A simple locking-free discrete shear triangular plate element

Transcription

1 Copyright 011 Tech Science Press CMES, vol.77, no.4, pp.1-38, 011 A simple locking-free discrete shear triangular plate element Y.C. Cai 1,,3, L.G. Tian 1 and S.N. Atluri 3 Abstract: A new three node triangular plate element, labeled here as DST-S6 (Discrete Shear Triangular element with 6 etra Shear degrees of freedom), is proposed for the analyses of plate/shell structures comprising of thin or thick members. The formulation is based on the DKT (Discrete Kirchhoff Technique) and an appropriate use of the independent shear DOF(Degrees Of Freedom). The shear locking is completely eliminated in the DST-S6, without any numerical epediencies such as the reduce integration, the use of assumed strains/stresses, or the need for the stabilization of the attendant zero energy modes. It is shown that the present DST- S6 is much simpler than the triangular shear-deformable plate elements currently available which pass the patch test for thick to very thin plates. The DST-S6 has two etra shear DOF per node, but the etra shear DOF are the rotations caused by the transverse shear deformation, and there is no difficulty in applying the essential boundary conditions for the present DST-S6. It is also demonstrated that the solution of a pressure-loaded simply-supported circular plate of the DST-S6 avoids the so-called polygon-circle Parado. Various numerical eamples indicate that the DST-S6 is a robust and high-performance element for thick and thin plates. Keywords: plate/shell element, locking-free, discrete Kirchhoff theory, triangular element, shear degrees of freedom 1 Introduction Plate/Shell structures play an important role in civil, naval and airspace engineering. There has been a great interest in high performance coupled with simple formulation, in the field of finite elements for plates and shells over the past several decades (Gal and Levy 006; Cai, Paik and Atluri 010; Iura and Atluri 003). 1 Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, Department of Geotechnical Engineering, Tongji University, Shanghai 0009, P.R.China Corresponding author. yc_cai@163.net. 3 Center for Aerospace Research & Education, University of California, Irvine

2 Copyright 011 Tech Science Press CMES, vol.77, no.4, pp.1-38, 011 The eisting Mindlin plate elements can be broadly divided into two main groups. The first group is the displacement model based on the assumed displacement function of the element, while the second group uses the mied/hybrid element method. The Mindlin-Reissner plate theory for the analyses of thin and thick plates requires only a C 0 continuity for the transverse displacement as well as the normal rotations, and the difficulties of C 1 continuity requirement for the transverse displacement for the Kirchhoff elements can be avoided. However, the Mindlin-Reissner plate elements based on the displacement approach give poor results in thin plate limit because of the shear locking phenomenon. In order to eliminate the shear locking in the Mindlin-Reissner plate elements, reduced integration, assumed natural stain (ANS) approach, and assumed stress elements (or mied/hybrid stress elements) are used. For eample, the reduced integration method by Zienkiewicz, Taylor and Too(1971), and Pugh, Hinton and Zienkiewicz (1978), the selective integration method by Malkus and Hughes(1978), and Hughes Cohen and Haroun (1978), the MITC family by Bathe and Dvorkin (1985), the MISC element by Nguyen, Rabczuk, Stephane and Debongnie (008), the DST family by Batoz and Lardeur (199), and Batoz and Katili (199), the RDKTM by Chen and Cheung (001), the hybrid-trefftz plate elements by Choo, Choi and Lee (010), the assumed stress/strain elements by Lee and Pian (1978), Katili(1993), and Brasile (008). Most of these elements are free from shear locking and are very useful for practical applications. However, these popular elements always involve very comple formulations to take the transverse shear effects into account for thick plates, and thus lead to compleity in the form, and difficulty in the programming, of the stiffness matri. In this work, we present a simple three node triangular plate element, labeled here as DST-S6 (Discrete Shear Triangular element with 6 etra Shear degrees of freedom), for the analyses of plate/shell structures comprising of thin or thick members. The formulation is based on the DKT (Discrete Kirchhoff Technique) (Batoz, Bathe and Ho 1980; Dhatt 1969) and an appropriate use of the independent shear DOF (Atluri 005; Li, Soric, Jarak and Atluri 005). The shear locking is completely eliminated in the DST-S6, without any numerical epediencies such as the reduced integration, the use of assumed strains/stresses, or the need for the stabilization of the attendant zero energy mode. It is shown that the present DST-S6 is much simpler than the triangular shear-deformable plate elements currently available, which also pass the patch test for thick to very thin plates.

3 A simple locking-free discrete shear triangular plate element 3 Interpolation functions.1 The DKT element in brief. Interpolation functions Consider a linear elastic triangular plate element undergoing infinitesimal deformation, as.1 shown The DKT inelement Fig.1. in The brief derivatives of the transverse displacement w around two independent aes (not including the influence of the transverse shear deformation) are Consider a linear elastic triangular plate element undergoing infinitesimal deformation, as shown in Fig.1. The derivatives of the transverse displacement w around two independent aes (not including θ = w, θ y = w y the influence of the transverse shear deformation) are ~ w ~ w θ =, θ y = (1) y (1) α 4 Fig.1 Triangular element The constraints of the zero-transverse-shear Kirchhoff plate theory are presented in the following discrete way: The constraints of the zero-transverse-shear Kirchhoff plate theory are presented in at the corner nodes i ( i =1,,3 ) the following discrete way: at the corner nodes ~ w w θi = i (i = ~, 1, θ, yi = 3) () = y i = i θ i = w at the mid-side points k ( k = 4,5,6, θ yi = w ) =i y () =i ~ θ ~ ~ ( θ + θ ) nk ni nj at the mid-side points k (k = 4,5,6) 1 = (3) ~ w θsk = s Figure 1: Triangular element 1.5 = l 1 ~ ~ ( w w ) ( θ + θ ) θ nk = 1 ( ) θ ni + θ n j =k k j i 4 si sj where i = ( i, y i ) θ sk = w s = 1.5 (w j w i ) 1 ( ) θ si + θ s j =k l k 4 are the coordinates of the node i ; s and n indicate the tangent and normal of the element side ij respectively; l k is the length of side ij ; i = 1,, 3 and j =,3, 1 when where k i = = 4,5,6 (. i,y i ) are the coordinates of the node i; s and n indicate the tangent and normal of the element side i j respectively; l 3 k is the length of side i j; i = 1,,3 and j =,3,1 when k = 4,5,6. (4) (3) (4)

4 4 Copyright 011 Tech Science Press CMES, vol.77, no.4, pp.1-38, 011 Along each side i j we have { } [ ]{ } θ sk mk n = k θ k θ nk n k m k θ yk (5) where m k = cosα k and n k = sinα k when k = 4,5,6. The rotations θ and θ y of the element can be defined as follows: θ = where 6 i=1 N i θ i, θ y = 6 i=1 N i θ yi (6) N j = (L j 1)L j ( j = 1,,3) N 4 = 4L 1 L,N 5 = 4L L 3,N 6 = 4L 3 L 1 (7) L i are the area coordinates of the three-node triangular plate element and can be epressed as L i = 1 A (a i + b i + c i y) (8) a i = j y m m y j, b i = y j y m, c i = j + m (9) where A is the area of the triangular element, and i = 1,,3; j =,3,1;m = 3,1,. Finally, the following DKT displacements in each plate element are obtained by using Eqs.() to (6): θ = θ y = 3 i=1 3 i=1 ( Ri w i + R i θ i + R yi θ yi ) ( Hi w i + H i θ i + H yi θ yi ) (10) where R 1 = 1.5(m 6 N 6 /l 6 m 4 N 4 /l 4 ), R = 1.5(m 4 N 4 /l 4 m 5 N 5 /l 5 ), R 3 = 1.5(m 5 N 5 /l 5 m 6 N 6 /l 6 ), ( R 1 = N 1 + N 4 0.5n 4 0.5m ( 4) + N6 0.5n 6 0.5m 6), ( R = N + N 4 0.5n 4 0.5m ( 4) + N5 0.5n 5 0.5m 5),

5 A simple locking-free discrete shear triangular plate element 5 H y H y H y ( R 3 = N 3 + N 5 0.5n 5 0.5m ( 5) + N6 0.5n 6 0.5m 6), R y1 = 0.75(m 4 n 4 N 4 + m 6 n 6 N 6 ), R y = 0.75(m 4 n 4 N 4 + m 5 n 5 N 5 ), R y3 = 0.75(m 5 n 5 N 5 + m 6 n 6 N 6 ); H 1 = 1.5(n 6 N 6 /l 6 n 4 N 4 /l 4 ), H = 1.5(n 4 N 4 /l 4 n 5 N 5 /l 5 ), H 3 = 1.5(n 5 N 5 /l 5 n 6 N 6 /l 6 ), H 1 = R y1, H = R y, H 3 = R y3, ( H y1 = 1 + N 4 0.5m 4 0.5n ( 1 = N1 + N4( 0.5m4 0.5n4 ) + N6( 0.5 4) + N6 m6 0.5m 0. 5 n 6 6 ), 0.5n 6), ( ) ( ) ( H y = + N 4 0.5m 4 0.5n ( 4) + N5 0.5m 5 0.5n = N + N4 0.5m4 0.5n4 + N5 0.5m5 5n5, 5), ( H y3 = N 3 + N 5 0.5m 5 0.5n ( 5) + N6 0.5m 6 0.5n ) 6 3 = N3 + N5( 0.5m5 0.5n5 ) + N6( 0.5m6 0. 5n6 ). The displacement functions of the proposed element DST-S6. The For the displacement three node element functions DST-S6 of the asproposed shown in Fig., element the shear DST-S6 strains are assumed as For the three 3 node element DST-S6 3 as shown in Fig., the shear strains are assumed as γ = (L i γ i ),γ y = (L i γ yi ) (11) 3 3 i=1 γ = Lγ, γ i=1 = Lγ (11) ( i i ) y ( i yi ) i= 1 i= 1 where γ i and γ yi are the independent shear DOF of node i. where γ and γ yi are the independent shear DOF of node i. i Figure : Three-node triangular plate element Fig. Three-node triangular plate element The total Therotations total rotations θ and θθ y of the thick plate (include the influence of the transverse shear and θ y of the thick plate (including the influence of the transverse shear deformation) can be given by deformation) can be given by θ ~ ~ = θ θ γ=, θ y = γ, θ y θ = γ y θ γ (1) (1) y Substituting Eqs.(10) and (11) into Eq.(1), and noticing that node i, one obtains θ = y 3 [ Ri wi + Riθ i + Ryiθ yi + ( Ri Li ) γ i + Ryiγ yi ] i= 1 y ~ θ = θ + γ i i i ~ and θ = θ + γ at yi yi yi

6 6 Copyright 011 Tech Science Press CMES, vol.77, no.4, pp.1-38, 011 Substituting Eqs.(10) and (11) into Eq.(1), and noticing that θ i = θ i + γ i and θ yi = θ yi + γ yi at node i, one obtains θ = 3 [R i w i + R i θ i + R yi θ yi + (R i L i )γ i + R yi γ yi ] i=1 θ y = 3 [H i w i + H i θ i + H yi θ yi + H i γ i + (H yi L i )γ yi ] i=1 (13) Thus, the matri form of the generalized displacement functions of the DST-S6 can be epressed as U = Φa = [ ] a 1 Φ 1 Φ Φ 3 a (14) where a 3 U = [ θ θ y γ γ y ] T (15) a i = [ ] T w i θ i θ yi γ i γ yi (16) R i R i R yi R i L i R yi Φ i = H i H i H yi H i H yi L i L i 0 (17) L i Notice that γ i and γ yi in Eq.(16) are retained as independent shear DOF of node i, which lead to a locking-free element which remains uniformly valid for either thick or thin plates/shells, without using such numerical epediencies as selective/reduced integrations and without the need for stabilizing the attendant spurious modes of zero-energy. 3 Stiffness matri of the DST-S6 For deriving the element stiffness matri of DST-S6 element, the strain energy in an element can be written as (Chen and Cheung 001) Π p = 1 ( χ T D b χ + γ T D s γ ) ddy (18) A e in which χ is the bending stain, γ is the shear strain χ = [ θ, θ y y, ] T θ y + θ [ ] y T,γ = γ γ y (19)

7 A simple locking-free discrete shear triangular plate element 7 and Eh 3 1 ν 0 D b = 1(1 ν ν 1 0,D s = ) 1 ν 0 0 5Eh 1(1 + ν) [ ] (0) where E is the elastic modulus, ν is the Poisson ratio, and h is the thickness of the plate element. Substituting Eq.(14) into Eq.(18), the element stiffness matri of the DST-S6 can be written as K e = K e b + K e s (1) in which the bending part and the shear transverse-strain part of the element stiffness matri can be respectively written as K e b = B T b D b B b ddy,k e s = A e B T s D s B s ddy A e () where B b = [ B b1 B b B b3 ] R i, R i, R yi, R γi, R yi, B bi = H i,y H i,y H yi,y H i,y H γi,y (4) R i,y + H i, R i,y + H i, R yi,y + H yi, R γi,y + H i, R yi,y + H γi, (3) where (), denotes a differentiation with respect to, R γi = R i L i and H γi = H yi L i. B s = [ B s1 B s B s3 ] [ ] Li 0 B si = L i 3.1 Comments on element DST-S6 (1) It is clear that if the transverse shear effects are not important (for thin plates) the DST-S6 will convergence to the DKT. The DST-S6 is completely locking free and passes all constant patch tests eactly. () It is clear from the above procedures that the formulation and the numerical implementation of the present stiffness matrices of the DST-S6 are much simpler than (5) (6)

8 8 Copyright 011 Tech Science Press CMES, vol.77, no.4, pp.1-38, 011 that of the eisting shear-deformable triangular plate elements currently available, which also pass the patch test for thin and thick plates. (3) The DST-S6 has five DOF per node, which increases the computational cost of the final set of algebraic equations in the present method. However, the computational cost for generating the element stiffness matri of the DST-S6 is much less than that of the other shear-deformable triangular plate elements, e.g., the DST-BL and DST-BK. In practical applications, the DST-S6 is more favorable than the other shear-deformable plate elements since it leads to a much simpler eplicit epression and more efficient evaluation of the stiffness matri. (4) The DST-S6 has five DOF per node, but the two etra shear DOF (γ i and γ yi ) are actually rotations causing by the transverse shear deformation, as shown in Fig.3, and thus there is no difficulty to apply the essential boundary condition for the present DST-S6. For eample, for the symmetric boundary shown in Fig.4, the boundary conditions of DST-S6 can be imposed in a weaker sense as follows = 0 : θ i = 0, γ i = 0 y = 0 : θ yi = 0, γ yi = 0 (7) γ θ ~ w θ = Figure 3: The transverse shear strain Fig. 3 The transverse shear strain 4 Numerical eamples In this section, several problems have been solved to demonstrate the performance of the present DST-S6. The T3, DST-BL and RDKTM elements have been selected for comparison with the proposed DST-S6, where

9 γ θ ~ w θ = A simple locking-free discrete shear triangular plate element 9 Fig. 3 The transverse shear strain Figure 4: Symmetric boundary condition Fig.4 Symmetric boundary condition 4. Numerical eamples T3: The triangular displacement element with full integration. DST-BL: In this section, Discrete several Mindlin problems triangular have been plate solved element to demonstrate proposed the performance by Baotz and of the Lardeur(1989). present DST-S6. The T3, DST-BL and RDKTM elements have been selected for comparison with the RDKTM: Re-constituting discrete Kirchhoff triangular plate element by Chen and proposed DST-S6, where Cheung (001). T3: The triangular displacement element with full integration. DST-BL: Discrete Mindlin triangular plate element proposed by Baotz and Lardeur(1989). 4.1RDKTM: Eigenvalues Re-constituting and rank discrete Kirchhoff triangular plate element by Chen and Cheung (001). Only three eigenvalues corresponding to the three rigid body modes (not including the 4.1 shear Eigenvalues DOF γ i andrank γ yi ) are always zero for various element shapes and for very thin to thick plates. The element has always a proper rank. Only three eigenvalues corresponding to the three rigid body modes (not including the shear DOF γ and i γ yi ) are always zero for various element shapes and for very thin to thick plates. The 4. Constant curvature patch tests element has always a proper rank. The element stiffness matri must satisfy the patch test in order to give reliable results. To check the performance of the element, we start with the patch test suggested in Batoz and Katili (199), and Katili (1993). We consider a patch of four elements as shown in Fig.5. The assumed material 8 properties of the plate are E = 1000 and ν = 0.3. The thickness of the plate is h. The characteristic length of the plate is l. The patch tests are performed by enforcing the following boundary conditions which lead to constant curvatures and zero transverse shear Case A: w = /; θ = ; θ y = 0; γ = 0; γ y = 0 Case B: w = y /; θ = 0; θ y = y; γ = 0; γ y = 0 Case C: w = y/; θ = y/; θ y = /; γ = 0; γ y = 0

10 4. Constant curvature patch tests The element stiffness matri must satisfy the patch test in order to give reliable results. To check the performance 30 of the element, Copyright we 011 start Tech with Science the patch Press test CMES, suggested vol.77, in no.4, Batoz pp.1-38, and Katili 011(199), and Katili (1993). Figure 5: Mesh for the patch test Fig. 5 Mesh for the patch test We consider a patch of four elements as shown in Fig.5. The assumed material properties of the plate are E = 1000 For and the element ν = DST-S6 The thickness eact results of the are plate obtained is h for. The anycharacteristic aspect ratio. But length for the of the plate is element DST-BL small errors appear for h/l > 0.1. The errors on the displacement l. The patch tests are performed by enforcing the following boundary conditions which lead to w at node 5 are reported on Fig.6. constant curvatures and zero transverse shear Case A: w 4.3 Constant / ; θ = shear ; θ = patch 0; γ tests = 0; γ = 0 = y y For verification of the constant shear deformation condition and zero curvature of Case B: wthe = ydst-bl / ; θ and = 0; DST-S6, θ y = y; the γ = following 0; γ y = 0fields have been imposed on the boundary nodes in Fig.5: Case A: w = /; θ = 1/; θ y = 0; γ = 1; γ y = 0 Case C: w = y / ; θ = y / ; θ y = / ; γ = 0; γ y = 0 Case B: w = y/; θ = 0; θ y = 1/; γ = 0; γ y = 1 For the element Fig.7DST-S6 shows the eact displacement results are wobtained of node 5for for any different aspect thickness/length ratio. But for the ration. element It DST-BL small errors can appear be seen for that the displacements h / l > 0.1. The errors approached on the displacement the pure shear line at w at high node values 5 are of reported on h/l, and thus the elements DST-BL and DST-S6 pass the constant shear patch tests. Fig Square plate under uniform load A simply supported or clamped square plate under uniform load q is considered for linear elastic analysis. The side length and the thickness of the square plate are l and h. A quarter of the plate is modeled due to the symmetry as shown in Fig.8. error in w 5 (%) The results for the central displacement w 0 and for the central bending moment M 0 for the simply supported plate are listed reported in Tabs.1 and. The results for the central displacement w 0 for the clamped plate are reported in Tab.3. As can be seen, the displacement solutions of the DST-S6 based on primal methods converge from "BELOW", and overall good results are obtained for the new locking-free element DST-S6 compared to T3, DST-BL and RDKTM. Case A

11 A simple locking-free discrete shear triangular plate element 31 error in w 5 (%) 0.5 Case A error in w 5 (%) DST-S6 DST-BL Case B log(h/l) error in w 5 (%) Case C Figure 6: Constant curvature patch tests

12 3 Copyright 011 Tech Science Press CMES, vol.77, no.4, pp.1-38, 011 Table 1: Central deflection ( ql 4 /100D ) for a simply supported square plate loaded by uniform load h/l 1.0e DST-S6( ) DST-S6(4 4) DST-S6(8 8) DST-S6(16 16) DST-BL(16 16) RDKTM(16 16) T3(16 16) Eact Table : Central moment ( ql /10 ) for a simply supported square plate loaded by uniform load h/l 1.0e DST-S6( ) DST-S6(4 4) DST-S6(8 8) DST-S6(16 16) DST-BL(16 16) RDKTM(16 16) T3(16 16) Eact Table 3: Central deflection ( ql 4 /100D ) for a clamped square plate loaded by uniform load h/l 1.0e DST-S6( ) DST-S6(4 4) DST-S6(8 8) DST-S6(16 16) DST-BL(16 16) RDKTM(16 16) T3(16 16) Eact

13 A simple locking-free discrete shear triangular plate element 33 error in w 5 (%) Case A error in w 5 (%) Case B Figure 7: Constant shear patch tests 4.5 Clamped circular plate under uniform load A clamped circular plate subjected to uniformly distributed load q is considered. The radius of the plate is r = 100 and the thickness of the plate is h. The material properties are E = 100 and ν = 0.3. Due to the double symmetry, only one quarter

14 34 Copyright 011 Tech Science Press CMES, vol.77, no.4, pp.1-38, Clamped circular plate Figure Fig.8 under 8: Model Model uniform of a ofclamped a clamped load square square plate plate A clamped circular plate subjected to uniformly a simply distributed supported load square q is considered. plate loaded The by radius uniform load Tab.1 Central deflection ( ql 4 100D) of the plate is h/l r = 100 and 1.0e-5 the thickness of the plate 0.01 is h. The 0.10 material properties 0.15 are of the plate is discretized as shown in Fig. 9. Four distributions of 5, and 89 E = 100 and nodes DST-S6( ) are ν = employed Due to the double symmetry, only one quarter of the plate is discretized for the convergence studies Similar good results are obtained as shown by DST-S6(4 4) the in Fig. present 9. Four element distributions DST-S6, of 5, as 81 listed and in 89 Tab. nodes 4. Numerical are employed results for the also indicate convergence that, DST-S6(8 8) although studies. Similar the primal good methods results are are obtained used, by the the displacement present element solutions DST-S6, ofas the eample Tab. 4. converge Numerical from results "UP" also for indicate the DST-S6, that, although DST-BL the and primal RDKTM methods (ecept are used, forthe the listed in DST-S6(16 16) displacement element solutions T3). of the eample converge from "UP" for the DST-S6, DST-BL and DST-BL(16 16) RDKTM (ecept for the element T3). RDKTM(16 16) T3(16 16) Eact Tab. Central moment ( ql 10) for a simply supported square plate loaded by uniform load h/l 1.0e DST-S6( ) DST-S6(4 4) DST-S6(8 8) DST-S6(16 16) DST-BL(16 16) RDKTM(16 16) T3(16 16) Figure 9: Mesh----- of one quarter of a clamped circular plate (81 nodes) Fig. 9 Mesh of one quarter of a clamped circular plate (81 nodes) Eact qr 100D for a clamped circular plate loaded by uniform load h/r ql 100D for a clamped square plate loaded by uniform load 1.0e DST-S6(5) h/l 1.0e DST-S6( ) DST-S6(81) DST-S6(4 4) DST-S6(89) DST-S6(8 8) DST-BL(89) Tab.4 Central deflection ( ) Tab.3 Central deflection ( )

15 A simple locking-free discrete shear triangular plate element 35 Table 4: Central deflection ( qr 4 /100D ) for a clamped circular plate loaded by uniform load h/r 1.0e DST-S6(5) DST-S6(81) DST-S6(89) DST-BL(89) RDKTM(89) T3(89) Eact Simply supported circular plate under uniform load A simply supported circular plate subjected to uniformly distributed load q is considered. The geometry and the material properties of the plate are the same as the clamped circular plate. The method proposed by Rhee and Atluri (1986) is employed to impose the geometrical boundary conditions for the simply supported circular plate when a polygonal domain approimation is used. Tab. 5 demonstrates that the solution of the present DST-S6 avoids the so-called Babuska Parado, by imposing the correct boundary condition. Numerical results also indicate that the displacement solutions of the DST-S6 based on primal methods converge from "BELOW", and overall good results are obtained for the new locking-free element DST-S6 compared to T3, DST-BL and RDKTM. Table 5: Central deflection ( qr 4 /100D ) for a simply supported circular plate loaded by uniform load h/r 1.0e DST-S6(5) DST-S6(81) DST-S6(89) DST-BL(89) RDKTM(89) T3(89) Eact

16 36 Copyright 011 Tech Science Press CMES, vol.77, no.4, pp.1-38, Conclusions A new locking-free triangular plate element DST-S6 having three nodes, and five DOF per node, has been proposed. The two etra transverse shear strains (γ i and γ yi ) are introduced as additional shear DOF to eliminate the shear locking phenomenon for shear-deformable plates, but the etra shear DOF of the DST-S6 can be treated as rotations caused by the transverse shear deformation, and there is no difficulty to impose the essential boundary condition for the present DST-S6. The most promising feature of the DST-S6 is the simplicity of its formulation and its numerical implementation. Numerical eamples indicate that the proposed DST- S6 passes the curvature patch tests and the shear patch tests eactly, and has high convergence rates and high accuracy. A whole family of shear-deformable triangular and quadrilateral plate elements can in fact be derived with the similar formulation of the DST-S6. The present element can be applied to the linear/nonlinear analysis of composite plates and shells as well. Acknowledgement: The authors gratefully acknowledge the support of National Basic Research Program of China (973 Program: 011CB013800), Program for Changjiang Scholars and Innovative Research Team in University(PCSIRT, IRT109), and Kwang-Hua Fund for College of Civil Engineering, Tongji University. This research was also supported in part by an agreement of UCI with ARL/ARO with D.Le and A.Ghoshal as cognizant collaborators, and by the World Class University (WCU) program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (Grant no.: R ). References Atluri, S.N. (005): Methods of Computer Modeling in Engineering & Science. Tech Science Press. Batoz, J.L.; Bathe, K.J.; Ho, L.W.(1980): A study of three-node triangular plate bending elements. International Journal for Numerical Methods in Engineering, Vol.15, pp Bathe, K.J.; Dvorkin, E.N.(1985): A four-node plate bending element based on Mindlin/Reissner plate theory and a mied interpolation. International Journal for Numerical Methods in Engineering, Vol.1, pp Batoz, J.L.; Katili, I.(199): On a simple triangular Reissner/Mindlin plate element based on incompatible modes and discrete constraints. International Journal

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