Phân tích dao động tư 5 do

1. Free vibration and buckling analysis of laminated composite plates using
the NURBS-based isogeometric finite element method
Saeed Shojaee a,⇑
, Navid Valizadeh a
, Ebrahim Izadpanah a
, Tinh Bui b
, Tan-Van Vu c
a
Department of Civil Engineering, Shahid Bahonar University, Kerman, Iran
b
Department of Civil Engineering, University of Siegen, Germany
c
School of Civil, Environmental and Architectural Engineering, Korea University, Republic of Korea
a r t i c l e i n f o
Article history:
Available online 20 January 2012
Keywords:
Isogeometric analysis
NURBS
Free vibration
Buckling
Composite laminates
a b s t r a c t
An isogeometric finite element method based on non-uniform rational B-splines (NURBS) basis functions
is developed for natural frequencies and buckling analysis of thin symmetrically laminated composite
plates based upon the classical plate theory (CPT). The approximation of the solution space for the deflec-
tion field of the plate and the parameterization of the geometry are performed using NURBS-based
approach. The essential boundary conditions are formulated separately from the discrete system equa-
tions by the aid of Lagrange multiplier method, while an orthogonal transformation technique is also
applied to impose the essential boundary conditions in the discrete eigen-value equation. The accuracy
and the efficiency of the proposed method are thus demonstrated through a series of numerical experi-
ments of laminated composite plates with different boundary conditions, fiber orientations, lay-up num-
ber, eigen-modes, etc. The obtained numerical results are then compared with either the analytical
solutions or other available numerical methods, and excellent agreements are found.
Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction
By owing the possession of the high strength to weight and stiff-
ness to weight ratios as compared to the conventional isotropic
materials, laminated composites have been increasingly and widely
used in a wide range of engineering structures and modern indus-
tries. Besides possessing the superior of the composite material
properties, the laminated composites also provide the convenient
design through tailoring of the stacking sequence and layer thick-
ness to optimize the desired characteristics for engineering applica-
tions. By a large requirement of using such laminated composite
plates to engineering applications, studies involving the stability
behaviors and natural vibrations of those structures are of great
importance and essential in predicting their structural responses.
Different analytical approaches have been introduced for the
buckling and free vibration analysis of composite plates, e.g. see
[1–7]. Although the analytical methods are able to provide deep
physical insights and their solutions are highly accurate, their appli-
cation is often restricted to problems having very simple geometries,
boundary and loading conditions, etc. In practical cases, solutions
based on numerical approaches are necessary. The Ritz methods
[8–18], the strip element method [19,20], the discrete singular con-
volution (DSC) method [21,22], etc. are some of typical numerical
methods successfully applied to the free vibration and buckling
analysis of laminated composite plates. Among them, the finite ele-
ment method (FEM) is one of the most powerful methods for solving
the complicated boundary and initial value problems e.g. see [23]. A
veryinterestingreview onapplicationoftheFEMtothoseissuesdur-
ing the past 20 years can be found in [24], and some advanced com-
putational aspects and more details in [25]. Further information on
the recent developments of the FEM for laminated composite plates
in various aspects including the buckling and vibration analysis can
also be found in Zhang and Yang [26]. Although the finite element
method is known as one of the most versatile and powerful tech-
niques in many problems, it has many inherent disadvantages
including, for instance, the cumbersome task of mesh generation;
a high order, i.e. C1
consistency, is not an easy task to construct con-
ventionally conformable plate elements; a time-consuming proce-
dure for the connectivity of elements; re-meshing in moving
boundary problems and so on. Recently, as an alternative to the
FEM, a family of the so-called meshless or meshfree methods, e.g.
see [27–31] have introduced to overcome the drawbacks of the
FEM. Some information on recent development of the meshless
methods for the vibration analysis of composite plate can be found
in Bui et al. [31].
Isogeometric analysis (IGA) is a recently developed computa-
tional approach that offers the possibility of integrating finite ele-
ment analysis (FEA) into conventional NURBS-based Computer
Aided Design (CAD) tools. The concept of the IGA in mechanic
problems is pioneered by Hughes and his co-workers [32] as a no-
vel technique for the discretization of partial differential equations.
0263-8223/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved.
doi:10.1016/j.compstruct.2012.01.012
⇑ Corresponding author.
E-mail address: saeed.shojaee@uk.ac.ir (S. Shojaee).
Composite Structures 94 (2012) 1677–1693
Contents lists available at SciVerse ScienceDirect
Composite Structures
journal homepage: www.elsevier.com/locate/compstruct

2. The method handles many great features shared by both the FEM
and the meshless methods. The basic idea behind IGA is to utilize
the basis functions that are able to model accurately the exact
geometries from the CAD point of view for numerical simulations
of physical phenomena. It can be achieved by using the B-splines
or NURBS for the geometrical description and invoke the isopara-
metric concepts to define the unknown field variables, e.g. deflec-
tion in our case. A distinct advantage over the FEM is that mesh
refinement is simply accomplished by re-indexing the parametric
space without interaction with the CAD system. An intriguing trait
of these functions is that they are typically smooth beyond the
classical C0
-continuity of the standard FEM. The IGA-based ap-
proaches have been constantly developed and have shown many
great advantages on solving many different problems in a wide
range of research areas such as fluid–structure interaction [33–
35], shells [36], structural analysis [37,38], fracture mechanics
[39,40] and so on.
Due to good inherent characteristics of the IGA method, the
development of an isogeometric finite element method associated
with the NURBS shape functions for free vibration and buckling
analysis of laminated composite plates is the prime scope of the
present work. In terms of plate analysis, the finite element formu-
lation based on the Kirchhoff theory requires elements with at
least C1
-inter-element continuity, which has gained many difficul-
ties to achieve for the free-form geometries when using the stan-
dard Lagrangian polynomials as basis functions. Higher-order
NURBS basis functions, however, with an increased inter-element
continuity can be easily obtained, thus the NURBS is well suited
for the Kirchhoff elements by means of the IGA. By using the
NURBS basis function, more accurate modeling of complex geome-
tries and exactly representation of common engineering shapes
such as circles, cylinders, spheres, and ellipsoids can be provided.
In IGA, even very coarse mesh can preserve the exactness of the
problem geometry. By employing the Lagrange multiplier in the
IGA, it makes the approach possible in the treatment of different
boundary conditions in the plate problem. In fact, some compared
methods in the open literature have been developed only for par-
ticular types of boundary condition, e.g. the MK method [31], in
which only the free and simply supported boundary conditions
are treated without any special techniques.
In this paper, the present formulation is followed the classical
plate theory (CPT). The Lagrange multiplier in conjugation with
the orthogonal transformation technique is employed for enforcing
the essential boundary conditions. The accuracy and the applicabil-
ity of the proposed method are subsequently justified on numerical
results of laminated composite plates through several numerical
examples in the class of eigenvalue problems. Based on the best
knowledge of the authors, no such identical task has been exam-
ined when this manuscript is being reported. After the introduc-
tion, deriving the NURBS shape functions is presented in the next
section. In Section 3, governing equations and discretization are
briefly introduced due to the sake of completeness. Comparative
studies and numerical applications for the free vibration problems
of laminated plates are presented in details in Section 4, whereas
the accuracy of the proposed approach for buckling analysis of
plates and composite laminates are discussed in Section 5. We
shall end with some conclusions derived from the present work
in the last section.
2. Derivation of the NURBS shape functions
The NURBS approximation employed for constructing the shape
functions and their derivatives is briefly presented in this section,
but one can refer to [41] for more details. Basically, a non-uniform
rational B-spline (NURBS) curve of order p is defined as
XðnÞ ¼
Xn
i¼1
Ri;pðnÞeXi ð1Þ
Ri;pðnÞ ¼
Ni;pðnÞwi
Pn
j¼1Nj;pðnÞwj
ð2Þ
in which, Ri,p stands for the univariate NURBS basis functions,
eXi ¼ ðxi; yiÞ; i ¼ 1; 2; . . . ; n are a set of n control points, wi are a set
of n weights corresponding to the control points that must be
non-negative and Ni,p represents the B-spline basis function of order
p. To construct a set of n B-spline basis functions of order p, a knot
vector N is defined in a parametric space as follows:
N ¼ fn1; n2; . . . ; nnþpþ1g ni 6 niþ1;
i ¼ 1; 2; . . . ; n þ p
ð3Þ
The parametric space is assumed to be n 2 [0,1]. The knot vectors
used for analysis purposes are generally open knot vectors to satisfy
the Kronecker-delta property at boundary points [42]. In an open
knot vector, the first and last knots are repeated p + 1 times. Given
a knot vector, the univariate B-spline basis function Ni,p can be con-
structed by the following Cox-de Boor recursive formula [41],
Ni;0ðnÞ ¼
1 if ni 6 n 6 niþ1
0 otherwise
&
ð4Þ
and
Ni;pðnÞ ¼
n À ni
niþp À ni
Ni;pÀ1ðnÞ þ
niþpþ1 À n
niþpþ1 À niþ1
Niþ1;pÀ1ðnÞ; p ¼ 1;2;3;...:
ð5Þ
The B-spline basis functions which are constructed from the open
knot vectors have the interpolation feature at the ends of the
parametric space. Quadratic B-spline basis functions with the inter-
polation feature at the ends of the parametric space are shown in
Fig. 1.
Generally, a NURBS surface of order p in n direction and order q
in g direction can be expressed as
Xðn; gÞ ¼
Xn
i¼1
Xm
j¼1
Rp;q
i;j ðn; gÞeXi;j
¼
Xn
i¼1
Xm
j¼1
Ni;pðnÞMj;qðgÞwi;j
Pn
^i¼1
Pm
^j¼1N^i;pðnÞM^j;qðgÞw^i;^j
eXi;j 0 6 n; g 6 1
ð6Þ
where Rp;q
i;j stand for the bivariate NURBS basis functions, eXi;j is a
control mesh of n  m control points, and wi,j are the corresponding
weights, while Ni,p and Mj,q are the B-spline basis functions defined
on the N and H knot vectors, respectively. The first derivative of
Fig. 1. Quadratic basis functions for an open knot vector N = {0,0,0,0.2,0.4,0.6,
0.8,1,1,1}.
1678 S. Shojaee et al. / Composite Structures 94 (2012) 1677–1693

3. Rp;q
i;j ðn; gÞ with respect to each parametric variable, e.g. n, is derived
by simply applying the quotient rule to Eq. (6) as
@Rp;q
i;j ðn; gÞ
@n
¼
@Ni;pðnÞ
@n
Mj;qðgÞwi;jWðn; gÞ À @Wðn;gÞ
@n
Ni;pðnÞMj;qðgÞwi;j
ðWðn; gÞÞ2
ð7Þ
and
Wðn; gÞ ¼
Xn
^i¼1
Xm
^j¼1
N^i;pðnÞM^j;qðgÞw^i;^j ð8Þ
@Wðn; gÞ
@n
¼
Xn
^i¼1
Xm
^j¼1
@N^i;pðnÞ
@n
M^j;qðgÞw^i;^j ð9Þ
Any higher order derivatives of the NURBS basis functions can be
obtained in a similar fashion. The important properties of the
NURBS basis functions can also be summarized as follows:
(1) Partition of unity, 8n;
Pn
i¼1Ri;pðnÞ ¼ 1.
(2) Non-negativity, "n, Ri,p(n) P 0.
(3) Basis functions of order p are p À mi times continuously dif-
ferentiable over a knot ni, where mi is the multiplicity of the
value of ni in the knot vector.
(4) Local support, i.e. the support of Ri,p(n) is compact and con-
tained in the interval [ni,ni+p+1]. In the two dimensional case,
for a specified knot span (element), there are only
(p + 1) Â (q + 1) number of non-zero basis functions. There-
fore, the total number of control points per element is
nen = (p + 1) Â (q + 1).
Based on the classical thin plate theory [41], only the deflection
of the plate w(x) is chosen as the independent variable while the
other two displacement components u(x) and v(x) can be obtained
from w(x). Therefore, in this paper, the NURBS basis is employed
for both the parameterization of the geometry and the approxima-
tion of the deflection field w(x) as follows:
wh
ðxðnÞÞ ¼
XnÂm
I¼1
/IðnÞwI ð10Þ
xðnÞ ¼
XnÂm
I¼1
/IðnÞeXi ð11Þ
In all the above equations, n = (n,g) is the parametric coordinates
vector, x = (x,y) is the physical coordinates vector, ~xi represents
the control points of a n  m control mesh, wI represents the deflec-
tion field at each control point, and /I(n) are the bivariate NURBS
basis functions of order p and q in n and g directions, respectively.
3. Governing equations and discretization
To make the paper self-contained, the governing equations and
the descretized equations for the free vibration and buckling prob-
lems of laminated composite plates are briefly given respectively
in this section.
3.1. Free vibration analysis
Consider a symmetrically laminated composite plate under
Cartesian coordinate system with the thickness h in the z-direction
and the fiber orientation h of a layer, as depicted in Fig. 2. The dis-
placements of the plate in the (x,y,z) directions are denoted by
(u,v,w). Based on the classical plate theory (CPT) [31,43,44], the
displacement fields can be assumed as follows:
u ¼ u v wf g
T
¼ Àz @
@x
Àz @
@y
1
n oT
w ¼ Tw ð12Þ
It should be noted that the in-plane displacements are disregarded
in the above equation and the assumed displacement field accounts
for the case of bending-twisting modes of symmetrically laminated
plates only. The pseudo-strains and pseudo-stresses of the plate are
given by,
ep ¼ À
@2
@x2
À
@2
@y2
À2
@2
@x@y
& 'T
w ¼ Lw ð13Þ
rp ¼ Mx My Mxyf g
T
ð14Þ
where Mx, My and Mxy are bending and twisting moments, respec-
tively. The relations between pseudo-strains and pseudo-stresses
can be denoted in the form of
rp ¼ Dep ð15Þ
Due to the assumption of classical plate theory (CPT) for a lami-
nated composite plate, D can be written as
D ¼
D11 D12 D16
D12 D22 D26
D16 D26 D66
2
6
4
3
7
5 ð16Þ
DIJ ¼
1
3
XN
k¼1
ðQIJÞk z3
k À z3
kÀ1
À Á
; I; J ¼ 1; 2; 6 ð17Þ
In the above equation, N is the number of layers of the composite
laminated plate and QIJ is implemented as the following
Q11 ¼ Q11 cos4
h þ 2ðQ12 þ 2Q66Þ sin
2
h cos2
h þ Q22 sin
4
h ð18Þ
Q12 ¼ ðQ11 þ Q22 À 4Q66Þ sin
2
h cos2
h þ Q12ðsin
4
h þ cos4
hÞ ð19Þ
Q16 ¼ ðQ11 À Q12 À 2Q66Þ sin h cos3
h
þ ðQ12 À Q22 þ 2Q66Þ sin
3
h cos h ð20Þ
Q22 ¼ Q11 sin
4
h þ 2ðQ12 þ 2Q66Þ sin
2
h cos2
h þ Q22 cos4
h ð21Þ
Q26 ¼ ðQ11 À Q12 À 2Q66Þ sin
3
h cos h
þ ðQ12 À Q22 þ 2Q66Þ sin h cos3
h ð22Þ
Q66 ¼ ðQ11 þ Q22 À 2Q12 À 2Q66Þ sin
2
h cos2
h
þ Q66ðsin
4
h þ cos4
hÞ ð23Þ
with
Q11 ¼
E1
1 À m12m21
; Q12 ¼
m12E2
1 À m12m21
;
Q22 ¼
E2
1 À m12m21
; Q66 ¼ G12; m21E1 ¼ m12E2 ð24Þ
In the above equations, h is the fiber orientation of each layer, E1
and E2 are the Young’s moduli parallel to and perpendicular to
the fibers orientation, G12 is the shear modulus, m12 and m21 are
the Poisson’s ratios.
The dynamic equations of free vibration of composite laminated
plates are obtained using the following Lagrangian equation [31,43]
Fig. 2. A schematic composite laminated plate.
S. Shojaee et al. / Composite Structures 94 (2012) 1677–1693 1679

4. d
dt
@H
@ _w
& '
À
@H
@w
& '
¼ 0 ð25Þ
where H = W À Pp defines the Lagrangian function in which W and
Pp are the kinetic energy and the potential energy, respectively. In
the above equation, the over-dot denotes the differentiation with
respect to time. The kinetic energy W of the plate is,
W ¼
1
2
Z
X
q _uT _udX ð26Þ
where q and X are the mass density of the material and plate
volume respectively. The potential energy Pp of the plate can be
written as,
Pp ¼
1
2
Z
X
eT
prp dX À
Z
Ct
uTtdC À
Z
X
uT
bdX ð27Þ
In Eq. (27), t and b are the prescribed boundary forces and the body
force vector, respectively. By substituting Eqs. (26) and (27) into Eq.
(25), the weak form of dynamic equations of the plate can be de-
rived as
d
dt
1
2
Z
X
q
@
@ _w
½ðT _wÞT
ðT _wÞŠdX þ
1
2
Z
X
@
@w
½ðLwÞT
DðLwÞŠdX
¼
Z
Ct
@
@w
ðTwÞTtdC þ
Z
X
@
@w
ðTwÞT
bdX ð28Þ
It should be noted that no external forces are required in the free
vibration problems, and the terms on the right-hand side of Eq.
(28) is thus omitted. By substituting the deflection w from Eq.
(10) into Eq. (28), and after some manipulations, the eigenvalue
equations of the composite laminated plates can be obtained as
ðK À x2
MÞQ ¼ 0 ð29Þ
where x is natural frequency, Q is the eigenvector of the form
fw1; w2; . . . ; wncp g consisting of deflections at all the control points
in the plate domain, M and K are the global mass and stiffness
matrices given by
KIJ ¼
Z
X
BT
I DBJ dX ð30Þ
MIJ ¼
Z
X
qeBT
I
eBJ dX ð31Þ
BI ¼ À/I;xx À/I;yy À2/I;xy
È ÉT
ð32Þ
eBI ¼ Àz
@/I
@x
Àz
@/I
@y
/I
'
ð33Þ
3.2. Buckling analysis
Consider a rectangular composite laminate subjected to in-
plane forces (Nx,Ny,Nxy) as depicted in Fig. 3. The in-plane forces
can be expressed as [43]
Nx ¼ ÀN0; Ny ¼ Àl1N0; Nxy ¼ Àl2N0 ð34Þ
where N0 is a constant and l1 and l2 can be functions of coordi-
nates. The total potential energy of composite laminates is thus de-
fined as,
P ¼ Pb þ Pi ð35Þ
where Pb and Pi represent the strain energy of bending and the
strain energy caused by in-plane forces, respectively. The strain en-
ergy of bending of the laminates is expressed as [43]
Pb ¼
1
2
Z
X
D11
@2
w
@x2
!2
þ 2D12
@2
w
@x2
@2
w
@y2
þ D22
@2
w
@y2
!2
2
4
þ4D66
@2
w
@x@y
!2
þ 4 D16
@2
w
@x2
þ D26
@2
w
@y2
!
@2
w
@x@y
3
5dX ð36Þ
and the strain energy caused by in-plane forces can be formulated
as,
Pi ¼
1
2
Z
X
Nx
@w
@x
2
þ Ny
@w
@y
2
þ 2Nxy
@w
@x
@w
@y
#
dX ð37Þ
By substituting the deflection field presented in Eq. (10) into Eqs.
(36) and (37), and applying the minimum total potential energy
principle [31,43,45,46] as
@P
@wi
¼ 0 ð38Þ
We obtain the discrete eigenvalue problem for composite laminated
plate as,
ðK À N0AÞQ ¼ 0 ð39Þ
where K is the global stiffness matrix which has the same form as
Eq. (30), and the matrix A is explicitly given by,
AIJ ¼
1
2
Z
X
2
@/I
@x
@/J
@x
þ 2l1
@/I
@y
@/J
@y
þ 2l2
@/I
@x
@/J
@y
þ
@/J
@x
@/I
@y
!
dX
ð40Þ
3.3. Essential boundary conditions
Due to the non-interpolating nature of NURBS basis functions,
similar to many meshfree shape functions, e.g. moving least square
approximation (MLS) [27], the properties of Kronecker Delta are
not satisfied. As a consequence, it induces a difficulty in imposing
the essential boundary conditions, which is not a straight forward
task as in the FEM. Therefore, several strategies have been devoted
[47,31] to overcome that drawback. In this study, we merely adopt
the Lagrange multiplier method [27,43,46,48] as it is a versatile
mean for imposing essential boundary conditions. In this way,
the weak form of the essential boundary conditions associated
with Lagrange multipliers is used to construct the discretized
boundary conditions as follows:
Z
Cu
dkdðw À wÞdC þ
Z
Cu
dkrðr À rÞdC ¼ 0 ð41Þ
where kd denotes the Lagrange multiplier related to the deflection,
kr is the Lagrange multiplier associated with the rotation to the
boundary, w represents the prescribed deflection, r is the rotation
to the boundary while r being the prescribed rotation to the
boundary.
To obtain the discretized equation from the Eq. (41), Lagrange
multipliers need to be interpolated on the essential boundaries.
In this study, the Lagrange shape functions are considered as the
interpolation space for the Lagrange multipliers and an image ofFig. 3. A rectangular plate subjected to in-plane forces.
1680 S. Shojaee et al. / Composite Structures 94 (2012) 1677–1693

5. Greville abscissas [47] in physical space are employed as the inter-
polation nodes on the essential boundaries. The Greville abscissas
are defined in parametric space as follows:
sp
i ¼
niþ1 þ Á Á Á þ niþp
p
; i ¼ 1; . . . ; n ð42Þ
where p is the NURBS order while n being the number of control
points.
Interpolating the Lagrange multipliers kd and kr using 1D La-
grange shape functions on the physical boundary points corre-
sponding to Greville abscissas, yields
kd ¼
Xnk
I¼1
NL
I kdI
ð43Þ
kr ¼
Xnk
I¼1
NL
I krI
ð44Þ
where NL
I is the 1D Lagrange basis function and nk is the number of
the essential boundary points applied for this interpolation. The
variations of Lagrange multipliers can be obtained as,
dkd ¼
Xnk
I¼1
NL
I dkdI
ð45Þ
dkr ¼
Xnk
I¼1
NL
I dkrI
ð46Þ
Substituting the deflection of the plate w from Eq. (10) into the
weak form (41) and considering that w ¼ 0 and r ¼ 0 for eigenvalue
analysis, leads to the following discretized equations
G2ncÂncp wncpÂ1 ¼ 0 ð47Þ
where nc is the number of constraint points on the boundaries,
w ¼ fw1; w2; . . . ; wncp gT
represents the deflections at all the control
points in the plate domain, and the matrix G is explicitly given
by, respectively, for the simply supported boundaries
GKI ¼
Z
Cu
NL
K /I NL
K /I;nn
h iT
dC ð48Þ
and for the clamped ones
GKI ¼
Z
Cu
NL
K /I NL
K /I;n
h iT
dC ð49Þ
where n is the unit normal to the boundary. Since G is a singular
matrix, the singular value decomposition [43,46,48] is used to
decompose G as
G2ncÂncp ¼ U2ncÂ2nc
SrkÂrk
0
0 0
!
2ncÂncp
VT
ncpÂncp
ð50Þ
In Eq. (50), U and V are orthogonal matrices, S is a diagonal matrix
which its elements are the singular values of the matrix G, rk de-
notes the rank of G which equals to the number of independent con-
straints. Using the orthogonal transformation technique [43,46,48],
V can be partitioned as
VT
¼ VncpÂrk
VncpÂðncpÀrkÞ
È ÉT
ð51Þ
Applying the coordinate transformation
Q ¼ VncpÂðncpÀrkÞ
eQ ð52Þ
Substituting the above equation in Eqs. (29) and (39), the eigen-
value equations for free vibration and buckling analysis in new
coordinate system can be written as
ðeK À x2fMÞ eQ ¼ 0 ð53Þ
ðeK À N0
eAÞ eQ ¼ 0 ð54Þ
where eK ¼ VT
ðncpÀrkÞÂncp
KVncpÂðncpÀrkÞ; fM ¼ VT
ðncpÀrkÞÂncp
MVncpÂðncpÀrkÞ and
eA ¼ VT
ðncpÀrkÞÂncp
AVncpÂðncpÀrkÞ are positive definite matrices which
their dimensions are reduced using the orthogonal transformation
technique.
4. Free vibration examples
In this section, the validity and the accuracy of the proposed ap-
proach is demonstrated by free vibration analysis of different
numerical examples with various boundary conditions. The results
obtained by the proposed isogeometric analysis are also compared
with other reference solutions available in the literature. For the
convenience of comparisons, the normalized natural frequency
parameters are used: b ¼ x2a4qh
D0
1=2
for isotropic rectangular
plates, whereas b ¼ x2a4qh
D0;1
1=2
for composite laminated rectangu-
lar plates, with D0 = Eh3
/12(1 À m2
) and D0;1 ¼ E1h
3
=12ð1 À m12m21Þ.
Unless otherwise stated, the following material properties and
geometrical parameters of square laminated plates are used: ratio
of elastic constants E1/E2 = 2.45 and G12/E2 = 0.48, Poisson’s ratio
m12 = 0.23, mass density q = 8000 kg/m3
, length a = b = 10 m and
thickness h = 0.06 m. For the isotropic square plates, the geometri-
cal parameters are the same as laminated plates but the material
properties: E = 2.45, m = 0.23 and q = 8000 kg/m3
are taken instead.
In examples, the boundaries of the plate are denoted as: simply
supported (S) and clamped (C). For example, notation CSCS means
the bottom and upper boundaries are clamped while the right and
left boundaries are simply supported, SSSS means fully simply sup-
ported and CCCC means fully clamped.
4.1. Thin isotropic and orthotropic square plates
To demonstrate the convergence of the proposed method, the
natural frequencies of single-layer thin isotropic and orthotropic
square plates with SSSS boundary conditions on all sides are stud-
ied. The exact solution natural frequencies for thin isotropic square
plate is obtained by the following equation [50]
xexact
mn ¼ p2
ðm2
þ n2
Þ; m; n ¼ 1; 2; 3; . . . ð55Þ
The exact solution of natural frequencies for thin orthotropic square
plate is obtained by the following equation [49]
xexact
mn ¼ p2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
D11
D0;1
m4 þ 2
D12
D0;1
þ 2
D66
D0;1
m2n2 þ
D22
D0;1
n4
s
ðm; n ¼ 1; 2; 3; . . .Þ ð56Þ
The percentage relative error is defined as follows
Relative errorð%Þ ¼
xnum
À xexact
xexact
 100 ð57Þ
Figs. 4 and 5, respectively, show the percentage relative errors
of the isogeometric analysis employing different order NURBS
basis functions and using linear and non-linear parameterization
for the fully simply supported thin isotropic square plates. The
results are obtained over a 40 Â 40 control mesh and the first
1000 frequencies are considered for comparison. Generally, it can
be observed from Figs. 4 and 5 that as the order of NURBS basis
functions increase, the discrepancies between the isogeometric
predictions and the exact solution decrease. The results of linear
parameterization are seen to be very good except at the end of
the spectrum where the relative error of quintic NURBS basis func-
tion suddenly increases. This problem first reported by Cottrell
et al. [52] in a rotation free isogeometric formulation and referred
as ‘‘outlier frequencies’’. They eliminated these outlier frequencies
S. Shojaee et al. / Composite Structures 94 (2012) 1677–1693 1681

6. using a non-linear parameterization. Our results of non-linear
parameterization, by using the Lagrange multiplier method and
the orthogonal transformation technique for imposing the essen-
tial boundary conditions, show that although the outlier frequen-
cies are eliminated by this parameterization, but the relative
errors of frequencies are larger than those obtained by linear
parameterization.
The percentage relative errors of the isogeometric analysis
using different order NURBS basis functions for SSSS supported
thin orthotropic square plates are also presented in Figs. 6 and 7
for a linear and non-linear parameterization, respectively. It can
be seen that the numerical results for orthotropic plate are almost
analogous to the ones obtained for the isotropic plate. Since in this
paper we focus on the lower frequencies, cubic NURBS basis func-
tions with linear parameterization are chosen by both computa-
tional efficiency and accuracy considerations for all the following
examples. It is worthwhile to note that the study of higher order
frequencies requires advanced and refined theories [53–55], while
the classical theories such as CPT and FSDT (first order shear defor-
mation theory) show some limitations for predicting these higher
frequencies [55]. Nevertheless, this subject is an independent work
which is beyond the scope of this research and is not discussed fur-
ther in this paper.
Table 1 lists the first six frequencies of thin orthotropic square
plate that are obtained employing cubic NURBS basis function with
linear parameterization on different meshes with 4, 8, 16, 32 and
64 elements in each direction. From this table, it is seen that there
are very good convergences of normalized natural frequencies ob-
tained by the present method in comparison with the analytical
solution [51]. For a better view, these results are also plotted in
Fig. 8.
4.2. Symmetrically laminated square plates
4.2.1. Convergence study of the frequency
Employing the cubic NURBS basis functions, we used different
densities of control points with linear parameterization to study
the convergence of the present method for a SSSS supported com-
posite laminate. An angle-ply (30°,À30°,30°) laminate is consid-
ered. Table 2 presents the normalized fundamental frequency for
different densities of control points which compared with the solu-
tions obtained by the element-free Galerkin (EFG) [48] and the
moving Kriging interpolation-based (MKI) meshfree methods
[31]. A good convergence of the frequency can be observed from
this table for the present method.
Fig. 4. Comparison of relative errors of isogeometric analysis using different order
NURBS basis functions for thin isotropic square plate (SSSS, linear
parameterization).
Fig. 5. Comparison of relative errors of isogeometric analysis using different order
NURBS basis functions for thin isotropic square plate (SSSS, non-linear
parameterization).
Fig. 6. Comparison of relative errors of isogeometric analysis using different order
NURBS basis functions for thin orthotropic square plate (SSSS, linear
parameterization).
Fig. 7. Comparison of relative errors of isogeometric analysis using different order
NURBS basis functions for thin orthotropic square plate (SSSS, non-linear
parameterization).
Table 1
Convergence of the normalized natural frequencies of the free vibration of a thin
orthotropic square plate using cubic order NURBS basis function (SSSS).
Number of elements
(number of control points)
Mode
1 2 3 4 5 6
4 Â 4 (7 Â 7) 15.17 33.39 44.64 60.97 67.58 92.94
8 Â 8 (11 Â 11) 15.17 33.25 44.40 60.69 64.54 90.20
16 Â 16 (19 Â 19) 15.17 33.25 44.38 60.68 64.45 90.13
32 Â 32 (35 Â 35) 15.17 33.25 44.38 60.68 64.45 90.13
64 Â 64 (67 Â 67) 15.17 33.25 44.38 60.68 64.45 90.13
Exact [51] 15.17 33.25 44.39 60.68 64.46 90.15
1682 S. Shojaee et al. / Composite Structures 94 (2012) 1677–1693

7. 4.2.2. Effect of layer numbers and boundary conditions
The effect of the layer number and the boundary conditions of
the composite laminates on the natural frequencies is also investi-
gated to demonstrate the validity of the present isogeometric code.
For this purpose, the normalized natural frequencies of three, and
five-ply square laminates with various orientations and different
boundary conditions are examined and their results are then com-
pared with the existing analytical and numerical solutions. For all
calculations, cubic order NURBS basis function over an 11 Â 11
control mesh is used. Tables 3 and 4, respectively, present the first
six modes of the normalized frequencies for square three-ply lam-
inate with SSSS and CCCC boundary conditions. Exact solutions
[56] and the results obtained by other numerical approaches such
as the EFG method by Chen et al. [48] and Dai et al. [57], the Ray-
leigh–Ritz method by Chow et al. [58], the Ritz method by Leissa
and Narita [51] and the MKI method by Bui et al. [31] are also pre-
sented for comparison. The tabulated frequencies computed by the
isogeometric analysis match well with references solutions. The
natural frequency for the five-ply laminates are compared in
Table 5, with those obtained by Chow et al. [58], Bui et al. [31]
and Leissa and Narita [51] showing an excellent agreement.
4.2.3. Effect of the aspect ratio
The effect of the aspect ratio is studied and the gained numerical
results are presented in Table 6 for a SSSS supported rectangular
five-layer laminate arranged as (30°,À30°,30°,À30°,30°) with dif-
ferent aspect ratios from 0.5 to 1.5, and compared with results of
the MKI method by Bui et al. [31]. For this problem, cubic NURBS ba-
sis function over an 11 Â 11 control mesh is employed. The results
Table 2
Convergence of the normalized fundamental frequency for a fully simply supported
laminated composite square plate.
Method Number of control points/nodes
9 Â 9 11 Â 11 13 Â 13 15 Â 15 17 Â 17
IGA 15.96 15.93 15.92 15.91 15.90
EFG [48] 20.02 16.05 15.88 15.89 15.86
MKI [31] 18.29 16.32 15.88 15.89 15.88
Fig. 8. Convergence study of the first six normalized eigenvalues xnum
xexact using
isogeometric analysis with cubic order NURBS basis function for thin orthotropic
square plate (SSSS, linear parameterization).
Table 3
The normalized natural frequencies of fully simply supported square three-ply laminate with various orientations.
Ply angle Method Mode
1 2 3 4 5 6
(0°,0°,0°) Present: IGA 15.17 33.25 44.39 60.69 64.54 90.20
Chen et al. [48] 15.18 33.34 44.51 60.79 64.80 90.39
Chow et al. [58] 15.19 33.31 44.52 60.78 64.55 90.31
Leissa and Narita [51] 15.19 33.30 44.42 60.77 64.53 90.29
Dai et al. [57]: CLPT 15.17 33.32 44.51 60.78 64.79 90.42
Dai et al. [57]: TSDT 15.22 33.76 44.79 61.11 66.76 91.69
Bui et al. [31] 15.06 33.30 44.36 59.52 65.95 89.53
Exact [56] 15.17 33.25 44.39 60.68 64.46 90.15
(15°,À15°,15°) Present: IGA 15.42 34.08 43.87 60.86 66.70 91.54
Chen et al. [48] 15.41 34.15 43.93 60.91 66.94 91.74
Chow et al. [58] 15.37 34.03 43.93 60.80 66.56 91.40
Leissa and Narita [51] 15.43 34.09 43.80 60.85 66.67 91.40
Dai et al. [57]: CLPT 15.40 34.12 43.96 60.91 66.92 91.76
Dai et al. [57]: TSDT 15.45 34.54 44.25 61.36 68.68 92.99
Bui et al. [31] 15.39 34.56 44.23 61.32 66.91 91.40
(30°,À30°,30°) Present: IGA 15.93 35.90 42.67 61.62 71.84 85.91
Chen et al. [48] 15.88 35.95 42.63 61.54 72.12 86.32
Chow et al. [58] 15.86 35.77 42.48 61.27 71.41 85.67
Leissa and Narita [51] 15.90 35.86 42.62 61.45 71.71 85.72
Dai et al. [57]: CLPT 15.87 35.92 42.70 61.53 71.10 86.31
Dai et al. [57]: TSDT 15.92 36.28 43.00 62.05 73.55 87.37
Bui et al. [31] 15.86 36.05 42.70 61.33 71.48 86.01
(45°,À45°,45°) Present: IGA 16.19 37.00 41.90 62.09 77.26 80.17
Chen et al. [48] 16.11 37.04 41.80 61.94 78.03 80.11
Chow et al. [58] 16.08 36.83 41.67 61.65 76.76 79.74
Leissa and Narita [51] 16.14 36.93 41.81 61.85 77.04 80.00
Dai et al. [57]: CLPT 16.10 37.00 41.89 61.93 77.99 80.11
Dai et al. [57]: TSDT 16.15 37.33 42.20 62.45 78.96 81.55
Bui et al. [31] 16.01 37.05 41.68 61.40 78.20 81.12
(0°,90°,0°) Present: IGA 15.17 33.73 44.03 60.69 65.85 90.97
Chen et al. [48] 15.18 33.82 44.14 60.79 66.12 91.16
Bui et al. [31] 15.18 33.49 44.52 61.39 66.96 91.73
S. Shojaee et al. / Composite Structures 94 (2012) 1677–1693 1683

8. show an acceptable agreement with the MKI approach. It can be seen
that higher aspect ratios leads to higher natural frequencies.
4.2.4. Verification of the mode shapes
In order to verify the mode shapes of composite laminates, a
square five-layer angle-ply (h,Àh,h,Àh,h) symmetric laminate with
different orientation angles is investigated. For the ply-orientation
angles of (0°,0°,0°,0°,0°) and (15°,À15°,15°,À15°,15°), SSSS and
for the ply-orientation angles of (30°,À30°,30°,À30°,30°) and
(45°,À45°,45°,À45°,45°), CCCC boundary conditions is enforced.
To compare our results with those obtained by Chow et al. [58],
the geometry and material parameters are chosen as length of
the laminate a = b = 10 m, thickness h = 0.06 m, mass density
q = 8000 kg/m3
, ratio of elastic constants E1/E2 = 15.4 and G12/
E2 = 0.79, Poisson’s ratios m12 = 0.3 and m21 = 0.0195. Again, cubic
order NURBS basis function with linear parameterization over an
11 Â 11 control mesh is applied. The first six dimensionless natural
frequencies are given in Table 7 and compared with those of Chow
et al. [58]. The corresponding mode shapes are also shown in Fig. 9.
These mode shapes fits well with eigenmodes reported in [58].
4.2.5. More verification examples
As another verification example, a square three-layer laminate
arranged as (0°,90°,0°) is discretized employing cubic NURBS basis
function over a 19 Â 19 control mesh. This problem has been also
investigated previously, e.g. the radial basis function (RBF) pseudo
Table 4
The normalized natural frequencies of fully clamped square three-ply laminate with various orientations.
Ply angle Method Mode
1 2 3 4 5 6
(0°,0°,0°) Present: IGA 29.09 50.80 67.30 85.65 87.30 118.63
Chen et al. [48] 29.27 51.21 67.94 86.25 87.97 119.3
Chow et al. [58] 29.13 50.82 67.29 85.67 87.14 118.6
Dai et al. [57]: CLPT 29.27 51.21 67.94 86.25 87.97 119.3
Dai et al. [57]: TSDT 30.02 54.68 70.41 89.36 92.58 123.6
(15°,À15°,15°) Present: IGA 28.90 51.41 65.94 84.57 89.88 119.37
Chen et al. [48] 29.07 51.82 66.54 85.17 90.56 120.0
Chow et al. [58] 28.92 51.43 65.92 84.55 89.76 119.3
Dai et al. [57]: CLPT 29.07 51.83 66.55 85.17 90.56 120.0
Dai et al. [57]: TSDT 29.85 55.25 69.14 88.53 94.92 124.3
(30°,À30°,30°) Present: IGA 28.52 53.14 62.71 83.88 95.37 114.46
Chen et al. [48] 28.69 53.57 63.24 84.43 96.13 115.4
Chow et al. [58] 28.55 53.15 62.71 83.83 95.21 114.1
Dai et al. [57]: CLPT 28.69 53.57 63.26 84.43 96.15 115.5
Dai et al. [57]: TSDT 29.51 56.84 66.17 87.83 100.5 118.9
(45°,À45°,45°) Present: IGA 28.34 54.64 60.46 83.73 102.23 105.88
Chen et al. [48] 28.50 55.11 60.91 84.25 103.2 106.7
Chow et al. [58] 28.38 54.65 60.45 83.65 102.0 105.6
Dai et al. [57]: CLPT 28.50 55.11 60.94 84.25 103.2 106.7
Dai et al. [57]: TSDT 29.34 58.19 64.14 87.67 107.4 110.6
(0°,90°,0°) Present: IGA 29.09 51.51 66.77 85.65 89.06 119.63
Chen et al. [48] 29.27 51.93 67.40 86.25 89.76 120.3
Table 5
The normalized natural frequencies of fully simply supported and fully clamped square five-ply laminate with various orientations.
Boundary condition (ply angle) Method Mode
1 2 3 4 5 6
SSSS:
(15°,À15°,15°,À15°,15°) Present: IGA 15.49 34.27 43.92 61.59 66.50 91.62
Chow et al. [58] 15.46 34.24 43.88 61.59 66.42 91.52
Leissa and Narita [51] 15.50 34.30 43.93 61.62 66.48 91.51
Bui et al. [31] 15.45 34.72 44.26 61.81 66.04 91.50
(30°,À30°,30°,À30°,30°) Present: IGA 16.11 36.64 42.63 63.51 71.65 85.99
Chow et al. [58] 15.98 36.58 42.53 63.37 71.43 85.86
Leissa and Narita [51] 16.10 36.64 42.62 63.45 71.60 85.88
Bui et al. [31] 15.99 36.70 42.63 62.69 71.26 85.02
(45°,À45°,45°,À45°,45°) Present: IGA 16.42 38.38 41.43 64.51 78.04 79.30
Chow et al. [58] 16.29 38.30 41.32 64.35 77.77 79.09
Leissa and Narita [51] 16.40 38.37 41.40 64.41 77.94 79.23
Bui et al. [31] 16.24 38.31 41.18 63.15 77.72 80.41
CCCC:
(15°,À15°,15°,À15°,15°) Present: IGA 28.97 51.63 66.02 85.57 89.52 120.54
Chow et al. [58] 29.00 51.65 66.01 85.55 89.40 120.5
(30°,À30°,30°,À30°,30°) Present: IGA 28.74 53.96 62.76 86.14 95.17 114.65
Chow et al. [58] 28.78 53.98 62.76 86.09 95.04 114.4
(45°,À45°,45°,À45°,45°) Present: IGA 28.62 56.32 59.94 86.53 103.20 105.07
Chow et al. [58] 28.68 56.34 59.94 86.48 103.0 104.9
1684 S. Shojaee et al. / Composite Structures 94 (2012) 1677–1693

9. spectral method, [59], the moving least squares-differential quad-
rature method [60], the p-Ritz approach [61], the MK meshfree
[61]. The laminate parameters for this example are E1/E2 = 40,
G12/E2 = 0.6, m12 = 0.25, m21 = 0.00625, h = 0.001 m and h/a = 0.001.
In this case, the natural frequency is normalized by b2 ¼
ffiffiffiffiffiffiffiffiffiffiffiffi
qhx2a4
p4D0;2
q
using another rigidity expression as D0;2 ¼ E2h
3
=ð12ð1 À m12m21ÞÞ.
Table 8 presents the first eight normalized natural frequencies
obtained by isogeometric analysis and other reference studies for
SSSS and CCCC laminates. Again, a good agreement between the
proposed isogeometric analysis and other numerical approaches
can be observed.
As an another example, we compare the results obtained by iso-
geometric analysis with some solutions based on refined theories for
free vibration of thin symmetric composite laminated plates with
Table 7
The normalized natural frequencies of fully simply supported and fully clamped square five-ply laminate with different orientations.
Ply angle Method Mode
1 2 3 4 5 6
SSSS:
(0°,0°,0°,0°,0°) Present: IGA 11.29 17.13 28.71 40.75 45.15 46.02
Chow et al. [58] 11.30 17.13 28.70 40.77 45.18 46.23
(15°,À15°,15°,À15°,15°) Present: IGA 11.91 19.89 33.22 39.76 47.72 51.86
Chow et al. [58] 11.82 19.76 32.93 39.53 47.42 52.73
CCCC:
(30°,À30°,30°,À30°,30°) Present: IGA 22.73 36.59 54.10 57.34 70.33 83.86
Chow et al. [58] 22.72 36.54 54.02 57.17 70.09 83.37
(45°,À45°,45°,À45°,45°) Present: IGA 22.40 41.70 48.42 65.27 78.12 84.49
Chow et al. [58] 22.40 41.64 48.32 65.09 77.76 84.06
Table 6
The normalized natural frequencies of a fully simply supported rectangular five-ply laminate for different length-to-length ratios a/b.
a/b Method Mode
1 2 3 4 5 6
1.5 Present: IGA 24.92 51.56 71.21 93.22 101.54 142.00
Bui et al. [31] 24.76 50.99 73.45 93.88 101.07 139.58
1.2 Present: IGA 19.25 45.08 49.57 75.96 89.75 98.31
Bui et al. [31] 19.13 45.02 49.89 75.39 91.10 101.56
1.0 Present: IGA 16.11 36.64 42.63 63.51 71.65 85.99
Bui et al. [31] 15.99 36.70 42.63 62.69 71.26 85.02
0.8 Present: IGA 13.50 26.94 39.65 48.43 54.35 75.21
Bui et al. [31] 13.23 26.16 39.07 47.48 51.34 69.95
0.5 Present: IGA 10.58 16.06 24.80 36.41 37.04 42.32
Bui et al. [31] 10.71 16.28 24.87 36.11 36.34 40.91
ModeB.C.
654321Angle
SSSS
o
0θ=
SSSS
o
15θ=
CCCC
o
30θ=
CCCC
o
45θ=
Fig. 9. The first six mode shapes of fully simply supported and fully clamped supported five-ply laminate with different orientations (h,Àh,h,Àh,h).
S. Shojaee et al. / Composite Structures 94 (2012) 1677–1693 1685

10. different lamination sequences. For this purpose, the non-dimen-
sionalized fundamental frequency parameter b3 ¼ x2a4q
E1h2
1=2
, with
the thickness ratio a/h = 100, of simply supported square plates with
regular symmetric angle-ply lamination scheme is considered. This
problem has been solved in [53] for an ED222 refined model with dif-
ferent methodologies such as Ritz (RM), Galerkin (GM) and General-
ized Galerkin (GGM). In the acronym ED222, E indicates that the
Equivalent Single Layer (ESL) theory have been used, D means that
the Principle of Virtual Displacements has been employed and the
subscript 222 shows the three different expansion orders used in
the displacements field. The material properties for this problem
are the same as [53] and the values of half-wave m, n waves are
assumed to be 12. The results of isogeometric analysis using cubic
NURBS and 19 Â 19 control points are presented in Table 9 and com-
pared with ED222 refined model with different methodologies. It is
seen that IGA result for these three lamination schemes is between
the Ritz-ED222 and the Generalized Galerkin-ED222 solutions, while
the results of Galerkin-ED222 model are higher than the others.
Next, the first six natural frequencies obtained by IGA for the
lamination sequence (30°,À30°,30°) are presented in Table 10
and also compared with ED444 and EDZ 444 refined models with dif-
ferent methodologies [53]. It should be noted that in EDZ 444 model
the Zig-Zag effects are accounted and the Z letter demonstrates this
issue. From Table 10, it can be observed that IGA results are in very
good agreement with ED444 and EDZ 444 models based on Ritz,
Galerkin and Generalized Galerkin methodologies.
4.3. Composite elliptical plate
In this example, the natural frequencies are calculated for ellip-
tical laminates. The radii of the elliptical laminates are a = 5 m and
b = 2.5 m, respectively. Other geometrical and material parameters
are set the same as square laminate in Section 4.2. The natural
frequency is normalized by b ¼ x2a4qh
D0;1
1=2
. Cubic order NURBS ba-
sis function with 196 control points is used for discretization of the
problem. In Fig. 10, control points and physical mesh are shown.
This problem has been investigated by Bui et al. [31] for three-
and five-ply composite plate with SSSS boundary condition, and
by Chen et al. [48] for three-ply composite plate with CCCC
boundary condition. The normalized first nine frequencies of the
three-ply laminated plate for SSSS and CCCC boundaries are pre-
sented in Tables 11 and 12 with various fiber orientations, respec-
tively. The results are also compared with reference solutions
[31,48] and an excellent agreement is again found as expected. In
addition, the first eight mode shapes of CCCC three-ply elliptical
composite plate with (45°,À45°,45°) are illustrated in Fig. 11.
The effect of the stiffness ratio E1/E2 on the natural frequencies
is investigated for a SSSS three-layer laminate arranged as
(45°,À45°,45°) which has been studied in [31]. Different stiffness
ratios E1/E2 ranging from 1 to 200 are considered and other param-
eters remain unchanged. The results of isogeometric analysis are
given in Table 13 and compared with results obtained by Bui
et al. [31] using MKI method. Generally, the results from both
methods are in good agreement except modes 3 and 5, where
the difference between two methods is slightly high. One reason
for these differences could be that by increasing, the eigen-modes
between two solutions become different and our comparison is
only between the first six natural frequencies provided by both
Table 8
The normalized natural frequencies of fully simply supported and fully clamped square three-ply laminate arranged as (0°, 90°,0°).
Ply angle (0°,90°,0°) Mode
1 2 3 4 5 6 7 8
SSSS:
Present: IGA 6.6254 9.4474 16.2067 25.1185 26.5020 26.6654 30.3183 37.7943
Ferreira and Fasshauer [59] 6.6180 9.4368 16.2192 25.1131 26.4938 26.6667 30.2983 37.7850
Lanhe et al. [60] 6.632 9.464 16.364 25.325 26.886 – – –
Liew [61] 6.6252 9.4470 16.2051 25.1146 26.4982 26.6572 30.3139 37.7854
Bui et al. [31] 6.6829 9.5017 16.745 24.8703 26.8623 26.7819 30.5408 37.6930
Exact [56] 6.6254 9.4473 16.2056 25.1181 26.5017 26.6585 30.3175 37.7892
CCCC:
Present: IGA 14.6680 17.6164 24.5165 35.5491 39.1734 40.7847 44.8037 50.3559
Ferreira and Fasshauer [59] 14.8138 17.6181 24.1145 36.0900 39.0170 40.8323 44.9457 49.0715
Lanhe et al. [60] 14.674 17.668 24.594 35.897 39.625 – – –
Liew [61] 14.6655 17.6138 24.5114 35.5318 39.1572 40.7685 44.7865 50.3226
Table 9
The normalized fundamental natural frequency b3 ¼ x2 a4q
E1h2
1=2
of a fully simply
supported square three-ply laminated plate with different lamination schemes.
Method Mode
(15°,À15°,15°) (30°,À30°,30°) (45°,À45°,45°)
Present: IGA 4.5024 4.6410 4.7110
Ritz-ED222 [53] 4.5007 4.6384 4.7081
Galerkin-ED222 [53] 4.5174 4.6834 4.7648
Generalized Galerkin-
ED222 [53]
4.5034 4.6458 4.7174
Table 10
The normalized natural frequencies of fully simply supported square three-ply laminated plate arranged as (30°,À30°,30°).
Ply angle (30°,À30°,30°) Mode
1 2 3 4 5 6
Present: IGA 4.6410 10.4654 12.4369 17.9310 20.9270 25.0135
Ritz-ED444 [53] 4.6382 10.4568 12.4235 17.9064 20.8942 24.9618
Ritz-EDZ 444 [53] 4.6382 10.4568 12.4235 17.9064 20.8942 24.9617
Galerkin-ED444 [53] 4.6832 10.2221 12.6339 17.5717 20.9957 25.0451
Galerkin-EDZ 444 [53] 4.6832 10.2221 12.6339 17.5717 20.9957 25.0451
Generalized Galerkin-ED444 [53] 4.6455 10.4640 12.4049 17.8887 20.8900 24.9146
Generalized Galerkin-EDZ 444 [53] 4.6455 10.4640 12.4049 17.8887 20.8900 24.9146
1686 S. Shojaee et al. / Composite Structures 94 (2012) 1677–1693

11. solutions but not considering the mode shapes. Tabulated fre-
quency parameters computed by the isogeometric analysis are also
shown in Fig. 12 for different stiffness ratios. It can be seen that by
increasing of stiffness ratio, the natural frequencies decrease,
although their reduction is very slow for higher stiffness ratios.
Next, we study the computational efficiency (CPU-time) of the
present IGA method for the free vibration analysis of a fully simply
supported three-layer laminate arranged as (0°,0°,0°). For serving
the comparison purpose, a solution based on the standard EFG
method is also performed for this problem in such a way. The
Fig. 10. An elliptical plate with 196 control points and 64 elements. (Left) Control mesh. (Right) Physical mesh.
Table 11
The normalized natural frequencies of fully simply supported elliptical three-ply laminate with various orientations.
Angle ply Method Mode
1 2 3 4 5 6 7 8 9
(0°,0°,0°) IGA 8.78 17.83 30.11 31.71 43.33 50.65 60.69 63.98 75.27
MKI [31] 8.97 18.11 31.77 32.12 44.14 50.85 – – –
(15°,À15°,15°) IGA 8.96 18.03 30.79 31.40 44.64 49.40 62.28 65.35 72.52
MKI [31] 9.12 18.26 31.92 32.30 44.92 49.90 – – –
(30°,À30°,30°) IGA 9.49 18.39 30.80 32.96 47.28 47.44 64.94 68.36 70.06
MKI [31] 9.61 18.55 31.35 34.12 47.06 47.61 – – –
(45°,À45°,45°) IGA 10.34 18.66 30.08 36.43 45.23 50.42 64.71 67.09 77.92
MKI [31] 10.41 18.76 30.57 37.36 45.04 49.96 – – –
(0°,90°,0°) IGA 8.95 17.89 30.78 31.62 43.92 50.38 61.18 65.46 74.83
MKI [31] 9.14 18.16 32.03 32.45 44.73 50.56 – – –
Table 12
The normalized natural frequencies of fully clamped elliptical three-ply laminate with various orientations.
Angle ply Method Mode
1 2 3 4 5 6 7 8 9
(0°,0°,0°) IGA 18.44 29.16 44.81 45.57 60.01 65.35 78.81 85.83 91.31
EFG [48] 18.48 29.38 44.97 45.72 60.44 65.33 79.24 85.31 91.50
(15°,À15°,15°) IGA 18.78 29.48 44.56 46.57 61.67 64.06 80.67 87.64 88.56
EFG [48] 18.83 29.70 44.73 46.72 62.06 64.07 81.09 87.14 88.90
(30°,À30°,30°) IGA 19.85 30.22 44.16 49.81 61.88 65.45 84.04 84.16 93.88
EFG [48] 19.89 30.44 44.34 49.95 61.94 65.77 84.63 84.81 93.36
(45°,À45°,45°) IGA 21.57 31.17 43.92 55.07 60.06 69.96 80.18 87.61 104.42
EFG [48] 21.60 31.38 44.11 55.17 60.19 70.21 81.64 88.25 103.7
(0°,90°,0°) IGA 18.77 29.36 44.83 46.58 60.91 65.14 79.55 87.82 90.87
EFG [48] 18.81 29.58 44.99 46.72 61.34 65.14 79.99 87.23 91.16
Fig. 11. The first eight mode shapes of fully clamped supported three-ply elliptical composite plate with (45°,À45°,45°).
S. Shojaee et al. / Composite Structures 94 (2012) 1677–1693 1687

12. programs are compiled on a personal computer with Intel Core 2
CPU-2.40 GHz and 4 GB of RAM. Three different densities of 121,
196 and 289 control points or nodes are considered. Basically,
the estimation here is to measure the CPU-time spent in construct-
ing the global stiffness matrix, imposing the essential boundary
conditions and solving the algebraic equations. The fundamental
frequency and the computational time for isogeometric analysis
using quadratic and cubic NURBS and the EFG employing quadratic
basis function are reported in Table 14. The computational time
versus number of control points or nodes are also plotted in
Fig. 13. It can be observed from the results that the quadratic
and cubic IGA are much more efficient than the quadratic EFG.
Obviously, there may have different reasons that make the EFG
method costs higher than the IGA significantly. The key point be-
hind this result may lie in the fact that an expensive price of gen-
erating the meshless shape functions is often required. From the
algorithm of the meshless method, it can also be seen that the
meshless shape functions vary from quadrature point to point gen-
erated during the implementation process, which totally differs
from the IGA whose basis functions are predefined before
simulation.
4.4. Composite square plate with a hole of complicated shape
A laminated composite square plate with a hole of complicated
shape is analyzed as the last numerical example of this section.
This example is chosen to demonstrate the applicability of the
present method for analysis of laminated plates with complicated
shapes. The geometry of the plate and its dimensions is shown in
Fig. 14 (left). The geometrical and material parameters are consid-
ered as follows: ratio of elastic constants E1/E2 = 2.45 and G12/E2 =
0.48, Poisson’s ratio m12 = 0.23, mass density q = 8000 kg/m3
and
thickness h = 0.06 m. The natural frequency is normalized by
b ¼ x2a4qh
D0;1
1=2
in which a = 10 m. As shown in Fig. 14 (right),
the plate is divided into 8 NURBS patches. For analysis of this
problem, the bending strip method proposed by Kiendl et al.
[62] is applied to maintain C1
continuity between patches. The
bending strip stiffness is chosen as Es = 102
. Fig. 15 shows two
different meshes used for analysis of this problem. As shown in
this figure, quadratic and cubic NURBS basis functions with 880
and 576 control points are applied for modeling this problem,
respectively. The normalized first six frequencies of the three-
ply laminated plate with SSSS boundary conditions are presented
in Table 15 for various fiber orientations. For the sake of simplic-
ity, the simply supported boundary conditions in this example are
imposed by simply fixing deflection DOFs of outer boundary
Table 13
The normalized natural frequencies of fully simply supported elliptical three-ply
laminate arranged as (45°,À45°,45°) with different stiffness ratios.
E1/E2 Method Mode
1 2 3 4 5 6
1 IGA 13.35 23.64 38.59 47.27 58.93 63.91
MKI [31] 13.52 23.54 38.54 49.29 58.03 63.96
2.45 IGA 10.34 18.66 30.08 36.43 45.23 50.42
MKI [31] 10.41 18.76 30.57 37.36 45.04 49.96
5 IGA 9.00 15.74 24.95 31.84 37.10 43.36
MKI [31] 9.00 16.46 27.07 31.66 39.05 43.26
10 IGA 8.24 13.67 21.25 29.40 31.30 38.75
MKI [31] 8.19 15.07 25.12 28.31 35.57 39.05
30 IGA 7.65 11.75 17.69 25.56 27.70 34.62
MKI [31] 7.58 13.92 23.64 25.61 32.71 35.14
50 IGA 7.52 11.28 16.78 24.08 27.31 32.81
MKI [31] 7.45 13.66 23.30 25.00 32.02 34.10
70 IGA 7.46 11.06 16.35 23.39 27.15 31.81
MKI [31] 7.39 13.54 23.15 24.72 31.70 33.61
90 IGA 7.42 10.93 16.11 22.98 27.05 31.22
MKI [31] 7.35 13.47 23.07 24.56 31.51 33.33
200 IGA 7.35 10.67 15.61 22.17 26.86 30.03
MKI [31] 7.24 13.25 22.77 24.01 30.82 32.28
Fig. 12. Convergence of the normalized natural frequencies with the stiffness ratio
E1/E2 for a fully simply supported elliptical three-ply laminate arranged as
(45°,À45°,45°).
Table 14
Comparison of the computational time and the normalized fundamental frequency of
fully simply supported (SSSS) elliptical three-ply laminate arranged among IGA and
EFG methods.
Method Number of control
points/nodes
Fundamental
frequency
Computational
time (s)
IGA-Quadratic 121 8.866 2.353
IGA-Cubic 8.786 3.042
EFG 8.940 3.464
IGA-Quadratic 196 8.829 4.093
IGA-Cubic 8.781 5.486
EFG 8.924 6.862
IGA-Quadratic 289 8.812 6.588
IGA-Cubic 8.780 9.000
EFG 8.809 15.690
Fig. 13. Comparison of the computational time between the IGA and the EFG.
1688 S. Shojaee et al. / Composite Structures 94 (2012) 1677–1693

13. control points. In Table 15, isogeometric results employing
quadratic NURBS functions is denoted by IGA-Q and the results
obtained by cubic NURBS functions is denoted by IGA-C. The iso-
geometric solutions are also compared with results obtained by
Fig. 14. A laminated composite square plate with a hole of complicated shape: model geometry (left) and patches discretization (right).
Fig. 15. Control mesh and physical mesh of the plate with a hole of complicated shape. (Left) Quadratic NURBS basis functions with 880 control points and 640 elements.
(Right) Cubic NURBS basis functions with 576 control points and 192 elements. Blue dash lines mark the element boundaries and red circles are control points. (For
interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Table 15
The normalized natural frequencies of fully simply supported three-ply laminate with a complicated shaped hole for various orientations.
Angle ply Method Mode
1 2 3 4 5 6
(0°,0°,0°) IGA-Q 18.288 31.110 35.735 55.499 62.464 82.033
IGA-C 18.194 30.932 35.678 55.383 62.164 81.795
EFG [31] 18.226 31.127 36.237 56.874 62.390 83.565
MKI [31] 18.169 30.303 36.581 57.429 64.145 85.656
(15°,À15°,15°) IGA-Q 19.020 32.260 36.078 56.480 63.727 83.830
IGA-C 18.912 32.045 36.004 56.345 63.370 83.629
EFG [31] 19.177 32.445 37.238 58.716 63.994 86.500
MKI [31] 18.323 31.472 37.617 63.077 66.538 86.486
(30°,À30°,30°) IGA-Q 20.448 34.184 37.179 58.660 66.310 88.107
IGA-C 20.316 33.933 37.074 58.484 65.895 87.973
EFG [31] 20.926 34.915 39.101 62.222 67.054 92.715
MKI [31] 20.310 33.987 39.898 58.111 69.699 92.099
(45°,À45°,45°) IGA-Q 21.128 35.122 37.692 59.545 67.948 91.196
IGA-C 20.982 34.848 37.559 59.325 67.518 91.220
EFG [31] 21.736 36.079 39.975 63.897 68.525 96.767
MKI [31] 20.987 34.897 39.269 63.375 69.017 96.588
(0°,90°,0°) IGA-Q 18.284 31.267 35.713 55.567 62.892 82.631
IGA-C 18.190 31.087 35.655 55.452 62.582 82.383
EFG [31] 18.278 32.264 36.134 57.151 65.853 90.678
MKI [31] 18.027 32.506 37.268 57.698 70.768 92.998
S. Shojaee et al. / Composite Structures 94 (2012) 1677–1693 1689

14. EFG and MKI methods [31]. It clearly observed that IGA results
are in good agreement with solutions computed using EFG and
MKI methods for all considered orientations. Fig. 16 shows the
first nine mode shapes of SSSS three-ply complicated shaped
composite plate with (30°,À30°,30°). These results have been
obtained by cubic NURBS basis functions with 576 control
points.
5. Buckling examples
A number of numerical examples showing the performance of
the proposed isogeometric approach in buckling analysis of isotro-
pic plates and composite laminates are presented in this section. In
all following numerical examples cubic order NURBS basis function
with linear parameterization is employed.
5.1. Thin isotropic rectangular plates
The material properties and geometrical parameters for
isotropic rectangular plates are assumed as: Young’s modulus
E = 200 Â 109
N/m2
, Poisson’s ratio m = 0.3, width b = 1 m, thickness
h = 0.01 m. The uniaxial in-plane compression load is applied in
x-direction and the normalized buckling load factor is defined as
b ¼ N0b2
p2D0
.
Firstly, the convergence of the uniaxial buckling load of an iso-
tropic square plate with various boundaries is investigated. The
results obtained by the isogeometric analysis for different densities
of control points are presented in Table 16 and also compared with
the analytical solutions [63]. The percentage relative errors of pres-
ent method with respect to analytical solution are also given in
parentheses. It can be seen that the achieved buckling loads
Fig. 16. The first nine mode shapes of SSSS three-ply complicated shaped composite plate with (30°,À30°,30°).
Table 16
Convergence of uniaxial buckling load of an isotropic square plate.
Boundary condition Number of control points Exact [63]
7 Â 7 9 Â 9 11 Â 11 13 Â 13 15 Â 15 17 Â 17
SSSS 4.001 4.000 4.000 4.000 4.000 4.000 4.000
(0.025) (0.000) (0.000) (0.000) (0.000) (0.000)
CCCC 10.255 10.107 10.084 10.078 10.076 10.075 10.070
(1.837) (0.367) (0.139) (0.079) (0.060) (0.050)
SCSC 6.811 6.754 6.746 6.744 6.744 6.744 6.740
(1.053) (0.208) (0.089) (0.059) (0.059) (0.059)
CSCS 7.772 7.707 7.696 7.693 7.692 7.692 7.690
(1.066) (0.221) (0.078) (0.039) (0.026) (0.026)
1690 S. Shojaee et al. / Composite Structures 94 (2012) 1677–1693

15. converges very well to the exact solution for all considered bound-
ary conditions. The convergence of the normalized buckling load
Nnum=Nexact for different boundary conditions is also shown in
Fig. 17. Since the results of present method using 11 Â 11 control
points are sufficiently accurate and also computationally efficient,
this control mesh is employed in following computations.
The critical uniaxial buckling load of the isotropic square plate
with different boundaries is listed in Table 17 to make a compari-
son with other numerical approaches such as the FEM and the
boundary element method (BEM) [64], the spline finite strip meth-
od (SFSM) [65], the differential quadrature element method
(DQEM) [66], the radial point interpolation method (RPIM) [67]
and the MKI method [68]. From this table, it is observed that the
present IGA method, the MKI method and the SFSM approach pre-
dict critical buckling load more accurate than the other methods
for all the considered boundaries. Additionally, in Fig. 18 the first
eight buckling modes of the fully simply supported (SSSS) square
plate subjected to uniaxial compression are illustrated. The effect
of the aspect ratio on the uniaxial buckling load factor is also stud-
ied in Table 18 for a fully clamped rectangular plate with different
aspect ratios a/b from 0.75 to 4 and compared with the analytical
solutions [63]. As expected, higher aspect ratios lead to lower
buckling load factors. Generally, the results of the buckling load
factors calculated by the present method are in good agreement
with exact solution, however, the percentage relative errors
increase slightly by increasing of aspect ratios.
5.2. Buckling analysis of symmetrically laminated plates
In this section, the effect of different boundary conditions on
the buckling load factor of a three-layer symmetrically laminated
composite plate is investigated. In order to compare our results
with those obtained by Liu et al. [46] and Liu [43] using the EFG
method, the material properties and geometrical parameters of
Fig. 17. Convergence of normalized buckling load Nnum=Nexact of an isotropic square
plate.
Table 17
Comparison of the critical uniaxial buckling load of an isotropic square plate with various boundaries among different approaches.
Boundary condition Method
Exact [63] RPIM [67] DQEM [66] MKI [68] FEM [64] BEM [64] SFSM [65] IGA
SSSS 4.000 4.017 3.9977 4.005 4.011 4.041 4.00 4.000
CCCC 10.070 10.308 10.052 10.108 10.392 10.387 10.08 10.084
SCSC 6.740 – – 6.780 6.882 6.972 – 6.746
CSCS 7.690 – – 7.702 7.796 7.757 7.70 7.696
Fig. 18. The first eight buckling modes of fully simply supported square plate subjected to uniaxial compression.
Table 18
The uniaxial buckling load factor of fully clamped rectangular plates with different
length-to-width ratios a/b subjected to uniaxial compression.
Method a/b
0.75 1 1.5 2 2.5 3 3.5 4
Present: IGA 11.68 10.08 8.37 7.92 7.63 7.47 7.45 7.39
Exact [63] 11.69 10.07 8.33 7.88 7.57 7.37 7.27 7.23
Error (%) 0.09 0.10 0.48 0.51 0.79 1.36 2.48 2.21
S. Shojaee et al. / Composite Structures 94 (2012) 1677–1693 1691

16. the laminates are set as follows: ratio of elastic constants E1/E2 =
2.45 and G12/E2 = 0.48, Poisson’s ratio m12 = 0.23, mass density q =
8000 kg/m3
, length a = b = 10 m and thickness h = 0.06 m. The uni-
axial in-plane compression load is applied in x-direction and the
normalized buckling load factor is defined as b ¼ N0b2
p2D0;1
.
An exact solution for the buckling of SSSS cross-ply plates,
which orientated to become specially orthotropic, subjected to
either uniaxial or biaxial loading was reported in [63] that is calcu-
lated by the following equation
Nexact
0 ¼
p2 D11
D0;1
m2
þ 2 D12
D0;1
þ 2 D66
D0;1
a
b
À Á2
n2
þ a
b
À Á4 D22
D0;1
n4
m2
h i
1 þ
Ny
Nx
a
b
À Á2 n
m
À Á2
ðm; n ¼ 1; 2; 3; . . .Þ ð58Þ
Table 19 presents the buckling load factors for this problem ob-
tained by isogeometric analysis employing cubic NURBS basis
function over an 11 Â 11 control mesh. The results are also com-
pared with those obtained by the above analytical solution and
the EFG method [43,46]. Again, a good agreement between the pro-
posed approach and other methods is observed.
6. Conclusions
We have developed the NURBS-based isogeometric finite ele-
ment method for analysis of natural frequencies and buckling phe-
nomena for the thin laminated composite plates. Several laminated
composite plates with different geometrical configurations and
boundary conditions are examined for such eigenvalue problems
considering various aspect ratios. The convergence of the natural
frequencies versus the nodal distributions is also analyzed. The
presented numerical results show that the proposed approach
can yield accurate solutions of the eigenvalue problems compared
to other existing methods available in the literature. The computa-
tional efficiency has also analyzed and the gained comparison
shows very much efficiency to the IGA compared with the EFG.
Due to many advantages of the IGA approach, e.g. in modeling ex-
act geometries by the aid of the NURBS method, the proposed
method is definitely efficient, robust and accurate. It is potential
and applicable to practical problems of engineering with inte-
grated advanced materials.
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Table 19
The uniaxial buckling load factor of square three-ply laminates with different
boundary conditions and various orientations subjected to uniaxial compression.
Boundary condition
(ply angle)
Method Boundary condition
SSSS CCCC CSCS SCCS
(0°,0°,0°) Present: IGA 2.36 6.71 4.27 3.93
EFG [46,43] 2.39 6.78 4.34 3.97
Exact [69] 2.36 – – –
(10°,À10°,10°) Present: IGA 2.40 6.64 4.33 3.95
EFG [46,43] 2.42 6.72 4.39 3.97
(15°,À15°,15°) Present: IGA 2.43 6.57 4.41 3.96
EFG [46,43] 2.45 6.64 4.46 3.96
(20°,À20°,20°) Present: IGA 2.48 6.48 4.51 3.97
EFG [46,43] 2.49 6.55 4.56 3.96
(30°,À30°,30°) Present: IGA 2.59 6.29 4.79 4.00
EFG [46,43] 2.57 6.36 4.84 3.96
(40°,À40°,40°) Present: IGA 2.65 6.15 4.90 4.01
EFG [46,43] 2.63 6.21 4.91 3.94
(45°,À45°,45°) Present: IGA 2.66 6.10 4.78 4.01
EFG [46,43] 2.64 6.16 4.79 3.93
(0°,90°,0°) Present: IGA 2.36 6.70 4.36 3.93
EFG [46,43] 2.39 6.78 4.43 3.97
Exact [69] 2.36 – – –
1692 S. Shojaee et al. / Composite Structures 94 (2012) 1677–1693

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