89-8 "$ 95 677 ;; =<5 (MLPG) 96:; + +, ( ) *$ % &'! " + +- +- &'! " - :+ > *? *5 A < ;! $ : + 9 8-7% "6 5 + + "- %! /* " 8-57-65/-((-/-(- ),-)*+'"( B + % - &A9 - + 9 %?, @ " :? > ; - H : 9 ; -&! @ FG - D E C! 9 %?, + 9 8- L (MLPG) ; 5 - + > % B * ") 5 DI 9 8- "6 + ; 8 * > 5H + +' 8? W V % @ - + - ' UE & > ; +S MLS - - R H< P? ) L MLS - R % - DE/ P Q & "- NO) M > - < ) : S * * > 5H % F - 9< 5 + > 5 * % H* X /* > ") Z 5 DE/P Q? * 9Y : ; - + >:)6)=B- Analysis of hin Isotropic and Orthotropic Plates Using Meshless Local Petrov- Galerkin Method With Various Geometric Shapes H Edalati B Soltani Department of Mechanical Engineering, Islamic Azad University, Jasb branch, Delijan, Iran Department of Mechanical Engineering, University of Kashan, Kashan, Iran Abstract Utilizing one of the mesh free methods, the present paper concerns static analysis of thin plates with various geometric shapes based on the mindlin classical plate theories In this numerical method, the domain of issue is solely expressed through a set of nodes and no mesh is required o express the domain of issues with various geometric shapes, first a set of nodes are defined in a standard rectangular domain, then via a map, these nodes are transferred to the main domain of the original issue herefore plates of various geometric shapes can be analyzed he meshless local petro galrkin method (MLPG) is utilized here he method is one of the weak form integral methods that uses MLS shape functions for approximation Regarding the absence of Delta feature in MLS functions, because of in MLPG, the system equation is constructed node by node therefore direct interpolation method can be used to enforce essential boundary conditions Outputs are compared with those of analytic and finite element methods, in order to validate the solution method, a few new examples will be solved Keywords : Mesh free, Plate s theory, Weak form integral, stress analysis B * $' (b "' 5 ; 6 (c : - e B) B % fe * 5 ( C - % - $' ; H ' (- go : ) "- $' ; * < " h - 5 + > i$< - B 5 + + + "- " % : 9 %?, " "- ; - * L UE * ") - R% : D 5 + > E? "- Q ' ; " @5! 5 + > [] 95 8! " - W S " A Q E >! 5 + 9 V W9 -& - [] " C< a C< B * 6 - > ; 5 + > % O Wa? - U O9 (D :%? 9 O "LE/ Difficulty in adaptive analysis mesh free or mesh less Finite element method * bsoltani@kashanuacir :
"6 [9]7% "6 Z [8]D"- Y - [] Q 6 L %G - FGM "6 RKPM > % Z9 C "6 O [] 5 '< 9 5 P, Z9 7$ @ MLPG >! O P, 5 R) Z9 []- []< - + "- % -!! MLPG - + * > % ; L 9I 9 9 Y B 5 ") -& 5 9 * X * > 6 + +' - - 5 + H< P? ) MLPG * > V UE & ' ( R % L P&9 8? W V % @ - + e! @Q 9 "6! $ : - + > % n - + M? 5 "),) % @ - % ; ; 8?! +O9 L MLS - L &9 MLS ' F5( ( R % ; - + > - B 99 8 " % E V ( R ; Ω u ( x ) e! - R ; - m h u x p x a x P x a x ( ) = j ( ) j ( ) = ( ) ( ) j = H (a [] -+ % V P ( x ) R R E m / 9% 6* - < + R ) ( / + U a( x ) ( ) = { ( ) ( ) L m ( )} a x a x a x a x x % E () E R x C () P ( x ) 8 my % )! z< B fe " MLS ( L R? % V () a( x ) 8", ( / ) - H* % R + W ( /; 9 n ( ) ( ) ( ) J = Wˆ x x i P x i a x u i i = W ˆ ( x x i ) x I C 5 < 9 E n E () E x = x i u e u i +% R n 9 E - +% 6 m U", E % (a - L MLS ( E h %? 9 P % u J = a () ( R M ' W a( x ) JE () - H* % UE & U E % % ( ) ( ) = B( x) Us A x a x { } Us = u u L un (5) (6) - m? - + "- O - L : 5! % ;5 % ) < " E % 5 - () "- % ; P, "- ; P ) UE, > - + > % > ; + (% ) - + : Q > : % < n! L UE * - R < % - m? - + "- O L O) " E o5 % ; % ;5 < " % G 5 /L > - + > ; * "- ; i E % []- + 98 97 8 - B 977 8 q > - + % 9,L +A9 P, W? 9 5 > ; 9E []- L % + B a 9 [,]- L ) U! B * % @ t5v R! q * > + [6,5] P * > + - B Z9 8 % DE/ P Q - + * 9 > E o5 8 - ) > ;S B, - v - 99 - + > 999 8 ; - + > 99 6 n + > 999 8 5 ; i 5 v [] 7 n +?E- > 8 6 +-W? ) "- ; w - 9< 5 + "- 5 6 P Q - + * "- % ; + * ; ; - P, O & i h % DE/ P Q > > - 5 + > DE/ P Q * > + ) - c5 % RH ' ) O) % "6 W" " 5 O DI RH "6 + 9 DI U* "6 Q A h< + & L UE @ ; M + " % 5 E 5 * UE G E 9 UE ; * +'? "- ; ) Q ( )? * "- - L O "6 O - - + > "6 O EFG - + > % Z9 A O ;A9 > ; [7] L 7% 9 finite difference method (FDM) Smoothed Particle Hydrodynamics (SPH) reproducing kernel particle method (RKPM) Element Free Galerkin (EFG) 5 Meshless Local Petrov-Galerkin method (MLPG) 6 Point Interpolation method (PIM) 7 Radial Point Interpolation method(rpim) l l O 7% "6? 5 8
C O"? * x ε xx ε = ε yy z = w = zl dw y γ xy x y σ xx σ = σ yy = zcldw σxy () - + % @ V O 9Z @ () [5] % V I F! O ϑ E c = ϑ ( ϑ ) ( ϑ) / (5) * - 8! V %G [6]- + % V I F 5LH Z Q Q Q6 Q = Q6 Q Q6 Q 6 Q6 Q 66-5 % 8 F % L cos θ sin θ sinθ cosθ = sin θ cos θ sinθ cosθ sinθ cosθ sinθ cosθ cos θ sin θ Q ij (6) (7) 5 < 9 9 e n A x W x P x P x i= ( ) = i ( ) ( i ) ( i ) - +% F A( x) W ( ) ( ) (8) i x = W x xi - D E % V (5) E B( x) F ( ) ( ) ( ) ( ) ( ) L ( ) ( ) ˆ ˆ ˆn n B x = W x P x W x P x W x P x (9) % C () E M&) (5) E * n h u ( x) = φ i ( x) ui = φ ( x) Us i= a ( x) U s (7) - H* () 5 * < n MLS- R φ ( x), - 5 % V x I C ( x) = { ( x ) ( x ) L n ( x) } ( n ) φ φ φ φ ( ) ( ) ( ) = P x A x B x & +' u, v, w,-)g( w () - ( x, y, z oc Ω () - Y! 95,) - R = (8) Q = [ ] [ Q ][ R ][ ][ R ] I F 9L Q ij (9) i ZA % θ? ") i ") ) H VI & Y OQ z< (* F ; - (i Q Q Q = Q Q Q66 [ ] E ϑ E Q = Q = ϑϑ ϑϑ E Q = Q66 = G ϑϑ ϑ E = ϑe - + % V H ") () + () C< R ZUE Q, + - + @ % - ( / Q ) 5LH? HQ (* D E % V Z - Z -, ε p = Ldw - z % { Y RC - () MJKL I-H-) ' HQ) z % E V u, v z u x u = v = z w = Luw y w D- / Q ) + + ( 5LH () W Z UE,) @ M&) 8
" % ) % @ - 8? M - L W > %, L DE/ P Q > % L * UE MLPG > Ωs ( σij, j i ) Ω α ( i i ) Vˆ + b d Vˆ w w d Γ = Γqu - + +% 96 / : % P 8 & +% R () : % * Γ qu : Ω q ) % @ - @ - 8? ( / α ) % @ - () E V * E - L > % % - L W > % o5 ; +S - Ωs ( ij, j bi ) Vˆ σ + dω = Vˆ % V * E () E % () E (7) () UE M&) Ωq ( ) - + % V "6 MLPG > * Vˆ D w b dω = () % () * E () ( E + 6 8E E P Q NO) NO) 8 &! % L [ K]{ W} = { F} 9 [ K] = [ K] + [ K] Ω i [ F] - H* % V () + e [ W] I F [ K] - 5 % V " % P Kij = Ω D D φj,xx Vˆ i,xx Vˆ i,yy Vˆ i,xy D D φj,yy dω Ω s D 66 φ j,xy Kij B i ij = dγ Γ si ˆ ( φ ( ) φ ) ( Dφ j,yyy ( D D66 ) φj,xxy ) n y Bij = Vi D j,xxx + D + D66 j,xyy nx + + + Vˆ i, x ( D φ j, xx + D φ j, yy ) nx + D66 φ j, xy n y Vˆ i, y D66 φ j, xy nx + ( D φ j, xx + D φ j, yy ) n y f i = ˆ V i fd Ω ˆ αv iwd Γ α V ˆ i, nθ d Γ Ωs Γsw Γsw () (5) (6) (7) (8) σ p = Dε p = DLdw () + % V ( %G! O D F ϑ Eh D = ϑ ( ϑ ) ( ϑ ) / n l D ij = ij Q k z k z k k = - () (5) )8) N <+- [5] - + % V 9 Y % @ - uγ = ub Γ u = Γ w Γ θ (6) - 9% '< S /?,) - u b - D E % V 9Y ) @ - ub = LbW (7) V DI 9 @ - L &? L b = n L b = - D E % { (8) VL (9) MLPG 96:; - " 8-5 7--5 > - L MLS ( R % OMLPG> [7] - B 998 8 Z9 @ MLPG! U! U B * +5 & @ 8 + ) 9 "6 5 L 7% 9 5 - E L 8* -! DE/ P Q * EFG - + >!? 8 & > ; - ; - + $ * > ; M % % F - P Q * > : - % c) > ; - B MLPG ; i 5 DE/ L 9 5 S 9 %? 8 & L MLS - R % O > ; - L C9 - H< ( R ; ; ) - l l O 7% "6? 5 essential boundary conditions Atluri 8
C O"? * )*+Q-7 + >? &&S + UE b I % F 5 * X > ; X 6 8y S * 5H % F + 5 +? > ; % G < " > ; % + B * ") - "' ) - R %, ; ) + 8* ; 5 - [9] - - ( OQ P @5 YQ > + MLPG - + > C Y! - - 5H + +' 6 ) + 5 * 8) VI' /6 /6 / / ( ) Y E 8(GPa) ; ( / ( ) Y I/ E + 9O< 'I? X + εexat ε p Error = εexat % V C< 8 Q 9 5 (6) % Q N / m *; 5 /m I/ 6 w 6 (VL) 5 i h "S H-S- =JKLR-<- ;A9 - + h C 6 5 O"( O5U - * )= ; ( 5 5 H- % V * ; - E + O< 8 Q - W max D W = qb [] (7) > ; I F - + EFG > I F H + O9 > ; 5 W,* M 8", e! 9 +S % @ - 8? ") 8? 9 % n - +O9 + M ) W L &9 % n - 8? > % ; &9 ; &9 % n - % W9 6 ) 9 6 % 9 " % ( 9 @ HQ W! * ) ϵ % 9,) % @ - FG HQ - 8? &9 9 o',) @ - ( &9 9 % 9 ") W* UE w w i = [ φ L φ n ] M = w i w n w w i φ φn w = i n L n n M = n w n % V <DOP UE * L UE w*$ C9 (9) () -6 5 * + ' 5 9 "6 C @5 @I5 @5 -&! @ ; - M - B ;O VI "6 % @ - + (a - R @ -& ; W& - U - R ) S 9 H 5 R! Y R/ x = N i ( ξ, η ) x i i = " O -& 6 9 + E - L [8] 8 ) () 85 Eh D = ( ϑ ) - YE b q YQC (8) @ 8y ; o6 * 8) H* X /6 5 + E + O< - P, - > % : ; * X ;A9 [ ] [ ] - /5 5 i h "S % YQ Y 5 X 8) @ 8y ; o6 * 9 +' 6 { /6 ; E + O< - P, - 5 - + * > ;, % + > * ;A9 [ ] [] H* /8 X 6 9' C9 ' 5 + X ; y = N i ( ξ, η) y i i = () VI x i, y i P Q : 9% VI ξ, η - R N i : Eh % - I' n - 5 % @ oh - N i = + i + i 9 + i =,,, ( ξ ξ )( η η ) ( ξ η ) ( ξ )( η η )( ξ ξ ) 9 N i = + i + 9 i i = 5,6,7,8 ( η )( ξ ξ )( η η ) 9 N i = + i + 9 i i = 9,,, () () (5)
& > 5 W,* M : - % Z9: - v 6 ;A9 - +' 9 @ @Q "6 * X % @ - * E Y Q - e P Q 9- - +' 9- ( - e P Q : - 9' - C9 { I + 9 - e : l l O 7% "6? 5 O"( *; ; ( 5 5 H- S ' X( WS-' Y+ OV *; )- 5 O"( O5U- *)=; ( 5? H- *; 8h DI 9 C "6 - E V X- - + []? R) C Y!,) 5 8) (7) C oh N / m 9 +' e P Q Z % - w i H - C Y Q - 9 @ - 5 ( 5 9 * X H* X - < /? 5 5 5 E - e Y P Q + - E Y C Y Q - - * ; ( 5' 5H-5-5 V-<- 7 7 /8 /5% 5 5 /98 /97% OV 5O"( / /7% /9 /9% E + O< O5U; ( 5' 5H-5-5V-<- 7 7 /7 /7% 5 5 /7 /7% OV 5O"( /8 /5% / /% E C< E + O< E C< K; ( 5'?H-55 5 <- /5 8h /? /89 /8 /5 / / / /77 /77 MLPG [ ] R) O5U; ( 5'?H-55 5 5 <- / /5 /8 /6 /5 /5 / / / /66 /7 8h /? EFG [ ] R) *; 5O"( N / m O5U - * )= ; ( 5 ' )* H- OQ VI' - Y! 5, 5! ve- () 8) 5 - + * > % X - < w max Eh α = pa - E % Coh ; i (9) 5 E Y Q - e P Q ' O"( * ; ( 55H-S' X( WS-' Y+OV *; )- 86
C O"? * + % * H 9 C6 DI e P Q 7 6 9- ;A9 f Y Q - +' DI 9 @ - 9+' - L : 8) OQ VI' - f Y! + 9 OQ P % H* - E X 8 7 8) a / b = z< * K; ( 5' Z5H-55 5-7 <- /98 / /8 /88 /5 /58 /5 / /6 9 a/b MLPG [ ] R) O5U ; ( 5 ' Z5 H- 5 5 5-8<- % - X C< % H ) O ; &C - -& [] R) /67 /56 - +' 5 ' ) * H- 5 5 5-6<- 5 5 /6 /55 MLPG* > /6 /59 /55 8 9 E VL { /5 a/b /6 /5 /98 /88 / /7 /958 /6 MLPG [ ] R) 5 O"( ' 5 -? -( ( H- *; 5! "6 Q 9< < { 9 @ - < VI'! Y5 %H* X(9) 8) ; o6 * 9 +' - E = 5 E G = 5 E, ϑ = 5 E + O< /697 B @ 8y [] * ; ( 5 ) * H- S ' X( WS-' Y+ OV *; )- O"( *; ; ( 5 ' Z5 H- S ' X( WS-6' Y+ OV *; )- O"( O"( *; ; ( 5 ) * H- S ' X( WS-5' Y+ OV *; )- *; 5 * X G % - H* X ) % * - 6 8) [] R) 5 " X 6 - L : 5 5 e P Q 5 ( 9 - Y Q - O"( O5U - * )=;( 5 ' Z5 H- -E (9) C! OQP % H* X- 87
( ^> 5) ')* H-6 5-<- [ ] R) /8 5 5 / MLPG* > / / 9 E { -( (--(- ':`=H- 6 + DI 8-5 6 ) f! m 6 { x + y = 9 E 9 @ - N / m!! O Y v X 8) ve-! < % - $9 5 -, - e P Q 9- - 6 8y OQ VI' oc 9 +' OQ P G % "6 Q W9 ( ^> 5) ':`=H-6 5-<- OQ P Abaqus +88 /6 / /898 /8 + > /9 /96 /7 /5 v VL { VL { Y v!o! *; ; ( 5 ' Z5 H- S ' X( WS-7' Y+ OV *; )- O"( ( 5' 5-(( H-5-5V-9<- 7 7 /6577 / 5 5 /6 /8 OV 5O"( * ; /657 /77 /66 /6 E E + O< C< " % L * B ; * 5 + X M /655 /655 ( oc a oc []- %/5 5 + X C< ; - + > 5 * X 8 - (/9/9/) ZS U "S %G C Y!, G = 5E, E = 5E ϑ = 5%? + 9 U z< - < 9 8 l l O 7% "6? 5 ); -8! O 7% 9 Y 5 ; /? M % ; 9 E + > ; O & % DE/ P Q "- O 5 + > - * * + P? EFG - { 5 + 9 -! X C9 - < ; - + > e ") - - P? > ; M A ; -im* - 5 : % "O 5 + > + ;A9 9 E E &9 X + - o6 6 " E w 7 6 5 ; ( 5 O[6 '? H- 5 5 5-8' O5U - Y!,) X, E = GPa, E = GPa, ϑ =, 5 5 a/b VI'! G = 6 GPa, R = m, h = mm, ' ) * -( ( H- 5 + ; - + > % N / m - < [] 88
C O"? * a: O5U ; ( (b * ; ( (a: b ' :`= H- S ' X( WS-9' [] Chen, XL, Liu, G R, and Lim, S P, An element free Galerkin method for the free vibration analysis of composite laminates of complicated shape Compos Struct, 59()(79-89) [] M Memar Ardestani, BS, Sh Shams, Analysis of functionally graded stiffened plates based on FSD utilizing reproducing kernel particle method Composite Structures, : p - [] Sladek, J, et al, Analysis of the bending of circular piezoelectric plates with functionally graded material properties by a MLPG method Engineering Structures, 7: p 8-89 [] Liew, KM, X Zhao, and AJM Ferreira, A review of meshless methods for laminated and functionally graded plates and shells Composite Structures, 9(8): p - [] Lancaster, PaS, K, Surfaces generated by moving least squares methods math comput, 98 7: p -58 [5] Krysl, P, Belytschko,, Analysis of thin plates by the element-free Galerkin method Computational Mechanics, 996 7: p 6-5 [6] Kaw, AK, Mechanics of composite materials, ed 6, CRC Press: aylor & Francis Group [7] Atluri, SNaZ, A new meshless local Petrov- Galerkin (MLPG) approach in computational mechanics Comput Mech, 998 : p 7-7 [8] shu, c, Differential Quadrature and Its Application in Engineering : springer ' + Matlab7 88: R) 9,i8c-,a[9] P? [] Liu, GR, Meshfree methods moving beyond the Finite element method CRC Press, [] imoshenko, S, heory of plates and shells, ed s edition 959 [] Zienkiewicz, OC, aylor, R L, he finite element method, ed f edition Vol aylor [] Reddy, JN, Mechanics of Laminated Composite Plates and shell theory and analysis, ed s edition : CRC PRESS [] Rao, CSaKP, LARGE DEFORMAION FINIE ELEMEN ANALYSIS OF LAMINAED CIRCULAR COMPOSIE PLAES Computers & Structures, 995:p59-6 % @ - > % L G ; ' X 6 >, % @ - 8? &9 C< E HQ 9 > B (a ( / +S - e M P%U - % ;I M X 6 { % &9 F-9 [] GR LIU, YG, AN INRODUCION O MESH FREE MEHODS AND HEIR PROGRAMMING 5: Springier [] Szafrana, Z, ELASIC ANALYSIS OF HIN FIBER- REINFORCED PLAES CIVIL AND ENVIRONMENAL ENGINEERING REPORS, 5 [] Colagrossi A, LM, Numerical simulation of interfacial flows by smoothed particle hydrodynamics Journal of Computational Physics, 9(): p 8-75 [] Johnson GR, SR, Beissel SR, SPH for high velocity impact computations Computer Methods in Applied Mechanics and Engineering 996 9( ): p 7 7 [5] Liu WK, JS, Li S, Adee J, Belytschko, Reproducing kernel particle method for structural dynamics Int J Numer Methods Eng, 995 8: p 655 79 [6] Liu WK, JS, Zhang YF, Reproducing kernel particle methods Int J Numer Methods Fluids, 995 : p 8-6 [7] Krysl, PaB,, Analysis of thin plates by the element-free Galerkin method Computmech, 996 7: p 5-6 [8] Ouatouati, AEaJ, D A, A new approach for numerical modal analysis using the element free method Numerical Methods Eng, 999 6(-7) [9] Liu, GRaC, X L, Bucking of symmetrically laminated composite plates using the element-free Galerkin method Struct Stability Dyn, (): p 8-9 89