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

Engineering Analysis with Boundary Elements 47 (2014) 68–81

Contents lists available at ScienceDirect

Engineering Analysis with Boundary Elements
journal homepage: www.elsevier.com/locate/enganabound

Isogeometric analysis of laminated composite plates based
on a four-variable refined plate theory
Loc V. Tran a, Chien H. Thai b, Hien T. Le c, Buntara S. Gan d, Jaehong Lee a,
H. Nguyen-Xuan e,n
a

Department of Architectural Engineering, Sejong University, 98 Kunja Dong, Kwangjin Ku, Seoul 143-747, South Korea
Division of Computational Mechanics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
c
Department of Naval Architecture and Marine Engineering, Ho Chi Minh City University of Technology, VNU-HCMC, 268 Ly Thuong Kiet Street,
Ho Chi Minh, Vietnam
d
Department of Architecture, College of Engineering, Nihon University, Koriyama City, Fukushima Prefecture, Japan
e
Department of Computational Engineering, Vietnamese-German University, Binh Duong New City, Vietnam
b

art ic l e i nf o

a b s t r a c t

Article history:
Received 15 October 2013
Received in revised form

19 May 2014
Accepted 30 May 2014

In this paper, a simple and effective formulation based on isogeometric approach (IGA) and a four
variable refined plate theory (RPT) is proposed to investigate the behavior of laminated composite plates.
RPT model satisfies the traction-free boundary conditions at plate surfaces and describes the non-linear
distribution of shear stresses without requiring shear correction factor (SCF). IGA utilizes basis functions,
namely B-splines or non-uniform rational B-splines (NURBS), which reveals easily the smoothness
of any arbitrary order. It hence handles easily the C1 requirement of the RPT model. Approximating the
displacement field with four degrees of freedom per each node, the present method retains the
computational efficiency while ensuring the reasonable accuracy in solution.
& 2014 Elsevier Ltd. All rights reserved.

Keywords:
Plate
Composite
Isogeometric analysis
Refined plate theory
Meshfree method

1. Introduction
Laminated composite plates are being increasingly used in
various fields of engineering such as aircrafts, aerospace, vehicles,
submarine, ships, buildings, etc., because they possess many
favorable mechanical properties such as high stiffness to weight
and low density. Therefore a lot of research about their behaviors
such as deformable characteristic, stress distribution, natural frequency and critical buckling load under various conditions haas
never been stopped. Pagano [1] initially investigated the analytical
three-dimensional (3D) elasticity method to predict the exact
solution of simple static problems. Noor et al. [2,3] further

developed 3D elasticity solution formulas for stress analysis of
composite structures. It is well known that such an exact 3D
approach is the most potential tool to obtain the true solution of
plates. However, it is not easy to solve practical problems with
complex (or even slightly complicated) geometries and boundary
conditions. In addition, each layer in the 3D elasticity theory is
modeled as one 3D solid and hence the computational cost of

n

Corresponding author.
E-mail address: (H. Nguyen-Xuan).

/>0955-7997/& 2014 Elsevier Ltd. All rights reserved.

laminated composite plate analyses is increased significantly.
Hence, many equivalent single layer (ESL) plate theories with
suitable assumptions [4] have been then proposed to transform
the 3D problem to a 2D one. Among the ESL plate theories, the
classical laminate plate theory (CLPT) based on the Love–Kirchoff
assumptions was first proposed. Due to ignoring the transverse
shear deformation, CLPT merely provides acceptable results for the
thin plate problems. The first order shear deformation theory
(FSDT) based on Reissner [5] and Mindlin [6], which takes into
account the shear effect, was therefore developed. In FSDT model,
with the linear in-plane displacement assumption through plate
thickness, the obtained shear strain/stress distributes inaccurately
and does not satisfy the traction free boundary conditions at the
plate surfaces. The shear correction factors (SCF) are therefore
required to rectify the unrealistic shear strain energy part. The

values of SCF are quite dispersed through many problems and may
be difficult to determine [7]. To bypass the limitations of the FSDT,
many kind of higher-order shear deformable theories (HSDT),
which include higher-order terms in the displacement approximation, have then been devised such as third-order shear deformation theory (TSDT) [8–10], trigonometric shear deformation theory
[11,12], exponential shear deformation theory (ESDT) [13–15],
refined plate theory (RPT) and so on. The RPT model was pointed


L.V. Tran et al. / Engineering Analysis with Boundary Elements 47 (2014) 68–81

out by Senthilnathan et al. [16] with four unknown variables
which is one variable lower than the TSDT model. Shimpi et al.
[17,18] proposed RPT with just only two unknown variables using
different distributed functions for the isotropic and orthotropic
plates. Recently, this model is deeply researched by Thai-Huu et al.
[19,20]. It is worth mentioning that the HSDT models provide
better results and yield more accurate and stable solutions (e.g.
inter-laminar stresses and displacements) [21,22] than the FSDT
ones without requiring the SCF. However, the HSDT requires the
C1-continuity of generalized displacement field leading to the
second-order derivative of the stiffness formulation. The enforcement of even C1 continuity across inter-element boundaries in
standard finite element method is not a trivial task. In the efforts
to address this difficulty, several C0 continuous elements [23–26]
were then proposed or Hermite interpolation function with the
C1-continuity was taken into account in the approximation of
transverse displacement [4]. Such elements may produce extra
unknown variables leading to an increase in the computational cost. In this paper, we show that C1-continuous elements will be naturally gained by using B-Spline or non-uniform
rational B-Spline (NURBS) shape functions without any additional
variables.
The NURBS basis functions are commonly used in the Computer Aided Design (CAD) software to describe the geometry domain

[27]. They are flexible to make refinement, de-refinement, and
degree elevation and gain easily the smoothness of arbitrary
continuous order. Also, NURBS can be used to approximate meshfree shape functions with a desired order of consistency [28] or to
merge into boundary element method to obtain the geometry and
traction fields around the boundary [29]. Another way, by coupling
geometry and approximations via NURBS, Hughes and co-workers
have introduced a new method so-called Isogeometric Analysis
(IGA) [31]. The core idea of IGA is to use same NURBS basis
functions for both describing the exact geometry and constructing
the finite element formulation [30]. The IGA has been well known
and widely applied to various practical problems [32–39], etc.
In this paper, a formulation based on the RPT model and
the isogeometric approach for static, free vibration and buckling
analysis of laminated composite plates is investigated. Some
higher-order distributed functions [8,13,14,17] are utilized to
describe the higher-order term in the displacement field. Several
numerical examples are given to show the performance of the
proposed method in comparison to others in the literature.
The paper is outlined as follows. Section 2 introduces the RPT
for composite plates. In Section 3, the formulation of plate theory
based on IGA is described. The numerical results and discussions
are provided in Section 4. Finally, this article is closed with some
concluding remarks.

et al. [16] proposed the refined plate theory model with one
reduced variable
w0 ¼ wb þws ;

uðx; y; zÞ ¼ u0 þ zβx þ gðzÞðβx þ w;x Þ
vðx; y; zÞ ¼ v0 þ zβy þ gðzÞðβy þ w;y Þ;

vðx; y; zÞ ¼ v0 À zwb;y þ gðzÞws;y
wðx; yÞ ¼ wb þ ws

wðx; yÞ ¼ w0
2

2

Á
r z r h2

ð1Þ

where gðzÞ ¼ À ð4z3 =3h Þ and the variables u0 ¼ fu0 v0 gT , w0 and
β ¼ fβx βy gT are the membrane displacements, the transverse displacement and the rotations in the y–z, x–z planes, respectively. By
making additional assumptions given in Eq. (2), Senthilnathan

ð3Þ

The relationships between strains and displacements are
described by
ε ¼ ½εxx εyy γ xy ŠT ¼ ε0 þ zκb þgðzÞκs
0

γ ¼ ½γ xz γ yz ŠT ¼ f ðzÞεs
where
2
6
ε0 ¼ 4

in which

ð4Þ
0

f ðzÞ ¼ g 0 ðzÞ þ 1

ð5Þ

2
3
2
3
"
#
wb;xx
ws;xx
ws;x
7
6 w
7
6 w
7
5; κb ¼ À 4 b;yy 5; κs ¼ 4 s;yy 5; εs ¼
ws;y
2wb;xy
u0;y þ v0;x
2ws;xy

3

u0;x
v0;y

ð6Þ

From Eq. (5), an additional condition is needed to satisfy
traction-free boundary condition at the top and bottom surfaces
0
of plate. It means that f ðzÞ ¼ 0 at z ¼ 7h=2. Based on this
condition, various distributed functions f ðzÞ in forms: third-order
polynomials by Reddy [8] and Shimpi [17], exponential function by
Karama [13], sinusoidal function by Arya [14] and are illustrated
in Table 1.
2.2. Weak form equations for plate problems
A weak form of the static model for the plates under transverse
loading f0 can be briefly expressed as
Z
Z
Z
δεT Db εdΩ þ δγT Ds γdΩ ¼ δwf 0 dΩ
ð7Þ
Ω

Ω

where
2

A
6
D ¼4B
E
b

B

E

Ω

3

D

7
F5

F

H

ð8Þ

and the material matrices are given as
Aij ; Bij ; Dij ; Eij ; F ij ; H ij
Z h=2
¼
ð1; z; z2 ; gðzÞ; zgðzÞ; g 2 ðzÞÞQ ij dz

Z

ÀÀh

ð2Þ

uðx; y; zÞ ¼ u0 À zwb;x þ gðzÞws;x

Dsij ¼

Regarding the effect of shear deformation, the higher-order
terms are incorporated into the displacement field. A simple and
famous theory for the bending plate is stated as [4]

β ¼ À ∇wb

where wb and ws are defined as the bending and shear components of deflection, respectively. Eq. (1) is taken in the simpler
form with four unknown variables

2. The refined plate theory
2.1. Displacement field

69

ði; j ¼ 1; 2; 6Þ

À h=2
h=2

À h=2

0

½f ðzފ2 Q ij dz

ði; j ¼ 4; 5Þ

ð9Þ

in which Q ij are transformed material constants of the kth lamina
(see [4] for more detail).
Table 1
The various forms of shape function.
0

Model

f ðzÞ

gðzÞ

f ðzÞ

Reddy [8]

2
À 43z3 =h
2
1
5 3

z
À
z
=h
4
3
À 2ðz=hÞ2

1À 4z2 =h

Karama [13]

2
z À 43z3 =h
2
5
5 3
z
À
z
=h
4
3
À 2ðz=hÞ2

Arya [14]

sin

Shimpi [17]

ze

Àπ Á
z
h

ze

sin

Àz

Àπ Á
z Àz
h

2

2
5
2
4ð1À 4z =h Þ



2
1 À 42 z2 e À 2ðz=hÞ
h
Àπ Á

π
cos hz
h


70

L.V. Tran et al. / Engineering Analysis with Boundary Elements 47 (2014) 68–81

The B-spline basis functions N i;p ðξÞ are defined by the following
recursion formula:

For the free vibration analysis, it can be derived from the
following dynamic equation:
Z
Z
Z
T
€~
δεT Db εdΩþ δγT Ds γdΩ ¼ δu~ mudΩ
ð10Þ
Ω

Ω

ξ

(

where m - the mass matrix is calculated according to the

consistent form
2
3
2
3
I0 0 0
I1 I2 I4
6
7
6
7
m ¼ 4 0 I0 0 5 where I0 ¼ 4 I 2 I 3 I 5 5
ð11Þ
I4 I5 I6
0 0 I0
Z
ðI 1 ; I 2 ; I 3 ; I 4 ; I 5 ; I 6 Þ ¼

h=2
À h=2

ρðzÞð1; z; z2 ; gðzÞ; zgðzÞ; g 2 ðzÞÞdz

as p ¼ 0; N i;0 ðξÞ ¼

8
>
<

9

u0 >
=
u1 ¼ À wb;x ;
>
>
: w
;
s;x

u2 ¼

8
>
<

v0

9
>
=

À wb;y ;
>
>
: w
;
s;y

ξi r ξ oξi þ 1

)

otherwise

ð15Þ

ð16Þ

Fig. 1 illustrates the set of one-dimensional and twodimensional B-spline basis functions according to open uniform
knot vector Ξ ¼ f0; 0; 0; 0; 0:5; 1; 1; 1; 1g.
To model exactly curved geometries (e.g. circles, cylinders,
spheres, etc.), each control point has additional value called
an individual weight ζ A [30]. We denote Non-uniform Rational
B-splines (NURBS) functions which are expressed as

8 9
>
<w>
=
u3 ¼ 0
>
: >
;
0

and a weak form of the plate under the in-plane forces can be
formed as
Z
Z
Z

δεT Db εdΩþ δγT Ds γdΩ þ ∇T δwN0 ∇wdΩ ¼ 0
ð14Þ
Ω

0

if

NA ðξ; ηÞ ¼ N i;p ðξÞM j;q ðηÞ

ð12Þ

ð13Þ

Ω

1

Using the tensor product of basis functions in two parametric
dimensions ξ and η with two knot vectors Ξ ¼ fξ1 ; ξ2 ; :::; ξn þ p þ 1 g
and Η ¼ fη1 ; η2 ; :::; ηm þ q þ 1 g, the two-dimensional B-spline basis
functions are obtained

and
8 9
>
< u1 >
=
~
u ¼ u2 ;

>
:u >
;
3

Àξ

N i;p ðξÞ ¼ ξi ξþÀp Àξi ξi N i;p À 1 ðξÞ þ ξi þi pþþp 1þÀ1 ξi þ 1 Ni þ 1;p À 1 ðξÞ

Ω

N ζ
RA ðξ; ηÞ ¼ mÂn A A
∑ N A ðξ; ηÞζ A

ð17Þ

A

Ω

where ∇T ¼ ½∂=∂x ∂=∂yŠT is the gradient operator and
2
3
N 0x N 0xy
4
5
N0 ¼
N 0xy N 0y

The NURBS function becomes the B-spline function when the
individual weight of control point is constant.

is a matrix related to the pre-buckling loads.

3.2. A novel RPT formulation based on NURBS approximation
Using the NURBS basis functions, the displacement field u of
the plate is approximated as

3. The composite plate formulation based on NURBS
basis functions

mÂn

uh ðξ; ηÞ ¼ ∑ RA ðξ; ηÞqA

where qA ¼ ½u0A v0A wbA wsA ŠT is the vector of nodal degrees of
freedom associated with the control point A.
Substituting Eq. (18) into Eq. (6), the in-plane and shear strains
become

A knot vector Ξ ¼ fξ1 ; ξ2 ; :::; ξn þ p þ 1 g is defined as a sequence of
knot value ξi A R, i ¼ 1; :::n þ p. An open knot, i.e, the first and the
last knots are repeated pþ 1 times, is used. A B-spline basis
function is C1 continuous inside a knot span and Cp À 1 continuous
at a single knot. Hence, as p Z 2 the present approach always
satisfies C1-requirement in based-RPT formulations.

mÂn

b1 T
b2 T
s T T
T
½εT0 κTb κTs εTs ŠT ¼ ∑ ½ðBm
A Þ ðBA Þ ðBA Þ ðBA Þ Š qA
A¼1

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

0

ð18Þ

A

3.1. A brief of NURBS functions

1/2

1

Fig. 1. 1D and 2D B-spline basis functions.

ð19Þ


L.V. Tran et al. / Engineering Analysis with Boundary Elements 47 (2014) 68–81

in which
2
3
2
0
0 0
RA;x
0 0 RA;xx
6 0
7 b1
60 0 R
m
0
0
R
A;y
A;yy
BA ¼ 4
5; BA ¼ À 4
RA;y RA;x 0 0
0 0 2RA;xy

2
3
"
#
0 0 0 RA;xx
0 0 0 RA;x
60 0 0 R
7 s
b2
A;yy 5; BA ¼
BA ¼ 4
0 0 0 RA;y
0 0 0 2RA;xy

Simply supported cross-ply:
0

3

07
5;

ð20Þ

ð21Þ

ðK À ω2 MÞq ¼ 0

ð22Þ

ðK À λcr Kg Þq ¼ 0

ð23Þ

where the global stiffness matrix K is given by
8 m 9T 2
38 m 9
Z >
Z
<B >
= A B E >
<B >
=
6
7
b1
b1
K¼
dΩ þ BsT Ds Bs dΩ
B
4B D F 5 B
>
Ω>
Ω
: b2 >
;
: b2 >
;
E F H
B

B

ð24Þ

ð25Þ

Ω

RA

where
8
9
2
3
RA 0
0
0
>
< R1 >
=
6
0 7
R~ ¼ R 2 ; R 1 ¼ 4 0 0 À RA;x
5;
>
:R >
;
0
0

0
R
A;x
3
2
3
2
0 RA
0
0
0 0 RA
6
6
0 7
R 2 ¼ 4 0 0 À RA;y
5; R 3 ¼ 4 0 0 0
0 0
0
RA;y
0 0 0
and the geometric stiffness matrix reads
Z
Kg ¼ ðBg ÞT N0 Bg dΩ

0

RA;x

RA;x

0

0

RA;y

RA;y

u0 ¼ w b ¼ w s ¼ 0

at left and right edges

v0 ¼ wb ¼ ws ¼ 0

at lower and upper edges

ð32Þ

Clamped:
u0 ¼ v0 ¼ wb ¼ ws ¼ wb;n ¼ ws;n ¼ 0

ð33Þ

4. Results and discussions
In this section, we show the performance of the present
method – RPT–IGA with various distributed functions as given in
Table 1 in analyzing the laminated composite plates. We illustrate
the present method using the cubic basis functions. The following
material properties are used for numerical tests:
Material I:

Material II:

Material III:
Face sheets
ð27Þ

E1 ¼ 131 GPa; E2 ¼ 1:5 GPa;
G12 ¼ 6:895 GPa; G13 ¼ 6:205 GPa;
ν12 ¼ 0:22; ρ ¼ 1627 kg=m3
Core property (Isotropic)

RA

E1 ¼ E2 ¼ 6:89 MPa;

3

7
0 5
0

G12 ¼ G13 ¼ 3:45 MPa;
ð28Þ

ν12 ¼ 0; ρ ¼ 97 kg=m3

ð29Þ

The ratio of the core thickness hc to the face sheet thickness hf

is equal to 4.
For convenience, the following normalized transverse displacement, in-plane stresses, shear stresses natural frequency and
bucking load are expressed as

ð30Þ

w¼

Ω

0

ð31Þ

ð26Þ

Ω

BgA ¼

at lower and upper edges

E1 =E2 ¼ varied; G12 ¼ G13 ¼ 0:6E2 ; G23 ¼ 0:5E2 ; ν12 ¼ 0:25; ρ ¼ 1:

Ã

the global mass matrix M is expressed as
Z
T
~

M ¼ R~ mRdΩ

where
"

u0 ¼ wb ¼ ws ¼ 0

E1 ¼ 25E2 ; G12 ¼ G13 ¼ 0:5E2 ; G23 ¼ 0:2E2 ; ν12 ¼ 0:25; ρ ¼ 1:

and the load vector is computed by
Z
F ¼ q0 RdΩ

RA

at left and right edges

Simply supported angle-ply:

Kq ¼ F

0

v0 ¼ w b ¼ w s ¼ 0

0

Substituting Eq. (19) into Eqs. (7), (10) and (14), the formulations of static, free vibration and buckling problem are formulated
by the following form:

where
Â
R¼ 0

71

#

in which ω; λcr A R þ are the natural frequency and the critical
buckling value, respectively.
It is observed from Eq. (24) that the SCF is no longer required in
b2
the stiffness formulation. Herein, Bb1
A and BA contain the secondorder derivative of the shape functions. Hence, it requires
C1-continuous element in approximate formulations. As expected,
our present formulation based on IGA (as p Z 2) matches well the
C1-continuity from the theoretical/mechanical viewpoint of plates
[22,37] and also the RPT model.
3.3. Essential boundary conditions
Various boundary conditions are applied for an arbitrary edge
with simply supported (S) and clamped (C) conditions including

pffiffiffiffiffiffiffiffiffiffi
102 wE2 h
σh
τh
3
; ω ¼ ωa2 =h ρ=E2 ; λcr ¼ λcr a2 =E2 h
;σ ¼
;τ¼

q0 a
q0 a4
q0 a2
3

2

4.1. Static analysis
4.1.1. Two-layer [0/90] anti-symmetric square plate
Let us consider a simply supported laminated [0/90] square
plate with material set I subjected to a sinusoidal pressure
q0 sin ðπx=aÞ sin ðπy=aÞ as shown in Fig. 2. We first investigate the
convergence of the normalization displacement and stresses with
length to thickness ratio a/h¼ 10. The plate is modeled with 7 Â 7,
11 Â 11 and 15 Â 15 cubic elements as shown in Fig. 3. The
obtained results based on RPT model using IGA (RPT–IGA) with
various distributed functions f(z) (as listed in Table 1) are tabulated
in Table 2. The relative error compared to analytical solution using


72

L.V. Tran et al. / Engineering Analysis with Boundary Elements 47 (2014) 68–81

HSDT given by Khdeir and Reddy [42] is given in the parentheses.
It is observed that the RPT models using third-order polynomials
gain the most accuracy displacement. It is seen that a slightly fine
mesh of 11 Â 11 cubic elements is enough to provide reasonably
accurate solutions. For illustration, this mesh is therefore used for
several following examples.

Next we investigate the aforementioned problem with various length to thickness ratios a/h ranging from 2 to 100. Table 3
lists the present results in comparison with the 3D solution
reported by Pagano [1] and the analytical approach [26] using
various plate models (e.g, FSDT, TSDT and RPT). It can be seen
that, using equivalent single layer (ESL) plate theories, the
present model is in very good agreement with the 3D solution
for the errors varying from 8% to 0.3% according to ratio a/h

varying from 2 to 100. Furthermore, using four-variable RPT, the
present method produces good results in both deflection and
axial stress compared to the results derived from the Reddy
model and improves significantly the Senthinathal model for
the accuracy of stresses.
In Table 4, we study the behavior of two-layer [0/90] laminate
square plate under two types of boundary condition (SSSS and
SFSF, F¼free edge). The present results are compared with those
published ones derived from the 3D approach of Vel and Batra
[41]; the analytical method based on CLPT, FSDT and HSDT models
reported by Khdeir and Reddy [42] and HOSNDPT using meshfree
method with 18DOFs/node by Xiao et al. [40]. It can be seen that
the obtained results agree very well with the 3D elastic solution
[41]. It is again observed that RPT–IGA using the third-order
distributed functions produces same results which match well
with the 3D solution. The transverse displacement of the plates is
illustrated in Fig. 4 according to SFSF and SSSS boundary conditions, respectively. Fig. 5 depicts the stress distribution through
the SSSS plate thickness using various f(z) functions such as TSDT
[8], HSDT [17], ESDT [13], SSDT [14]. Using RPT model, the in-plane
stresses are almost matched together while being slightly different
for the out-plane stresses.
4.1.2. Five-layer square sandwich plate [0/90/core/0/90]

under sinusoidal load
A five-layer square sandwich plate with material set III is
considered. For illustration, third-order distributed function
reported by Reddy [8] is adopted. Table 5 summarizes the results
of the present and analytical approaches. As seen, the normalized

Fig. 2. Square laminate plate under sinusoidal load.

Fig. 3. Meshing and control net (in red color) of the square plate using cubic elements: (a) 7 Â 7; (b) 11 Â 11; (c) 15 Â 15. (For interpretation of the references to color in this
figure legend, the reader is referred to the web version of this article.)
Table 2
The convergence of the normalized displacement and axial stress of a simply supported laminated [0/90] plate (a/h¼ 10) subjected to a sinusoidal load.
Method

Mesh
7Â7

RPT–IGA

Reddy

À Á
w 2a; 2b

Shimpi
Arya
Karama

11 Â 11

15 Â 15

1.2159 (0.01)

1.2161 ( À 0.01)

1.2161 ( À 0.01)

1.2159 (0.01)
1.2129 (0.25)
1.2093 (0.55)

1.2161 ( À 0.01)
1.2131 (0.24)
1.2096 (0.53)

1.2161 ( À 0.01)
1.2131 (0.24)
1.2096 (0.53)

À 0.7351 (1.57)

À 0.7421 (0.63)

À 0.7443 (0.33)

À 0.7351 (1.57)
À 0.7366 (1.37)
À 0.7379 (1.19)

À 0.7421 (0.63)
À 0.7436 (0.43)
À 0.7449 (0.25)

À 0.7443 (0.33)
À 0.7458 (0.13)
À 0.7471 ( À 0.04)
À 0.7468

Analytical solution [42]
RPT–IGA

1.216
Reddy [8]
Shimpi [17]
Arya [14]
Karama [13]

Analytical solution [42]
(*)The error in parentheses.

Àa

σ x 2; 2b;

À h2

Á

L.V. Tran et al. / Engineering Analysis with Boundary Elements 47 (2014) 68–81

73

Table 3
The normalized deflection and axial stress of a laminated [0/90] plate with a/h ratios.
a/h

3D solution [1]

Analytical approach [26]

RPT–IGA

Reddy model

Senthil model

Whitney model

Reddy

Shimpi

Arya

Karama

2

w
σx

4.9362
À 0.9070

4.5619
À 1.4277

4.5619
À 1.8199

5.4103
À 0.7151

4.5618
À 1.4215

4.5618
À 1.4215

4.4335
À 1.4423

4.2901
À 1.4565

5

w

σx

1.7287
À 0.7723

1.6670
À 0.8385

1.6670
À 1.4133

1.7627
À 0.7151

1.6669
À 0.8337

1.6669
À 0.8337

1.6538
À 0.8392

1.6382
À 0.8439

10

w
σx

1.2318
À 0.7317

1.2161
À 0.7468

1.2161
À 1.3500

1.2416
À 0.7151

1.2161
À 0.7421

1.2161
À 0.7421

1.2131
À 0.7436

1.2096
À 0.7449

20

w
σx

1.1060
À 0.7200

1.1018
À 0.7235

1.1018
À 1.3340

1.1113
À 0.7151

1.1018
À 0.7189

1.1018
À 0.7189

1.1011
À 0.7193

1.1002
À 0.7196

100

w
σx

1.0742

À 0.7219

1.0651
À 0.7161

1.0651
À 1.3288

1.0651
À 0.7151

1.0651
À 0.7115

1.0651
À 0.7115

1.0650
À 0.7115

1.0650
À 0.7115

Table 4
The non-dimensional deflection and stresses of a two-layer laminated [0/90] composite square plate subjected to a sinusoidal load.
BC

Plate model

SSSS

3D model [41]
HSDT[42]
FSDT[42]
CLPT [42]
HOSNDPT [40]
RPT–IGA

SFSF

3D model [41]
HSDT[42]
FSDT[42]
CLPT [42]
HOSNDPT [40]
RPT–IGA

Exact
FEM

MQ-MLPG
TPS-MLPG
Reddy
Shimpi
Arya
Karama
Exact
FEM

MQ-MLPG

TPS-MLPG
Reddy
Shimpi
Arya
Karama

À Á
w 2a; 2b

À
Á
σ x 2a; 2b; À h2

À
Á
σ y 2a; 2b; h2

À
Á
σ xy 0; 0; 2h

À
Á
σ yz 2a; 0; 0

1.227
1.216
1.214
1.237
1.064

1.220
1.213
1.2161
1.2161
1.2131
1.2096

À 0.7304
À 0.7468
À 0.6829
À 0.7157
À 0.7157
À 0.726
À 0.723
À 0.7421
À 0.7421
À 0.7436
À 0.7449

0.7309
0.7468
0.6829
0.7157
0.7157
0.727
0.724
0.7421
0.7421
0.7436
0.7449

0.0497
–
–
–
–
0.0494
0.0491
0.053
0.053
0.053
0.0531

–
0.319
–
0.2729
0
0.298
0.278
0.3181
0.3181
0.3252
0.3319

1.210
1.2295
1.189
1.1907
1.1849

1.21
1.21
1.2192
1.2192
1.221
1.2225

0.0119
–
–
–
–
0.0118
0.0119
0.0121
0.0121
0.0121
0.0122

–
0.4489
–
0.3882
0
0.488
0.499
0.4507
0.4507
0.46
0.4686

2.026
1.992
2.002
2.028
1.777
2.028
2.028
1.990
1.990
1.9851
1.9794

0.2503
0.2624
0.2212
0.2469
0.2403
0.249
0.249
0.25472
0.25472
0.2555
0.2562

Fig. 4. Deflection of two-layer [0/90] antisymmetric square plates: (a) SFSF and (b) SSSS.

displacement and stresses obtained are acceptable to the analytical solution [26]. It is indicated that the results derived from both
TSDT and RPT models are almost identical while FSDT model (with

SCF ¼5/6) leads to very poor results in both deflection and shear
stresses, especially for transverse shear stress. This conclusion is
clearly addressed in Fig. 6. Table 6 reveals that a good agreement


74

L.V. Tran et al. / Engineering Analysis with Boundary Elements 47 (2014) 68–81

0.05

0.05

TSDT
HSDT
SSDT
ESDT

0.04
0.03

0.03

0.02

0.02

0.01

0.01

0

0

−0.01

−0.01

−0.02

−0.02

−0.03

−0.03

−0.04

−0.04

−0.05
−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

TSDT
HSDT
SSDT
ESDT

0.04

−0.05
−0.06

0.8

−0.04

−0.02

0

0.02

0.04

0.06

0.05
TSDT
HSDT
SSDT
ESDT

0.04
0.03
0.02
0.01
0
−0.01
−0.02
−0.03
−0.04
−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Fig. 5. The stresses through the thickness of laminate composite plate under full simply supported condition with a/h ¼ 10 via several refined plate models.

Table 5
The non-dimensional deflection and stresses of a five-layer square sandwich plate [0/90/core/0/90] under sinusoidal load with a/h ¼ 10.
Method

Model

À Á
w 2a; 2b

À
Á
σ x 2a; 2b; À 2h

À
Á
σ y 2a; 2b; h2

À
Á
τxy 0; 0; h2

À
Á

τyz 2a; 0; À3h

Analytical solution [26]

TSDT
RPT
FSDT

2.3075
2.3075
1.3570

0.6815
0.7634
0.6200

À 0.6815
À 0.7631
À 0.6200

0.0787
0.0787
0.0693

–
–
–

IGA

TSDT
RPT
FSDT

2.2588
2.2588
1.3512

0.6834
0.6857
0.6308

À 0.6834
À 0.6857
À 0.6308

0.0710
0.0710
0.0626

1.8360
1.8352
0.6014

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.2

0.1

0.1

0

0

−0.1

−0.1

−0.1

−0.2

−0.2

−0.2

−0.3

−0.3

−0.3

−0.4

−0.4

−0.4

0.1

RPT
TSDT
FSDT

0

−0.5
−0.8 −0.6 −0.4 −0.2

0

0.2

0.4

0.6

0.8

−0.5
−0.08 −0.06 −0.04 −0.02

RPT
TSDT
FSDT

0

0.02 0.04 0.06 0.08

0.5
0.4
0.3

−0.5

RPT
TSDT
FSDT

0

0.5

1

1.5

2

À
Á
Fig. 6. The stresses through the thickness of a sandwich plate under full simply supported condition with a/h¼ 10, hc/hf ¼ 4 via various plate models. (a) axial stress σ x 2a; 2b; z .
Àa
Á
(b) Shear stress τxy ð0; 0; zÞ. (c) Shear stress τyz 2; 0; z .


L.V. Tran et al. / Engineering Analysis with Boundary Elements 47 (2014) 68–81

between two models TSDT and RPT is again observed when ratio
hc/hf varies from 4 to 200.
4.2. Free vibration analysis
4.2.1. The cross-ply laminated [0/90]N square composite plate
Let next us consider a cross-ply N layers laminated [0/90]N
composite plate with a/h ¼ 5 under simply supported boundary
conditions. Herein, material set II is used. The effects of the
number of layers N and elastic modulus ratios E 1 /E 2 are

Table 6
The non-dimensional deflection of a sandwich plate [0/90/core/0/90] via hc/hf ratio.
hc/hf

RPT

TSDT

FSDT

4
10
20
30
40
50
100
200

2.2588
8.9432
34.7064
67.8504
93.931
110.5829
137.3968
152.4244

2.2588
8.9432
34.7057
67.8456
93.9228
110.5745
137.3956

152.4252

1.3512
2.3429
4.0458
5.7579
7.4721
9.1866
17.7546
34.8463

75

tabulated in Table 7. A good agreement is found for the present
IGA–RPT models in comparison with three-dimensional elasticity approach proposed by Noor [2] and the analytical
method given by Kant [45]. It is in general observed that the
IGA–RPT based on the exponential function achieves the highest results which match very well with the 3D solution as E 1 /
E 2 r 10. As E 1 /E 2 ratio ranges from 20 to 40, the present results
are asymptotic to analytical solutions for 2D plate model using
HSDT and RPT [45].
Next, with constant E1/E2 ratio (¼ 40), the variation of natural
frequency of a two-layer laminated composite plate via length to
thickness ratio are listed in Table 8. It is again seen that the
obtained results match well with the analytical one using 12DOFs
published by Kant [45]. The difference reduces via the increase in
the ratio of a/h (approximate from 8% to 0.02% as changing of a/h
from 4 to 100). The first three mode shapes of a thick plate
(a/h ¼10) is then plotted in Fig. 7. It is clear that beside the full
mode shape of deflection (above), the mode shapes of four
unknown parameters along line y¼a/2 are illustrated.

To close this sub-section, the effect of boundary condition on
normalized frequency of ten-layer cross-ply composite plate is
plotted in Table 9. Compared with those reported by Reddy and
Khdeir [46], the present model again obtains good agreement. It
can be seen that, present model using RPT gains the closest results
to analytical solution using TSDT with slightly higher results. In

Table 7
The natural frequency of a simply supported [0/90]N composite plate.
N

1

3D elasticity [2]
Analytical solution [45]

RPT–IGA

2

3D elasticity [2]
Analytical solution [45]

RPT–IGA

3

3D elasticity [2]
Analytical solution [45]

RPT–IGA

5

E1/E2

Model

3D elasticity [2]
Analytical solution [45]

RPT–IGA

3

10

20

30

40

HSDT-12DOFs
HSDT-9DOFs
RPT
FSDT

6.2578
6.2336

6.1566
6.2169
6.149

6.9845
6.9741
6.9363
6.9887
6.9156

7.6745
7.714
7.6883
7.821
7.6922

8.1763
8.2775
8.257
8.505
8.3112

8.5625
8.7272
8.7097
9.0871
8.8255

Reddy
Shimpi

Arya
Karama

6.2169
6.2169
6.2189
6.2224

6.9887
6.9887
6.9965
7.0066

7.8211
7.8211
7.838
7.8585

8.5051
8.5051
8.5317
8.563

9.0872
9.0872
9.1237
9.1662

HSDT-12DOFs
HSDT-9DOFs

TSDT, RPT
FSDT

6.5455
6.5146
6.4319
6.5008
6.4402

8.1445
8.1482
8.1010
9.1954
8.1963

9.4055
9.4675
9.4338
9.6265
9.6729

10.165
10.2733
10.2463
10.5348
10.6095

10.6798
10.8221
10.7993

11.1716
11.2635

Reddy
Shimpi
Arya
Karama

6.5008
6.5008
6.5012
6.5034

8.1954
8.1954
8.1930
8.1939

9.6265
9.6265
9.6205
9.6201

10.5348
10.5348
10.5268
10.5261

11.1716
11.1716

11.1628
11.1629

HSDT-12DOFs
HSDT-9DOFs
TSDT, RPT
FSDT

6.6100
6.5711
6.4873
6.5552
6.4916

8.4143
8.3852
8.3372
8.4041
8.3883

9.8398
9.8346
9.8012
9.9175
9.9266

10.6958
10.7113
10.6853
10.8542

10.8723

11.2728
11.3051
11.2838
11.5007
11.5189

Reddy
Shimpi
Arya
Karama

6.5558
6.5558
6.5567
6.5596

8.4052
8.4052
8.4066
8.4122

9.9181
9.9181
9.9211
9.9313

10.8547
10.8547

10.8604
10.8758

11.5012
11.5012
11.5103
11.5314

HSDT-12DOFs
HSDT-9DOFs
TSDT, RPT
FSDT

6.6458
6.6019
6.5177
6.5842
6.5185

8.5625
8.5163
8.4680
8.5126
8.4842

10.0843
10.0438
10.0107
10.0674
10.0483

11.0027
10.9699
10.9445
11.0197
10.9959

11.6245
11.5993
11.5789
11.673
11.6374

Reddy
Shimpi
Arya
Karama

6.5842
6.5842
6.5854
6.5885

8.5126
8.5126
8.5156
8.5229

10.0674
10.0674

10.0741
10.0882

11.0197
11.0197
11.031
11.0523

11.673
11.673
11.6894
11.7182


76

L.V. Tran et al. / Engineering Analysis with Boundary Elements 47 (2014) 68–81

Table 8
The natural frequency of simply supported laminated [0/90] composite plate with E1/E2 ¼40.
Plate model

a/h
4

10

20

50

100

Analytical solution [45]

HSDT-12DOFs
HSDT-9DOFs
TSDT
RPT
FSDT

7.9081
7.8904
8.3546
8.3546
8.0889

10.4319
10.4156
10.568
10.568
10.461

11.0663
11.0509
11.1052
11.1052
11.0639

11.2688

11.2537
11.2751
11.2751
11.2558

11.2988
11.2837
11.3002
11.3002
11.2842

RPT–IGA

Reddy
Shimpi
Arya
Karama

8.3547
8.3547
8.4018
8.4564

10.5681
10.5681
10.5812
10.5965

11.1053
11.1053

11.109
11.1133

11.2752
11.2752
11.2758
11.2758

11.3003
11.3003
11.3004
11.3006

ω1 = 10.5681
0.6

0
−0.1

0.4

−0.2
−0.3

0.2

−0.4
−0.5
−0.6

y

0

y

y

ω 3 = 26.5015

ω 2 = 26.5015

−0.2

−0.7

−0.4

−0.8
−0.9

−0.6
0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

x

0.6

0.8

1

0.5
0.4
0.3
0.2
0.1
0
−0.1
−0.2
−0.3
−0.4
−0.5
0

0.2

: u0

: v0

0.4

0.6

0.8

1

x

x

: wb

: ws

Fig. 7. Vibration mode shapes: full plate (upper) and line y¼a/2 (lower) of simply supported laminated [0/90] composite plate with E1/E2 ¼ 40, a/h¼ 10.

Table 9
The natural frequency of ten-layer cross-ply [0/90]5 plate with a/h ¼5 and E1/E2 ¼40.
Plate model

Boundary conditions

Analytical solution [46]

TSDT
FSDT
CLPT

RPT–IGA

Reddy
Shimpi
Arya
Karama

SFSF

SFSC

SSSS

SSSC

SCSC

CCCC

8.155
8.139
11.459

8.966
8.919
13.618

11.673
11.644
12.167

12.514
12.197
23.348

13.568
12.923
30.855

–
–
–

11.673
11.673
11.6894
11.7182

13.0041
13.0041
13.0463
13.1062

14.1566
14.1566
14.2418
14.3513

15.2991
15.2991
15.4558
15.6438

8.1554
8.1554
8.1661
8.1853

9.0832
9.0832
9.0971
9.1201

addition, when the constrained edge changes from F to S and C,
the structural stiffness increases, the magnitudes of free vibration
thus increase, respectively. The mode shapes according to various
boundary conditions are illustrated in Fig. 8.

4.2.2. The sandwich plate with curved boundary: a comparison
of computational efficiency
Let us consider a plate with an annular geometry with a uniform
thickness h, outer radius R and inner one r as shown in Fig. 9. Material

L.V. Tran et al. / Engineering Analysis with Boundary Elements 47 (2014) 68–81

77

Fig. 8. Mode shape profile of ten layers [0/90]5 composite plate under various boundary conditions. (a) SFSF. (b) SFSC. (c) SSSS. (d) SSSC. (e) SCSC. (f) CCCC.

Table 10
The natural frequency _
ω of circular sandwich plate via R/h ratios and various plate
models.
R/h

Method

Mode number
1

2 f ace

thickness ratio R/h. A good agreement is observed for RPT and
TSDT, while FSDT model remains too stiffened as plate becomes
thicker. The first six mode shapes of a circular plate are depicted
in Fig. 10.
As the inner radius r a 0, a fully annular plate is obtained as
shown in Fig. 9. Due to symmetry, an upper haft of plate has been
modeled in Fig. 11 with the symmetric constraint: displacement
along y-direction equals to zero at y¼ 0. With data R/h¼10 and
R/r ¼2, the first six normalized frequencies _
ω ¼ ωðR À rÞ2 =

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h ðρ=E2 Þf ace are listed in Table 11. Now we show the

3

4

5

6

2

TSDT
RPT
FSDT

1.8408
1.8185
3.2749

2.9851
2.969
5.1587

2.9903
2.9709
5.1849

4.0577

4.0497
6.9426

4.2675
4.1186
6.9527

4.5676
4.422
7.1665

5

TSDT
RPT
FSDT

4.0777
4.0224
6.5314

6.5847
6.6477
11.0073

6.5921
6.6517
11.0291

8.8982

9.0267
15.2293

9.2296
9.1634
15.8679

9.8714
9.7342
16.9509

10

TSDT
RPT
FSDT

6.6153
6.5599
8.6196

11.274
11.5226
16.1005

11.2808
11.5286
16.1161

15.6121

16.0142
23.3139

16.3027
16.557
24.8155

17.4576
17.3499
26.6094

20

TSDT
RPT
FSDT

8.6768
8.6529
9.5541

16.2863
16.5898
19.1564

16.2912
16.5953
19.1648

23.6428

24.1341
28.8009

25.1854
25.8627
31.7216

27.0669
27.1200
34.0548

100

TSDT
RPT
FSDT

9.9093
9.9088
9.9258

20.4639
20.498
20.614

20.4643
20.4985
20.6145

31.3827

31.4395
31.6544

35.2143
35.3459
35.8107

37.7988
37.8367
38.3766

Fig. 9. The annular plate model.

III is used. The plate is clamped at the outer boundary. For illustration,
2
we use the distributed function f ðzÞ ¼ z À ð4=3h Þz3 for RPT model.
The analytical solution was not available. The aim of this study is to
estimate the solution of the plates involving curved edges.
By setting the inner radius r ¼ 0, the model becomes the
clamped circular sandwich plate. Table 10 shows the dependence
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
of the natural frequency _
ω ¼ ωR2 =h ðρ=E Þ
on the radius to

2

computational efficiency of the present approach. The program is
compiled by a personal computer with Intel (R) Core (TM) 2 Duo
CPU – 2 GHz and RAM – 4 GB. It can be seen that with the same

mesh, RPT model produces lowest degree of freedoms (DOFs).
Hence, it spends lowest computational cost with just 284 s
compared with 608 s and 520 s according to TSDT and FSDT ones,
respectively. However, RPT model also archives closed results to
TSDT than FSDT one. Table 12 tabulates the frequency parameter _
ω
of the annular plates via outer radius to inner radius ratio R/r and
radius to thickness ratio R/h. It is concluded that the frequency
parameters decrease sequentially following to increase in inner
radius to outer radius ratio r/R and decrease in radius to thickness


78

L.V. Tran et al. / Engineering Analysis with Boundary Elements 47 (2014) 68–81

Mode1

Mode2

Mode4

Mode3

Mode5

Mode6

Fig. 10. The first six mode shapes of circular sandwich plate with R/h ¼10.

Table 12
The dependent of natural frequency _
ω of annular plate on R/h and r/R ratios.
R/h

r/R

Mode number
1

Fig. 11. The mesh of a half annular plate.

Table 11
The natural frequency _
ω of annular sandwich plate with R/h ¼ 10 and R/r2 ¼2.
Model Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode 6

TSDT
RPT
FSDT

2.6579
2.5954
3.5775

2.9051
2.9295
4.0890

3.5579

3.7458
5.4463

4.4029
4.6822
7.2097

5.4377
5.7805
9.3348

DOFs CPU
time

6.5720 2340
6.9574 1872
11.5482 2340

608
284
520

ratio R/h. To enclose this section the first six mode shapes of the
plate are depicted in Fig. 12.
4.3. Buckling analysis
Â
Ã
4.3.1. The angle-ply laminated θ=À θ square composite plate
A simply supported two-layer angle-ply laminated ½θ= ÀθŠ
square plate is subjected to uniaxial compressive load along the

x-direction shown in Fig. 13a. Material set II is used. The results
tabulated in Table 13 are compared with that of Ren [50] and the
analytical solution [20] using FSDT, HSDT and RPT assumptions.
For all values of a/h ratio and fiber orientation, present model with
third-order functions give the closest buckling load to that of RPT
predicted by Thai et al. [20]. It can be again seen that all models
give the slightly same results for thin plates (a/h ¼100). Fig. 14
illustrates the buckling mode of two-layer angle-ply laminated
composite plate in case of θ¼ 45. It can be seen that as plate

2

3

4

5

6

2

0
0.2
0.5
0.8

1.8185
1.2536
0.7273

0.3353

2.9690
1.7450
0.7907
0.3379

2.9709
2.5472
0.9636
0.3529

4.0497
3.2135
1.1880
0.3632

4.1186
3.2771
1.4621
0.3819

4.4220
3.5679
1.7685
0.4060

5

0

0.2
0.5
0.8

4.0224
2.6684
1.5623
0.6663

6.6477
3.8843
1.7189
0.6717

6.6517
5.6464
2.1153
0.6951

9.0267
6.8217
2.6133
0.7174

9.1634
7.2075
3.1889
0.7537

9.7342

7.7229
3.8012
0.7987

10

0
0.2
0.5
0.8

6.5599
4.1002
2.5954
1.1875

11.5226
6.6405
2.9295
1.1985

11.5286
9.8901
3.7458
1.2416

16.0142
11.7335
4.6822
1.2875

16.5570
12.7742
5.7805
1.3585

17.3499
13.9715
6.9574
1.4461

20

0
0.2
0.5
0.8

8.6529
5.0797
3.5545
2.0288

16.5898
9.3630
4.1602
2.0500

16.5953
14.5846

5.6767
2.1495

24.1341
18.0392
7.2449
2.2306

25.8627
19.2986
9.2193
2.3665

27.1200
22.5169
11.4600
2.5383

100

0
0.2
0.5
0.8

9.9088
5.5779
4.1878
3.4620

20.4980
11.3311
5.0633
3.5002

20.4985
18.5059
7.4224
3.8409

31.4395
24.9014
9.7455
3.9826

35.3459
25.3016
13.0368
4.2738

37.8367
32.4963
17.2741
4.6740

thickness
reduces,
the
non-dimension
buckling

value
3
λcr ¼ λcr a2 =E2 h increases according to changing of mode shape
from two halves sine wave (a/h¼ 4) to a half sine wave (a/h ¼10;
100, respectively). Furthermore, the portion of shear deflection
components ws in transverse displacement reduces and tends to
zero as a/h¼ 100. The present models, hence, reduce to CLPT
model.

4.3.2. The three-layer symmetric cross-ply [0/90/0] composite plate
Finally, we investigate the biaxial buckling load of a symmetric cross-ply [0/90/0] simply supported plate as shown in
Fig. 13b. This test aims to show that the RPT–IGA is also existing
the deficiency in case of the symmetric laminated composite
plates. This fact is originated from the feature of the RPT model,
which was confirmed by Kant et al. [26,45]. Various length-tothickness a/h with elastic modulus ratios are studied. Table 14


L.V. Tran et al. / Engineering Analysis with Boundary Elements 47 (2014) 68–81

Mode 1

Mode 4

79

Mode 2

Mode 3

Mode 5

Mode 6

Fig. 12. The first six mode shapes of annular plate with R/h¼ 10 and R/r ¼2.

Fig. 13. Geometry of laminated composite plates under axial (a) and biaxial (b) compression.

0.6
0.4
0

z

z

0.2
−0.2
−0.4
−0.6
−0.8
0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

: u0

: v0

0.2

0.4

0.6

0.8

1

x

x

x

y

0.4

−0.3
−0.4
−0.5
−0.6
−0.7
−0.8

−0.9

y

0.2

y

0

0
−0.1
−0.2

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

z

0.8

: wb

: ws

Fig. 14. Buckling mode shapes: full plate (upper) and line y ¼a/2 (lower) of simply supported [45/ À 45] composite plate with various length to thickness ratios: (a) a/h¼ 4;
(b) a/h ¼10; and (c) a/h ¼100.


80

L.V. Tran et al. / Engineering Analysis with Boundary Elements 47 (2014) 68–81

Table 13
The normalized uniaxial buckling load λcr of simply supported two layer laminated ½θ= À θŠ composite plate with E1/E2 ¼40.
a/h

θ

Ren [50]

Analytical solution [20]
HSDT

RPT–IGA

FSDT

RPT

Reddy

Shimpi

Arya

Karama

4

30
45

9.5368
9.8200

9.3391
8.2377

7.5450
6.7858

9.3518
8.3963

9.3522
8.3966

9.3522
8.3966

9.6731

8.6472

9.9211
8.9414

10

30
45

15.7517
16.4558

17.1269
18.1544

16.6132
17.5522

17.2795
18.1544

17.2797
18.1545

17.2797
18.1545

17.3495
18.2383

17.4311
18.3354

100

30
45

20.4793
21.6384

20.5017
21.6663

20.4944
21.6576

20.504
21.6663

20.5042
21.6664

20.5042
21.6664

20.5052
21.6676

20.5063
21.6689

Table 14
The normalized biaxial buckling load λcr of simply supported three-layer laminated [0/90/0] composite plate under various a/h ratios and E1/E2 ¼40.
Plate model

a/h

HSDT–RPIM [48]
FSDT–RPIM [48]
HSDT–FEM [49]

2

5

10

15

20

1.457
1.419
1.465

5.519
5.484
5.526

10.251
10.189
10.259

12.239
12.213
12.226

13.164
13.132
13.185

HSDT[47]

Arya
Soldatos
Thai

1.3862
1.3641
1.4316

5.3668
5.3834
5.3236

9.9188
9.9495
9.8795

12.0205
12.0398
11.9978

13.0379
13.0504
13.0239

RPT–IGA

Reddy
Shimpi
Arya
Karama

1.6864
1.6864
1.7215
1.7679

6.1752
6.1752
6.1571
6.1500

10.8825
10.8825
10.8549
10.8336

12.7140
12.7140
12.6957
12.6808

13.5135
13.5135
13.5014
13.4916

0.8
0.6
0.4

z

0.2
0
−0.2
−0.4
−0.6
−0.8
x

0

0.2

0.4

0.6

0.8

1

y

: u0

: v0

: wb

: ws

Fig. 15. Buckling mode shapes: (a) a full plate model and (b) along line x ¼a/2 of simply supported [0/90/0] composite plate with E1/E2 ¼ 40, a/h ¼10 (note that u0  v0).

shows the critical buckling parameter λcr with respect to various
length-to-thickness ratios. The obtained results are compared
with those of the isogeometric approach based on HSDT [47],
the finite element method based on HSDT [49], the meshfree
method based on both FSDT and HSDT [48]. The present model
reflects well the true buckling mode shape as shown in Fig. 15. It
is seen that obtained results using RPT assumption are more
stiffened than those using other HSDT ones when the plate
thickness is increased. Therefore further research on the RPT
model is still necessary as observed in [26,45].

5. Conclusions
In this paper, we presented an effective formulation based on
isogeometric approach (IGA) and a four variable refined plate
theory (RPT) for static, free vibration and buckling analysis of
laminated composite plates. We addressed a general four-variable
refined plate theory with various distributed functions which is
used to approximate the high-order term in displacement field.
Utilizing NURBS basis function, the present method enables us to
achieve easily the smoothness with arbitrary continuous order and


L.V. Tran et al. / Engineering Analysis with Boundary Elements 47 (2014) 68–81

therefore naturally fulfills the C1-continuity of the present plate
model. Just using four DOFs per each node, the present method
retains the computational efficiency and gains the good agreement
results compared with other models in the literature especially for
anti-symmetric laminated plates. However, for symmetric one,
e.g, symmetric cross-ply [0/90/0], RPT overestimates the obtained
results. To overcome this disadvantage, we suggest using the
RPT–IGA in combination with the layerwise theory [51]. We
assume the four unknown variable refined plate theory in each
layer and the imposition of displacement continuity at the layers
interfaces. In our opinion, such an approach will be promising to
provide an effectively alternative finite element tool for modeling
and analysis of plate structures.
Acknowledgments
This research is funded by the Vietnam National Foundation for
Science and Technology Development (NAFOSTED) under grant
number 107.02-2012.17. The support is gratefully acknowledged.

The fifth author appreciates for the support from the Basic Science
Research Program through the National Research Foundation of
Korea (NRF) funded by the Ministry of Education, Science, and
Technology (2010-0019373 and 2012R1A2A1A01007405).
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