DSpace at VNU: Isogeometric analysis of laminated composite plates based on a four-variable refined plate theory

Engineering Analysis with Boundary Elements 47 (2014) 68–81 Contents lists available at ScienceDirect Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound Isogeometric analysis of laminated composite plates based on a four-variable refined plate theory Loc V Tran a, Chien H Thai b, Hien T Le c, Buntara S Gan d, Jaehong Lee a, H Nguyen-Xuan e,n a Department of Architectural Engineering, Sejong University, 98 Kunja Dong, Kwangjin Ku, Seoul 143-747, South Korea Division of Computational Mechanics, Ton Duc Thang University, Ho Chi Minh City, Vietnam c Department of Naval Architecture and Marine Engineering, Ho Chi Minh City University of Technology, VNU-HCMC, 268 Ly Thuong Kiet Street, Ho Chi Minh, Vietnam d Department of Architecture, College of Engineering, Nihon University, Koriyama City, Fukushima Prefecture, Japan e Department of Computational Engineering, Vietnamese-German University, Binh Duong New City, Vietnam b art ic l e i nf o a b s t r a c t Article history: Received 15 October 2013 Received in revised form 19 May 2014 Accepted 30 May 2014 In this paper, a simple and effective formulation based on isogeometric approach (IGA) and a four variable refined plate theory (RPT) is proposed to investigate the behavior of laminated composite plates RPT model satisfies the traction-free boundary conditions at plate surfaces and describes the non-linear distribution of shear stresses without requiring shear correction factor (SCF) IGA utilizes basis functions, namely B-splines or non-uniform rational B-splines (NURBS), which reveals easily the smoothness of any arbitrary order It hence handles easily the C1 requirement of the RPT model Approximating the displacement field with four degrees of freedom per each node, the present method retains the computational efficiency while ensuring the reasonable accuracy in solution & 2014 Elsevier Ltd All rights reserved Keywords: Plate Composite Isogeometric analysis Refined plate theory Meshfree method Introduction Laminated composite plates are being increasingly used in various fields of engineering such as aircrafts, aerospace, vehicles, submarine, ships, buildings, etc., because they possess many favorable mechanical properties such as high stiffness to weight and low density Therefore a lot of research about their behaviors such as deformable characteristic, stress distribution, natural frequency and critical buckling load under various conditions haas never been stopped Pagano [1] initially investigated the analytical three-dimensional (3D) elasticity method to predict the exact solution of simple static problems Noor et al [2,3] further developed 3D elasticity solution formulas for stress analysis of composite structures It is well known that such an exact 3D approach is the most potential tool to obtain the true solution of plates However, it is not easy to solve practical problems with complex (or even slightly complicated) geometries and boundary conditions In addition, each layer in the 3D elasticity theory is modeled as one 3D solid and hence the computational cost of n Corresponding author E-mail address: hung.nx@vgu.edu.vn (H Nguyen-Xuan) http://dx.doi.org/10.1016/j.enganabound.2014.05.013 0955-7997/& 2014 Elsevier Ltd All rights reserved laminated composite plate analyses is increased significantly Hence, many equivalent single layer (ESL) plate theories with suitable assumptions [4] have been then proposed to transform the 3D problem to a 2D one Among the ESL plate theories, the classical laminate plate theory (CLPT) based on the Love–Kirchoff assumptions was first proposed Due to ignoring the transverse shear deformation, CLPT merely provides acceptable results for the thin plate problems The first order shear deformation theory (FSDT) based on Reissner [5] and Mindlin [6], which takes into account the shear effect, was therefore developed In FSDT model, with the linear in-plane displacement assumption through plate thickness, the obtained shear strain/stress distributes inaccurately and does not satisfy the traction free boundary conditions at the plate surfaces The shear correction factors (SCF) are therefore required to rectify the unrealistic shear strain energy part The values of SCF are quite dispersed through many problems and may be difficult to determine [7] To bypass the limitations of the FSDT, many kind of higher-order shear deformable theories (HSDT), which include higher-order terms in the displacement approximation, have then been devised such as third-order shear deformation theory (TSDT) [8–10], trigonometric shear deformation theory [11,12], exponential shear deformation theory (ESDT) [13–15], refined plate theory (RPT) and so on The RPT model was pointed L.V Tran et al / Engineering Analysis with Boundary Elements 47 (2014) 68–81 out by Senthilnathan et al [16] with four unknown variables which is one variable lower than the TSDT model Shimpi et al [17,18] proposed RPT with just only two unknown variables using different distributed functions for the isotropic and orthotropic plates Recently, this model is deeply researched by Thai-Huu et al [19,20] It is worth mentioning that the HSDT models provide better results and yield more accurate and stable solutions (e.g inter-laminar stresses and displacements) [21,22] than the FSDT ones without requiring the SCF However, the HSDT requires the C1-continuity of generalized displacement field leading to the second-order derivative of the stiffness formulation The enforcement of even C1 continuity across inter-element boundaries in standard finite element method is not a trivial task In the efforts to address this difficulty, several C0 continuous elements [23–26] were then proposed or Hermite interpolation function with the C1-continuity was taken into account in the approximation of transverse displacement [4] Such elements may produce extra unknown variables leading to an increase in the computational cost In this paper, we show that C1-continuous elements will be naturally gained by using B-Spline or non-uniform rational B-Spline (NURBS) shape functions without any additional variables The NURBS basis functions are commonly used in the Computer Aided Design (CAD) software to describe the geometry domain [27] They are flexible to make refinement, de-refinement, and degree elevation and gain easily the smoothness of arbitrary continuous order Also, NURBS can be used to approximate meshfree shape functions with a desired order of consistency [28] or to merge into boundary element method to obtain the geometry and traction fields around the boundary [29] Another way, by coupling geometry and approximations via NURBS, Hughes and co-workers have introduced a new method so-called Isogeometric Analysis (IGA) [31] The core idea of IGA is to use same NURBS basis functions for both describing the exact geometry and constructing the finite element formulation [30] The IGA has been well known and widely applied to various practical problems [32–39], etc In this paper, a formulation based on the RPT model and the isogeometric approach for static, free vibration and buckling analysis of laminated composite plates is investigated Some higher-order distributed functions [8,13,14,17] are utilized to describe the higher-order term in the displacement field Several numerical examples are given to show the performance of the proposed method in comparison to others in the literature The paper is outlined as follows Section introduces the RPT for composite plates In Section 3, the formulation of plate theory based on IGA is described The numerical results and discussions are provided in Section Finally, this article is closed with some concluding remarks et al [16] proposed the refined plate theory model with one reduced variable w0 ¼ wb ỵws ; ux; y; zị ẳ u0 ỵ zx ỵ gzịx ỵ w;x ị vx; y; zị ẳ v0 þ zβy þ gðzÞðβy þ w;y Þ; vðx; y; zÞ ẳ v0 zwb;y ỵ gzịws;y wx; yị ẳ wb ỵ ws wx; yị ẳ w0 2 r z r h2 1ị where gzị ẳ 4z3 =3h Þ and the variables u0 ¼ fu0 v0 gT , w0 and β ¼ fβx βy gT are the membrane displacements, the transverse displacement and the rotations in the y–z, x–z planes, respectively By making additional assumptions given in Eq (2), Senthilnathan ð3Þ The relationships between strains and displacements are described by ẳ ẵxx yy xy T ẳ ỵ zb ỵgzịs ẳ ẵ xz yz T ẳ f zịs where ẳ in which 4ị f zị ẳ g zị ỵ 5ị 3 " # wb;xx ws;xx ws;x w w 5; κb ¼ À b;yy 5; κs ¼ s;yy 5; s ẳ ws;y 2wb;xy u0;y ỵ v0;x 2ws;xy u0;x v0;y ð6Þ From Eq (5), an additional condition is needed to satisfy traction-free boundary condition at the top and bottom surfaces of plate It means that f zị ẳ at z ẳ 7h=2 Based on this condition, various distributed functions f ðzÞ in forms: third-order polynomials by Reddy [8] and Shimpi [17], exponential function by Karama [13], sinusoidal function by Arya [14] and are illustrated in Table 2.2 Weak form equations for plate problems A weak form of the static model for the plates under transverse loading f0 can be briefly expressed as Z Z Z T Db d ỵ T Ds d ẳ wf dΩ ð7Þ Ω Ω where A D ¼4B E b B E Ω D F5 F H ð8Þ and the material matrices are given as Aij ; Bij ; Dij ; Eij ; F ij ; H ij Z h=2 ¼ ð1; z; z2 ; gðzÞ; zgðzÞ; g ðzÞÞQ ij dz Z ÀÀh ð2Þ ux; y; zị ẳ u0 zwb;x ỵ gzịws;x Dsij ¼ Regarding the effect of shear deformation, the higher-order terms are incorporated into the displacement field A simple and famous theory for the bending plate is stated as [4] β ¼ À ∇wb where wb and ws are defined as the bending and shear components of deflection, respectively Eq (1) is taken in the simpler form with four unknown variables The refined plate theory 2.1 Displacement field 69 ði; j ẳ 1; 2; 6ị h=2 h=2 h=2 ẵf zị2 Q ij dz i; j ẳ 4; 5Þ ð9Þ in which Q ij are transformed material constants of the kth lamina (see [4] for more detail) Table The various forms of shape function Model f ðzÞ gðzÞ f ðzÞ Reddy [8] À 43z3 =h z À z =h À 2ðz=hÞ2 1À 4z2 =h Karama [13] z À 43z3 =h 5 z À z =h À 2ðz=hÞ2 Arya [14] sin Shimpi [17] ze Àπ Á z h ze sin Àz Àπ Á z Àz h 2 4ð1À 4z =h Þ   À 42 z2 e À 2ðz=hÞ h Àπ Á π cos hz h 70 L.V Tran et al / Engineering Analysis with Boundary Elements 47 (2014) 68–81 The B-spline basis functions N i;p ðξÞ are defined by the following recursion formula: For the free vibration analysis, it can be derived from the following dynamic equation: Z Z Z T ~ T Db dỵ T Ds d ¼ δu~ mudΩ ð10Þ Ω Ω ξ ( where m - the mass matrix is calculated according to the consistent form 3 I0 0 I1 I2 I4 7 m ¼ I0 where I0 ¼ I I I 5 ð11Þ I4 I5 I6 0 I0 Z ðI ; I ; I ; I ; I ; I ị ẳ h=2 À h=2 ρðzÞð1; z; z2 ; gðzÞ; zgðzÞ; g zịịdz as p ẳ 0; N i;0 ị ẳ > < u0 > = u1 ¼ À wb;x ; > > : w ; s;x u2 ¼ > < v0 > = À wb;y ; > > : w ; s;y ξi r ξ oξi þ ) otherwise ð15Þ ð16Þ Fig illustrates the set of one-dimensional and twodimensional B-spline basis functions according to open uniform knot vector Ξ ¼ f0; 0; 0; 0; 0:5; 1; 1; 1; 1g To model exactly curved geometries (e.g circles, cylinders, spheres, etc.), each control point has additional value called an individual weight ζ A [30] We denote Non-uniform Rational B-splines (NURBS) functions which are expressed as > = u3 ¼ > : > ; and a weak form of the plate under the in-plane forces can be formed as Z Z Z T Db dỵ T Ds d ỵ T wN0 wd ¼ ð14Þ Ω if NA ðξ; ηÞ ¼ N i;p ðξÞM j;q ðηÞ ð12Þ ð13Þ Ω Using the tensor product of basis functions in two parametric dimensions ξ and η with two knot vectors Ξ ¼ fξ1 ; ; :::; n ỵ p ỵ g and ẳ f1 ; ; :::; m ỵ q ỵ g, the two-dimensional B-spline basis functions are obtained and > < u1 > = ~ u ¼ u2 ; > :u > ; Àξ N i;p ị ẳ i ỵp i i N i;p ị ỵ i ỵi pỵỵp 1ỵ1 i ỵ Ni ỵ 1;p ị N RA ; ị ẳ mn A A N A ; ị A 17ị A where T ẳ ẵ=x ∂=∂yŠT is the gradient operator and N 0x N 0xy N0 ¼ N 0xy N 0y The NURBS function becomes the B-spline function when the individual weight of control point is constant is a matrix related to the pre-buckling loads 3.2 A novel RPT formulation based on NURBS approximation Using the NURBS basis functions, the displacement field u of the plate is approximated as The composite plate formulation based on NURBS basis functions mÂn uh ; ị ẳ RA ; ịqA where qA ẳ ½u0A v0A wbA wsA ŠT is the vector of nodal degrees of freedom associated with the control point A Substituting Eq (18) into Eq (6), the in-plane and shear strains become A knot vector Ξ ¼ fξ1 ; ξ2 ; :::; n ỵ p ỵ g is dened as a sequence of knot value ξi A R, i ¼ 1; :::n ỵ p An open knot, i.e, the rst and the last knots are repeated pỵ times, is used A B-spline basis function is C1 continuous inside a knot span and Cp À continuous at a single knot Hence, as p Z the present approach always satisfies C1-requirement in based-RPT formulations mÂn b1 T b2 T s T T T ẵT0 Tb Ts Ts T ẳ ẵBm A ị BA ị BA ị BA ị Š qA A¼1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 ð18Þ A 3.1 A brief of NURBS functions 1/2 Fig 1D and 2D B-spline basis functions ð19Þ L.V Tran et al / Engineering Analysis with Boundary Elements 47 (2014) 68–81 in which 0 RA;x 0 RA;xx b1 60 R m 0 R A;y A;yy BA ¼ 5; BA ¼ À RA;y RA;x 0 0 2RA;xy " # 0 RA;xx 0 RA;x 60 0 R s b2 A;yy 5; BA ¼ BA ¼ 0 RA;y 0 2RA;xy Simply supported cross-ply: 07 5; ð20Þ ð21Þ K Mịq ẳ 22ị K cr Kg ịq ẳ 23ị where the global stiffness matrix K is given by m 9T 38 m Z > Z = A B E > = b1 b1 K¼ dΩ þ BsT Ds Bs dΩ B 4B D F B > Ω> Ω : b2 > ; : b2 > ; E F H B B ð24Þ ð25Þ Ω RA where RA 0 > < R1 > = R~ ¼ R ; R ¼ 0 À RA;x 5; > :R > ; 0 R A;x 3 RA 0 0 RA 6 R ¼ 0 À RA;y 5; R ¼ 0 0 0 RA;y 0 and the geometric stiffness matrix reads Z Kg ẳ Bg ịT N0 Bg dΩ RA;x RA;x 0 RA;y RA;y u0 ¼ w b ¼ w s ¼ at left and right edges v0 ¼ wb ¼ ws ¼ at lower and upper edges 32ị Clamped: u0 ẳ v0 ¼ wb ¼ ws ¼ wb;n ¼ ws;n ¼ ð33Þ Results and discussions In this section, we show the performance of the present method – RPT–IGA with various distributed functions as given in Table in analyzing the laminated composite plates We illustrate the present method using the cubic basis functions The following material properties are used for numerical tests: Material I: Material II: Material III: Face sheets 27ị E1 ẳ 131 GPa; E2 ẳ 1:5 GPa; G12 ¼ 6:895 GPa; G13 ¼ 6:205 GPa; ν12 ¼ 0:22; ρ ¼ 1627 kg=m3 Core property (Isotropic) RA E1 ¼ E2 ¼ 6:89 MPa; G12 ẳ G13 ẳ 3:45 MPa; 28ị 12 ẳ 0; ẳ 97 kg=m3 29ị The ratio of the core thickness hc to the face sheet thickness hf is equal to For convenience, the following normalized transverse displacement, in-plane stresses, shear stresses natural frequency and bucking load are expressed as 30ị wẳ 31ị 26ị BgA ẳ at lower and upper edges E1 =E2 ¼ varied; G12 ¼ G13 ¼ 0:6E2 ; G23 ¼ 0:5E2 ; ν12 ¼ 0:25; ρ ¼ 1: à the global mass matrix M is expressed as Z T ~ M ¼ R~ mRdΩ where " u0 ¼ wb ¼ ws ¼ E1 ¼ 25E2 ; G12 ¼ G13 ¼ 0:5E2 ; G23 ¼ 0:2E2 ; ν12 ¼ 0:25; ρ ¼ 1: and the load vector is computed by Z F ¼ q0 RdΩ RA at left and right edges Simply supported angle-ply: Kq ¼ F v0 ¼ w b ¼ w s ¼ 0 Substituting Eq (19) into Eqs (7), (10) and (14), the formulations of static, free vibration and buckling problem are formulated by the following form: where  R¼ 71 # in which ω; λcr A R ỵ are the natural frequency and the critical buckling value, respectively It is observed from Eq (24) that the SCF is no longer required in b2 the stiffness formulation Herein, Bb1 A and BA contain the secondorder derivative of the shape functions Hence, it requires C1-continuous element in approximate formulations As expected, our present formulation based on IGA (as p Z 2) matches well the C1-continuity from the theoretical/mechanical viewpoint of plates [22,37] and also the RPT model 3.3 Essential boundary conditions Various boundary conditions are applied for an arbitrary edge with simply supported (S) and clamped (C) conditions including pffiffiffiffiffiffiffiffiffiffi 102 wE2 h σh τh ; ω ¼ ωa2 =h ρ=E2 ; λcr ¼ λcr a2 =E2 h ;σ ¼ ;τ¼ q0 a q0 a4 q0 a2 4.1 Static analysis 4.1.1 Two-layer [0/90] anti-symmetric square plate Let us consider a simply supported laminated [0/90] square plate with material set I subjected to a sinusoidal pressure q0 sin ðπx=aÞ sin ðπy=aÞ as shown in Fig We first investigate the convergence of the normalization displacement and stresses with length to thickness ratio a/h¼ 10 The plate is modeled with  7, 11  11 and 15  15 cubic elements as shown in Fig The obtained results based on RPT model using IGA (RPT–IGA) with various distributed functions f(z) (as listed in Table 1) are tabulated in Table The relative error compared to analytical solution using 72 L.V Tran et al / Engineering Analysis with Boundary Elements 47 (2014) 68–81 HSDT given by Khdeir and Reddy [42] is given in the parentheses It is observed that the RPT models using third-order polynomials gain the most accuracy displacement It is seen that a slightly fine mesh of 11  11 cubic elements is enough to provide reasonably accurate solutions For illustration, this mesh is therefore used for several following examples Next we investigate the aforementioned problem with various length to thickness ratios a/h ranging from to 100 Table lists the present results in comparison with the 3D solution reported by Pagano [1] and the analytical approach [26] using various plate models (e.g, FSDT, TSDT and RPT) It can be seen that, using equivalent single layer (ESL) plate theories, the present model is in very good agreement with the 3D solution for the errors varying from 8% to 0.3% according to ratio a/h varying from to 100 Furthermore, using four-variable RPT, the present method produces good results in both deflection and axial stress compared to the results derived from the Reddy model and improves significantly the Senthinathal model for the accuracy of stresses In Table 4, we study the behavior of two-layer [0/90] laminate square plate under two types of boundary condition (SSSS and SFSF, F¼free edge) The present results are compared with those published ones derived from the 3D approach of Vel and Batra [41]; the analytical method based on CLPT, FSDT and HSDT models reported by Khdeir and Reddy [42] and HOSNDPT using meshfree method with 18DOFs/node by Xiao et al [40] It can be seen that the obtained results agree very well with the 3D elastic solution [41] It is again observed that RPT–IGA using the third-order distributed functions produces same results which match well with the 3D solution The transverse displacement of the plates is illustrated in Fig according to SFSF and SSSS boundary conditions, respectively Fig depicts the stress distribution through the SSSS plate thickness using various f(z) functions such as TSDT [8], HSDT [17], ESDT [13], SSDT [14] Using RPT model, the in-plane stresses are almost matched together while being slightly different for the out-plane stresses 4.1.2 Five-layer square sandwich plate [0/90/core/0/90] under sinusoidal load A five-layer square sandwich plate with material set III is considered For illustration, third-order distributed function reported by Reddy [8] is adopted Table summarizes the results of the present and analytical approaches As seen, the normalized Fig Square laminate plate under sinusoidal load Fig Meshing and control net (in red color) of the square plate using cubic elements: (a)  7; (b) 11  11; (c) 15  15 (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Table The convergence of the normalized displacement and axial stress of a simply supported laminated [0/90] plate (a/h¼ 10) subjected to a sinusoidal load Method Mesh 7Â7 RPT–IGA Reddy À Á w 2a; 2b Shimpi Arya Karama 11  11 15  15 1.2159 (0.01) 1.2161 ( À 0.01) 1.2161 ( À 0.01) 1.2159 (0.01) 1.2129 (0.25) 1.2093 (0.55) 1.2161 ( À 0.01) 1.2131 (0.24) 1.2096 (0.53) 1.2161 ( À 0.01) 1.2131 (0.24) 1.2096 (0.53) À 0.7351 (1.57) À 0.7421 (0.63) À 0.7443 (0.33) À 0.7351 (1.57) À 0.7366 (1.37) À 0.7379 (1.19) À 0.7421 (0.63) À 0.7436 (0.43) À 0.7449 (0.25) À 0.7443 (0.33) À 0.7458 (0.13) À 0.7471 ( À 0.04) À 0.7468 Analytical solution [42] RPT–IGA 1.216 Reddy [8] Shimpi [17] Arya [14] Karama [13] Analytical solution [42] (*)The error in parentheses Àa σ x 2; 2b; À h2 Á L.V Tran et al / Engineering Analysis with Boundary Elements 47 (2014) 68–81 73 Table The normalized deflection and axial stress of a laminated [0/90] plate with a/h ratios a/h 3D solution [1] Analytical approach [26] RPT–IGA Reddy model Senthil model Whitney model Reddy Shimpi Arya Karama w σx 4.9362 À 0.9070 4.5619 À 1.4277 4.5619 À 1.8199 5.4103 À 0.7151 4.5618 À 1.4215 4.5618 À 1.4215 4.4335 À 1.4423 4.2901 À 1.4565 w σx 1.7287 À 0.7723 1.6670 À 0.8385 1.6670 À 1.4133 1.7627 À 0.7151 1.6669 À 0.8337 1.6669 À 0.8337 1.6538 À 0.8392 1.6382 À 0.8439 10 w σx 1.2318 À 0.7317 1.2161 À 0.7468 1.2161 À 1.3500 1.2416 À 0.7151 1.2161 À 0.7421 1.2161 À 0.7421 1.2131 À 0.7436 1.2096 À 0.7449 20 w σx 1.1060 À 0.7200 1.1018 À 0.7235 1.1018 À 1.3340 1.1113 À 0.7151 1.1018 À 0.7189 1.1018 À 0.7189 1.1011 À 0.7193 1.1002 À 0.7196 100 w σx 1.0742 À 0.7219 1.0651 À 0.7161 1.0651 À 1.3288 1.0651 À 0.7151 1.0651 À 0.7115 1.0651 À 0.7115 1.0650 À 0.7115 1.0650 À 0.7115 Table The non-dimensional deflection and stresses of a two-layer laminated [0/90] composite square plate subjected to a sinusoidal load BC Plate model SSSS 3D model [41] HSDT[42] FSDT[42] CLPT [42] HOSNDPT [40] RPT–IGA SFSF 3D model [41] HSDT[42] FSDT[42] CLPT [42] HOSNDPT [40] RPT–IGA Exact FEM MQ-MLPG TPS-MLPG Reddy Shimpi Arya Karama Exact FEM MQ-MLPG TPS-MLPG Reddy Shimpi Arya Karama À Á w 2a; 2b À Á σ x 2a; 2b; À h2 À Á σ y 2a; 2b; h2 À Á σ xy 0; 0; 2h À Á σ yz 2a; 0; 1.227 1.216 1.214 1.237 1.064 1.220 1.213 1.2161 1.2161 1.2131 1.2096 À 0.7304 À 0.7468 À 0.6829 À 0.7157 À 0.7157 À 0.726 À 0.723 À 0.7421 À 0.7421 À 0.7436 À 0.7449 0.7309 0.7468 0.6829 0.7157 0.7157 0.727 0.724 0.7421 0.7421 0.7436 0.7449 0.0497 – – – – 0.0494 0.0491 0.053 0.053 0.053 0.0531 – 0.319 – 0.2729 0.298 0.278 0.3181 0.3181 0.3252 0.3319 1.210 1.2295 1.189 1.1907 1.1849 1.21 1.21 1.2192 1.2192 1.221 1.2225 0.0119 – – – – 0.0118 0.0119 0.0121 0.0121 0.0121 0.0122 – 0.4489 – 0.3882 0.488 0.499 0.4507 0.4507 0.46 0.4686 2.026 1.992 2.002 2.028 1.777 2.028 2.028 1.990 1.990 1.9851 1.9794 0.2503 0.2624 0.2212 0.2469 0.2403 0.249 0.249 0.25472 0.25472 0.2555 0.2562 Fig Deflection of two-layer [0/90] antisymmetric square plates: (a) SFSF and (b) SSSS displacement and stresses obtained are acceptable to the analytical solution [26] It is indicated that the results derived from both TSDT and RPT models are almost identical while FSDT model (with SCF ¼5/6) leads to very poor results in both deflection and shear stresses, especially for transverse shear stress This conclusion is clearly addressed in Fig Table reveals that a good agreement 74 L.V Tran et al / Engineering Analysis with Boundary Elements 47 (2014) 68–81 0.05 0.05 TSDT HSDT SSDT ESDT 0.04 0.03 0.03 0.02 0.02 0.01 0.01 0 −0.01 −0.01 −0.02 −0.02 −0.03 −0.03 −0.04 −0.04 −0.05 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 TSDT HSDT SSDT ESDT 0.04 −0.05 −0.06 0.8 −0.04 −0.02 0.02 0.04 0.06 0.05 TSDT HSDT SSDT ESDT 0.04 0.03 0.02 0.01 −0.01 −0.02 −0.03 −0.04 −0.05 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Fig The stresses through the thickness of laminate composite plate under full simply supported condition with a/h ¼ 10 via several refined plate models Table The non-dimensional deflection and stresses of a five-layer square sandwich plate [0/90/core/0/90] under sinusoidal load with a/h ¼ 10 Method Model À Á w 2a; 2b À Á σ x 2a; 2b; À 2h À Á σ y 2a; 2b; h2 À Á τxy 0; 0; h2 À Á τyz 2a; 0; À3h Analytical solution [26] TSDT RPT FSDT 2.3075 2.3075 1.3570 0.6815 0.7634 0.6200 À 0.6815 À 0.7631 À 0.6200 0.0787 0.0787 0.0693 – – – IGA TSDT RPT FSDT 2.2588 2.2588 1.3512 0.6834 0.6857 0.6308 À 0.6834 À 0.6857 À 0.6308 0.0710 0.0710 0.0626 1.8360 1.8352 0.6014 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0 −0.1 −0.1 −0.1 −0.2 −0.2 −0.2 −0.3 −0.3 −0.3 −0.4 −0.4 −0.4 0.1 RPT TSDT FSDT −0.5 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 −0.5 −0.08 −0.06 −0.04 −0.02 RPT TSDT FSDT 0.02 0.04 0.06 0.08 0.5 0.4 0.3 −0.5 RPT TSDT FSDT 0.5 1.5 À Á Fig The stresses through the thickness of a sandwich plate under full simply supported condition with a/h¼ 10, hc/hf ¼ via various plate models (a) axial stress σ x 2a; 2b; z Àa Á (b) Shear stress τxy ð0; 0; zÞ (c) Shear stress τyz 2; 0; z L.V Tran et al / Engineering Analysis with Boundary Elements 47 (2014) 68–81 between two models TSDT and RPT is again observed when ratio hc/hf varies from to 200 4.2 Free vibration analysis 4.2.1 The cross-ply laminated [0/90]N square composite plate Let next us consider a cross-ply N layers laminated [0/90]N composite plate with a/h ¼ under simply supported boundary conditions Herein, material set II is used The effects of the number of layers N and elastic modulus ratios E /E are Table The non-dimensional deflection of a sandwich plate [0/90/core/0/90] via hc/hf ratio hc/hf RPT TSDT FSDT 10 20 30 40 50 100 200 2.2588 8.9432 34.7064 67.8504 93.931 110.5829 137.3968 152.4244 2.2588 8.9432 34.7057 67.8456 93.9228 110.5745 137.3956 152.4252 1.3512 2.3429 4.0458 5.7579 7.4721 9.1866 17.7546 34.8463 75 tabulated in Table A good agreement is found for the present IGA–RPT models in comparison with three-dimensional elasticity approach proposed by Noor [2] and the analytical method given by Kant [45] It is in general observed that the IGA–RPT based on the exponential function achieves the highest results which match very well with the 3D solution as E / E r 10 As E /E ratio ranges from 20 to 40, the present results are asymptotic to analytical solutions for 2D plate model using HSDT and RPT [45] Next, with constant E1/E2 ratio (¼ 40), the variation of natural frequency of a two-layer laminated composite plate via length to thickness ratio are listed in Table It is again seen that the obtained results match well with the analytical one using 12DOFs published by Kant [45] The difference reduces via the increase in the ratio of a/h (approximate from 8% to 0.02% as changing of a/h from to 100) The first three mode shapes of a thick plate (a/h ¼10) is then plotted in Fig It is clear that beside the full mode shape of deflection (above), the mode shapes of four unknown parameters along line y¼a/2 are illustrated To close this sub-section, the effect of boundary condition on normalized frequency of ten-layer cross-ply composite plate is plotted in Table Compared with those reported by Reddy and Khdeir [46], the present model again obtains good agreement It can be seen that, present model using RPT gains the closest results to analytical solution using TSDT with slightly higher results In Table The natural frequency of a simply supported [0/90]N composite plate N 3D elasticity [2] Analytical solution [45] RPT–IGA 3D elasticity [2] Analytical solution [45] RPT–IGA 3D elasticity [2] Analytical solution [45] RPT–IGA E1/E2 Model 3D elasticity [2] Analytical solution [45] RPT–IGA 10 20 30 40 HSDT-12DOFs HSDT-9DOFs RPT FSDT 6.2578 6.2336 6.1566 6.2169 6.149 6.9845 6.9741 6.9363 6.9887 6.9156 7.6745 7.714 7.6883 7.821 7.6922 8.1763 8.2775 8.257 8.505 8.3112 8.5625 8.7272 8.7097 9.0871 8.8255 Reddy Shimpi Arya Karama 6.2169 6.2169 6.2189 6.2224 6.9887 6.9887 6.9965 7.0066 7.8211 7.8211 7.838 7.8585 8.5051 8.5051 8.5317 8.563 9.0872 9.0872 9.1237 9.1662 HSDT-12DOFs HSDT-9DOFs TSDT, RPT FSDT 6.5455 6.5146 6.4319 6.5008 6.4402 8.1445 8.1482 8.1010 9.1954 8.1963 9.4055 9.4675 9.4338 9.6265 9.6729 10.165 10.2733 10.2463 10.5348 10.6095 10.6798 10.8221 10.7993 11.1716 11.2635 Reddy Shimpi Arya Karama 6.5008 6.5008 6.5012 6.5034 8.1954 8.1954 8.1930 8.1939 9.6265 9.6265 9.6205 9.6201 10.5348 10.5348 10.5268 10.5261 11.1716 11.1716 11.1628 11.1629 HSDT-12DOFs HSDT-9DOFs TSDT, RPT FSDT 6.6100 6.5711 6.4873 6.5552 6.4916 8.4143 8.3852 8.3372 8.4041 8.3883 9.8398 9.8346 9.8012 9.9175 9.9266 10.6958 10.7113 10.6853 10.8542 10.8723 11.2728 11.3051 11.2838 11.5007 11.5189 Reddy Shimpi Arya Karama 6.5558 6.5558 6.5567 6.5596 8.4052 8.4052 8.4066 8.4122 9.9181 9.9181 9.9211 9.9313 10.8547 10.8547 10.8604 10.8758 11.5012 11.5012 11.5103 11.5314 HSDT-12DOFs HSDT-9DOFs TSDT, RPT FSDT 6.6458 6.6019 6.5177 6.5842 6.5185 8.5625 8.5163 8.4680 8.5126 8.4842 10.0843 10.0438 10.0107 10.0674 10.0483 11.0027 10.9699 10.9445 11.0197 10.9959 11.6245 11.5993 11.5789 11.673 11.6374 Reddy Shimpi Arya Karama 6.5842 6.5842 6.5854 6.5885 8.5126 8.5126 8.5156 8.5229 10.0674 10.0674 10.0741 10.0882 11.0197 11.0197 11.031 11.0523 11.673 11.673 11.6894 11.7182 76 L.V Tran et al / Engineering Analysis with Boundary Elements 47 (2014) 68–81 Table The natural frequency of simply supported laminated [0/90] composite plate with E1/E2 ¼40 Plate model a/h 10 20 50 100 Analytical solution [45] HSDT-12DOFs HSDT-9DOFs TSDT RPT FSDT 7.9081 7.8904 8.3546 8.3546 8.0889 10.4319 10.4156 10.568 10.568 10.461 11.0663 11.0509 11.1052 11.1052 11.0639 11.2688 11.2537 11.2751 11.2751 11.2558 11.2988 11.2837 11.3002 11.3002 11.2842 RPT–IGA Reddy Shimpi Arya Karama 8.3547 8.3547 8.4018 8.4564 10.5681 10.5681 10.5812 10.5965 11.1053 11.1053 11.109 11.1133 11.2752 11.2752 11.2758 11.2758 11.3003 11.3003 11.3004 11.3006 ω1 = 10.5681 0.6 −0.1 0.4 −0.2 −0.3 0.2 −0.4 −0.5 −0.6 y y y ω = 26.5015 ω = 26.5015 −0.2 −0.7 −0.4 −0.8 −0.9 −0.6 0.2 0.4 0.6 0.8 0.2 0.4 x 0.6 0.8 0.5 0.4 0.3 0.2 0.1 −0.1 −0.2 −0.3 −0.4 −0.5 0.2 : u0 : v0 0.4 0.6 0.8 x x : wb : ws Fig Vibration mode shapes: full plate (upper) and line y¼a/2 (lower) of simply supported laminated [0/90] composite plate with E1/E2 ¼ 40, a/h¼ 10 Table The natural frequency of ten-layer cross-ply [0/90]5 plate with a/h ¼5 and E1/E2 ¼40 Plate model Boundary conditions Analytical solution [46] TSDT FSDT CLPT RPT–IGA Reddy Shimpi Arya Karama SFSF SFSC SSSS SSSC SCSC CCCC 8.155 8.139 11.459 8.966 8.919 13.618 11.673 11.644 12.167 12.514 12.197 23.348 13.568 12.923 30.855 – – – 11.673 11.673 11.6894 11.7182 13.0041 13.0041 13.0463 13.1062 14.1566 14.1566 14.2418 14.3513 15.2991 15.2991 15.4558 15.6438 8.1554 8.1554 8.1661 8.1853 9.0832 9.0832 9.0971 9.1201 addition, when the constrained edge changes from F to S and C, the structural stiffness increases, the magnitudes of free vibration thus increase, respectively The mode shapes according to various boundary conditions are illustrated in Fig 4.2.2 The sandwich plate with curved boundary: a comparison of computational efficiency Let us consider a plate with an annular geometry with a uniform thickness h, outer radius R and inner one r as shown in Fig Material L.V Tran et al / Engineering Analysis with Boundary Elements 47 (2014) 68–81 77 Fig Mode shape profile of ten layers [0/90]5 composite plate under various boundary conditions (a) SFSF (b) SFSC (c) SSSS (d) SSSC (e) SCSC (f) CCCC Table 10 The natural frequency _ ω of circular sandwich plate via R/h ratios and various plate models R/h Method Mode number f ace thickness ratio R/h A good agreement is observed for RPT and TSDT, while FSDT model remains too stiffened as plate becomes thicker The first six mode shapes of a circular plate are depicted in Fig 10 As the inner radius r a 0, a fully annular plate is obtained as shown in Fig Due to symmetry, an upper haft of plate has been modeled in Fig 11 with the symmetric constraint: displacement along y-direction equals to zero at y¼ With data R/h¼10 and R/r ¼2, the first six normalized frequencies _ ẳ R rị2 = q h =E2 Þf ace are listed in Table 11 Now we show the TSDT RPT FSDT 1.8408 1.8185 3.2749 2.9851 2.969 5.1587 2.9903 2.9709 5.1849 4.0577 4.0497 6.9426 4.2675 4.1186 6.9527 4.5676 4.422 7.1665 TSDT RPT FSDT 4.0777 4.0224 6.5314 6.5847 6.6477 11.0073 6.5921 6.6517 11.0291 8.8982 9.0267 15.2293 9.2296 9.1634 15.8679 9.8714 9.7342 16.9509 10 TSDT RPT FSDT 6.6153 6.5599 8.6196 11.274 11.5226 16.1005 11.2808 11.5286 16.1161 15.6121 16.0142 23.3139 16.3027 16.557 24.8155 17.4576 17.3499 26.6094 20 TSDT RPT FSDT 8.6768 8.6529 9.5541 16.2863 16.5898 19.1564 16.2912 16.5953 19.1648 23.6428 24.1341 28.8009 25.1854 25.8627 31.7216 27.0669 27.1200 34.0548 100 TSDT RPT FSDT 9.9093 9.9088 9.9258 20.4639 20.498 20.614 20.4643 20.4985 20.6145 31.3827 31.4395 31.6544 35.2143 35.3459 35.8107 37.7988 37.8367 38.3766 Fig The annular plate model III is used The plate is clamped at the outer boundary For illustration, we use the distributed function f zị ẳ z 4=3h ịz3 for RPT model The analytical solution was not available The aim of this study is to estimate the solution of the plates involving curved edges By setting the inner radius r ¼ 0, the model becomes the clamped circular sandwich plate Table 10 shows the dependence qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of the natural frequency _ ω ¼ ωR2 =h ðρ=E Þ on the radius to computational efficiency of the present approach The program is compiled by a personal computer with Intel (R) Core (TM) Duo CPU – GHz and RAM – GB It can be seen that with the same mesh, RPT model produces lowest degree of freedoms (DOFs) Hence, it spends lowest computational cost with just 284 s compared with 608 s and 520 s according to TSDT and FSDT ones, respectively However, RPT model also archives closed results to TSDT than FSDT one Table 12 tabulates the frequency parameter _ ω of the annular plates via outer radius to inner radius ratio R/r and radius to thickness ratio R/h It is concluded that the frequency parameters decrease sequentially following to increase in inner radius to outer radius ratio r/R and decrease in radius to thickness 78 L.V Tran et al / Engineering Analysis with Boundary Elements 47 (2014) 68–81 Mode1 Mode2 Mode4 Mode3 Mode5 Mode6 Fig 10 The first six mode shapes of circular sandwich plate with R/h ¼10 Table 12 The dependent of natural frequency _ ω of annular plate on R/h and r/R ratios R/h r/R Mode number Fig 11 The mesh of a half annular plate Table 11 The natural frequency _ ω of annular sandwich plate with R/h ¼ 10 and R/r2 ¼2 Model Mode Mode Mode Mode Mode Mode TSDT RPT FSDT 2.6579 2.5954 3.5775 2.9051 2.9295 4.0890 3.5579 3.7458 5.4463 4.4029 4.6822 7.2097 5.4377 5.7805 9.3348 DOFs CPU time 6.5720 2340 6.9574 1872 11.5482 2340 608 284 520 ratio R/h To enclose this section the first six mode shapes of the plate are depicted in Fig 12 4.3 Buckling analysis  à 4.3.1 The angle-ply laminated θ=À θ square composite plate A simply supported two-layer angle-ply laminated ½θ= ÀθŠ square plate is subjected to uniaxial compressive load along the x-direction shown in Fig 13a Material set II is used The results tabulated in Table 13 are compared with that of Ren [50] and the analytical solution [20] using FSDT, HSDT and RPT assumptions For all values of a/h ratio and fiber orientation, present model with third-order functions give the closest buckling load to that of RPT predicted by Thai et al [20] It can be again seen that all models give the slightly same results for thin plates (a/h ¼100) Fig 14 illustrates the buckling mode of two-layer angle-ply laminated composite plate in case of θ¼ 45 It can be seen that as plate 0.2 0.5 0.8 1.8185 1.2536 0.7273 0.3353 2.9690 1.7450 0.7907 0.3379 2.9709 2.5472 0.9636 0.3529 4.0497 3.2135 1.1880 0.3632 4.1186 3.2771 1.4621 0.3819 4.4220 3.5679 1.7685 0.4060 0.2 0.5 0.8 4.0224 2.6684 1.5623 0.6663 6.6477 3.8843 1.7189 0.6717 6.6517 5.6464 2.1153 0.6951 9.0267 6.8217 2.6133 0.7174 9.1634 7.2075 3.1889 0.7537 9.7342 7.7229 3.8012 0.7987 10 0.2 0.5 0.8 6.5599 4.1002 2.5954 1.1875 11.5226 6.6405 2.9295 1.1985 11.5286 9.8901 3.7458 1.2416 16.0142 11.7335 4.6822 1.2875 16.5570 12.7742 5.7805 1.3585 17.3499 13.9715 6.9574 1.4461 20 0.2 0.5 0.8 8.6529 5.0797 3.5545 2.0288 16.5898 9.3630 4.1602 2.0500 16.5953 14.5846 5.6767 2.1495 24.1341 18.0392 7.2449 2.2306 25.8627 19.2986 9.2193 2.3665 27.1200 22.5169 11.4600 2.5383 100 0.2 0.5 0.8 9.9088 5.5779 4.1878 3.4620 20.4980 11.3311 5.0633 3.5002 20.4985 18.5059 7.4224 3.8409 31.4395 24.9014 9.7455 3.9826 35.3459 25.3016 13.0368 4.2738 37.8367 32.4963 17.2741 4.6740 thickness reduces, the non-dimension buckling value λcr ¼ λcr a2 =E2 h increases according to changing of mode shape from two halves sine wave (a/h¼ 4) to a half sine wave (a/h ¼10; 100, respectively) Furthermore, the portion of shear deflection components ws in transverse displacement reduces and tends to zero as a/h¼ 100 The present models, hence, reduce to CLPT model 4.3.2 The three-layer symmetric cross-ply [0/90/0] composite plate Finally, we investigate the biaxial buckling load of a symmetric cross-ply [0/90/0] simply supported plate as shown in Fig 13b This test aims to show that the RPT–IGA is also existing the deficiency in case of the symmetric laminated composite plates This fact is originated from the feature of the RPT model, which was confirmed by Kant et al [26,45] Various length-tothickness a/h with elastic modulus ratios are studied Table 14 L.V Tran et al / Engineering Analysis with Boundary Elements 47 (2014) 68–81 Mode Mode 79 Mode Mode Mode Mode Fig 12 The first six mode shapes of annular plate with R/h¼ 10 and R/r ¼2 Fig 13 Geometry of laminated composite plates under axial (a) and biaxial (b) compression 0.6 0.4 z z 0.2 −0.2 −0.4 −0.6 −0.8 0.6 0.8 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 : u0 : v0 0.2 0.4 0.6 0.8 x x x y 0.4 −0.3 −0.4 −0.5 −0.6 −0.7 −0.8 −0.9 y 0.2 y 0 −0.1 −0.2 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 z 0.8 : wb : ws Fig 14 Buckling mode shapes: full plate (upper) and line y ¼a/2 (lower) of simply supported [45/ À 45] composite plate with various length to thickness ratios: (a) a/h¼ 4; (b) a/h ¼10; and (c) a/h ¼100 80 L.V Tran et al / Engineering Analysis with Boundary Elements 47 (2014) 68–81 Table 13 The normalized uniaxial buckling load λcr of simply supported two layer laminated ½θ= À θŠ composite plate with E1/E2 ¼40 a/h θ Ren [50] Analytical solution [20] HSDT RPT–IGA FSDT RPT Reddy Shimpi Arya Karama 30 45 9.5368 9.8200 9.3391 8.2377 7.5450 6.7858 9.3518 8.3963 9.3522 8.3966 9.3522 8.3966 9.6731 8.6472 9.9211 8.9414 10 30 45 15.7517 16.4558 17.1269 18.1544 16.6132 17.5522 17.2795 18.1544 17.2797 18.1545 17.2797 18.1545 17.3495 18.2383 17.4311 18.3354 100 30 45 20.4793 21.6384 20.5017 21.6663 20.4944 21.6576 20.504 21.6663 20.5042 21.6664 20.5042 21.6664 20.5052 21.6676 20.5063 21.6689 Table 14 The normalized biaxial buckling load λcr of simply supported three-layer laminated [0/90/0] composite plate under various a/h ratios and E1/E2 ¼40 Plate model a/h HSDT–RPIM [48] FSDT–RPIM [48] HSDT–FEM [49] 10 15 20 1.457 1.419 1.465 5.519 5.484 5.526 10.251 10.189 10.259 12.239 12.213 12.226 13.164 13.132 13.185 HSDT[47] Arya Soldatos Thai 1.3862 1.3641 1.4316 5.3668 5.3834 5.3236 9.9188 9.9495 9.8795 12.0205 12.0398 11.9978 13.0379 13.0504 13.0239 RPT–IGA Reddy Shimpi Arya Karama 1.6864 1.6864 1.7215 1.7679 6.1752 6.1752 6.1571 6.1500 10.8825 10.8825 10.8549 10.8336 12.7140 12.7140 12.6957 12.6808 13.5135 13.5135 13.5014 13.4916 0.8 0.6 0.4 z 0.2 −0.2 −0.4 −0.6 −0.8 x 0.2 0.4 0.6 0.8 y : u0 : v0 : wb : ws Fig 15 Buckling mode shapes: (a) a full plate model and (b) along line x ¼a/2 of simply supported [0/90/0] composite plate with E1/E2 ¼ 40, a/h ¼10 (note that u0  v0) shows the critical buckling parameter λcr with respect to various length-to-thickness ratios The obtained results are compared with those of the isogeometric approach based on HSDT [47], the finite element method based on HSDT [49], the meshfree method based on both FSDT and HSDT [48] The present model reflects well the true buckling mode shape as shown in Fig 15 It is seen that obtained results using RPT assumption are more stiffened than those using other HSDT ones when the plate thickness is increased Therefore further research on the RPT model is still necessary as observed in [26,45] Conclusions In this paper, we presented an effective formulation based on isogeometric approach (IGA) and a four variable refined plate theory (RPT) for static, free vibration and buckling analysis of laminated composite plates We addressed a general four-variable refined plate theory with various distributed functions which is used to approximate the high-order term in displacement field Utilizing NURBS basis function, the present method enables us to achieve easily the smoothness with arbitrary continuous order and L.V Tran et al / Engineering Analysis with Boundary Elements 47 (2014) 68–81 therefore naturally fulfills the C1-continuity of the present plate model Just using four DOFs per each node, the present method retains the computational efficiency and gains the good agreement results compared with other models in the literature especially for anti-symmetric laminated plates However, for symmetric one, e.g, symmetric cross-ply [0/90/0], RPT overestimates the obtained results To overcome this disadvantage, we suggest using the RPT–IGA in combination with the layerwise theory [51] We assume the four unknown variable refined plate theory in each layer and the imposition of displacement continuity at the layers interfaces In our opinion, such an approach will be promising to provide an effectively alternative finite element tool for modeling and analysis of plate structures Acknowledgments This research is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.02-2012.17 The support is gratefully acknowledged The fifth author appreciates for the support from 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four-variable. .. [25] Sankara CA, Igengar NGR A C0 element for free vibration analysis of laminated composite plates J Sound Vib 1996;191:721–38 [26] Kant T, Swaminathan K Analytical solutions for the static analysis. .. [32–39], etc In this paper, a formulation based on the RPT model and the isogeometric approach for static, free vibration and buckling analysis of laminated composite plates is investigated Some higher-order