DSpace at VNU: Nonlinear mechanical, thermal and thermo-mechanical postbuckling of imperfect eccentrically stiffened thin FGM cylindrical panels on elastic foundations

Thin-Walled Structures 96 (2015) 155–168

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Thin-Walled Structures
journal homepage: www.elsevier.com/locate/tws

Nonlinear mechanical, thermal and thermo-mechanical postbuckling
of imperfect eccentrically stiffened thin FGM cylindrical panels on
elastic foundations
Nguyen Dinh Duc a,n, Ngo Duc Tuan b, Tran Quoc Quan a, Nguyen Van Quyen a,
Tran Van Anh a
a
b

Vietnam National University, Hanoi, 144 Xuan Thuy-Cau Giay, Hanoi, Vietnam
The University of Melbourne, Parkville, VIC 3010, Australia

art ic l e i nf o

a b s t r a c t

Article history:
Received 24 April 2015
Received in revised form
19 July 2015
Accepted 3 August 2015

This paper presents an analytical approach to investigate the nonlinear stability analysis of eccentrically
stiffened thin FGM cylindrical panels on elastic foundations subjected to mechanical loads, thermal loads
and the combination of these loads. The material properties are assumed to be temperature-dependent

and graded in the thickness direction according to a simple power law distribution. Governing equations
are derived basing on the classical shell theory incorporating von Karman–Donnell type nonlinearity,
initial geometrical imperfection, the Lekhnitsky smeared stiffeners technique and Pasternak type elastic
foundations. Explicit relations of load–deflection curves for FGM cylindrical panels are determined by
applying stress function and Galerkin method. The effects of material and geometrical properties, imperfection, elastic foundations and stiffeners on the buckling and postbuckling of the FGM panels are
discussed in detail. The obtained results are validated by comparing with those in the literature..
& 2015 Elsevier Ltd. All rights reserved.

Keywords:
Nonlinear mechanical and thermal postbuckling
Eccentrically stiffened FGM cylindrical panels
Imperfection
Elastic foundations

1. Introduction
Composite panels are commonly used in aerospace, mechanics,
naval and other high-performance engineering applications due to
their light weight, high specific strength and stiffness, excellent
thermal characteristics. At high temperatures, composite panels
are found to buckle without the application of mechanical loads.
Therefore, the buckling and postbuckling response of composite
panels have to be well understood. Recently, a new class of composite materials known as functionally graded materials (FGMs)
attracts special attention of a lot of authors in the world. FGM is a
new generation of composite material in which its mechanical
properties vary smoothly and continuously from one surface to the
other. Functionally graded structures such as cylindrical panels in
recent years, play the important part in the modern industries. As
a result, static response of FGM cylindrical panels has been the
subject of many studies for a long period of time. Shen and Wang
[1] presented thermal postbuckling analysis for FGM cylindrical

panels resting on elastic foundations. They [2] also studied the
nonlinear bending analysis of simply supported FGM cylindrical
panel resting on an elastic foundation in thermal environments.
n

Corresponding author.
E-mail address: (N.D. Duc).

/>0263-8231/& 2015 Elsevier Ltd. All rights reserved.

Lee et al. [3] investigated the thermomechanical behaviors of FGM
panels in hypersonic airflows. Alibeigloo and Chen [4] developed
the three-dimensional elasticity solution for static analysis of a
FGM cylindrical panel with simply supported edges. Tung and Duc
[5] studied the nonlinear response of thick FGM doubly curved
shallow panels resting on elastic foundations and subjected to
some conditions of mechanical, thermal, and thermomechanical
loads. They [6] also investigated the nonlinear response of pressure-loaded FGM cylindrical panels with temperature effects.
Aghdam et al. [7] considered bending of moderately thick clamped
FGM conical panels subjected to uniform and non-uniform distributed loadings. Du et al. [8] studied the nonlinear forced vibration of infinitely long functionally graded cylindrical shells is
using the Lagrangian theory and the multiple scale method. A
semi-analytical solution for static response of fully clamped sheardeformable FGM doubly curved panels is presented by Shahmansouri et al. [9]. Kiani et al. [10] focused on the static, dynamic and
free vibration analysis of a FGM doubly curved panel. Bich et al.
[11] researched the linear buckling of FGM truncated conical panels subjected to axial compression, external pressure and the
combination of these loads. Static and dynamic stabilities of FGM
panels which are subjected to combined thermal and aerodynamic
loads are investigated in work of Sohn and Kim [12] based on the
first order shear deformation theory. Yang et al. [13] published the

156

N.D. Duc et al. / Thin-Walled Structures 96 (2015) 155–168

results on thermo-mechanical postbuckling analysis of FGM cylindrical panels with temperature-dependent properties. Recently,
in 2014, Duc [14] published a valuable book “ Nonlinear static and
dynamic stability of functionally graded plates and shells”, in which
the results about nonlinear static stability of shear deformable
FGM panels are presented. Tung [15] introduced an analytical
approach to investigate the effects of tangential edge constraints
on the buckling and postbuckling behavior of FGM flat and cylindrical panels subjected to thermal, mechanical and thermomechanical loads and resting on elastic foundations.
However, since this area is relatively new, there are very little
researches on nonlinear static problems of FGM cylindrical panels
and cylindrical shells reinforced by stiffeners. Duc and Quan [16]
investigated the nonlinear response of eccentrically stiffened FGM
cylindrical panels on elastic foundations subjected to mechanical
loads. Najafizadeh et al. [17] considered the elastic buckling of
FGM stiffened cylindrical shells by rings and stringers subjected to
axial compression loading. Dung et al. [18] analyzed the nonlinear
buckling and postbuckling of FGM stiffened thin circular cylindrical shells surrounded by elastic foundations in thermal environments and under torsional load. Bich et al. [19] presented an
analytical approach to investigate the nonlinear static and dynamic buckling of imperfect eccentrically stiffened FGM thin circular cylindrical shells subjected to axial compression.
To the knowledge of the authors, there is limited publication on
the stability of FGM structures reinforced by eccentrically stiffeners in thermal environments. The most difficult part in this type of
problem is to calculate the thermal mechanism of FGM structures
as well as stiffeners under thermal loads. Duc et al. [20,21] investigated the nonlinear postbuckling of an eccentrically stiffened
thin FGM plate and circular cylindrical shell resting on elastic
foundation in thermal environments. Development of the results
in these researches, this paper deals with the nonlinear postbuckling of imperfect eccentrically stiffened thin FGM cylindrical
panels on elastic foundations under mechanical loads, thermal
loads and the combination of these loads. The material properties

are assumed to be temperature-dependent and graded in the
thickness direction according to a simple power law distribution.
Both of the panels and the stiffeners are assumed to be deformed
due to the presence of temperature. Using Galerkin method and
stress function, the effects of geometrical and material properties,
imperfection, elastic foundations and stiffeners on the nonlinear
response of the imperfect eccentrically stiffened FGM cylindrical
panels are analyzed.

2. Problem statement
Consider an eccentrically stiffened functionally graded cylindrical panel with the radii of curvature, thickness, axial length and
arc length of the panel are R , h, a and b, respectively and is defined
in coordinate system (x, y, z ) , as shown in Fig. 1. The panel is reinforced by eccentrically longitudinal and transversal stiffeners.
The width and thickness of longitudinal and transversal stiffeners
are denoted by dx , hx and dy , hy respectively; sx , sy are the spacings
of the longitudinal and transversal stiffeners. The quantities Ax , Ay
are the cross-section areas of stiffeners and Ix, Iy, zx , z y are the
second moments of cross-section areas and the eccentricities of
stiffeners with respect to the middle surface of panel, respectively.
E0 is Young's modulus of ring and stringer stiffeners. In order to
provide continuity between the panel and stiffeners, suppose that
stiffeners are made of full metal (E0 = Em ) .
The panel is made from a mixture of ceramic and metal, and
the material constitution is varied gradually by a simple power law
distribution, in which the volume fractions of the ceramic and
metal are expressed as

⎛ 2z + h ⎞ N
Vm (z ) = ⎜
⎟ ,

⎝ 2h ⎠

Vc (z ) = 1 − Vm (z ),

(1)

where N is volume fraction index ( 0 ≤ N < ∞), subscripts m
and c stand for the metal and ceramic constituents, respectively.
Effective properties Preff of FGM panel, such as the elastic modulus
E and the thermal expansion coefficient α are determined by linear
rule of mixture as

Preff (z ) = Prc Vc (z ) + Prm Vm (z ),

(2)

in which Pr denotes a temperature-dependent material property. The effective properties of the FGM panel are obtained by
substituting Eq. (1) into Eq. (2) as

[E (z, T ), α (z, T )] = ⎡⎣ Ec (T ), αc (T ) ⎤⎦
⎛ 2z + h ⎞ N
⎟ ,
+ ⎡⎣ Emc (T ), αmc (T ) ⎤⎦ ⎜
⎝ 2h ⎠

(3)

where

Emc (z, T ) = Em (z, T ) − Ec (z, T ), αmc (z, T ) = αm (z, T ) − αc (z, T ),

(4)

and the Poisson's ratio is assumed to be constant
ν (z ) = v = const .
A material property Pr can be expressed as a nonlinear function
of temperature [1,2,13]

Fig. 1. Configuration and the coordinate system of an eccentrically stiffened cylindrical panel on elastic foundations.


N.D. Duc et al. / Thin-Walled Structures 96 (2015) 155–168

(

)

Pr = P0 P−1T−1 + 1 + P1T + P2 T2 + P3 T 3 ,

(5)

in which T = T0 + ΔT, ΔT is the temperature increment in the
environment containing the panel and T0 = 300 K (room temperature), P0, P−1, P1, P2 and P3 are coefficients characterizing of the
constituent materials.
The panel–foundation interaction of Pasternak model is given
by

qe = k1w − k2 ∇2w,

Nx = (A11 +

157

E0 A xT 0
) εx + A12 εy0 + (B11 + CxT ) k x + B12 k y + Φ1,
sxT

Ny = A12 εx0 + (A22 +

E0 A yT

) εy0 + B12 k x + (B22 + CyT ) k y + Φ1,

syT

0
+ 2B66 k xy,
Nxy = A66 γxy

Mx = (B11 + CxT ) εx0 + B12 εy0 + (D11 +

E0 IxT
) k x + D12 k y + Φ2,
sxT

(6)

where ∇2 = ∂ 2/∂x 2 + ∂ 2/∂y2, w is the deflection of the panel, k1 is
Winkler foundation modulus and k2 is the shear layer foundation
stiffness of Pasternak model.

My = B12 εx0 + (B22 + CyT ) εy0 + D12 k x + (D22 +

E0 I yT
syT

) k y + Φ 2,

0
+ 2D66 k xy.
Mxy = B66 γxy

(11)

where
3. Theoretical formulation
Taking into account the von Karman–Donnell geometrical
nonlinearity terms, the strains at the middle surface and curvatures relating to the displacement components u, v, w based on
the classical thin shell theory are [22,23]

⎛ ε 0⎞ ⎛
⎞
u, x + w ,2x/2
⎜ x⎟ ⎜
⎟
⎜ εy0 ⎟ = ⎜
2 ⎟,
⎜ ⎟ ⎜ v, y − w /R + w , y/2⎟
⎜ γ 0 ⎟ ⎜⎝ u, y + v, x + w , x w , y ⎟⎠
⎝ xy ⎠

⎛ kx ⎞ ⎛ − w ⎞
, xx
⎜ ⎟ ⎜
⎟
⎜ k y ⎟ = ⎜ − w , yy ⎟,
⎜
⎜ ⎟ ⎝ − w , xy ⎟⎠
⎝ k xy ⎠

E1
E1ν
E1
, A12 =
, A66 =
,
2 (1 + ν )
1 − ν2
1 − ν2
E2
E2 ν
E2
B11 = B22 =
, B12 =
, B66 =
,
2 (1 + ν )
1 − ν2
1 − ν2
E3

E3 ν
E3
D11 = D22 =
, D12 =
, D66 =
,
2 (1 + ν )
1 − ν2
1 − ν2
h /2
1
E (z ) α (z )ΔT (1, z ) dz,
(Φ1, Φ2 ) = −
1 − ν −h /2
A11 = A22 =

∫

(7)

and

0
where εx0 and εy0 are normal strains, γxy
is the shear strain at the
middle surface of the panel and k ij , ij = x, y, xy are the curvatures.
The strain components across the panel thickness at the distance z from the mid-plane are given by

dyT (hyT )3
dxT (hxT )3

+ A xT (z xT )2 , I yT =
+ A yT (z yT )2 ,
12
12
E0 A yT z yT
E AT zT
CxT = 0 Tx x , CyT =
,
sx
syT

⎛ 0⎞
⎛ kx ⎞
⎛ εx ⎞ ⎜ εx ⎟
⎜ εy ⎟ = ⎜ ε 0 ⎟ + z ⎜⎜ k ⎟⎟.
y
⎜γ ⎟ ⎜ y ⎟
⎜
⎟
⎝ xy ⎠ ⎜ γ 0 ⎟
2k xy ⎠
⎝
⎝ xy ⎠

hyT + hT
hxT + hT
, z yT =
,
2
2

⎛ 1
⎞
E h
1
−
E1 = Ec h + mc , E2 = Emc h2 ⎜
⎟,
⎝N+2
N+1
2 (N + 1) ⎠

sh
x ,

sh
σxy

)

σysh =

IxT =

z xT =

(8)

Hooke's law for cylindrical panel taking into account the temperature-dependent properties is defined as

(σ

(12)

E3 =

⎡ 1
⎤
Ec h3
1
1
+ Emc h3 ⎢
−
+
⎥,
⎣N+3
N+2
12
4 (N + 1) ⎦

(13)

E (z, T )
[(εx, εy )
1 − ν2

with the geometric shapes of stiffeners after the thermal deformation process in Eq. (13) can be determined as the follows:

+ ν ( εy, εx ) − (1 + ν ) α (z, T ) ΔT (1, 1)],

dxT = dx (1 + αm T (z )), dyT = dy (1 + αm T (z )),

E (z, T )
γ ,
=
2 (1 + ν ) xy

(9)

hyT = hy (1 + αm T (z )), z xT = z x (1 + αm T (z )),

and for stiffeners

(σxst , σyst ) = E0 (T )(εx, εy ) −

hxT = hx (1 + αm T (z )),

E0 (T )
α0 (T )ΔT (1, 1),
1 − 2ν

z yT = z y (1 + αm T (z )), sxT = sx (1 + αm T (z )),
(10)

where E0 (T ) , α0 (T ) are the Young's modulus and thermal expansion coefficient of the stiffeners, respectively. Unlike other
publications, in this paper, material properties of the eccentrically
outside stiffeners are assumed to depend on temperature.
All elastic moduli of FGM panels and stiffeners are assumed to
be temperature dependence and they are deformed in the presence of temperature. Therefore, the geometric parameters, the
panel's shape and stiffeners vary through the deforming process
due to the temperature change. However, because the thermal

stress of stiffeners is subtle which distributes uniformly through
the whole panel structure, we can ignore it. The contribution of
stiffeners can be accounted for using the Lekhnitsky smeared
stiffeners technique. Then integrating the stress–strain equations
and their moments through the thickness of the panel, the expressions for force and moment resultants of an eccentrically
stiffened FGM cylindrical panel are obtained as

syT = sy (1 + αm T (z )),

(14)

The nonlinear equilibrium equations of FGM cylindrical panels
based on classical shell theory are given as [22,23]

Nx, x + Nxy, y = 0

(15a)

Nxy, x + Ny, y = 0

(15b)

Mx, xx + 2Mxy, xy + My, yy +
− k1w + k2 ∇2w = 0,

Ny
+ Nx w , xx + 2Nxy w , xy + Ny w , yy + q
R
(15c)

where q is an external pressure uniformly distributed on the
surface of the panel.
The geometrical compatibility equation for an imperfect FGM
cylindrical panel is written as [22,23]


158

N.D. Duc et al. / Thin-Walled Structures 96 (2015) 155–168

0
εx0, yy + εy0, xx − γxy
w ,2xy − w , xx w , yy + 2w , xy w ,⁎xy − w , xx w ,⁎yy
, xy =

− w , yy w ,⁎xx −

w , xx
.
R

⁎
⁎
⁎
⁎
⁎
− 2A12
) f, xxyy + B21
A11
f, xxxx + A22

f, yyyy + (A66
w , xxxx

(16)

The first two equations of the nonlinear motion Eqs. (15a),
(15b) are automatically satisfied by choosing the stress function
f (x, y ) as

Nx = f, yy , Ny = f, xx , Nxy = − f, xy .

(17)

Substituting relation (17) into Eq. (11), we obtain

⁎
⁎
⁎
⁎
+ B12
+ B22
− 2B66
) w , xxyy
w , yyyy + (B11

⎛
− ⎜ w ,2xy − w , xx w , yy + 2w , xy w ,⁎xy − w , xx w ,⁎yy − w , yy w ,⁎xx
⎝
−

w , xx ⎞
⎟ = 0.
R ⎠

(23)

Eqs. (22) and (23) are nonlinear equations in terms of variables
w and f and they are used to investigate the nonlinear stability of
FGM eccentrically stiffened cylindrical panels on elastic
foundations.

⁎
⁎
⁎
⁎
⁎
⁎
εx0 = A22
f, yy − A12
f, xx + B11
w , xx + B12
w , yy − (A22
− A12
) Φ1,
⁎
⁎
⁎
⁎
⁎
⁎

εy0 = A11
f, xx − A12
f, yy + B21
w , xx + B22
w , yy − (A11
− A12
) Φ1,
⁎
⁎
0
γxy
f, xy + 2B66
w , xy,
= − A66

4. Solution procedures

(18)
In the present study, the edges of eccentrically stiffened FGM
cylindrical panel are assumed to be simply supported. Depending
on the in-plane restraint at the edges, three cases of boundary
conditions, labeled as Cases 1, 2 and 3 will be considered [5].

where
⁎
A11
=

E0 Ay ⎞
E A ⎞

A
1⎛
1⎛
⁎
⁎
⎟,
= 12 , A22
= ⎜ A22 +
⎜ A11 + 0 x ⎟, A12
sx ⎠
sy ⎠
Δ⎝
Δ
Δ⎝

⁎
A66
=

⎛
E0 Ay ⎞
1
E A ⎞⎛
2
⎟ − A12
, Δ = ⎜ A11 + 0 x ⎟ ⎜ A22 +
⎝
A66
sx ⎠ ⎝
sy ⎠

Case 1. Four edges of the FGM cylindrical panel are simply supported and freely movable (FM). The associated boundary conditions are

⁎
⁎
B11
= A22
( B11 + Cx ) − A12⁎ B12, B22⁎ = A11⁎ ( B22 + Cy ) − A12⁎ B12,
⁎
⁎
⁎
B12
B12 − A12
= A22
( B22 + Cy ), B21⁎ = A11⁎ B12 − A12⁎ ( B11 + Cx ),
⁎
B66
=

B66
.
A66

w = Nxy = Mx = 0, Nx = Nx0 at x = 0, a

(19)

Substituting once again Eq. (18) into the expression of
Mx, My, Mxy in Eq. (11), then Mx, My, Mxy into the Eq. (15c) leads to

w = Nxy = My = 0, Ny = Ny0 at y = 0, b,

(24)

Case 2. Four edges of the FGM cylindrical panel are simply supported and immovable (IM). In this case, boundary conditions are

⁎
⁎
⁎
⁎
⁎
B21
f, xxxx + B12
f, yyyy + (B11
+ B22
− 2B66
) f, xxyy

w = u = Mx = 0, Nx = Nx0 at x = 0, a

⁎
⁎
− D11
w , xxxx − D22
w , yyyy

w = v = My = 0, Ny = Ny0 at y = 0, b,

⁎
⁎

⁎
− (D12
+ D21
+ 4D66
) w , xxyy + Nx w , xx + 2Nxy w , xy

+ Ny w , yy +

Ny
+ q − k1w + k2 ∇2w = 0,
R

(20)

where
⁎
D11
= D11 +

⁎
D22
= D22 +

E0 IxT
⁎
⁎
− B11 + CxT B11
− B12 B21
,
sxT

(

E0 I yT
syT

(

w = Nxy = Mx = 0, Nx = Nx0 at x = 0, a

)

)
− (B

w = v = My = 0, Ny = Ny0 at y = 0, b,

⁎
⁎
⁎
D12
= D12 − B11 + CxT B12
− B12 B22
,
⁎
⁎
D21
= D12 − B12 B11
⁎
D66

= D66 −

22

)

⁎
+ CyT B21
,

⁎
B66 B66
.

(21)

For an imperfect cylindrical panel, Eq. (20) is modified into
form as

⁎
⁎
⁎
⁎
⁎
− D11
w , xxxx − D22
w , yyyy − (D12
+ D21
+ 4D66

) w , xxyy

+ f, yy w , xx − 2f, xy w , xy + f, xx w , yy +

⁎

f, xx
R

(26)

where Nx0, Ny0 are in-plane compressive loads at movable
edges (i.e. Case 1 and the first of Case 3) or are fictitious compressive edge loads at immovable edges (i.e. Case 2 and the second
of Case 3).
The mentioned conditions (24)–(26) can be satisfied identically
if the panel deflection w is chosen by [5,15,16]

w = W sin λm x sin δ n y ,

(27)

where λm = mπ /a , δn = nπ /b , m , n = 1, 2, ...are natural numbers
representing the number of half waves in the x and y directions,
respectively; W is the amplitude of deflection.
Concerning with the initial imperfection w ⁎, we introduce an
assumption it has the same form like the panel deflection w , i.e.

⁎
⁎
⁎

⁎
⁎
B21
f, xxxx + B12
f, yyyy + (B11
+ B22
− 2B66
) f, xxyy

+ k2 ∇2w = 0,

Case 3. All edges of the FGM cylindrical panel are simply supported. Two edges x = 0, a are freely movable, whereas the remaining two edges y = 0, b are immovable. For this case, the
boundary conditions are defined as

)

⁎
⁎
− B12 B12
− B22 + CyT B22
,

(

(25)

+ q − k1w
(22)

where w (x, y ) is a known function representing initial small imperfection of the panel.

Setting Eq. (18) into Eq. (16) gives the compatibility equation of
an imperfect eccentrically stiffened FGM cylindrical panel as

w ⁎ (x, y) = μh sin λm x sin δ n y ,

(28)

where the coefficient μ varying between 0 and 1 represents imperfection size.
Introduction of Eqs. (27) and (28) into the compatibility Eq.
(23), we define the stress function as


N.D. Duc et al. / Thin-Walled Structures 96 (2015) 155–168

f = A1 cos 2λm x + A2 cos 2δ n y + A3 sin λm x sin δ n y +

4.1. Mechanical stability analysis

1
Nx0 y2
2

1
Ny0 x2 ,
2

+

(29)

with

λ m2
δn2
W (W + 2μh), A2 =
W (W + 2μh)
⁎ 2
⁎
δn2
32A11λ m
32A22

A1 =

λ m2

A3 =

−

(

(B

⁎ 4
21λ m

(

⁎

⁎
⁎
⁎
δn4 + (B11
+ B12
+ B22
− 2B66
) λ m2

⁎ 4
λm
A11

+

⁎
δn4
A22

+

⁎
(A66

−

⁎
2A12
) λ m2 δn2

Consider a simply supported eccentrically stiffened
FGM cylindrical panel with all movable edges and resting on
elastic foundations. Two cases of mechanical loads will be
analyzed.
4.1.1. Eccentrically stiffened FGM cylindrical panel under uniform
external pressure
Consider an eccentrically stiffened FGM cylindrical panel with
movable edges and only subjected to uniform external pressure on
the upper surface of the panel. In this case, Nx0 = Ny0 = 0, and Eq.
(31) leads to

W

)
δ )
W.
)

⁎ 4
⁎
⁎
⁎
λ m + A22
δn4 + (A66
R A11
− 2A12
) λ m2 δn2

2
n

(30)

Setting Eqs. (25)–(27) into Eq. (22) and applying the Galerkin
procedure for the resulting equation we obtain equation for determining nonlinear static analysis of eccentrically stiffened FGM
cylindrical panels on elastic foundations.

(

(

−

+

)

)

+

⎤
⎥
⎥
⎥
⎥
⎥
⎥⎦

(

16Bh4

(m B

2 2
a

+ n2

)

2 2 2
⎡ ⁎ 4 4
⁎
⁎
⁎
⁎ 4⎤
m3nπ 4Ba2 Rb ⎣ B21m Ba + (B11 + B22 − 2B66 ) m n Ba + B12 n ⎦
⎡ A ⁎ m4 B 4 + (A ⁎ − 2A ⁎ ) m2n2B 2 + A ⁎ n4 ⎤
8Bh3
⎣ 11
⎦
a
a
22
66
12

⁎

16Bh2 ⎡⎣ A11
m4 Ba4

m5nπ 2Ba4 Rb2

⁎
⁎
⁎
+ (A66
− 2A12
) m2n2Ba2 + A22
n4 ⎤⎦

2m4 n2π 2Ba4 Rb

⁎
⁎
⁎
⁎
m4 Ba4 + (A66
n4 ⎤⎦
3Bh3 ⎡⎣ A11
− 2A12
) m2n2Ba2 + A22

⁎
⁎
⁎
⁎
⁎ 4⎤

m4 Ba4 + (B11
n ⎦
2m2n2π 4Ba2 ⎡⎣ B21
+ B22
− 2B66
) m2n2Ba2 + B12
⁎
⁎
⁎
⁎
m4 Ba4 + (A66
n4 ⎤⎦
3Bh4 ⎡⎣ A11
− 2A12
) m2n2Ba2 + A22

,

⁎
B⁎ ⎞
m2n2π 4Ba2 ⎛ B21
π 2n2Rb
⎜⎜ ⁎ + 12
⎟,
+
⁎ ⎟
4
⁎ 3
A22
24A11 Bh

12Bh ⎝ A11
⎠

mnπ 6 ⎛ m4 Ba4
n4 ⎞
⎜
+ ⁎ ⎟⎟,
4 ⎜ A⁎
A11 ⎠
256Bh ⎝ 22

(33)

k1a4
k a2
⁎
⁎
⁎
⁎
⁎
⁎
= hA11
= hA22
= hA12
, K2 = 2 ⁎ , A11
, A22
, A12
,
⁎
D11

D11
⁎
⁎
⁎
A66
= hA66
=
, B11

⁎
B11

⁎
B⁎
B⁎
B22
⁎
⁎
= 12 , B21
= 21 ,
, B12
h
h
h
⁎
⁎
D12
D22
⁎
= 3 , D12 = 3 ,

h
h

⁎
=
, B22

h
⁎
D11
⁎
=
= 3 , D22
,
h
h
⁎
D⁎
D66
⁎
⁎
¯ = W /h,
= 21
=
D21
,
D
, Bh = b/h, Ba = b/a, W
66
h3

h3

⁎
B66

⁎
B66

⁎
D11

Rb = b/ R.

δ4 ⎞
mnπ 2 ⎛ λ m4
⎜ ⁎ + n⁎ ⎟ W (W + μh)(W + 2μh)
A11 ⎠
64λm δ n ⎝ A22

mnπ
Nx0 λ m2 + Ny0 δn2 (W + μh)
4λ m δ n
4 Ny0
4q
+
+
= 0,
λm δ n R
λm δ n

⁎
mnπ 4Ba2 D11
K2

16Bh4

K1 =

W (W + μh)
⁎
⎡
⎤
B⁎
2 B
δn
+⎢
− ( 21
+ 12
) λm δ n ⎥ W (W + 2μh)
⁎
⁎
⁎
A22
3 A11
⎣ 6A11Rλm
⎦

2

+

and

)

(

(32)

⁎
⁎
⁎
⁎
⁎
+ mn5π 6D22
+ m3n3π 6Ba2 (D12
+ D21
+ 4D66
)
m5nπ 6Ba4 D11

b41 =

−

16Bh4

b21 = −

+

−

⁎
mnπ 2Ba4 D11
K1

b31 = −

(

)

2
2 2 2
⎡ ⁎ 4 4
⁎
⁎
⁎
⁎ 4⎤
mnπ 6 ⎣ B21m Ba + (B11 + B22 − 2B66 ) m n Ba + B12 n ⎦
,
+
⎡ A ⁎ m4 B 4 + (A ⁎ − 2A ⁎ ) m2n2B 2 + A ⁎ n4 ⎤
16Bh4
⎣ 11
⎦
a
a
22

66
12

W

(

)(

)

where

)

8λ m δ n
3
⎡
λ m2
1
⎢
⁎ 4
⁎
⁎
⁎
⎢ A11
λ m + A22
δn4 + (A66
− 2A12
) λ m2 δn2 R

⎢
⁎ 4
⁎
⁎
⁎
⁎
⎢
B21
λ m + B12
δn4 + (B11
+ B22
− 2B66
) λ m2 δn2
⎢−
⁎ 4
⁎
⁎
⁎
4
2 2
⎢⎣
A11λ m + A22 δn + (A66 − 2A12 ) λ m δn

(

¯ W
¯ +μ W
¯ + 2μ ,
+ b41 W

+

mnπ 2
4λ m δ n
⎧
⁎
⁎
⁎
⁎
4
2 2⎤ ⎫
2 ⎡ ⁎ 4
⎪ 2 λ m ⎣ B21λ m + B12 δn + (B11 + B22 − 2B66 ) λ m δn ⎦ ⎪
⎪ R
⎡ A ⁎ λ 4 + A ⁎ δ 4 + (A ⁎ − 2A ⁎ ) λ 2 δ 2 ⎤ ⎪
⎣ 11 m
22 n
66
12 m n ⎦
⎪
⎪
⎪
⎪
2
⁎ 4
⁎
⁎
⁎
⁎
λ m + B12

δn4 + (B11
+ B22
− 2B66
) λ m2 δn2 ⎤⎦ ⎪
⎪ ⎡⎣ B21
⎪−
⎪
⎡ A ⁎ λ 4 + A ⁎ δ 4 + (A ⁎ − 2A ⁎ ) λ 2 δ 2 ⎤
⎪
⎪
⎣ 11 m
22 n
66
12 m n ⎦
⎨
⎬
⎪ λ4
⎪
1
⎪− m
⎪
2 ⎡ ⁎ 4
⁎
⁎
⁎
⎪ R ⎣ A11λ m + A22
⎪
δn4 + (A66
− 2A12
) λ m2 δn2 ⎤⎦

⎪
⎪
⎪ −D ⁎ λ 4 − D ⁎ δ 4 − (D ⁎ + D ⁎ + 4D ⁎ ) λ 2 δ 2
⎪
m
m n
22 n
11
12
21
66
⎪
⎪
2
2
⎪ − k2 λ m + δn − k1
⎪
⎩
⎭

)

¯ + b21W
¯ W
¯ + μ + b31W
¯ W
¯ + 2μ
q = b11W

b11 =

(

159

(34)

Eq. (32) may be used to trace postbuckling load–deflection
curves of FGM cylindrical panels resting on elastic foundations
subjected to uniform external pressure.
For a perfect panel (μ = 0) , Eq. (32) leads to

)

(31)

where m , n are odd numbers. Hereafter, we will consider in detail
three problems corresponding to three mentioned loading types.

¯ + (b21 + b31) W
¯ 2 + b41 W
¯ 3.
q = b11W

(35)


160

N.D. Duc et al. / Thin-Walled Structures 96 (2015) 155–168

4.1.2. Eccentrically stiffened FGM cylindrical panel under axial
compressive loads
A movable edges eccentrically stiffened cylindrical panel supported by elastic foundations and subjected to axial compressive
loads Fx uniformly distributed at two curved edges x = 0, a in the
absence of external pressure and thermal loads is considered.
In this case, the prebuckling force resultants are

q = 0, Ny0 = 0, Nx0 = − Fx h.

(36)

(

)

¯ W
¯ + 2μ
¯
W
W
¯ + b32
¯ W
¯ + 2μ ,
+ b22 W
+ b42 W
¯
¯
W+μ
W+μ

(

)

(37)

where

b12 =
+
π

⎧⎡
⎤
2
λm
1
⎪⎢
⎥
δ n2
⎪⎢
⎥
⁎ 4
⁎ 4
⁎
⁎
2 2 R
⎪ ⎢ A11λm + A22 δ n + (A66 − 2A12 ) λm δ n
⎥

⎪⎢
⎥
⎪⎢
4 + B ⁎ δ 4 + (B ⁎ + B ⁎ − 2 B ⁎ ) λ 2 δ 2
⎥
B ⁎ λm
n
m n
4 ⎪⎢
21
12
11
22
66
2
⎨ −
Nx0 = Φ1 +
δn ⎥
⎢
2
⎥
⁎ λ 4 + A ⁎ δ 4 + (A ⁎ − 2A ⁎ ) λ 2 δ 2
mnπ ⎪
A
⎥⎦
⎪ ⎢⎣
11 m
22 n
66
12 m n

⎪
⎡
A⁎ ⎤
⎪
1
2 + (B ⁎ A ⁎ + B ⁎ A ⁎ ) δ 2 − 12 ⎥
⎢ (B ⁎ A ⁎ + B ⁎ A ⁎ ) λm
⎪
21 12
12 11
22 12 n
⁎ 2 ⎢⎣ 11 11
R ⎥⎦
⎪ ⁎ ⁎
⎩ (A11A22 − A12 )

(

)

(

⁎
m2π 2Ba2 D11

Bh2

+

⁎

n4 π 2D22

+

m2Ba2 Bh2

4 4
⁎
⎣ A11 m Ba

+

Bh2

⁎
⁎
n4 ⎤⎦
− 2A12
) m2n2Ba2 + A22

⁎
(A66

)

(

⁎
⁎
⁎

n2π 2 (D12
+ D21
+ 4D66
)

m2Ba2 Rb2

2⎡

(41)

Substitution of Eqs. (27)–(29) into Eq. (41) and then the result
into Eq. (40) give fictitious edge compressive loads

The introduction of Eq. (36) into Eq. (31) gives

Fx = b12

∂u
⁎
⁎
⁎
⁎
⁎
⁎
f, yy − A12
f, xx + B11
w, xx + B12
w, yy − (A22
= A22

− A12
) Φ1
∂x
1
− w ,2x − w , x w ,⁎x,
2
∂v
⁎
⁎
⁎
⁎
⁎
⁎
f, xx − A12
f yy + B22
w, yy + B21
w, xx − (A11
= A11
− A12
) Φ1
∂y
1
w
.
− w ,2y − w , y w ,⁎y +
2
Ry

+

)

1
2 + A ⁎ δ 2 ) W (W + 2μh) ,
(A ⁎ λm
11
12 n
8 (A ⁎ A ⁎ − A ⁎ 2 )
11 22
12

⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎬W
⎪
⎪
⎪
⎪
⎪
⎪
⎭

(42)

2

+

2 2 2
⎡ ⁎ 4 4
⁎
⁎
⁎
⁎ 4⎤
π 2 ⎣ B21m Ba + (B11 + B22 − 2B66 ) m n Ba + B12 n ⎦
⎡ A ⁎ m4 B 4 + (A ⁎ − 2A ⁎ ) m2n2B2 + A ⁎ n4 ⎤
m2Ba2 Bh2
⎣ 11
⎦
a
a
22
66
12

⎧⎡
⎤
4
λm
1
⎪⎢
⎥
⎪⎢
⎥
⁎ 4

⁎ 4
⁎
⁎
2 2 R
⎪ ⎢ A11λm + A22 δ n + (A66 − 2A12 ) λm δ n
⎥
⎪⎢
⎥
⎪⎢
⁎
⁎
⁎
⁎
⁎
4
4
2
2
⎥
B λ m + B δ n + (B + B − 2 B ) λ m δ n
4 ⎪⎢
21
12
11
22
66
2
⎨ −
Ny0 = Φ1 +
λm⎥

⎥
⁎ λ 4 + A ⁎ δ 4 + (A ⁎ − 2A ⁎ ) λ 2 δ 2
mnπ 2 ⎪ ⎢
A
⎥⎦
⎪ ⎢⎣
11 m
22 n
66
12 m n
⎪
⎡
A⁎ ⎤
⎪
1
⁎
⁎
⁎
⁎
⁎
⁎
2
⎢
(B A + B A ) λm + (B A + B ⁎ A ⁎ ) δ n2 − 22 ⎥
⎪
21 22
12 12
22 22
R ⎥⎦
⎪ (A ⁎ A ⁎ − A ⁎ 2 ) ⎢⎣ 11 12

⎩ 11 22
12
1
2 + A ⁎ δ 2 ) W (W + 2μh) .
+
(A ⁎ λm
12
22 n
8 (A ⁎ A ⁎ − A ⁎ 2 )
11 22
12

(

+

(

)

⁎
D11
m2Ba2 + n2 K2

m2Bh2

b22 = −

+

⁎
Ba2 D11
K1

m2π 2Bh2

,

32mnBa2 Rb

b42 =

+

⁎
B⁎ ⎞
8n ⎛ B21
⎜
⎟,
+ 12
⁎
2 A⁎
A22
3mBh ⎝ 11
⎠

⎛ m4 B 4
π2
n4 ⎞
⎜ ⁎ a + ⁎ ⎟⎟.

2 2 2⎜
A11 ⎠
16m Ba Bh ⎝ A22

(38)

Eq. (37) is employed to trace postbuckling load–deflection
curves of the imperfect eccentrically stiffened FGM panel subjected
to axial compressive loads.
For a perfect cylindrical panel (μ = 0) only subjected to axial
compressive load Fx , Eq. (37) leads to

¯ + b42 W
¯ 2,
Fx = b12 + (b22 + b32 ) W

(39)

From which upper buckling compressive load may be obtained
with W → 0 as Fx = b12.
4.2. Thermal stability analysis
A simply supported eccentrically stiffened FGM cylindrical panel on elastic foundations with all immovable edges is considered.
The panel is subjected to uniform external pressure q and simultaneously exposed to temperature environments. The in-plane
condition on immovability at all edges, i.e. u = 0 at x = 0, a and
v = 0 at y = 0, b, is fulfilled in an average sense as [5]
b

∫0 ∫0

a

∂u
dxdy = 0,
∂x

a

∫0 ∫0

b

∂v
dydx = 0.
∂y

)

(43)

Introducing Nx0, Ny0 at Eqs. (42), (43) into Eq. (31) gives

⁎
⁎
⁎
⁎
m4 Ba4 + (A66
n4 ⎤⎦
3π 2Bh ⎡⎣ A11
− 2A12
) m2n2Ba2 + A22

2nRb
⁎
3m3π 2Ba2 Bh A11

)

(

2 2 2
⎡ ⁎ 4 4
⁎
⁎
⁎
⁎ 4⎤
32n ⎣ B21m Ba + (B11 + B22 − 2B66 ) m n Ba + B12 n ⎦
,
+
⁎
⁎
⁎
⁎
3mBh2 ⎡⎣ A11
m4 Ba4 + (A66
n4 ⎤⎦
− 2A12
) m2n2Ba2 + A22

b32 = −

)

(

2 2 2
⎡ ⁎ 4 4
⁎
⁎
⁎
⁎ 4⎤
2Rb ⎣ B21m Ba + (B11 + B22 − 2B66 ) m n Ba + B12 n ⎦
−
⎡ A ⁎ m4 B 4 + (A ⁎ − 2A ⁎ ) m2n2B2 + A ⁎ n4 ⎤
Bh
⎣ 11
⎦
a
a
22
66
12

⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎬W

⎪
⎪
⎪
⎪
⎪
⎪
⎭

(40)

From Eqs. (7) and (18) one can obtain the following expressions
in which initial imperfection has been included

⎧
⎡ ⁎ 4
⁎ 4
⁎
⁎
⁎
2 2⎤
⎪ λ 2 ⎣⎢ B21λm + B12 δ n + (B11 + B22 − 2B66 ) λm δ n ⎦⎥
⎪2 m
⎡ ⁎ 4
⁎ δ 4 + (A ⁎ − 2 A ⁎ ) λ 2 δ 2 ⎤
⎪ R
+
A
A
λ
⎣⎢ 11 m

22 n
66
12 m n ⎦⎥
⎪
⎪
2
⎪ ⎡ ⁎ 4
⁎ δ 4 + (B ⁎ + B ⁎ − 2 B ⁎ ) λ 2 δ 2 ⎤
B
B
λ
+
⎪ ⎣⎢ 21 m
12 n
11
22
66 m n ⎦⎥
1 ⎪−
⎡
⎤
⎨
⁎ 4
⁎ 4
⁎
⁎
Φ1 =
2 2
2⎪
⎣⎢ A11λm + A22 δ n + (A66 − 2A12 ) λm δ n ⎦⎥
λm

⎪
4
⎪ λm
1
⎪−
4 + A ⁎ δ 4 + (A ⁎ − 2A ⁎ ) λ 2 δ 2 ⎤
⎪ R 2 ⎡⎢ A ⁎ λm
⎣ 11
22 n
66
12 m n ⎦⎥
⎪
⎪
⁎
4
2δ2 − k λ2 + δ2 − k
⎪ − D λm − D ⁎ δ n4 − (D ⁎ + D ⁎ + 4D ⁎ ) λm
2 m
1
n
n
11
22
12
21
66
⎩

(

⎡
⎡
2
λm
1
⎢
⎢
⎢
⎢
4 + A ⁎ δ 4 + (A ⁎ − 2A ⁎ ) λ 2 δ 2 R
A ⁎ λm
n
m
n
⎢
⎢
22
66
12
32δ n2 ⎢ 11
+⎢
4 + B ⁎ δ 4 + (B ⁎ + B ⁎ − 2B ⁎ ) λ 2 δ 2
⎢ 3mnπ 2 ⎢
B ⁎ λm
66 m n
21
12 n
11
22
⎢

⎢−
⎢
⎢
⁎ λ 4 + A ⁎ δ 4 + (A ⁎ − 2A ⁎ ) λ 2 δ 2
A
m
n
⎢⎣
⎢⎣
11
22
66
12 m n

(

)

(

(

)

)

)

⎤
⎥

⎥
⎥
⎥
⎥
⎥
⎥
⎥⎦

⎧⎡
⎤
2
λm
1
⎪⎢
⎥
δ n2
⎪⎢
⎥
⁎ 4
⁎ 4
⁎
⁎
2 2 R
⎪ ⎢ A11λm + A22 δ n + (A66 − 2A12 ) λm δ n
⎥
⎪⎢
⎥
⎪⎢
⁎
⁎

⁎
⁎
⁎
4
4
2
2
⎥
B λm + B δ n + (B + B
− 2B ) λ m δ n
4 ⎪⎢
21
12
11
22
66
2
⎨ −
−
δn ⎥
⎥
⁎ λ 4 + A ⁎ δ 4 + (A ⁎ − 2A ⁎ ) λ 2 δ 2
mnπ 2 ⎪ ⎢
A
⎥⎦
⎪ ⎢⎣
11 m
22 n
66
12 m n

⎪
⎡
⎪
A⁎ ⎤
1
⁎
⁎
⁎
⁎
⁎
⁎
2
⎢
⎪+
(B A + B A ) λm + (B A + B ⁎ A ⁎ ) δ n2 − 12 ⎥
21 12
12 11
22 12
R ⎥⎦
⎪ (A ⁎ A ⁎ − A ⁎ 2 ) ⎢⎣ 11 11
⎩
11 22
12
⎡
⎤
⁎
B⁎
δ n2
⎥ W (W + 2μh)
4 ⎢

2 B
+
− ( 21
+ 12
) δ n2 ⎥
⎢
⁎
2
A⁎
(W + μh)
3 A⁎
mnπ 2 ⎣⎢ 6A11Rλm
11
22
⎦⎥
⎡
⎤
⎛ ⁎ 4
⎞
⁎
⁎
⁎
4
2
(A λm + A δ n2 ) ⎥
⎢ 1 ⎜ A11λm + A22 δ n ⎟
11
12
−⎢
⁎ A⁎

⎥ W (W + 2μh) .
⎟⎟ +
⁎
⁎
⁎
2 ⎜⎜
2
A
λ
16
⎢⎣
m⎝
11 22
⎠ 8 (A11A22 − A12 ) ⎥⎦

(

)

(

)

(

)

⎫
⎪
⎪

⎪
⎪
⎪
⎪
⎪
⎪
W
⎬
⎪ (W + μh)
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎭

⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎬W
⎪
⎪
⎪
⎪

⎪
⎪
⎭

(44)

In this paper, the eccentrically stiffened FGM cylindrical panel is exposed to temperature environments uniformly raised
from stress free initial state Ti to final value Tf and temperature
increment ΔT = Tf − Ti is considered to be independent from
thickness variable. The thermal parameter is obtained from Eq.
(12) as

Φ1 = − PhΔT,
where

(45)


N.D. Duc et al. / Thin-Walled Structures 96 (2015) 155–168

P=

1 ⎡
Emc αc + Ec αmc
E α ⎤
+ mc mc ⎥.
⎢ Ec αc +
1 −v⎣
N+1
2N + 1 ⎦

161

(46)

Setting Eq. (45) into Eq. (44) gives

ΔT = b13

(

)

¯ W
¯ + 2μ
¯
W
W
¯ + b33
¯ W
¯ + 2μ ,
+ b23 W
+ b43 W
¯
¯
W+μ
W+μ

(

)

(47)

bi3

in which specific expressions of coefficients
(i = 1, 4) are
given in Appendix A.
Eq. (47) shows the relationship of thermal load–deflection of
the eccentrically stiffened FGM panel in postbuckling state and
used to trace postbuckling curves of the FGM panel under
thermal load. The two sides of Eq. (47) are temperature dependence which makes it very complex. The iterative algorithm
is used to determine the deflection–load relations in the buckling period of the FGM panel. To be more specific, given the
volume fraction index
N , the geometrical parameters
(b /a, b /h, b /R ) and the value of W /h, we can use these values to
determine ΔT in Eq. (47) as the follows: we choose an initial
step for ΔT1 on the right side in Eq. (47) with ΔT = 0
( T = T0 = 300 K ). In the next iterative step, we replace the known
value of ΔT found in the previous step to determine the right
side of Eq. (47), ΔT2. This iterative procedure will stop at the
kth -steps if ΔTk satisfies the condition |ΔT − ΔTk | ≤ ε . Here, ΔT is a
desired solution for the temperature and ε is a tolerance used in
the iterative steps.
If the imperfection μ = 0 and W → 0, from above expression
(47) gives ΔT = b13.
Fig. 2. Comparisons of nonlinear load–deflection curves with results of Tung [15]
for the unstiffened FGM cylindrical panel under axial compressive loads.

4.3. Thermo-mechanical stability analysis
The simply supported FGM cylindrical panel with tangentially
restrained edges is assumed to be subjected to external pressure q
uniformly distributed on the outer surface of the panel and exposed to uniformly raised temperature field.
Subsequently, setting Eq. (45) into Eqs. (42) and (43) then the
result into Eq. (31) give

(

)

(

¯ + b24 W
¯ W
¯ + μ + b34 W
¯ W
¯ + 2μ
q = b14 W
+

¯
b44 W

( W¯ + μ)( W¯ + 2μ) −

b54 P ΔT ,

)
(48)

in which specific expressions of coefficients bi4 (i = 5) are given in
Appendix B.
Eq. (48) expresses explicit relation of pressure-deflection
curves for eccentrically stiffened FGM cylindrical panels
rested on elastic foundations and under combined action of
uniformly raised temperature field and uniform external
pressure.

5. Numerical results and discussion
5.1. Validation of the present approach
To validate the present study, firstly, Fig. 2 compares the
results of this paper for an unstiffened FGM cylindrical panel
under axial compressive loads with the results given in work of
Tung [15] with different values of elastic foundation stiffness K1
and K2 in the case of temperature independent properties. Fx is
found from Eq. (37) and the data base in this case is taken:
b /a = 1, b /h = 50, b /R = 0.1, N = 1, μ = 0.1, K1 = K2 = 0.
Secondly, Fig. 3 compares the present results with those of Duc
et al. [16] for stiffened and unstiffened FGM cylindrical panel under uniform external pressure based on classical shell theory in
the case of temperature independent properties. The input

Fig. 3. Comparisons of nonlinear load–deflection curves with results of Duc et al.
[16] for the unstiffened and stiffened FGM cylindrical panel under uniform external
pressure.


162

N.D. Duc et al. / Thin-Walled Structures 96 (2015) 155–168

Fig. 4. Comparisons of thermal postbuckling behavior with results of Shen and
Wang [1] for the unstiffened FGM cylindrical panels under uniform temperature
rise.

parameters are: b/a ¼1, b/h¼ 50, b/R ¼0.5, K1 ¼10, K2 ¼30 where q
is found from Eq. (32).
Finally, the thermal postbuckling behavior of Si3N4/SUS304
unstiffened FGM cylindrical panels under uniform temperature
rise with two values of Winkler elastic foundation stiffness are
compared in Fig. 4 with the theoretical results of Shen and
Wang [1] based on the higher order shear deformation shell
theory. The temperature dependent properties are taken
into account. The input parameters are chosen as:
N = 0.5, m = n = 1, h = 5 mm , b /h = 40, a/b = 1.2, a/R = 0.5, .

Fig. 5. Effects of temperature increment on the nonlinear response of the eccentrically stiffened FGM cylindrical panels under uniform external pressure (movable
edges).

5.2. Effects of temperature
Figs. 5–7 show effects of temperature increment ΔT on the
nonlinear response of the eccentrically stiffened FGM cylindrical
panels under uniform external pressure (movable edges), axial
compressive loads (movable edges) and uniform external pressure (immovable edges), respectively. Obviously, the load-carrying capacity of the panel with temperature independent
properties is higher than the one of the panel with temperature
dependent properties. Moreover, the increase of temperature
increment leads to the reduction of load-carrying capacity of the
panel.

μ = 0.05

The elastic foundation stiffness in this comparison are identified as
K1⁎ =

k1b4
Em h3

, K2⁎ =

k 2 b2
Em h3

with Em is determined at room temperature.

As can be seen that good agreements are obtained in these
three comparisons.
Next, we will investigate the effects of the volume fraction index, the geometrical dimensions, elastic foundations, imperfections and stiffeners on the nonlinear response of the eccentrically
stiffened FGM cylindrical panel.
The effective material properties with dependent temperature
in Eq. (5) are listed in Table 1 [1,2,13]. The Poisson's ratio is v = 0.3.
The parameters for the stiffeners are [16]

5.3. Effects of elastic foundations and initial imperfection
Figs. 8–11 indicate effects of elastic foundations on the
nonlinear response of the eccentrically stiffened FGM cylindrical panels under uniform external pressure (movable edges),
axial compressive loads (movable edges), uniform temperature rise (immovable edges) and uniform external pressure
(immovable edges), respectively. Obviously, the load-carrying
capacity of the panel becomes considerably higher due to the
support of elastic foundations. In addition, the beneficial effect

s1 = s2 = 0.4 m, z1 = z2 = 0.0225 m, h1 = h2 = 0.003 m,

d1 = d2 = 0.004 m.

Table 1
Material properties of the constituent materials of the considered FGM panel.
Material

Property

P0

PÀ1

P1

P2

P3

Si3N4 (Ceramic)

E (Pa)
ρ (kg/m3)

348.43e9
2370
5.8723eÀ 6

0
0
0

À 3.70eÀ 4
0
9.095e À 4

2.160e À 7
0
0

À 8.946e À 11
0
0

13.723

0

0

0

0

201.04e9
8166
12.330eÀ 6

0
0
0

3.079eÀ 4
0
8.086e À 4

À 6.534e À 7
0
0

0
0
0

15.379

0

0

0

0

α (K−1)
k (W/mK)
SUS304 (Metal)

E (Pa)
ρ (kg/m3)
α (K−1)

k (W/mK)


N.D. Duc et al. / Thin-Walled Structures 96 (2015) 155–168

Fig. 6. Effects of temperature increment on the nonlinear response of the eccentrically stiffened FGM cylindrical panels under axial compressive loads (movable
edges).

163

Fig. 8. Effects of elastic foundations on the nonlinear response of the eccentrically
stiffened FGM cylindrical panels under uniform external pressure (movable edges).

Fig. 7. Effects of temperature increment on the nonlinear response of the eccentrically stiffened FGM cylindrical panels under uniform external pressure (immovable edges).

of the Pasternak foundation on the postbuckling response of
the eccentrically stiffened FGM cylindrical panels is better
than the Winkler one. The effects of initial imperfection with
the coefficient μ on the nonlinear response of the eccentrically
stiffened FGM cylindrical panels under different type of loads
are also shown in Figs. 8–11. It can be seen that the perfect
cylindrical panel has a better mechanical and thermal loading
capacity than those of the imperfect panel.
5.4. Effects of volume fraction index
Figs. 12–15 show effect of volume fraction index N on the

Fig. 9. Effects of elastic foundations on the nonlinear response of the eccentrically
stiffened FGM cylindrical panels under axial compressive loads (movable edges).

nonlinear response of the imperfect and perfect eccentrically

stiffened FGM cylindrical panels under uniform external pressure
(movable edges), axial compressive loads (movable edges), uniform temperature rise (immovable edges) and uniform external
pressure (immovable edges), respectively. As expected, the loadcarrying capacity of the FGM panel gets better if the volume N
increases. This is reasonable because when N is increased, the
ceramic volume fraction is increased; however, elastic module of
ceramic is higher than metal ( Ec > Em ). The results from these


164

N.D. Duc et al. / Thin-Walled Structures 96 (2015) 155–168

Fig. 12. Effect of volume fraction index N on the nonlinear response of the eccentrically stiffened FGM cylindrical panels under uniform external pressure
(movable edges).

Fig. 10. Effects of elastic foundations on the nonlinear response of the eccentrically
stiffened FGM cylindrical panels under uniform temperature rise (immovable
edges).

Fig. 13. Effect of volume fraction index N on the nonlinear response of the eccentrically stiffened FGM cylindrical panels under axial compressive loads (movable edges).

figures also show that the buckling and postbuckling load is very
sensitive to the change of initial imperfection.
5.5. Effects of stiffeners

Fig. 11. Effects of elastic foundations on the nonlinear response of the eccentrically
stiffened FGM cylindrical panels under uniform external pressure (immovable
edges).

The influences of stiffeners as well as initial imperfection on

the nonlinear postbuckling response of FGM cylindrical panels
under uniform external pressure (movable edges), axial compressive loads (movable edges), uniform temperature rise (immovable edges) and uniform external pressure (immovable edges)


N.D. Duc et al. / Thin-Walled Structures 96 (2015) 155–168

165

Fig. 14. Effect of volume fraction index N on the nonlinear response of the eccentrically
stiffened FGM cylindrical panels under uniform temperature rise (immovable edges).

Fig. 16. Effects of stiffeners on the nonlinear response of FGM cylindrical panels
under uniform external pressure (movable edges).

Fig. 15. Effect of volume fraction index N on the nonlinear response of the eccentrically stiffened FGM cylindrical panels under uniform external pressure (immovable edges).

are shown in Figs. 16–19, respectively. We can see that postbuckling load-carrying capability of the stiffened panel is higher than
that of unstiffened panel. In other words, the stiffeners can enhance the loading capacity for the cylindrical FGM panel. Moreover, from these figures, the initial imperfection considerably impact on the nonlinear response of stiffened and unstiffened FGM
cylindrical panel.

Fig. 17. Effects of stiffeners on the nonlinear response of FGM cylindrical panels
under axial compressive loads (movable edges).


166

N.D. Duc et al. / Thin-Walled Structures 96 (2015) 155–168

Fig. 20. Effects of ratio b/a on the nonlinear response of the eccentrically stiffened
FGM cylindrical panels under uniform external pressure (movable edges).

Fig. 18. Effects of stiffeners on the nonlinear response of FGM cylindrical panels
under uniform temperature rise (immovable edges).

Fig. 21. Effects of ratio b/h on the nonlinear response of the eccentrically stiffened
FGM cylindrical panels under axial compressive loads (movable edges).

Fig. 19. Effects of stiffeners on the nonlinear response of FGM cylindrical panels
uniform external pressure (immovable edges).

5.6. Effects of geometrical parameters
Figs. 20–22 analyze the effects of geometrical parameters on
the nonlinear response of eccentrically stiffened FGM cylindrical
panels. Specifically, Figs. 20 and 21 indicate the influences of ratios

b /a, b /h on the nonlinear response of eccentrically stiffened FGM
panels under uniform external pressure (movable edges) and axial
compressive loads (movable edges), respectively. The thermal
postbuckling behavior of the eccentrically stiffened FGM panels
with various values of ratio b /R is illustrated in Fig. 22. As can be
observed, the load-carrying capacity of the eccentrically stiffened
FGM panel increases when increasing the ratios b /a, b /h, and b /R.
The influences of initial imperfection on the nonlinear postbuckling of the eccentrically stiffened FGM cylindrical panels also


N.D. Duc et al. / Thin-Walled Structures 96 (2015) 155–168

167

Appendix A

⎡ ⁎ 4 4
⎤
⎧
⎫
⁎
⁎
⁎
B m B + B ⁎ n4 + (B11
+ B22
− 2B66
) m2n2Ba2 ⎦
⎪ 2 Rb ⎣ 21 a 12
⎪
⎡
⎤
⁎
⁎
⁎
⁎
4
4
2
2
4
2
B
⎪ h ⎣ A11 m Ba + A22 n + (A66 − 2A12 ) m n Ba ⎦
⎪
⎪
⎪

2
⎡ ⁎ 4 4
⁎ 4
⁎
⁎
⁎
⎪
2 2 2⎤
⎪
2
⎣ B21m Ba + B12 n + (B11 + B22 − 2B66 ) m n Ba ⎦
π
⎪−
⎪
2 2 2
⎡ ⁎ 4 4
⁎ 4
⁎
⁎
2 2 2⎤
m
B
B
⎪
⎪
a h
⎣ A11 m Ba + A22 n + (A66 − 2A12 ) m n Ba ⎦
⎪
⎪
⎪

1 ⎪ m2B 2 R 2
3
⎨
⎬
1
a b
b1 = −
P ⎪ − π 2 ⎡ A⁎ m4B 4 + A⁎ n4 + (A ⁎ − 2A⁎ ) m2n2B 2 ⎤
⎪
a
a
⎣ 11
⎦
22
66
12
⎪
⎪
2 2⎪
⎪
π2
4 4
⁎
⁎ 4
⁎
⁎
⁎ n π
⎪ − m2B 2 B 2 D11m Ba + D22 n − (D12 + D21 + 4D66 ) B 2 ⎪
a h
h ⎪

⎪
⎪ D ⁎ K2 m2Ba2 + n2
⎪
⁎
D11
K1Ba2
)
11 (
⎪
⎪
− 2 2
2
⎪
⎪
B
m Bh
⎩
⎭
h

(

)

, b23

Fig. 22. Effects of ratio b/R on the nonlinear response of the eccentrically stiffened
FGM cylindrical panels under uniform temperature rise (immovable edges).

shown in Figs. 20–22.

6. Concluding remarks
The paper presents an analytical investigation on the buckling and
postbuckling of imperfect eccentrically stiffened thin FGM cylindrical
panels resting on elastic foundations subjected to mechanical loads,
thermal loads and the combination of these loads. The formulations
are based on the classical shell theory taking into account geometrical
nonlinearity, initial imperfection, the Lekhnitsky smeared stiffeners
technique and Pasternalk elastic foundations. The Galerkin method is
used to obtain explicit expressions of load–deflection curves. The results show that elastic foundations and stiffeners have a beneficial
influence on the buckling loads and postbuckling load carrying capacity of the FGM cylindrical panels. The study also shows the effects of
volume fraction index, imperfection and geometrical parameters on
the nonlinear response of FGM cylindrical panels.

⎧ ⎡ 32mnB 2 R
⎫
⎤
1
a b
⎪⎢
⎪
⎥
2
⁎
⁎ 4
⁎
⁎
4 4
2 2 2
⎪ ⎢ 3π Bh ( A11 m Ba + A22 n + (A66 − 2A12 ) m n Ba ) ⎥

⎪
⎪⎢
⎪
⁎
⁎ 4
⁎
⁎
⁎
4 4
2 2 2 ⎥
⎪ ⎢ 32n ( B21m Ba + B12 n + (B11 + B22 − 2B66 ) m n Ba ) ⎥
⎪
−
⎪ ⎢ 3mBh2 A⁎ m4B 4 + A⁎ n4 + (A ⁎ − 2A⁎ ) m2n2B 2 ⎥
⎪
a
a
(
)
11
22
66
12
⎦
⎪⎣
⎪
⎪⎧⎡
⎪
⎫
2

⎤
4m n Ba Rb
⎪⎪
⎪
⎪
⎥
⎢
2
4
4
2
2
⁎
⁎
⁎
⁎
⎪ ⎪ π Bh ( A11 m Ba4 + A22 n + (A66 − 2A12 ) m n Ba2 )
⎪⎪
⎥
⎢
1⎪ ⎢
⎪⎪
⎥
⁎
⁎
⁎
⎬ 3
= − ⎨⎪
4n B ⁎ m4B 4 + B ⁎ n4 + (B11
+ B22

− 2B66
) m2n2Ba2 )
⎪⎪ , b2
⎥
P ⎪ ⎪ ⎢ − ( 21 a 12
⎪
⎪
4
4
2
2
⁎
⁎
⁎
⁎
4
2
⎪ ⎢⎣ mBh2 ( A11 m Ba + A22 n + (A66 − 2A12 ) m n Ba ) ⎥⎦
⎪
⎪⎪
⎪⎪
⁎
⎬
⎨
4A12 Rb
⎪
⎪
⎪⎪
⎪ ⎪ − mnπ 2B (A⁎ A⁎ − A⁎ 2)
h 11 22

⎪
⎪
12
⎪
⎪
⎪⎪
4
⎪⎪+
⎪
⎪
2
⁎
⁎
⁎
⎪
⎪
mnBh2 (A11 A22 − A12 )
⎪⎪
⎪⎪
⎪
⎪
2
2
2
⁎
⁎
⁎
⁎
⁎
⁎

⁎
⁎
⎡
⎤
⎪ ⎪ ⎣ (B A + B A ) m Ba + (B A + B22 A ) n ⎦⎪⎪
12 11
12
11 11
21 12
⎭⎭
⎩⎩
⎫
⎧ ⎡ 32mnB 2 R
⎤
1
a b
⎪
⎪⎢
⎥
2
4 4
2 2 2
⁎
⁎ 4
⁎
⁎
3
B
π
2

A
m
B
A
n
A
A
m
n
B
+
+
(
−
)
h
a) ⎥
( 11 a 22
⎪
⎪⎢
66
12
⎪
⎪⎢
4 4
2 2 2 ⎥
⁎
⁎ 4
⁎
⁎

⁎
2
B
m
B
B
n
B
B
B
m
n
B
+
+
(
+
−
)
a)
⎪
⎪ ⎢ 32n ( 21 a 12
11
22
66
⎥
−
⎪
⎪ ⎢ 3mBh2 A⁎ m4B 4 + A⁎ n4 + (A ⁎ − 2A⁎ ) m2n2B 2 ⎥
a

a)
(
11
22
66
12
⎦
⎣
⎪
⎪
⎪⎧⎡
⎫⎪
⎤
4m n Ba2 Rb
⎪⎪
⎪⎪
⎥
⁎
⁎ 4
⁎
⁎
⎪ ⎪ ⎢ π 2Bh ( A11
m4Ba4 + A22
n + (A66
− 2A12
) m2n2Ba2 )
⎪⎪
⎥
⎢
1⎪⎪⎢

⎪⎪
⎥
⎬ 3
⁎
⁎
⁎
= − ⎨⎪
4n B ⁎ m4B 4 + B ⁎ n4 + (B11
+ B22
− 2B66
) m2n2Ba2 )
⎪⎪ , b3
⎥
P ⎪ ⎢ − ( 21 a 12
⎪⎪
⁎
⁎ 4
⁎
⁎
m4Ba4 + A22
n + (A66
− 2A12
) m2n2Ba2 ) ⎥
⎪ ⎪ ⎢⎣ mBh2 ( A11
⎦
⎪⎪
⎪⎪
⁎
⎬⎪
4A12 Rb

⎪⎨
−
⎪⎪
⎪⎪
⁎ ⁎
⁎ 2
mnπ 2Bh (A11
A22 − A12
)
⎪⎪
⎪
⎪
⎪⎪
⎪
4
⎪ +
⎪⎪
⎪ ⎪ mnB 2 (A⁎ A⁎ − A⁎ 2)
12
h 11 22
⎪⎪
⎪⎪
⎡ (B ⁎ A ⁎ + B ⁎ A ⁎ ) m2B2 + (B ⁎ A ⁎ + B ⁎ A ⁎ ) n2⎤⎪
⎪
⎪⎪
⎪
⎦⎪
a
22 12
21 12

12 11
⎭⎭
⎩ ⎩ ⎣ 11 11
= −

=

⁎
⁎
⁎
⁎ ⎤
⎡
1⎢
2n R b
8n B21A22 + B12 A11 ⎥ 3
−
(
) , b4
⁎
⁎
⁎ 2
P ⎢⎣ 3m3π 2A11
⎥⎦
A11
A22
3mBh2
Ba Bh

⎡
⁎ 2

⁎ 2 ⎤
⎛ A ⁎ m4 B 4 + A ⁎ n4 ⎞
π 2 (A11
Ba + A12
n) ⎥
π2
1⎢
a
22
⎟⎟ +
⎜⎜ 11
.
⁎
⁎
2
2
2
⁎
⁎
⁎ 2 ⎥
P ⎢⎣ 16m Ba Bh ⎝
A11 A22
⎠ 8Bh2 (A11
− A12
)⎦
A22


168

N.D. Duc et al. / Thin-Walled Structures 96 (2015) 155–168

Appendix B

b14
⎤
⎡ 3 4 2 ⎡ B ⁎ m4B 4 + B ⁎ n4 + (B ⁎ + B ⁎ − 2B ⁎ ) m2n2B 2 ⎤
a⎦
11
22
66
⎥
⎢ m nπ Ba Rb ⎣ 21 a 12
3
⎡ ⁎ 4 4
⁎ 4
⁎
⁎
2 2 2⎤
8
B
⎥
⎢
h
⎣ A11 m Ba + A22 n + (A66 − 2A12 ) m n Ba ⎦
⎥
⎢
2
⎡ ⁎ 4 4
⁎ 4

⁎
⁎
⁎
2 2 2⎤
⎥
⎢
6 B m Ba + B12 n + (B11 + B22 − 2B66 ) m n Ba ⎦
⎥
⎢ − mnπ4 ⎣ 21
⎡
⎤
⁎
⁎ 4
⁎
⁎
m4Ba4 + A22
n + (A66
− 2A12
) m2n2Ba2 ⎦
⎥
⎢ 16Bh ⎣ A11
⎥
⎢
5
2
2
2
= − ⎢ m nπ Ba Rb
1
⎥

−
⎡ ⁎ 4 4
⁎ 4
⁎
⁎
2 2 2⎤
⎥
⎢
16Bh2
⎣ A11 m Ba + A22 n + (A66 − 2A12 ) m n Ba ⎦
⎥
⎢
⎢ mnπ 6
mnπ 6
4 4
2 2 2⎥
⁎
⁎ 4
⁎
⁎
⁎
⎢ − 16B 4 D11m Ba + D22 n − 16B 4 (D12 + D21 + 4D66 ) m n Ba ⎥
h
h
⎥
⎢
⁎ 2
⁎ 4
Ba
mnπ 2K1D11

Ba
⎥
⎢ mnπ 4K2 D11
2 2
2
+
−
m
B
n
a
⎥⎦
⎢⎣
16Bh4
16Bh4

(

)

(

)

⎡ ⎡
⎤⎤
m4n2π 2Ba4 Rb
⎢ ⎢
⎥⎥
3 A⁎ m4B 4 + A⁎ n4 + (A ⁎ − 2A⁎ ) m2n2B 2

B
a)
⎢ 5 ⎢ h ( 11 a 22
66
12
⎥⎥
⎢ 12 ⎢
⎥⎥
2 2 4 2 ( B ⁎ m4Ba4 + B ⁎ n4 + (B ⁎ + B ⁎ − 2B ⁎ ) m2n2Ba2 )
12
11
22
66
⎢ ⎢ m n π Ba 21
⎥⎥
−
4
⁎
⁎ 4
⁎
⁎
⎥
⎢ ⎢
Bh
A11
m4Ba4 + A22
n + (A66
− 2A12
) m2n2Ba2 ) ⎥
(

⎦⎥
⎣
4
, b24 = − ⎢⎢
⎥, b3
⎡ 4 (B ⁎ A⁎ + B ⁎ A⁎ ) m4B 4 + (B ⁎ A⁎ + B ⁎ A⁎ ) m2n2B 2
1
π
a
a
11
11
21
12
12
11
22
12
⎥
⎢ ⎢
⁎ ⁎
⁎ 2
⎥
⎢ 4 ⎣ Bh4
A22 − A12
(A11
)
⎥
⎢
⎤

2 2 2
⁎
⎥
⎢
A12 m π Ba Rb
⎥
⎥
⎢ − 3 ⁎ ⁎
⁎ 2
Bh (A11 A22 − A12 ) ⎦
⎦
⎣
⁎
⎡ n2π 2R
B ⁎ m2n2π 4Ba2 ⎤ 4
1 B
b
⎥, b4
= −⎢
− ( 21
+ 12
)
⁎
⁎
3
⁎
⎢⎣ 24A11 Bh
⎥⎦
6 A11
A22

Bh4

⎡
4 4
2 2 2 ⎤
⁎
⁎
mnπ 6 ⎛ m4 Ba4
n4 ⎞
mnπ 6 (A11 m B + A12 m n Ba ) ⎥ 4
⎜⎜ ⁎ + ⁎ ⎟⎟ +
=⎢
,b
⎢⎣ 256Bh4 ⎝ A22
⎥⎦ 5
⁎
⁎
⁎ 2
A11 ⎠ 128Bh4
A22
(A11
− A12
)
=

m3nπ 4Ba2
(W + μh).
16Bh2

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