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Dynamic response of composite plate on pastalnak foundation based on element smooth discrete shear joint (cs-fem-dsg3)

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https://www.eduzhai.net International Journal of Composite M aterials 2013, 3(6A): 19-27 DOI: 10.5923/s.cmaterials.201309.03 Dynamic Responses of Composite Plates on the Pasternak Foundation Subjected to a Moving Mass by a Cell-based Smoothed Discrete Shear Gap (CS-FEM-DSG3) Method T. Nguyen-Thoi1,2, H. Luong-Van3, P. Phung-Van2, T. Rabczuk4,*, D. Tran-Trung5 1Faculty of M athematics and Computer Science, University of Science, Vietnam National University - HCM C, Vietnam 2Division of Computational M echanics, Ton Duc Thang University, Vietnam 3Faculty of Civil Engineering, Ho Chi M inh City University of Technology (HCMUT), Vietnam 4Institute of Structural M echanics, Bauhaus-University Weimar, M arienstrasse 15, 99423, Weimar, Germany 5Faculty of Construction & Electricity, Ho Chi M inh City Open University, Vietnam Abstract A cell-based smoothed discrete shear gap method (CS-FEM-DSG3) using triangular elements was recently proposed to improve the performance of the discrete shear gap method (DSG3) for static and dynamics analyses of Mindlin plates. In this paper, the CS-FEM-DSG3 is incorporated with spring systems for dynamic analyses of composite plates on the Pasternak foundation subjected to a moving mass. The composite plate-foundation system is modeled as a d iscretizat ion of triangular plate elements supported by discrete springs at the nodal points representing the Pasternak foundation. The position of the moving mass with specified velocity on triangular elements at any time is defined, and then the moving mass is transformed into loads at nodes of elements. The accuracy and reliability of the proposed method is verified by comparing its numerical solutions with those of other available numerica l results. A parametric e xa mination is also conducted to determine the effects of various parameters on the dynamic response of the composite plates on the Pasternak foundation subjected to the moving mass. Keywords Smoothed Finite Element Methods (S-FEM ), Co mposite Plate, Cell-based Smoothed Discrete Shear Gap Technique (CS-FEM -DSG3), Pasternak Foundation, Moving Mass 1. Introduction Dynamic response of plates on foundations subjected to a moving mass can be found in various types of engineering structures and real life applications such as basement foundations of building, traffic highways, airport runways, raft foundations, etc. For nu merical analysis of p lates on elastic foundations subjected to a moving mass, Thompson[1] first carried out an an alys is o f d yn amic b eh av io r o f roads su b ject ed to longitudinally moving loads by assuming the pavement as an infinitely long thin p late resting on elastic foundations. This analysis however cannot be used effectively for pavements of fin ite d imensions. Gbadeyan and On i[2] contributed a closed form solution by using double Fourier sine integral transformat ion to analyse a simp ly supported rectangular plate resting on elastic Pasternak foundation traversed by an arbitrary nu mber of mov ing concentrated masses. Kim and Roesset[3] have studied an infinite plate resting on elastic * Corresponding author: timon.rabczuk@uni-weimar.de (T. Rabczuk) Published online at https://www.eduzhai.net Copyright © 2013 Scientific & Academic Publishing. All Rights Reserved Winkler foundation subjected to moving loads with transformed field do main analyses using Fourier transform. Further, the elastic foundation is also represented as a Pasternak model and characterized by two moduli, one is the vertical spring modulus of foundation and the other is the shear modulus of foundation. In vibrations of continuous systems, types of support conditions have direct effect on the natural frequencies. Cheng and Kit ipornchai[4] proposed a memb rane analogy to derive an exact explicit eigenvalues for v ibration and buckling of simply supported FG p lates resting on elastic foundation using the first order shear deformation theory (FSDT). Chiena and Chen[5] studied the effect of Pasternak foundation on non-linear vibrat ion of laminated plates. Aiello and Ombres[6] used a Ray leigh – Ritz method to evaluate the vibrations of laminates resting on a Pasternak foundation. Omurtag et al.[7] investigated the vibration of Kirchoff plates on Winkler and Pasternak foundations. Malekzadeh et al.[8] used ANSYS software to analyze the vibration of non-ideal simp ly supported la minated plate on an e lastic foundation subjected to in-plane s tres s es . In the other frontier of developing advanced finite element technologies, Liu and Nguyen Thoi Trung[9] have applied a strain smoothing technique[10] into the conventional FEM 20 T. Nguyen-Thoi et al.: Dynamic Responses of Composite Plates on the Pasternak Foundation Subjected to a M oving M ass by a Cell-based Smoothed Discrete Shear Gap (CS-FEM -DSG3) M ethod using linear interpolations to formulate a series of smoothed fin ite element methods (S-FEM ) including the cell-based smoothed FEM (CS-FEM )[11-14] a node-based smoothed FEM (NS-FEM )[15-17], an edge-based smoothed FEM (ES-FEM)[18,19] and a face-based smoothed FEM (FS-FEM )[20]. Each of these smoothed FEM has different properties and has been used to produce desired solutions for a wide class of benchmark and practical mechanics problems. The S-FEM models have also been further investigated and applied to various problems such as plates and shells[21-26], u (= x, y, z ) u0 ( x, y ) + zβ x ( x, y ) v (= x, y, z ) v0 ( x, y ) + zβ y ( x, y ) (1) w( x, y, z) = w( x, y) where u0, v0, w are the d isplacements of the mid-plan of plate; βx , β y are the rotations of the middle p lane around y-axis and x–axis, respectively, with the positive directions defined in Figure 1. piezoelectricity[27,28], v isco-elastoplasticity[29,30], limit and shakedown analysis for solids[31], and so me other applications[32-34], etc. Extending the idea of the CS-FEM to plate structures, Nguyen-Thoi et al.[35] have recently formulated a cell-based smoothed stabilized discrete shear gap element (CS– FEM-DSG3) for static, and free vibration analyses of isotropic Mindlin plates by incorporating the CS-FEM with the original DSG3 element[36]. In the CS-FEM -DSG3, each triangular element will be div ided into three sub-triangles, and in each sub-triangle, the stabilized Figure 1. Reissner-Mindlin thick plate and positive directions of the displacement u, v, w and βx , β y DSG3 is used to compute the strains. Then the strain smoothing technique on whole the triangular element is used The linear strain can be given as to smooth the strains on these three sub-triangles. The numerical results showed that the CS–FEM -DSG3 is free of shear locking and achieves the high accuracy compared to the exact solutions and others existing elements in the literature.   ε x   =  ε y  γ x y    u0, x     β x,x    v0, y  + z  β y, y  u0, x + v0, y  β x , y + β y,x  (2) This paper hence extends the CS-FEM-DSG3 to dynamic responses of composite plates on the Pasternak foundation subjected to a moving mass. The composite plate-foundation system is modeled as a discretization o f triangular plate elements supported by discrete springs at the nodal points representing the Pasternak foundation. The position of the moving mass with specified velocity on triangular elements at any time is defined, and then the moving mass is transformed into loads at nodes of elements. The accuracy and reliability of the proposed method is verified by comparing its numerical solutions with those of others available nu merical results. A parametric examination is conducted to determine the effects of various parameters on the dynamic response of the plates on the Pasternak foundation subjected to the moving mass. = ε0 + zκ = γγ xyzz  = ww,,yx ++ ββ xy  γ (3) In the laminate composite plate, the constitutive equation of a kth orthotropic layer in local coordinate is derived from Hook’s law for plane stress as σ  xx σ yy  τ xy τ xz (k )       = Q11 Q21 Q61 0 Q12 Q22 Q62 0 Q16 Q26 Q66 0 0 0 0 Q55 0 0 0 Q54       ( k ) ε  ε γ γ xx yy xy xz (       k ) (4) τ yz   0 0 0 Q45 Q44  γ yz  where materia l constants are given by 2. Weak Form for the Laminate Composite Plate Consider a laminate composite plate under = Q11 bending 1= −νE112ν 21 , Q12 1= −ν1ν21E2ν2 21 , Q22 E2 1 −ν12ν 21 (5) deformation as shown in Figure 1. The middle (neutral) = Q66 G= 12 , Q55 G= 13 , Q44 G23 surface of plate is chosen as the reference plane that occupies in wh ich E1, E2 are the Young modulus in the 1 and 2 a domain Ω ⊂ R2 . The displacement field according to directions, respectively, and G12, G23, G13 are the shear Reissner–Mindlin model wh ich based on the first-order modulus in the 1-2, 2-3, 3-1 p lanes, respectively, and ν12 are shear deformation theory[37] can be expressed by Poisson’s ratios. International Journal of Composite M aterials 2013, 3(6A): 19-27 21 The la minate is usually made of several orthotropic layers in which the stress-strain relat ion for the kth orthotropic di = [ui vi wi β xi β yi ]T is the displacement vector of lamina (with the arbitrary fiber orientation compared to the the nodal degrees of freedom of uh associated to ith node, reference axes) is co mputed by res p ectiv ely . σ  xx σ yy  τ xy (k )     =  Q11 Q21 Q61 Q12 Q22 Q62 Q16 Q26 Q66 0 0 0 0 0 0 (k    ) ε ε γ xx yy xy ( k     ) (6) τ xz     0 0 0  Q55 Q54  γ xz   The memb rane, bending and shear strains can be then expressed in the matrix forms as ∑ ∑ ∑ ε0 == Bimdi ; κ = Bbi di ; γ B is d i (11) i i i where τ yz   0 0 0 Q45 Q44  γ yz  where Qij are transformed material constants of= the kth Bim = N0i,x N0i,y 00 00 00 ; Bib 0 0 0 0 0 0 Ni,x 0 0 Ni,y   ; lamina as in Ref[37].  Ni,y Ni,x 0 0 0 0 0 0 Ni,y Ni,x  (12) The Galerkin weakform of transient analysis of composite plates on Pasternak foundation can be written as[8]: ∫ ∫ ∫ Ω δε T p Dε p dΩ + Ω δγT Dsγ dΩ + Ω δ uT m udΩ (7) ∫ ∫ ∫ + Ωδ wT kww dΩ − δ wT Ω kg ∇2wdΩ = δ uT b dΩ Ω where kw and kg contains the elastic moduli of Pasternak foundation; m is the matrix containing the mass density of the material; [ ε p = ε0 κ]T ; D and Ds are material Bis = 0 0 0 Ni,x 0 Ni,y Ni 0 0 Ni   in which Ni,x and Ni,y are the derivatives of the shape functions in x-direction and y-direction, respectively The discretized system of equations of composite plates on Pastarnak foundation using the FEM for transient analysis then can be expressed as Md + Kd = F (13) constant matrices given in the form of where K is the global stiffness matrix g iven by D = = DBm DBb  Ds in wh ich ∫t/2 k= −t /2 Qij dz ; i, j ( ) ∫ ∫ ∫ 4, 5 k = 5 6 = K (8) BT DB dΩ + Ω Ω BsT DsBsdΩ + Ω N T w kwNw dΩ (14) ∫ ∫ − Ω N w, T x kg Nw,x dΩ − Ω Nw, yT kg Nw, y dΩ ∫ ∫ t/2 t/2 = Dmij = −t /2 Qij dz ; Bij −t /2 zQij dz; ∫ = Dbij = −t /t2/2 z 2Qij dz (i, j 1, 2, 6) in which Nw = [0 0 N1 0 0 0 0 N2 0 0 0 0 N3 0 0] ; B (9) =[B m Bb] and F, M are the load vector defined by ∫ ∫ =F pΝ dΩ + f b= ; M NT mN dΩ (15) Ω Ω 3. FEM Formulation for Composite Plates on Pasternak Foundation Now, by discretizing the bounded domain Ω of the composite plate into Ne fin ite elements such that  Ω= Ne Ωe and Ωi ∩ Ω j = ∅ , i ≠ j , then the finite e=1 element solution uh = u v w βx β y T of a displacement model for the co mposite plate is expressed as Ni 0 0 0 0  ∑ uh Nn   0 Ni 0 0 0   =  0 0 Ni 0 0 di Nd (10) i=1   0 0 0 Ni 0    0 0 0 0 Ni  where Nn is the total nu mber of nodes of problem domain discretized; Ni is shape function at ith node; 4. Formulation of the CS-FEM-DSG3 for Composite Plates on Pasternak Foundation In the CS-FEM-DSG3[35], the domain discretization is the same as that of the DSG3[36] using Nn nodes and Ne triangular elements. However in the formu lation of the CS-FEM -DSG3, each triangular element is divided into three sub-triangles by connecting the central point O of the element to three field nodes as shown in Figure 2. Using the DSG3 formu lation[35] for each sub-triangle, the memb rane, bending and shear strains in 3 sub-triangles are then obtained, respectively, by = ε e∆ 0 j Bm∆ j de , j = 1, 2, 3 (16) κe∆ j = Bb∆ j de , j = 1, 2, 3 (17) γe∆ j = Bs∆ j de , j = 1, 2, 3 (18) where de is the vector containing the nodal degrees of freedom of the element; m∆ B j , Bb∆ j , Bs∆ j , j=1,2,3, are 22 T. Nguyen-Thoi et al.: Dynamic Responses of Composite Plates on the Pasternak Foundation Subjected to a M oving M ass by a Cell-based Smoothed Discrete Shear Gap (CS-FEM -DSG3) M ethod memb rane, bending and shearing gradient matrices by the DSG3[36] of jth sub-triangle, respectively. sub-triangle 2 1 central point ∆1 O ∆3 ∆2 3 triangular elements. In this model, the mass moves along the line inclined an angle θ compared with x axis. Suppose that at the time point t, the position of the moving mass is (a,b) in the Cartesian coordinate system Oxy. Then, the position of the moving mass ( x , y ) at the time t = t + ∆t are defined as x = v∆t cosθ + a ; y = v∆t sin θ + b (22) where v is velocity of the moving mass and ∆t is step time. ∆ ∆ Figure 2. Three sub-triangles ( 1 , 2 and ∆ 3 ) created from the triangle 1-2-3 in the CS-MIN3 by connecting the central point O with three field nodes 1, 2 and 3 Now, applying the cell-based strain smoothing operation in the CS-FEM [11], the constant membrane, bending and shear strains e∆ ε0 j , e∆ κ j , γ e∆ j , j = 1, 2, 3 are, respectively, used to create element smoothed strains ε e 0 , κ e and γ e on the triangular element Ω , such as: e = ε0e Β= mde ; κ e Β= bde ; γ e Β sde (19) where Β m , Β b and Β s are the smoothed strain gradient matrices, respectively, given by ∑ ∑ = = Β m A= 1e j31 = A∆ j Βm∆ j ; Β b A1e j31 A∆ j Βb∆ j ; (20) ∑ Β s = 1 Ae 3 A∆ j Βs∆ j j =1 Therefore the g lobal stiffness matrix of the CS-FEM -DSG3[35] is co mputed by ∫ ∫ ∫ = K B T DB Ω dΩ + Ω B s T DsB s dΩ + Ω N T w kw N w dΩ (2 1) ∫ ∫ − Ω N w, T x kg N w, x dΩ − Ω N w, T y kg Nw, y dΩ where B = B m B b  Figure 3. Position of a moving mass crossing triangular elements The force vector F is transformed fro m the moving mass at the position (x , y ) into the load at nodes of elements is defined by F = MgNw (23) Note that in the moving mass problem, it is necessary to add a numerical scheme for defin ing which elements containing the moving mass. 6. Numerical Results In this section, various numerical examp les are performed to show the accuracy and stability of the CS-FEM -DSG3 co mpared to the others existing numerical solutions. The section will include three parts. The first two-part aims to verify the accuracy of the CS-FEM-DSG3 by comparing its numerical solutions with those of others available nu merical results for the static and free vibration analyses of composite plates on the Pasternak foundation. The third part aims to illustrate the performance of the present method for the dynamic analysis of composite plates on Pasternak foundation subjected to a moving mass. 5. Transformation of Moving Mass into the Load at Nodes of Elements In the mov ing mass problem, the mass M is considered as a concentrated load which has magnitude P=Mg, where g is the acceleration of gravity. The discretization of problem domain into triangular elements is arbitrary, and hence when a concentrated mass (or a concentrated load) moves with velocity v on a line along the longitudinal direction of the plate, this concentrated load will cross triangular elements arbitrarily. We hence need to define the position of the moving mass crossing triangular elements and to transform the moving mass into the load at nodes of elements at any time t. Figure 3 shows a model of a moving mass crossing Table 1. Non-dimensional defections of composite plate under SSL load wit h a/t = 10, 20, 100 P ly [0/90/0] Met ho d FEM-T3[38] FEM-Q4[38] CS-FEM-DSG3 Reddy[39] 10 0.6281 0.6458 0.6604 0.6693 SSL 20 0.4516 0.4666 0.4909 0.4921 100 0.3714 0.4073 0.4425 0.4337 [0/90/90/0] FEM-T3[38] FEM-Q4[38] CS-FEM-DSG3 Reddy[39] 0.6211 0.6387 0.653 0.6627 0.4503 0.4655 0.4898 0.4912 0.3675 0.4073 0.4423 0.4337 [0/90/0/90/0] FEM-T3[38] FEM-Q4[38] CS-FEM-DSG3 Reddy[39] 0.5868 0.6034 0.6164 0.6277 0.4406 0.4556 0.4792 0.4814 0.3661 0.4069 0.4394 0.4333 International Journal of Composite M aterials 2013, 3(6A): 19-27 23 6.1. Static and Free Vi brati on Anal ysis of Composite Plates 6.1.1. Static Analysis We now consider a simp ly supported square laminate plate (length a, thickness t) subjected to sinusoidally distributed load (SSL) and uniform distributed load (UDL) shown in Figure 4. Material properties are given by E2=1, E1=25E2, G23=0.2E2, G12=G13=0.5E2, ν12=0.25. A non-dimensional w = 100E2wt3 / (qa4 ) is used. Table 1 and Table 2 display the non-dimensional central node deflection of the simp ly supported composite plate subjected to SSL and UDL load with ratios length-to-thickness a/t = 10, 20, 100. It is seen that the results by the CS-FEM-DSG3 agree well with those by the Reddy[39] and are better than those of FEM-T3[38] and FEM-Q4[38]. 6.1.2. Free Vib ration Analysis We analyze a clamped square plates (CCCC) (length a, thickness t) shown in Figure 5 with the material properties are E1 = 40E2, G12 = G13 = 0.6E2, G23 = 0.5E2, ν 12 = 0.25 . A n o n -d imen s io n al frequency parameter is also used, where plate. is the flexu ral rigidity of the Figure 5. A three layers (0/90/0) square composite laminated plate model (a) Sinusoidally distributed load (SSL) Table 3 shows five lowest non-dimensional frequency parameters of a CCCC co mposite plate. Again, it is seen that the results by the CS-FEM-DSG3 agree well with those of[40-42]. In addit ion, Figure 6 plots the shape of six lo west eigenmodes of co mposite plate using the CS-FEM-DSG3. It is seen that the shapes of eigen-modes reveal the real physical modes. Table 3. Five lowest non-dimensional frequency parameters of a CCCC composite plate a/t Met ho d 1 2 5 Liew[40] 4.44 6.64 Zhen[41] 4.54 6.52 Ferreira[42] 4.44 6.64 CS-FEM-DSG3 4.68 7.00 Mode 3 4 7.70 9.18 8.17 9.47 7.69 9.18 8.00 9.62 5 9.74 9.49 9.74 10.11 (b) Uniform distributed load (SSL) 10 Figure 4. Model of a simply supported square laminate plate subjected to sinusoidally distributed load (SSL) and uniform distributed load (UDL) Liew[40] Zhen[41] Ferreira[42] 7.41 10.39 13.91 15.43 15.81 7.48 10.21 14.34 14.86 16.07 7.41 10.39 13.91 15.43 15.81 Table 2. Non-dimensional defections of composite plate under UDL load CS-FEM-DSG3 7.31 10.53 13.41 15.55 15.83 wit h a/t = 10, 20, 100 P ly [0/90/0] Met ho d FEM-T3[38] 10 0.9639 UDL 20 0.6989 100 0.5744 20 Liew[40] Zhen[41] 10.9 5 14.03 20.39 23.20 24.99 11.0 0 14.06 20.32 23.50 25.35 FEM-Q4[38] CS-FEM-DSG3 0.9874 1.0127 0.7195 0.7565 0.6307 0.6787 Ferreira[42] 10.9 5 14.03 20.39 23.20 24.98 Reddy[39] 1.0219 0.7572 0.6697 CS-FEM-DSG3 11.0 3 14.18 20.73 23.36 25.25 [0/90/90/0] FEM-T3[38] 0.9641 0.7085 0.5795 FEM-Q4[38] CS-FEM-DSG3 0.9883 1.0111 0.7302 0.7676 0.643 0.6965 10 0 Liew[40] 14.6 6 17.61 24.51 35.53 39.15 Reddy[39] [0/90/0/90/0] FEM-T3[38] FEM-Q4[38] CS-FEM-DSG3 Reddy[39] 1.025 0.912 0.935 0.9553 0.9727 0.7694 0.6966 0.7182 0.7547 0.7581 0.6833 0.5812 0.6465 0.6961 0.6874 Zhen[41] 14.6 0 17.81 25.23 37.16 38.52 Ferreira[42] 14.4 3 17.39 24.31 35.40 37.78 CS-FEM-DSG3 13.7 1 16.83 23.93 33.78 35.06 24 T. Nguyen-Thoi et al.: Dynamic Responses of Composite Plates on the Pasternak Foundation Subjected to a M oving M ass by a Cell-based Smoothed Discrete Shear Gap (CS-FEM -DSG3) M ethod Mode 1 Mode 3 Mode 2 Mode 4 of various foundation coefficients. Figure 8, Figure 9 and Figure 10 p lot the deflection of the first free v ibration modes of the plate on the Pasternak foundation at the middle line along the longitudinal direction x. It can be seen that when the stiffness of foundation becomes stiffer, the deflect ions of modeshape of the plate on the elastic foundation change significantly co mparing with those of the plate without foundation. Table 4. Three lowest non-dimensional frequency paramet ers of a simply supported composite plate on Pasternak foundation Fo un dat io n k1 = 100, k2 = 0 Met ho d Ref[43] Ref[8] CS-FEM-DSG3 1 21.60 21.38 21.64 Mode 2 - 25.27 26.23 3 66.83 61.70 k1 = 100, k2 = 10 Ref[43] Ref[8] CS-FEM-DSG3 25.76 25.57 25.20 41.68 41.21 72.83 70.74 Mode 5 Mode 6 Figure 6. Shape of six lowest eigenmodes of composite plate[0/90/0] by the CS-FEM-DSG3. 6.2. Free Vi brati on Analysis of Composite Plates on Pasternak Foundati on We now analy ze a simp ly supported composite plate on Pasternak foundation (length a, thickness t) shown in Figure 7 with the material properties are E2=10.3e9; E1=40E2; G23=0.5E2; G13=0.6E2; ν12 = 0.25 ; G12=0.6E2; a/t = 100. A n o n -d imen s io n al frequency parameter ( ) ϖ = ω a2 /t ρ / E2 is also used. Figure 8. Deflection of the first mode of the plate on the Pasternak foundation at middle line along the longitudinal direction x Figure 7. Model of composite laminated plate on Pasternak foundation Table 4 shows three lowest frequencies of composite plate on Pasternak foundation. It is observed that the results of CS-FEM -DSG3 agree well with the reference solutions in[43,8]. We next study the deflection of free vibrat ion modes of the plate on the Pasternak foundation corresponding to three sets Figure 9. Deflection of the second mode of the plate on the Pasternak foundation at middle line along the longitudinal direction x International Journal of Composite M aterials 2013, 3(6A): 19-27 25 Now, a parametric examination by the CS-FEM-DSG3 is conducted to determine the effects of various parameters on the dynamic response of the composite plates on the Pasternak foundation subjected to a moving mass. The variation of the deflect ion of middle line along the length of plate by the CS-FEM-DSG3 with various foundation coefficients, is shown in Figure 11 and Figure 12. The results show that when the stiffness of foundation becomes stiffer, the deflection of the plate becomes smaller, as expected. 7. Conclusions Figure 10. Deflection of the third mode of the plate on the Pasternak foundation at middle line along the longitudinal direction x 6.3. Dynamic Analysis of Composite Pl ates on Pasternak Foundati on Subjected to a Moving Mass In this section, model of co mposite plate is similar to section 6.2. We consider a concentrated mass M=1000kg moving with velocity v=20m/s on the middle line along the longitudinal direction x o f a co mposite plate with the simp ly supported boundary. The paper presents an incorporation of the orig inal CS-FEM -DSG3 with spring systems for dynamic analyses of composite plates on the Pasternak foundation subjected to a moving mass. The composite plate-foundation system is modeled as a discretization of triangular plate elements supported by discrete springs at the nodal points representing the Pasternak foundation. The position of the moving mass with specified horizontal velocity on triangular elements at any time is defined and transformed into loads at nodes of elements. The accuracy and reliability of the proposed method is verified by comparing its numerica l solutions with those of others available nu merical results. A examination of effects of various parameters on the dynamic response of the composite plates on the Pasternak foundation subjected to a moving mass is conducted and gives the expected results. ACKNOWLEDGEMENTS This research is funded by Vietnam National Un iversity HoCh iMinh City (VNU-HCM ) under grant number B2013-20-07. Figure 11. Effect of k1 to deflection of middle line of the composite plate on Pasternak foundation when the mass moves to the middle position of the composite plate REFERENCES [1] Thompson, W. E., 1986, Analysis of dynamic behavior of roads subject to longitudinally moving loads., HRB, 39, 1-24. [2] Gbadeyan, J. A., Oni, S. 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