Analysis of Thin Isotropic and Orthotropic Plates Using Meshless Local Petrov- Galerkin Method With Various Geometric Shapes

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1 89-8 "$ ;; =<5 (MLPG) 96:; + +, ( ) *$ % &'! " &'! " - :+ > *? *5 A < ;! $ : % " "- %! /* " /-((-/-(- ),-)*+'"( B + % - &A " :? > ; - H : 9 ; FG - D E C! 9 %?, L (MLPG) ; > % 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 : ) "- $' ; * < " h > i$< - B "- " % : 9 %?, " "- ; - * L UE * ") - R% : D 5 + > E? "- Q ' ; 5 + > [] 95 8! " - W S " A Q E >! 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 :

2 "6 [9]7% "6 Z [8]D"- Y - [] Q 6 L %G - FGM "6 RKPM > % Z9 C "6 O [] 5 '< 9 5 P, Z9 MLPG >! O P, 5 R) Z9 []- []< - + "- % -!! MLPG - + * > % ; L 9I 9 9 Y B 5 ") -& 5 9 * X * > 6 + +' H< P? ) MLPG * > V UE & ' ( R % L P&9 8? W V "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 % [] B 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 6 n + > ; i 5 v [] 7 n +?E- > W? ) "- ; w - 9< 5 + "- 5 6 P Q - + * "- % ; + * ; ; - P, O & i h % DE/ P Q > > > DE/ P Q * > + ) - c5 % RH ' ) O) % "6 W" " 5 O DI RH "6 + 9 DI U* "6 Q A h< + & L ; 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

3 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 () [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 M&) 8

4 " % ) - 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 - L &? L b = n L b = - D E % { (8) VL (9) MLPG 96:; - " > - L MLS ( R % OMLPG> [7] - B MLPG! U! U B * ) 9 "6 5 L 7% 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

5 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 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 " % ( HQ W! * ) ϵ % 9,) - FG HQ - 8? &9 9 - ( &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 @5 ; - M - B ;O VI "6 - + (a - -& ; W& - U - R ) S 9 H 5 R! Y R/ x = N i ( ξ, η ) x i i = " O -& E - L [8] 8 ) () 85 Eh D = ( ϑ ) - YE b q YQC 8y ; o6 * 8) H* X /6 5 + E + O< - P, - > % : ; * X ;A9 [ ] [ ] - /5 5 i h "S % YQ Y 5 X 8y ; o6 * 9 +' 6 { /6 ; E + O< - P, * > ;, % + > * ;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)

6 & > 5 W,* M : - % Z9: - v 6 ;A9 - "6 * X - * E Y Q - e P Q ' 9- ( - e P Q : - 9' - C9 { I 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 ( 5 9 * X H* X - < /? 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 <- / /5 /8 /6 /5 /5 / / / /66 /7 8h /? EFG [ ] R) *; 5O"( N / m O5U - * )= ; ( 5 ' )* H- OQ VI' - Y! 5, 5! ve- () 8) * > % 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

7 C O"? * + % * H 9 C6 DI e P Q ;A9 f Y Q - +' DI - 9+' - L : 8) OQ VI' - f Y! + 9 OQ P % H* - E X 8 7 8) a / b = z< * K; ( 5' Z5H <- /98 / /8 /88 /5 /58 /5 / /6 9 a/b MLPG [ ] R) O5U ; ( 5 ' Z5 H <- % - X C< % H ) O ; &C - -& [] R) /67 /56 - +' 5 ' ) * H <- 5 5 /6 /55 MLPG* > /6 /59 / E VL { /5 a/b /6 /5 /98 /88 / /7 /958 /6 MLPG [ ] R) 5 O"( ' 5 -? -( ( H- *; 5! "6 Q 9< < { - < VI'! Y5 %H* X(9) 8) ; o6 * 9 +' - E = 5 E G = 5 E, ϑ = 5 E + O< /697 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

8 ( ^> 5) ')* H-6 5-<- [ ] R) /8 5 5 / MLPG* > / / 9 E { -( (--(- ':`=H- 6 + DI ) f! m 6 { x + y = 9 E - N / m!! O Y v X 8) ve-! < % - $9 5 -, - e P Q y 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 - { ! X C9 - < ; - + > e ") - - P? > ; M A ; -im* - 5 : % "O 5 + > + ;A9 9 E E &9 X + - o6 6 " E w ; ( 5 O[6 '? H ' 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

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