ˆ a ˆ a ˆ 1:a Fig.. E ct of thicknss ratio on th critical tmpratur of antismmtric angl-pl 45 = 45 =45 = 45 skw laminats (a) SSSS and (b) CCCC mshš. whr, E f rfrs to th fac shts. Numrical rsults ar prsntd in Tabl for simpl supportd (SSSS) panls with W ˆ 15 and 45. Two paramtrs, a=h and h f =h ar arid, whr h f is thicknss of ach of th fac shts. For th alidation of th prsnt modls, -D lasticit solution rsults [1,14] ar also gin for panls with W ˆ. It ma b notd that th HSDT rsults match wll with th -D lasticit solution for all alus of h f =h, whras FSDT or-stimats th buckling tmpratur b a signi cant margin at highr alus of h f =h. In th cas of skw panls, for all alus of W and a=h ratios considrd, th FSDT and HSDT rsults ar almost idntical for

7 T. Kant, C.S. Babu / Composit Structurs 49 () ±85 8 panls with r thin fac shts (h f =h ˆ :5. Howr, with incrasing fac sht thicknss, FSDT clarl or-stimats th critical tmpratur. Th di rnc btwn FSDT and HSDT incrass with skw angl. For modratl thick panls a=h ˆ with h f =h ˆ :15, th di rnc incrass from about.4% to 18.1% as W incrass from to 45. For a panl with a=h ˆ 1 and h f =h ˆ :15, th di rnc incrass from about.% to 41.% as th skw angl incrass from to Conclusions Two C isoparamtric nit lmnt formulations ar usd for thrmal buckling analsis of skw br-rinforcd laminatd composit plats and composit sandwich plats. Th accurac of th prsnt formulations is dmonstratd for isotropic plats. Nw rsults ar prsntd for laminatd anisotropic and sandwich plats with arious skw gomtris. Th snsitiit of th critical buckling tmpratur to ariations in skw angl, width-to-thicknss ratio and boundar conditions is studid. In gnral th critical tmpratur alus incras with incras in skw angl and th incras is mor pronouncd in thin laminats than in thick laminats. Through-thicknss shar dformation is r larg in thick laminats, and this ffct incrass with incras in th skw angl. Th rsults show that th di rncs in prdictions of FSDT and HSDT ar small for composit laminats. Howr, for sandwich panls, in comparison to HSDT, FSDT or-stimats th critical tmpratur b a signi cant margin and th margin incrass as th skw angl incrass. Th rsults prsntd hr for both thin and thick laminats ar th rst of thir kind and it is blid that th ma sr as bnchmark alus for othr dsignrs and rsarchrs to tst th alidit of thir numrical tchniqus and softwar for similar kinds of problms. Appndi A Th mid-surfac strain ctor, is prssd in trms of linar ( ) and non-linar componnts L as ˆ L A:1 whr n ˆ L ˆ / / w T w / / w w / / w o w A: n L L L L L L L L L L L L L L L L L L L L L / L / L w L T: w L / L / L w L w L / L / L w L Lo w A: Th componnts of linar strain ctor ar: Tabl Critical tmpratur paramtr k T for simpl supportd (SSSS) smmtric skw sandwich plats with composit cross-pl fac shts mshš a=h W Thor h f =h D Elas [1,14] HSDT FSDT HSDT FSDT HSDT FSDT HSDT FSDT D Elas HSDT FSDT HSDT FSDT HSDT FSDT HSDT FSDT

8 84 T. Kant, C.S. Babu / Composit Structurs 49 () ±85 ˆ ou o ˆ o o ˆ ou o ˆ ou o ˆ o o ˆ ou o o o ˆ ˆ ˆ ˆ ˆ ˆ ˆ oh o ˆ oh o ˆ oh o ˆ oh o ˆ ˆ ˆ / ˆ h ow o / ˆ h ow o ˆ oh o oh o ˆ oh o oh o w ˆ u w ˆ / ˆ h / ˆ h w ˆ w ˆ / ˆ / ˆ w ˆ w ˆ : Th componnts of non-linar strain ctor L ar: L ˆ 1 ou 1 o 1 ow o o o L ˆ 1 ou 1 o 1 ow o o o L ˆ ou o ou o o o o o ow ow o o L ˆ 1 oh 1 oh ou o o o L ˆ 1 oh 1 oh ou o o o L ˆ ou o oh o ou o o o o o ou o o o o o ou o ou ou o o o o o o o o o o oh oh o o oh o L ˆ 1 ou 1 o L ˆ 1 ou 1 o o o o o oh o oh o oh o oh oh o o oh o oh oh o o L ˆ ou ou o o o o o o oh oh o o oh oh o o oh oh o o oh o oh o L ˆ 1 oh 1 o L ˆ 1 oh o oh 1 oh o o L ˆ oh oh o o oh oh o o L ˆ ou oh o o o oh o o L ˆ ou o L ˆ ou o L ˆ ou o L ˆ ou o L ˆ ou o o o oh o o oh o o oh o ou oh o o o oh o o o oh o o oh o ou oh o o o oh o o o oh o o oh o ou oh o o o oh o o o oh o o oh o ou oh o o ou oh o o ou oh o o o oh o o L ˆ ou oh o o o oh o o oh o o oh o o o oh o o L ˆ ou oh o o o oh o o L ˆ ou oh o o ou oh o o o oh o o o oh o o / L ˆ ou o h o o h / L ˆ ou o h o o h w L ˆ ou u o o o h oh o h oh o w L ˆ ou u o o o h oh o h oh o / L ˆ oh u o oh h o o / L ˆ oh u o h o o w L ˆ h oh o h oh h oh o o h ou o h o o h ou o oh o h ou o h o o ou h o o u w L ˆ h oh o h oh o u h oh o ou o o o oh h o ou o o o oh h o

9 T. Kant, C.S. Babu / Composit Structurs 49 () ±85 85 / L ˆ oh u o oh o o h o ou h o / L ˆ oh u o oh o o h o ou h o w L ˆ oh h o oh h w L o ˆ oh h o oh h o : Th rigidit matrics in Eq. (1) ar Q ij H 1 Q ij H Q ij H 5 Q ij H D M ˆ XNL Q ij H 5 Q ij H Q ij H 9 4 Q ij H 9 Q ij H 11 5 Sm: Q ij H 1 Q ij H Q ij H 4 Q ij H D C ˆ XNL Q ij H 4 Q ij H Q ij H 8 4 Q ij H Q ij H 8 Q ij H 1 5 Q ij H 8 Q ij H 1 Q ij H 1 Q ij H Q ij H 5 Q ij H D B ˆ XNL 4 Q ij H Q ij H 9 5 Sm: Q ij H 11 Q ml H 1 Q ml H Q ml H Q ml H 4 Q ml H 5 Q ml H Q ml H Q ml H 4 Q ml H 5 Q ml H Q ml H D S ˆ XNL Q ml H 5 Q ml H Q ml H Q ml H 8 Q ml H Q ml H 8 Q ml H 9 4 Sm: Q ml H 9 Q ml H 1 5 Q ml H 11 A:4 whr i j ˆ 1 and l m ˆ 4 5 H i ˆ zi L 1 zi L : i Rfrncs [1] Rdd ARK, Palaninathan R. Buckling of laminatd skw plats. Thin-Walld Struct 1995(4):41±59. [] Jaunk N, Knight NF, Ambur DR. Buckling of arbitrar quadrilatral anisotropic plats. AIAA J 1995(5):98±44. [] Wang S. Buckling analsis of skw br-rinforcd composit laminats basd on rst-ordr shar dformation plat thor. Compos Struct 199(1):5±19. [4] Babu CS, Kant T. Two shar dformabl nit lmnt modls for buckling analsis of skw br-rinforcd composit and sandwich panls. Compos Struct 19994():115±4. [5] Prabhu MSS, Durasula S. Elastic stabilit of thrmall strssd clampd-clampd skw plats. ASME J Appl Mch 19441(): 8±1. [] Prabhu MSS, Durasula S. Thrmal buckling of rstraind skw plats. ASCE J Eng Mch 1941():19±5. [] Prabhu MSS, Durasula S. Thrmal post-buckling charactristics of clampd skw plats. Comput Struct 19():1±85. [8] Kant T. Numrical analsis of thick plats. Comput Mth Appl Mch Eng 1981(1):1±18. [9] Kant T, Own DRJ, Zinkiwicz OC. A r nd highr-ordr C plat bnding lmnt. Comput Struct 19815():1±8. [1] Kant T, Panda BN. A simpl nit lmnt formulation of a highr-ordr thor for unsmmtricall laminatd composit plats. Compos Struct 19889():15±4. [11] Zinkiwicz OC, Talor RL. Th nit lmnt mthod. 4th d., ol. 1. Singapor: McGraw-Hill, [1] Noor AK, Burton WS. Thr-dimnsional solutions for thrmal buckling of multilard anisotropic plats. ASCE J Eng Mch (4):8±1. [1] Noor AK, Ptrs JM, Burton WS. Thr-dimnsional solutions for initiall strssd structural sandwichs. ASCE J Eng Mch 19941():84±. [14] Noor AK, Burton WS, Brt CW. Computational modls for sandwich panls and shlls. Appl Mch R 19949():155±99.

NONLINEAR ANALYSIS OF PLATE BENDING

NONLINEAR ANALYSIS OF PLATE BENDING CONTENTS Govrning Equations of th First-Ordr Shar Dformation thor (FSDT) Finit lmnt modls of FSDT Shar and mmbran locking Computr implmntation Strss calculation Numrical