Vietnam Journal of Mechanics, VAST, Vol.43, No. 3 (2021), pp. 221 – 235
DOI: https://doi.org/10.15625/0866-7136/15503
AN EDGE-BASED SMOOTHED FINITE ELEMENT FOR
BUCKLING ANALYSIS OF FUNCTIONALLY GRADED
MATERIAL VARIABLE-THICKNESS PLATES
Tran Trung Thanh1, Tran Van Ke1, Pham Quoc Hoa2,
Tran The Van2, Nguyen Thoi Trung3,∗
1Le Quy Don Technical University, Hanoi, Vietnam
2Tran Dai Nghia University, Ho Chi Minh City, Vietnam
3Ton Duc Thang University, Ho Chi Minh City, Vietnam
∗E-

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-mail: nguyenthoitrung@tdtu.edu.vn
Received: 22 September 2020 / Published online: 10 August 2021
Abstract. The paper aims to extend the ES-MITC3 element, which is an integration of the
edge-based smoothed ﬁnite element method (ES-FEM) with the mixed interpolation of
tensorial components technique for the three-node triangular element (MITC3 element),
for the buckling analysis of the FGM variable-thickness plates subjected to mechanical
loads. The proposed ES-MITC3 element is performed to eliminate the shear locking phe-
nomenon and to enhance the accuracy of the existing MITC3 element. In the ES-MITC3
element, the stiffness matrices are obtained by using the strain smoothing technique over
the smoothing domains formed by two adjacent MITC3 triangular elements sharing the
same edge. The numerical results demonstrated that the proposed method is reliable and
more accurate than some other published solutions in the literature. The inﬂuences of
some geometric parameters, material properties on the stability of FGM variable-thickness
plates are examined in detail.
Keywords: buckling analysis, critical load, variable thickness plate, edge-based ﬁnite ele-
ment method, ES-MITC3.
1. INTRODUCTION
The functionally graded materials (FGMs) can change the material properties grad-
ually, continuously, and smoothly in different directions. Therefore, the delamination
in laminated composites can be eliminated in these materials. They are made of two
components, mainly metal with high toughness and ceramic with outstanding heat and
corrosive resistance properties. Due to such excellent properties, they are applied in var-
ious high-tech industries such as automotive, nuclear, civil engineering, and aerospace.
There have been many studies on the mechanical behavior of FGM structures including
the buckling problem. Some typical studies can be summarized as follows. Ramu et
â 2021 Vietnam Academy of Science and Technology
222 Tran Trung Thanh, Tran Van Ke, Pham Quoc Hoa, Tran The Van, Nguyen Thoi Trung
al. [1] studied the stability of FGM under uniaxial and biaxial compression load using
the ﬁnite element method (FEM) based on classical plate theory (CPT). Rohit et al. [2]
used third-order shear deformation theories (TSDT) to analyze the buckling of the sim-
ple supported FGM plates under uniaxial load. Wu et al. [3] studied the stability of FGM
plates subjected to thermal and mechanical loads using FSDT. Javaheri et al. [4] based
on the analytical method (AM) for the stability analysis of FGM plates subjected to in-
plane compressive load. Zenkour [5] calculated the free vibration and buckling of FGM
constant-thickness sandwich plates. Shariat et al. [6] studied the buckling of thick FGM
plate by AM. Thai et al. [7] used an efﬁcient and simple reﬁned theory for buckling anal-
ysis of FGM plates. Reddy [8] combined an analytical method and TSDT to analyze the
buckling of the FGM plate. Thinh et al. [9] proposed an eight unknown higher-order
shear deformation theory for vibration and buckling analysis of constant-thickness FGM
plates.
Variable-thickness structures are extensively used in many types of high-
performance surfaces like aircraft, civil engineering, and other engineering ﬁelds. Using
these structures will help adjust the weight of structural, and hence help maximize the
capacity of the material. For example, Thang et al. [10] investigated the effects of variable-
thickness on buckling and post-buckling of imperfect sigmoid FGM plates on elastic
foundation (EF) subjected to compressive loading. Eisenberger et al. [11] investigated the
buckling of variable-thickness thin isotropic plates by using the extended Kantorovich
method. Naei et al. [12] analyzed the buckling of the FGM variable-thickness circular-
plate using FEM. Jalali et al. [13] investigated thermal buckling of the FGM nonuniform-
thickness circular sandwich plates employed the pseudo-spectral method. Alipour and
co-workers used semi-analytical to studied buckling of heterogeneous variable-thickness
viscoelastic circular-plates lying on the EF [14], and variable-thickness bi-directional FGM
circular-plates placed on nonuniform-EF [15]. Alinaghizadeh et al. [16] applied the gener-
alized differential quadrature (GDQ) method for buckling analysis of variable-thickness
radially FGM annular sector plates located on two parameters EF. Bouguenina et al. [17]
conducted analyses of FGM variable-thickness plates under thermal loads using ﬁnite
difference method. Benlahcen et al. [18] employed an analytical solution to examine
buckling of simply supported FGM plates with parabolic-concave thickness variation.
Minh and Duc [19] investigated the effect of cracks on the stability of the FGM variable-
thickness plates using TSDT and phase-ﬁeld theory. In addition, Zenkour [20] presented
the hygrothermal mechanical bending of variable-thickness plates using the AM. Allam
et al. [21] investigated thermoelastic stresses in FG variable-thickness rotating annular
disks using inﬁnitesimal theory. Thien et al. [22] developed the isogeometric analysis
(IGA) to analyze the buckling of non-uniform thickness nanoplates resting on the EF.
To improve the convergence and accuracy for classical triangular elements, the origin
MITC3 element [23] is proposed to combine with the ES-FEM [24] to give the so-called
ES-MITC3 element [25–30]. In the formulation of the ES-MITC3 element, the system stiff-
ness matrix is employed using strains smoothed over the smoothing domains associated
with the edges of the triangular elements. The numerical results of the present study
demonstrated that the ES-MITC3 element has the following superior properties: (1) the
ES-MITC3 element can avoid the transverse shear locking phenomenon even with the
An edge-based smoothed ﬁnite element for buckling analysis of functionally graded material variable-thickness plates 223
ratio of the thickness to the length of the structures reach 10−8 (readers can see detail in
Ref. [25]; (2) the ES-MITC3 element has higher accuracy than the existing triangular ele-
ments such as MITC3 element [23], DSG3 element [31] and CS-DSG3 element [32]; and is
a good competitor with the MITC4 element [33].
According to the best of authors’ knowledge, the stability of FGM variable-thickness
plates using the ES-MITC3 element has not yet been studied. Therefore, this paper aims
to extend the ES-MITC3 element for the buckling analysis of FGM variable-thickness
plates. The formulation is based on the FSDT due to its simplicity and computational
efﬁciency. The accuracy and reliability of the present approach are veriﬁed by comparing
the present numerical results with those of other available methods. Finally, the inﬂuence
of geometrical parameters, and material properties on the buckling of FGM plates are
fully studied.
2. THEORETICAL FORMULATION
2.1. FGM material
The FGM is made up of two components: ceramic and metal. The mechanical prop-
erties of FGM are assumed to vary smoothly through the thickness of plates as follows [6]
P (z) = (Pc − Pm) Vc (z) + Pm, (1)
z p h (x, y) h (x, y)
V (z) = + 0.5 with z 2 − ; , (2)
c h(x, y) 2 2
in which P (z) represents for Young’s modulus E(z), Poisson’s ratio u(z); subscripts m
and c denotes the metal and ceramic constituents; Vc(z) is the volume fraction of ceramic
which according to a power-law function with p is the power-law index. The value of p
equals to 0 and +Ơ represents a fully ceramic and fully metal plate, respectively. Note
that, the thickness of plate is different at various positions on the plate and depends on
the law of thickness variation (h is the function of x- and y-variables).
2.2. Mindlin’s plate theory
According to Mindlin’s plate theory, the displacement ﬁeld of the plate is given by [1]
8
u (x, y, z) = u (x, y) + zq (x, y)
0 x
v (x, y, z) = v0 (x, y) + zqy (x, y) (3)
>
: w (x, y, z) = w0 (x, y)
in which u, v, w, qx, qy are ﬁve unknown displacements of the mid-surface of the plate.
The strain ﬁeld can be expressed as follows
8 9 8 9 8 9 8 9
> #x > > u,x > > u0,x > > qx,x >
> > > > > > > >
#y => v,y => v0,y => qy,y =>
# = #xy = u,y + v,x = u0,y + v0,x + z qx,y + qy,x . (4)
> gxz > > w,x + u,z > > v0.x + qx > > 0 >
> > > > > > > >
: gyz ; : w,y + v,z ; : w0,y + qy ; : 0 ;
224 Tran Trung Thanh, Tran Van Ke, Pham Quoc Hoa, Tran The Van, Nguyen Thoi Trung
Eq. (3) may be written by
# # + zk
= 1 = m . (5)
#2 g
From Hooke’s law, the linear stress-strain relations can be determined by a formulation
8 9 2 3 8 9
s Q Q 0 0 0 #
> x > 11 12 > x >
> sy > 6 Q Q 0 0 0 7 > #y >
< = 6 21 22 7 < =
sxy = 6 0 0 Q66 0 0 7 #xy , (6)
6 7
> txz > 4 0 0 0 Q55 0 5 > gxz >
> > > >
: tyz ; 0 0 0 0 Q44 : gyz ;
in which
E(z) u(z)E(z) E(z)
Q = Q = , Q = Q = , Q = Q = Q = . (7)
11 22 1 − u(z)2 12 21 1 − u(z)2 44 55 66 2(1 + u(z))
The force and moment resultants are obtained as follows [30]
T
Nx Ny Nxy = A#m + Bk, (8a)
T
Mx My Mxy = B#m + Ck, (8b)
T s
Qxz Qyz = A g, (8c)
with
2 3
Z h(x,y)/2 Q11 Q12 0
2
(A, B, C) = 4 Q21 Q22 0 5 1, z, z dz, (9)
−h(x,y)/2
0 0 Q66
Z h(x,y)/2 Q 0
As = 55 dz. (10)
−h(x,y)/2 0 Q44
It should be noted that compared to uniform thickness plates, all the matrices in Eqs. (9)
and (10) depend on the law of thickness variation and thus the limits of integrations also
depend on the position of points on plates.
2.3. Finite element formulation for buckling analysis of FGM variable-thickness plates
The bounded domain W of the FGM plate is discretized into ne three-node triangular
ne
n
elements with n nodes such that y ≈ ∑ ye and yi \ yj = ?, i 6= j. Then the gen-
e=1
T
e h e e e e e i
eralized displacements at any point u = uj , vj , wj , qxj, qyj of the element ye can be
approximated as [23]
2 3
NI (x) 0 0 0 0
nne 6 0 NI (x) 0 0 0 7 nne
e 6 7 e e
u (x) = 6 0 0 NI (x) 0 0 7 d = N (x) d , (11)
∑ 6 7 j ∑ j
j=1 4 0 0 0 NI (x) 0 5 j=1
0 0 0 0 NI (x)
An edge-based smoothed ﬁnite element for buckling analysis of functionally graded material variable-thickness plates 225
ne e
where n is the number of nodes of ye; N (x) is the shape function matrix; and dj =
T
h e e e e e i th
uj , vj , wj , qxj, qyj are the nodal degrees of freedom (DOF) associated with the j node
of ye.
The membrane bending strains of MITC3 element can be expressed in the matrix
form as follows [23]
e e e e e e e
#m = Bm1 Bm2 Bm3 d = Bmd , (12a)
e e e e e e e
k = Bb1 Bb2 Bb3 d = Bbd . (12b)
The smoothing domains yk is constructed based on the edges of the triangular el-
nk k k k
ements such that Any =edge[-kbased=1y smoothedand yi \ finiteyj =element? for fori 6buckling= j. An analysis edge-based smoothing
domain yk for the inner of functionally edge k is grade formedd material by connecting variable- twothickness end-nodes plates of the edge to the
centroids of adjacent triangular MITC3 elements as shown in Fig.1.
ả
Fig.Fig. 1 1. .The The smoothing smoothing domain domain •yk isis formedformed byby triangular triangular elements elements.
Applying the edge-based smooth technique [24], the smoothed membrane, bending and shear
ả Applyingả ả the edge-based smooth techniqueả [24], the smoothed membrane, bending
strain Mòđ, [â , \â over thek smoothingk k domain • can be created by:k
and shear strain# ˜m,k ˜ ,g ˜ over the smoothing domain y can be created by
Z
k k
ỷ #˜m = #mỷF (x) dy, (13a)
Mò, = y My,k ™ (x)dψ, (13a)
´† Z
k˜ k = kFk (x) dy, (13b)
yk
ỷ Z ỷ
[â g=˜ k y= [ ™gF(xk )(dxψ) d, y, (13c) (13b)
´† yk
where #m, k and g the compatible membrane, bending and the shear strains, respectively;
Z
k k
F (x) is a given smoothing functionỷ that satisﬁesỷ( ) at least the unity property F (x) dy (13c)
\â = y \ ™ x dψ, yk
†
= 1. ´
ỷ
where M,, [ and \ the compatible membrane, bending and the shear strains, respectively; ™ (x) is a
ỷ( )
given smoothing function that satisfies at least the unity property∫´† ™ x dψ = 1.
In this study, we use the constant smoothing function [24]:
1
x ∈ ψỷ
™ỷ(x) = E≠ỷ (14)
0 x ∉ ψỷ
in which ≠ỷ is the area of the smoothing domain ψỷ and is given by
1 ẫỏ†
≠ỷ = y dψ = ∞ ≠õ (15)
´† 3 õàX
where Åầỷ is the number of the adjacent triangular elements in the smoothing domain ψỷ; and ≠õ is the
area of the ỗth triangular element attached to the edge Đ.
The stiffness matrix of the FGM plate using the ES-MITC3 is assembled by [24]:
†
ẫà∂
ỷ
≥Ơ 8 = ∞ ≥Ơ ầ (16)
ỷàX
ỷ ỷ
where Ơ≥ầ is the ES-MITC3 stiffness matrix of the smoothing domain ψ and given by
226 Tran Trung Thanh, Tran Van Ke, Pham Quoc Hoa, Tran The Van, Nguyen Thoi Trung
In this study, we use the constant smoothing function [24]
8 1
< , x 2 yk
Fk (x) = Ak (14)
: 0, x 2/ yk
in which Ak is the area of the smoothing domain yk and is given by
ek
Z 1 n
Ak = dy = Ai, (15)
k ∑
y 3 i=1
where nek is the number of the adjacent triangular elements in the smoothing domain yk;
and Ai is the area of the ith triangular element attached to the edge k.
The stiffness matrix of the FGM plate using the ES-MITC3 element is assembled by
[24]
k
nsh
˜ ˜ k
Kp = ∑ Ke , (16)
k=1
k k
where Kee is the ES-MITC3 element stiffness matrix of the smoothing domain y and given
by
Z
˜ k ˜ kT AB ˜ k ˜ kT s ˜ k ˜ kT AB ˜ k k ˜ kT s ˜ k k
Ke = B B + Bs A Bs dy = B B A + Bs A Bs A , (17)
yk BC BC
in which
h i
˜ kT ˜ k ˜ k
B = Bmj Bbj , (18)
and the strain-displacement matrices are presented in detail in [30].
The geometric stiffness matrix of the FGM plate using the ES-MITC3 element is de-
termined by [28]
nk
sh Z
˜ ˜ ek ˜ e ˜ T ¯ ˜
Kg = Kg with Kg = Y NYi dy, (19)
∑ k i
k=1 y
where
N N
N = x xy , (20)
Nxy Ny
with
Z h(x,y)
Nx, Ny, Nxy = sx, sy, sxy dz, (21)
−h(x,y)
and Y˜i is presented in [28]. It is noted that the integrations in Eq. (21) also depend on the
law of thickness variation, therefore the limits of integrations will depend on the position
of points on plates.
Apply the principle of minimum total potential energy, the stability problem in-
volves the solution of the following eigen problem in which Pcr is the critical load
K˜ p + PcrK˜ g = 0. (22)
An edge-based smoothed ﬁnite element for buckling analysis of functionally graded material variable-thickness plates 227
3. CONVERGENCE AND ACCURACY OF THE PROPOSED METHOD
In order to evaluate the convergence and accuracy of the proposed method, the au-
thors consider the following two examples:
Example 1. Firstly, we consider a fully clamped (CCCC) FGM constant-thickness
plate with material properties given by the metal (Al) Em = 70 GPa, um = 0.3 and ce-
∗
ramic (Al2O3) Ec = 380 GPa, uc = 0.3. The non-dimensional critical load Pcr of FGM
plates with different mesh-size are listed in Table1. It can be seen that, in all cases, the
An edgeAn- basededge- basedsmoothed smoothed finite elementfinite element for buckling for buckling analysis analysis
results by the ES-MITC3 element converge faster and are more accurate than those by
of functionally of functionally grade dgrade materiald material variable variable-thickness-thickness plates plates
the MITC3 element. Speciﬁcally, at the 18ì18 mesh-size, the ES-MITC3 element gives
∗ ∗ ; ;ừ ừ
TableTable 1. The 1. convergence The convergence of mesh of -meshsize of-size non of-dimensional non-dimensional critical critical load %load*ứ = %%*ứ*ứ=Œ %/*ứ?*ŒℎI/ ?of* ℎCCCCI of CCCC
∗ 2 3
Table 1. The convergence of mesh-sizesquaresquare ofFGM non-dimensional FGM plate splate. s. critical load Pcr = Pcrb /Ech0
of CCCC square FGM plates
a/h a/h p pMesh Mesh size sizeES -MITC3ES-MITC3 Error Error(%) (%) MITC3MITC3 Error Error(%) (%) Wu [3]Wu [3]
100a/h 100 p1 Mesh1 12x12 size12x12 ES-MITC34.62124.6212 0.12 Error 0.12 (%) 4.6265 MITC34.6265 Error0.23 0.23 (%) Wu [3]
14x1414x14 4.61854.6185 0.06 0.06 4.62104.6210 0.11 0.11
12ì12 4.6212 0.12 4.6265 0.23
1416x16ì1416x16 4.6167 4.61854.6167 0.02 0.06 0.02 4.6194 4.62104.6194 0.08 0.11 0.08 4.61584.6158
100 1 1618x18ì1618x18 4.6160 4.61674.6160 0.01 0.02 0.01 4.6180 4.61944.6180 0.05 0.08 0.05 4.6158
1820x20ì1820x20 4.6160 4.61604.6160 0.01 0.01 0.01 4.6176 4.61804.6176 0.04 0.05 0.04
20ì20 4.6160 0.01 4.6176 0.04
40 40 5 5 12x1212x12 3.00413.0041 0.15 0.15 3.00553.0055 0.20 0.20
1214x14ì1214x14 3.0020 3.00413.0020 0.08 0.15 0.08 3.0030 3.00553.0030 0.11 0.20 0.11
14ì14 3.0020 0.08 3.0030 0.11
16x1616x16 3.00063.0006 0.03 0.03 3.00163.0016 0.07 0.07 2.99962.9996
40 5 16ì16 3.0006 0.03 3.0016 0.07 2.9996
1818x18ì1818x18 2.9998 2.99982.9998 0.01 0.01 0.01 3.0005 3.00053.0005 0.03 0.03 0.03
2020x20ì2020x20 2.9998 2.99982.9998 0.01 0.01 0.01 3.0001 3.00013.0001 0.02 0.02 0.02
(a) The FGM plate with a/h = 100 and p = 1 (b) The FGM plate with a/h = 40 and p = 5
a) Thea) FGM The FGM plate platewith a/hwith=100 a/h =100and p and=1. p=1. b) Theb) FGM The FGM plate platewith a/hwith=40 a/h and=40 p and=5. p=5.
Fig. 2. The convergence of mesh-size to non-dimensional∗ critical∗ ; load;ừ ừ
Fig. 2.Fig. The 2. convergenceThe convergence of mesh of mesh-size -tosize non to-dimensional non-dimensional critical critical load %load= %%*ứ =Œ %/*ứ?Œℎ/ ?of* ℎsquareI of square
∗ 2 3 *ứ *ứ * I
Pcr = Pcrb /FGMEch0FGM plateof square .plate . FGM plate
∗ ∗ ; ;ừ ừ
TableTable 2. Comparison 2. Comparison of nondimensional of nondimensional critical critical load %load*ứ = %%*ứ*ứ=Œ %/*ứ?*ŒℎI/ ?of* ℎrectangularI of rectangular FGM FGM plate splate. s.
(h=a/40;(h= aa/40;=1 is a =1fixed) is fixed). .
b/a b/a p p Wu [3]Wu [3] MITC3MITC3 Error Error(%) (%) PresentPresent Error Error(%) (%)
1.5 1.5 0 0 11.851611.8516 11.891311.8913 0.33 0.33 11.863311.8633 0.10 0.10
2 2 0 0 17.529917.5299 17.568617.5686 0.22 0.22 17.548117.5481 0.10 0.10
3 3 0 0 35.123935.1239 35.216835.2168 0.26 0.26 35.153035.1530 0.08 0.08
1.5 1.5 2 2 4.64004.6400 4.63344.6334 0.14 0.14 4.64304.6430 0.06 0.06
2 2 2 2 6.858106.85810 6.84466.8446 0.20 0.20 6.84986.8498 0.12 0.12
3 3 2 2 13.769713.7697 13.720313.7203 0.36 0.36 13.753813.7538 0.12 0.12
228 Tran Trung Thanh, Tran Van Ke, Pham Quoc Hoa, Tran The Van, Nguyen Thoi Trung
the converging results with the maximum error of 0.01% compared to those by Wu et
al. [3] using the analytical method (AM). In contrast, the MITC3 element at the18x18
mesh-size has not yet converged as shown in Fig.2. Furthermore, the obtained results
by the ES-MITC3 element are compared to those of other published results as shown
in Table2. It should be noted that the error is determined by the following formula:
jPresent − [3]j
Error (%) = 100 ì and types of boundary conditions are deﬁned as fol-
j[3]j
lows: 1) Simply supported edge boundary condition (S): u0 = w = jx = 0 at y = 0,
y = b or v0 = w = jy = 0 at x = 0, x = a; and 2) Clamped edge boundary condition
(C): at y = 0, y = b or v0 = w = jx = jy = 0 at x = 0, x = a.
∗ 2 3
Table 2. Comparison of non-dimensional critical load Pcr = Pcrb /Ech0 of rectangular FGM plates.
(h = a/40; a = 1 is ﬁxed)
b/a p Wu [3] MITC3 Error (%) Present Error (%)
1.5 0 11.8516 11.8913 0.33 11.8633 0.10
2 0 17.5299 17.5686 0.22 17.5481 0.10
3 0 35.1239 35.2168 0.26 35.1530 0.08
1.5 2 4.6400 4.6334 0.14 4.6430 0.06
2 2 6.8581 6.8446 0.20 6.8498 0.12
3 2 13.7697 13.7203 0.36 13.7538 0.12
Example 2. Secondly, a simply supported (SSSS) isotropic plate with linearly variable
y
thickness h = h (1 + a ) is considered. The non-dimensional critical load is calculated
0 b
∗ 2 2 3
by Pcr = Pcrb /(p D) with D = Eh0/12. The obtained results of the present work are
∗
Table 3. Comparison of non-dimensional critical load Pcr of SSSS isotropic plates
with variable thickness
a
a/b Method
0.125 0.25 0.5 0.75 1
IGA-FSDT [22] 7.4621 8.7531 11.5687 14.6953 18.1368
0.5 Kantorovich method [11] 7.4645 8.7633 11.6112 14.7942 18.3175
ES-MITC3 7.4625 8.7601 11.5989 16.6987 18.2981
IGA-FSDT [22] 5.4194 6.3869 8.5627 11.0657 13.9017
0.7 Kantorovich method [11] 5.4199 6.3891 8.5741 11.0979 13.9730
ES-MITC3 5.4198 6.3885 8.5738 11.0889 13.9865
IGA-FSDT [22] 4.8428 5.7224 7.7327 10.0858 12.7877
0.9 Kantorovich method [11] 4.8413 5.7165 7.7111 10.0460 12.7381
ES-MITC3 4.8418 5.7203 7.7198 10.0683 12.7524
Tran Trung Thanh, Tran Van Ke, Pham Quoc Hoa, Tran The Van, Nguyen Thoi Trung
Example 2. Secondly, a simply supported (SSSS) isotropic plate with linearly variable thickness
4
ℎ = ℎ (1 + α ) is considered. The non-dimensional buckling load is calculated by %∗ =
I ự *ứ
; ; ừ
%*ứŒ /(“ ”) with ” = ?ℎI/12. The obtained results of the present work are compared to those by
Thien et al. [22] using the IGA based on FSDT and Eisenberger et al. [11] employed Kantorovich
method. These results are listed in Table 3. It is observed that the obtained results by the proposed
method are in a good agreement with those published in the literature. From the above two examples,
it can be concluded that the proposed method is reliable for further analyses.
∗
Table 3. Comparison of non-dimensional critical load %*ứ of SSSS isotropic plates with variable
thickness.
α
a/b Method
0.125 0.25 0.5 0.75 1
0.5 IGA-FSDT [22] 7.4621 8.7531 11.5687 14.6953 18.1368
Kantorovich
7.4645 8.7633 11.6112 14.7942 18.3175
method [11]
ES-MITC3 7.4625 8.7601 11.5989 16.6987 18.2981
0.7 IGA-FSDT [22] 5.4194 6.3869 8.5627 11.0657 13.9017
Kantorovich
5.4199 6.3891 8.5741 11.0979 13.9730
method [11]
ESAn-MITC3 edge-based smoothed ﬁnite5.4198 element for buckling6.3885 analysis of functionally8.5738 graded material variable-thickness11.0889 plates 22913.9865
0.9 IGA-FSDT [22] 4.8428 5.7224 7.7327 10.0858 12.7877
compared to those by Thien et al. [22] using the IGA based on FSDT and Eisenberger et
Kantorovich
al. [11] employed Kantorovich4.8413 method.5.7165 These results are7.7111 listed in Table10.04603. It is observed 12.7381
thatmethod the obtained [11] results by the proposed method are in a good agreement with those
publishedES-MITC3 in the literature.4.8418 From the above5.7203 two examples,7.7198 it can be concluded10.0683 that the12.7524
proposed method is reliable for further analyses.
4. BUCKLING4. BUCKLING ANALYSIS ANALYSIS OF OF FGM FGM VARIABLE VARIABLE-THICKNESS-THICKNESS PLATES PLATES
a
In this section, we consider the FGM variable-thickness plate’ ( = 100, a is ﬁxed)
In this section, we consider the FGM variable-thickness plate ( = h100, a is fixed) as shown in
1ữ 0
as shown in Fig.3. The plate thickness varies along the x-direction following the law 2
Fig. 3. The plate thickness variesx along the x-direction following the law ℎ = ℎ(G) = ℎ (1 + ). The
h = h (x) = h (1 + ). The material parameters of the FGM plate are given by:I metal ’
0 a
material parameters(Al) Em = of70 the GPa, FGMum =plate0.3 are and given ceramic by: (Al metal2O3) (Al)Ec = ?380, = GPa,70 ◊%œuc =, @0.3., = The0.3 non-and ceramic
2 l
∗ Pcrb ∗ ÀÿŸự
(Al2O3) ?*dimensional= 380 ◊%œ critical, @* = load0.3. isThe introduced non-dimensional by Pcr = critical. load is introduced by %*ứ = ⁄ .
3 Jÿ1
Ech0 ữ
y
p p
a
hx()
ceramic
x
h0
x metal
b
z
Fig. 3. Fig.The 3 .FGM The FGM variable variable-thickness-thickness plate plate under under in in-plane-plane force force along along the thex-direction x-direction.
4.1. Effect of power-law index p
In order to study the effect of the power-law index p on buckling of FGM plates, we
consider a square FGM plate with different boundary conditions (BCs), and the power-
law index p is changed from 0 to 100. The non-dimensional critical load of the FGM
plates is listed in Table4 and displayed in Fig.4. It can be seen that the critical force of
the plate depends not only on the BC but also on the power-law index p. The rich ceramic
FGM plates have a higher hardness than the rich metal FGM plates, so the critical force
is higher. The critical force decreases when the power-law index p increases, and the rate
decreases faster when the index p increases from 0 to 1, and slower when p > 1.
An edgeAn edge-based-based smoothed smoothed finite finite element element for buckling for buckling analysis analysis
of functionally of functionally grade graded materiald material variable variable-thickness-thickness plates plates
4.1. Effect4.1. Effect of power of power-law- lawindex index p p
In orderIn order to study to study the effectthe effect of the of powerthe power-law- lawindex index p on p buckling on buckling of FGM of FGM plates, plates, we considerwe consider a a
squaresquare FGM FGM plate plate with with different different boundary boundary conditions conditions (BCs) (BCs), and, andthe powerthe power-law- lawindex index p is pchange is changed d
fromfrom 0 to 0100. to 100. The Thenon -ndimensionalon-dimensional critical critical load load of the of FGMthe FGM plate plates is slisted is listed in T inable Table 4 and 4 anddisplayed displayed
in Fig.in Fig.4. It 4can. It becan seen be seen that thatthe criticalthe critical force force of the of platethe plate depends depends not onlynot only on the on BCthe BCbut alsobut also on theon the
powerpower-law- lawindex index p. The p. Therich richceramic ceramic FGM FGM plates plates have have a higher a higher hardness hardness than than the richthe richmetal metal FGM FGM
plates,plates, so the so criticalthe critical force force is higher. is higher. The The critical critical force force decreases decreases when when the the power power-law- law index index p p
increaseincreases, ands, andthe ratethe ratedecreases decreases faster faster when when the indexthe index p increases p increases from from 0 to 01 ,to and 1, andslower slower when when C > C >
1. 1.
TableTable 4. The 4. Thecritical critical load load of the of squarethe square FGM FGM variable variable-thickness-thickness plate plate. .
NonNon-dimensional-dimensional critical critical load load %∗ %∗
p p *ứ *ứ
SSSSSSSS SCSCSCSC CSCSCSCS CCCCCCCC
0 0 5.35085.3508 10.215810.2158 9.24229.2422 13.796713.7967
0.5 0.5 3.47693.4769 6.62996.6299 6.01426.0142 8.95598.9559
1 1 2.67902.6790 5.10485.1048 4.63754.6375 6.89656.8965
2 2 2.09412.0941 3.98723.9872 3.62673.6267 5.38665.3866
2305 5 Tran1.7712 Trung1.7712 Thanh, Tran Van Ke,3.3705 Pham3.3705 Quoc Hoa, Tran The Van,3.0662 Nguyen3.0662 Thoi Trung 4.55234.5523
10 10 1.61191.6119 3.06833.0683 2.78812.7881 4.14324.1432
Table 4. The critical load of the square FGM variable-thickness plate
20 20 1.42311.4231 2.7112.711 2.45982.4598 3.66053.6605
50 50 1.21091.2109 2.30912.3091 2.09212.0921 ∗ 3.11803.1180
Non-dimensional critical load Pcr
100 100p 1.11151.1115 2.86392.8639
SSSS2.1208 SCSC2.1208 CSCS1.92021.9202 CCCC
4.2. Effect4.2. Effect of 0length of length to width to 5.3508width ratio ratio b/a b/a 10.2158 9.2422 13.7967
0.5 3.4769 6.6299 6.0142 8.9559
NextNext, a rectangular,1 a rectangular FGM 2.6790FGM plate plate with with the powerthe 5.1048 power-law- lawindex index p=2 4.6375 pand=2 anddifferent different BC BCis 6.8965 considered. is considered. The The
lengthlength to width to width2 ratio ratio b/a isb/a taken 2.0941 is taken from from 0.5 to0.5 5 towhile 3.9872 5 while the widththe width of 3.6267the of platethe plate a is aassumed is assumed 5.3866 to be to constant. be constant.
The Thenon -nondimensional-dimensional5 critical critical 1.7712 load loads ares providedare provided 3.3705 in T inable Table 5 and 5 andpresented 3.0662 presented in Fig. in Fig.5. It 4.5523 5 can. It canbe seen be seen that that
the lengththe length to10 width to width ratio ratio b/a 1.6119 b/astrongly strongly alters alters the 3.0683 criticalthe critical load load of 2.7881 the of theFGM FGM variable variable 4.1432-thickness-thickness plate. plate.
Specifically,Specifically, 20 as ratio as ratio b/a b/aincrease 1.4231 increases, thes, criticalthe critical 2.7110 load load dec reases decreases 2.4598 rapidly. rapidly. Moreover, Moreover, 3.6605 Tables Tables 4, 54 ,and 5 and
FiguresFigures 4, 5 4show50, 5 show that thatthe fully 1.2109the fully clamped clamped FGM FGM 2.3091 plate plate has hasthe greatestthe 2.0921 greatest critical critical force, force, 3.1180while while the criticalthe critical
forceforce is smallest is 100smallest in the in casethe 1.1115case of the of fullythe fully simple simple 2.1208 support. support. This This is 1.9202easy is easy to understand to understand 2.8639because because the fullythe fully
clampedclamped plate plate leads leads to an to increase an increase in the in stiffnessthe stiffness of the of FGMthe FGM plate plate. .
Fig. 4. TheFig. effectFig.4. The 4. of Theeffectp on effect theof p critical ofon p on buckling Fig.Fig. 5. The Fig.5. The effect 5. Theeffect of effectb /ofa b/aratio of ratiob/a on ratio theon the criticalon criticalthe critical
the criticalthe critical bucklingload buckling of load the FGMload of the plateof FGMthe FGM plate. plate. buckingbuckingbucking load load of load theof the FGM of FGMthe plate FGM plate. plate.
4.2. Effect of length to width ratio b/a
Next, a rectangular FGM plate with the power-law index p = 2 and different BCs
are considered. The length to width ratio b/a is taken from 0.5 to 5 while the width of
the plate a is assumed to be constant. The non-dimensional critical loads are provided in

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