an edge-based smoothed finite element method for

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International Journal of Computational Methods Vol. 10, No. 1 (2013) 1340005 (27 pages) c World Scientific Publishing Company DOI: 10.1142/S0219876213400057

AN EDGE-BASED SMOOTHED FINITE ELEMENT METHOD FOR ANALYSIS OF LAMINATED COMPOSITE PLATES

Int. J. Comput. Methods 2013.10. Downloaded from www.worldscientific.com by 113.161.90.253 on 03/27/13. For personal use only.

H. H. PHAN-DAO∗,† , H. NGUYEN-XUAN‡,§,¶ , C. THAI-HOANG† and T. NGUYEN-THOI†,‡ ∗Faculty

of Civil Engineering Ton Duc Thang University Ho Chi Minh City, Vietnam †Division of Computational Mechanics Ton Duc Thang University Ho Chi Minh City, Vietnam ‡Department

of Mechanics, Faculty of Mathematics and Computer Science, University of Science VNU-HCM, Vietnam §[email protected] T. RABCZUK

Institute of Structural Mechanics Bauhaus-University Weimar Marienstr. 15, D-99423 Weimar, Germany Received 20 February 2011 Accepted 25 August 2011 Published 28 February 2013

This paper promotes a novel numerical approach to static, free vibration and buckling analyses of laminated composite plates by an edge-based smoothed finite method (ESFEM). In the present ES-FEM formulation, the system stiffness matrix is established by using the strain smoothing technique over the smoothing domains associated with the edges of the triangular elements. A discrete shear gap (DSG3) technique without shear locking is combined into the ES-FEM to give a so-called edge-based smoothed discrete shear gap method (ES-DSG3) for analysis of laminated composite plates. The present method uses only linear interpolations and its implementation into finite element programs is quite simple. Numerical results for analysis of laminated composite plates show that the ES-DSG3 performs quite well compared to several other published approaches in the literature. Keywords: FEM; SFEM; discrete shear gap method (DSG); stabilized method; laminated composite plates.

¶Corresponding

author. 1340005-1


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1. Introduction Composite materials with fiber-reinforced have been widely used in various engineering such as aircrafts, aerospace, vehicles, buildings, etc. Laminated composites are made of two or several lamina layers with different materials stacked together to achive desired properties (e.g., high stiffness and strength-to-weight ratios, long fatigue life, wear resistance, damping, etc.) [Reddy (2004)]. Most of developed models for laminated composite plates are the Equivalent Single-Layer (ESL) models such as Classical Laminated Plate Theory (CLPT) [Liu et al. (2007)], First Order Shear Deformation Theory (FSDT) [Liew et al. (2003); Ferreira et al. (2005); Cen et al. (2006); Liu et al. (2008)], and High Order Shear Deformation Theory (HSDT) [Phan and Reddy (1985); Khdeir and Librescu (1988); Chakrabarti and Sheikh (2003)]. The FSDT is simple to implement and gives better results than CLPT since 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. Such the FSDT only requires C 0 -continuity of generalized displacements. However, it requires so-called shear correction factors (SCFs) to take into account the nonlinear distribution of shear stress terms. When the laminated plates becomes thicker, layer-wise (LW) model [Roque et al. (2005); Ferreira et al. (2008)] can be recommended to improve the accuracy of transverse shear stresses. Between ESL and LW models, the ESL has been widely used because of some advantages such as the simplicity in modeling and formulating constitutive equations, and the low computational cost. Especially, in analyzing thin or moderately thick laminated plates, the ESL models often offer relatively accurate results. In this paper, we focus on the low-order elements using the FSDT of ESL. Due to the limitation of analytical solutions for laminated composite plates in practical applications, many numerical methods have been developed with various degrees of success. One of the most popular methods is the finite element method (FEM). In development of improved finite element technologies, based on a strain smoothing technique, Liu and his research team have recently developed a family of smoothed finite element methods (SFEM) [Liu and Nguyen-Thoi (2010)] such as the cell/element-based SFEM (CS-FEM) [Liu et al. (2007)], the node-based SFEM (NS-FEM) [Liu et al.(2009)], the edge-based SFEM (ES-FEM) [Liu et al. (2008)] and the face-based SFEM (FS-FEM) [Nguyen-Thoi et al. (2009)]. Concerning on the ES-FEM using triangular meshes (ES-FEM-T3) [Liu et al. (2008)], numerical results showed that (1) ES-FEM-T3 model is often found superconvergent and even more accurate than those of the FEM using quadrilateral elements (FEM-Q4) using the same sets of nodes; (2) It does not exist spurious nonzeros energy modes which cause instability for vibration analysis and (3) The implementation of the method is simple and no penalty parameter is required, and the computational efficiency (computation time for the same accuracy) is much better than the FEM-T3 when the same sets of nodes are used. The ES-FEM-T3 1340005-2


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ES-FEM Formulation for Analysis of Laminated Composite Plates

has been recently developed for analysis of isotropic plates and shells [NguyenXuan et al. (2010); Cui et al.(2010)]. In this paper, a novel numerical approach using the ES-FEM-T3 is presented for static, free vibration and buckling analyses of laminated composite plates. In order to avoid shear locking, the ES-FEM-T3 is hence incorporated with the discrete shear gap (DSG) method [Bletzinger et al. (2000)] to give a so-called edge-based smoothed discrete shear gap method (ES-DSG3). It can be found by numerical examples that the ES-DSG3 triangular element agrees well with analytical and other published approaches. The paper is organized as follows. Next section describes the weak form and FEM formulation of governing equations of laminated composite plates for static, free vibration and buckling problems. In Sec. 3, a formulation of the ES-FEM-T3 with the discrete shear technique is introduced. Numerical results are presented in Sec. 4. Finally, we close our paper with some concluding remarks. 2. Governing Equations, Weak Form and FEM Formulation Based on FSDT, the displacement field u = [u v w]T is expressed as [Reddy (2004)]  u(x, y, z, t) = u0 (x, y, t) + zβx (x, y, t)    v(x, y, z, t) = v 0 (x, y, t) + zβy (x, y, t) (1)    w(x, y, z, t) = w0 (x, y, t), where (u0 , v 0 , w0 ) are the displacements of a point on the midplane (i.e., z = 0), and βx , βy denote the transverse rotations about the y, x axes, respectively. The in-plane strain vector = [x y γxy ]T can be written as = 0 + zκ,

(2)

where 0 , κ are the membrane strain and the bending strain, respectively.       b   u0,x m βx,x x x         0    m   0 =  y  =  v,y  , κ =  =  βy,y by           m 0 b γxy u0,y + v,x β + β x,y y,x γ

(3)

xy

and the transverse shear vector γ forms γxz βx − w,x γ= = . γyz βy − w,y

(4)

The weak form of the static models based on Reissner–Mindlin assumption for laminated plates can be described as

¯ + δγ T Ds γ)dΩ = (δT D δuT pdΩ + δuT tdΓ, (5) Ω

Ω

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Γ


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where p = [0 0 p]T , p is the transverse loading per area unit, t represents prescribed traction on the natural boundary and ¯ = A B , D (6) B D where Aij , Dij , Bij are the extensional, bending, bending-extension coupling stiffness tensors, respectively, which are defined as follows:

h/2 ¯ ij dz, (i, j = 1, 2, 6), (1, z, z 2)Q (7) (Aij , Bij , Dij ) = −h/2


Dijs =

h/2

−h/2

¯ ∗ dz, Q ij

(i, j = 4, 5).

(8)

The weak form is applied for the free vibration of Reissner–Mindlin plates can be obtained as

¯ + δγ T Ds γ)dΩ = (δT D δuT m¨ udΩ. (9) Ω

Ω

In the case of in-plane buckling problem under pre-buckling stresses σ ˆ 0 , the weak form writes

T s T ¯ δ D + δγ D γ dΩ + h ∇T δw σ ˆ 0 ∇wdΩ Ω

Ω

σ ∇δu0 ˆ0 0 T T +h dΩ (10) ∇ δu0 ∇ δv0 ∇δv0 0 σ ˆ0 Ω

T σ h3 ∇δβx ˆ0 0 T + dΩ = 0, ∇ δβx ∇ δβy ∇δβy 0 σ ˆ0 12 Ω 0 0 σx 0.5τxy ˆ0 = are the gradient vector and where ∇T = [∂/∂x ∂/∂y] and σ 0 0.5τxy σy0 in-plane pre-buckling stress, respectively. The geometry domain Ω is discretized into Ne finite triangular elements such e as Ω ≈ ∪N e=1 Ωe and Ωi ∩ Ωj = for i = j. With Reissner–Mindlin assumption, the generalized displacement field of laminated composite plates is approximated by standard finite element solutions as   0 0 0 0 NI (x) 0 0 0  NI (x) Nn    h  (11) u = 0 0  NI (x)   dI I=1  0  NI (x) syms NI (x)

where Nn is the total number of nodes, NI (x) and dI = [u0 v 0 w0 βx βy ] are the shape function and the nodal degrees of freedom of uh associated to node I, respectively. 1340005-4


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The membrane, bending, shear, and geometrical strains of laminated plates are expressed as 0 = Bm κ= BbI dI , γ = BsI dI , (12) I dI , I

= g

I

I

BgI dI ,

(13)

I

where




 0 0 0 0 NI,x  0 Bm NI,y 0 0 0 , I = NI,y NI,x 0 0 0 0 0 NI,x NI 0 s BI = , 0 0 NI,y 0 NI 

NI,x  0  BgI =   0  0 0

NI,y 0 0 0 0

0 NI,x 0 0 0

0 NI,y 0 0 0



0 0 b  BI = 0 0 0 0

0 NI,x 0 0 0 NI,y

 0 NI,y  , NI,x (14)

0 0 NI,x 0 0

0 0 NI,y 0 0

0 0 0 NI,x 0

0 0 0 NI,y 0

0 0 0 0 NI,x

T 0 0   0   . 0  NI,y (15)

The system equation of the Reissner–Mindlin plate for static analysis can be obtained as Kd = F,

(16)

where K is the global system stiffness matrix:

T K= Bm BBb + BTb BBm dΩ + BTm ABm dΩ + BTb DBb dΩ Ω Ω Ω

+

Km

Kmb +KT mb

BTs Ds Bs dΩ = Km + Kmb + KTmb + Kb + Ks Ω

Kb

(17)

Ks

and F is the load vector:

pNT dΩ + f b ,

F=

(18)

Ω

where f b is the remaining part of F subjected to prescribed boundary loads. For free vibration problems, we obtain K − ω 2 M d = 0, (19) 1340005-5


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where ω is the natural frequency, and M is the  1 0  0

 T M= N mNdΩ, m = ρh  0 Ω    0

global mass matrix:  0 0 0 0 1 0 0 0  0 1 0 0  . h2 0 0 0  12   h2 0 0 0 12

(20)

For the buckling problems, we have


(K − λcr Kg )d = 0,

(21)

where λcr is the critical buckling load and Kg is the geometrical stiffness matrix:

Kg = BTg τ Bg dΩ, (22) Ω

where



hˆ σ0  0    0 τ =   0   0

0 hˆ σ0 0

0 0 hˆ σ0

0

0

0

0

0 0 0 3 h σ ˆ0 12 0

0 0 0



    .  0   h3 σ ˆ0  12

(23)

3. A Formulation of ES-FEM-T3 with Discrete Shear Technique 3.1. Brief on the DSG3 formulation The approximated displacement field uh of 3-node triangular elements as shown in Fig. 1 associated to node I through linearly shape functions NI is defined by N1 = 1 − ξ − η,

N2 = ξ,

N3 = η.

Fig. 1. 3-node triangular element. 1340005-6

(24)


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The membrane strains are computed by 0 = Bm de ,

(25)

where Bm

2 b−c 1 4 = 0 2Ae d−a

0 d−a b−c

0 0 0

0 0 0

0 0 0

c 0 −d

0 −d c

0 0 0

0 0 0

0 0 0

−b 0 a

0 a −b

0 0 0

3 0 05. 0

0 0 0

(26)


The curvatures are written as κ = Bb de ,

(27)

where 2 0 1 4 0 Bb = 2Ae 0

0 0 0

0 0 0

b−c 0 d−a

0 d−a b−c

0 0 0

0 0 0

0 0 0

c 0 −d

0 −d c

0 0 0

0 0 0

0 0 0

−b 0 a

3 0 a 5. −b

(28)

The geometrical strains are approximated by g = Bg de ,

(29)

where Bg 2

b−c 6d − a 6 6 0 6 6 0 6 1 6 6 0 = 6 0 2Ae 6 6 6 0 6 6 0 6 4 0 0

0 0 b−c d−a 0 0 0 0 0 0

0 0 0 0 b−c d−a 0 0 0 0

0 0 0 0 0 0 b−c d−a 0 0

0 0 0 0 0 0 0 0 b−c d−a

c −d 0 0 0 0 0 0 0 0

0 0 c −d 0 0 0 0 0 0

0 0 0 0 c −d 0 0 0 0

0 0 0 0 0 0 c −d 0 0

0 0 0 0 0 0 0 0 c −d

−b a 0 0 0 0 0 0 0 0

0 0 −b a 0 0 0 0 0 0

0 0 0 0 −b a 0 0 0 0

0 0 0 0 0 0 −b a 0 0

3 0 0 7 7 0 7 7 0 7 7 7 0 7 7 0 7 7 0 7 7 0 7 7 −b5 a (30)

in which a = x2 − x1 , b = y2 − y1 , c = y3 − y1 , d = x3 − x1 and Ae is the area of the triangular element. Since we used only low-order 3-node triangular elements to formulate stiffness matrices, it is simply interpolated using the linear basis functions for membrane displacements, deflections and rotations without any additional variables. Hence, the membrane, bending and geometrical strains of the standard FEM are constants and unchanged while the transverse shear strains remain linearly interpolated functions. To regularize these transverse shear strains, Bletzinger et al. [2000] have proposed 1340005-7


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the DSG3. As a result, the shear strains then become constants and shear locking problem is solved. One writes γ DSG3 = BDSG3 de , s

(31)

where BDSG3 s


2 0 1 6 = 4 2Ae 0

0

b−c

Ae

0

0

0

c

0

d−a

0

Ae

0

0

−d

ac 2 ad − 2

bc 2 bd − 2

0

0

−b

0

0

a

bd 2 ad 2

−

bc 3 2 7. 5 ac 2 (32)

−

Substituting Eqs. (26), (28), and (32) into (17), and Eq. (30) into (22), the global stiffness matrices are obtained as KDSG3 =

Ne

KeDSG3 ,

(33)

KeDSG3 , g

(34)

e=1

KDSG3 = g

Ne e=1

where the element stiffness matrix, KeDSG3 and the element geometrical stiffness of the DSG3 element are given as follows: matrix, KeDSG3 g

eDSG3 K = [BTm ABm + (BTm BBb + BTb BBm ) Ωe

+ BTb DBb + (BDSG3 )T Ds BDSG3 ]dΩ s s = BTm ABm Ae + (BTm BBb + BTb BBm )Ae + BTb DBb Ae

KeDSG3 g

+ (BDSG3 )T Ds BDSG3 Ae , s s

= BTg τ Bg dΩ = BTg τ Bg Ae .

(35) (36)

Ωe

3.2. Formulation of ES-DSG3 Similar to the FEM, the ES-FEM [Liu et al. (2008)] also uses a mesh of elements. When 3-node triangular elements are used, the shape functions used in the ES-FEM are also identical to those in the FEM, and hence the displacement field in the ES-FEM is also ensured to be continuous on the whole problem domain. However, being different from the FEM which computes the stiffness matrix K and geometrical matrix Kg based on the elements, the ES-FEM uses the strain smoothing technique [Chen et al. (2001)] to compute these matrices based on the edges. The stiffness matrix K in the ES-FEM hence are called the smoothed stiffness matrix ˜ The geometrical matrix in the ES-FEM hence are called the and symbolized K. 1340005-8


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edge-based smoothing cell

3

1 Γk

Ωk

centroid

element i

4 element j

2


Fig. 2. (Color online) Smoothing domain associated with interior edge k.

˜ g . In this process, the finite elesmoothed geometrical matrix and symbolized K ment mesh is further divided into the smoothing domains Ωk based on edges of ed elements such as Ω ≈ ∪N k=1 Ωk and Ωi ∩ Ωj = for i = j, in which Ned is the total number of edges of all elements in the entire problem domain. For triangular elements, the smoothing domain Ωk associated with the edge k is created by connecting two endpoints of the edge to centroids of adjacent elements as shown in Fig. 2. The strain smoothing formulation is defined by following equation:

˜hk = h (x)φk (x)dΩ, (37) Ωk

where φk (x) is a smoothing function that is positive and normalized to unity

φk (x) dΩ = 1. (38) Ωk

For simplicity, the smoothing function φk is chosen to be a step function 1/Ak , x ∈ Ωk φk (x) = 0, x∈ / Ωk ,

(39)

where Ak is the area of the smoothing domain Ωk . From Eq. (37), the smoothed membrane strains, curvatures, shear strains and geometrical strains over the smoothing domain Ωk are defined by

1 1 1 0 0 ˜k = ˜k = ˜k = dΩ, κ κdΩ, γ γ DSG3 dΩ, (40) Ak Ωk Ak Ωk Ak Ωk

1 ˜ gk = g (x)dΩ. (41) Ak Ωk Substituting Eqs. (25), (27), and (31) into (40), and Eq. (29) into (41), the average strains at edge k can be obtained by the following form (k) Nn

˜0k

=

I=1

(k) Nn

˜m B I (xk )dI ,

˜k = κ

(k) Nn

˜ bI (xk )dI , B

I=1

1340005-9

˜k = γ

I=1

˜ sI (xk )dI , B

(42)


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H. H. Phan-Dao et al. (k) Nn

˜gk

=

˜ g (xk )dI , B I

(43)

I=1 (k)

where Nn is the number of nodes belonging to elements directly connected to (k) (k) ˜ m (xk ), edge k (Nn = 3 for boundary edges and Nn = 4 for inner edges) and B I g b s ˜ ˜ ˜ BI (xk ), BI (xk ) and BI (xk ) are the average gradient matrices corresponding to the smoothing domain Ωk and given by (k)

(k)


Ne 1 1 m ˜ BI = Ai Bm i , Ak i=1 3 (k)

Ne 1 ˜ sI = 1 Ai Bsi , B Ak i=1 3

Ne 1 1 b ˜ Ai Bbi , BI = Ak i=1 3

(44)

(k)

Ne 1 ˜g = 1 Ai Bgi , B I Ak i=1 3

(k)

(45) (k)

where Ne is the number of elements attached to the edge k (Ne = 1 for the (k) boundary edges and Ne = 2 for inner edges). The global stiffness and geometrical matrices of the ES-DSG3 element [NguyenXuan et al. (2010)] are obtained by ˜ = K

Ned

˜ (k) , K

(46)

˜ (k) K g

(47)

k=1

˜g = K

Ned k=1

where

˜ (k) = K

˜ Tm AB ˜ s dΩ ˜ m + (B ˜ Tm BB ˜b + B ˜ Tb BB ˜ m) + B ˜ Tb DB ˜b + B ˜ Ts Ds B B

Ωk

˜ Tm AB ˜ s Ak , ˜ m Ak + (B ˜ Tm BB ˜b + B ˜ Tb BB ˜ m )Ak + B ˜ Tb DB ˜ b Ak + B ˜ Ts Ds B =B (48)

˜ Tg τ B ˜ g dΩ = B ˜ Tg τ B ˜ g Ak . B

˜ (k) K g =

(49)

Ωk

It can be observed from Eqs. (48) and (49) that the stiffness matrices are constants over smoothing domains and can be computed by straight-forward integration. Note that the rank of the ES-DSG3 element is similar to that of the DSG3 element and the stability of the ES-DSG3 element is also ensured. 4. Numerical Examples In this section, we examine the performance of the ES-DSG3 element for laminated composite plates through a number of benchmark problems. In the computation, the material properties and the thickness of all layers are assumed to be the same 1340005-10


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(a) Sinusoidally distributed load (SSL)

(b) Uniform distributed load (UDL)


Fig. 3. Simply supported cross-ply square laminated plates.

but the ply’s angle may be different among the layers. Two sets of simply supported boundary conditions considered in this paper are defined below [Reddy (2004)] with the coordinate shown in Fig. 3: SS-1: At x = 0 and x = a, v0 = w0 = βy = 0. At y = 0 and y = b, u0 = w0 = βx = 0. SS-2: At x = 0 and x = a, u0 = w0 = βy = 0. At y = 0 and y = b, v0 = w0 = βx = 0. The results of the present approach are compared with some existing analytical solutions and those of several other published methods in the literature. 4.1. Static analysis 4.1.1. Cross-ply laminated square plates subjected to sinusoidally and uniformly distributed loads First, we consider three symmetric cross-ply laminated square plates (length a and thickness h) made of material M 1 as shown in Table 1 subjected to sinusoidally (SSL) and uniformly (UDL) distributed loads q0 = 1 (see Fig. 3) with the simply supported boundary SS-1. An used mesh of 13 × 13 nodes is depicted in Fig. 4. The convergence of the normalized central deflection w∗ = 100E2 wh3 /(q0 a4 ) with various ratios a/h is shown in Table 2 and Fig. 5. Figure 6 also depicts the error Table 1. Material properties. HM graphite epoxy Moduli

M1

M2

M3

E11 (GPa) E22 (GPa) G12 (GPa) G13 (GPa) G23 (GPa) ν12 = ν23 = ν13

25.0 1.0 0.5 0.5 0.2 0.25

40.0 1.0 0.6 0.6 0.5 0.25

40.0 1.0 0.5 0.5 0.5 0.25

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Fig. 4. Geometry data for symmetric cross-ply square laminated composite plate and a typical mesh.

Table 2. Normalized central deflection w ∗ = 100E2 wh3 /(q0 a4 ) of simply supported cross-ply square laminated plates under sinusoidally and uniformly distributed loads, respectively. SSL

UDL

Methods

10

20

100

10

20

100

[0/90/0] FEM-T3 FEM-Q4 FEM-Q9 DSG3 ES-DSG3 Exact‡

0.6281 0.6458 − 0.6629 0.6728 0.6693

0.4516 0.4666 − 0.4865 0.4942 0.4921

0.3714 0.4073 − 0.4277 0.4352 0.4337

0.9639 0.9874 1.0219 1.0159 1.0287 1.0219

0.6989 0.7195 0.7573 0.7521 0.7624 0.7572

0.5744 0.6307 0.6697 0.6642 0.6743 0.6697

[0/90/90/0] FEM-T3 FEM-Q4 FEM-Q9 DSG3 ES-DSG3 Exact‡

0.6211 0.6387 − 0.6544 0.6647 0.6627

0.4503 0.4655 − 0.4846 0.4924 0.4912

0.3675 0.4073 − 0.4272 0.4347 0.4337

0.9641 0.9883 1.0250 1.0136 1.0276 1.0250

0.7085 0.7302 0.7694 0.7604 0.7716 0.7694

0.5795 0.6430 0.6829 0.6744 0.6854 0.6833

[0/90/0/90/0] FEM-T3 FEM-Q4 FEM-Q9 DSG3 ES-DSG3 Exact‡

0.5868 0.6034 − 0.6164 0.6261 0.6277

0.4406 0.4556 − 0.4701 0.4777 0.4814

0.3661 0.4069 − 0.4214 0.4288 0.4333

0.9120 0.9350 0.9727 0.9554 0.9687 0.9727

0.6966 0.7182 0.7581 0.7404 0.7515 0.7581

0.5812 0.6465 0.6868 0.6688 0.6799 0.6874

‡

The exact solutions given by Reddy [2004].

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Fig. 5. (Color online) Convergence of normalized central deflection w ∗ = 100E2 wh3 /(q0 a4 ) with two loading cases.



Fig. 6. (Color online) Simply supported cross-ply [00 /900 /900 /00 ] square plate with various mesh sizes: Percent error of normalized central deflections in two loading cases.

percentages of normalized central deflections with various span-to-thickness ratios a/h and mesh sizes. Numerical results show that the ES-DSG3 solutions converge well to the exact value [Reddy (2004)]. The present element outperforms the FEMT3 (three-node triangular element), FEM-Q4 (four-node quadrilateral element with a reduced integration for the shear terms), FEM-Q9 (nine-node quadrilateral element) and DSG3 elements for this problem. Now we consider the computational time of the present method compared with FEM-T3, FEM-Q4 and higher-order methods FEM-Q9 elements. The formulation was implemented in MATLAB (version 7.8.0.347) and the program is compiled by a personal computer with Intel Core 2 Duo CPU-1.8GHz and RAM-2GB. The computational cost is evaluated for setting up the system matrices and solving the 1340005-13


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(a) Percent errors on normalized central deflections.

(b) CPU time versus number of node perside.

Fig. 7. (Color online) Simply supported cross-ply [00 /900 /900 /00 ] square plate under uniformly distributed load with various mesh sizes (Material M 1, a/h = 100).

algebraic equations. In Fig. 7, it can be seen that the additional CPU time by the smoothing operations of the present method leads to much computational time than that of the FEM-T3 and FEM-Q4. However, the performance of the ES-DSG3 is better than the FEM-T3 and even much better than the FEM-Q4 with refined meshes. Note that the present formulation uses only a linear approximation without any additional DOF. 4.1.2. Angle-ply laminated square plates subjected to sinusoidally distributed load Next, antisymmetric 2-layer [−45◦ /45◦] and 8-layer [−45◦ /45◦ ]4 angle-ply square plates subjected to doubly sinusoidal load q = q0 sin(x/a) sin(y/a) are depicted in Fig. 3(a). The plates are made of material M 3 with length a and thickness h. Boundary conditions SS-2 [Reddy (2004)] are used. For comparison the plates are modeled with 10 × 10 nodes. Normalized central deflection w∗ ( a2 , a2 ) and the normalized stress σx∗ ( a2 , a2 , h2 ) at point (a/2, a/2, h/2) are calculated. The present results are compared with some numerical ones of other published elements such as CTMQ20 [Cen et al. (2002)], RDKQ-L20/L24 [Zhang and Kim (2006)] and MFE [Singh et al. (2000)]. It is observed from Table 3 that the results derived from the ES-DSG3 element show good agreement with the analytical solution and those of other high-order elements for all ratios a/h. This implies that the present method can perform well for antisymmetric laminated composite plates. Due to the influence of stacking sequence of lamina layers, the anisotropic characteristic of 2-layer [−45◦/45◦ ] square plate is more dominated than 8-layer plate. Hence, the normalized central deflection of 8-layer antisymmetric laminated composite plate is smaller than 2-layer one. The 1340005-14


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ES-FEM Formulation for Analysis of Laminated Composite Plates Table 3. Simply supported angle-ply [−45◦ /45◦ ] and [−45◦ /45◦ ]4 square laminated plates under sinusoidally distributed load: Comparison of normalized central deflection and normalized stresses.


[−45◦ /45◦ ] a/h

Methods

w ∗ ( a2 , a2 )

10

CTMQ20 (8 × 8) RDKQ-L20 (10 × 10) RDKQ-L24 (10 × 10) MFE (8 × 8) Q4 (10 × 10) ES-DSG3 (10 × 10) Exact (FSDT)‡

0.8218 0.8241 − 0.8257 0.7890 0.8294 0.8284

0.2543 0.2517 − − − 0.2433 0.2498

0.4157 0.4171 0.4173 0.4189 0.4069 0.4157 0.4198

0.1507 0.1512 0.1513 − − 0.1412 0.1445

20


0.6906 0.6931 0.6960 − 0.6565 0.6963 0.6981

0.2523 0.2513 0.2516 − − 0.2432 0.2498

0.2846 0.2861 0.2863 − 0.2745 0.2829 0.2896

0.1496 0.1506 0.1506 − − 0.1412 0.1445

100


0.6519 0.6533 0.6546 0.6558 0.6141 0.6533 0.6564

0.2474 0.2488 0.2500 − − 0.2431 0.2498

0.2463 0.2466 0.2467 0.2472 0.2321 0.2393 0.2479

0.1459 0.1464 0.1465 − − 0.1408 0.1445

‡The

σx∗ ( a2 , a2 ,

[−45◦ /45◦ ]4 h ) 2

w ∗ ( a2 , a2 )

σx∗ ( a2 , a2 ,

h ) 2

exact solutions given in Reddy [2004].

convergence of approximate solutions with respect to various mesh sizes is depicted in Fig. 8. The ES-DSG3 works well with this problem while the Q4 behaves too stiff. 4.2. Free vibration analysis In this subsection, three numerical examples are illustrated to determine eigenvalues of problems with various modulus ratios and span-to-thickness ratios. The mass density ρ is equal to one. 4.2.1. Effect of modulus ratios Let us consider a simply supported 4-layer cross-ply [0◦ /90◦ /90◦ /0◦ ] square laminated composite plate with various modulus ratios. In the computation, the spanto-thickness a/h is chosen to be five, the material parameters of each layer are M 2 and the modulus ratios E1 /E2 are changed from 10, 20, 30 to 40. The typical mesh of 15 × 15 nodes is used. Normalized fundamental frequencies are presented in Table 4. The obtained results are compared with exact solutions in Reddy (2004) and available solutions of other published methods such as MISQ20 [Nguyen-Van 1340005-15


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Fig. 8. (Color online) Simply supported angle-ply [−45◦ /45◦ ] and [−45◦ /45◦ ]4 square laminated plates under sinusoidally distributed load (a/h = 100): Percent errors of normalized central deflection.

composite plate: Table 4. Simply supported cross-ply [0◦ /90◦ /90◦ /0◦ ] square laminated p Convergence of normalized fundamental frequency (ω ∗ = (ωa2 /h) ρ/E2 , a/h = 5). E1 /E2 Methods

Nodes

10

20

30

40

ES-DSG3

7×7 11×11 15×15 15×15

8.6245 8.3898 8.3295 8.3094 8.2924 8.3101 8.2982

9.8376 9.6362 9.5849 9.5698 9.5613 9.5801 9.5671

10.5446 10.3730 10.3299 10.3224 10.3200 10.3490 10.3260

11.0297 10.8827 10.8465 10.8471 10.8490 10.8640 10.8540

MISQ20 MLSDQ RBF Exact‡ ‡The

exact solutions given by Reddy (2004).

et al. (2008)], MLSDQ [Liew et al. (2003)] and RBF [Ferreira et al. (2005)]. The effect of modulus ratios E1 /E2 on the accuracy of the fundamental frequencies is schematically shown in Fig. 9. It can be seen that the results of the ES-DSG3 element converge to exact solutions and are compared well other numerical results.

4.2.2. Effect of span-to-thickness ratios The simply supported 4-layer cross-ply [0◦ /90◦ /90◦ /0◦ ] square laminated composite plate with various ratios a/h is considered in this subsection. The material made of plate is M 1 as given in Table 1 and the modulus ratio E1 /E2 is taken 40. 1340005-16


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(a) Convergence of the present solution.

(b) Relative error percentages of normalized fundamental frequencies with various modulus rations.

Fig. 9. (Color online) Simply supported cross-ply [00 /900 /900 /00 ] square laminated plate.

Table 5. Simply supported cross-ply [0◦ /90◦ /90◦ /0◦ ] square laminated composite plate with ` various ´ pratios a/h: The normalized fundamental frequency (E1 /E2 = ρ/E2 ). 40, ω ∗ = ωa2 /h a/h Methods

Nodes

ES-DSG3 15×15 MISQ20 15×15 HSDT p-Ritz RBF-pseudospectral HOIL theory Local theory Global theory Global-local theory

5

10

20

25

50

100

10.8465 10.8461 10.9891 10.8550 10.8074 10.6730 10.6820 10.6876 10.7294

15.1353 15.1658 15.2689 15.1434 15.1007 15.0660 15.0690 15.0721 15.1658

17.6454 17.7192 17.6669 17.6583 17.6338 17.5350 17.6360 17.6369 17.8035

18.0550 18.1380 18.0490 18.0718 18.0490 18.0540 18.0550 18.0557 18.2404

18.6564 18.7535 18.4624 18.6734 18.6586 18.6700 18.6700 18.6702 18.9022

18.8183 18.9189 18.7561 18.8359 18.8223 18.8350 18.8350 18.8352 19.1566

The results of the ES-DSG3 element are presented in Table 5 and compared with those of the other numerical methods such as MISQ20 [Nguyen-Van et al. (2008)], HSDT [Phan and Reddy (1985)], p-Ritz [Liew (1996)], RBF-pseudospectral [Ferreira and Fasshauer (2007)], HOIL (Higher-order individual-layer theory) [Cho et al. (1991)], Local high-order theory [Wu and Chen (1994)], Global higher-order theory [Matsunaga (2000)] and Global-local higher-order theory [Zhen and Wanji (2006)]. The effect of span-to-thickness ratios on the normalized fundamental frequency is considered. The reliability of the present method compared with the other numerical models is reported in Table 5. Also, the subplot (a) in Fig. 10 gives an illustration of the normalized fundamental frequencies of the above numerical approaches with E1 /E2 = 40. In this case, the present method is relatively compared with p-Ritz 1340005-17


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(a) The normalized fundamental frequencies ω ∗

(b) Percent error of normalized fundamental frequencies relatively with p-Ritz solution.

◦ ◦ ◦ ◦ Fig. 10. (Color online) ` Cross-ply ´ p [0 /90 /90 /0 ] square laminated plate with various a/h ratios ρ/E2 ). (E1 /E2 = 40, ω ∗ = ωa2 /h

solution [Liew (1996)] in Fig. 10(b) and it is clear that the reasonable results are obtained using the ES-DSG3. Furthermore, we consider a clamped cross-ply [0◦ /90◦/0◦ ] square laminated composite plate with various span-to-thickness ratios a/h. First six natural frequencies of 3-layer [0◦ /90◦ /0◦ ] laminated plate are resulted in Table 6. For comparison, several other methods such as p-Ritz, global-local higher-order theory, MLSDQ [Lanhe et al. (2005)] and Galerkin approach [Shi et al. (2004)] are also reported in this table. It is again observed that the results of the present method are acceptable with available solutions. The first six mode shapes are plotted in Fig. 11. 4.2.3. Effect of ply’s angle A clamped circular symmetric 4-layer [θ/ − θ/ − θ/θ] laminated plate with the diameter D and the thickness h is considered for eigenvalues analysis. The spanto-thickness D/h is assumed to be 10 and the modulus ratio E1 /E2 = 40. The geometry domain of plate is discretized irregular mesh with using the same set of 212 nodes as shown in Fig. 12. The effect of various ply’s angles θ on normalized fundamental frequency is presented in Table 7. The obtained solutions are compared with those of MISQ20 and MLSDQ methods. It can be seen that the ES-DSG3 results agree well with two other published solutions. The first six mode shapes of natural frequencies are also displayed in Fig. 13. 4.3. Buckling analysis In this subsection, the performance of the ES-DSG3 is demonstrated through some numerical examples for buckling analysis of symmetric/antisymmetric laminated 1340005-18


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ES-FEM Formulation for Analysis of Laminated Composite Plates Table 6. Clamped cross-ply [0◦ /90◦ /0◦ ] square laminated composite plate: The normalized p fundamental frequency (E1 /E2 = 40, D0 = E2 h3 /(12(1 − ν12 ν21 )), ω ∗ = (ωa2 /π 2 ) ρh/D0 ). Modes a/h

1

2

3

4

ES-DSG3 p-Ritz Global-local theory ES-DSG3 p-Ritz Global-local theory Galerkin method MLSDQ

4.4722 4.447 4.540 7.4779 7.411 7.484 7.451 7.432

6.7465 6.642 6.524 10.6486 10.393 10.207 10.451 10.399

7.7576 7.700 8.178 14.0799 13.913 14.340 13.993 13.958

9.3663 9.185 9.473 16.0029 15.429 14.863 15.534 15.467

10.0061 9.738 9.492 16.3540 15.806 16.070 15.896 15.838

20

ES-DSG3 p-Ritz Global-local theory Galerkin method

11.0964 10.953 11.003 11.015

14.5191 14.028 14.064 14.152

21.6658 20.388 20.321 20.691

23.6490 23.196 23.498 23.323

26.0711 24.978 25.350 25.142

100

ES-DSG3 p-Ritz Global-local theory Galerkin method MLSDQ

14.7278 14.666 14.601 14.583 14.674

18.3390 17.614 17.812 17.762 17.668

26.7313 24.511 25.236 25.004 24.594

39.2865 35.532 37.168 36.644 35.897

40.8216 39.157 38.528 38.073 39.625

5


10

Methods

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Mode 6

5

Fig. 11. (Color online) Clamped cross-ply [0◦ /90◦ /0◦ ] square laminated plate (E1 /E2 = 40, a/h = 10): first six mode shapes. 1340005-19


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Fig. 12. Geometry data of circular laminated plate.

Table 7. Clamped circular 4-layers [θ/−θ/−θ/θ] p laminated plate: The normalized fundamental frequencies of first five modes (ω ∗ = (ωD 2 /4h) ρ/E2 , E1 /E2 = 40, D/h = 10). Modes θ

Methods

0◦

1

2

3

4

5

ES-DSG3 MISQ20/24 MLSDQ

22.254 22.123 22.211

31.032 29.768 29.651

43.516 41.726 41.101

44.338 42.805 42.635

53.436 50.756 50.309

15◦


22.563 22.698 22.774

32.519 31.568 31.455

43.630 43.635 43.350

46.451 44.318 43.469

55.197 53.468 52.872

30◦


23.941 24.046 24.071

37.056 36.399 36.153

44.134 44.189 43.968

53.633 52.028 51.074

58.715 57.478 56.315

45◦


24.637 24.766 24.752

39.798 39.441 39.181

43.798 43.817 43.607

59.057 57.907 56.759

59.195 57.945 56.967

plates. The effect of modulus ratios E1 /E2 and span-to-thickness ratios a/h are considered. 4.3.1. Effect of modulus ratios Let us consider a simply supported cross-ply [0◦ /90◦ /90◦/0◦ ] square laminated plate. The geometry and uniaxial in-plane compression are illustrated in Fig. 14 and the span-to-thickness a/h is taken to be equal 10. The geometry domain is discretized using 7 × 7, 13 × 13, 17 × 17 nodes. The present results are given in Table 8. The effect of various modulus ratios E1 /E2 on normalized critical buckling 1340005-20


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ES-FEM Formulation for Analysis of Laminated Composite Plates Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Mode 6

Fig. 13. (Color online) Clamped circular angle-ply [45◦ / − 45◦ / − 45◦ /45◦ ] laminated plate (E1 /E2 = 40, a/h = 10): the first six mode shapes.

loads is schematically shown in Fig. 15 and the relative error percentages compared with 3D elasticity solutions [Noor (1975)] are given in a parentheses. It is seen that the results of the ES-DSG3 element agree very well with those of other published methods such as RPIM [Liu et al. (2007)], HSDT [Phan and Reddy (1985), Khdeir and Librescu (1988)]. 4.3.2. Cross-ply and angle-ply subjected to uniaxial in-plane compression This problem considers the influence of stacking sequence of symmetric/ antisymmetric square laminated plates on the normalized critical buckling load with several thickness ratios a/h. The problem model is described in Fig. 14. The material

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Fig. 14. Geometry data of square laminated plate under uniaxial in-plane compression.

Table 8. Simply supported cross-ply [0◦ /90◦ /90◦ /0◦ ] square laminated composite plate: Convergence of normalized critical buckling loads with various modulus ratios E1 /E2 (λ∗ = Nx a2 /(E2 h3 ), a/h = 10). E1 /E2 Methods

Nodes

3

10

20

30

40

ES-DSG3

7×7 13×13 17×17

5.814 5.434 5.382 (1.66%)

10.425 9.968 9.903 (1.44%)

15.546 15.242 15.195 (1.17%)

19.515 19.499 19.493 (0.98%)

22.714 23.035 23.078 (0.86%)

5.401 (2.02%)

9.985 (2.28%)

15.374 (2.36%)

19.537 (1.21%)

23.154 (1.19%)

5.114 (−3.40%)

9.774 (0.12%)

15.298 (1.86%)

19.957 (3.38%)

23.340 (2.01%)

5.442 (2.80%)

10.026 (2.70%)

15.418 (2.66%)

19.813 (2.64%)

23.489 (2.66%)

5.294

9.762

15.019

19.304

22.881

RPIM [Liu et al. (2007)] HSDT [Phan and Reddy (1985)] HSDT [Khdeir and Librescu (1988)] 3D elasticity [Noor (1975)]

parameters M 1 are employed. The problem domain is discretized into 17 × 17 nodes and simply supported boundary boundaries SS1 for cross-ply plate and SS2 for angle-ply plate [see, e.g., Reddy (2004))] are exploited. The obtained results of the ES-DSG3 element along with those of several methods such as FSDT [Chakrabarti and Sheikh (2003); Phan and Reddy (1985)] and HSDT [Phan and Reddy (1985)] are reported in Tables 9 and 10. Because an exact solution is unknown, the results of the present element are relatively compared with above numerical ones. The difference between ES-DSG3 and FSDT [Chakrabarti and Sheikh (2003)] is about 1% to 2% with cross-ply [0◦ /90◦ ] and [0◦ /90◦ /90◦/0◦ ] cases, while it is about 1% to 4% with angle-ply [45◦ / − 45◦ ] and [45◦ / − 45◦ /45◦ / − 45◦ ] cases. These differences may be acceptable because our method uses only linear approximations. 1340005-22


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Fig. 15. (Color online) Simply supported cross-ply [0◦ /90◦ /90◦ /0◦ ] square laminated plate with various modulus ratios: Percent error of normalized critical buckling load (λ∗ = Nx a2 / (E2 h3 ), a/h = 10).

Table 9. Simply supported cross-ply [0◦ /90◦ ] and [0◦ /90◦ /90◦ /0◦ ] square laminated composite plate: The normalized critical buckling loads with various ratios a/h (E1 /E2 = 40, λ∗ = Nx a2 /(E2 h3 )). a/h Methods

10

20

50

100

[0◦ /90◦ ] ES-DSG3 FSDT [Chakrabarti and Sheikh (2003)] FSDT [Phan and Reddy (1985)] HSDT [Phan and Reddy (1985)]

11.191 11.349 11.353 11.563

12.552 12.510 12.515 12.577

13.005 12.879 12.884 12.895

13.075 12.934 12.939 12.942

[0◦ /90◦ /90◦ /0◦ ] ES-DSG3 FSDT [Chakrabarti and Sheikh (2003)] FSDT [Phan and Reddy (1985)] HSDT [Phan and Reddy (1985)]

23.078 23.409 23.471 23.349

31.394 31.625 31.707 31.637

35.107 35.254 35.356 35.419

35.721 35.851 35.955 35.971

4.3.3. Effect of modulus ratios of cross-ply square plate under bi-axial in-plane compression The last example is a simply supported cross-ply [0◦ /90◦ /0◦ ] square laminated plate as shown in Fig. 16. The material properties M 1 is used and the span-to-thickness ratio a/h is assumed to be 10. The obtained results of the ES-DSG3 with 17 × 17 1340005-23


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H. H. Phan-Dao et al. Table 10. Simply supported angle-ply [45◦ / − 45◦ ] and [45◦ / − 45◦ /45◦ / − 45◦ ] square laminated composite plate: The normalized critical buckling load with various ratios a/h, (E1 /E2 = 25, λ∗ = Nx a2 /(E2 h3 )). a/h


Methods

10

20

50

100

[45◦ / − 45◦ ] ES-DSG3 FSDT [Chakrabarti and Sheikh (2003)] HSDT [Phan and Reddy (1985)]

12.035 12.600 12.622

14.484 14.629 14.644

15.376 15.329 15.336

15.514 15.435 15.441

[45◦ / − 45◦ /45◦ / − 45◦ ] ES-DSG3 FSDT [Chakrabarti and Sheikh (2003)] HSDT [Phan and Reddy (1985)]

19.453 19.593 21.962

30.792 30.949 31.032

34.820 35.084 35.120

35.486 35.769 35.795

Fig. 16. Geometry data of square laminated plate under biaxial in-plane compression.

Table 11. Simply supported cross-ply [0◦ /90◦ /0◦ ] square laminated composite plate: The normalized critical bi-axial buckling load with various modulus ratios E1 /E2 (λ∗ = Nx a2 /(E2 h3 )). E1 /E2 Methods ES-DSG3 FSDT [Fares and Zenkour (1999)] HSDT [Khdeir and Librescu (1988)]

10

20

30

40

4.965 4.963 4.963

7.445 7.588 5.516

9.154 8.575 9.056

10.350 10.202 10.259

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nodes are presented in Table 11. When the modulus ratio E1 /E2 becomes larger, the normalized critical bi-axial buckling loads are also increased. It is finally confirmed from Table 11 that the ES-DSG3 performs well for this problem.


5. Conclusions An improved finite element model that uses the ES-DSG3 has been presented for static, free vibration and buckling analyses of laminated composite plates. The present formulation is simply obtained from linear approximations using 3-node triangular elements with degrees of freedom at vertex nodes without any additional degrees of freedom and hence no more requirement of high computational cost. Through various numerical examples, the present results are in good agreement with the analytical solution and are compared well with results of several other published elements in the literature. Due to its simple and effective implementation, the present method is thus promising to provide a reliable numerical tool for analysis of laminated composite plates. Acknowledgments This work was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED), the Basic Research Project of Vietnam National University Hochiminh City; (Grant No. B2011-18-09). These supports are gratefully acknowledged. References Bletzinger, K., Bischoff, M. and Ramm, E. [2000] “A unified approach for shear-locking-free triangular and rectangular shell finite elements,” Comput. Struct. 75(3), 321–334. Chen, J. S., Wu, C. T., Yoon, S. and You, Y. [2001] “A stabilized conforming nodal integration for Galerkin mesh-free methods,” Int. J. Numer. Method Eng. 50, 435–466. Cen, S., Long, Y. Q. and Yao, Z. H. [2002] “A new hybrid-enhanced displacement-based element for the analysis of laminated composite plates,” Comput. Struct. 80, 819–833. Cen, S., Long, Y.-Q., Yao, Z.-H. and Chiew, S.-P. [2006] “Application of the quadrilateral area co-ordinate method: A new element for Mindlin-Reissner plate,” Int. J. Numer. Method Eng. 66, 1–45. Chakrabarti, A. and Sheikh, A. H. [2003] “Buckling of laminated composite plates by a new element based on higher order shear deformation theory,” Mech. Compos. Mater. Struct. 10(4), 303–317. Cho, K. N., Bert, C. W. and Striz, A. G. [1991] “Free vibrations of laminated rectangular plates analyzed by higher order individual-layer theory,” J. Sound Vib. 145(3), 429–442. Cui, X. Y., Liu, G. R., Li, G., Zhang, G. Y. and Zheng, G. [2010] “Analysis of plates and shells using an edge-based smoothed finite element method,” Comput. Mech 45, 141–156. Fares, M. E. and Zenkour, A. M. [1999] “Buckling and free vibration of non-homogeneous composite cross-ply laminated plates with various plate theories,” Compos. Struct. 44, 279–287. 1340005-25


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Ferreira, A. J. M. and Fasshauer, G. E. [2007] “Analysis of natural frequencies of composite plates by an RBF-pseudospectral method,” Compos. Struct. 79(2), 202–210. Ferreira, A. J. M., Roque, C. M. C. and Jorge, R. M. N. [2005] “Free vibration analysis of symmetric laminated composite plates by FSDT and radial basis functions,” Comput. Method Appl. Mech. Eng. 194(39–41), 4265–4278. Ferreira, A. J. M., Fasshauer, G. E., Batra, R. C. and Rodrigues, J. D. [2008] “Static deformations and vibration analysis of composite and sandwich plates using a layerwise theory and RBF-PS discretizations with optimal shape parameter,” Compos. Struct. 86, 328–343. Khdeir, A. A. and Librescu, L. [1988] “Analysis of symmetric cross-ply elastic plates using a higher-order theory. Part II: buckling and free vibration,” Compos. Struct. 9, 259–277. Lanhe, W., Hua, L. and Daobin, W. [2005] “Vibration analysis of generally laminated composite plates by the moving least squares differential quadrature method,” Compos. Struct. 68(3), 319–330. Liew, K. M. [1996] “Solving the vibration of thick symmetric laminates by Reissner/Mindlin plate theory and the p-Ritz method,” J. Sound Vib. 198(3), 343–360. Liew, K. M., Huang, Y. Q. and Reddy, J. N. [2003] “Vibration analysis of symmetrically laminated plates based on FSDT using the moving least squares differential quadrature method,” Comput. Method. Appl. Mech. Eng. 192(19), 2203–2222. Liu, G. R. and Nguyen-Thoi, T. [2010] Smoothed Finite Element Methods (CRC Press, Taylor and Francis Group, NewYork). Liu, G. R., Dai, K. Y. and Nguyen-Thoi, T. [2007] “A smoothed finite element for mechanics problems,” Comput. Mech. 39, 859–877. Liu, G. R., Nguyen-Thoi, T. and Lam, K. Y. [2008] “An edge-based smoothed finite element method (ES-FEM) for static and dynamic problems of solid mechanics,” J. Sound Vib. 320, 1100–1130. Liu, G. R., Zhao, X., Dai, K. Y., Zhong, Z. H., Li, G. Y. and Han, X. [2008] “Static and free vibration analysis of laminated composite plates using the conforming radial point interpolation method,” Compos. Sci. Technol. 68(2), 354–366. Liu, G. R., Nguyen-Thoi, T., Nguyen-Xuan, H. and Lam, K. Y. [2009] “A node-based smoothed finite element method (NS-FEM) for upper bound solutions to solid mechanics problems,” Comput. Struct. 87, 14–26. Liu, L., Chua, L. P. and Ghista, D. N. [2007] “Mesh-free radial basis function method for static, free vibration and buckling analysis of shear deformable composite laminates,” Compos. Struct. 78, 58–69. Matsunaga, H. [2000] “Vibration and stability of cross-ply laminated composite plates according to a global higher-order plate theory,” Compos. Struct. 48(4), 231–244. Nguyen-Thoi, T., Liu, G. R., Lam, K. Y. and Zhang, G. Y. [2009] “A face-based smoothed finite element method (FS-FEM) for 3D linear and nonlinear solid mechanics problems using 4-node tetrahedral elements,” Int. J. Numer. Method Eng. 78, 324–353. Nguyen-Van, H., Mai-Duy, N. and Tran-Cong, T. [2008] “Free vibration analysis of laminated plate/shell structures based on FSDT with a stabilized nodal-integrated quadrilateral element,” J. Sound Vib. 313(1–2), 205–223. Nguyen-Xuan, H., Liu, G. R., Thai-Hoang, C. and Nguyen-Thoi, T. [2010] “An edge-based smoothed finite element method (ES-FEM) with stabilized discrete shear gap technique for analysis of Reissner-Mindlin plates,” Comput. Method Appl. Mech. Eng. 199, 471–489. Noor, A. K. [1975] “Stability of multilayered composite plates,” Fibre Sci. Technol. 8(2), 81–89.

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Phan, N. D. and Reddy, J. N. [1985] “Analysis of laminated composite plates using a higher-order shear deformation theory,” Int. J. Numer. Method Eng. 21, 2201–2219. Reddy, J. N. [2004] Mechanics of Laminated Composite Plates and Shells Theory and Analysis, 2nd edition (CRC Press, New York). Roque, C. M. C., Ferreira, A. J. M. and Jorge, R. M. N. [2005] “Modelling of composite and sandwich plates by a trigonometric layerwise deformation theory and radial basis functions,” Compos. Part B: Eng. 36, 559–572. Shi, J. W., Nakatani, A. and Kitagawa, H. [2004] “Vibration analysis of fully clamped arbitrarily laminated plate,” Compos. Struct. 63(1), 115–122. Singh, G., Raveendranath, P. and Rao, G. V. [2000] “An accurate four-node shear flexible composite plate element,” Int. J. Numer. Method Eng. 47, 1605–1620. Wu, C. and Chen, W. [1994] “Vibration and stability of laminated plates based on a local high order plate theory,” J. Sound Vib. 177(4), 503–520. Zhang, Y. X. and Kim, K. S. [2006] “Geometrically nonlinear analysis of laminated composite plates by two new displacement-based quadrilateral plate elements,” Compos. Struct. 72, 301–310. Zhen, W. and Wanji, C. [2006] “Free vibration of laminated composite and sandwich plates using globallocal higher-order theory,” J. Sound Vib. 298(1–2), 333–349.

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an edge-based smoothed finite element method for

an edge-based smoothed finite element method for

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