The Interlaminar Stress of Laminated Composite under Uniform Axial Deformation

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1 Moling an umrical Simulation of Matrial Scinc, 23, 3, Publish Onlin April 23 ( 49 Th Intrlaminar Strss of Laminat Composit unr Uniform Axial Dformation Chuin Yang, ubing Chn 2, Shxu Zhao 2 Bing Institut of Mchanical Equipmnt, Bing, China 2 Shanghai iao Tong Univrsit, Shanghai, China jbchn@sjtu.u.cn Rciv ovmbr 27, 22; rvis Dcmbr 28, 22; accpt anuar, 23 Copright 23 Chuin Yang t al. This is an opn accss articl istribut unr th Crativ Commons Attribution Licns, which prmits unrstrict us, istribution, an rprouction in an mium, provi th original wor is proprl cit. ABSTRACT Th intrlaminar strsss ar anal b combining th first shar thor with th larwis thor mtho. An th plat subjct to a uniform axial strain is stui b th simplifi isplacmnt fil. Using th simplifi isplacmnt fil, th quations of finit lmnt mtho ar vlop b th principl of virtual wor. An th amount of calculation is ruc b using th linar lmnt. Thn, som numrical xampls ar givn to vrif th accurac of th mtho an anal th istribution of intrlaminar strsss along -axis an -axis. Th shaps of th strsss curvs in th vicinit of th fr g ar vr iffrnt from th intrior ara. Morovr, th influnc of th pl angl on th intrlaminar strsss is anal for th plat [θ/ θ] s. It can b foun that th shaps of th strsss along -axis ar similar whn th angl is iffrnt, whil th valus of th intrlaminar strsss ar chang apparntl with th pl angl. Kwors: Intrlaminar Strsss; Finit Elmnt Mtho; Laminat Composit Plat. Introuction Th composit matrials ar wil us in th aviation, spac inustris an mchanical nginring bcaus of thir goo comman of mchanical proprt. Th laminat composit plat is ma up of multilar lamination. Thir intrlaminar strsss can significantl contribut to lamination vn whn th ar much lowr than th failur strngth of th classical lamination thor. Th ma ma th potntial of laminat composit plat can not b wor out for its carring capacit triorat bcaus of th lamination. In th vicinit of th fr g, th intrlaminar strsss ar vari fast, which is th main rason of lamination. An th lamination of th laminat composit plats is th most common struction form of laminat composit plats. Thrfor, th rsarch of intrlaminar strsss is of grat significanc to practical applications. Man rsarchrs hav on a lot of wor about it. Th strsss in th vicinit of fr g ar xprss as a two-imnsional stat b th classical lamination thor [,2]. Aftrwar, it was prov to b a thr-imnsional stat b man rsarchrs [3-5]. In rcnt ars, man mor popl hav start to rsarch th intrlaminar strsss. Hiroui Matsunaga anal th strsss an isplac- mnts in th laminat composit bams subjct to latral prssurs b using th mtho of powr sris xpansion of isplacmnt componnts [6]. Asghar osir an Arash Bahrami stui th intrlaminar strsss in antismmtric angl-pl laminats b vloping a ruc form of isplacmnt fil for long antismmtric angl-pl composit laminats subjct to xtnsional an/or torsional loas [7]. Thofamis S. Plagianaos an Dimitris A. Saravanos propos a highr-orr Larwis thortical framwor to calculat th static rspons of thic composit an sanwich composit plat [8]. Th isplacmnt fil th assum in ach iscrt lar inclu quaratic an cubic polnomial istributions of in-plan isplacmnts. Furthrmor, a Rit-tp xact solution [8] was implmnt to il th structural rspons of thic composit an sanwich composit plats. Hung Soo Kim t al. vlop a strss functionbas variational mtho to invstigat th intrlaminar strsss nar th ropp plis [9]. M. Amabili an.. R vlop a consistnt highr-orr shar formation non-linar thor for shlls of gnric shap []. Using th vlop thor, a simpl support, laminat circular clinrical shlls subjct a larg amplitu forc vibrations ar stui. Amir K. Miri an Asghar osir invstigat fr-g ffcts in antism- Copright 23 SciRs. MSMS

2 5 C.. YAG ET AL. mtric angl-pl laminat shll panls unr uniform axial xtnsion b using larwis thor []. An th problm was analticall solv for spcific bounar conitions along th gs. Rn Xiaohui t al vlop a highr-orr ig-ag thor for laminat composit an sanwich plats [2]. Th propos thor can prict mor accurat in-plan isplacmnts an strsss in comparison with othr ig-ag thoris.. L. Mantari t al. vlop a nw shar formation thor for sanwich an composit plats [3]. Th propos isplacmnts fil was assss b prforming svral computations of th plats govrning quations an th rsults wr rlativl clos to 3D lasticit bning solutions. Th first shar thor is combin with th Larwis thor (LWT) [4] to anal th intrlaminar strsss of th laminat composit plats in this papr. Th first shar thor assum th plat as an quit-singl lar as to buil th isplacmnt fil whos componnt is C continuit. Th Larwis thor buils th isplacmnt fil b isprsing th plat to man numrical lars. In this papr, th intrlaminar strsss ar anal b suprimpos th first shar thor on th Larwis thor. Thn, th isplacmnt fil is simplifi for th smmtric pl composit plat which subjct to a uniform axial strain. Th finit lmnt quation is riv b th principl of virtual wor. Thn, th linar lmnt is us to solv th problm. Of cours, it ruc th amount of calculation whil th accurac is nsur. At last, th rsults of th intrlaminar strsss ar givn for iffrnt pl conitions of laminat composit plats. 2. Th Displacmnt Fil Suprimposing th first shar thor on th Larwis thor, th isplacmnt fil can b xprss as:,,, x,, u x u x x U x,,,,, v x v x x V x,,,, w x w x W x whr, is th numbr of numrical lars through th thicnss. is th global intrpolation function which is linar or quaratic Lagrang intrpolation function in gnral. An th iscrtiation of th isplacmnts is fulfill b it (s Figur ). Consiring th plat which is smmtric pli an subjct to a uniform axial strain (s Figur 2), th isplacmnt fil can b simplifi as:,, x u x x U (),,,, v x v V w x V whr, is th uniform strain along x-axis an th isplacmnt v is inpnnt of variabl x. Thrfor, this situation can b solv as th problm of plan strain. In this papr, th linar Lagrang intrpolation function [4] is us as th intrpolation function, an it can b xprss as blow [4]: 2 (3) whr,, an h is th thicnss h 2, h of th th lar. Th isplacmnt fil aopt hr is satisfi th isplacmnt continuit conition an th shar strsss continuit conition btwn lars. 3. Th Finit Elmnt Equation Consiring th laminat composit plat which is smmtric pli an subjct to a uniform axial strain along x-axis. Substitut Formula (2) into th principl of virtual wor as blow: V V v f u i i v t u i i s v v (2) (4) Thn th finit lmnt quation can b riv an wrot simpl as: K F (5) whr, K is th lmnt stiffnss matrix, rprsnts th isplacmnt vctor of lmnt no an F rprsnts th noal loa vctor. Th finit xprssion of th finit lmnt quation can b sn in th appnix at th + th lar + U + U - U - x U 4 4 U 3 3 U 2 2 u U U + U + - U U - U + U Φ () Figur. Th iscrtiation of isplacmnts through th thicnss. Copright 23 SciRs. MSMS

3 C.. YAG ET AL. 5 2a ε h h th lar th lar h h 2n lar st lar + + 2h x ε θ Fr g b Intrfac b Fr g Figur 2. Gomtrical mol. n of this papr. It shoul b notic hr that th lmnt stiffnss matrix in Formula (5) is unsmmtric, ξ which is iffrnt from th gnral finit lmnt quation. Hr th Lagrang lmnt with 3 nos (s Fig- 2 3 ur 3) is appli to solv th problm as is us Figur 3. Th Lagrang lmnt with 3 nos. to intrpolat th isplacmnts along -axis, which simplif th ucing procss of finit lmnt quation. mol is stui hr as th laminat composit plat is Using th natur coorinats, th intrpolation shap assum to b smmtrical about th x- plan an -axis. function can b xprss as blow: As to compar th rsults of this papr with th rsults of othr rsarchrs, th strsss ar normali as blow: 2 (8) 2 2 (6) Cross Pl Laminats: [/9]s an [9/]s Th Strsss along -Axis whr,. Th istributions of strsss (σ, σ x, σ ) along -axis at Th rsults at th Gauss points hav th highst accurac bcaus of th Gauss intgral is us in th calcula- = an = h/2 of th cross-pl laminat composit plat ([/9] s ) ar shown in Figurs 4 an 5. Th rsults tion procss. Thrfor, th strsss at th Gauss points ar in accoranc with th quasi-3d lmnt mtho [5]. ar us to plot th curvs in this papr. It shoul b notic that th rsults of th nos which connct two if- As shown in th Figurs, At th intrfac = or = h/2, σ grows fast nar th fr g. Manwhil, σ x is vr frnt lmnts ar not qual in th two lmnts. clos to ro which is sam as th thortical rsult. An th strss σ approachs to ro grauall in th vicinit 4. umrical Rsults of th g, which is accoranc with th bounar conition that σ is ro at th g. A smmtric pli composit plat subjct to a uniform axial strain is stui hr. An its lngth, Th istributions of strsss (σ, σ x, σ ) along -axis at with an hight ar 2a, 2b an 2h rspctivl (s = an = h/2 of th cross-pl laminat composit Figur 2). At th sam tim, it is assum that ach matrial lar is orthotropic an qual thicnss. An th sn that σ an σ grow fast in th vicinit of th g, plat ([9/] s ) ar shown in Figurs 6 an 7. It can b lasticit moulus, shar moulus an Poisson s ratio ar whil th valu of σ x is ro at th g. as follows: It can b obsrv that th failur of matrial is occurr asil in th vicinit of th fr g of th cross- 6 E 38 GPa 2 psi pl laminat composit plat. 6 E2 E3 4.5 GPa 2. psi (7) 6 G2 G3 G GPa.85 psi Th Strsss along -Axis Th istributions of strsss (σ v2 v3 v23.2, σ x, σ ) along -axis of th cross-pl laminat composit plat ([9/] s ) ar whr th subscripts, 2, 3 rprsnt th thr principal shown in Figurs 8-. It can b obsrv that th valus axis of th matrial, an psi = Pa. Th /8 of th strsss ar ro on th surfac. Th intrlaminar Copright 23 SciRs. MSMS

4 52 C.. YAG ET AL. Figur 4. Th strsss along -axis, =, ([/9]s). Figur 7. Th strsss along -axis, = h/2, ([9/]s). Figur 5. Th strsss along -axis, = h/2, ([/9]s). Figur 8. Th strsss along -axis, =.25b. Figur 6. Th strsss along -axis, =, ([9/]s). pa nar th intrfac =.7h an it changs sunl at =.5h, whil its valu is clos to ro at =. At th cross sction =.99b, strss σ changs from tn- strsss at th cross sction =.25b ar almost sam as =.5b. Along -axis, th strss σ grows slowl an rachs th maximum at =. Th strss σ x is clos to ro an harl chang. Bsis, th strss σ rachs a Figur 9. Th strsss along -axis, =.5b. Copright 23 SciRs. MSMS

5 C.. YAG ET AL. 53 sion to prssur sunl at th intrfac =.5h. Li σ, strss σ changs from prssur to tnsion sunl at =.5h whil its valu is clos to ro whn =. Bsis, it can b obsrv from th thr figurs that th strsss ar much highr whn =.99b than th othrs. It mans that th strsss ar much highr nar th fr g than th intrior ara. An th struction is asir to b happn in th vicinit of th fr g Angl Pl Laminats: [45/-45]s Th Intrlaminar Strsss along -Axis Th rsults of th intrlaminar strsss of th laminat composit plat ([45/-45] s ) ar shown in Figurs an 2. It can b obsrv that σ grows vr fast in th vicinit of th g whil a minimum is happn at th g. Strss σ rachs a pa in th vicinit of th g follow a sharp cras. It is iffrnt from th lastic solution propos b Wang an Choi which shows that strss σ is vanish at th g. Strss σ x grows as a smooth c urv along -axis an rachs a maximum at th g Th Intrlaminar Strsss along -Axis Th istributions of strsss (σ, σ x, σ ) along -axis of th cross-pl laminat composit plat ([45/-45] s ) ar shown in Figurs 3-5. It can b obsrv that th val us of thr strsss ar ro on th surfac. An at th cross sctions =.25b an =.5b, th istributions of intrlaminar strsss ar similar. Both th thr strsss ta a sharp turn at th intrfac of th cross sction =.99b. Strss σ changs from tnsion to prssur whil strsss σ an σ x rach a sharp pa at =.5h. It also can b sn that th valu of th strsss nar th fr g is much highr than th strss at th intrior ara Angl Pl Laminats: [45/-45//9/9//-45/45]s Th Strsss along -Axis Th istributions of strsss (σ, σx, σ ) along -axis of Figur. Th strsss along -axis, =.99b. Figur 2. Th strsss along -axis, = h/2. Figur. Th strsss along -axis, =. Figur 3. Th strsss along -axis, =.25b. Copright 23 SciRs. MSMS

6 54 C.. YAG ET AL. th angl pl laminats ([45/-45//9/9//-45/45] s) ar shown in Figurs It can b obsrv that th curvs of th strsss chang sharpl in th vicinit of Figur 7. Th strsss along -axis, = h/8. Figur 4. Th strsss along -axis, =.5b. Figur 8. Th strsss along -axis, = h/4. Figur 5. Th strsss along -axis, =.99b. Figur 9. Th strsss along -axis, = 3h/8. Figur 6. Th strsss along -axis, =. th g. Strss σ rachs a pa nar th g thn to ro whil strss σ an σ x both rach a maximum or minimum at th g. Copright 23 SciRs. MSMS

7 C.. YAG ET AL. 55 Figur 2. Th strsss along -axis, = h/2. Figur 23. Th strsss along -axis, = 7h/8. Figur 2. Th strsss along -axis, = 5h/8. Figur 24. Th strsss along -axis, =.25b. Figur 22. Th strsss along -axis, = 3h/ Th Strsss along -Axis Th istributions of strsss (σ, σ x, σ ) along -axis of th angl pl laminats ([45/-45//9/9//-45/45] s ) ar shown in Figur Th intrlaminar strsss ar Figur 25. Th strsss along -axis, =.5b. ro on th surfac. An th curvs at cross sctions =.25b an =.5b ar vr similar. That strss σ rachs a pa nar th intrfac =.5h. Th irction of Copright 23 SciRs. MSMS

8 56 C.. YAG ET AL. strss σ x is opposit btwn th uppr an lowr half. Furthrmor, strss σ x tas turns at th intrfacs =.25h, =.25h, =.75h an =.875h. Similar to strss σ x, strss σ ta turns at th intrfacs =.25h, =.375h, =.625h an =.75h. An its irction is opposit btwn th uppr an lowr half, whil it is clos to ro at th intrfacs = an =.5h. At th cross sction =.99b, strss σ ta turns at th intrfacs =.25h, =.375h, =.625h an =.875h. It is a tnsion strss btwn intrfacs =.25h an =.75h whil it is a comprssion in th othrs. Th curv of strss σ x is antismmtric to th intrfac =.5h. Morovr, it tas turns at th intrfacs =.25h, =.25h, =.75h an =.875h. Similar to strss σ x, th curv of strss σ is antismmtric to =.5h an it tas turns at th intrfacs xcpt =.5h Th Influnc of th Pl Angl on th Intrlaminar Strsss Th laminat composit plat ([θ/ θ] s ) is consir to anal th influnc of th pl angl on th intrlaminar strsss. Strss σ along -axis at th cross sction =.99b is shown in Figur 27 whn th angl is iffrnt (θ = 5, 5, 3, 45, 6, 75, 85 ). It can b obsrv that strss σ is chang with th pl angl apparntl whil th shaps of th curvs in Figur 27 ar similar. At =.5h of cross sction =.99b, th rlationship btwn th absolut valu of strss σ an th pl angl is shown in Figur 28. That whn th pl angl rang btwn an 3, th absolut valu of strss σ is incras. Morovr, it rachs a maximum as th an gl is qual to 3. Thn whn th pl angl rang btwn 3 an 6, it is cras with th angl. Bsis, it closs to ro whn th angl rang btwn 6 an 9. Strss σ x along -axis at th cross sction =.99b is shown in Figur 29 whn th pl angl is iffrnt Figur 26. Th strsss along -axis, =.99b. Figur 27. Influnc of θ to strss σ along -axis, =.99b. Figur 28. Strss σ changs with θ, =.99b, = h/2. (θ = 5, 5, 3, 45, 6, 75, 85 ). Li strss σ, strss σ x is chang with th pl angl apparntl whil th shaps of th curvs in Figur 29 ar similar. An at =.5h of cross sction =.99b, th rlationship btwn th absolut valu of strss σ x an th pl angl is shown in Figur 3. Th maximum is happn whn th angl is about 2, which is iffrnt from Figur 28. Strss σ along -axis at th cross sction =.99b is shown in Figur 3 whn th pl angl is iffrnt (θ = 5, 5, 3, 45, 6, 75, 85 ). Li th othr two strsss, strss σ is chang apparntl with th pl angl whil th shaps of th curvs in Figur 3 ar similar. An at =.5h of cross sction =.99b, th rlationship btwn th absolut valu an strss σ with th pl angl is shown in Figur 32 which is similar to Figur Rsults an Discussion Th intrlaminar strsss ar not uniform along -axis of th laminat composit plat subjct to a uniform axial strain. But it changs sharpl in th vicinit of th Copright 23 SciRs. MSMS

9 C.. YAG ET AL. 57 Figur 29. Influnc of θ to strss σ x along -axis, =.99b. Figur 3. Strss σ x changs with θ, =.99b, = h/2. Figur 3. Influnc of θ to strss σ x along -axis, =.99b. g an rachs a minimum or maximum at th fr g which mas th lamination phnomnon oc- Figur 32. Strss σ x changs with θ, =.99b, = h/2. curr to fail th matrial. In this papr, th intrlaminar strsss ar calculat for laminat composit plat with iffrnt pl mannrs. Firstl, th istributions of th intrlaminar strsss along -axis ar stui. Strsss σ an σ x chang sharpl in th vicinit of th fr g, such as incras or cras sharpl or a pa happn nar th fr g. Morovr, it rachs a maximum (or minimum) at th g. Strss σ an σ x ar vr small in th intrior ara in comparison with strsss in th vicinit of th fr g. Also strss σ is vr small at th ara far from th fr g, but its valu is clos to ro aftr a pa happn nar th g, which is iffrnt from strsss σ an σ x. In a wor, th valus of th intrlaminar strsss nar th fr g ar much highr in th intrior ara. So th failur of matrial asir happn in th vicinit of fr g than in th intrior ara. Sconl, th istributions of th intrlaminar strsss alon g -axis ar stui. Th intrlaminar strsss ar clos to ro on th surfac. An th strss σ is vr small at th intrfac =. At th cross sction far awa from th fr g, strss σ osn t chang vr much an strsss σ x an σ both ta a turn at th intrfac. In contrar, th intrlaminar strsss chang vr sharpl at th vicinit of th fr g. Strsss σ an σ x both rach a pa at th intrfac whil th irction of strss σ changs to opposit sunl at th intrfac ±θ. At last, th influnc of th pl angl on th intrlaminar strsss is anal for th laminat composit plat ([θ/ θ] s ). Th valus of th intrlaminar strsss ar chang apparntl with th pl angl. But th curvs of th strsss along -axis ar similar. Th absolut valu of th intrlaminar strsss is incras as th pl angl rang btwn an a spcific valu which is about 2 or 3, thn a cras procss is happn as th pl angl rang btwn th spcific valu an 6. Bsis, it closs to ro whn th angl rang btwn 6 an Copright 23 SciRs. MSMS

10 58 C.. YAG ET AL. 9. So whn th laminats is angl pli as [θ/ θ] s,th angl is suggst btwn 6 an 9 to lowr th fr g ffct. Th accurac can b nsur b using th isplacmnt fil prsnt in this papr. An th valiation of using th linar finit lmnt is monstrat b som numrical rsults. Also it is convnint that onl th linar finit lmnt is us to calculat th intrlaminar strsss as th intrpolation along -axis is inclu in th isplacmnt fil. It shoul b notic that th finit lmnt us hr can b onl appli to th laminats subjct to a uniform strain. But it can b as to b xpan to anal th othr laminats. 6. Acnowlgmnts Th authors woul li to xprss thir gratitu for th support provi b th ational atural Scinc Founation of China (O. 725). REFERECES [] S. B. Dong, K. S. Pistr an R. L. Talor, On th Thor of Laminat Anisotropic Shlls an Plats, ournal of th Aronautical Scincs, Vol. 29, o. 8, 962, pp [2] E. Rissnr an Y. Stavs, Bning an Strtching of Crtain Tps of Htrognous Alotropic Elastic Plats, ournal of Appli Mchanics, Vol. 28, o. 3, 96, pp oi:.5/ [3].. Pagano, Strss Fils in Composit Laminats, Intrnational ournal of Solis an Structurs, Vol. 4, o. 4, 978, pp oi:.6/2-7683(78)92-3 [4].. Pagano, Fr Dg Strss Fils in Composit Laminats, Intrnational ournal of Solis an Structurs, Vol. 4, o. 5, 978, pp oi:.6/2-7683(78)92-5 [5] C.-P. Wu an H.-C. Kuo, An Intrlaminar Strss Mix Finit Elmnt Mtho for th Analsis of Thic Laminat Composit Plats, Composit Structurs, Vol. 24, o., 993, pp oi:.6/ (93)952-r [6] H. Matsunaga, Intrlaminar Strss Analsis of Laminat Composit Bams Accoring to Global Highr- Orr Dformation Thoris, Composit Structurs, Vol. 55, o., 22, pp oi:.6/s ()34-9 [7] A. osir an A. Bahrami, Intrlaminar Strsss in Antismmtric Angl-Pl Laminats, Composit Structurs, Vol. 78, o., 27, pp oi:.6/j.compstruct [8] T. S. Plagianaos an D. A. Saravanos, Highr-Orr Larwis Laminat Thor for th Priction of Intrlaminar Shar Strsss in Thic Composit an Sanwich Composit Plats, Composit Structurs, Vol. 87, o., 29, pp oi:.6/j.compstruct [9] H. S. Kim, S. Y. Rh an M. Cho, Simpl an Efficint Intrlaminar Strsss Analsis of Composit Plats with Intrnal Pl-Drop, Composit Structurs, Vol. 84, o., 28, pp oi:.6/j.compstruct [] M. Amabili an.. R, A w on-linar Highr-Orr Shar Dformation Thor Larg Amplitu Vibrations of Laminat Doubl Curv Shlls, Intrnational ournal of on-linar Mchanics, Vol. 45, o. 4, 2, pp oi:.6/j.nonlinmc [] A. K. Miri an A. osir, Intrlaminar strsss in Antismmtric Angl-Pl Clinrical Shll Panls, Composit Structurs, Vol. 93, o. 2, 2, pp oi:.6/j.compstruct [2] X. H. Rn, W.. Chn an Z. Wu, A w Zig-Zag Thor an C Bning Elmnt for Composit an Sanwich Plats, Archiv of Appli Mchanics, Vol. 8, o. 2, 2, pp oi:.7/s [3]. L. Mantari, A. S. Otm an C. Gus Soars, A w Highr Orr Shar Dformation Thor for Sanwich an composit Laminat Plats, Composits Part B: Enginring, Vol. 43, o. 3, 22, pp [4] A. osir an M. Mali, Fr-Eg Strsss in Gnral Composit Plats, Intrnational ournal of Mchanical Scincs, Vol. 5. o. -, 28, pp oi:.6/j.mcsci [5] C.-F. Liu an H.-S. ou, A w Finit Elmnt Formulation for Intrlaminar Strss Analsis, Computr an Structurs, Vol. 48. o., 993, pp oi:.6/ (93)9464-o Copright 23 SciRs. MSMS

11 C.. YAG ET AL. 59 Appnix Th finit xprssion of th finit lmnt quation K = F can b writtn as blow: m j 2 3 K vj K x K 3 32 K U j K V j Fi 3 3 K W j w hr i, 2,, m., 2,,3 6. m is th numbr of th no of th lmnt. is th numbr of th nu- of K mrical lars along -axis. Th xprssions an F i i, j,2,, m.,,2,,3 6 can b writtn as blow: K A 22, K K B K K B22 22 K D66 A55 33 K D22 A K K D26 A K K A K K A i 23 j K K A K K B A K K B A ˆ i j K B26 j B45 i 3 32 j i K B26i A45 j K K B A K K B A K Bˆ B 33 3 i 23 j 44 i K B A j 3 33 j i 23i 44 j K A B K 3I3 I I I32 3I23 K I I A26 B45 K A D 3I33 I i I j 36j 45i K A B 3I232 I I K A D 3I233 I i I j 23j 44i K A Aˆ 3I33 I i I j 45j 36 i K A Aˆ 3I332 I i I j 44j 23 i K B B 3I333 I I i 2 i Fi A2 F i B6 3 i 3 Fi B2 F i i A6 32 i Fi A2 33 Fi A3 i whr, i, j,2,, m. I,,2,,. i an j ar th intrpolation shap function of th no. is th uniform axial strain along x-axis. An th cofficint xprssions in K an F i can b writtn as: A C 2 D C B C A A C B C B C Copright 23 SciRs. MSMS

12 6 C.. YAG ET AL. B ˆ B C I I I A B C A C I I I I I Aˆ D C B I C I whr, is th coorinat of -axis. is -coorinat of th uppr surfac of th th lar whil + is -coorinat of th lowr surfac of th th lar. C is th offaxis stiffnss cofficint of th th lar ( i, j =, 2, 3, 4, 5, 6). I an ar th intrpolation function of th Equa tions () an (2). Copright 23 SciRs. MSMS