DSpace at VNU: Corrigendum to ‘‘Nonlinear dynamic response of imperfect eccentrically stiffened FGM double curved shallow shells on elastic foundation’’ [Compos. Struct. 99 (2013) 88–96]

Composite Structures 102 (2013) 306–314

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Composite Structures
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Corrigendum

Corrigendum to ‘‘Nonlinear dynamic response of imperfect eccentrically
stiffened FGM double curved shallow shells on elastic foundation’’
[Compos. Struct. 99 (2013) 88–96]
Nguyen Dinh Duc ⇑
University of Engineering and Technology, Vietnam National University, 144 Xuan Thuy, Cau Giay, Hanoi, Viet Nam

a r t i c l e

i n f o

Article history:
Available online 10 April 2013
Keywords:
Nonlinear dynamic
Eccentrically stiffened FGM double curved
shallow shells
Imperfection
Elastic foundation

a b s t r a c t
This paper presents an analytical investigation on the nonlinear dynamic response of eccentrically stiffened functionally graded double curved shallow shells resting on elastic foundations and being subjected
to axial compressive load and transverse load. The formulations are based on the classical shell theory

taking into account geometrical nonlinearity, initial geometrical imperfection and the Lekhnitsky
smeared stiffeners technique with Pasternak type elastic foundation. The non-linear equations are solved
by the Runge–Kutta and Bubnov–Galerkin methods. Obtained results show effects of material and geometrical properties, elastic foundation and imperfection on the dynamical response of reinforced FGM
shallow shells. Some numerical results are given and compared with ones of other authors.
Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction
Functionally Graded Materials (FGMs), which are microscopically composites and made from mixture of metal and ceramic
constituents, have received considerable attention in recent years
due to their high performance heat resistance capacity and excellent characteristics in comparison with conventional composites.
By continuously and gradually varying the volume fraction of constituent materials through a specific direction, FGMs are capable of
withstanding ultrahigh temperature environments and extremely
large thermal gradients. Therefore, these novel materials are chosen to use in temperature shielding structure components of aircraft, aerospace vehicles, nuclear plants and engineering
structures in various industries. As a result, in recent years important studies have been researched about the stability and vibration
of FGM plates and shells.
The research on FGM shells and plates under dynamic load is
attractive to many researchers in the world.
Firstly we have to mention the research group of Reddy et al.
The vibration of functionally graded cylindrical shells has been
investigated by Loy, Lam and Reddy [1]. Lam and Hua has taken
into account the influence of boundary conditions on the frequency
characteristics of a rotating truncated circular conical shell [2]. In
[3] Pradhan et al. studied vibration characteristics of FGM cylindrical shells under various boundary conditions. Ng et al. studied the
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dynamic stability analysis of functionally graded cylindrical shells

under periodic axial loading [4]. The group of Ng and Lam also published results on generalized differential quadrate for free vibration
of rotating composite laminated conical shell with various boundary conditions in 2003 [5]. In the same year, Yang and Shen [6]
published the nonlinear analysis of FGM plates under transverse
and in-plane loads.
In 2004, Zhao et al. studied the free vibration of two-side simply-supported laminated cylindrical panel via the mesh-free kpRitz method [7]. About vibration of FGM plates Vel and Batra [8]
gave three dimensional exact solution for the vibration of FGM
rectangular plates. Also in this year, Sofiyev and Schnack investigated the stability of functionally graded cylindrical shells under
linearly increasing dynamic tensional loading in [9] and obtained
the result for the stability of functionally graded truncated conical
shells subjected to a periodic impulsive loading [10], and they published the result of the stability of functionally graded ceramic–
metal cylindrical shells under a periodic axial impulsive loading
in 2005 [11]. Ferreira et al. received natural frequencies of FGM
plates by meshless method [12], 2006. In [13], Zhao et al. used
the element-free kp-Ritz method for free vibration analysis of conical shell panels. Liew et al. studied the nonlinear vibration of a
coating-FGM-substrate cylindrical panel subjected to a temperature gradient [14] and dynamic stability of rotating cylindrical
shells subjected to periodic axial loads [15]. Woo et al. investigated
the non linear free vibration behavior of functionally graded plates
[16]. Kadoli and Ganesan studied the buckling and free vibration
analysis of functionally graded cylindrical shells subjected to a
temperature-specified boundary condition [17]. Also in this year,


307

N.D. Duc / Composite Structures 102 (2013) 306–314

Wu et al. published their results of nonlinear static and dynamic
analysis of functionally graded plates [18]. Sofiyev has considered
the buckling of functionally graded truncated conical shells under
dynamic axial loading [19]. Prakash et al. studied the nonlinear axisymmetric dynamic buckling behavior of clamped functionally

graded spherical caps [20]. In [21], Darabi et al. obtained the nonlinear analysis of dynamic stability for functionally graded cylindrical shells under periodic axial loading. Natural frequencies and
buckling stresses of FGM plates were analyzed by Matsunaga using
2-D higher-order deformation theory [22]. In 2008, Shariyat also
obtained the dynamic thermal buckling of suddenly heated temperature-dependent FGM cylindrical shells under combined axial
compression [23] and external pressure and dynamic buckling of
suddenly loaded imperfect hybrid FGM cylindrical with temperature-dependent material properties under thermo-electromechanical loads [24]. Allahverdizadeh et al. studied nonlinear free
and forced vibration analysis of thin circular functionally graded
plates [25]. Sofiyev investigated the vibration and stability behavior of freely supported FGM conical shells subjected to external
pressure [26], 2009. Shen published a valuable book ‘‘Functionally
Graded materials, Non linear Analysis of plates and shells’’, in which
the results about nonlinear vibration of shear deformable FGM
plates are presented [27]. Last years, Zhang and Li published the
dynamic buckling of FGM truncated conical shells subjected to
non-uniform normal impact load [28], Bich and Long studied the
non-linear dynamical analysis of functionally graded material shallow shells subjected to some dynamic loads [29], Dung and Nam
investigated the nonlinear dynamic analysis of imperfect FGM
shallow shells with simply supported and clamped boundary conditions [30]. Bich et al. has also considered the nonlinear vibration
of functionally graded shallow spherical shells [31].
In fact, the FGM plates and shells, as other composite structures, ussually reinforced by stiffening member to provide the
benefit of added load-carrying static and dynamic capability with
a relatively small additional weight penalty. Thus study on static
and dynamic problems of reinforced FGM plates and shells with
geometrical nonlinearity are of significant practical interest. However, up to date, the investigation on static and dynamic of eccentrically stiffened FGM structures has received comparatively little
attention. Recently, Bich et al. studied nonlinear dynamical analysis of eccentrically stiffened functionally graded cylindrical panels
[32].
This paper presents an dynamic nonlinear response of double
curved shallow eccentrically stiffened shells FGM resting on elastic
foundations and being subjected to axial compressive load and
transverse load. The formulations are based on the classical shell
theory taking into account geometrical nonlinearity, initial geometrical imperfection and the Lekhnitsky smeared stiffeners technique with Pasternak type elastic foundation. The nonlinear

transients response of doubly curved shallow shells subjected to
excited external forces obtained the dynamic critical buckling
loads are evaluated based on the displacement response using
the criterion suggested by Budiansky–Roth. Obtained results show
effects of material, geometrical properties, eccentrically stiffened,
elastic foundation and imperfection on the dynamical response of
FGM shallow shells.
2. Eccentrically stiffened double curved FGM shallow shell on
elastic foundations
Consider a ceramic–metal stiffened FGM double curved shallow
shell of radii of curvature Rx, Ry length of edges a, b and uniform
thickness h resting on an elastic foundation.
A coordinate system (x, y, z) is established in which (x, y) plane
on the middle surface of the panel and z is thickness direction (Àh/
2 6 z 6 h/2) as shown in Fig. 1.

z
b

a

h

y

x

Ry
Rx

Fig. 1. Geometry and coordinate system of an eccentrically stiffened double curved
shallow FGM shell on elastic foundation.

The volume fractions of constituents are assumed to vary
through the thickness according to the following power law
distribution

V m ðzÞ ¼


N
2z þ h
;
2h

V c ðzÞ ¼ 1 À V m ðzÞ

ð1Þ

where N is volume fraction index (0 6 N < 1). Effective properties
Preff of FGM panel are determined by linear rule of mixture as

Preff ðzÞ ¼ Prm V m ðzÞ þ Prc V c ðzÞ

ð2Þ

where Pr denotes a material property, and subscripts m and c stand
for the metal and ceramic constituents, respectively. Specific
expressions of modulus of elasticity E(z) and q(z) are obtained by
substituting Eq. (1) into Eq. (2) as

½EðzÞ; qðzފ ¼ ðEm ; qm Þ þ ðEcm ; qcm Þ


N
2z þ h
2h

ð3Þ

where

Ecm ¼ Ec À Em ; qcm ¼ qc À qm ;

mðzÞ ¼ const ¼ m

ð4Þ

It is evident from Eqs. (3), (4) that the upper surface of the panel
(z = Àh/2) is ceramic-rich, while the lower surface (z = h/2) is metal-rich, and the percentage of ceramic constituent in the panel is
enhanced when N increases.
The panel–foundation interaction is represented by Pasternak
model as

qe ¼ k1 w À k2 r2 w
2

2

2

ð5Þ
2

2

where r = @ /@x + @ /@ y ,w is the deflection of the panel, k1 is
Winkler foundation modulus and k2 is the shear layer foundation
stiffness of Pasternak model.
3. Theoretical formulation
In this study, the classical shell theory and the Lekhnitsky
smeared stiffeners technique are used to obtain governing equations and determine the nonlinear dynamical response of FGM
curved panels. The strain across the shell thickness at a distance
z from the mid-surface are

0

1

0

1

0

1

e0x
ex
kx

B e C B e0 C
B
C
@ y A ¼ @ y A À z@ ky A
0
cxy
2kxy
cxy

ð6Þ

where e0x ; e0x and c0xy are normal and shear strain at the middle surface of the shell, and kx, ky,kxy are the curvatures. The nonlinear
strain–displacement relationship based upon the von Karman theory for moderately large deflection and small strain are:


308

0

N.D. Duc / Composite Structures 102 (2013) 306–314

1

0

1

0

1

0

1

e0x
u;x À w=Rx þ w2;x =2
wx;x
kx
B e0 C B
C B
C B
C
@ y A ¼ @ v ;y À w=Ry þ w2;y =2 A; @ ky A ¼ @ wy;y A
w;xy
kxy
c0xy
u;y þ v ;x þ w;x w;y

ð7Þ

In which u, v are the displacement components along the x, y
directions, respectively.
The force and moment resultants of the FGM panel are determined by

ðNi ; Mi Þ ¼

Z

x2

h

z

z2
x1
2

O

h=2

ri ð1; zÞ dz i ¼ x; y; xy

ð8Þ

Àh=2

The constitutive stress–strain equations by Hooke law for the
shell material are omitted here for brevity. The shell reinforced
by eccentrically longitudinal and transversal stiffeners is shown
in Fig. 1. The shallow shell is assumed to have a relative small rise
as compared with its span. The contribution of stiffeners can be accounted for using the Lekhnitsky smeared stiffeners technique.
Then intergrading the stress–strain equations and their moments
through the thickness of the shell, the expressions for force and
moment resultants of an eccentrically stiffened FGM shallow shell
are obtained [32]:





E1
EA1 0
E m
E2
E2 m
þ
ex þ 1 2 e0y À
þ C 1 kx À
ky
2
2
1Àm
s1
1Àm
1Àm
1 À m2




E1 m 0
E1
EA2 0
E m
E2
Ny ¼
e þ
þ
ey À 2 2 kx À

þ C 2 ky
1 À m2 x
1 À m2
s2
1Àm
1 À m2




E2
E2 m 0
E3
EI1
E3 m
kx À
Mx ¼
þ C 1 e0x þ
e À
þ
ky
1 À m2
1 À m2 y
1 À m2 s1
1 À m2




E2 m 0

E2
E3 m
E3
EI2
My ¼
e þ
þ C 2 e0y À
kx À
þ
ky
1 À m2 x
1 À m2
1 À m2
1 À m2 s2


1
N xy ¼
E1 c0xy À 2E2 kxy
2ð1 þ mÞ


1
M xy ¼
E2 c0xy À 2E3 kxy
2ð1 þ mÞ
ð9Þ

s1

s1

s1

s1

s2

a

Fig. 2. Configuration of an eccentrically stiffened shallow shells

Mx;xx þ 2Mxy;xy þ My;yy þ

@2w
@t 2

ð11Þ

where


 


Z h
2
A
A
q

A
A
q¼
qðzÞ dz þ q0 1 þ 2 ¼ qm þ cm h þ q0 1 þ 2
s1 s2
Nþ1
s1 s2
À2h

ð12Þ
2

2

in which q0 is the mass density of stiffeners; q @@t2u ! 0 and q @@t2v ! 0
into consideration because of u ( w,v ( w the Eq. (11) can be
rewritten as:

Mx;xx þ 2Mxy;xy þ My;yy þ

Nx Ny
þ
þ Nx w;xx þ 2Nxy w;xy þ Ny w;yy þ q
Rx Ry

À k1 w þ k2 r2 w ¼ q



Ecm

h
E1 ¼ Em þ
Nþ1

@2w
@t 2

ð13Þ

Calculating from Eq. (9), obtained:

2

E2 ¼

ð10Þ

In above relations (9) and (10), the quantities A1, A2 are the cross
section areas of the stiffeners and I1, I2, z1, z2 are the second moments of cross section areas and eccentricities of the stiffeners with
respect to the middle surface of the shell respectively, E is elasticity
modulus in the axial direction of the corresponding stiffener witch
is assumed identical for both types of stiffeners (Fig. 2). In order to
provide continuity between the shell and stiffeners, suppose that
stiffeners are made of full metal (E = Em) if putting them at the metal-rich side of the shell, and conversely full ceramic stiffeners
(E = Ec) at the ceramic-rich side of the shell [32].
The nonlinear dynamic equations of a FGM shallow shells based
on the classical shell theory are [33]

N xy;x þ Ny;y

Nx Ny
þ
þ Nx w;xx þ 2Nxy w;xy þ Ny w;yy þ q
Rx Ry

À k1 w þ k2 r2 w ¼ q

where

@2u
@t 2
@2v
¼q 2
@t

s2

1



N x;x þ N xy;y ¼ q

s2
b

z1

Nx ¼

Ecm Nh
2ðN þ 1ÞðN þ 2Þ

!
Em
1
1
1
3
E3 ¼
À
þ
h
þ Ecm
N þ 3 N þ 2 4N þ 4
12
EA1 z1
EA2 z2
C1 ¼
; C2 ¼
s1
s2

s2

0
x
0
y

e ¼ A22 Nx À A12 Ny þ B11 kx þ B12 ky
e ¼ A11 Ny À A12 Nx þ B21 kx þ B22 ky
c0xy ¼ A66 Nxy þ 2A66 kxy

ð14Þ

where




1 EA1
E1
1 EA2
E1
þ
¼
þ
;
A
22
D s1
1 À m2
D s2
1 À m2
1 E1 m
2ð1 þ mÞ
; A66 ¼
A12 ¼
E1

D 1 À m2


 
2
EA1
E1
EA2
E1
E1 m
D¼
þ
þ
À
2
2
2
s1
1Àm
s2
1Àm
1Àm




E2
E2 m
E2
E2 m

B11 ¼ A22 C 1 þ
;
B
À
A
À A12
12
22 ¼ A11 C 2 þ
2
2
2
1Àm
1Àm
1Àm
1 À m2




E2 m
E2
E2 m
E2
B12 ¼ A22
À A12
þ C 2 ; B21 ¼ A11
À A12
þ C1
2
2

2
2
1Àm
1Àm
1Àm
1Àm
E2
B66 ¼
E1
ð15Þ
A11 ¼

Substituting once again Eq. (14) into the expression of Mij in (9)
leads to:

M x ¼ B11 Nþx B21 NÀy D11 kx À D12 ky
M x ¼ B12 Nþx B22 NÀy D21 kx À D22 ky
M xy ¼

B66 NÀxy 2D66 kxy

ð16Þ


309

N.D. Duc / Composite Structures 102 (2013) 306–314

The approximate solutions of w, w⁄ and f satisfying boundary
conditions (23) are assumed to be [27–31]

where

D11
D22
D12
D21
D66



EI1
E3
E2
E2 m
B11 À
¼
þ
À
C
þ
B21
1
s1 1 À m 2
1 À m2
1 À m2


EI2
E3

E2
E2 m
¼
þ
À C2 þ
B12
B22 À
s2 1 À m 2
1 À m2
1 À m2


E3 m
E2
E2 m
¼
À C1 þ
B22
B12 À
1 À m2
1 À m2
1 À m2


E3 m
E2
E2 m
B21 À
¼
À C2 þ

B11
1 À m2
1 À m2
1 À m2
E3
E2
¼
À
B66
2ð1 þ mÞ 2ð1 þ mÞ

ð17Þ

Ny ¼ f;xx ;

Nxy ¼ Àf;xy

ð24aÞ

wà ¼ W0 sin km x sin dn y

ð24bÞ

f ¼ gðtÞ½sin km x sin dn y À hðxÞ À xðyފ

ð24cÞ

where km = mp/a, dn = np/b and W is the maximum deflection; W0 is
a constant; h(x) and x(y) chosen such that:

gh00 ðxÞ ¼ p0 h gx00 ðyÞ ¼ r0 h

Then Mij into Eq. (13) and f(x, y) is stress function defined by

Nx ¼ f;yy ;

w ¼ WðtÞ sin km x sin dn y

ð18Þ

Subsequently, substitution of Eqs. (24a and b) into Eqs. (22) and
(24c) into Eq. (19) and applying the Galerkin procedure for the
resulting equation yield leads to:

Â

g A11 m4 þ ðA66 À 2A12 Þm2 n2 k2 þ A22 n4 k4

þ

þ f;xx w;yy þ

þ ðD12 þ D21 þ

þ f;yy w;xx À 2f ;xy w;xy

f;yy f;xx
@2w
þ
þ q À k1 w þ k2 r2 w ¼ q 2

Rx Ry
@t

ð19Þ

g

a4

e0x ¼ A22 f;yy À A12 f;xx þ B11 w;xx þ B12 w;yy
e0y ¼ A11 f;xx À A12 f;yy þ B21 w;xx þ B22 w;yy
c0xy ¼ ÀA66 f;xy þ 2A66 w;xy Þ

D11 m4 þ ðD12 þ D21 þ 4D66 Þn2 m2 k2 þ D22 n4 k4

a4

À k2 W

p2
a2

ðm2 þ k2 n2 Þ ¼ q

W

p2 h
a2

ðm2 r 0 þ n2 p0 k2 Þ À k1 À k2

2

A11 f;xxxx þ ðA66 À 2A12 Þf;xxyy þ A22 f;yyyy þ B21 w;xxxx

ð22Þ

@2W
@t2

ð27Þ

where m,n are odd numbers, and k ¼ ab.
Eliminating g from two obtained equations leads to non-linear
second-order ordinary differential equation for f(t):
"

ð21Þ

Ã

32
k2 p2 hW 2
W gmnp2 4 þ
ðm r 0 þ n2 p0 k2 Þ
3
a ! a2


p2 m2 n2 k2
16h r 0 p0

16q
À
þ
þ
þ
À k1 W
À 2g
mnp2 Rx Ry
mnp2
a
Ry
Rx

ð20Þ

Setting Eq. (21) into Eq. (20) gives the compatibility equation of
an imperfect eccentrically stiffened shallow FGM shells as

þ ðB11 þ B22 À 2B66 Þw;xxyy þ B12 w;yyyy ¼ w2;xy À w;xx w;yy
w;yy À wÃ;yy w;xx À wÃ;xx
Ã
Ã
À wÃ2
À
;xy þ w;xx w;yy À
Rx
Ry

p4 Â

þ

Ã
Ã
e0x;yy þ e0y;xx À c0xy;xy ¼ w2;xy À w;xx w;yy À wÃ2
;xy þ w;xx w;yy

From the constitutive relations (18) in conjunction with Eq. (14)
one can write

ð26Þ

Ã
B21 m4 þ ðB11 þ B22 À B66 Þn2 m2 k2 þ B12 n4 k4 À ðW

À W0Þ

in which w (x, y) is a known function representing initial small
imperfection of the eccentrically stiffened shallow shells. The geometrical compatibility equation for an imperfect shallow shells is
written

w;yy À wÃ;yy w;xx À wÃ;xx
À
:
Rx
Ry

n2 k2 m2
ðW À W 0 Þ
þ

Rx
Ry

p4 Â

⁄

À

p

2

Ã

!

Â
Ã
þ W B21 m4 þ ðB11 þ B22 À 2B66 Þm2 n2 k2 þ B12 n4 k4

16 mnk2  2
þ
W À W 20 ¼ 0
2
3 p

B21 f;xxxx þ B12 f;yyyy þ ðB11 þ B22 À 2B66 Þf;xxyy À D11 w;xxxx
À D22 w;yyyy À ðD12 þ D21 þ 4D66 Þw;xxyy þ D11 wÃ;xxxx
4D66 ÞwÃ;xxyy

a2

À

For an imperfect FGM curved panel, Eq. (13) are modified into
form

D22 wÃ;yyyy

ð25Þ

!

a2

ðm2 þ k2 n2 Þ À

p4 P2
a4

P1

þ

2 2

p2 m2
a2

!2

Ry
3

þ

m
n k P2 p
m
n k
15
þ
À
P3 À
þ
P1
a 2 Ry
Rx
P 1 a4
Ry
Rx
"
!
#
1 m2 n2 k2 16mnk2 16mnp2 k2 P2
þ ðW 2 À W 20 Þ 2
þ
À
4

a
Ry
Rx
3P 1
a
P1
!
2
2 2
2
2 2
32mn
p
k
P
32mnk
m
n
k
1
2
þ ÀW 2
þ WðW À W 0 Þ
þ
4
2
P1
3a
P1
3a

Ry
Rx


512m2 n2 k 1
16h r 0 p0
16q
@2W
þ
À WðW 2 À W 20 Þ
À
þ
¼q 2
4
2
2
9a
P1 mnp Rx Ry
mnp
@t
þ ðW À W 0 Þ4

p

2 2

p2

2

2

4

2

n 2 k2
Rx

!

P2
P1

#

ð28Þ

Eqs. (19) and (22) are nonlinear equations in terms of variables
w and f and used to investigate the nonlinear dynamic and nonlinear stability of thick imperfect stiffened FGM double curved panels
on elastic foundations subjected to mechanical, thermal and thermo mechanical loads.

where

4. Nonlinear dynamic analysis

The obtained Eq. (28) is a governing equation for dynamic
imperfect stiffened FGM doubly-curved shallow shells in general.
_
The initial conditions are assumed as Wð0Þ ¼ W 0 ; Wð0Þ

¼ 0. The
nonlinear Eq. (28) can be solved by the Newmark’s numerical integration method or Runge–Kutta method.

In the present study, suppose that the stiffened FGM shallow
shell is simply supported at its all edges and subjected to a transverse load q(t), compressive edge loads r0(t) and p0(t). The boundary conditions are

w ¼ N xy ¼ M x ¼ 0;

Nx ¼ Àr 0 h at x ¼ 0; a

w ¼ N xy ¼ M y ¼ 0;

Ny ¼ p0 h at y ¼ 0; b:

ð23Þ

where a and b are the lengths of in-plane edges of the shallow shell.

P1 ¼ A11 m4 þ ðA66 À 2A12 Þm2 n2 k2 þ A22 n4 k4
P2 ¼ B21 m4 þ ðB11 þ B22 À 2B66 Þm2 n2 k2 þ B12 n4 k4
4

2 2 2

ð29Þ

4 4

P3 ¼ D11 m þ ðD12 þ D21 þ 4D66 Þm n k þ D22 n k

4.1. Nonlinear vibration of eccentrically stiffened FGM shallow shell
Consider an imperfect stiffened FGM shallow shell acted on by
uniformly distributed excited transverse q(t) = Qsin Xt, i.e.
p0 = r0 = 0, from (28) we have


310

N.D. Duc / Composite Structures 102 (2013) 306–314



Q 1 W þ Q 2 ðW À W 0 Þ þ Q 3 W 2 À W 20 À Q 4 W 2 þ Q 5 WðW


@2W
À W 0 Þ À Q 6 W W 2 À W 20 þ Q 7 sin Xt ¼ q 2
@t

ð30Þ

where

Q4 ¼
Q5 ¼

p2
a2

ðm2 þ k2 n2 Þ þ

32mnp2 k2 P2
3a4
P1

p4 P2
a4 P1

À

p2 m2
a2

Ry

!
p4 P2 p2 m2 P2
2 2
2
ðm
þ
k
n
Þ
À
þ
a2
a2
a4 P1
a2 RP 1

!
2 2
4
4
p m P2 p
m
À 4 P3 À 2
þ ðW À W 0 Þ 2
a RP 1
a
R P1
"
#

 1 16m3 nk2 16mnp2 k2 P
2
2
2
À
þ W À W0
þ ÀW 2
a2 3P1 R
a4
P1

p2 h

W

!

n2 k2 P2
Rx P1
!
!2
p2 m2 n2 k2 P2 p4
m2 n2 k2
1
Q2 ¼ À 2
þ
þ 4 P3 þ
þ
P1
a
Ry
Rx P 1 a
Ry
Rx
!
1 m2 n2 k2 16mnk2 16mnp2 k2 P 2
Q3 ¼ 2
þ
À
a
Ry
Rx
3P1
a4
P1
Q 1 ¼ k1 þ k2

4.2.1. Imperfect eccentrically stiffened FGM cylindrical panel acted on
by axial compressive load
Eq. (28) in this case Rx ? 1, Ry = R, p0 = q = 0;r0 – 0 can be
rewritten as:

þ

m2 r 0 À k1 À k2

32mnp2 k2 P2
32mnk2 m2 1
þ WðW À W 0 Þ
4
3a
P1
3a2
R P1
2

 512m2 n2 k 1
@
W
À W W 2 À W 20
¼q 2
9a4
P1
@t
Â

ð31Þ
!

32mnk2 m2 n2 k2 1
þ
3a2
Ry
Rx P 1

512m2 n2 k 1
9a4
P1
16Q 0
Q7 ¼
mnp2

W

p2 h
a2

m2 r 0 ¼ W k1 þ k2

q

À

ð32Þ

ð33Þ

where denoting

H1 ¼ x2L ¼
H2 ¼
H3 ¼

1

q

Q4 À Q3 À Q5

q

xNL

m2 r0 ¼ k1 þ k2

ð34Þ

q

ð35Þ

where xNL is the nonlinear vibration frequency and s is the amplitude of nonlinear vibration.
4.2. Nonlinear dynamic buckling analysis of imperfect eccentrically
stiffened FGM shallow shell
The aim of considered problems is to search the critical dynamic
buckling loads. They can be evaluated based on the displacement

responses obtained from the motion Eq. (28). This criterion suggested by Budiansky and Roth is employed here as it is widely accepted. This criterion is based on that, for large values of loading
speed the amplitude-time curve of obtained displacement response increases sharply depending on time and this curve obtained a maximum by passing from the slope point, and at the
time t = tcr a stability loss occurs, and here t = tcr is called critical
time and the load corresponding to this critical time is called dynamic critical buckling load.

p m P2
2

À

p

2

p4 P 2
a4 P1

m4
P3 þ 2
a4
R P1

þ

4

!

512m2 n2 k 1
9a4

P1

ð37Þ

p2
a2

ðm2 þ k2 n2 Þ þ
"

p4 P2
a4 P1

À

p2 m2 P2
a2 RP 1

À

p2 m 2 P 2
a2 RP 1

1 16m3 nk2 16mnp2 k2 P2
À
2
a
3P 1 R
a4
P1

R P1
#
32mnp2 k2 P2 32mnk2 m2 1
512m2 n2 k 1
þ
þ W2
À
9a4
P1
3a4
P1
3a2
R P1
þ

Q6


1
8H2
3H3 2 2
¼ xL 1 þ
s
þ
s
3px2L
4x2L

ðm2 þ k2 n2 Þ þ

Taking of W – 0, i.e. considering the shell after the loss of stability we obtain

a2

Seeking solution as W(t) = scos xt and applying procedure like
Garlerkin method to Eq (33), the frequency-amplitude relation of
nonlinear free vibration is obtained

a2

2

þ W3

p2 h

ðQ 1 þ Q 2 Þ

p2

a2 RP 1
a2 RP 1
"
1 16m3 nk2 16mnp2 k2 P2
À W2 2
À
a
3P1 R
a4
P1

#
32mnp2 k2 P2 32mnk2 m2 1
À
þ
3a4
P1
3a2
R P1

The equation of nonlinear free vibration of a perfect FGM shallow panel can be obtained from:

€ þ H1 W þ H2 W 2 þ H 3 W 3 ¼ 0
W

p m P2
2

From Eq. (30) the fundamental frequencies of natural vibration
of the imperfect stiffened FGM shell can be determined by the
relation:

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
ðQ 1 þ Q 2 Þ

ð36Þ

The static critical load can be determined by the equation to be
€ ¼ 0; W 0 ¼ 0
reduced from Eq. (36) by putting W

Q6 ¼

xL ¼

p2

p4
a4

m4

P3 þ

2

ÀW

ð38Þ
From Eq. (38) the upper buckling load can be determined by
W=0
rupper ¼

a2
m2 h

p

2

k1 þ k2

p2
a2

ðm2 þ k2 n2 Þ þ

p4 P2 p2 m2 P2 p2 m2 P2 p4
a4 P

À

1

a2 RP

À

1

a2 RP

þ

1

a4

P3 þ

m4

!

2

R P1
ð39Þ

dr0
And the lower buckling load is found using the condition dW
¼ 0,
it follows:

rlower ¼

a2
2

k1 þ k2

p2 hm
p2 m2 P2 p4

À
"

a2 RP 1

þ

a4

p2
a2

ðm2 þ k2 n2 Þ þ

P3 þ

m4
R2 P 1

À

p4 P 2 p2 m 2 P 2
a4 P 1

À

a2 RP1

9a4 P 1
1024m2 n2 k

#2
1 16m3 nk2 16mnp2 k2 P 2 32mnp2 k2 P 2 32mnk2 m2 1
À
þ
À

a2 3P 1 R
a4
P1
3a4
P1
3a2
R P1
"


2
512m2 n2 k 1
1 16m3 nk2 16mnp2 k2 P 2
þþ4
À
9a4
P1
a2 3P 1 R
a4
P1
##
32mnp2 k2 P 2 32mnk2 m2 1
þ
À
ð40Þ
3a4
P1
3a2
R P1

311

N.D. Duc / Composite Structures 102 (2013) 306–314

4.2.2. Imperfect eccentrically stiffened shallow FGM cylindrical panel
subjected to transverse load
Eq. (28) in this case Rx ? 1,Ry = R, p0 = r0 = 0 can be rewritten as:

"

W Àk1 À k2

p2
a2

ðm2 þ k2 n2 Þ À

p m P2

p4 P2
a4

P1

þ

p2 n2 k2 P2
!

a2 RP

#

þ ðW À W 0 Þ

2

4

4

xL(rad/s)

N

1

0
1
2
5

p
m
À 4 P3 À
2 RP
a
a
RP

1
1
"
#

 1 16m3 nk2 16mnp2 k2 P
2
2
2
À
þ W À W0
a2 3RP 1
a4
P1
2

Table 1
The dependence of the fundamental frequencies of nature vibration of spherical FGM
double curved shallow shell on volume ratio N.

32mnp2 k2 P2
32m3 nk2 1
þ
WðW
À
W
Þ
0
3a4
P1

3Ra2 P1


2 2
16q
@2W
2
2 512m n k 1
À W W À W0
þ
¼q 2
4
2
9a
P1 mnp
@t

Reinforced

Unreinforced

56.130 Â 105
39.034 Â 105
31.982 Â 105
24.047 Â 105

55.667 Â 105
38.515 Â 105
31.441 Â 105
23.477 Â 105

þ ÀW 2

ð41Þ

The static critical load can be determined by the equation to be re€ ¼ 0; W 0 ¼ 0 and using condition
duced from Eq. (41) by putting W
dq
¼ 0.
dW
4.2.3. Imperfect eccentrically stiffened FGM shallow spherical panel
under transverse load
Eq. (28) in this case Rx = Ry = R, p0 = r0 = 0 can be rewritten as:

"

W Àk1 À k2

p2

2

2 2

p4 P2

!

p2 m2 þ n2 k2 P2

#

ðm þ k n Þ À 4
þ
a P1 a2
R
P1
2
!
!2 3
2
2
2 2
4
2
2 2
p
m
þ
n
k
P
p
m
þ
n
k
15
2
þ ðW À W 0 Þ4 2

À P3 À
P1
a
R
P1 a4
R
!
"
#

 1 m2 þ n2 k2 16mnk2 16mnp2 k2 P
2
þ W 2 À W 20
À
a2
R
3P1
a4
P1
!
2
2 2
2
2 2
32mnk m þ n k
1
2 32mnp k P 2
þ ÀW
þ WðW À W 0 Þ
P1

3a4
P1
3a2
R

 512m2 n2 k 1
16q
@2W
À W W 2 À W 20
þ
¼q 2
ð42Þ
4
2
9a
P1 mnp
@t
a2

The static critical load can be determined by the equation to be
€ ¼ 0; W 0 ¼ 0 and using condireduced from Eq. (42) by putting W
dq
tion dW
¼ 0.
5. Numerical results and discussion
The eccentrically stiffened FGM shells considered here are shallow shell with in-plane edges:

Table 2
The dependence of the fundamental frequencies of nature vibration of spherical FGM
double curved shallow shell on elastic foundations.

K1, K2

xL (rad/s)
Reinforced

Unreinforced

K1 = 200, K2 = 0
K1 = 200, K2 = 10
K1 = 200, K2 = 20
K1 = 200, K2 = 30
K1 = 0, K2 = 10
K1 = 100, K2 = 10
K1 = 150, K2 = 10
K1 = 200, K2 = 10

33.574 Â 10 5
39.034 Â 10 5
44.079 Â 10 5
48.535 Â 10 5
26.734 Â 105
31.534 Â 10 5
35.585 Â 10 5
39.034 Â 10 5

32.865 Â 105
38.515 Â 105
43.273 Â 105
46.371 Â 105
25.646 Â 105

30.078 Â 105
35.033 Â 105
38.515 Â 105

Table 3
Comparison of - with result reported by Bich et. al. [32], Alijani et. al. [34], Chorfi and
Houmat [35] and Matsunaga [36].
(a/Rx, b/Ry)

N

Present

Ref. [32]

Ref. [34]

Ref. [35]

Ref. [36]

0
0.5
1
4
10

0.0562
0.0502
0.0449

0.0385
0.0304

0.0597
0.0506
0.0456
0.0396
0.0381

0.0597
0.0506
0.0456
0.0396
0.0380

0.0577
0.0490
0.0442
0.0383
0.0366

0.0588
0.0492
0.0403
0.0381
0.0364

FGM cylindrical panel
(0, 0.5)
0

0.0624
0.5
0.0528
1
0.0494
4
0.0407
10
0.0379

0.0648
0.0553
0.0501
0.0430
0.0409

0.0648
0.0553
0.0501
0.0430
0.0408

0.0629
0.0540
0.0490
0.0419
0.0395

0.0622
0.0535

0.0485
0.0413
0.0390

FGM plate
(0, 0)

a ¼ b ¼ 2 m; h ¼ 0:01 m;
Ec ¼ 380 Â 109 N=m2 ;

3

qm ¼ 2702 kg=m ; qc ¼ 3800 kg=m3 ;
s1 ¼ s2 ¼ 0:4;

z1 ¼ z2 ¼ 0:0225 ðmÞ;

ð43Þ

4

5.7419

x 10

Reinforced, Rx =Ry =3(m), N=5

m ¼ 0:3

Table 1 presents the dependence of the fundamental frequencies

of nature vibration of spherical FGM shallow shell on volume ratio N
in which m ¼ n ¼ 1; a ¼ b ¼ 2 ðmÞ, h ¼ 0:01 ðmÞ; K 1 ¼ 200; K 2 ¼
10, R¼
x Ry ¼ 3 ðmÞ; W 0 ¼ 1e À 5
From the results of Table 1, it can be seen that the increase of
volume ration N will lead to the decrease of frequencies of nature
vibration of spherical FGM shallow shell.
Table 2 presents the frequencies of nature vibration of spherical
double curved FGM shallow shell depending on elastic foundations. These results show that the increase of the coefficients of
elastic foundations will lead to the increase of the frequencies of
nature vibration. Moreover, the Pasternak type elastic foundation
has the greater influence on the frequencies of nature vibration
of FGM shell than Winkler model does.
Based on (28) the nonlinear vibration of imperfect eccentrically
stiffened shells under various loading cases can be performed.

Reinforced, Rx =R(y)=3(m), N=0

5.7418

Unreinforced, Rx =Ry =3(m), N=5
Unreinforced, Rx =Ry =3(m), N=0

5.7417

ωNL (rad/s)

Em ¼ 70 Â 109 N=m2 ;

5.7416

5.7415

5.7414

5.7413

0

0.05

τ

0.1

Fig. 3. Frequency-amplitude relation.

0.15


312

N.D. Duc / Composite Structures 102 (2013) 306–314

Fig. 4. Effect of eccentrically stiffeners on nonlinear dynamic response of the
shallow spherical FGM shell.

Fig. 6. Influence of elastic foundations on nonlinear dynamic response of the
eccentrically stiffened shallow spherical FGM shell.

Fig. 5. Deflection-velocity relation of the eccentrically stiffened shallow spherical
FGM shell.
Fig. 7. Effect of volume metal-ceramic on nonlinear response of the eccentrically
stiffened shallow spherical FGM shell.

Particularly for spherical panel we put R1x ¼ R1y in (28), for cylindrical
shell R1x ¼ 0 and for a plate R1x ¼ R1y ¼ 0.
Table 3 presents the comparison on the fundamental frequency
qffiffiffiffi
parameter - ¼ xL h qEcc (In the Tables 1–3, xL is calculated from
Eq. (32)) given by the present analysis with the result of Alijani
et al. [34] based on the Donnell’s nonlinear shallow shell theory,
Chorfi and Haumat [35] based on the first–order shear deformation
theory and Matsunaga [36] based on the two-dimensional (2D)
higher order theory for the perfect unreinforced FGM cylindrical
panel. The results in Table 3 were obtained with m = n = 1,
a = b = 2 (m), h = 0.02 (m), K1 = 0, K2 = 0; W⁄ = 0 and with the chosen material properties in (43). As in Table 3, we can observe a very
good agreement in this comparison study.

Fig. 3 shows the relation frequency-amplitude of nonlinear free
vibration of reinforced and unreinforced spherical shallow FGM
shell on elastic foundation (calculated from Eq. (35)) with m ¼
n ¼ 1; a ¼ b ¼ 2 ðmÞ, h ¼ 0:01 ðmÞ; K 1 ¼ 200, K 2 ¼ 10; R¼
x Ry ¼ 3
ðmÞ; W 0 ¼ 1e À 5. As expected the nonlinear vibration frequencies
of reinforced spherical shallow FGM shells are greater than ones of
unreinforced shells.
The nonlinear Eq. (28) is solved by Runge–Kutta method. The
below figures, except Fig. 6, are calculated basing on k1 = 100;
k2 = 10.

Fig. 4 shows the effect of eccentrically stiffeners on nonlinear
respond of the FGM shallow shell on elastic foundation. The FGM


N.D. Duc / Composite Structures 102 (2013) 306–314

313

Fig. 8. Effect of dynamic loads on nonlinear response.
Fig. 10. Influence of initial imperfection on nonlinear dynamic response of the
eccentrically stiffened spherical panel.

Fig. 9. Effect of Rx on nonlinear dynamic response.

shell considered here is spherical panel Rx = Ry = 5 m. Clearly, the
stiffeners played positive role in reducing amplitude of maximum
deflection. Relation of maximum deflection and velocity for spherical shallow shell is expressed in Fig. 5.
Fig. 6 shows influence of elastic foundations on nonlinear dynamic response of spherical panel. Obviously, elastic foundations
played negative role on dynamic response of the shell: the larger
k1 and k2 coefficients are, the larger amplitude of deflections is.
Fig. 7 shows effect of volume metal-ceramic on nonlinear dynamic response of the eccentrically stiffened shallow spherical
FGM shell.
Figs. 8 and 9 show effect of dynamic loads and Rx on nonlinear
dynamic response of the eccentrically stiffened shallow spherical
FGM shell.

Fig. 11. Nonlinear dynamic response of eccentrically stiffened spherical and
cylindrical FGM panel.

Fig. 10 shows influence of initial imperfection on nonlinear dynamic response of the eccentrically stiffened spherical panel. The

increase in imperfection will lead to the increase of the amplitude
of maximum deflection.
Fig. 11 shows nonlinear dynamic response of shallow eccentrically stiffened spherical and eccentrically stiffened cylindrical FGM
panels. For eccentrically stiffened cylindrical FGM panel, in this
case, the obtained results is identical to the result of Bich in [32].
6. Concluding remarks
This paper presents an analytical investigation on the nonlinear
dynamic response of eccentrically stiffened functionally graded
double curved shallow shells resting on elastic foundations and


314

N.D. Duc / Composite Structures 102 (2013) 306–314

being subjected to axial compressive load and transverse load. The
formulations are based on the classical shell theory taking into account geometrical nonlinearity, initial geometrical imperfection
and the Lekhnitsky smeared stiffeners technique with Pasternak
type elastic foundation. The nonlinear equations are solved by
the Runge–Kutta and Bubnov-Galerkin methods. Some results
were compared with the ones of the other authors.
Obtained results show effects of material, geometrical properties, eccentrically stiffened, elastic foundation and imperfection
on the dynamical response of reinforced FGM double curved shallow shells. Hence, when we change these parameters, we can control the dynamic response and vibration of the FGM shallow shells
actively.
Acknowledgments
This work was supported by Project in Mechanics of the National
Foundation for Science and Technology Development of VietnamNAFOSTED. The author is grateful for this financial support.
References
[1] Loy CT, Lam KY, Reddy JN. Vibration of functionally graded cylindrical shells.
Int Mech Sci 1999;41:309–24.

[2] Lam KY, Li Hua. Influence of boundary conditions on the frequency
characteristics of a rotating truncated circular conical shell. J Sound Vib
1999;223(2):171–95.
[3] Pradhan SC, Loy CT, Lam KY, Reddy JN. Vibration characteristics of FGM
cylindrical shells under various boundary conditions. J Appl Acoust
2000;61:111–29.
[4] Ng TY, Lam KY, Liew KM, Reddy JN. Dynamic stability analysis of functionally
graded cylindrical shells under periodic axial loading. Int J Solids Struct
2001;38:1295–309.
[5] Ng TY, Hua Li, Lam KY. Generalized differential quadrate for free vibration of
rotating composite laminated conical shell with various boundary conditions.
Int J Mech Sci 2003;45:467–587.
[6] Yang J, Shen HS. Non-linear analysis of FGM plates under transverse and inplane loads. Int J Non-Lin Mech 2003;38:467–82.
[7] Zhao X, Ng TY, Liew KM. Free vibration of two-side simply-supported
laminated cylindrical panel via the mesh-free kp-Ritz method. Int J Mech Sci
2004;46:123–42.
[8] Vel SS, Batra RC. Three dimensional exact solution for the vibration of FGM
rectangular plates. J Sound Vib 2004;272(3):703–30.
[9] Sofiyev AH, Schnack E. The stability of functionally graded cylindrical shells
under linearly increasing dynamic tensional loading. Eng Struct
2004;26:1321–31.
[10] Sofiyev AH. The stability of functionally graded truncated conical shells
subjected to a periodic impulsive loading. Int Solids Struct 2004;41:3411–24.
[11] Sofiyev AH. The stability of compositionally graded ceramic–metal cylindrical
shells under a periodic axial impulsive loading. Compos Struct
2005;69:247–57.
[12] Ferreira AJM, Batra RC, Roque CMC. Natural frequencies of FGM plates by
meshless method. J Compos Struct 2006;75:593–600.

[13] Zhao X, Li Q, Liew KM, Ng TY. The element-free kp-Ritz method for free

vibration analysis of conical shell panels. J Sound Vib 2006;295:906–22.
[14] Liew KM, Yang J, Wu YF. Nonlinear vibration of a coating-FGM-substrate
cylindrical panel subjected to a temperature gradient. Comput Methods Appl
Mech Eng 2006;195:1007–26.
[15] Liew KM, Hu GY, Ng TY, Zhao X. Dynamic stability of rotating cylindrical shells
subjected to periodic axial loads. Int J Solids Struct 2006;43:7553–70.
[16] Woo J, Meguid SA, Ong LS. Non linear free vibration behavior of functionally
graded plates. J Sound Vib 2006;289:595–611.
[17] Ravikiran Kadoli, Ganesan N. Buckling and free vibration analysis of
functionally graded cylindrical shells subjected to a temperature-specified
boundary condition. J Sound Vib 2006;289:450–80.
[18] Tsung-Lin Wu, Shukla KK, Huang Jin H. Nonlinear static and dynamic analysis
of functionally graded plates. Int J Appl Mech Eng 2006;11:679–98.
[19] Sofiyev AH. The buckling of functionally graded truncated conical shells under
dynamic axial loading. J Sound Vib 2007;305(4-5):808–26.
[20] Prakash T, Sundararajan N, Ganapathi M. On the nonlinear axisymmetric
dynamic buckling behavior of clamped functionally graded spherical caps. J
Sound Vib 2007;299:36–43.
[21] Darabi M, Darvizeh M, Darvizeh A. Non-linear analysis of dynamic stability for
functionally graded cylindrical shells under periodic axial loading. Compos
Struct 2008;83:201–11.
[22] Hiroyuki Matsunaga. Free vibration and stability of FGM plates according to a
2-D high order deformation theory. J Compos Struct 2008;82:499–512.
[23] Shariyat M. Dynamic thermal buckling of suddenly heated temperaturedependent FGM cylindrical shells under combined axial compression and
external pressure. Int J Solids Struct 2008;45:2598–612.
[24] Shariyat M. Dynamic buckling of suddenly loaded imperfect hybrid FGM
cylindrical with temperature-dependent material properties under thermoelectro-mechanical loads. Int J Mech Sci 2008;50:1561–71.
[25] Allahverdizadeh A, Naei MH, Nikkhah Bahrami M. Nonlinear free and forced
vibration analysis of thin circular functionally graded plates. J Sound Vib
2008;310:966–84.

[26] Sofiyev AH. The vibration and stability behavior of freely supported FGM
conical shells subjected to external pressure. Compos Struct 2009;89:356–66.
[27] Shen Hui-Shen. Functionally graded materials. Non linear analysis of plates
and shells. London, New York: CRC Press, Taylor & Francis Group; 2009.
[28] Zhang J, Li S. Dynamic buckling of FGM truncated conical shells subjected to
non-uniform normal impact load. Compos Struct 2010;92:2979–83.
[29] Bich DH, Long VD. Non-linear dynamical analysis of imperfect functionally
graded material shallow shells. Vietnam J Mech, VAST 2010;32(1):1–14.
[30] Dung DV, Nam VH. Nonlinear dynamic analysis of imperfect FGM shallow
shells with simply supported and clamped boundary conditions. In:
Proceedings of the tenth national conference on deformable solid mechanics,
Thai Nguyen, Vietnam; 2010. p. 130–41.
[31] Bich DH, Hoa LK. Nonlinear vibration of functionally graded shallow spherical
shells. Vietnam J Mech 2010;32(4):199–210.
[32] Bich DV, Dung DV, Nam VH. Nonlinear dynamical analysis of eccentrically
stiffened functionally graded cylindrical panels. Compos Struct
2012;94:2465–73.
[33] Brush DD, Almroth BO. Buckling of bars, plates and shells. McGraw-Hill; 1975.
[34] Alijani F, Amabili M, Karagiozis K, Bakhtiari-Nejad F. Nonlinear vibrations of
functionally graded doubly curved shallow shells. J Sound Vib
2011;330:1432–54.
[35] Chorfi SM, Houmat A. Nonlinear free vibration of a functionally graded doubly
curved shallow shell of elliptical plan-form. Compos Struct 2010;92:2573–81.
[36] Matsunaga H. Free vibration and stability of functionally graded shallow shells
according to a 2-D higher-order deformation theory. Compos Struct
2008;84:132–46.