Nguyen-Xuane,n a Department of Architectural Engineering, Sejong University, 98 Kunja Dong, Kwangjin Ku, Seoul 143-747, South Korea b Division of Computational Mechanics, Ton Duc Thang U
Trang 1Isogeometric analysis of laminated composite plates based
Loc V Trana, Chien H Thaib, Hien T Lec, Buntara S Gand, Jaehong Leea,
H Nguyen-Xuane,n
a
Department of Architectural Engineering, Sejong University, 98 Kunja Dong, Kwangjin Ku, Seoul 143-747, South Korea
b
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
a r t i c l e i n f o
Article history:
Received 15 October 2013
Received in revised form
19 May 2014
Accepted 30 May 2014
Keywords:
Plate
Composite
Isogeometric analysis
Refined plate theory
Meshfree method
a b s t r a c t
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 C1requirement 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
1 Introduction
Laminated composite plates are being increasingly used in
variousfields 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
freq-uency 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
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
Hence, many equivalent single layer (ESL) plate theories with
the 3D problem to a 2D one Among the ESL plate theories, the
shear deformation, CLPT merely provides acceptable results for the
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 approxima-tion, have then been devised such as third-order shear deforma-tion theory (TSDT)[8–10], trigonometric shear deformation theory
[11,12], exponential shear deformation theory (ESDT) [13–15],
Contents lists available atScienceDirect
Engineering Analysis with Boundary Elements
http://dx.doi.org/10.1016/j.enganabound.2014.05.013
0955-7997/& 2014 Elsevier Ltd All rights reserved.
n Corresponding author.
E-mail address: hung.nx@vgu.edu.vn (H Nguyen-Xuan).
Trang 2out 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
ones without requiring the SCF However, the HSDT requires the
second-order derivative of the stiffness formulation The
standardfinite element method is not a trivial task In the efforts
to address this difficulty, several C0continuous elements[23–26]
were then proposed or Hermite interpolation function with the
unknown variables leading to an increase in the
ele-ments 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
Compu-ter Aided Design (CAD) software to describe the geometry domain
degree elevation and gain easily the smoothness of arbitrary
continuous order Also, NURBS can be used to approximate
mesh-free shape functions with a desired order of consistency[28]or to
merge into boundary element method to obtain the geometry and
geometry and approximations via NURBS, Hughes and co-workers
have introduced a new method so-called Isogeometric Analysis
functions for both describing the exact geometry and constructing
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
numerical examples are given to show the performance of the
proposed method in comparison to others in the literature
for composite plates InSection 3, the formulation of plate theory
based on IGA is described The numerical results and discussions
are provided inSection 4 Finally, this article is closed with some
concluding remarks
Regarding the effect of shear deformation, the higher-order
uðx; y; zÞ ¼ u0þzβxþgðzÞðβxþw;xÞ
vðx; y; zÞ ¼ v0þzβyþgðzÞðβyþw;yÞ;
wðx; yÞ ¼ w0
h
2 rzrh
ð1Þ
where gðzÞ ¼ ð4z3=3h2Þ and the variables u0¼ fu0v0gT, w0 and
β ¼ fβxβygT are the membrane displacements, the transverse
dis-placement and the rotations in the y–z, x–z planes, respectively By
reduced variable
form with four unknown variables uðx; y; zÞ ¼ u0zwb ;xþgðzÞws ;x
vðx; y; zÞ ¼ v0zwb ;yþgðzÞws;y
The relationships between strains and displacements are described by
where
ε0¼
u0;x
v0;y
u0 ;yþv0 ;x
2 6
3 7 5; κb¼
wb;xx
wb;yy 2wb;xy
2 6
3 7 5; κs¼
ws ;xx
ws ;yy
2ws ;xy
2 6
3 7 5; εs¼ ws;x
ws ;y
ð6Þ
traction-free boundary condition at the top and bottom surfaces
condition, various distributed functions f ðzÞ in forms: third-order
inTable 1
2.2 Weak form equations for plate problems
A weak form of the static model for the plates under transverse loading f0can be briefly expressed as
Z
ΩδεTDbεdΩþZ
ΩδγTDsγdΩ ¼Z
where
Db¼
2 6
3
and the material matrices are given as
Aij; Bij; Dij; Eij; Fij; Hij
h=2ð1; z; z2; gðzÞ; zgðzÞ; g2ðzÞÞQijdz ði; j ¼ 1; 2; 6Þ
Dsij¼Z h=2
h=2½f0ðzÞ2Qijdz ði; j ¼ 4; 5Þ ð9Þ
(see[4]for more detail)
Table 1 The various forms of shape function.
Reddy [8] z 4 z 3 =h 2 4 z 3 =h 2
1 4z 2 =h 2 Shimpi [17] 5 z 5 z 3 =h 2 1 z 5 z 3 =h 2 5 ð14z 2 =h 2 Þ Karama [13] ze 2ðz=hÞ 2
ze 2ðz=hÞ2z 1 4
2 z 2
e 2ðz=hÞ2
h z
sin π
h z
h cos π
h z
Trang 3
For the free vibration analysis, it can be derived from the
following dynamic equation:
Z
ΩδεTDbεdΩþZ
ΩδγTDsγdΩ ¼Z
Ωδ ~uT
where m - the mass matrix is calculated according to the
consistent form
m ¼
2
6
3
7
5 where I0¼
I1 I2 I4
I2 I3 I5
I4 I5 I6
2 6
3
ðI1; I2; I3; I4; I5; I6Þ ¼Zh =2
h=2ρðzÞð1; z; z2; gðzÞ; zgðzÞ; g2ðzÞÞdz ð12Þ and
~u ¼
u1
u2
u3
8
>
>
9
>
>; u1¼
u0
wb ;x
ws;x
8
>
>
9
>
>; u2¼
v0
wb ;y
ws;y
8
>
>
9
>
>; u3¼
w 0 0
8
>
>
9
>
>
ð13Þ and a weak form of the plate under the in-plane forces can be
formed as
Z
ΩδεTDb
εdΩþ
Z
ΩδγTDs
γdΩþ Z
Ω∇T
where∇T¼ ½∂=∂x ∂=∂yT
is the gradient operator and
0
x N0
xy
N0xy N0
2
4
3
5
is a matrix related to the pre-buckling loads
3 The composite plate formulation based on NURBS
basis functions
3.1 A brief of NURBS functions
A knot vectorΞ ¼ fξ1; ξ2; :::; ξn þ p þ 1g is defined as a sequence of
knot valueξiAR, i ¼ 1; :::nþp An open knot, i.e, the first and the
last knots are repeated pþ 1 times, is used A B-spline basis
function is C1continuous inside a knot span and Cp 1continuous
satisfies C1
-requirement in based-RPT formulations
The B-spline basis functions Ni ;pðξÞ are defined by the following recursion formula:
Ni;pð Þ ¼ξ ξ ξ i
ξ i þ p ξ iNi;p 1ð Þþξ ξi þ p þ 1 ξ
ξ i þ p þ 1 ξ i þ 1Ni þ 1;p 1ð Þξ
as p ¼ 0; Ni;0ð Þ ¼ξ 1 if ξirξoξi þ 1
ð15Þ
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 1g and Η ¼ fη1; η2; :::; ηm þ q þ 1g, the two-dimensional B-spline basis functions are obtained
Fig 1 illustrates the set of one-dimensional and two-dimensional B-spline basis functions according to open uniform knot vectorΞ 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
B-splines (NURBS) functions which are expressed as
RAðξ; ηÞ ¼ NAζA
∑
mn
A
NAðξ; ηÞζA
ð17Þ
The NURBS function becomes the B-spline function when the individual weight of control point is constant
3.2 A novel RPT formulation based on NURBS approximation
the plate is approximated as
uhðξ; ηÞ ¼ ∑mn
A
where qA¼ ½u0Av0AwbAwsAT
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
½εT
0κT
bκTεTT¼ ∑mn
A ¼ 1
½ðBm
AÞTðBb1
AÞTðBb2
A ÞTðBs
AÞTTqA ð19Þ
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Trang 4in which
BmA¼
RA ;y RA ;x 0 0
2
6
3 7 5; Bb1
A ¼
2 6
3 7 5;
Bb2A ¼
2
6
3 7 5; Bs
A¼ 0 0 0 RA;x
ð20Þ
Substituting Eq.(19)into Eqs.(7), (10) and (14), the
formula-tions of static, free vibration and buckling problem are formulated
by the following form:
where the global stiffness matrix K is given by
K ¼
Z
Ω
Bm
Bb1
Bb2
8
>
>
9
>
>
T
2
6
3
m
Bb1
Bb2
8
>
>
9
>
>dΩþ
Z
ΩBsTDsBsdΩ ð24Þ and the load vector is computed by
F ¼
Z
where
the global mass matrix M is expressed as
M ¼
Z
Ω
~RT
where
~R ¼
R1
R2
R3
8
>
>
9
>
>; R1¼
2 6
3 7 5;
R2¼
2
6
3 7 5; R3¼
2 6
3
and the geometric stiffness matrix reads
Kg¼Z
where
BgA¼ 00 00 RRA;x RA;x
A ;y RA ;y
ð30Þ
in which ω; λcrARþ
are the natural frequency and the critical buckling value, respectively
It is observed from Eq.(24)that the SCF is no longer required in
the stiffness formulation Herein, Bb1
second-order derivative of the shape functions Hence, it requires
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
Simply supported cross-ply:
v0¼ wb¼ ws¼ 0 at left and right edges
Simply supported angle-ply:
u0¼ wb¼ ws¼ 0 at left and right edges
Clamped:
4 Results and discussions
In this section, we show the performance of the present
Table 1in 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:
E1¼ 25E2; G12¼ G13¼ 0:5E2; G23¼ 0:2E2; ν12¼ 0:25; ρ ¼ 1:
Material II:
E1=E2¼ varied; G12¼ G13¼ 0:6E2; G23¼ 0:5E2; ν12¼ 0:25; ρ ¼ 1: Material III:
Face sheets
E1¼ 131 GPa; E2¼ 1:5 GPa;
G12¼ 6:895 GPa; G13¼ 6:205 GPa;
ν12¼ 0:22; ρ ¼ 1627 kg=m3
Core property (Isotropic)
E1¼ E2¼ 6:89 MPa;
G12¼ G13¼ 3:45 MPa;
ν12¼ 0; ρ ¼ 97 kg=m3
The ratio of the core thickness hcto the face sheet thickness hf
is equal to 4
For convenience, the following normalized transverse displace-ment, in-plane stresses, shear stresses natural frequency and bucking load are expressed as
2
wE2h3
q0a4 ; σ ¼ σh
2
q0a2; τ ¼qτh
0a; ω ¼ ωa2=h ffiffiffiffiffiffiffiffiffiffiρ=E2
p
; λcr¼ λcra2=E2h3
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 inFig 2 Wefirst investigate the convergence of the normalization displacement and stresses with length to thickness ratio a/h¼ 10 The plate is modeled with 7 7,
various distributed functions f(z) (as listed inTable 1) are tabulated
inTable 2 The relative error compared to analytical solution using
Trang 5HSDT given by Khdeir and Reddy[42]is given in the parentheses.
It is observed that the RPT models using third-order polynomials
mesh of 11 11 cubic elements is enough to provide reasonably
accurate solutions For illustration, this mesh is therefore used for
several following examples
Next we investigate the aforementioned problem with
lists the present results in comparison with the 3D solution
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
axial stress compared to the results derived from the Reddy
the accuracy of stresses
InTable 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
the obtained results agree very well with the 3D elastic solution
distributed functions produces same results which match well with the 3D solution The transverse displacement of the plates is
condi-tions, 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
considered For illustration, third-order distributed function
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 7; (b) 11 11; (c) 15 15 (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Table 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.
Trang 6displacement and stresses obtained are acceptable to the
analy-tical solution[26] It is indicated that the results derived from both
TSDT and RPT models are almost identical while FSDT model (with
stresses, especially for transverse shear stress This conclusion is
Table 3
The normalized deflection and axial stress of a laminated [0/90] plate with a/h ratios.
Table 4
The non-dimensional deflection and stresses of a two-layer laminated [0/90] composite square plate subjected to a sinusoidal load.
σ x a; b ; h
σ y ; b ; h
σ xy 0; 0; h
σ yza; 0; 0
Fig 4 Deflection of two-layer [0/90] antisymmetric square plates: (a) SFSF and (b) SSSS.
Trang 7−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8
−0.05
−0.04
−0.03
−0.02
−0.01
0 0.01
0.02
0.03
0.04
0.05
TSDT HSDT SSDT ESDT
−0.05
−0.04
−0.03
−0.02
−0.01 0 0.01 0.02 0.03 0.04
0.05
TSDT HSDT SSDT ESDT
−0.05
−0.04
−0.03
−0.02
−0.01 0 0.01 0.02 0.03 0.04
0.05
TSDT HSDT SSDT ESDT
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/h¼10.
σ x a; b ; h
σ y ; b ; h
τ xy 0; 0; h
τ yz a; 0; h 3
−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
RPT TSDT
−0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08
−0.5
−0.4
−0.3
−0.2
−0.1 0 0.1 0.2 0.3 0.4
0.5
RPT TSDT
−0.5
−0.4
−0.3
−0.2
−0.1 0 0.1 0.2 0.3 0.4 0.5
RPT TSDT
Fig 6 The stresses through the thickness of a sandwich plate under full simply supported condition with a/h¼ 10, h c /h f ¼4 via various plate models (a) axial stress σ xa; b ; z (b) Shear stress τ ð0; 0; zÞ (c) Shear stress τ a ; 0; z.
Trang 8between two models TSDT and RPT is again observed when ratio
hc/hfvaries from 4 to 200
4.2 Free vibration analysis
4.2.1 The cross-ply laminated [0/90]Nsquare composite plate
composite plate with a/h ¼ 5 under simply supported boundary
conditions Herein, material set II is used The effects of the
E2r 10 As E1/E2ratio ranges from 20 to 40, the present results are asymptotic to analytical solutions for 2D plate model using
Next, with constant E1/E2ratio (¼ 40), the variation of natural frequency of a two-layer laminated composite plate via length to
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
(a/h ¼10) is then plotted in Fig 7 It is clear that beside the full
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
can be seen that, present model using RPT gains the closest results
to analytical solution using TSDT with slightly higher results In
Table 6
The non-dimensional deflection of a sandwich plate [0/90/core/0/90] via h c /h f ratio.
Table 7
The natural frequency of a simply supported [0/90] N composite plate.
Trang 9addition, 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 inFig 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 inFig 9 Material
Table 8
The natural frequency of simply supported laminated [0/90] composite plate with E 1 /E 2 ¼40.
ω =
−0.9
−0.8
−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
x
: u0
ω =
−0.6
−0.4
−0.2 0 0.2 0.4 0.6
x
: v0 : wb
ω =
−0.5
−0.4
−0.3
−0.2
−0.1 0 0.1 0.2 0.3 0.4 0.5
x
: ws
Fig 7 Vibration mode shapes: full plate (upper) and line y¼a/2 (lower) of simply supported laminated [0/90] composite plate with E 1 /E 2 ¼40, a/h¼10.
Table 9
The natural frequency of ten-layer cross-ply [0/90] 5 plate with a/h ¼5 and E 1 /E 2 ¼40.
Trang 10III is used The plate is clamped at the outer boundary For illustration,
we use the distributed function f ðzÞ ¼ z ð4=3h2Þz3 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
of the natural frequency _ω ¼ ωR2
=hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðρ=E2Þf ace
on the radius to thickness ratio R/h A good agreement is observed for RPT and
TSDT, while FSDT model remains too stiffened as plate becomes
inFig 10
along y-direction equals to zero at y¼ 0 With data R/h¼10 and
=
h ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðρ=E2Þf ace
q
compiled by a personal computer with Intel (R) Core (TM) 2 Duo
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
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
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.
Fig 9 The annular plate model.
Table 10 The natural frequency _ ω of circular sandwich plate via R/h ratios and various plate models.
R/h Method Mode number
FSDT 6.5314 11.0073 11.0291 15.2293 15.8679 16.9509
FSDT 8.6196 16.1005 16.1161 23.3139 24.8155 26.6094
20 TSDT 8.6768 16.2863 16.2912 23.6428 25.1854 27.0669 RPT 8.6529 16.5898 16.5953 24.1341 25.8627 27.1200 FSDT 9.5541 19.1564 19.1648 28.8009 31.7216 34.0548
100 TSDT 9.9093 20.4639 20.4643 31.3827 35.2143 37.7988
FSDT 9.9258 20.614 20.6145 31.6544 35.8107 38.3766