Kỷ yếu hội nghị khoa học công nghệ toàn quốc khí - Lần thứ IV ISOGEOMETRIC APPROACH FOR STATIC ANALYSIS OF LAMINATED COMPOSITE PLATES PHƯƠNG PHÁP ĐẲNG HÌNH HỌC CHO PHÂN TÍCH TĨNH TẤM COMPOSITE NHIỀU LỚP Nguyen Thi Bich Lieu1a, Nguyen Xuan Hung2b HCMC University of Technology and Education, Vietnam Vietnamese-German University, Binh Duong Province, Vietnam a lieuntb@hcmute.edu.vn; bhung.nx@vgu.edu.vn ABSTRACT A generalized higher-order shear deformation theory for static analysis of laminated composite plates using isogeometric analysis (IGA) is presented The present theory not only is derived from the classical plate theory (CPT) but also includes the first-order shear deformation theory (FSDT) The displacement field depends on arbitrary distributed function The shear locking phenomenon can be ignored and hence the shear stress free surface conditions are naturally satisfied Although it has same number of degrees of freedom as the FSDT, it does not require shear correction factors Galerkin weak form of static analysis model for laminated composite plates is used to obtain the discreted system of equations It can be solved by isogeometric approach based on the non-uniform rational B-splines (NURBS) basic functions Two numerical examples of the laminated composite plates including symmetric and non-symmetric with various boundary conditions are presented to illustrate the effectiveness of the proposed method compared to other methods reported in the literature Keywords: Laminated composite plates, isogeometric analysis (IGA), higher-order shear deformation theory, NURBS, static analysis TÓM TẮT Một lý thuyết biến dạng cắt tổng quát bậc cao cho phân tích tĩnh composite nhiều lớp sử dụng phương pháp phân tích đẳng hình học (IGA) đưa Lý thuyết không suy từ lý thuyết cổ điển (CPT) mà bao gồm lý thuyết biến dạng cắt bậc (FSDT) Nó định nghĩa cho trường chuyển vị phụ thuộc vào hàm phân bố Không có tượng khóa cắt điều kiện ứng suất cắt không tự nhiên thỏa mãn Mặc dù lý thuyết đề xuất có bậc tự với FSDT không yêu cầu hệ số hiệu chỉnh cắt Dạng yếu Galerkin mô hình phân tích tĩnh cho composite nhiều lớp sử dụng để đạt hệ thống phương trình rời rạc Nó giải phương pháp đẳng hình học dựa hàm sở NURBS Hai ví dụ số composite nhiều lớp bao gồm đối xứng bất đối xứng với điều kiện biên khác đưa để minh họa hiệu phương pháp đề xuất so với phương pháp khác Từ khóa: composite nhiều lớp, phân tích đẳng hình học (IGA), lý thuyết biến dạng cắt bậc cao, hàm NURBS, phân tích tĩnh INTRODUCTION Due to its dominant role in many engineering structures and modern industries, laminated composite and sandwich plates are widely used in adiverse array of structures in areas such as aviation, shipbuilding, civil engineering and so on [1] Therefore, the development of efficient and reliable mathematical models, deformation theories, and analysis 758 Kỷ yếu hội nghị khoa học công nghệ toàn quốc khí - Lần thứ IV methods to predict the short and long-term behavior of the multilayer composite structures is extremely important A large number of plate theories have been developed to significantly contribute to advances in computational mechanics of the plate problem For instance, the classical plate theory (CPT) [2] is a highly effective means of analyzing of thin plates with no accounting transverse shear strains Emergence of the first-order shear deformation theory (FSDT) [3] has generally been viewed as an improvement over the CPT for both moderately thick and thin plates which takes shear effect into account The generalized displacement field in FSDT is quite simple However, a shear correction factor (SCF) is required to rectify the unrealistic shear strain energy component To overcome the limitations of the FSDT, the higher-order shear deformation theories (HSDTs) have been then developed These theories disinterest SCFs and give more accurate and stable solutions (e.g., inter-laminar stresses and displacements) than the FSDT ones These higher-order shear deformation theories include, third-order shear deformation theory (TSDT) [4], fifth-order shear deformation theory (FiSDT) [5], trigonometric shear deformation theory (TrSDT) [6], exponential shear deformation theory (ESDT) [7]and others The HSDT model requires C1-continuity of generalized displacement field leading to the second-order derivative of the stiffness formulation and thus causes difficulties in the standard finite element formulations In recent years, C1-continous elements based on mesh free method [8] were proposed to solve the plate and shell problems In this paper, we demonstrate that the C1-continuous elements can be easily achieved by adopting isogeometric analysis without any additional variables As we knew, finite element method (FEM) is an efficient computational tool for various classes of engineering problems and is the reliable choice for solving partial differential equations in the complex domains However, it also existssomedis advantages depended on meshing process Therefore, Hughes et al.developed a highly effective numerical technique so called isogeometric analysis (IGA)[9] that is capable of integrating finite element analysis (FEA) into conventional NURBS-based CAD design tools Following a decade of development, isogeometric analysis has surpassed the standard finite elements in terms of effectiveness and reliability for problems from simple to complex In IGA, the non-uniform rational B-splines (NURBS) basis functions are used not only for geometric description but also for approximation of the displacement field, subsequently allowing us to describe the curved geometry precisely by using only a few elements and yielding solutions with a higher accuracy Some remarkable references for plates can be listed as Kirchhoff plates [10], Mindlin-Reissner plates [11], plates based on HSDT [12] and plates based on the layer wise theory [13] In this paper, a higher-order displacement fields in which displacement field is defined in general form of distributed functions varying across the plate thickness The finite element formulation based on the HSDT requires elements with at least C1-inter-element continuity It is difficult to achieve such elements for free-form geometries when using the standard Lagrangian polynomials as basis functions However, in IGA can be easily obtained because NURBS basis functions are Cp-1 continuous Two numerical examples are provided to illustrate the effectiveness and reliability of the present method in comparison with other results from the literature 759 Kỷ yếu hội nghị khoa học công nghệ toàn quốc khí - Lần thứ IV A GENERALIZED HIGHER-ORDER SHEAR DEFORMATION PLATE THEORY Figure Plate geometry and coordinate system Let Ω be the domain in R2 occupied by the mid-plane of the plate and u , v , w and β =(β x ;β y) T denote the displacement components in the x; y; z directions and the rotations in the x-z and y-z planes (or the-y and the-x axes), respectively Figure shows the geometry of plate and coordinate system A higher-order shear deformation theory derived from the classical plate theory is defined as follows [14]: u ( x, y , z , t ) = u0 ( x, y , t ) − z v ( x, y , z , t ) = v0 ( x, y , t ) − z ∂w ( x, y , t ) ∂x ∂w ( x, y , t ) ∂y + f ( z ) β x ( x, y , t ) + f ( z ) β y ( x, y , t ) w ( x, y , z , t ) = w ( x, y , t ) ; h  −h ≤z≤   2  (1) where f (z) is shape function determining the distribution of the transverse shear strains and stresses through the thickness of plates This distributed function is chosen so that tangential stress-free boundary conditions at the top and bottom surfaces of the plates are satisfied In the present formulation, if distributed function f(z) is chosen to be equal zero, the higher-order shear deformation theory will take the form of classical plate theory (CPT) (see in Eq.(2)) u (= x , y , z ) u0 ( x , y ) − z ∂w ∂x h ∂w ;  − h v(= x, y , z ) v0 ( x, y ) − z  ≤z≤  2 ∂y  (2) w( x, y , z ) = w ( x, y ) By defining f(z) = z and substituting φ x = −w, x + β x into Eq (1), the first order shear deformation theory (FSDT) is obtained as u (= x, y , z ) u0 ( x, y ) + zφ x −h h v(= x, y , z ) v0 ( x, y ) + zφ y ;  ≤z≤  2  w( x, y , z ) = w ( x, y ) In this paper, we use Eq.(1) with some functions f(z) in Table 760 (3) Kỷ yếu hội nghị khoa học công nghệ toàn quốc khí - Lần thứ IV Table 1:Various forms of shape function and its derivatives Model f(z) Reddy (TSDT)[4] z− Arya (TrSDT) [6] f'(z) 4z2 z 1− 4  h 3h πz    h  π sin  Karama (ESDT)[7]  ze Nguyen (FiSDT) [5] z− 2 h Thai [14]  −2 z  h h 2 πz    h  cos  z   z   −2 h  − e     h   z + h z − h z + 10 h z (1 − ( z )2 ) / (1 + ( z )2 ) h tan −1 ( z ) − z h h h The in-plane strain vector ε p is thus expressed by the following equation ε= [ε xx ε yy γ xy ]T= ε + zε1 + f ( z )ε p (4) and the transverse shear strain vector γ has the following form γ = γ xz (5) T γ yz  = f ' ( z )ε1s where f ' ( z ) is derivative of f ( z ) function and  u0, x    ε =  v0, y  , ε1= v0, x + u0, y     − w, xx   β x, x  βx       − w, yy  , ε =  β y , y  , ε s =  β   y  −2 w, xy     β y, x + β x, y    (6) By neglecting σ z for each orthotropic layer, the constitutive equation of kth layer in the local coordinate system derived from Hooke’s law for a plane stress is given by σ 1k   Q Q12  k   11 σ  Q12 Q22  k τ 12  =   k  0 τ 13    τ k    23  0 Q66 0 0 Q55 Q45     Q54  Q44  k  ε1k   k ε2   k γ 12   k γ 13  γ k   23  (7) in which reduced stiffness components, Qijk , are expressed by k Q11 = E1k k ; Q12 = k − ν 12k ν 21 ν 12k E2k E2k k k k k k k k ; Q ; Q33 = = G= G= G23 22 12 ; Q55 13 ; Q 44 k k − ν 12k ν 21 − ν 12k ν 21 (8) where E1 , E2 are the Young’s modulus in the and directions, respectively; G12 , G13 , G23 are the shear modulus in the 1-2, 3-1, 2-3 planes, respectively and ν ij are the Poisson’s ratios 761 Kỷ yếu hội nghị khoa học công nghệ toàn quốc khí - Lần thứ IV The stress - strin relationship in the global reference system (x,y,z) is computed by k σ xx    k   Q11 Q12 σ yy  Q12 Q22  k    τ xy  = Q61 Q62  k   0  τ xz    k τ   0  yz  0   0  0   Q55 Q54  Q45 Q44  Q16 Q26 Q66 0 k k ε xx   k  ε yy   k  γ xy   k γ xz  γ k   yz  (9) where Qijk is the transformed material constant matrix A weak form of the static model for the plates under transverse loading q can be written as ∫ δε Ω T p Dε p dΩ + D γdΩ ∫ δ wq dΩ ∫ δγ = (10) T s Ω Ω where q0 is the transverse loading per unit area and A B E  D =  B D F   E F H  (11) in which ( Aij , Bij , Dij , Eij , Fij , H ij ) = ∫ h/2 −h/2 ( Dsij ) = ∫ h/2 (1, z , z , f ( z ), zf ( z ), f ( z ))Qij dz i, j = 1, 2,6 (12)  f ' ( z )  Qij dz i, j = 4,5 −h/2  AN ISOGEOMETRIC LAMINATED PLATE FORMULATION GENERALIZED HIGER-ORDER SHEAR DEFORMATION THEORY USING By using the NURBS basis functions defined in [9,14], both the description of the geometry (or the physical point) and the displacement field u of the plate are approximated as follows m× n m× n A A x h (ξ ,η ) = ∑ RA (ξ ,η )PA and u h (ξ ,η ) = ∑ RA (ξ ,η )q A (13) where n×m is the number of basis functions, and xT = ( x y ) is the physical coordinate vector InEq.(13), RA (ξ ,η ) is a rational basic function, q A = u0 A v0 A wA PA is the control point and T β xA β yA  is the vector of nodal degrees of freedom associated with control point A By substituting Eq (13)with Eq.(4), the in-plane and shear strains can be rewritten as m× n ε p γ  = ∑ B mA T A =1 BbA1 BbA2 T B sA  q A in which 762 (14) Kỷ yếu hội nghị khoa học công nghệ toàn quốc khí - Lần thứ IV 0 − R 0 A, xx   B = 0 − RA, yy 0  ,   0 −2 RA, xy 0  b1 A B b2 A 0 0 R  A, x   R A, y  , = 0 0   0 0 RA, y RA, x  R  A, x  0 RA  s m , = B A = B   A   0 0 RA   RA, y R A, y RA, x 0 0  0 0  0  (15) By substituting Eq (14)with Eq.(10),the formulation of static analysis is obtained in the following form Kq = F (16) where the global stiffness matrix K is given by  B m T    K ∫  Bb1  = Ω    Bb   m   A B E  B         b1 s T s s B D F B B D B + ( ) dΩ       E F H  Bb      (17) and the load vector F is calculated as F = ∫ Ω q0 RdΩ (18) in which R = 0 RA 0  (19) RESULTS AND DISCUSSIONS This section, due to the limitation of number of pages of the conference, we consider two examples through a series of benchmark problems for laminated composite plates including one for symmetric plate and one for anti-symmetric plate 4.1 Four-layer [00/900/900/00] square laminated plate under a sinusoidally distributed load A four-layer fully simply supported square laminated plate subjected to a sinusoidal πx πy pressure defined as q(x, y) = q0 sin( )sin( ) is considered, as shown in Figure All layers a b of the laminated plate are assumed to be of the same thickness and made of the same linearly elastic composite materials The length to width ratio is a/b = and the length to thickness ratios are a/h = 4, 10, 20 and 100, respectively Material is used = E1 25 E2 , G= G= 0.5 E2 , G= 0.2 E2 ,ν= 0.25 12 13 23 12 The normalized displacement and stresses are defined as a a h2 a a h h2 a a h w (100 E2 h3 )= w( , , 0) / qa ; σ xx = = σ ( , , ); σ σ yy ( , , ) xx yy 2 qa qa 2 2 2 = σ xy h2 h h a h a = σ xy (0, 0, ); σ xz = σ xz (0, , 0); σ yz σ yz ( , 0, 0) qa qa qa 2 763 (20) Kỷ yếu hội nghị khoa học công nghệ toàn quốc khí - Lần thứ IV Figure Geometry of a laminated plate under a sinusoidally distributed load Table 2: Normalized displacement and stresses of a simply supported [00/900/900/00] square laminated plate under a sinusoidally distributed load a/h Method w σ xx σ yy σ xy τ xz τ yz Exact 3D [15] 1.9540 0.7200 0.6660 0.0467 0.2700 0.2910 IGA-Reddy[4] 1.8936 0.6607 0.6300 0.0440 0.2064 0.2389 IGA-Arya[6] 1.9088 0.6796 0.6332 0.0450 0.2162 0.2462 IGA-Thai[14] 1.9258 0.7164 0.6381 0.0467 0.2396 0.2624 IGA-Soldatos[12] 1.8920 0.6644 0.6316 0.0439 0.2055 0.2382 Exact 3D [15] 0.7430 0.5590 0.4030 0.0276 0.3010 0.1960 IGA-Reddy[4] 0.7147 0.5440 0.3881 0.0267 0.2640 0.1530 IGA-Arya[6] 0.7198 0.5486 0.3905 0.0270 0.2787 0.1588 IGA-Thai[14] 0.7272 0.5552 0.3937 0.0273 0.3133 0.1704 IGA-Soldatos[12] 0.7142 0.5449 0.3881 0.0267 0.2627 0.1526 Exact 3D [15] 0.5170 0.5430 0.3090 0.0230 0.3280 0.1560 IGA-Reddy[4] 0.5060 0.5383 0.3038 0.0228 0.2825 0.1234 IGA-Arya[6] 0.5070 0.5395 0.3090 0.0228 0.2989 0.1272 IGA-Thai[14] 0.5098 0.5412 0.3058 0.0229 0.3372 0.1366 IGA-Soldatos[12] 0.5059 0.5385 0.3038 0.0228 0.2810 0.1231 100 Exact 3D [15] 0.4347 0.5390 0.2710 0.0214 0.3390 0.1410 IGA-Reddy[4] 0.4342 0.5379 0.2704 0.0213 0.2897 0.1116 IGA-Arya[6] 0.4344 0.538 0.2705 0.0213 0.3069 0.1148 IGA-Thai[14] 0.4345 0.538 0.2705 0.0213 0.3467 0.1229 IGA-Soldatos[12] 0.4343 0.5379 0.2704 0.0213 0.2882 0.1114 10 20 764 Kỷ yếu hội nghị khoa học công nghệ toàn quốc khí - Lần thứ IV Figure Comparison of the normalized stress distributions through the thickness of a four-layer [00/900/900/00] laminated composite square plate (a/h = 4) Table displays the obtained results using various distributed functions with IGA for the normalized displacement and stresses The obtained results including some f(z) functions of Reddy[4] (IGA-Reddy), Arya [6] (IGA-Arya), Thai et al.[14] (IGA-Thai), Soldatos [12] (IGA- Soldatos) and the exact 3D elasticity approach of Pagano [15] According to this table, IGA conforms well to the exact 3D elasticity solution for all ratios a/h, especially for thick plates For a thick plate with a/h = and 10, the IGA-Thai [14] is accurate than other solutions using IGA It even moves beyond TSDT by Reddy [4] Figure plots the distribution of stresses through the thickness of a four-layer square plate with a/h = Obviously, obtained results have the similarity with each other 4.2 Two-layer [00/900] square laminated plate under a sinusoidally distributed load Let us consider a two-layer (00/900) squareplate subjected to sin load and material as same above example witha/b = 1; a/h = 10 Three boundary conditions are presented SCSC, SSSS SFSF, where S = simply supported, C= clampedand F = free 765 Kỷ yếu hội nghị khoa học công nghệ toàn quốc khí - Lần thứ IV The normalized displacement is defined as: w = (100 E2 h3 ) w ( a / 2, a / 2,0 ) qa (21) Table 3: Normalized displacement w and stresses of a [00/900] square laminated plate under a sinusoidally distributed load with various boundary conditions BC w σ xx σ yy τ xy τ yz 0.6490 -0.4653 0.3888 0.0221 - IGA-Reddy[4] 0.6146 -0.4917 0.3778 0.0183 0.1530 IGA-Nguyen[5] 0.5971 -0.5009 0.3733 0.0075 0.1472 IGA-Thai[14] 0.5980 -0.4993 0.3732 0.0076 0.1457 1.2270 -0.7304 0.7309 0.0497 - IGA-Reddy[4] 1.2161 -0.7446 0.7446 0.0533 0.2837 IGA-Nguyen[5] 1.2044 -0.7492 0.7492 0.0534 0.2766 IGA-Thai[14] 1.2049 -0.7482 0.7482 0.0533 0.2735 2.0260 -0.2503 1.2100 0.0119 - IGA-Reddy[4] 1.9925 -0.2606 1.2262 0.0098 0.3993 IGA-Nguyen[5] 1.9736 -0.2632 1.2315 0.0032 0.3872 IGA-Thai[14] 1.9744 -0.2626 1.2301 0.0031 0.3829 Method SCSC AS-Vel[16] SSSS AS-Vel[16] SFSF AS-Vel[16] Figure 4: The normalized stress distributions through the thickness of a two-layer [00/900] laminated composite square plate (a/h = 10) with IGA-Reddy 766 Kỷ yếu hội nghị khoa học công nghệ toàn quốc khí - Lần thứ IV Table displays the obtained results from IGA-Reddy, IGA-Nguyen and IGA-Thai These results are compared with analytical solutions of Vel et al [16] It can be seen that the normalized displacement and stresses are close to those of the analytical solutions for all various boundary conditions, especially IGA-Reddy solution Figure illustrates the normalized stress distributions through the thickness with a/h = 10 using IGA-Reddy for three differential boundary conditions CONCLUSIONS This work presents an isogeometric finite element method for static of laminated composite plates The results obtained using the generalized higher-order shear deformation theory showed high reliability and matched well for all test cases from thin to thick plates when compared with analytical solutions and exact three-dimensional elasticity As described in this paper, the distributed function f(z) along the thickness of plate can be easily modified to obtain the most optimum solution for the targeted engineering problem The choice of the distributed function f(z) is still an open question The authors want to emphasize the higherorder continuity of NURBS basic functions and the flexibility and generality of generalized displacements when combining IGA with arbitrary distribution functions in this paper REFERENCES [1] Reddy, J N, Mechanics of Laminated Composite Plates and Shells Theory and Analysis Second edition, CRC Press, New York, 2004 [2] Whitney, J M., The effect of boundary conditions on the response of laminated composites Journal of Composite Materials, 1970, Vol 4, p 192–203 [3] Nguyen-Xuan, H., Rabczuk, T., Nguyen-Thanh, N., Nguyen-Thoi, T.&Bordas, T, A node-based smoothed finite element method (NS-FEM) for analysis of Reissner– Mindlin plates Computational Mechanics, 2010, Vol 46 p 679-701 [4] Reddy, J.N., A simple higher-order theory for laminated composite plates Journal of Applied Mechanics 1984, Vol 51, p 745–752 [5] H Nguyen-Xuan, H C Thai, and T Nguyen-Thoi Isogeometric finite element analysis of composite sandwich plates using a higher order shear deformation theory Composite Part B, 2013, Vol 55, p 558–574 [6] Arya, H., Shimpi, R P.,& Naik, N K., A zig-zag model for laminated composite beams Composite Structures, 2002, Vol 56, p 21–24 [7] Karama, M., Afaq, K S.,&Mistou, S., Mechanical behavior of laminated composite beam by new multi-layered laminated composite structures model with transverse shear stress continuity International Journal of Solids and Structures, 2003, Vol p 1525– 1546 [8] Rabczuk, T., Areias, P M A.,& Belytschko, T., A meshfree thin shell method for nonlinear dynamic fracture International Journal for Numerical Methods in Engineering, 2007, Vol 72, p 524–548 [9] Hughes T.J.R., Cottrell J.A., &Bazilevs Y., Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Computer Methods in Applied Mechanics and Engineering 2005, Vol 194, p.4135–95 [10] Shojae, S., Izadpanah, E., Valizade, N., & Kiendl, J., Free vibration analysis of thin plates by using a NURBS-based isogeometric approach, Finite Elements in Analysis and Design, 2012, Vol 61, p 23–34 767 Kỷ yếu hội nghị khoa học công nghệ toàn quốc khí - Lần thứ IV [11] Chien, H T., Nguyen-Xuan, H., Nguyen-Thanh, N., Le, T-H., Nguyen-Thoi, T., & Rabczuk, T., Static, free vibration, and buckling analysis of laminated composite Reissner–Mindlin plates using NURBS-based isogeometric approach, International Journal for Numerical Methods in Engineering, 2012, Vol 91, p 571–603 [12] Nguyen-Xuan, H., Thai H.C., &Nguyen-Thoi, T., Isogeometric finite element analysis of composite sandwich plates using a new higher order shear deformation theory, Composite part B, 2013,Vol 55, p 558-574 [13] Chien, H.T., Ferreira, A.J.M., Carrera, E., &Nguyen-Xuan, H., Isogeometric analysis of laminated composite and sandwich plates using a layerwise deformation theory, Composite Structures, 2013, Vol 104, p.196–214 [14] Thai, H.C., Ferreira, A.J.M., Rabczuk T., Bordas S.P.A., &Nguyen-Xuan, H., Isogeometric analysis of laminated composite and sandwich plates using a new inverse trigonometric shear deformation theory European Journal of Mechanics - A/Solids 2014, Vol 43, p 89-108 [15] Pagano, N.J., Exact solutions for rectangular bidirectional composites and sandwich plates, Journal of Composite Materials, 1970, Vol 4, p 20–34 [16] Vel, S.S., Batra, R.C., Analytical solutions for rectangular thick laminated plates subjected to arbitrary boundary conditions American Institute of Aeronautics and Astronautics Journal 1999, Vol 37, p 1464–1473 AUTHOR’S INFORMATION Nguyen Thi Bich Lieu, Faculty of Civil and Applied Mechanics, HCMC University of Technology and Education, lieuntb@hcmute.edu.vn, 0938.839.657 Nguyen Xuan Hung, Department of Computational Engineering, Vietnamese-German University, Binh Duong Province, hung.nx@vgu.edu.vn, 0906.682.393 768 [...]... Nguyen-Xuan, H., Thai H.C., &Nguyen-Thoi, T., Isogeometric finite element analysis of composite sandwich plates using a new higher order shear deformation theory, Composite part B, 2013,Vol 55, p 558-574 [13] Chien, H.T., Ferreira, A.J.M., Carrera, E., &Nguyen-Xuan, H., Isogeometric analysis of laminated composite and sandwich plates using a layerwise deformation theory, Composite Structures, 2013, Vol 104,... &Nguyen-Xuan, H., Isogeometric analysis of laminated composite and sandwich plates using a new inverse trigonometric shear deformation theory European Journal of Mechanics - A/Solids 2014, Vol 43, p 89-108 [15] Pagano, N.J., Exact solutions for rectangular bidirectional composites and sandwich plates, Journal of Composite Materials, 1970, Vol 4, p 20–34 [16] Vel, S.S., Batra, R.C., Analytical solutions for rectangular...Kỷ yếu hội nghị khoa học và công nghệ toàn quốc về cơ khí - Lần thứ IV [11] Chien, H T., Nguyen-Xuan, H., Nguyen-Thanh, N., Le, T-H., Nguyen-Thoi, T., & Rabczuk, T., Static, free vibration, and buckling analysis of laminated composite Reissner–Mindlin plates using NURBS-based isogeometric approach, International Journal for Numerical Methods in Engineering, 2012, Vol 91,... solutions for rectangular thick laminated plates subjected to arbitrary boundary conditions American Institute of Aeronautics and Astronautics Journal 1999, Vol 37, p 1464–1473 AUTHOR’S INFORMATION 1 Nguyen Thi Bich Lieu, Faculty of Civil and Applied Mechanics, HCMC University of Technology and Education, lieuntb@hcmute.edu.vn, 0938.839.657 2 Nguyen Xuan Hung, Department of Computational Engineering,