INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int J Numer Meth Engng (2012) Published online in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/nme.4282 Static, free vibration, and buckling analysis of laminated composite Reissner–Mindlin plates using NURBS-based isogeometric approach Chien H Thai1 , H Nguyen-Xuan1,2, * ,† , N Nguyen-Thanh3 , T-H Le4 , T Nguyen-Thoi1,2 and T Rabczuk3 Division of Computational Mechanics, Ton Duc Thang University, Ho Chi Minh, Vietnam of Mechanics, Faculty of Mathematics and Computer Science, Ho Chi Minh City University of Science, Vietnam National University–Ho Chi Minh, 227 Nguyen Van Cu Street, Ho Chi Minh, Vietnam Institute of Structural Mechanics, University Bauhaus, Marienstrae 15, 99423 Weimar, Germany Department of Naval Architecture and Marine Engineering, Ho Chi Minh City University of Technology, Vietnam National University–Ho Chi Minh, 268 Ly Thuong Kiet Street, Ho Chi Minh, Vietnam Department SUMMARY This paper presents a novel numerical procedure based on the framework of isogeometric analysis for static, free vibration, and buckling analysis of laminated composite plates using the first-order shear deformation theory The isogeometric approach utilizes non-uniform rational B-splines to implement for the quadratic, cubic, and quartic elements Shear locking problem still exists in the stiffness formulation, and hence, it can be significantly alleviated by a stabilization technique Several numerical examples are presented to show the performance of the method, and the results obtained are compared with other available ones Copyright © 2012 John Wiley & Sons, Ltd Received May 2011; Revised 15 November 2011; Accepted 19 December 2011 KEY WORDS: isogeometric analysis; laminated composite Reissner–Mindlin plates; stabilization technique INTRODUCTION Composite materials with fiber reinforcement have been widely used in various engineering structures such as aircrafts, aerospace, vehicles, buildings, etc Laminated composites are made of two or several lamina layers with different materials stacked together to achieve desired properties (e.g., high stiffness and strength-to-weight ratios, long fatigue life, wear resistance, damping, etc.) [1] There are a number of approaches to model laminated composite plates One of the popular approaches is to rely on equivalent single-layer (ESL) models such as classical laminated plate theory [2–7], first-order shear deformation theory (FSDT) [8–18], and high-order shear deformation theory (HSDT) [19–24] The FSDT is simple to implement and gives better results than the classical laminated plate theory because the generalized displacement field not only requires any derivative but also includes transverse shear strains Also, the computational cost using the FSDT is cheaper than that using the HSDT The FSDT only requires C -continuity of generalized displacements However, it requires the so-called shear correction factors to take into account the non-linear *Correspondence to: H Nguyen-Xuan, Division of Computational Mechanics, Ton Duc Thang University, Ho Chi Minh, Vietnam † E-mail: nxhung@hcmus.edu.vn Copyright © 2012 John Wiley & Sons, Ltd C H THAI ET AL distribution of shear stress terms When the laminated plates become thicker, a layer-wise model [25, 26] can be recommended to improve the accuracy of transverse shear stresses Between the ESL and layer-wise models, the ESL has been widely used because of some advantages such as simplicity in modeling and formulating constitutive equations and low computational cost Especially in analyzing thin or moderately thick laminated plates, the ESL models often offer relatively accurate results In order to predict the interlaminar stress of a composite laminate, the 3D continuum-based theory [27, 28] is commonly used in the calculation, but the computational cost using 3D models is more expensive It is also worth mentioning that there are some other theories such as the global–local [29, 30], the sinusoidal shear deformation plate theory by Touratier [31], the exponential shear deformation plate theory by Karama [32], the hyperbolic sin deformation theory by Soldatos [33], the exponential shear deformation plate theory by Aydogdu [34], the hyperbolic shear deformation theory by Meiche et al [35], the variable refined theory [36–38], etc In this paper, we focus on non-uniform rational B-spline (NURBS) elements using the FSDT of ESL for static, free vibration, and buckling analysis of composite plates Because of the limitation of analytical approaches, various numerical methods have been developed such as FEM [39, 40], BEM [41, 42], smoothed FEM [43–49], mesh-free methods [50–56], etc In the development of advanced computational methodologies, Hughes et al [57] have recently proposed a NURBS-based isogeometric analysis to bridge the gap between computer-aided design (CAD) and finite element analysis In contrast to the standard FEM with Lagrange polynomial basis, the isogeometric approach utilized more general basis functions such as NURBS that are common in CAD approaches Isogeometric analysis is thus very promising because it can directly use CAD data to describe both exact geometry and approximate solution For structural mechanics, isogeometric analysis has been extensively studied for nearly incompressible linear and non-linear elasticity and plasticity problem [58], structural vibrations [59], the Reissner–Mindlin shell [60], the Kirchhoff–Love shell [61, 62], the large deformation with rotation-free [63] and structural shape optimization [64], etc We present in this paper a NURBS-based isogeometric approach, where the same shape functions are used to describe the field variables as the geometry of the domain, for static, free vibration, and buckling analysis of laminated composite Reissner–Mindlin plate structures We formulate the isogeometric stiffness matrices for the quadratic, cubic, and quartic elements A simply stabilized technique is then proposed to alleviate significantly the shear locking phenomenon without any modification of shear strains, and full integration is still used to calculate isogeometric stiffness matrices Several numerical results are illustrated to demonstrate the effectiveness of the present method The paper is arranged as follows A brief of the B-spline and NURBS surface is described in Section Section describes an isogeometric approximation for laminated composite Reissner– Mindlin plates An improved technique on shear terms is given in Section Several numerical examples are illustrated in Section Finally, we close our paper with some concluding remarks NURBS-BASED ISOGEOMETRIC ANALYSIS FUNDAMENTALS 2.1 Knot vectors and basis functions Let „ D , , : : :, nCpC1 be a non-decreasing sequence of parameter values, i i C1 , i D 1, : : :, n C p The i are called knots, and „ is the set of coordinates in the parametric space If all knots are equally spaced, the knot vector is called uniform Otherwise, they are called a non-uniform knot vector The first and last knots are repeated p C times, which are called open A B-spline basis function is C inf continuous inside a knot span and C p continuous at a single knot A knot value can appear more than once and is then called a multiple knot At a knot of multiplicity k, the continuity is C p k Given a knot vector, the B-spline basis functions Ni ,p / of order p D are defined recursively on the corresponding knot vector as follows: ² if i 6 i C1 , Ni ,0 / D (1) otherwise Copyright © 2012 John Wiley & Sons, Ltd Int J Numer Meth Engng (2012) DOI: 10.1002/nme NURBS-BASED ISOGEOMETRIC ANALYSIS FOR LAMINATED COMPOSITE PLATES 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 1/2 1/2 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1/2 Figure This figure illustrates quadratic, cubic, and quartic B-spline basis functions The basis functions are defined by the following recursion formula: i Ni ,p / D i Cp Ni ,p /C i i CpC1 i CpC1 Ni C1,p / with p D 1, 2, 3, : : :/ (2) i C1 It is well known that, for p D and 1, the basis functions of isogeometric analysis are identical to those of standard piecewise constant and linear finite elements, respectively Nevertheless, for p > 2, they are different [57] Therefore, the present work will consider the basis functions with p > Figure illustrates a set of one-dimensional quadratic, cubic, and quartic B-spline basis functions for open uniform knot vectors „ D ¹0, 0, 0, 1=2, 1, 1, 1º, „ D ¹0, 0, 0, 0, 1=2, 1, 1, 1, 1º, and „ D ¹0, 0, 0, 0, 0, 1=2, 1, 1, 1, 1, 1º, respectively 2.2 NURBS surface The B-spline curve is defined as C / D n X Ni ,p /Pi , (3) i D1 where Pi are the control points and Ni ,p / is the pth-degree B-spline basis function defined on the open knot vector Copyright © 2012 John Wiley & Sons, Ltd Int J Numer Meth Engng (2012) DOI: 10.1002/nme C H THAI ET AL The B-spline surfaces are defined by the tensor ® product of basis ¯ functions® in two parametric¯ dimensions and Á with two knot vectors „ D , : : :, nCpC1 and H D Á1 , Á2 : : :, ÁmCqC1 expressed as S , Á/ D n X m X Ni ,p /Mj ,q Á/ Pi ,j , (4) i D1 j D1 where Pi ,j is the bidirectional control net and Ni ,p / and Mj ,q Á/ are the B-spline basis functions defined on the knot vectors over an n m net of control points Pi ,j To note the same FEM, we identify the logical coordinates (i, j) of the B-spline surface with the traditional notation of node A [60] Equation (4) is rewritten as S , Á/ D n m X NA , Á/ PA , (5) A where NA , Á/ D Ni ,p /Mj ,q Á/ is the shape function associated with node A Non-uniform rational B-splines are obtained by augmenting every point in the control mesh PA with the homogenous coordinate wA The weighting function is constructed as follows: w , Á/ D n m X NA , Á/ wA (6) AD1 The NURBS surfaces are then defined by nP m S , Á/ D NA , Á/ wA PA A w , Á/ D n m X NN A , Á/PA , (7) AD1 where NN A , Á/ D NA , Á/ wA =w , Á/ are the rational basis functions An example of a cubic NURBS surface with three elements is presented in Figure AN ISOGEOMETRIC FORMULATION FOR LAMINATED COMPOSITE REISSNER-MINDLIN PLATES 3.1 The displacements, strains, and stresses of plates Let be the domain in R2 occupied by the mid-plane of the plate and u0 , v0 , and w0 and ˇ D ˇx , ˇy /T denote the displacement components in the x, y, and ´ directions and the rotations in the y–´ and x–´ planes (Figure 3), respectively The displacement field based on the FSDT Figure Physical mesh and control mesh with cubic non-uniform rational B-spline surface Copyright © 2012 John Wiley & Sons, Ltd Int J Numer Meth Engng (2012) DOI: 10.1002/nme NURBS-BASED ISOGEOMETRIC ANALYSIS FOR LAMINATED COMPOSITE PLATES Figure Geometry of a typical Mindlin–Reissner plate is defined as u x, y, ´/ D u0 x, y/ C ´ˇx x, y/ , v x, y, ´/ D v x, y/ C ´ˇy x, y/ , (8) w x, y, ´/ D w x, y/ The in-plane strain vector "p D 6 "p D "xx @u0 @x @v @y @u0 @y C @v T "yy xy 7 7C´6 @x can be rewritten as @ˇ x @x @ˇy @y @ˇx @y C @ˇy @x 7 D "m C ´"b , and the transverse shear strain vector has the following form: " s # " @w # C ˇx "x´ @x s " D D @w "sy´ C ˇy @y (9) (10) The constitutive equation of an orthotropic layer in local coordinate is derived from Hooke’s law for plane stress by 9.k/ 9.k/ 3.k/ Q11 Q12 Q16 0 "xx > xx > ˆ ˆ ˆ > ˆ > ˆ > > ˆ ˆ > ˆ > ˆ > ˆ > Q Q Q 0 " ˆ > ˆ yy > 21 22 26 yy > ˆ ˆ > < < = = 0 D Q61 Q62 Q66 , (11) xy xy ˆ ˆ > > ˆ > ˆ > ˆ > ˆ > ˆ x´ > > ˆ x´ > > 0 Q55 Q54 ˆ ˆ ˆ > ˆ > ˆ > ˆ > : ; : ; 0 Q45 Q44 y´ y´ where material constants are given as Q11 D E1 12 21 , Q12 D 12 E2 , Q22 D 12 21 E2 12 21 (12) Q66 D G12 , Q55 D G13 , Q44 D G23 , where E1 and E2 are the Young modulus in the and directions, respectively; G12 , G23 , and G13 are the shear modulus in the 1–2, 2–3, and 3–1 planes, respectively; and 12 and 21 are Poisson ratios Copyright © 2012 John Wiley & Sons, Ltd Int J Numer Meth Engng (2012) DOI: 10.1002/nme C H THAI ET AL The laminate is usually made of several orthotropic layers in which the stress–strain relation for the kth orthotropic lamina with the arbitrary fiber orientation compared with the reference axes is given by 3.k/ 8 9.k/ N 9.k/ Q11 QN 12 QN 16 0 "xx > xx > ˆ ˆ ˆ > ˆ > ˆ > > ˆ ˆ > ˆ > ˆ > ˆ N N N Q 0 Q Q "yy > ˆ ˆ > 21 22 26 yy > ˆ > ˆ > < = < = 7 D QN 61 QN 62 QN 66 , (13) 0 xy xy ˆ > > ˆ > ˆ > ˆ ˆ > ˆ > ˆ ˆ x´ > 0 QN 55 QN 54 ˆ > > x´ > ˆ > ˆ > ˆ ˆ > ˆ > : ; : ; 0 QN 45 QN 44 y´ y´ where QN ij is a transformed material constant of the kth lamina (see [40] for more detail) 3.2 Weak form equation of plates A weak form of the static model for Reissner–Mindlin composite plates can be briefly expressed as Z Z Z T N sT s s ı"p D"p d C ı" D " d D ıwpd , (14) where p is the transverse loading per unit area and " # A B N D D B Db (15) in which A, B, Db , and Ds are matrices of extensional stiffness, bending–extensional coupling stiffness, bending stiffness, and transverse shearing stiffness, respectively, defined as [1]: Á Aij , Bij , Db ij D h=2 R 1, ´, ´2 QN ij d´ i, j D 1, 2, , (16) h=2 Ds ij D h=2 R QN ij d´ i, j D 4, (17) h=2 For the free vibration analysis of a Mindlin–Reissner plate model, a weak form may be derived from the dynamic form of the principle of virtual work under the assumptions of the FSDT: Z Z Z T N sT s s R ı"p D"p d C ı" D " d D ıuT mud (18) In the case of in-plane buckling analyses and assuming pre-buckling stresses O , non-linear strains appear, and the weak form can be reformulated as R R R N p d C ı"s T Ds "s d C h r T ıw O rwd ı"Tp D" " #" # O0 ru0 R T T r ıu0 r ıv0 Ch d O0 rv0 (19) " #" # O0 rˇx R r T ıˇx r T ıˇy C h12 d D 0, O0 rˇy Ä 0 x xy T are the gradient operator and in-plane where r D Œ@=@x @=@y and O D 0 xy y pre-buckling stresses, respectively Copyright © 2012 John Wiley & Sons, Ltd Int J Numer Meth Engng (2012) DOI: 10.1002/nme NURBS-BASED ISOGEOMETRIC ANALYSIS FOR LAMINATED COMPOSITE PLATES Using the same NURBS basis functions, we interpolate the displacement field as n m X h u D NN A , Á/qA , (20) A where n m is the number basis functions and NN A , Á/ and qA D ŒuA vA wA ˇxA ˇyA T are the rational basic functions and the degrees of freedom of uh associated with control point A, respectively The membrane bending, shear strains, and geometrical strains are written as X X X X g b "m D Bm BbA qA , "s D BsA qA , "g D BA qA , (21) A qA , " D A A A A where 6 Bm A D4 " Bsi D NN A,x 0 NN A,y NN A,y NN A,x 0 NN A,x 0 NN A,y 0 0 NN A,x NN A,y , NN A,x b 0 0 , BA D 0 0 0 0 NN A,y # NN A , NN A (22) and 6 6 g BA D 6 6 NN A,x NN A,y 0 0 0 0 0 NN A,x NN A,y 0 0 0 0 0 NN A,x NN A,y 0 0 0 0 0 NN A,x NN A,y 0 0 0 0 0 NN A,x NN A,y 3T 7 7 7 (23) For the static analysis, the formulation of a Mindlin–Reissner plate can then be obtained as Kq D f (24) and, for the free vibration analysis, can be rewritten as !2M q D K (25) and, for the buckling analysis, one writes, cr Kg K q D 0, where K is the global stiffness matrix and the load vector, Z Z Z ÁT Bm /T ABm d C Bm /T BBb d C KD Bb BBm d Z C (26) Z C Bb ÁT Db Bb d (27) s T s s B / D B d , Z fD Copyright © 2012 John Wiley & Sons, Ltd pNd C fb , (28) Int J Numer Meth Engng (2012) DOI: 10.1002/nme C H THAI ET AL in which fb is the remaining part of f subjected to prescribed boundary loads and M is the global mass matrix, Z MD h 0 6 h 6 0 h with m D 6 6 0 NT mNd 0 7 7 7, 7 0 h3 12 (29) h3 12 Figure Square full clamped and simply supported plate model and its control points 0.98 Normalized deflection w Normalized deflection w Exact Quadratic Cubic Quartic 0.96 0.94 0.92 0.9 0.88 Exact Quadratic Cubic Quartic 0.99 0.98 0.97 0.96 0.95 10 15 20 25 30 35 40 Number control points per side (a) clamped plate 45 50 0.94 10 15 20 25 30 35 40 45 50 Number control points per side (b) simple supported plate Figure Normalized deflection of simply supported and clamped square isotropic plates subjected to uniformly distributed load (a) Clamped plate (b) Simple supported plate Copyright © 2012 John Wiley & Sons, Ltd Int J Numer Meth Engng (2012) DOI: 10.1002/nme NURBS-BASED ISOGEOMETRIC ANALYSIS FOR LAMINATED COMPOSITE PLATES and Kg is the global geometrical stiffness matrix, Z Kg D Bg /T Bg d (30) with hO0 6 6 D6 6 0 0 hO0 0 0 hO0 0 h3 O 12 0 0 h3 O 12 0 0 7 7 7, 7 (31) in which , h, !, and cr are the density mass, the thickness, the natural frequency, and the critical buckling load, respectively AN IMPROVED TECHNIQUE ON SHEAR TERMS In FEM, locking appears in low-order elements Shear locking will disappear for the interpolation functions of order or higher [65] Similar to FEM, the NURBS-based quadratic, cubic, and quartic elements are still locking when the plate becomes very thin This is due to shear affects remaining in the stiffness formulation To overcome this drawback, we adopt a stabilization technique proposed by Lyly et al [66] into the shear terms for these elements A material matrix related to shear terms given in Equation (17) is modified as N s D Ds D h2 , h2 C ˛l (32) where l is the longest length of the edges of the NURBS element, and ˛ is a positive constant given in the interval of 0.05 ˛ 0.15 It is found from numerical experiments of NURBS-based isogeometric plate elements that this parameter can be fixed at 0.1 [66], which can provide reasonable accuracy in solutions 0.98 0.96 Exact Quadratic Cubic Quartic 0.94 0.92 0.9 0.88 10 15 20 25 Number control points per side (a) clamped plate 30 Normalized strain energy Normalized strain energy 0.995 Exact Quadratic Cubic Quartic 0.99 0.985 0.98 0.975 10 15 20 25 30 Number control points per side (b) simple supported plate Figure Normalized strain energy of simply supported and clamped square isotropic plates subjected to uniformly distributed load (a) Clamped plate (b) Simple supported plate Copyright © 2012 John Wiley & Sons, Ltd Int J Numer Meth Engng (2012) DOI: 10.1002/nme C H THAI ET AL The global stiffness matrix from Equation (28) therefore changes as follows: Z KD Z C Z Bm /T ABm d C Z Bm /T BBb d C Ns BBm d Z C 0.1 0.08 Exact p=2(α=0.05) p=2(α=0.1) p=2(full integration) p=2(reduce integration) 0.06 0.04 0.02 102 103 104 105 106 0.4 0.35 Exact p=2(α=0.05) p=2(α=0.1) p=2(full integration) p=2(reduce integration) 0.3 0.25 0.2 0.15 0.1 0.05 101 102 103 Ratio L/t 105 106 105 106 105 106 (b) A supported isotropic plate with p=2 0.16 0.45 0.14 0.12 Exact p=3(α=0.05) p=3(α=0.1) p=3(full integration) p=3(reduce integration) 0.1 0.08 0.06 0.04 0.02 Central deflection (100wcDt3/pL4) Central deflection (100wcDt3/pL4) 104 Ratio L/t (a) A clamped isotropic plate with p=2 102 103 104 105 106 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 Exact p=3(α=0.05) p=3(α=0.1) p=3(full integration) p=3(reduce integration) 101 102 103 Ratio L/t 104 Ratio L/t (c) A clamped isotropic plate with p=3 (d) A supported isotropic plate with p=3 0.16 0.45 0.14 0.12 Exact p=4(α=0.05) p=4(α=0.1) p=4(full integration) p=4(reduce integration) 0.1 0.08 0.06 0.04 0.02 102 103 104 Ratio L/t (e) A clamped isotropic plate with p=4 105 106 Central deflection (100wcDt3/pL4) Central deflection (100wcDt3/pL4) Db Bb d 0.45 0.12 101 ÁT B / D B d 0.14 101 Bb s Central deflection (100wcDt3/pL4) Central deflection (100wcDt3/pL4) ÁT (33) s T 0.16 101 Bb 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 101 Exact p=4(α=0.05) p=4(α=0.1) p=4(full integration) p=4(reduce integration) 102 103 104 Ratio L/t (f) A supported isotropic plate with p=4 Figure Performance of present element with various L= h ratios of clamped and simply supported isotropic plates (a) A clamped isotropic plate with p D (b) A supported isotropic plate with p D (c) A clamped isotropic plate with p D (d) A supported isotropic plate with p D (e) A clamped isotropic plate with p D (e) A supported isotropic plate with p D Copyright © 2012 John Wiley & Sons, Ltd Int J Numer Meth Engng (2012) DOI: 10.1002/nme NURBS-BASED ISOGEOMETRIC ANALYSIS FOR LAMINATED COMPOSITE PLATES by Liew and Chen [72], the solution of Leissa and Ayoub [73], and the solution of Timoshenko and James [74] Normalized buckling load factors are presented in Table II The results of the present method are close to those published solutions Mode 1, Omega = 7.4147 Mode 2, Omega = 10.4075 0.5 0.5 z z 1 0 −0.5 0.8 0.6 0.2 0.4 0.4 0.6 x 0.8 0.4 y x 0.2 0.8 0.6 0.2 0.6 y 0.2 0.8 Mode 4, Omega = 15.4592 Mode 3, Omega = 13.9162 0.5 z z 0.4 −1 −0.5 0.8 0.6 0.2 0.4 0.4 0.6 x 0.8 0.4 x 0.2 0.8 0.6 0.2 y 0.6 y 0.2 0.8 Mode 6, Omega = 19.5967 Mode 5, Omega = 15.8164 0.5 0.5 z z 0.4 −0.5 0.8 0.6 0.2 0.4 x 0.4 0.6 0.2 0.8 y −0.5 0.8 0.6 0.2 0.4 x 0.4 0.6 y 0.2 0.8 Figure 15 Mode shapes 1–6 of a [0=90=0] clamped laminated plate with b= h D 10 and a=b D using the cubic element Figure 16 Geometry and control points of the five-layer skew laminated and square laminated plates Copyright © 2012 John Wiley & Sons, Ltd Int J Numer Meth Engng (2012) DOI: 10.1002/nme C H THAI ET AL 5.2 Static analysis of laminated composite plates 5.2.1 Three-layer square sandwich plate, under uniform load Let us consider a simply supported sandwich square plate proposed by Srinivas [75] subjected to a uniform transverse load q The ratio of length to thickness, a= h, is taken as 10 The sandwich laminated plate is made of one inner layer (core), with the following core properties 6 N Qcore D 6 0.999781 0.231192 0 0.231192 0.524886 0 0 0.262931 0 0 0.266810 0 0 0.159914 7 7, (34) Table VIII A non-dimensional frequency parameter $ D !a2 = h =E2 /1=2 of simply supported and clamped cross-ply Œ90ı =0ı =90ı =0ı =90ı skew plate with various ˛ angles ˛ Method 0ı 15ı 30ı 45ı 60ı SSSS Quadratic Cubic Quartic MISQ20 [83] MLSDQ [13] RBF [12] FSDT [84] 1.5669 1.5668 1.5668 1.5733 1.5709 1.5791 1.5699 1.6819 1.6816 1.6815 1.6896 1.6886 1.6917 — 2.0692 2.0681 2.0675 2.0820 2.1026 2.0799 2.0844 2.8595 2.8563 2.8545 2.8855 2.8798 2.8228 2.8825 4.4741 4.4654 4.4609 4.5412 4.4998 4.3761 — CCCC Quadratic Cubic Quartic MISQ20 [83] MLSDQ [13] RBF [12] FSDT [84] 2.3734 2.3727 2.3724 2.3869 2.3790 2.4021 2.3820 2.4659 2.4651 2.4648 2.4803 2.4725 2.4932 — 2.7813 2.7803 2.7799 2.7998 2.7927 2.8005 2.7921 3.4595 3.4580 3.4573 3.4893 3.4723 3.4923 3.4738 4.9328 4.9297 4.9282 4.9989 4.9430 4.9541 — Boundary FSDT, first-order shear deformation theory Table IX A non-dimensional frequency parameter $ D !a2 = h =E2 /1=2 of simply supported and clamped angle-ply Œ45ı = 45ı =45ı = 45ı =45ı skew plate with various ˛ angles ˛ Method 0ı 15ı 30ı 45ı 60ı SSSS Quadratic Cubic Quartic MISQ20 [83] MLSDQ [13] RBF [12] FSDT [84] 1.8759 1.8410 1.8408 1.8413 1.8248 1.8357 1.8792 1.9264 1.8906 1.8905 1.8889 1.8838 1.8586 — 2.1444 2.1008 2.1001 2.0955 2.0074 2.0382 2.0002 2.6679 2.6054 2.6029 2.5672 2.5028 2.4862 2.4788 4.1514 4.0248 4.0211 3.9718 4.0227 3.8619 — CCCC Quadratic Cubic Quartic MISQ20 [83] MLSDQ [13] RBF [12] FSDT [84] 2.2790 2.2780 2.2776 2.2908 2.2787 2.3324 2.2857 2.3446 2.3436 2.3432 2.3570 2.3504 2.3962 — 2.6543 2.6532 2.6527 2.6708 2.6636 2.6981 2.6626 3.3403 3.3389 3.3383 3.3683 3.3594 3.3747 3.3523 4.8319 4.8293 4.8281 4.8982 4.8566 4.8548 — Boundary CCCC, clamped on all boards; FSDT, first-order shear deformation theory; SSSS, simply supported on all boards Copyright © 2012 John Wiley & Sons, Ltd Int J Numer Meth Engng (2012) DOI: 10.1002/nme NURBS-BASED ISOGEOMETRIC ANALYSIS FOR LAMINATED COMPOSITE PLATES and two outside layers (skins) are derived from this core QN skin D RQN core (35) The normalized displacement and stresses of the square sandwich plate is defined as $ D 0.999781w.a=2, a=2, 0/=.hq/I D N xx xx a=2, a=2, 2h=5/=qI N xx D xx a=2, a=2, h=2/=q D N xx xx a=2, a=2, 2h=5/=q In this problem, we investigate the convergence of the normalization deflection of the present method with different R values The plate is modeled with 9, 17 17, 25 25, and 33 33 control points per side Figure 11a–c illustrates the normalization deflection for R D 5, 10, 15 The present results converge quickly to the exact solution Figure 11d depicts the relative error of center deflection of the present results with the exact solution [75] It is seen that the results of the present method are in excellent agreement with the high-order collocation (wavelets) method reported by Ferreira et al [11] The obtained results are compared with the exact solution [75] and published results based on FSDT [76], HSDT [76], RBF pseudospectral (RBF-PS) method [77], and wavelets method [11] as shown in Table III Figure 17 Geometry of a circular plate 0.5 0.8 0.4 0.6 0.3 0.4 0.2 0.1 0.2 0 −0.1 −0.2 −0.2 −0.4 −0.3 −0.6 −0.4 −0.5 −0.5 −0.8 −0.8 −0.6 −0.4 −0.2 0.5 (a) 0.2 0.4 0.6 0.8 (b) Figure 18 Mesh and control net for a disk of radius 0.5: (a) coarse mesh and (b) control net Table X Control points and weights for a disk of radius 0.5 i xi yi wi p p 2=4 2=4 p 2=2 p 2/2 p p2=4 2=4 Copyright © 2012 John Wiley & Sons, Ltd p0 p2=2 2/2 0 p0 p 2=2 2/2 p p2=4 2=4 p 2=2 p0 2/2 p p2=4 2=4 Int J Numer Meth Engng (2012) DOI: 10.1002/nme C H THAI ET AL 5.2.2 Four-layer [0=90=90=0] square laminated plate under sinusoidal load Let us consider a simply supported square laminated plate subjected to a sinusoidal load q as shown in Figure 12 The length-to-width ratio is a=b D 1, and the length-to-thickness ratios are a= h D 4, 10, 20, 100 The plate is modeled with 33 33 control points per side The material properties are given by E1 D 25E2 , G12 D G13 D 0.5E2 , G23 D 0.2E2 , and 12 D 0.25 The normalized displacement and the normal stresses of the center of four-layer Œ0=90=90=0 square plate is given by $ D 100E2 h3 w.a=2, a=2, 0/= qa4 , N xx D N yy D h2 qb yy a=2, a=2, h=4/I Nxy D h2 qb h2 qb xx a=2, a=2, h=2/, xy 0, 0, h=2/I Nx´ D h qb xy 0, b=2, 0/ As shown in Table IV, the results of the present method are compared with those of the HSDT approach proposed by Reddy [78], the FSDT formulation by Akhras et al [79], the RBF-PS reported by Ferreira et al [77], the high-order collocation method [11], and the elasticity solution given by Pagano [27] The present solution agrees very well with other published results For transverse shear stress analysis, the present approach provides solutions in good agreement with finite strip method using FSDT by Akhras et al [80] Figure 13 plots the distribution of stresses through thickness plate with a= h D 4, 10 5.3 Free vibration analysis of laminated composite plates Now, we examine the accuracy and the efficiency of the quadratic, cubic, and quartic elements in analyzing natural frequencies of laminated composite plates In practice, composite plates with various boundary conditions, span-to-thickness ratios, fiber orientation angles, and modulus ratios are analyzed The following material parameters of a layer are used in all plate examples unless otherwise stated: E1 =E2 D 10, 20, 30, 40, G12 D G13 D 0.6E2 , G23 D 0.5E2 , 12 D 0.25, and D The shear correction factors k12 D k22 D =12 are used for all computations The plate may have free (F), simply supported (S) or clamped (C) edges 5.3.1 Square laminated plates Let us consider a four-layer [0=90=90=0] simply supported plate, where all layers of the laminated plate are assumed to be of the same thickness and mass density and made of the same linearly elastic composite material The thickness-to-length ratio is h=a D 0.2 The first normalized frequencies derived from the present method are listed in Table V The obtained results are compared with exact solutions using HSDT [22, 40], the moving least squares differential quadrature method [13], RBFs [77], and wavelets [11] The presented results approach the wavelet ones and are close to the exact values [22, 40] that are based on HSDT Figure 19 Meshes produced by h refinement (knot insertion) Copyright © 2012 John Wiley & Sons, Ltd Int J Numer Meth Engng (2012) DOI: 10.1002/nme NURBS-BASED ISOGEOMETRIC ANALYSIS FOR LAMINATED COMPOSITE PLATES Next, we consider a three-layer laminated [0=90=0] rectangular clamped plate with E1 =E2 D 40 The normalized frequencies are defined as $ D !b = h=D0 /1=2 with D0 D E2 h3 =12 12 21 / The geometry is described in Figure 14 The length-to-width ratios are a=b D 1, 2, and the width-to-thickness ratios are b= h D 5, 10, 20, 100 The plate is modeled with 17 control points per side The present results are compared with solutions reported by Liew et al [13] based on the FSDT and global–local higher-order theory used by Zhen and Wanji [30], as well as RBFs and wavelet functions given by Ferreira et al [11, 77] The first eight normalized frequencies of the present method are given in Tables VI and VII for a=b D and 2, Table XI A non-dimensional frequency parameter $ D !a2 = h =E2 /1=2 of a circular four-layer ŒÂ= Â= Â= clamped laminated plate Mode  15 30 45 Method Quadratic (8 8) Cubic (8 8) Quartic (8 8) 22.1126 22.1562 22.2146 29.6536 29.6252 29.7010 41.4386 41.0117 41.0574 42.5865 42.6636 42.7554 50.4566 50.4487 50.5616 57.1676 55.5807 55.5172 Quadratic (16 16) Cubic (16 16) Quartic (16 16) 22.0989 22.1110 22.1227 29.5409 29.5550 29.5735 40.8126 40.8150 40.8410 42.5447 42.5650 42.5854 50.2975 50.3201 50.3478 54.7732 54.7332 54.7609 MISQ20 [83] MLSDQ [13] 22.123 22.211 29.768 29.651 41.726 41.101 42.805 42.635 50.756 50.309 56.950 54.553 Quadratic (8 8) Cubic (8 8) Quartic (8 8) 22.6611 22.7078 22.7666 31.3978 31.3894 31.4626 43.3495 43.4310 43.5254 43.9221 43.5886 43.6444 53.0311 53.0438 53.1510 59.9002 58.5805 58.5645 Quadratic (16 16) Cubic (16 16) Quartic (16 16) 22.6500 22.6626 22.6751 31.3012 31.3166 31.3359 43.3124 43.3335 43.3550 43.3833 43.3899 43.4165 52.8952 52.9197 52.9486 57.8347 57.8064 57.8349 MISQ20 [83] MLSDQ [13] 22.698 22.774 31.568 31.455 43.635 43.350 44.318 43.469 53.468 52.872 60.012 57.386 Quadratic (8 8) Cubic (8 8) Quartic (8 8) 23.9507 24.0029 24.0627 36.0530 36.0942 36.1742 43.8305 43.9090 43.9999 51.3057 51.1856 51.2685 56.7897 56.8376 56.9421 66.2664 66.3023 66.4101 Quadratic (16 16) Cubic (16 16) Quartic (16 16) 23.9428 23.9565 23.9703 35.9896 36.0085 36.0298 43.7948 43.8164 43.8390 50.9574 50.9745 51.0024 56.6770 56.7038 56.7337 66.0745 66.1011 66.1316 MISQ20 [83] MLSDQ [13] 24.046 24.071 36.399 36.153 44.189 43.968 52.028 51.074 57.478 56.315 67.099 66.220 Quadratic (8 8) Cubic (8 8) Quartic (8 8) 24.6405 24.6953 24.7560 38.9814 39.0541 39.1444 43.4536 43.5211 43.6056 57.0442 57.0962 57.2030 57.1069 57.1182 57.2287 65.4762 65.4778 65.5658 Quadratic (16 16) Cubic (16 16) Quartic (16 16) 24.6335 24.6478 24.6622 38.9379 38.9591 38.9814 43.4120 43.4330 43.4559 56.8708 56.8937 56.9205 56.9251 56.9531 56.9844 65.2751 65.3002 65.3320 MISQ20 [83] MLSDQ [13] 24.766 24.752 39.441 39.181 43.817 43.607 57.907 56.759 57.945 56.967 66.297 65.571 Copyright © 2012 John Wiley & Sons, Ltd Int J Numer Meth Engng (2012) DOI: 10.1002/nme C H THAI ET AL respectively The results of the present method are very close to those given by Ferreira et al [11] using high-order wavelet functions The first six shape modes of the three-layer plate with b= h D 10 are plotted in Figure 15 5.3.2 Skew laminated composite plates Let us consider the five-layer skew laminated Œ90=0=90=0=90 and Œ45= 45=45= 45=45 square plates with full simply supported (SSSS) and clamped (CCCC) condition boundary as shown in Figure 16 In this problem, various skew angles are considered The length-to-thickness ratio a= h is taken to be 10 The normalized frequencies are defined as $ D !a2 = h =E2 /1=2 For comparison, the plate is modeled with 17 17 control points per side The normalized frequencies with various skew ˛ angles from 0ı to 60ı are Mode 2, Omega = 38.9814 −0.5 0.5 −1 z z Mode 1, Omega = 24.6622 −1.5 0.5 −2 −0.5 0 x −0.5 x 1 0.5 −1 0 0 x y 0.5 −0.5 Mode 6, Omega = 65.332 0.5 z z −0.5 −1 0.5 0.5 −0.5 y Mode 5, Omega = 56.9844 x 0.5 −1 −0.5 0.5 −0.5 0.5 −0.5 −0.5 0.5 −1.5 −0.5 y Mode 4, Omega = 56.9205 z z Mode 3, Omega = 43.4559 x y 0.5 −0.5 −2 −0.5 0.5 −1 −0.5 −1 0.5 −2 −0.5 y x y 0.5 −0.5 Figure 20 The first six mode shapes of a circular four-layer clamped laminated plate with a= h D 10 and  D 45ı Copyright © 2012 John Wiley & Sons, Ltd Int J Numer Meth Engng (2012) DOI: 10.1002/nme NURBS-BASED ISOGEOMETRIC ANALYSIS FOR LAMINATED COMPOSITE PLATES shown in Tables VIII and IX The present results are compared with those of the Mixed Interpolation Smoothing Quadrilateral element with 20 DOFs (MISQ20) [83], the moving least squares differential quadrature method (MLSDQ) [13], radial basis approach [12], and FSDT [84] The obtained solutions are in good agreement with other existing ones 5.3.3 Circular plates A circular four-layer ŒÂ= Â= Â= laminated plate with clamped boundary and various fiber orientation angles ˛ D 0ı , 15ı , 30ı , 45ı is studied The geometry is illustrated in Figure 17 A rational quadratic basis is used to describe a circle The coarsest mesh, „ H , is Figure 21 Geometry and control points of the triangular plate Table XII A non-dimensional frequency parameter $ D !a2 = h =E2 /1=2 of a triangular clamped laminated plate, E1 =E2 D 25 and a= h D 100 Mode Lay-up Method 30ı Quadratic Cubic Quartic A˛-DSG3 [49] MISQ20 [83] LS12 [85] 54.128 52.776 52.400 52.928 52.222 51.735 88.264 85.191 84.455 85.348 84.840 83.376 118.917 113.480 112.431 113.726 113.245 110.71 133.474 126.954 125.263 127.797 127.063 123.05 171.616 158.427 156.016 158.514 159.778 153.62 189.548 178.318 174.664 178.559 178.549 169.81 Œ45ı = 45ı Quadratic Cubic Quartic A˛-DSG3 [49] MISQ20 [83] LS12 [85] 54.467 53.109 52.735 53.213 52.571 52.057 89.363 86.450 85.769 86.761 86.269 84.715 119.769 114.415 113.283 114.555 113.882 111.25 135.727 129.688 128.456 130.590 130.911 126.73 172.529 159.417 156.815 159.222 160.286 153.86 193.778 183.922 181.145 183.982 186.491 177.31 Œ0ı =90ı Quadratic Cubic Quartic A˛-DSG3 [49] MISQ20 [83] LS12 [85] 54.119 52.865 52.535 53.102 52.453 51.933 93.769 91.068 90.474 91.540 91.179 89.391 110.664 105.376 104.335 105.973 105.284 102.88 147.467 142.199 141.182 143.571 144.020 139.28 168.088 154.681 152.260 155.169 156.290 149.94 194.620 179.324 175.514 179.378 178.941 172.06 Œ0ı =90ı =0ı Quadratic Cubic Quartic A˛-DSG3 [49] MISQ20 [83] LS12 [85] 71.474 70.310 69.996 70.719 70.000 69.252 112.740 109.286 108.395 109.721 109.088 106.73 152.148 146.779 145.870 147.345 148.223 143.88 169.032 161.409 158.977 161.216 160.771 155.06 213.669 200.530 197.812 202.288 203.320 193.84 237.668 223.777 218.363 221.885 221.969 210.11 Œ30ı = Copyright © 2012 John Wiley & Sons, Ltd Int J Numer Meth Engng (2012) DOI: 10.1002/nme C H THAI ET AL defined by the knot vectors as follows: ® ¯ „D 0 1 I HD ® 0 1 ¯ The exact geometry is represented with a single element using nine control points, as shown in Figure 18 The geometric data are given in Table X The first four meshes used in the analysis are plotted in Figure 19 Table XI summarizes the frequencies of circular plate with a span-tothickness ratio of D= h D 10 derived from the present element in comparison with the MISQ20 element [83] using FSDT and the moving least squares differential quadrature method [13] using FSDT It can be seen from Table XI that the present results give a slightly smaller value than those obtained by all previous results Figure 20 provides the first six mode shapes of a circular four-layer Œ45= 45= 45=45 clamped laminated plate Mode 1, Omega = 69.9961 Mode 2, Omega = 108.3953 0.2 0.6 0.1 0.4 z z 0.2 −0.1 −0.2 −0.3 −0.2 0 0.2 0.2 0.4 0.4 0.6 x 0.8 0.2 0.4 0.6 0.8 0.6 x 0.8 y 0.2 0.4 0.6 0.8 y Mode 4, Omega = 158.9771 Mode 3, Omega = 145.8699 0.3 0.4 0.2 0.2 z z 0.1 −0.2 −0.1 −0.2 −0.4 0 0.2 0.2 0.4 0.4 0.6 x 0.8 0.2 0.4 0.6 0.8 0.6 x 0.8 y 0.4 0.6 0.8 y Mode 6, Omega = 218.3627 0.2 0.2 0.1 0.1 0 z z Mode 5, Omega = 197.8121 0.2 −0.1 −0.1 −0.2 −0.2 0 0.2 0.2 0.4 x 0.4 0.6 0.8 0.2 0.4 y 0.6 0.8 x 0.6 0.8 0.2 0.4 0.6 0.8 y Figure 22 The first six mode shapes of a triangle Œ0=90=0 clamped laminated plate with a= h D 100 using quartic element Copyright © 2012 John Wiley & Sons, Ltd Int J Numer Meth Engng (2012) DOI: 10.1002/nme NURBS-BASED ISOGEOMETRIC ANALYSIS FOR LAMINATED COMPOSITE PLATES 5.3.4 Triangular laminated plates Consider clamped triangular plates with various cross-ply and angle-ply shown in Figure 21 The following material properties are used in the analysis: E1 =E2 D 25, G12 D G13 D 0.5E2 , G23 D 0.5E2 , 12 D 0.25, and D Table XII shows a comparison of frequency parameter $ using the quadratic, cubic, and quartic elements with the solutions using the LS12 higher-order element of Haldar and Sengupta [85], the MISQ20 element [83], and the A˛-DSG3 element [49] based on FSDT The present results give a slightly larger value than those obtained by the LS12 element and the MISQ20 element for this problem The first six mode shapes are also displayed in Figure 22 5.4 Buckling analysis of composite plate In this subsection, the buckling load factor is defined as N D cr a2 = E2 h3 , where a, h, and cr are the edge length, the thickness of the composite plate, and the critical buckling load, respectively The material parameters are assumed to be as follows: E1 =E2 D 3, 10, 20, 30, 40, G12 D G13 D 0.6E2 , G23 D 0.5E2 , and 12 D 0.25 The plate may have free (F), simply (S) Figure 23 Geometry of laminated composite plates under axial and biaxial compression Table XIII A normalized critical buckling load of simply supported cross-ply Œ0ı =90ı =90ı =0ı square plate with various E1 =E2 ratios E1=E2 Method Control mesh 10 20 30 40 Quadratic 13 17 13 17 5.3263 5.3198 5.3183 9.8371 9.8245 9.8215 15.1591 15.1408 15.1365 19.5125 19.4906 19.4854 23.1653 23.1410 23.1353 Cubic 13 17 13 17 5.3220 5.3184 5.3177 9.8279 9.8216 9.8203 15.1452 15.1365 15.1346 19.4957 19.4854 19.4832 23.1465 23.1353 23.1328 Quartic 13 17 13 17 5.3192 5.3178 5.3175 9.8238 9.8207 9.8200 15.14 15.1353 15.1342 19.4897 19.4839 19.4826 23.1401 23.1337 23.1322 5.401 5.114 5.442 5.294 9.985 9.774 10.026 9.762 15.374 15.298 15.418 15.019 19.537 19.957 19.813 19.304 23.154 23.340 23.489 22.881 Liu et al [56] Phan and Reddy [20] Khdeir and Librescu [23] Noor and Mathers [28] Copyright © 2012 John Wiley & Sons, Ltd Int J Numer Meth Engng (2012) DOI: 10.1002/nme C H THAI ET AL supported, or clamped (C) edges The shear correction factors k12 D k22 D all computations =12 are used for 5.4.1 Square plate under uniaxial compression A simply supported four-layer cross-ply Œ0ı =90ı =90ı =0ı square laminated plate is subjected to uniaxial compression shown in Figure 23 The span-to-thickness ratio of the plate a= h is taken to be 10 The convergence of the normalized critical buckling load of a simply supported four-layer crossply square laminated with the various modulus ratios is shown in Table XIII The results of the present method are compared with the 3D elasticity solution [28], radial point interpolation method with RBF based on HSDT [56], and FEM solution based on HSDT [20, 23] The present results are Table XIV A normalized critical buckling load of simply supported cross-ply square plate with various a= h ratios a= h Number layer Œ0ı =90ı 00 =900 =900 =00 Method 10 20 50 100 Quadratic Cubic Quartic MISQ20 [86] FSDT [24] FSDT [19] HSDT [19] 11.0985 11.097 11.0966 11.169 11.349 11.353 11.563 12.4322 12.4301 12.4294 12.520 12.510 12.515 12.577 12.8765 12.8724 12.8631 12.967 12.879 12.884 12.895 12.9526 12.9415 12.9389 13.033 12.934 12.939 12.942 Quadratic Cubic Quartic MISQ20 [86] FSDT [24] FSDT [19] HSDT [19] 23.1353 23.1328 23.1322 23.236 23.409 23.471 23.349 31.5558 31.5511 31.5498 31.747 31.625 31.707 31.637 35.3366 35.3288 35.3258 35.561 35.254 35.356 35.419 35.9716 35.9566 35.9499 36.190 35.851 35.955 35.971 FSDT, first-order shear deformation theory; HSDT, high-order shear deformation theory Table XV A normalized critical buckling load of cross-ply Œ0ı =90ı and Œ0ı =90ı 5 square plate with various mixed boundaries (E1 =E2 D 40; a= h D 10) Boundary condition Number layer Method SSSS SSFF SSCC SSSC SSFC SSFS Œ0ı =90ı Quadratic Cubic Quartic MISQ20 [86] MLSDQ [87] RKPM [88] FSDT [21] HSDT [21] 11.4932 11.4917 11.4913 11.291 11.301 11.582 11.353 11.562 5.0156 5.0149 5.0146 4.860 4.823 4.996 4.851 4.940 19.5048 19.4925 19.4883 20.082 19.871 20.624 20.067 21.464 16.4553 16.4453 16.4420 16.470 — 16.872 16.437 17.133 6.3075 6.3052 6.3043 6.140 — 6.333 6.166 6.274 5.4842 5.4832 5.4828 5.342 — 5.502 5.351 5.442 Œ0ı =90ı 5 Quadratic Cubic Quartic MISQ20 [86] MLSDQ [87] RKPM [88] FSDT [21] HSDT [21] 25.5305 25.5276 25.5269 25.525 25.338 25.703 25.450 25.423 12.2357 12.2342 12.2338 12.131 12.030 12.224 12.092 12.077 35.1905 35.1813 35.1784 35.105 34.604 35.162 34.837 35.376 32.6899 32.6890 32.6882 32.870 — 32.950 32.614 32.885 14.4854 14.4820 14.4828 14.352 — 14.495 14.358 14.351 12.6197 12.6179 12.6174 12.541 — 12.658 12.524 12.506 FSDT, first-order shear deformation theory; HSDT, high-order shear deformation theory Copyright © 2012 John Wiley & Sons, Ltd Int J Numer Meth Engng (2012) DOI: 10.1002/nme NURBS-BASED ISOGEOMETRIC ANALYSIS FOR LAMINATED COMPOSITE PLATES in good agreement with those solutions The normalized critical buckling loads become larger with the increase of the E1 =E2 modulus ratios Next, the effect of the span-to-thickness ratio (a= h) on the uniaxial compression load is also considered for two-layer and four-layer simply supported cross-ply square plates Table XIV illustrates the normalized critical buckling load of two-layer and four-layer simply supported plates The present result is in fair agreement with available solutions In addition, the influence of the mixed boundary conditions of two-layer and 10-layer cross-ply square plates is studied We consider the normalized critical buckling load factors of SSSS, SSFF, SSCC, SSFC, and SSFS plates with length-to-thickness ratios of a= h D 10 and modulus ratios of E1 =E2 D 40 Given as in 0 −0.05 −0.2 z −0.1 z −0.4 −0.6 −0.8 −1 −0.15 0.8 0.2 −0.2 0.6 0.4 x 0.4 0.6 0.8 0.2 1 0.5 0.8 0.2 y 0 z z 0.4 0.6 (b) −0.5 −0.5 0.8 0.2 −1 0.6 0.4 x 0.4 0.6 0.8 0.2 1 0.8 0.2 y 0.6 0.4 x (c) 0.4 0.6 0.8 0.2 y (d) 1 0.8 0.8 0.6 0.6 z z 0.4 x 0.5 0.4 0.4 0.2 0 0.2 y (a) −1 0.8 0.6 0.8 0.2 0.6 0.4 x 0.4 0.6 0.8 0.2 0 y 0.8 0.2 0.6 0.4 x (e) 0.2 0.4 0.6 0.8 0.2 y (f) Figure 24 Fundamental buckling modes of 10-layer Œ0ı =90ı 5 : (a) SSSS, (b) SSFF, (c) SSCC, (d) SSSC, (e) SSFC, and (f) SSFS Copyright © 2012 John Wiley & Sons, Ltd Int J Numer Meth Engng (2012) DOI: 10.1002/nme C H THAI ET AL Table XVI Biaxial buckling of simply supported cross-ply Œ0ı =90ı =0ı square plate with various modulus ratios E1 =E2 Method Quadratic Cubic Quartic MISQ20 [86] FSDT [89] HSDT [23] 10 20 30 40 4.9966 4.9960 4.9958 4.939 4.963 4.963 7.5165 7.5157 7.5155 7.488 7.588 7.516 8.8756 8.8724 8.8712 9.016 8.575 9.056 10.0573 10.0537 10.0525 10.252 10.202 10.259 FSDT, first-order shear deformation theory; HSDT, high-order shear deformation theory Table XV, the results of the present method agrees well with the solution of several other methods such as MISQ20 [86], MLSDQ [87], the reproducing kernel particle method (RKPM) [88] and the finite element method based on FSDT and HSDT [23] Figure 24 illustrates buckling shape modes of 10-layer Œ0ı =90ı 5 plate with various boundary conditions 5.4.2 Square plate under biaxial compression Finally, we consider the three-layer symmetric cross-ply Œ0ı =90ı =0ı simply supported plate subjected to the biaxial buckling load shown in Figure 24 The span-to-thickness ratio a= h is taken to be 10 The plate is modeled with control net mesh of 17 17 The effect of modulus ratio E1 =E2 on the critical biaxial buckling load is studied in this section Table XVI shows the normalized critical buckling loads When the modulus ratio E1 =E2 increases, the normalized critical biaxial buckling loads are also increased It can be seen that the present method works well compared with several other methods CONCLUSIONS An isogeometric formulation has been developed for static, free vibration, and bucking analysis of the laminated composite Reissner–Mindlin plates One of the advantages of the present approach is to increase easily the order of basic functions without any difficulty whereas the classical FEM requires redistribution of meshing basis Therefore, the quadratic, cubic, and quartic elements have been investigated, and their results show very good agreement compared with those of other existing methods The other advantage of isogeometric analysis lies in the exact description of complicated geometries although the geometry of benchmark problems tested in this work is rather simple Although the present method can produce highly accurate solution, its computational cost is still significant because of an excessive overhead of control points with increasing refinement An adaptive approach based on T-spline [90] or polynomial splines over hierarchical T-meshes [91, 92] will be therefore very promising in enhancing computational effect In the present approach, we only develop isogeometric analysis for laminated composite plates based on FSDT Hence, it will be necessary to apply the third-order shear deformation theory to predict more effectively the interlaminar stress of laminated plates In other aspects of eliminating the locking issue, we will also study the MITC approach for isogeometric plate/shell approach ACKNOWLEDGEMENT The support of the Vietnam National Foundation for Science and Technology Development (NAFOSTED) is gratefully acknowledged REFERENCES Reddy JN Mechanics of Laminated Composite Plates and Shells Theory and Analysis, 2nd edn CRC Press: New York, 2004 Reissner E, Stavsky Y Bending and stretching of certain types of aeolotropic elastic plates Journal of Applied Mechanics 1961; 28:402–408 Copyright © 2012 John Wiley & Sons, Ltd Int J Numer Meth Engng (2012) DOI: 10.1002/nme NURBS-BASED ISOGEOMETRIC ANALYSIS FOR LAMINATED COMPOSITE PLATES Stavsky Y Bending and stretching of laminated aeolotropic plates Journal of Engineering Mechanics Division 1961; 87:31–56 Dong SB, Pister KS, Taylor RL On the theory of laminated anisotropic plates and shells Journal of Aeronautical Science 1962; 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analysis of composite plates, four examples for free vibration analysis of composite plates, and two examples for buckling analysis of composite plates In... geometry of the domain, for static, free vibration, and buckling analysis of laminated composite Reissner–Mindlin plate structures We formulate the isogeometric stiffness matrices for the quadratic,... orthotropic and laminated plates according to a higher order shear deformation theory Journal of Sound and Vibration 1985; 89:157–170 20 Phan ND, Reddy JN Analysis of laminated composite plates using