BENDING ANALYSIS OF COMPOSITE LAMINATED PLATES USING HIGHER-ORDER SHEAR DEFORMATION THEORY WITH ZIG-ZAG FUNCTION

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1 VO. 6 NO. DECEMBER ISSN ARPN Jrnal f Engineering and Applied Science 6- Aian Reearch Pblihing Netr ARPN. All right reered..arpnjrnal.cm BENDING ANAYSIS OF COMPOSITE AMINATED PATES USING HIGHER-ORDER SHEAR DEFORMATION THEORY WITH ZIG-ZAG FUNCTION T. Dharma Raj and J. Sreh mar Department f Mechanical Engineering Gdaari Intitte f Engineering and Technlg Rajahmndr India Department f Mechanical Engineering JNTU Cllege f Engineering Hderabad India raj_thmmala@ah.cm ABSTRACT In thi paper an analtical prcedre i deelped t inetigate the bending characteritic f laminated cmpite plate baed n higher rder hear diplacement mdel ith ig-ag fnctin. Thi ig-ag fnctin impre lpe dicntinitie at the interface f laminated cmpite plate. The eqatin f mtin i btained ing the dnamic erin f Hamiltn principle. The ltin are btained ing Naier and nmerical methd fr anti-mmetric cr-pl and angle-pl laminate ith a pecific tpe f impl pprted bndar cnditin SS- and SS- repectiel. In thi paper the nmerical relt are preented fr bending f anti-mmetric cr-pl and angle-pl laminated plate. All the ltin preented are cle agreement ith the ther f elaticit and cled frm ltin aailable in the literatre. erd: laminated cmpite plate bending anali higher rder ther ig-ag fnctin Hamiltn principle Naier methd.. INTRODUCTION The plate are traight and plane rface trctre he thicne i light cmpared t ther dimenin gemetricall. Staticall plate hae impl pprted and fied bndar cnditin inclding elatic pprt and elatic retraint r in me cae pint pprt. The tatic r dnamic lad are carried b plate are predminantl perpendiclar t the plate rface. The accrate predictin f the repne characteritic f laminated trctre i a challenging ta becae f their intrinic anitrp hetergeneit and l rati f the tranere hear mdl t the in-plane Yng mdl. Hence it i necear t anale the bending characteritic f laminated cmpite plate. Seeral plate therie hae been deelped b Reiner E; Sta Y [] impred the CPT b inclding the inflence f bending-eteninal cpling in n-mmetrical laminate. Pagan N. J. [] tdied the mechanical repne f cmpite laminate b cnidering the prblem f clindrical bending f bi-directinal laminate. Eact ltin ithin the frame r f the linear ther f elaticit are deelped and cmpared ith the repectie ltin f CPT. Pagan N. J. and Hatfield S. J. [] amed a nifrm hear train thrgh the thicne f the plate and neglected lcal effect fr the finite element frmlatin baed n the nn cnfirming rectanglar plate bending element. ant T. et al. [4] preented a methd fr the nmerical anali f elatic plate ith t ppite impl pprted end. Redd J.N. [5] deelped a higher rder hear defrmatin ther f laminated cmpite plate. The ther cntain the ame dependent nnn a in the firt rder hear defrmatin ther f Whitne and Pagan 97 bt accnt fr parablic ditribtin f the tranere hear train thrgh the thicne f the plate. Redd J. N. [6] etablihed a reie f all third rder t dimeninal technical thei f plate that atif anihing f tranere hear tree n bnding plane f the plate i preented and their eqialence. Ch M. and im J. S. [7] deelped a higher rder ig-ag ther fr laminated cmpite plate ith mltiple delaminatin. M. arama. S. Afaq S. Mit [8] preent a ne mlti-laer laminated cmpite trctre mdel t predict the mechanical behair f mlti-laered laminated cmpite trctre. A a cae td the mechanical behair f laminated cmpite beam i eamined frm bth a tatic and dnamic pint f ie. Sanjib Gami [9] preented a imple C finite element frmlatin ith embedded higher-rder hear defrmatin ther and a - dimeninal tate f tre and train fr thic and thin laminated cmpite plate. Peman hrai Rajamhan Ganean and Ramin Sedaghati [] deelped an efficient facet hell element fr the gemetricall nnlinear anali f laminated cmpite trctre ing the crtatinal apprach. M. E. Fareand M. and h. Elmarghan [] preented a refined nnlinear igag hear defrmatin ther f cmpite laminated plate ing a mdified mied ariatinal frmlatin. Dieg Amade F. Trre et al. [] implemented a frmlatin fr the bending anali f cmpite laminated plate ith pieelectric laer ing the generalied finite element methd. In thi paper a Higher -rder hear defrmatin ther ith Zig Zag fnctin i prped and deelp the analtical prcedre t anale the bending characteritic f laminated cmpite plate hich tae care f the dden change f prpertie frm lamina t lamina.. THE HIGHER-ORDER SHEAR DEFORMATION THEORY WITH ZIG-ZAG FUNCTION A rectanglar plate f a; b and h h i cnidered. The elaticit ltin 6

2 VO. 6 NO. DECEMBER ISSN ARPN Jrnal f Engineering and Applied Science 6- Aian Reearch Pblihing Netr ARPN. All right reered..arpnjrnal.cm 7 indicate that the tranere hear tre ar parablicall thrgh the plate thicne. Thi reqire the e f diplacement field in hich the in-plane diplacement are epanded a cbic fnctin f the thicne crdinate in additin the tranere nrmal train ma ar in nn linearl thrgh the plate thicne. The higher-rder diplacement field ith ig-ag fnctin hich atif the abe criteria i amed in the flling frm: Where and dente the diplacement f a pint n the mid-plane are rtatin f the nrmal t the midplane abt -ai and ai are the higher rder defrmatin term defined at the mid-plane i the Zig-Zag fnctin defined a: h Z Z i the lcal tranere crdinate ith it rigin at the center f the th laer h i the crrepnding laer thicne. The train cmpnent are: The tre-train relatinhip in the glbal -- crdinate tem can be ritten a: The gerning eqatin f diplacement mdel ill be deried ing the principle f irtal r a dt V U T δ δ δ 4 The irtal r tatement hn in Eq. 4 integrating thrgh the thicne f laminate the in-plane and tranere frce and mment reltant relatin in the frm f matri btained a: D D B B A M M N N b t 5 Eqating the cefficient f each f irtal diplacement δ δ δ δ δ δ δ δ δ δ δ t er the eqatin f mtin ill be btained and are epreed in term f diplacement b btitting fr the frce and mment reltant. The Naier ltin f impl pprted anti mmetric cr pl laminated plate: The SS- bndar cnditin fr the anti mmetric cr pl laminated plate are: At edge and a M M N N 6 a At edge and b M M N N 6b The SS- bndar cnditin fr the anti mmetric angle pl laminated plate are:

3 VO. 6 NO. DECEMBER ISSN ARPN Jrnal f Engineering and Applied Science 6- Aian Reearch Pblihing Netr ARPN. All right reered..arpnjrnal.cm At edge and a N M M N 7a At edge and b N M M N 7b The diplacement at the mid plane ill be defined t atif the bndar cnditin in Eq. 6 and 7. Thee diplacement ill be btitted in gerning eqatin t btain the eqatin in term f A B D parameter. The btained eqatin ill be led t find the behair f the laminated cmpite plate.. RESUTS AND DISCUSSIONS The impl pprted bndar cnditin SS- hn in Eq. 6 are cnidered fr ltin f antimmetric cr-pl laminate here a Eq. 7 fr ltin f anti-mmetric angle-pl laminate ing a higher rder hear defrmatin ther ith ig- ag fnctin. The material prpertie f graphite ep ed fr each lamina f the laminated cmpite plate are: E 4 E E E G 6 X N / cm G G 6.6 X N / cm µ µ µ.5 G 6.5 X N / cm The deflectin internal tre-reltant and tree are preented here in nn-dimeninal frm ing the flling mltiplier: Eh h m m m m4 m5 4 qa qa qa qa Figre- h the ariatin f nndimeninalied maimm nrmal tree a a fnctin f n f laer and different therie fr anti-mmetric cr-pl laminated plate nder inidal tranere lad. The -laered plate eperience larger tree than the f 4 6 and 8 laered plate and the tre cncentratin i redced in the latter. Th the effect f bending-tretching cpling preent in -laered plate n tree i t increae the magnitde f tree. Figre- h the maimm tranere hear tree fr impl pprted angle-pl laminate a a fnctin f thicne n f laer. Frm figre it i bered that the lpe dicntinitie in HSDT are taen care b the f ig-ag fnctin in preent ther. Figre- h the ariatin f nn-dimeninalied in plane and tranere tree a a fnctin f thicne crdinate /h. Frm the figre it i bered that the preent mdel are in cle agreement ith D elaticit ltin. h qa N.Nrmal Stre N f aer HSDT HSDTWZF Figre. Nn-dimeninalied ma. tranere nrmal tre Vn f laer fr impl pprted anti-mmetric cr-pl laminated qare plate SS SS 8

4 VO. 6 NO. DECEMBER ISSN ARPN Jrnal f Engineering and Applied Science 6- Aian Reearch Pblihing Netr ARPN. All right reered..arpnjrnal.cm. SS.5 ma tranere hear tret /45hsdt -45/454HSDT -45/45 HSDTWZF -45/454 HSDTWZF thicne crdinate/h Figre. Nn-dimeninalied ma. tranere hear tre t V thicne crdinate fr impl pprted antimmetric angle-pl laminated qare plate N.Nrmal tre. SS SS Thicne crdinate/h -.8 ant&panda Pagan HSDTWZF HSDT Figre. Variatin f the nrmal trethrgh the laminated thicne fr a impl pprted cr-pl qare plate nder inidal tranere lad 4. CONCUSIONS The ariatin ith mth lpe-cntinitie at the interface i ecnmicall achieed b empling cntin linear and nn-linear fnctin. The ig-ag fnctin i a alable tl t enhance the perfrmance f bth claical and adanced therie. The effect f bending-tretching-cpling i ignificant fr all mdl rati ecept fr the cle t nit n anti-mmetric angle-pl laminated cmpite plate f ame thicne fr an nmber f laer. The effect f bending tretching cpling preent in t laered plate n tree and deflectin i t increae the magnitde than the f 4 6 and laered plate. REFERENCES [] Reiner. E On bending f elatic plate. arterl f applied Mathematic. 5: [] Pagan N.J Eact ltin fr cmpite laminate in clindrical bending. J. f cmpite material. :

5 VO. 6 NO. DECEMBER ISSN ARPN Jrnal f Engineering and Applied Science 6- Aian Reearch Pblihing Netr ARPN. All right reered. [] Pagan. J.N. and J.H. Sharn. 97. Elatic Behair f Mltilaered Bidirectinal Cmpite. J. Strctral Cmpite Material. : 9-9. [4] ant. T. 98. Nmerical Anali f Elatic Plate ith T ppite impl pprted end b egmentatin methd. J. f Cmpter and Strctre. 4-4: 95-. [5] Redd J.N A imple Higher-Order Ther fr aminated Cmpite Plate. J. f Applied Mechanic. 5: [6] Redd J.N. 99. A General Nn-linear Third-rder ther f plate ith Mderate Thicne. Int. J. f Nn-linear Mechanic. 56: [7] Ch. M and J.S. im.. Higher-Order Zig-Zag Ther fr aminated Cmpite ith Mltiple Delaminatin. J. f Applied Mechanic. 68: [8] M. arama et al.. Mechanical behair f laminated cmpite beam b the ne mlti-laered laminated cmpite trctre mdel ith tranere hear tre cntinit. J. f Slid and Strctre. 4: [9] Sanjib Gami. 6. A C plate bending element ith refined hear defrmatin ther fr cmpite trctre. J. f Cmpite Strctre. 7: [] Peman hrai et al. 8. An efficient facet hell element fr crtatinal nnlinear anali f thin and mderatel thic laminated cmpite trctre. J f Cmpter and Strctre. 869: [] M.E. Fareand and M.h. Elmarghan. 8. A refined igag nnlinear firt-rder hear defrmatin ther f cmpite laminated plate. J. f Cmpite Strctre. pp [] Dieg Amade F. Trre et al.. Ealatin and erificatin f an HSDT-aerie generalied finite element frmlatin fr adaptie pieelectric laminated plate. J f Cmpter Methd in Applied Mechanic and Engineering. 5-8: arpnjrnal.cm