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1 This rticle ws downloded y:[knt, Trun] On: 8 Jnury 2008 Access Detils: [suscription numer ] Pulisher: Tylor & Frncis Inform Ltd Registered in Englnd nd Wles Registered Numer: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Journl of Therml Stresses Puliction detils, including instructions for uthors nd suscription informtion: An Efficient Semi-Anlyticl Model for Composite nd Sndwich Pltes Sujected to Therml Lod Trun Knt ; Sndeep S. Pendhri ; Yogesh M. Desi Deprtment of Civil Engineering, Indin Institute of Technology Bomy, Powi, Mumi, Indi Online Puliction Dte: 01 Jnury 2008 To cite this Article: Knt, Trun, Pendhri, Sndeep S. nd Desi, Yogesh M 'An Efficient Semi-Anlyticl Model for Composite nd Sndwich Pltes Sujected to Therml Lod', Journl of Therml Stresses, 31:1, To link to this rticle: DOI: / URL: PLEASE SCROLL DOWN FOR ARTICLE Full terms nd conditions of use: This rticle mye used for reserch, teching nd privte study purposes. Any sustntil or systemtic reproduction, re-distriution, re-selling, lon or su-liceng, systemtic supply or distriution in ny form to nyone is expressly foridden. The pulisher does not give ny wrrnty express or implied or mke ny representtion tht the contents will e complete or ccurte or up to dte. The ccurcy of ny instructions, formule nd drug doses should e independently verified with primry sources. The pulisher shll not e lile for ny loss, ctions, clims, proceedings, demnd or costs or dmges whtsoever or howsoever cused rig directly or indirectly in connection with or rig out of the use of this mteril.

2 Journl of Therml Stresses, 31: , 2008 Copyright Tylor & Frncis Group, LLC ISSN: print/ x online DOI: / Downloded By: [Knt, Trun] At: 12:05 8 Jnury 2008 AN EFFICIENT SEMI-ANALYTICAL MODEL FOR COMPOSITE AND SANDWICH PLATES SUBJECTED TO THERMAL LOAD Trun Knt, Sndeep S. Pendhri, nd Yogesh M. Desi Deprtment of Civil Engineering, Indin Institute of Technology Bomy, Powi, Mumi, Indi A simple, semi-nlyticl model with mixed stresses nd displcements fundmentl vriles strting from the exct three dimensionl 3D governing prtil differentil equtions PDEs of lminted composite nd sndwich pltes for thermo-mechnicl stress nlysis hs een presented in this pper. The plte is ssumed simply supported on ll four edges. Two different temperture vritions through the thickness of pltes re considered for numericl investigtion. The ccurcy nd the effectiveness of the proposed model re ssessed y compring numericl results from the present investigtion with the ville elsticity solutions. Some new results for sndwich lmintes re lso presented for future reference. Keywords: Composites; Lmintes; Sndwich; Semi-nlyticl; Therml lod INTRODUCTION Lminted composite nd sndwich pltes re extensively used due to their high specific strength nd high specific stiffness. With the dvncement of the technology of lminted mterils, it is now possile to use these mterils in high temperture situtions. However, composites hve no yield-limit, unlike metls nd hve vriety of filure modes, such s fier filure, mtrix crcking, inter fier filure nd delmintion, which give rise to dmge growing in service. Moreover, composite nd sndwich pltes re sujected to significnt therml stresses due to different therml properties of the djcent lmins nd therefore ccurte predictions of thermlly induced deformtions nd stresses represent mjor concern in design of conventionl structures. Behvior of composite nd sndwich pltes cn e chrcterized y complex 3D stte of stress. In mny instnces, these lminted structurl elements re modertely thick in reltion to their spn dimensions. For thick or modertely thick structurl elements, the norml to the mid surfce is distorted due to inhomogeneity in the trnsverse sher moduli, which is smller thn in-plne Young s moduli, Received 30 April 2007; ccepted 11 August Prtil support of USIF Indo-US Collortive Sponsored Reserch Project IND104 95IU001 is grtefully cknowledged. Suggestions of the reviewers, which hve een incorported in the finl version of the pper, re very much pprecited. Address correspondence to Trun Knt, Deprtment of Civil Engineering, Indin Institute of Technology Bomy, Powi, Mumi , Indi. E-mil: tknt@civil.iit.c.in 77

3 78 T. KANT ET AL. Downloded By: [Knt, Trun] At: 12:05 8 Jnury 2008 resulting in significnt effects of trnsverse sher deformtion nd lso trnsverse norml deformtion. The 3D elsticity nlysis of lmintes with lrge numer of orthotropic/ isotropic lyers ecomes very complex [1 3]. Therefore, reserchers hve their ttention on two dimensionl 2D nlyticl models y introducing some ssumptions concerning the deformtion of the trnsverse normls tht re dependent on the nture of prolem under considertion. Clssicl lmintion plte theory CLPT is sed on the min ssumption tht the lminte is thin. As consequence it is ssumed tht the norml to the lminte mid surfce remins stright, inextensile nd norml during the deformtion. Muletsch [4] seems to hve written the first pper, ville in the literture, on therml stresses in isotropic pltes nd Pell [5] is the first who studied therml deflections of nisotropic thin pltes under ritrry temperture loding. On the other hnd, the first-order sher deformtion theory FOST, sed on Reissner [6] nd Mindlin [7] pproches, considers effects of the trnsverse sher deformtion y ssuming it to e constnt through the thickness of lmintes, hs een used y Reddy nd Cho [8], Weinstein et l. [9], Rolfes et l. [10] nd Argyris nd Tenek [11]. Due to the constnt sher ssumption, FOST is indequte to ccount for ccurte sher distortion nd fictitious sher correction coefficient to correct the sher strin energy is normlly used. Further, severl higher-order sher deformtion theories HOSTs with Tylor series-type expnsion in the thickness direction for the displcements hve een developed for composite nd sndwich pltes under therml loding [12 14]. CLPT, FOST nd HOST re the equivlent gle lyer ESL theories in which slope discontinuity in the inplne displcements nd sher stress continuity t the lmine interfces re not stisfied. To overcome the discrepncy of ESL, discrete lyer theories DLTs nd zig-zg theories hve een developed for thermomechnicl nlysis of composite nd sndwich pltes [15, 16]. The present rticle which strts from 3D equtions nd does not mke ny kinetic or kinemtic ssumptions is minly concerned with the formultion of twopoint oundry vlue prolem BVP governed y set of coupled first-order ODEs, d yz = Azyz pz 1 dz in the intervl h/2 z h/2 with ny hlf of the dependent vriles prescried t the edges z =±h/2 under therml loding. Here, yz is n n-dimensionl vector of fundmentl vriles whose numer n equls the order of PDE, Az is n n coefficient mtrix which is function of mteril properties in the thickness direction nd pz is n n-dimensionl vector of non-homogenous loding terms. It is clerly seen tht mixed nd/or non-homogeneous oundry conditions re esily dmitted in this formultion. THEORETICAL FORMULATION A plte composed of numer of isotropic/orthotropic, liner elstic lmine of uniform thickness with pln dimension nd thickness h is considered Figure 1. The ngle etween the fier direction nd reference xis, x is mesured

4 Downloded By: [Knt, Trun] At: 12:05 8 Jnury 2008 A SEMI-ANALYTICAL MODEL FOR COMPOSITE AND SANDWICH PLATES 79 Figure 1 Lminte geometry with positive set of lmin/lminte reference xes nd fier orienttion. in nticlockwise direction s shown in Figure 1. Simply diphrgm supported end conditions on ll four edges re considered Tle 1. Plte is sujected to only therml lod nd ll surfces re free from ny externl stresses. Further, it is ssumed tht the therml lod is distriuted linerly through the thickness Figure 2. Tx y z = T 0 x y 2z h T 1x y 2 Constitute Reltions Ech lmin in the lminte hs een considered to e in 3D stte of stress so tht the constitutive reltion for typicl ith lmin with reference to the principl mteril coordinte xes 1 2 nd 3 cn e written s, 1 i = 1 E E 2 i 31 2 E 3 t1 T 3

5 80 T. KANT ET AL. Tle 1 Boundry conditions BCs Downloded By: [Knt, Trun] At: 12:05 8 Jnury 2008 BC imposed on displcement field BC imposed on stress field Fce x = 0, v = w = 0 Fce x = /2 u = 0 xz = 0 Fce y = 0, u = w = 0 Fce y = /2 v = 0 yz = 0 Top fce z = h/2 xz = yz = 0 nd z = 0 Bottom fce z = h/2 xz = yz = z = 0 12 i = 12 G 12 2 i = 3 i = i 13 i = 12 E E E G 13 i E 3 t2 T 3 E 2 1 i 2 E 3 t3 T 3 3 i i nd 23 i 23 = in which t1 T, t2 T, nd t3 T re the free therml strins tht rise due to temperture vrition. These cn lso e written s, i = C 11 C 12 C C 22 C C C Sym C 55 0 C 66 G 23 i i 1 t1 T 2 t2 T 3 t3 T where 1, 2 3, 12, 13, 23 re stresses nd 1, 2, 3, 12, 13, 23 re liner strin components with reference to the lmin coordintes 1 2 nd 3. C mn s Figure 2 Through thickness temperture distriution.

6 A SEMI-ANALYTICAL MODEL FOR COMPOSITE AND SANDWICH PLATES 81 Downloded By: [Knt, Trun] At: 12:05 8 Jnury 2008 m n = 16 re elsticity constnts of the ith lmin with reference to the fier xes defined in Appendix A. Stress-strin reltions for the ith lmin in lminte coordintes x y z cn e written s, x y z xy xz yz = Q 11 Q 12 Q 13 Q Q 22 Q 23 Q Q Q Sym Q 55 Q 56 Q 66 x tx T y ty T z tz T xy txy T xz where x, y, z, xy, xz, yz re stresses; x, y, z, xy, xz, yz re strin components nd tx T, ty T, tz T, txy T re free therml strins with respect to lminte xes x y z nd Q mn s m n = 16 re the trnsformed elsticity constnts of the ith lmin with reference to the lminte xes. Elements of mtrix [Q] re defined in Appendix B. Strin-Displcement Reltionship Generl 3D liner strin-displcement reltions cn e written s, yz 5 x = u x xy = u y v x y = v y xz = u z w x z = w z yz = v z w y 6 Equtions of Equilirium The 3D differentil equtions of equilirium re, x x yx y zx z B x = 0 xy x y y zy z B y = 0 7 xz x yz y z z B z = 0 Here, B x, B y nd B z re components of ody force in x, y nd z directions, respectively. Prtil Differentil Equtions Equtions 5 7 hve totl of 15 unknowns, six stresses x y z xy xz yz, 6 strins x y z xy xz yz nd 3 displcements u v w in 15 equtions. After simple lgeric mnipultions, system of PDEs involving

7 82 T. KANT ET AL. only 6 fundmentl vriles u v w xz yz nd z clled primry vriles re otined s follows: Downloded By: [Knt, Trun] At: 12:05 8 Jnury 2008 u z = 1 [ ] w Q65 Q 55 Q 66 Q 56 Q 65 yz Q 66 xz x v z = 1 w z = 1 Q 55 Q 66 Q 56 Q 65 [ u z Q 31 [ Q55 yz Q 56 xz ] w y ] x Q u 34 y Q v 32 y Q v 34 x T Q31 Q tx Q 32 ty tz Q 34 txy 33 xz z = Q 11 Q 13Q 31 Q 44 Q 43Q 34 2 u x 2 2 u y 2 Q 12 Q 44 Q 13Q 32 Q 42 Q 43Q 32 [ Q 11 Q 13Q 31 [ Q 41 Q 43Q 31 tx yz z = Q 41 Q 43Q 31 2 u Q 24 Q 23Q 34 Q 41 Q 14 Q 13Q 34 Q 14 Q 13Q 34 Q 43Q 31 2 v x 2 Q 43Q 34 2 v xy 2 v y Q13 z 2 x Q43 z y B x tx Q 12 Q 13Q 32 ty x 2 2 u y 2 Q 24 Q 42 Q 23Q 34 Q 22 Q 23Q 32 [ Q 21 Q 23Q 31 [ Q 41 Q 43Q 31 Q 42 Q 43Q 32 ty Q 21 Q 44 Q 23Q 31 Q 44 Q 43Q 34 2 u xy Q 14 Q 13Q 34 Q 44 Q 43Q 34 Q 43Q 34 2 v x 2 Q 43Q 32 2 v xy 2 v y Q43 z 2 x Q23 z y B y tx Q 22 Q 23Q 32 ty tx Q 42 Q 43Q 32 2 u xy Q 24 Q 23Q 34 ty Q 44 Q 43Q 34 txy ] T x txy ] T y txy ] T y txy ] T x z z = xz x yz y B z 8

8 A SEMI-ANALYTICAL MODEL FOR COMPOSITE AND SANDWICH PLATES 83 Downloded By: [Knt, Trun] At: 12:05 8 Jnury 2008 Inplne Vrition of Primry Vriles The ove PDEs defined y Eq. 8 cn e reduced to coupled firstorder ODEs y ug doule Fourier trigonometric series for primry vriles stisfying completely the simple diphrgm end conditions t ll 4 edges, x = 0, nd y = 0,, s follows: ux y z = mn vx y z = mn wx y z = mn xz x y z = mn yz x y z = mn z x y z = mn u mn z cos mx v mn z mx w mn z mx xzmn z cos mx yzmn z mx zmn z mx ny ny cos ny ny ny cos ny 9 in the ove oth m, n re Further, temperture vritions long the inplne directions re lso expressed in usoidl form s Tx y z = m n T m n z m x n x 10 in which oth m, n lso ssume integer vlues Liner First-Order Ordinry Differentil Equtions ODEs On sustitution of Eqs. 9 nd 10 in Eq. 8, the following 6 coupled firstorder ODEs corresponding to ech set of modl vlues m nd n re otined. u mn z 0 0 B 13 B u mn z 0 v mn z 0 0 B 23 0 B 25 0 v mn z 0 d w mn z B = 31 B B 36 w mn z p 3 dz xzmn z B 41 B B 46 xzmn z p 4 yzmn z B 51 B B 56 yzmn z p 5 zmn z B 64 B 65 0 zmn z which cn e written in compct form s, d dz yz = B ijzyz pz 11 The elements of mtrices B ij z nd vector pz re given in the Appendix C. p 6

9 84 T. KANT ET AL. Downloded By: [Knt, Trun] At: 12:05 8 Jnury 2008 Tle 2 Trnsformtion of BVP into IVPs Strting edge; z = h/2 Finl edge; z = h/2 Intg. u v w xz yz z u v w xz yz z Lod term Y 11 Y 21 Y 31 Y 41 Y 51 Y 61 Include ssumed ssumed ssumed known known known Y 12 Y 22 Y 32 Y 42 Y 52 Y 62 Delete unity Y 13 Y 23 Y 33 Y 43 Y 53 Y 63 Delete unity Y 14 Y 24 Y 34 Y 44 Y 54 Y 64 Delete unity Finl X 1 X 2 X 3 known known known u T v T w T Include

10 A SEMI-ANALYTICAL MODEL FOR COMPOSITE AND SANDWICH PLATES 85 Downloded By: [Knt, Trun] At: 12:05 8 Jnury 2008 Eqution 11, defines the governing two-point BVP in ODEs through thickness of the lminte in the domin h/2 <z<h/2 with stress components known t the top nd ottom fces. The sic pproch to the numericl integrtion of the BVP defined in Eq. 11 is to trnsform the given BVP into set of IVPs one non-homogeneous nd n/2 homogeneous. The solution of BVP defined y Eq. 11 is then otined y forming liner comintion of one non-homogeneous nd n/2 homogeneous solutions so s to stisfy the oundry conditions t z = h/2 [17]. This gives rise to system of n/2 liner lgeric equtions, the solutions of which determines the unknown n/2 components, X 1, X 2 nd X 3 Tle 2 t the strting edge z = h/2. Then finl numericl integrtion of Eq. 11 produces the desired results. Avilility of efficient, ccurte nd roust ODE numericl integrtors for IVPs helps in computing relile vlues of the primry vriles through the thickness. Chnge in mteril properties re incorported y chnging coefficients of mteril mtrix ppropritely for ech lmin. Secondry Reltions Secondry vriles, x, y nd xy cn e expressed in terms of primry vriles with the help of constitutive nd strin-displcement reltion s, Q13 Q x = 31 m Q Q 11 u mn z mx ny 33 mn Q13 Q 32 n Q Q 12 v mn z mx ny 33 mn Q 14 Q 13Q 34 n u Q mn z cos mx ny cos 33 mn Q 14 Q 13Q 34 m v mn z cos mx cos ny Q13 mn zmn z mx mn tx { Q13 Q 31 Q 11 m n Tz m x n y my Q13 Q 32 Q 12 } Q13 Q ty 34 Q Q 14 xty Q23 Q y = 31 m Q Q 11 u mn z 33 mn Q23 Q 32 n Q Q 22 v mn z 33 mn Q 24 Q 23Q 34 n u Q mn z 33 mn mx mx cos mx ny ny cos ny

11 Downloded By: [Knt, Trun] At: 12:05 8 Jnury T. KANT ET AL. Q 24 Q 23Q 34 Q23 mn v mn z zmn z mx mn tx { Q23 Q 31 Q 21 m n Tz m x n y mn m cos mx cos ny my Q23 Q 32 Q 22 Q43 Q xy = 31 m Q Q 11 u mn z 33 mn Q43 Q 32 n Q Q 42 v mn z 33 mn Q 44 Q 43Q 34 n u Q mn z 33 mn Q 44 Q 43Q 34 m v mn z Q43 zmn z mx mn tx { Q43 Q 31 Q 41 m n Tz m x n y } Q23 Q ty 34 Q Q 24 xty mx my mx cos mx cos mx Q43 Q 32 Q 42 ny ny ny cos cos ny } Q43 Q ty 34 Q Q 44 xty NUMERICAL INVESTIGATION A computer code is developed y incorporting the present formultion in FORTRAN 90 for the nlysis of composite nd sndwich pltes under therml lod. Numericl investigtions on vrious exmples hve een performed including vlidtion of the present semi-nlyticl formultion nd solution of new prolems. The 3D elsticity solution presented y Bhskr et l. [3] nd Rohwer et l. [14] nd other nlyticl solutions ville in the literture hve een used for proper comprison of the otined results. Mteril properties used here hve een tulted in Tle 3. Two therml lod cses re considered here for numericl studies. 1. Equl temperture rise of the ottom nd the top surfce of the plte with usoidl inplne vritions: Tx y ±h/2 = T 0 x y, Cse A. 2. Equl rise nd fll of temperture of the top nd ottom surfce of the plte with usoidl inplne vritions: Tx y h/2 = Tx y h/2 = T 0 x y, Cse B.

12 A SEMI-ANALYTICAL MODEL FOR COMPOSITE AND SANDWICH PLATES 87 Tle 3 Mteril properties Downloded By: [Knt, Trun] At: 12:05 8 Jnury 2008 Set Source Property I Rohwer E 1 = 1500 GP E 2 = 100 GP E 3 = 100 GP et l. [14] 12 = = = 030 G 12 = 50 GP G 13 = 50 GP G 23 = 3378 GP 1 = 0139E-6 k 1 2 = 90E-6 k 1 3 = 90E-6 k 1 II Bhskr E 1 = 1724 GP E 2 = 689 GP E 3 = 689 GP et l. [3] 12 = = = 025 G 12 = 345 GP G 13 = 345 GP G 23 = 1378 GP 1 = 10k 1 2 = 11250k 1 3 = 11250k 1 Fce Sheet III Khre E 1 = 1724 GP E 2 = 689 GP E 3 = 689GP et l. [18] 12 = = = 025 G 12 = 345 GP G 13 = 345 GP G 23 = 1378 GP 1 = 01E-5 k 1 2 = 20E-5 k 1 3 = 01E-5 k 1 Core Sheet E 1 = 0276 GP E 2 = 0276 GP E 3 = 3450 GP 12 = = = 025 G 12 = GP G 13 = 0414 GP G 23 = 0414 GP 1 = 01E-6 k 1 2 = 02E-5 k 1 3 = 01E-6 k 1 Following normliztions hve een used in ll exmples considered here for the comprison of the results. s = h u v = 1 h 1 T 0 s u v w = h3 w 3 1 T 0 4 z = z E 2 1 T x y xy = x E 2 1 T 0 s 2 y xy xz yz = xz E 2 1 T 0 s yz 15 in which r over the vrile defines its normlized vlue. A convergence study on numer of steps required for numericl integrtion in the thickness direction of the lminte is performed first for ll exmples. It is oserved in ll exmples tht steps re enough for converged solution. Detils of the convergence studies re not presented here for the ske of revity. Illustrtive exmples considered in the present work re discussed next. Exmple 1 A homogeneous, orthotropic plte with simple support end conditions Tle 1 on ll four edges nd sujected to therml lod hs een considered to study the effect of the temperture distriution nd vlidte the present methodology. Mteril properties re presented in Tle 3I. The normlized mximum stresses x y xy xz yz nd trnsverse displcement w for vrious spect rtios rnging from thick to thin plte re presented in Tle 4 for oth type of therml lods. Moreover, through thickness vritions of trnsverse sher stress xz, trnsverse norml stress z, in-plne norml stress x nd trnsverse displcement w for n spect rtio of 5 re shown in Figures 3 nd 4 for

13 88 T. KANT ET AL. Downloded By: [Knt, Trun] At: 12:05 8 Jnury 2008 Tle 4 Mximum stresses x y xy xz nd yz nd the trnsverse displcement w of squre homogeneous orthotropic pltes under therml lod s Source x 2 2 ± h 2 y 2 2 ± h 6 Cse A: Tx y ±h/2 = T 0 x 10 xy 0 0 ± h 2 y 10 xz 0 2 ±03h 10 2 yz 2 0 ±03h 10 2 w 2 2 ± h 2 4 Present nlysis ± ± ± Present nlysis ± ± ± Present nlysis ± ± ± s Source x 2 2 ± h 2 Cse B: Tx y h/2 = Tx y h/2 = T 0 x y 2 2 ± h 2 10 xy 0 0 ± h 2 y xz 0 2 ±03h yz 2 0 ±03h 10 2 w 2 2 ± h 2 4 Present nlysis ± Present nlysis ± Present nlysis ±

14 Downloded By: [Knt, Trun] At: 12:05 8 Jnury 2008 A SEMI-ANALYTICAL MODEL FOR COMPOSITE AND SANDWICH PLATES 89 Figure 3 Vrition of normlized trnsverse sher stress xz trnsverse norml stress z c inplne norml stress x d trnsverse displcement w through thickness of homogeneous orthotropic plte sujected to therml lod, Tx y ±h/2 = T 0 x y Cse A. cse A nd cse B, respectively. 3D elsticity nd HOST solutions given y Rohwer et l. [14] re lso plotted on sme trce for comprison of the present solution. This comprison clerly indictes tht the present results re very close to the elsticity solutions compred to HOST nd thus proves the superiority of the present model. Lrge vlue of xz s compred to yz Tle 4 is due to higher modulus vlues of G 13 nd E 1 s compred to G 23 nd E 2. Trnsverse norml stress z shows compression t the plte center Figure 3 nd roughly cuic distriution of trnsverse sher stress xz through the thickness of plte Figure 3 is oserved for constnt temperture Cse A. Moreover, in cse B, the trnsverse norml stress z is found to e too smll s compred to cse A with compressive vlue in the

15 Downloded By: [Knt, Trun] At: 12:05 8 Jnury T. KANT ET AL. Figure 4 Vrition of normlized trnsverse sher stress xz trnsverse norml stress z c inplne norml stress x d trnsverse displcement w through thickness of homogeneous orthotropic plte sujected to therml lod, Tx y h/2 = Tx y h/2 = T 0 x y Cse B. upper hlf nd tensile vlue in the lower hlf of plte. And trnsverse sher stresses xz nd yz re found to e nerly sme with opposite signs. Exmple 2 Vrious three-lyered, symmetric, cross-ply 0 /90 /0, squre lmintes with spect rtios, s = 4, 10 nd 20 nd simple support end conditions on ll four edges Tle 1 sujected to constnt Cse A nd vried Cse B temperture distriution through thickness nd usoidl vritions long the inplne directions re considered here to show the ility of the present model to hndle lyered structure. Mteril properties re presented in Tle 3II. Results for spect rtios, s = 4, 10 nd 20 hve een compred in Tle 5 with elsticity solutions given y

16 A SEMI-ANALYTICAL MODEL FOR COMPOSITE AND SANDWICH PLATES 91 Downloded By: [Knt, Trun] At: 12:05 8 Jnury 2008 Tle 5 Mximum stresses x y xy xz nd yz nd the trnsverse displcement w of three lyered symmetric 0 /90 /0 squre composite pltes under therml lod s Source x 2 2 ± h 2 Cse A: Tx y ±h/2 = T 0 x y 2 2 ± h 6 xy 0 0 ± h 2 y xz 0 2 ±04h yz 2 0 ±04h 10w 2 2 ± h 2 4 Present nlysis ± ± Present nlysis ± ± Present nlysis ± ± Cse B: Tx y h/2 = Tx y h/2 = T 0 x y s Source x 2 2 ± h 2 y 2 2 ± h 2 xy 0 0 ± h 2 xz 0 2 ±04h yz 2 0 ±04h w 2 2 ± h 2 4 Present nlysis ± Exct solution ± Present nlysis ± Exct solution ± Present nlysis ± Exct solution ± Bhskr et l. [3].

17 92 T. KANT ET AL. Downloded By: [Knt, Trun] At: 12:05 8 Jnury 2008 Bhskr et l. [3]. Present results re seen to e closest to the elsticity solutions. Through thickness vritions of trnsverse sher stress xz, trnsverse norml stress z, inplne norml stress x nd inplne displcement u for n spect rtio of 5 hve een presented in Figures 5 nd 6 for cse A nd cse B, respectively. Solutions re only ville for vried temperture Cse B nd solutions with constnt temperture Cse A will e useful s enchmrk solution in future. Vrition of trnsverse sher stress xz for constnt temperture Cse A is found to e smooth curved profile in the top nd ottom lmin 0 ut lmost liner profile is oserved in the middle lmin 90 with zero vlue t the mid-surfce Figure 5, wheres for vried temperture Cse B, xz vries smoothly in curved fshion through the thickness Figure 6. Interesting distriution of trnsverse Figure 5 Vrition of normlized trnsverse sher stress xz trnsverse norml stress z c inplne norml stress x d inplne displcement u through thickness of 0 /90 /0 symmetric composite plte sujected to therml lod, Tx y ±h/2 = T 0 x Cse A. y

18 Downloded By: [Knt, Trun] At: 12:05 8 Jnury 2008 A SEMI-ANALYTICAL MODEL FOR COMPOSITE AND SANDWICH PLATES 93 Figure 6 Vrition of normlized trnsverse sher stress xz trnsverse norml stress z c inplne norml stress x d trnsverse displcement w through thickness of 0 /90 /0 symmetric composite sujected to therml lod, Tx y h/2 = Tx y h/2 = T 0 x y Cse B. norml stress z is oserved for this configurtion in oth types of lodings. In cse of constnt temperture Cse A, z in top nd ottom lmin 0 shows compression wheres, z in middle lmin 90 shows tension t the plte center Figure 5 nd in cse of vried temperture Cse B, z in top lmin 0 is compressive, z in ottom lmin 0 is tensile nd z in middle lmin 90 hs mixed ehvior of compression nd tension elow nd ove the mid-surfce Figure 6 which proves the necessity of refined model to model the ccurtely such highly non-liner ehviour. All vritions re oserved to e symmetric out the mid-surfce s expected.

19 94 T. KANT ET AL. Downloded By: [Knt, Trun] At: 12:05 8 Jnury 2008 Tle 6 Mximum stresses x y xy xz nd yz nd the trnsverse displcement w of four lyered unsymmetric 0 /90 /0 /90 squre composite pltes under therml lod s Source x 2 2 ± h 2 Cse A: Tx y ±h/2 = T 0 x y 2 2 ± h 6 10 xy 0 0 ± h 2 y xz 0 2 ±025h yz 2 0 ±025h 10 2 w 2 2 ± h 2 4 Present nlysis ± Present nlysis ± Present nlysis ± s Source x 2 2 ± h 2 Cse B: Tx y h/2 = Tx y h/2 = T 0 x y 2 2 ± h 2 10 xy 0 0 ± h 2 y xz 0 2 ±02h yz 2 0 ±02h 10 2 w 2 2 ± h 2 4 Present nlysis Present nlysis Present nlysis

20 A SEMI-ANALYTICAL MODEL FOR COMPOSITE AND SANDWICH PLATES 95 Downloded By: [Knt, Trun] At: 12:05 8 Jnury 2008 Exmple 3 A 4-lyered, unsymmetric, cross-ply 0 /90 /0 /90, squre composite plte with equl thickness under the therml lod is considered in this exmple with simple support end conditions Tle 1. Mteril properties re presented in Tle 3I. Results of the mximum normlized stresses x y xy xz yz nd trnsverse displcement w re presented in Tle 6 for vrious spect rtios nd through thickness vritions of trnsverse sher stress xz, trnsverse norml stress z, inplne norml stress x nd trnsverse displcement w re depicted in Figure 7 Vrition of normlized trnsverse sher stress xz trnsverse norml stress z c inplne norml stress x d trnsverse displcement w through thickness of 0 /90 /0 /90 unsymmetric composite plte sujected to therml lod, Tx y ±h/2 = T 0 x Cse A. y

21 96 T. KANT ET AL. Downloded By: [Knt, Trun] At: 12:05 8 Jnury 2008 Figures 7 nd 8 for n spect rtio of 5 for cse A nd cse B, respectively. 3D elsticity solution given y Rohwer et l. [14] is used for comprison of the results otined through present investigtions. Excellent greements of present results with elsticity solutions suggest tht the formultion is cple to hndle such unsymmetric lminte configurtions. It is lso seen tht trnsverse sher stress xz hs symmetry out mid plne with sher stress yz Tle 6. Zig-zg vrition of trnsverse sher stress xz through the thickness of plte is oserved Figure 7 nd 8 nd this is due to the rupt chnge in stiffness etween 0 nd 90 lyers Figure 8 Vrition of normlized trnsverse sher stress xz trnsverse norml stress z c inplne norml stress x d trnsverse displcement w through thickness of 0 /90 /0 /90 unsymmetric composite sujected to therml lod, Tx y h/2 = Tx y h/2 = T 0 x y Cse B.

22 A SEMI-ANALYTICAL MODEL FOR COMPOSITE AND SANDWICH PLATES 97 Downloded By: [Knt, Trun] At: 12:05 8 Jnury 2008 Tle 7 Mximum stresses x y xy xz nd yz nd the trnsverse displcement w of symmetric three lyered 0 /core/0 squre sndwich pltes under therml lod s Source x 2 2 ± h 2 y 2 2 ± h 6 Cse A: Tx y ±h/2 = T 0 x 10 xy 0 0 ± h 2 y 10 2 xz 0 2 ±04h 10 2 yz 2 0 ±04h 10 3 w 2 2 ± h 2 4 Present nlysis ± Present nlysis ± Present nlysis ± s Source x 2 2 ± h 2 Cse B: Tx y h/2 = Tx y h/2 = T 0 x y 2 2 ± h 2 10 xy 0 0 ± h 2 y xz 0 2 ±04h yz 2 0 ±04h 10 2 w 2 2 ± h 2 4 Present nlysis ± Present nlysis ± Present nlysis ±

23 98 T. KANT ET AL. for oth cses A nd B. Vrition of trnsverse norml stress z is seen to e ntisymmetric out mid plne Figure 8. Downloded By: [Knt, Trun] At: 12:05 8 Jnury 2008 Exmple 4 A symmetric squre sndwich plte 0 /core/0 with simple support end conditions Tle 1 on ll four edges nd sujected to therml lod hs een considered here. Exct solution of this exmple is not ville in the literture. Mteril properties re presented in Tle 3III. Thickness of ech fce sheets is one tenth of the totl thickness of the plte. The normlized mximum stresses Figure 9 Vrition of normlized trnsverse sher stress xz trnsverse norml stress z c inplne norml stress x d inplne displcement u through thickness of 0 /core/0 symmetric sndwich plte sujected to therml lod, Tx y ±h/2 = T 0 x Cse A. y

24 Downloded By: [Knt, Trun] At: 12:05 8 Jnury 2008 A SEMI-ANALYTICAL MODEL FOR COMPOSITE AND SANDWICH PLATES 99 Figure 10 Vrition of normlized trnsverse sher stress xz trnsverse norml stress z c inplne norml stress x d trnsverse displcement w through thickness of 0 /core/0 symmetric sndwich sujected to therml lod, Tx y h/2 = Tx y h/2 = T 0 x y Cse B. x y xy xz yz nd trnsverse displcement w for vrious spect rtios, 4, 10 nd 20 re presented in Tle 7. Figures 9 nd 10 show the through thickness vritions of trnsverse sher stress xz, trnsverse norml stress z, inplne norml stress x nd trnsverse displcement w for n spect rtio of 4 for cse A nd cse B, respectively. Theses results should serve s enchmrk solutions in future. GENERAL DISCUSSION The defined vritions of tempertures through the thickness of plte re considered here so tht present solutions cn e compred with the ville

25 100 T. KANT ET AL. Downloded By: [Knt, Trun] At: 12:05 8 Jnury D elsticity results. However, the technique is cple to hndle ny kind of temperture vritions. Further, the present model mintins the continuity of trnsverse stresses nd displcements t the lmine interfces without involving ny complexity in the formultion nd solution technique. It is oserved in ll exmples considered in the present study tht vrition in trnsverse displcement w long the thickness is very smll for n spect rtio equl/greter thn 10 thin plte. However, for thick pltes with spect rtios less thn 5, w vries significntly Figures 4d, 8d nd 10d. The vrition of trnsverse norml stress z here is quite different from wht is oserved in the cse of mechnicl loding. CONCLUDING REMARKS An efficient, simple semi-nlyticl model sed on solution of two-point BVP governed y set of coupled first-order ODEs through the thickness of plte is proposed in this rticle for thermo-mechnicl stress nlysis. The sher trction free conditions t the top nd ottom of plte nd continuity of trnsverse stresses nd displcement t the lyer interfces re exctly stisfied which is one of the importnt fetures of the developed model. Moreover, the solution lso ensures the fundmentl elsticity reltionship etween stress, strin nd displcement fields within the elstic continuum. It is shown through numericl investigtions tht results otined y present pproch re highly ccurte. Another importnt feture of this pproch is tht oth displcements nd stresses re computed simultneously with the sme degree of ccurcy. APPENDIX A Coefficients of [C] Mtrix C 11 = E C 22 = E C 44 = G 12 C 55 = G 13 C 66 = G 23 C 12 = E C 23 = E C 13 = E C 33 = E where = APPENDIX B Coefficients of [Q] Mtrix Q 11 = C 11 c 4 2C 12 2C 44 c 2 s 2 C 22 s 4 Q 12 = C 12 c 4 s 4 C 11 C 22 4C 44 c 2 s 2 Q 13 = C 13 c 2 C 23 s 2

26 A SEMI-ANALYTICAL MODEL FOR COMPOSITE AND SANDWICH PLATES 101 Downloded By: [Knt, Trun] At: 12:05 8 Jnury 2008 Q 14 = C 11 C 12 2C 44 c 3 s C 12 C 22 2C 44 cs 3 Q 22 = C 22 c 4 2C 12 2C 44 c 2 s 2 C 11 s 4 Q 23 = C 23 c 2 C 13 s 2 Q 24 = C 12 C 22 2C 44 c 3 s C 11 C 12 2C 44 cs 3 = C 33 Q 34 = C 31 C 32 cs Q 44 = C 11 2C 12 C 22 2C 44 c 2 s 2 C 44 c 4 s 4 Q 55 = C 55 c 2 C 66 s 2 Q 56 = C 55 C 66 cs Q 66 = C 55 s 2 C 66 c 2 APPENDIX C Coefficients of [B] Mtrix B 13 = m B 13 = n B 31 = Q 31 m B 41 = B 42 = B 51 = B 52 = B 64 = m Q 66 B 14 = Q 55 Q 66 Q 56 Q 65 Q 55 B 15 = Q 55 Q 66 Q 56 Q 65 Q 11 Q 13Q 31 [ Q 12 [ Q 21 B 32 = Q 32 n m B 36 = 1 Q 44 Q 43Q 34 n 2 2 ] Q13 Q 32 Q43 Q 34 mn 2 Q Q 44 B = ] Q31 Q 23 Q43 Q 34 mn 2 Q Q n 2 2 Q 2 44 Q 43Q 34 m 2 2 B 2 56 = Q 22 Q 23Q 32 B 65 = n Coefficients of p Vector 2 Q13 Q23 m n p 3 = 1 Q31 tx Q 32 ty tz Q 34 txy Tz

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