Analysis of Rectangular Stiffened Plates Based on FSDT and Meshless Collocation Method

Transcription

1 Joural of Solid Mechaic Vol. 9, No. 3 (07) Aalyi of Rectagular Stiffeed Plate Baed o FSD ad Mehle Collocatio Method Sh. Hoeii, B. Soltai * Faculty of Mechaical Egieerig, Uiverity of Kaha, Kaha, Ira Received 0 Jue 07; acceted 8 Augut 07 ABSRAC I thi aer, bedig aalyi of cocetric ad eccetric beam tiffeed quare ad rectagular late uig the mehle collocatio method ha bee ivetigated. For detectig the goverig equatio of late ad beam, Midli late theory ad imoheko beam theory have bee ued, reectively, with the tiffe matrice of the late ad the beam obtaied earately. he tiffe matrice of the late ad the beam were combied together uig traformatio equatio to obtai a total tiffe matrix. Beig ideedet of the meh alog with it imler imlemetatio roce, comared to the other umerical method, the mehle collocatio method wa ued for aalyzig the beam tiffeed late. I order to roduce mehle hae fuctio, radial oit iterolatio method wa ued where momet matrix igularity roblem of the olyomial iterolatio method wa fixed. Alo, the Multiquadric radial bai fuctio wa ued for oit iterolatio. Ued to have olutio of icreaed accuracy ad tability were olyomial with the radial bai fuctio. Several examle are reeted to demotrate the accuracy of the method ued to aalyze tiffeed late with the accuracy of the reult howig accetable accuracy that the emloyed method i aalyzig cocetric ad eccetric beam tiffeed quare ad rectagular late. 07 IAU, Arak Brach. All right reerved. Keyword: Beam tiffeed late; Cocetric ad eccetric tiffeer; Mehle collocatio method; Radial oit iterolatio. INRODUCION S IFFENED late are widely ued i differet tructure. High tregth to weight ratio ha made tiffeed late exteively ued acro uch idutrial alicatio a aeroace tructure, road bridge, hi hull, etc. Exteive reearche have bee doe for the aalyi of tiffeed late. he earliet model for the aalyi of tiffeed late iclude grillage model [] ad orthotroic model []. Defiig a equivalet thicke of tiffeed late, thee model uffered from iadequate accuracy ad efficiecy. herefore, ew model are develoed where the late ad tiffeer are coidered a earate tructure with comatibility equatio itroduced to coider iteractio betwee the late ad tiffeer ([3], [4] ad [5]). Differet method are ued to aalyze tiffeed late, uch a Rayleigh-Ritz method [6], fiite differece method [7], fiite elemet method ([5], [8], ad [9]), cotrai method [4], fiite tri method [0], ad emi aalytical fiite differece method []. We et al. [] ued boudary elemet method to aalyze tiffeed late. hey ued couled boudary elemet formulatio of hear deformable late ad two dimeioal lae tre * Correodig author. el.: ; Fax: addre: boltai@kahau.ac.ir (B.Soltai). 07 IAU, Arak Brach. All right reerved.

2 Sh. Hoeii ad B. Soltai 569 elaticity. It i eceary to ue umerical method itead of aalytical aroache for uch comlex egieerig tructure a tiffeed late. Amog other umerical method, fiite elemet method wa take ito coideratio due to it tability ad efficiecy. However, oe of the mai roblem aociated with thi method i it reliace o meh. herefore, the idea of elimiatig the mehe where the olutio are jut deedet o the ode, caued ocalled mehle or meh-free method to emerge ([3] ad [4]). Peg et al. [3] ivetigated a rectagular tiffeed late baed o elemet-free Galerki method. Memar Ardetai et al. [5] tudied a FGM tiffeed late baed o reroducig kerel article method. hey tudied the tability of RKPM for the aalyi of tiffeed late. Oe of the ueful mehle method i a mehle collocatio method baed o radial bai fuctio. With o backgroud mehe required to be itegrated ito the mehle collocatio method, it i truly mehle. Furthermore, it i imler to formulate ad imlemet comared to other mehle method uch a MLPG or EFG. Recetly, radial bai fuctio have ejoyed coiderable ucce ad reearch a a techique for iterolatig data ad fuctio. A radial bai fuctio ca be ee a a lie that deed o the Euclidia ditace betwee ditict data ceter alo called odal or collocatio oit. hee fuctio reolve the igularity matrix i the olyomial oit iterolatio method. Alo, Kroecker delta fuctio roertie imly that boudary coditio ca be imoed with o ecial method required. Kaa [6] itroduced the cocet of olvig PDE by a uymmetric RBF collocatio method baed uo Multiquadric iterolatio fuctio. Hardy [7] itroduced multiquadric for the aalyi of cattered geograhical data. I 990, Kaa [6] ued multiquadric to olve artial differetial equatio. he RBF are ieitive to atial dimeio, makig the imlemetatio of the method much eaier tha that of, e.g., fiite elemet. A imortat feature of RBF method i that it doe ot require a meh. he oly geometric roerty eeded i a RBF aroximatio i the ditace betwee the two oit. Workig with higher dimeioal roblem i ot difficult a ditace are eay to comute i ace of ay umber of dimeio. Oe of the mai roblem of collocatio method i rereeted by it itability i the reece of derivative boudary coditio which may ot oly caue ome itabilitie i the olutio of the roblem, but may alo lead to lower accuracy ad eve wrog reult i ome cae. here are, however, may way to tabilize roblem with derivative boudary coditio. Uig Hermite-tye collocatio method, uig fictitiou oit outide the roblem domai, uig regular grid withi the roblem domai alog the derivative boudary, uig mehle weak-trog form ad the ue of dee ode ear the derivative boudary are ome of uch way, to ame a few. Uig dee ode ear the derivative boudary igificatly icreae the accuracy ad tability of the reult [3]. I thi method, the more umber of ode are ued acro area ear the derivative boudary coditio. Solid oit i Fig. how uiform ode ditributio i a arbitrary domai with the cro ig howig additioal oit itroduced to icreae the accuracy ad tability of the reult. Additioal ode Field ode Fig. Uig comact ode i regio ear to the derivative boudarie for icreaig the tability ad accuracy of the roblem. I thi aer, the deflectio of tiffeed quare ad tiffeed rectagular late are tudied uig the collocatio method baed o radial bai fuctio. Poit iterolatio are erformed via a multiquadric iterolatio fuctio. he boudary coditio are directly alied becaue of Kroecker delta fuctio roerty of the hae fuctio. he iteractio betwee the late ad tiffeer are imoed through the comatibility equatio i which deflectio of tiffeer i coidered to be equal to that of the late alog the lie of iteractio. 07 IAU, Arak Brach

3 570 Aalyi of Rectagular Stiffeed Plate Baed MESHLESS COLLOCAION FORMULAION BASED ON RPIM. Radial oit iterolatio formulatio Radial bai aroximatio fuctio are mehle umerical cheme that ca exloit accurate rereetatio of the boudary, are eay to imlemet ad ca be ectrally accurate. I thi aer, the formulatio of a global uymmetrical collocatio RBF-baed method to comute ellitic oerator i reeted. Coider a liear ellitic artial differetial oerator L ad a bouded regio i R withi ome boudary.. I a tatic roblem, oe may eek the comutatio of dilacemet u from a global ytem of equatio. Lu f i (a) LBu g i (b) where L ad L B are liear oerator i the domai ad o the boudary, reectively. he right-had ide of Eq. () rereet the exteral force alied i the iteral domai ad the boudary coditio alied alog the erimeter of the roblem, reectively (Fig. ). he radial bai fuctio aroximatio of a fuctio u i m () u ( x ) R ( x ) a ( x ) b R ( x ) a ( x ) b i i j j i j where Ri ( x ) i a radial bai fuctio, i the umber of RBF, j ( x ) i moomial i the ace coordiate x x, y, ad m i the umber of olyomial bai fuctio. Whe m=0, ure RBF are ued. Otherwie, the RBF i augmeted with m olyomial bai fuctio. Coefficiet a i ad b j are cotat. I the radial bai fuctio Ri ( x ), the variable are hae arameter c ad the ditace betwee the oit of iteret x ad a ode at x i, that defied a: r x x i for -D roblem (3a) r ( x x i ) ( y y i ) for -D roblem (3b) he mot commo RBF are: 3 Cubic: ( r) r hi late lie: Gauia: Multiquadric: ( r) e ( r) r log( r) ( cr ) ( r) c r Ivere Multiquadric: ( r) ( c r ) where the Euclidia ditace r i real ad o-egative. 07 IAU, Arak Brach

4 Sh. Hoeii ad B. Soltai 57 Fig. Node ditributio o ier ad boudary regio of global domai. RPIM hae fuctio with ure RBF uually caot a the tadard atch tet. Addig olyomial term u to the liear order ca eure the C coitecy that i eeded to a the tadard atch tet. I geeral, addig olyomial ca alway imrove the accuracy of the reult while reducig the eitivity of the hae arameter, o that it will rovide the uer with much more freedom ad a wider rage of chooig hae arameter [9]. Coefficiet ai ad b j i Eq. () ca be determied by eforcig Eq. () to be atified at all ode urroudig the oit of iteret x. hi lead to liear equatio, oe for each ode. he matrix form of thee equatio ca be exreed a follow U R a P b 0 m (4a) where the vector of fuctio value U i U u, u,..., u (4b) he momet matrix of RBF i R( r ) R( r ) R ( r ) R( r ) R( r ) R ( r ) R0 R( r ) R ( r ) R ( r ) ( ) (4c) he olyomial momet matrix i Pm x y m ( x) x y m ( x ) x y m ( x ) ( m) (4d) he vector of coefficiet of RBF i a a, a,..., a (4e) ad the vector of coefficiet for olyomial i b b, b,..., b m (4f) here are +m variable i Eq. (4a). he additioal m equatio ca be added uig the followig m cotrait coditio. 07 IAU, Arak Brach

5 57 Aalyi of Rectagular Stiffeed Plate Baed j ( x i ) ai Pm a 0 (5) i By combiig Eq. (4a) ad Eq. (5), yield the followig et of equatio i the matrix form U where U R0 P m a G a 0 P m 0 b 0 (6a) a a, a,..., a, b, b,..., b 0 U u, u,..., u,0,...,0 m (6b) (6c) By combiig Eq. (6a) ad Eq. (4a) ad ome imlificatio, RPIM hae fuctio ca be exreed a: R ( x ) ( x ) G ( x ), ( x ),..., ( x ), ( x ),..., m ( x ) (7) Fially, the RPIM hae fuctio correodig to the odal dilacemet vector ( x ) are obtaied a: ( x ) ( x ), ( x ),..., ( x ) (8) Eq. () ca be rewritte a: u ( x ) ( x ) U i ui (9) i he derivative of u(x) are obtaied a: u x x U, l ( ), l ( ) (0) where l deote either the coordiate x or y. he choice of hae arameter directly ifluece the accuracy of reult. Hardy [7] ugget the ue of N c 0.85d, where d di N, ad d i i the ditace from the data oit x i to it earet eighbor. Frake [8], i.5d o the other had, recommed c, where D i the diameter of the mallet circle cotaiig all data oit. N Fahauer [9] argue the ue of c, where N a i the umber of collocatio oit i either x- or y- directio. N a a Ferreira et al. [0] ugget c for late with 5 N a h. I thi aer, we ued the Multiquadric radial bai fuctio to iterolate cattered oit. he hae arameter D ha bee choe to be c for late with uit legth of ide ad c for late of o-idetical legth N a N of the ide, where α i a uer defied arameter that i differet i differet roblem. 07 IAU, Arak Brach

6 Sh. Hoeii ad B. Soltai 573. Stiffe matrix he global ytem of equatio metioed i Eq. (). By ubtitutig Eq. (9) ito Eq. (), reult L( U ) f (a) L ( U ) g (b) B By uttig Eq. () i a matrix form, the global tiffe matrix i formed K N N U N FN (a) where tiffe matrix, K, i K L L L L ( ( )),..., ( ( )), ( ( )),..., ( ( )) (b) NI B NI B NB Nodal value vector, U, i,,..., NI, NI,..., (c) NB U u u u u u ad the global ource vector F coit of F f f f g g ( ), ( ),..., ( ), ( ),..., ( ) (d) NI NI NB he Eq. (a) ca be olved by uig commo method for olvig the ytem of equatio. 3 COMPAIBILIY EQUAIONS raformatio equatio were ued to combie the tiffe matrix of late ad beam. he tiffeer i coidered to be attached to the lower ide of the late. he mehle model of a tiffeed late i how i Fig.. Some of the ode are ubcriber betwee late ad tiffeer, o that the comatibility coditio of the tiffeed late ca be writte a follow: w w z h z h (3a) x x z h z h (3b) y y z h z h (3c) where h i the thicke of the late ad h i the deth of the x-tiffeer. Subcrit ad are geeric otatio which refer to x- or y- tiffeer ad late, reectively. By uig RPIM hae fuctio, Eq. (3) ca be reformulated a: 07 IAU, Arak Brach

7 574 Aalyi of Rectagular Stiffeed Plate Baed 07 IAU, Arak Brach ( ) 0 0 ( ) ( ) 0 0 ( ) 0, (,..., ) 0 0 ( ) 0 0 ( ) i j j j i j i j i xj j xj j j j i yj i j yj w w i (4) where ad are the umber of ode o the late domai ad tiffeer domai, reectively. he matrix form of Eq. (4) exreed a: (5a) where,, ad defied a follow (3 3 ) ( ) 0 0 ( ) ( ) 0 0 ( ) ( ) 0 0 ( ), ( ) 0 0 ( ) ( ) 0 0 ( ) ( ) 0 0 ( ) x y w w (3 ) x y (5b) (3 3 ) ( ) 0 0 ( ) ( ) 0 0 ( ) ( ) 0 0 ( ), ( ) 0 0 ( ) ( ) 0 0 ( ) ( ) 0 0 ( ) x y x w w (3 ) y (5c) Fig.3 Node ditributio i tiffeed late.

8 Sh. Hoeii ad B. Soltai 575 I geeral, the traformatio equatio ca be obtaied a: ( ) (6) For x- ad y-tiffeer, Eq. (6) ca be reformulated a: x x x ( ) I (7a) y y y ( ) I (7b) where x I y, I (3 3 ) (3 3 ) x x y y (7c) where x ad y are the umber of ode o x- ad y-tiffeer, reectively. Fially, by uig the traformatio equatio for beam tiffeed late, total tiffe matrix defied a: N N x y x x i x x y y i y y i x i i y i i i (8) K K (( ) I ) K (( ) I ) (( ) I ) K (( ) I ) where N x ad N y deote the umber of tiffeer i x- ad y-directio, reectively. 4 GOVERNING EQUAIONS OF SIFFENED PLAE 4. Goverig equatio of imoheko beam Dilacemet field for imoheko beam i x u u0 z (9a) w w 0 (9b) I-lae dilacemet of the beam (u 0 ) i very mall ad it ca be eglected. he trai field defied a: x u x z x x (0a) xz w u w 0 x x z x (0b) he virtual trai eergy with hear trai virtual eergy i 07 IAU, Arak Brach

9 576 Aalyi of Rectagular Stiffeed Plate Baed L L x x x x U ( xx z, x xz [ w 0, x ]) da dx [ M xx, xx Qx ( w 0, x ] dx 0 A 0 () where xx, xz, M xx ad Qx are ormal tre, hear tre, bedig momet ad hear force, reectively ad defied a: M xx z xx da Q A da x xz A Virtual otetial eergy from ˆq load, i L V qˆ( x ) w 0 dx 0 (a) (b) (3) By ubtutig virtual trai eergy ad virtual otetial eergy i virtual work ricile ad earatig the x factor of w 0 ad, equilibrium equatio obtaied a: Mxx, x Qx 0 (4a) Qxx, qˆ (4b) By combiig bedig momet, hear force ad equilibrium equatio, goverig equatio of the x-tiffeer obtaied a: x Axz (, x w 0, xx ) qˆ ( x ) (5a) x x Dxx, xx K Axz ( w 0, x ) 0 (5b) where K i hear correctio factor ad Axz ad Dxx defied a: A G da G A xz xz xz A D E z da E I xx A yy (6a) (6b) Boudary coditio for imoheko beam are x w 0 0, 0 Clamed (7a) w0 0, M xx 0 Simly Suorted (7b) Q x 0, M 0 Free (7c) xx Similarly, goverig equatio of y-tiffeer are y yz (, y 0, yy ) ˆ( ) (8a) A w q y 07 IAU, Arak Brach

10 Sh. Hoeii ad B. Soltai 577 where y y yy, yy yz ( 0, y ) 0 (8b) D K A w Ayz ad D yy are A G da G A yz yz yz A D E z da E I yy A xx (9a) (9b) 4. Goverig equatio of Midli late Dilacemet field for Midli late i x 0 (30a) u u z y 0 (30b) v v z w w 0 (30c) x y where ad deote the rotatio aroud y- ad x- axi, reectively. I-lae dilacemet (u 0 ad v 0 ) are very mall ad egligible. Liear trai comoet are xx u x z x x (3a) yy v y z y y (3b) xy x y u v z ( ) y x y x (3c) xz yz w u w 0 x x z x w v w 0 y y z y (3d) (3e) Equilibrium equatio of Midli late obtaied from virtual work ricile ( W U V 0 ). Virtual trai eergy ad virtual otetial eergy are h ( xx xx yy yy xy xy xz xz yz yz ) U dz dx dy A h (3a) 07 IAU, Arak Brach

11 578 Aalyi of Rectagular Stiffeed Plate Baed V q( x, y ) w 0 dx dy A (3b) Fially the equilibrium equatio obtaied by earatig the coefficiet of 0, x y w ad. Qx, x Qy, y q( x, y ) (33a) M M Q xx, x xy, y x 0 (33b) M M Q yy, y xy, x y 0 (33c) where the reultat momet ad force are h xx yy xy xx yy xy h (34a) [ M, M, M ] [,, ] z dz h x y xz yz h (34b) [ Q, Q ] K [, ] dz Goverig equatio of Midli late i term of dilacemet, obtaied a: K Eh ( w x y 0, xx w 0, yy, x, y ) q ( x, y ) ( ) D( ) x x D( ) x y K Eh x (, xx, yy ) (, xx, yx ) ( w 0, x ) 0 ( ) D( ) y y D( ) x y K Eh y (, xx, yy ) (, yx, yy ) ( w 0, y ) 0 ( ) (35a) (35b) (35c) where 3 Eh D, i bedig tiffe. ( ) 5 NUMERICAL RESULS AND DISCUSSION I thi ectio everal umerical examle are give to demotrate the ability of the mehle RPIM collocatio method i the aalyi of beam tiffeed late. 5. Simly uorted quare late with oe rectagular tiffeer I the firt tudy a imly uorted quare late with oe tiffeer of rectagular cro ectio alog x-directio (y=0.5 i.) i coidered a how i Fig. 4. A uiformly ditributed load of.0 i i alied to the to urface of the late. Elatic roertie of the late ad tiffeer are aumed to be E.7e 7 i ad IAU, Arak Brach

12 Sh. Hoeii ad B. Soltai 579 Fig.4 Square late with oe tiffeer alog y=0.5 i. hi examle ha bee olved by McBea [8] ad Roow [4] uig the FEM ad become a bechmark roblem i the field of tiffeed late. 5 5 ode are coidered to dicretize the late ad 5 ode are ued for x- tiffeer. he coidered hae arameter are c for late ad c 7.7 e( 3) ad c 5.8 e( 3) for 5 cocetric ad eccetric tiffeer, reectively. he aalyi reult of cocetric ad eccetric tiffeed late are how i Fig. 5 ad 6, reectively, where they are alo comared to the reult of Peg et al. [3], Roow [4], McBea [8] ad Memar Ardetai et al. [5]. Alo, able. reort the reult of the ceter deflectio of the tiffeed late comared to thoe give by Peg et al. [3], Roow [4], McBea [8] ad Memar Ardetai et al. [5]. (a) (b) Fig.5 Deflectio of cocetrically quare tiffeed late with imly uorted boudary coditio alog (a)x=0.5 i. ad (b) y=0.5i. (a) (b) Fig.6 Deflectio of eccetrically quare tiffeed late with imly uorted boudary coditio alog (a) x=0.5 i. ad (b) y=0.5i. 07 IAU, Arak Brach

13 580 Aalyi of Rectagular Stiffeed Plate Baed able Ceter deflectio of quare tiffeed late for cocetric ad eccetric cofiguratio. Ref. Cocetric RD * (%) Eccetric RD (%) Peg et al [3] 4.845e e Roow [4] 4.556e e McBea [8] 4.557e e Memar Ardetai et al. [5] 4.750e e-4.59 Preet e e-4 *Relative Differece Due to the reult of the able, maximum ad miimum relative differece are 6.53%, with that reult of Roow [4] ad 0.80% with that reult of Peg et al. [3], reectively. he earet referece from the aect of aumtio i the formulatio of the roblem ad umerical method are Ref. [3] ad Ref. [5]. he relative differece betwee the reet reult ad thee referece are i the accetable rage. Alo, due to Fig. 5 ad 6, the reult for the other ode have accetably accurate comared to the referece. Fig. 7 how correodig deflectio cotour to the tiffeed late by oe cocetric rectagular beam. Fig.7 Deflectio cotour of tiffeed late by oe cocetric rectagular beam. 5. Simly uorted ad clamed quare late with oe rectagular tiffeer hi examle i the ame a before, excet that it ha chaged boudary coditio. Edge alog y ad x-axi coidered to be clamed ad imly uorted, reectively. Shae arameter are coidered ame a before. Fig. 8 how the dimeio of the late ad tiffeer ad the oitio of boudary coditio. Fig.8 Square late with oe tiffeer alog y=0.5 i. ad variou boudary coditio. he reult are comared to thoe reult of Abaqu aalyi. he umber of elemet ha bee coidered a 4 4 (5 5 ode). he elemet tye of the late ad tiffeer are S4R ad B3, reectively. he reult of eccetric tiffeed late with imly uorted ad clamed edge are how i Fig. 9, where they are alo comared to thoe reult of Abaqu. By icreaig the umber of ode i the area cloe to the clamed edge, the accuracy of the reult will be icreaed. he reult of Fig. 9 how that the MCM ha a accetably accurate i variou boudary coditio. 07 IAU, Arak Brach

14 Sh. Hoeii ad B. Soltai 58 (a) (b) Fig.9 Deflectio of eccetrically quare tiffeed late with imly uorted ad clamed boudary coditio alog (a) y=0.5 i. ad (b) x=0.5 i. 5.3 Simly uorted quare late with oe I-hae tiffeer hi roblem i imilar to the roblem i Sectio 5., excet for the fact that cro ectio of beam i I-hae ad the beam i cocetric with the late. Alo, the thicke of the late wa coidered to be 0. i. Fig. 0 how the dimeio of the late ad the tiffeer. Fig.0 Square late with oe I-hae tiffeer alog y=0.5 i. For olvig thi roblem, 5 5 ode are coidered ad hae arameter for late, c, ad for 5 tiffeer, c 7.7 e( 3), have bee iteded. he reult are comared to thoe reult of Peg et al. [3] ad are how i Fig.. Due to the Fig., bedig of the tiffeed late i ymmetric ad the maximum deflectio i haeig i the ceter of the late, becaue the thicke of the late i more tha that of the reviou roblem; therefore the tiffe matrix i icreaig or late get harder. 5.4 Simly uorted quare late with oe -hae tiffeer I thi roblem, elatic roertie of the late ad the tiffeer are ame a thoe i the roblem reeted i Sectio 5., excet for the fact that the beam i coidered to be cocetric, with a -hae cro ectio. For the ake of olvig thi roblem, 5 5 ode were coidered ad the hae arameter were choe to be c for the late ad c 7.7 e( 3) for the tiffeer IAU, Arak Brach

15 58 Aalyi of Rectagular Stiffeed Plate Baed (b) (a) Fig. Deflectio of cocetrically quare I-hae tiffeed late with imly uorted boudary coditio alog (a) x=0.5 i. ad (b) y=0.5 i. Fig. Square late with oe -hae tiffeer alog y=0.5 i. Fig. 3 comare the umerical reult of the reet aer to thoe of Peg et al. [3]. Similar to the reviou examle, bedig of the tiffeed late i ymmetric ad the maximum deflectio i haeig i the ceter of the late. Alo, due to Fig. 3 the reult are correodig to thoe of Peg et al. [3]. 5.5 Simly uorted rectagular late cetrally tiffeed by two tiffeer A imly uorted rectagular late cetrally tiffeed by two tiffeer i ubjected to a uiformly ditributed load of 0.0 i. Dimeio of the late ad the tiffeer are how i Fig. 4. he late ad the tiffeer are made of the ame material, with the elatic modulu of E 37 e i ad Poio ratio of 0.3. Cocetric ad eccetric tiffeer were coidered. hi roblem ha bee olved by Roow et al. [4] (uig FEM) ad Chag [] (uig FEM ad aalytical olutio). Becaue i thi cae, the ditace betwee the ode are idetical, 9 ode were coidered to olve thi roblem. he hae arameter for the late, x-tiffeer,.5d ad y-tiffeer were take a c, cx.4 ad cy.5, reectively. N he deflectio of cocetric ad eccetric tiffeed late at x = 7.5 i. ad x = 5 i. are how i Fig. 5 ad 6, reectively. he deflectio cotour of the cocetric tiffeed late by the two cocetrically tiffeer are how i Fig. 7. A ca be ee i Fig. 5 to 7, maximum deflectio i ot haeig i the ceter of the late ad the area of the late i divided ito four art. Fig. 8 demotrate the correodig momet diagram for thi roblem at y = 5 i. for the cocetric ad eccetric tiffeer. 07 IAU, Arak Brach

16 Sh. Hoeii ad B. Soltai 583 (b) (a) Fig.3 Deflectio of cocetrically quare -hae tiffeed late with imly uorted boudary coditio alog (a) x=0.5 i. ad (b) y=0.5 i. Fig.4 Simly uorted rectagular late cetrally tiffeed by two tiffeer. Mior differece i the deflectio of the rectagular tiffeed late were oberved betwee the rooed method i the reet aer ad referece, due to differet theoretical bae ad aumtio. he mai target i to determie the deflectio of the ode o the boudarie ad o the tiffeer. A ca be ee i Fig. 5 ad 6, zero deflectio i determied at boudarie with the deflectio of the ode o the tiffeer ejoyig accetable accuracy. Similar to the reviou cae, differece i momet tem from differece i the aumtio ad olutio method. 5.5 Simly uorted quare late orthogoally tiffeed by equiditat tiffeer A imly uorted quare late of ide legth.0m ad thicke 0.0m i orthogoally tiffeed with equally aced tiffeer of rectagular ectio (deth 0.m, width 0.0m). he late i ubjected to a uiformly ditributed load of q=6 t/m. he umber of the tiffeer are 4+4, +, 0+0, 8+8, 6+6 ad 4+4. A quare late orthogoally tiffeed by 4+4 tiffeer ha bee how i Fig. 9. he late ad tiffeer are made of the ame material, with the elatic modulu of E=.e7 t/m ad the Poio ratio of ν=0.3. he umber of ode for each roblem have bee choe to be (N +4) (N +4) (with N deotig the umber of tiffeer i x- or y-directio), excet for the 4+4 tiffeer, a the roblem fail to rovide tability with (8 8) ode at which ode cofiguratio the reult reder iaccurate. Coequetly, the umber of ode for 4+4 tiffeer wa choe to be (3 3). D he hae arameter wa take a c for the late; however, regardig variou umber of ode, the N 3 hae arameter wa choe to be c for the tiffeer. 07 IAU, Arak Brach

17 584 Aalyi of Rectagular Stiffeed Plate Baed (b) (a) Fig.5 Deflectio of cocetrically rectagular tiffeed late with imly uorted boudary coditio alog (a) x=7.5 i. ad (b) y=7.5 i. (a) (b) Fig.6 Deflectio of eccetrically rectagular tiffeed late with imly uorted boudary coditio alog (a) x=7.5 i. ad (b) y=7.5 i. Fig.7 Deflectio cotour of the cocetric tiffeed late by two beam i x- ad y- directio. 07 IAU, Arak Brach

18 Sh. Hoeii ad B. Soltai 585 (a) (b) Fig.8 Diagram of momet of rectagular tiffeed by two tiffeer i y=5 i. for (a) cocetric ad (b) eccetric tiffeer. Fig.9 A quare late orthogoally tiffeed by four tiffeer i each directio. able. reort the reult of the maximum deflectio of the tiffeed late comared to thoe give by Deb ad Booto [], Biwal ad Ghoh [3], Sadek ad awfik [5] ad Peg et al. [3]. he table demotrate the maximum deflectio reductio by icreaig the umber of tiffeer. he differece betwee the referece come from differet aumtio ad formulatio ued to olve the roblem. I term of the aumtio take, Ref. [3] rereet the cloet referece to the reet reearch. he miimum ad maximum relative differece betwee the reet reult ad thoe reult of referece [3] are 0.490% ad %, reectively where, they are i a accetably rage. able Maximum deflectio of tiffeed late by equal ditace beam i two directio (m 0 3 ). Number of tiffeer i two directio FEM Ref. [] Orthotroic Ref. [] Ref. [3] Ref. [5] Ref. [3] Preet reult CONCLUSIONS I thi tudy, a mehle collocatio method wa ued to aalyze a imly uorted rectagular tiffeed late baed o FSD. Radial oit iterolatio method baed o multiquadric fuctio wa ued for roducig hae fuctio. wo cae of cocetric ad eccetric tiffeer were coidered. he total otetial eergy ricile wa ued to derive goverig equatio of the tiffeed late. 07 IAU, Arak Brach

19 586 Aalyi of Rectagular Stiffeed Plate Baed Same a the referece, the reult how that the maximum deflectio of thier tiffeed late i ot haeig i the ceter of the late ad for thicker tiffeed late, it hae i the ceter of the late. Ulike FEM, uig meh-free RPIM method, o meh geeratio i eeded with the oitio of the tiffeer beig arbitrary. Alo, meh-free RPIM method i le comlex ad truly mehle a it doe ot eed ay itegratio. herefore, the CPUru-time of thi method i igificatly le tha the other method. Due to elimiatig of the itegratio, the error of the umerical itegratio i removed ad the accuracy of the reult i icreaed. Ejoyig accetable accuracy, the reult roved thi method to be a efficiet aroach for tiffeed late. REFERENCES [] Kedrick S., 995, he aalyi of a flat lated grillage, Euroea Shibuildig 5: 4-0. [] Schade H., 940, he orthogoally tiffeed late uder uiform lateral load, Joural of Alied Mechaic ASME 6: [3] Peg L., Kitiorchai S., Liew K., 005, Aalyi of rectagular tiffeed late uder uiform lateral load baed o FSD ad elemet-free Galerki method, Iteratioal Joural of Mechaical Sciece 47(): [4] Roow M., Ibrahimkhail A., 978, Cotrait method aalyi of tiffeed late, Comuter ad Structure 8(): [5] Sadek E. A., awfik S. A., 000, A fiite elemet model for the aalyi of tiffeed lamiated late, Comuter ad Structure 75(4): [6] Liew K. M., Lam K. Y., Chow S.., 990, Free vibratio aalyi of rectagular late uig orthogoal late fuctio, Comuter ad Structure 34(): [7] Aku G., Ali R., 976, Free vibratio aalyi of tiffeed late uig fiite differece method, Joural of Soud ad Vibratio 48(): 5-5. [8] McBea R., 968, Aalyi of Stiffeed Plate by the Fiite Elemet Method, hei, Staford Uiverity. [9] Nguye-hoi., Bui-ua., Phug-Va P., Nguye-ua H., Ngo-hah P., 03, Static, free vibratio ad bucklig aalye of tiffeed late by CS-FEM-DSG3 uig triagular elemet, Comuter ad Structure 5: [0] Azizia Z., Dawe D., 985, he aalytical tri method of olutio for tiffeed rectagular late uig fiite tri method, Comuter ad tructure (3): [] Mukhoadhyay M., 989, Vibratio ad tability of aalyi of tiffeed late by emi-aalytic fiite differece method, Part II: Coideratio of bedig ad axial dilacemet, Joural of Soud ad Vibratio 30: [] We P., Aliabadi M., Youg A., 00, Boudary elemet aalyi of hear deformable tiffeed late, Egieerig Aalyi with Boudary Elemet 6(6): [3] Liu G., 005, A Itroductio to Mehfree Method ad heir Programmig, Sriger. [4] Liu G., 009, Meh Free Method: Movig Beyod the Fiite Elemet Method, CRC Pre. [5] Ardetai M. M., Soltai B., Sham S., 04, Aalyi of fuctioally graded tiffeed late baed o FSD utilizig reroducig kerel article method, Comoite Structure : [6] Kaa E. J., 990, Multiquadric-a cattered data aroximatio cheme with alicatio to comutatioal fluiddyamic-i, Comuter ad Mathematic with Alicatio 9(8): [7] Hardy R. L., 97, Multiquadric equatio of toograhy ad other irregular urface, Joural of Geohyical Reearch 78(8): [8] Frake R., 98, Scattered data iterolatio:tet of ome method, Mathematic of Comutatio 38(57): [9] Fahauer G., 997, Solvig artial differetial equatio by collocatio with radial bai fuctio, Proceedig of the 3rd Iteratioal Coferece o Curve ad Surface, Surface Fittig ad Multireolutio Method. [0] Ferreiraa A. J. M., Batrab R. C., Roquea C. M. C., Qiac L. F., Marti P. A. L. S., 005, Static aalyi of fuctioally graded late uig third-order hear deformatio theory ad a mehle method, Comoite Structure 69(4): [] Chag S., 973, Aalyi of Eccetrically Stiffeed Plate, hei, Uiverity of Miouri, Columbia. [] Deb A., Booto M., 988, Fiite elemet model for tiffeed late uder travere loadig, Comuter ad Structure 8(3): [3] Biwal K. C., Ghoh A. K., 994, Fiite elemet aalyi for tiffeed lamiated late uig higher order hear deformatio theory, Comuter ad Structure 53(): IAU, Arak Brach