Meshless Local Petrov-Galerkin Formulation for Static Analysis of Composite Plates Reinforced by Unidirectional Fibers

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1 Amerc Jourl of Mechcl Egeerg, 0, Vol., o. 7, Avlble ole t Scece d Educto Publshg DOI:0.69/jme76 Meshless Locl PetrovGler Formulto for Sttc Alyss of Composte Pltes Reforced by Udrectol Fbers Ml Žmdá,*, Del Recy, Zor Pelgć, Mrt Dudsy Deprtmet of Appled Mechcs, Uversty of Žl, Žl, Slov Republcy Plstc Omum Auto Exterors, Ltd., Slov Republc *Correspodg uthor: ml.zmd@fstroj.uz.s Receved October 8, 0; Revsed October, 0; Accepted ovember, 0 Abstrct Ths pper dels wth the pplcto of meshless methods for the lyss of composte pltes. The m tteto s focused o the mplemetto of the Meshless Locl Petrov Gler (MLPG) formulto for multlyered orthotropc pltes. At frst for ths purpose the mplemetto of homogezto theory ws eeded d lyzes were mde to obt homogezed mterl propertes of composte pltes. The softwre for homogezto of mterl propertes uses drect homogezto method tht s bsed o volume verge of stresses o the represettve volume elemet (RVE). Homogezto s performed by mult softwre pproch, by lg MATLAB d ASYS softwre. The dt obted re used lyzes performed user ow softwre, whch s bsed o the MLPG method. Str, stress d dsplcemet felds were lyzed. Results obted by MLPG were compred wth those obted by FEM progrms, ASYS d ABAQUS. Keywords: composte pltes, homogezto, meshless, locl PetrovGler method Cte Ths Artcle: Ml Žmdá, Del Recy, Zor Pelgć d Mrt Dudsy, Meshless Locl PetrovGler Formulto for Sttc Alyss of Composte Pltes Reforced by Udrectol Fbers. Amerc Jourl of Mechcl Egeerg, o. 7 (0): do: 0.69/jme76.. Itroducto Plte d shell structures re the most wdely used structurl members mechcl d cvl egeerg ths to ther good weght to lod crryg cpblty rto. Ppes, eroples, hulls of shps [], brdge reforcemets s well s reforced roofs d membres re good exmples. Ths to good mterl propertes, especlly to good weght to stregth rto fberreforced mterls cme to prctce. Lmtes re composed of multple lyers where fber oretto lyers c be dfferet []. By ths wy t s possble to obt requred mterl propertes requred drecto. The fte elemet method (FEM) s oe of the most wdely used d most populr umercl methods for lyzg plte structures []. Although the method s stble, well developed d hs reched extesve developmet durg lst decdes, t lso hs some lmttos. Oe of the ssumptos FEM s exstece of the mesh creted from the fte elemets. Mesh qulty ffects the ccurcy of obted soluto d stblty of covergece. Cretg well defed meshes c be very tme cosumg some cses. Geerlly t s recommeded to use fte elemets wth shpes whch re the most smlr to del shpes [4,5]. Degeerted geometry or deformed shpe c hve egtve effect o the soluto ccurcy. Ths s very mportt lyzg oler problems wth lrge deflectos or lrge strs such s metl formg, frgmetto fter mpct, etc. I these cses mproper mesh cuses serous decrese of ccurcy or flure of computto. I lst yers crese of terest ew type of umercl methods ow s meshless methods ws observed [6,7,8,9,0]. These methods re terestg due to ther flexblty d blty of solvg boudry vlue problems wthout predefed mesh. Computtol model these methods s represeted by set of odes dstrbuted wth globl dom d ts boudry. These odes do ot hve to be coected to explctly defed elemets. Iformto bout reltos betwee the odes s theoretclly ot ecessry before lyss. By ths wy some problems wth the mesh FEM c be overpssed. Problems wth remeshg c be overpssed by ddg or removg some odes s ecessry. I some methods the resultt stress felds re globlly cotuous d ths smplfes cosequet lyses. Oe of the res where meshless methods re coveet to use s lyss of plte d shell structures. These methods re useful due to flexblty of meshless lgorthms d blty of meshless pproxmto fuctos to obt terpoltg feld wth hgh order of cotuty by smple wy. I some cses t s possble to overcome some locg effects more smple wy th FEM. Meshless methods re reltvely ew cocept computtol mechcs. Compred to FEM

2 46 Amerc Jourl of Mechcl Egeerg formultos there re less meshless formultos vlble for plte d shell structures. Addtol reserch d developmet of geerl meshless methods ble to successfully solve vrous problems plte structurl lyzes re therefore ecessry. Meshless pproches for problems of cotuum mechcs hve ttrcted much tteto durg the pst decde especlly owg to ther hgh dptvty d low costs to prepre put dt for umercl lyses. represettve volume elemet (RVE). For the lyss of the mterl propertes ow softwre ws progrmmed MATLAB lguge d prt of the soluto ws crred out ASYS softwre. The RVE cossts of volume elemets d s the loded by ut strs vrous drectos. The effectve lm propertes re obted from the volume frctos of stress vlues obted by lodg the RVE.. Composte Homogezto Pltes c be reforced by log glss, crbo or evlr fbres. The frst step s to obt homogezed lm mterl propertes. These c be obted by usg homogezto techques... Homogezto Techques There re vrous homogezto methods [,]. Drect homogezto s bsed o the volume verge of feld vrbles, s stress, str d eergy desty. Effectve propertes c be clculted from effectve propertes deftos. The dmeter d the clculto of feld vrbles c be clculted umerclly, for exmple by FEM, BEM d the geometry d mcrostructurl propertes c be geerlzed. Idrect homogezto s bsed o the Eshelby soluto of selfdeformto for oe cluso fte mtrx equvlet cluso method []. Ths method does ot use vergg feld vrbles d the effectve propertes c be obted by deducg the volume frctos d the cluso geometry lso s the compoet propertes. Accordg to ths ptter these methods were developed: selfcosstet scheme [4], geerlzed selfcosstet scheme Chrstese d Lo [5], dfferetl methods [6], MorT method [7]. They re ofte used to clculte vrous propertes of compostes. But the geerlzed mcrostructurl morphology, whch s ofte preset rel mterls, cot be hdled out determstclly these models. Costtutve resposes of the compoet phses re lmted d the estmted results wth lrge dsgreemets re ot relble eough. These models cot for they suffcet represetto of mcroscopc stresses d strs ctch the effects of locl ohomogeetes. A lterte pproch to drect d drect homogezto s the vrtol method, whch c determe the upper d lower borders of the elstcty modulus [8]. A reltvely ew pproch or homogezto of mcrostructures cosstg of mthemtcl homogezto bsed o twoscle exteso of the dsplcemet feld. Ths comes from the lyss of physcl systems cotg two or more scles [9]. Ths pproch s good for multphse mterls, whch re turl scles of mcroscopc scles, chrcterstc heterogeety or locl dscotuty spcg. Mcroscopc scles re chrcterzed by the body dmesos. Ths method c be clled mthemtcl homogezto... Homogezto Results Ths prt descrbes the procedure of homogezto of mterl propertes of compostes of usg the method of Fgure. Fber rrgemet mtrx: )hexgol rry, b) squre rry Homogezed lm RVE cossts of fber d epoxy mtrx. The fbers re from three mterl types, crbo, glss, polyrmd. We ssumed cyldrcl fber shpes d del coheso betwee the fber d the mtrx. Used crbo fbers re dustrlly lbeled s T00 d M40J. The glss fber lbel s EGlss d SGlss. Polyrmde fbers hve the lbel K49. Fber mterl propertes re lsted Tble. d the mtrx propertes re lsted Tble [0]. The RVE dmesos re clculted for the hexgol fber cofgurto Fgure from the reltos () d for squre cofgurto the RVE dmesos re Fgure b, clculted from the reltos (). f 0 πd =, =.tg(60 ), = 0,5., () 8 0 V tg(60 ) f d f π 4 V f =, =, = 0,5., () Where s the xdrecto, ths cse the fber drecto, s the y drecto, orthogol to the fber drecto, drecto z, trsverse vertcl to the fber drecto. Tble. Fber mterl propertes Crbo M40J Fber mterl Fberglss SGlss Kevlr K49 E f [GP] 77 85,5 5,5 F t [GP] ν ρ f [g/m ] d f [μm] E f [GP]

3 Amerc Jourl of Mechcl Egeerg 46 Mtrx mterl Tble. Mtrx mterl propertes E m F t ρ ν m [GP] [MP] [g/m ] G m [GP] Epoxy Tble. RVE Mterl propertes for volume frcto Vf=0.4 V f=0.4 Crbo M40J Fberglss SGlss Kevlr K49 h s h s h s E [GP] E [GP] G [GP] G [GP] v v Goverg Equtos Clsscl lmted plte theory ws formulted by dervg clsscl plte theory for composte pltes. I ths theory the plte composes of orthotropc lyers wth totl thcess h. The mdsurfce of lyered plte s locted the rego Ω, ple (x, x ). Axs x z s perpedculr to the mdsurfce Fgure. th lyer s locted betwee coordtes from z = z to z = z + thcess drecto x. Deformto of the plte s descrbed by the Resser Mdl plte theory []. I ths theory the sher strs thcess drecto re costt d correcto coeffcets re ecessry for clculto of trsverse sher forces. Sptl dsplcemet feld cused by trsverse lodg c be expressed, terms of for dsplcemet compoets u, u, u the form of u (x,x,t) = xw (x), u (x,x,t) = xw (x), u (x,t) = w (x), () Where x =[ x, x ] T s the posto vector, w α ( x, x, t) re ple rottos bout the xes for dces α =, d w (x, x, t) s the deflecto of the z ple (x, x ). ε (x,x,t) = xw,(x), ε (x,x,t) = xw,(x), ε (x, x,t) = x w, (x) + w,(x) /, ε (x,t) = w (x) + w,(x) /, ε (x,t) = w (x) + w, (x) /. (4) If th s lm mde of orthotropc mterl, the the relto betwee stresses σ j d strs ε m s expressed by the costtutve equto for stress tesor σ j (x,x,t) = c jm ε m (x,x,t), (5) Wth ssumpto of homogeeous coeffcets of costtutve tesor c jm for th lm. From the equto (4) t c be see tht strs re cotuous through the plte thcess. Dscotuous mterl coeffcets cuse the stffess chge the terfces d hece the stresses lm terfces re dscotuous. Equto (5) for plte problems s usully wrtte s tesor of elstc costts of the secod order. Costtutve equtos for orthotropc mterl d ple stress hve the the form of σ ε σ ε σ G (x), = ε (6) σ ε σ ε Where E /e E v /e E v /e E /e G (x) = 0 0 G, G G ( ) ( ) ( ) e = v Where v, E α s Youg's elstcty G modulus xα (α =, ) xs drecto, G, d G v re sher elstcty modul d αβ re Posso's rtos of th lm. Fgure. Composte plte ) geometry d dsplcemets, b) momets d sher forces Ler deformto c be expressed Fgure. Lyer coordte otto

4 464 Amerc Jourl of Mechcl Egeerg Bedg momets M αβ d Q α c be expressed tegrl form s M h/ σ z σ + M σ xdx σ = = xdx, M h/ z σ = σ (7) h/ z Q σ + σ κ dx κ dx. Q = σ = h/ σ = z (8) Where κ = 5/6 the ReserMdl plte theory, z coordte s cosdered s s show Fgure. By substtutg (4) d (6) to resultt momets d forces (7), (8) t s possble to express bedg momets M αβ d sher forces Q α, α, β=, for orthotropc plte terms of dsplcemets d rottos. I the cse of lyerwse cotuous mterl propertes the followg reltos re obted M = D (w + w ) + C w αβ αβ α,β β,α αβ Q = C (w + w ). α α α,α γ,γ (9) For dces α β (9) Este summto rule does ot pply d mterl prmeters D αβ d C αβ re gve by reltos h/ z + v v D = ( ) z E z dz = E z dz e h/ = z e ( + ) v = E z z, ( ) = e ( + ) h/ v v D = ( ), z E z dz = E z z ( ) e h/ = e h/ D = z G ( z) dz = G z z, + h/ = ( ) ( + ) ( + ) h/ v v C = ( ), z E z dz = E z z e h/ = e ( + ) h/ v v C = ( ), z E z dz = E z z e h/ = e h/ C = κ G ( z) dz = κ G z z. α α α h/ = (0) For homogeeous plte reltos (0) re smplfed to the form of D D Gh D = ( v ), D = ( v ), D = D =, C = D v, C = D v, C = C = 0, Eα h Dα =, D v = D v, Cα = κ hgα. e () Plte s subjected to trsverse dstrbuted lodg q(x, t). If ech lm hs homogeeous desty thcess drecto, equtos of moto for Resser ler theory for thc pltes c be wrtte the form of M ( xt, ) Q (x,t) = I w ( xt, ) () αβ,β α M α Qα,α ( x,t) + q(x,t) = IQw α(x,t) () Where the upper dot deotes the tme dervtve by t, wth the order equl to the dot umber, the dexes α, β =, ( + ) h/ z + = M = = h/ = z = I z ρ(z)dz ρ z dz ρ z z ( + ) h/ z + = Q = = h/ = z = I ρ(z)dz ρ dz ρ z z,, (4) re the globl ertl chrcterstcs of the lmte plte. For plte wth costt desty through the whole thcess re these chrcterstcs expressed s I M =ρh /, I Q = ρh. I MLPG method the locl we form s ssembled o locl subdom Ω s, whch s smll dom for ech ode wth the globl dom [6]. These locl subdoms overlp ech other d cover the whole globl dom Ω, Fgure 4. Locl we form of the goverg equtos (9) for X Ω q x c be wrtte s (5) * Mαβ, β ( x, t) Qα ( x, t) IM wα ( x, t) wαγ ( x) dω= 0, (6) * Qαα, (,) x t q(,) x t IQw(,) x t w () x dω= 0, Ω Q Fgure 4. Locl boudres for we formulto, the support dom Ω for MLS pproxmto of the trl fucto, d support dom of weght fucto roud ode X, Ω Q s tegrto dom roud gve ode where w * αγ d w * s the weght or testg fucto. By pplyg to the equtos () d (4) the Guss dvergece theorem we obt * * Mα ( x, tw ) αγ ( x) dγ Mαβ ( x, tw ) αγ, β ( x) dω * * * Qα ( x, tw ) αγ ( x) dω IM w αγ ( x, tw ) αγ ( x) dω= 0, * * Qα( x) α( x) w ( x) dγ Qα( x) w, α( x) dω ( ) ( ) * * * IQw ( x, tw ) ( x) dω+ q x w x dω= 0, (7) (8)

5 Amerc Jourl of Mechcl Egeerg 465 Ω Q s boudry of locl subdom, M α (x)=m αβ (x) β (x) s the orml bedg momet d α s the outwrd orml o the boudry Ω Q. Locl we forms (7) d (8) re strtg pots for dervg locl boudry tegrl equtos o the bss of proper test fucto. Ut step fucto s chose ech subdom for test fuctos w * αγ (x) d w * (x) * αγ w δ v αγ ( x) = 0 v x ( Ω Ωs ) x ( Ω Ωs ) ( s ) ( s ) v x Ω Ω * w ( x) = 0v x Ω Ω (9) The the locl we form (5) d (6) trsforms o the followg locl tegrl equtos Mα ( x) dγ Qα ( x) dω IM w α ( x) dω= 0 (0) ( ) () Qα ( x) α ( x) dγ IQ w ( x) dω+ q x dω= 0 where w α (x;t) s the trl fucto correspodg to rottos α =,, the trl fucto w (x; t) correspods to the trsverse dsplcemet. These trl fuctos re ssembled MLS pproxmto over odes locl subdom roud the pot x. 4. umercl Implemetto of MLPG I geerl, meshless method uses locl terpolto to represet the trl fucto wth the vlues (or the fcttous vlues) of the uow vrble t some rdomly locted odes []. To pproxmte the dstrbuto of the geerlzed dsplcemets (rottos d deflecto) ΩX over umber of rdomly locted h odes X j (j =,,..., ), the MLS pproxmt w ( x, t) of w ( x, t) s defed by h (, ) ( ) ˆ w xt = ϕ xw ( t), () Where vector w h = [w h, w h, w h ] T, ϕ α s the MLS pproxmted shpe fucto for the correspodg x ode. The pproxmto of drectol dervtves w(x,t) for correspodg ode vlues re gve s w ( x, t, ) = ˆ () t ϕ, ( ), w x () Where ϕ α,(x) s the prtl shpe fucto dervtve ϕ α (x) for the = x, x drecto. Substtutg equtos () d () to the bedg momet defto equto (9) by usg the expresso M α (x; t) = M αβ (x; t) β (x), becomes * * Mx ( ) = B( xw ) + B( xw ) (4) * = α( x) Bα( xw ) Where vector w *α s defed s colum vector w *α * =[ w ˆ * ; w ˆ ] T, the mtrces α ( x ) re relted to the orml vector (x) t the re boudry Ωs defed by 0 ( x) = d 0 C 0 ( x) =, 0 C (5) mtrx whch defes the grdets of shpe fuctos whch s wrtte s B D ϕ, 0 ( x) = 0 D ϕ, D ϕ, D ϕ, ϕ, 0 B ( x) =, 0 ϕ, d (6) The fluece of mterl oretto for the composte s cluded C αβ d D αβ, whch re defed equtos (0), the sher force equtos re * * Qx ( ) = Cx ( ) ϕ ( xw ) + F ( x ) wˆ (7) where Q(x) = [Q (x) ; Q (x)] T d C ( x) 0 ϕ, Cx ( ) =, F ( x) =, 0 C ( x) ϕ, (8) Substtutg the MLS dscretzed momet d force feld (4) d (7) to the locl tegrl equtos (0) d () locl dscrete tegrl equtos wll be formed * α ( ) α ( ) dγ ( ) ϕ ( ) dω xb x Cx x w L Q +Γ QM Ω Q (9) * I ( ) ( ) ˆ M w t ϕ xdω w CxF ( ) ( x) dω = Mx ( ) dγ, ΓQM (0) * C ( ) ( ) ˆ x ϕ x dγ w w C( x) F ( x) dγ ˆ IQ w ( t) ϕ ( ) dω = q( ) dω, x x

6 466 Amerc Jourl of Mechcl Egeerg where M ( x) s the prescrbed momet o the ΓQM boudry d C 0 Cx ( ) = (, ) = ( C, C ) () 0 C The equtos (9) d (0) re for the subdoms the eghborhood of the er ode x d lso for the ode o the boudry Γ QM. The pot x locted o the globl boudry Γ the boudry subdom Ω Q s splt to L Q d Γ QM (ths prt of the globl boudry o whch re the bedg momets prescrbed), Fgure I the locl we form (5) d (6) re ot ssocted wth pelzto prmeters or Lgrge multpler, becuse the boudry codtos o Γ QM (prt of the globl boudry, o whch dsplcemets d rottos re prescrbed) c be drectly formulted, by usg the terpolto formulto (). * α ( ) α ( ) dγ ( ) ϕ ( ) dω xb x Cx x w L Q +Γ QM Ω Q wˆ ( ) ( ) dω = ( ) dγ, CxF x Mx ΓQM * ˆ ϕ = q( x) dω, (5) C ( x) ( x) dγw w C ( x) F ( x) dγ (6) whch s the locl tegrl equto form, whch s solved by the softwre for plte bedg problems. The homogezed dt obted re used lyzes performed user ow softwre, whch s bsed o the MLPG method. A smple flowchrt represetg process of clculto by MLPG s gve Fgure 5. QM ϕ ( x ) wˆ = wx ( ) for( x) Γ, () where wx ( ) s the geerlzed dsplcemet vector descrbed o the ΓQM boudry. For clmped plte the frst three elemets of the vector (dsplcemet d rottos) re zero for the clmped edge d equto () s used ll boudry odes ths cse. For smply supported plte oly the thrd member of the dsplcemet vector s prescrbed the rottos re uow. The equto (9) d the thrd member of the equto () re ppled o the odes lyg o the globl boudry. O the odes o the globl boudry, o whch o boudry codto re prescrbed, both locl tegrto equtos (9) d (0) re ppled. 4.. Smplfcto of Formulto for Sttc Alyss Equtos (9) d (0) were smplfed for sttc lodg, the tme hs o physcl meg, but ebles to defe the stepwse cresg lodg. () ˆ IM w (t), φ (x)dω = 0 Ω Q (4) ˆ I. Q w (t) φ (x)dω = 0 Ω Q The resultg system of equtos (9) d (0) hs the form Fgure 5. Flowchrt represetg process of clculto 5. umercl Results I the preseted lyzes we cosdered composte plte wth dmesos L x d L y, where L x = 0,4m d L y = 0.m, Fgure 6. It cossts from sx ples, every ply hs thcess Δz = m, the totl thcess of the plte s h = 0.005m. The mterl oretto of the s [45 / 90 / (0) / 90 / 45]. The mterl propertes of lyzed mterls re preseted Tble. Ths prt presets the results for the deformtos d stresses. Both re preseted grphclly d tbles. The verge percetge error (APE) ws clculted ccordg to equto ref MLPG ref ( ) ( ) APE = 00 u ( x ) u ( x ) / u ( x ) Where s totl umber of odes gve dom, u ref (x ) s the referece vlue ode x, u MLPG (x ) s vlue clculted by mes of MLPG ode x. A rectgulr plte wth two types of boudry codtos ws cosdered: ) clmped plte d b) smply supported plte.

7 Amerc Jourl of Mechcl Egeerg 467 Fgure 6. Plte dmesos d pot dstrbuto, whch the results were compred 5.. Alyss : M40J Fber Fgure 7 shows the deformto course through the pltes thcess the pot of mxmum deflecto of the plte mde from the M40J mterl wth vf04, the results re preseted Tble 4. Fgure 8. Stress dstrbuto the mddle of ech ply M40J vf04 ) σ, b) σ. Tble 5. Comprso of stresses σ d σ from MLPG d FEM, M40J vf04 Ply σ MLPG [P].8e+6.6e+6 5.4e+6 5.4e+6.6e+6.e+6 σ REF [P].5e+6.4e+6 5.0e+6 5.0e+6.4e+6.5e+6 err [%] σ MLPG [P] 4.45e+6.04e+6 0.6e+6 σ REF [P] 4.5e+6.9e+6 0.6e+6 0.6e+6 0.6e+6.04e+6.9e e 4 4.5e 4 err [%] Fgure 7. Deformto course through the pltes thcess. ) deformto ε, b) deformto ε. Tble 4. Comprso of deformtos ε d ε from MLPG d FEM, M40J vf04 Ply Alyss : SGlss Fber Fgure 9 shows the deformto course through the pltes thcess the pot of mxmum deflecto of the plte mde from the SGlss mterl wth vf04, the results re preseted Tble 6. ε MLPG [].6e4 0.97e4.4e5.e5 0.97e4.6e4 ε REF.58e4 0.95e4.e5.e5 0.95e4.58e4 err [%] ε MLPG.56e4.4e4 7.e5 7.e5.4e4.56e4 ε REF.65e4.9e4 7.e5 7.e5.9e4.65e4 err [%] Fgure 8 shows the stress course the mddle of ech ply the pot of mxmum deflecto of the plte. The results re preseted Tble 5. Fgure 9. Deformto course through the pltes thcess. ) deformto ε, b) deformto ε

8 468 Amerc Jourl of Mechcl Egeerg Tble 6. Comprso of deformtos ε d ε from MLPG d FEM, SGlss vf04 Ply ε MLPG [] 4.48e4.69e4 0.90e4 0.90e4.69e4 4.48e4 ε REF 4.7e4.6e4 0.87e4 0.87e4.6e4 4.7e4 err [%] ε MLPG 8.5e4 5.e4.70e4.70e4 5.e4 8.5e4 ε 8.7e4 4.96e4.65e4.65e4 4.96e4 8.7e4 err [%] Fgure 0 shows the stress course the mddle of ech ply the pot of mxmum deflecto of the plte. The results re preseted Tble 7. Tble 8. Comprso of deformtos ε d ε from MLPG d FEM, K49 vf04 Ply ε MLPG [].4e4.05e4 0.68e4 0.68e4.05e4.4e4 ε REF.5e4.0e4 0.67e4 0.67e4.0e4.5e4 err [%] ε MLPG 6.5e4.9e4.e4.e4.9e4 6.5e4 ε REF 6.7e4.99e4.e4.e4.99e4 6.7e4 err [%] Fgure 0. Stress dstrbuto the mddle of ech ply SGlss vf04 ) σ. b) σ Tble 7. Comprso of stresses σ d σ from MLPG d FEM, SGlss vf04 Ply σ MLPG [P] 7.4e+6.95e+6.6e+6 σ REF [P] 7.e+6.87e+6.5e+6.6e+6.5e+6.95e+6.87e+6 7.4e+6 7.e+6 err [%] σ MLPG [P] 9.74e+6 9.e+6.4e+6 σ REF [P] 9.45e e+6.8e+6.4e+6.8e+6 9.e+6 8.8e e e+6 err [%] Alyss : K49 Fber Fgure shows the stress course the mddle of ech ply the pot of mxmum deflecto of the plte. The results re preseted Tble 9. Fgure. Stress dstrbuto the mddle of ech ply K49vf04 ) σ, b) σ Tble 9. Comprso of stresses σ d σ from MLPG d FEM, K49 vf04 Ply σ MLPG [P] 6.e+6.5e+6 4.4e+6 σ REF [P] 6.08e+6.48e+6 4.6e+6 4.4e+6 4.6e+6.5e+6.48e+6 6.e e+6 err [%] σ MLPG [P] 7.9e+6.7e+6.4e+6.4e+6.7e+6 7.9e+6 σ REF [P] 8.04e+6.e+6.e+6.e+6.e e+6 err [%] Cocluso Fgure. Deformto course through the pltes thcess. ) deformto ε, b) deformto ε Ths pper ws prepred order to mplemet d pply "ew" umercl methods whch re used to solve problems mechcs, specfclly lyzg the stress codtos of composte plte structures. The preseted methods re very ttrctve but the theoretcl uderstdg of these methods d ther pplctos re

9 Amerc Jourl of Mechcl Egeerg 469 ot s extesve s the owledge of the fte elemet method. The meshless method s ew tool tht fter suffcet debuggg of the method prmeters (le tegrto, the tegrto of sze, sze of weght fucto) provdes ccurte results. Errors the vlues of str d stress my hve severl sources. The ccurcy of the MLPG method s flueced by severl fctors le roudg errors whe complg pproxmto d roudg errors cused by umercl tegrto. Metoed errors re reflected the dffereces betwee the referece vlues d clculted vlues. Both FEM progrms were clcultg deflecto the mddle lyer s the rthmetc verge of the deformto, or str, the lower d upper terfce of the plte. Acowledgemet The uthors grtefully cowledge for support the Slov d Techology Assstce Agecy regstered uder umber APVV06907, Slov Grt Agecy VEGA /6/. Refereces [] Dves, P., Rjpse, Yp, D.S., Durblty of Compostes Mre Evromet, Sprger Scece + Busess Med Dordrecht, 04. [] Povr, S., Kormov, E., Sttcl d Dymcl Alyss of Composte Sdwch Pltes, Bullet of the Trslv Uversty of Brsov, Seres I: Egeerg Sceces, 4 (5), o., 0. [] Zmd, M., Dudsy, M., Fte Elemet Implemetto of Flure d Dmge Smulto Composte Pltes, I: HU,., eds: Compostes d Ther Propertes, ITech, Rje, 0, 5. [4] Lu, G.R., Que,S.S., The Fte Elemet Method: Prctcl Course, Elsever Scece, Ltd., 00. [5] Sg, M., Kops, P., Vso, M., Some Computtol Aspects of Vehcle Shell Frmes Optmzto Subjected to Ftgue Lfe Commuctos, (4), 00, p [6] Atlur, S.., The Meshless Method, (MLPG) For Dom & BIE Dscretztos, Tech Scece Press, 004. [7] Slde, J., Slde, V., Krvce, J., We, P.; Zhg, Ch., Meshless Locl PetrovGler (MLPG) Method for Resser Mdl Pltes uder Dymc Lod Computer Meth. Appl. Mech. Eg., 96, 007, [8] Slde, J., Slde, V., Atlur, S.., Applcto of the Locl Boudry Itegrl Equto Method to Boudryvlue Poblems.,It. Appl. Mech., 8 (), 00, [9] Sores, D. Jr., Slde, J., Slde, V., Zmd, M., Medvecy, S : Porous Med Alyss by Modfed MLPG Formultos, CMC: Computers, Mterls,& Cotu, 7(), 0, pp [0] Zmd, M., Recy, D., Souup, J., Flure of Compostes wth Short Fbres. Commuctos, ( 4), 00, 9. [] Qu, Q.H., Yg., Q,S., McroMcro Theory o Multfeld Couplg Behvor of Heterogeous Mterls, Sprger, 008. [] M. Sejoh, J. Zem, Mcromechcs Prctce, WIT Press, Southmpto, UK, 0. [] Eshelby, J. D., The Determto of Elstc Feld of Ellpsodl Icluso d Relted Problems, I:Proc. R. Soc. Lodý, [4] Hll, R., A selfcosstet Mechcs of Composte Mterls, J. Mech. Phys. Solds,, 965,. [5] Chrstese, R. M., Lo. K. H. (979), Solutos for Effectve Sher Propertes Three Phse Sphere d Cylder Models, J. Mech. Phys. Solds, 7, 50. [6] orrs, A.., A Dfferetl Scheme for the Effectve Modul of Compostes, Mechcs of Mterls, 985, 6. [7] Mor, T. T, K., Averge Stress Mtrx d Averge Elstc Eergy of Mterls wth Msfttg Icluso. Act Mt.,, 97, [8] Hsh, Z., Shtrm, S., O Some Vrtol Prcples Asotropc d ohomogeeous Elstcty, J. Mech. Phys. Solds, vol. 0, 96, 54. [9] Besouss, A., Lo, J. L., Ppcolou, G., Asymptotc Alyss for Perodc Structures, Amsterdm Holld, 978 [0] Wlleberger, F. T., Bghm, P. A., Fberglss d Glss Techology, Sprger, 000. [] Reddy J.., Mechcs of Lmted Composte Pltes: Theory d Alyss, CRC Press, Boc Rto, 997.