DSpace at VNU: Nonlinear dynamic response of imperfect eccentrically stiffened FGM double curved shallow shells on elastic foundation

Composite Structures 99 (2013) 88–96

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Composite Structures
journal homepage: www.elsevier.com/locate/compstruct

Nonlinear dynamic response of imperfect eccentrically stiffened FGM double
curved shallow shells on elastic foundation
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 1 December 2012
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 2012 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 Lam and Reddy (1999) in [1]. Lam and Li 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
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

E-mail address:
0263-8223/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved.
/>
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 nonlinear 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,
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


89

N.D. Duc / Composite Structures 99 (2013) 88–96

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 thermoelectro-mechanical 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, Nonlinear 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
(2010) 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, usually 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.
The volume fractions of constituents are assumed to vary
through the thickness according to the following power law
distribution

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.

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 + @ / oy , 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 0 1
0
1
ex
ex
kx
B e C B e0 C
B
C

@ y A ¼ @ y A À z@ ky A
cxy
2kxy
c0xy

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:

0

1 0
1
e0x
u;x À w=Rx þ w2;x =2
B e0 C B
C
@ y A ¼ @ v ;y À w=Ry þ w2;y =2 A;

c

0
xy

u;y þ v ;x þ w;x w;y

0

1 0

1
wx;x
kx
B
C B
C
@ ky A ¼ @ wy;y A
w;xy
kxy

ð7Þ

In which u, v are the displacement components along the x, y
directions, respectively.


90

N.D. Duc / Composite Structures 99 (2013) 88–96

The force and moment resultants of the FGM panel are determined by

ðNi ; Mi Þ ¼

Z

x2

h=2

ri ð1; zÞdz 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




E1
EA1 0
E1 m 0
E2
E2 m
kx À
Nx ¼
þ
e
þ
e

À
þ
C
ky
1
x
y
1 À m2
s1
1 À m2
1 À m2
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 1 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
EA1 0
E m
E3
EI2

ky
My ¼
e þ
þ
ey À 3 2 kx À
þ
1 À m2 x
1 À m2
s1
1Àm
1 À m2 s2


1
N xy ¼
E1 c0xy À 2E2 kxy
2ð1 þ mÞ


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

h

z

z2

x1
2

O

s1

s1

s1

s2



Ecm
h
E1 ¼ Em þ
Nþ1

s2

1

Fig. 2. Configuration of an eccentrically stiffened shallow shells.

where

q¼

Z

h
2

À2h



qðzÞdz ¼ qm þ

qcm
Nþ1

2


h

ð12Þ

2

in which 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
Rx Ry
@2w
@t 2

ð13Þ

Calculating from Eq. (9), obtained:

2

Ecm Nh
E2 ¼
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

ð10Þ

ð14Þ





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

; A66 ¼
E1
D 1 À m2


 
2
EA1
E1
EA2
E1
E1 m
À
D¼
þ
þ
s1
1 À m2
s2
1 À m2
1 À m2


E2
E2 m
À A12
B11 ¼ A22 C 1 þ
;
1 À m2
1 À m2



E2
E2 m
À A12
B22 ¼ A11 C 2 þ
1 À m2
1 À m2


E2 m
E2
;
B12 ¼ A22
À
A
þ
C
12
2
1 À m2
1 À m2


E2 m
E2
À A12
þ C1
B21 ¼ A11
1 À m2

1 À m2
E2
B66 ¼
E1
A11 ¼

@2u
@t 2
@2v
¼q 2
@t

N x;x þ N xy;y ¼ q

ð15Þ

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

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

þ q À k1 w þ k2 r2 w ¼ q

e0x ¼ A22 Nx À A12 Ny þ B11 kx þ B12 ky
e0y ¼ A11 Ny À A12 Nx þ B21 kx þ B22 ky
c0xy ¼ A66 Nxy þ 2A66 kxy
where

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. 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
The nonlinear dynamic equations of a FGM shallow shells based
on the classical shell theory are [33]

M x;xx þ 2M xy;xy þ My;yy þ

s2
b

þ q À k1 w þ k2 r2 w ¼ q

where:

s2

z1



N xy;x þ Ny;y

s1

a

M x ¼ B11 Nx þ B21 Ny À D11 kx À D12 ky

@2w
@t 2

M x ¼ B12 Nx þ B22 Ny À D21 kx À D22 ky
ð11Þ

M xy ¼ B66 Nxy À 2D66 kxy

ð16Þ


91

N.D. Duc / Composite Structures 99 (2013) 88–96

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

where:

D11
D22
D12
D21



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
B22 À
¼
þ
À
C
þ

B12
2
s2 1 À m 2
1 À m2
1 À m2


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


E3 m
E2
E2 m
B21 À
¼
À
C
þ

B11
2
1 À m2
1 À m2
1 À m2

D66 ¼

ð17Þ

ð24aÞ

wà ¼ W 0 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

E3
E2
À
B66
2ð1 þ mÞ 2ð1 þ mÞ

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

Nx ¼ f;yy ; Ny ¼ f;xx ; Nxy ¼ Àf;xy

w ¼ WðtÞ sin km x sin dn y

ð18Þ

Subsequently, substitution of Eq. (24a,b) into Eq. (22), (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

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

À

B21 f;xxxx þ B12 f;yyyy þ ðB11 þ B22 À 2B66 Þf;xxyy À D11 w;xxxx À D22 w;yyyy
þ ðD12 þ D21 þ 4D66 ÞwÃ;xxyy þ f;yy w;xx À 2f ;xy w;xy þ f;xx w;yy
f;yy f;xx
@2w
þ
þ q À k1 w þ k2 r2 w ¼ q 2
Rx Ry
@t

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

w;yy À wÃ;yy w;xx À wÃ;xx
À
:
Rx
Ry

ð20Þ

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

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

ð19Þ

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

À

a2

p2

ð21Þ

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

!
n2 k2 m2
ðW À W 0 Þ
þ
Rx
Ry

g

16 mnk2
ðW 2 À W 20 Þ ¼ 0
3 p2

p4 h
a4

ð26Þ

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

i

i
D11 m4 þ ðD12 þ D21 þ 4D66 Þn2 m2 k2 þ D22 n4 k4
!
2
32
p2 hW 2
p2 m2 n2 k2
2
2k
2
ðm r 0 þ n p0 k Þ À 2 g
þ
þ W gmnp 4 þ
3
a2
a
a
Ry
Rx


2
16h r 0 p0
16q
p
@2 W
À
þ
À k1 W À k2 W 2 ðm2 þ k2 n2 Þ ¼ q 2

þ
a
mnp2 Rx Ry
mnp2
@t
À ðW À W 0 Þ

p4 h
a4

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):
"
W

p2 h
a2

ðm2 r 0 þ n2 p0 k2 Þ À k1 À k2
2

þ ðW À W 0 Þ4

A11 f;xxxx þ ðA66 À 2A12 Þf;xxyy þ A22 f;yyyy þ B21 w;xxxx
þ ðB11 þ B22 À 2B66 Þw;xxyy þ B12 w;yyyy
w;yy À wÃ;yy w;xx À wÃ;xx
¼ w2;xy À w;xx w;yy À wÃ;xy2 þ wÃ;xx wÃ;yy À

À
Rx
Ry

Ã

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

À ðD12 þ D21 þ 4D66 Þw;xxyy þ D11 wÃ;xxxx þ D22 wÃ;yyyy

þ

ð25Þ

! #
p4 P2 p2 m2 n2 k2 P2
2 2
2
ðm
þ
k
n
Þ
À
þ
þ
a2
a4 P1 a2 Ry

Rx P 1

p2

!

p2 m2 n2 k2 P2 p4
a2

Ry

þ

Rx

P1

À

a

P À
4 3

m2 n2 k2
þ
Ry
Rx

!2

3
15
P1

"
þ ðW 2 À W 20 Þ

ð22Þ

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.

À W2

!
#
1 m2 n2 k2 16mnk2 16mnp2 k2 P2
þ
À
a2 Ry
Rx
3P 1
a4
P1

!

32mnp2 k2 P2
32mnk2 m2 n2 k2 1
þ
WðW
À
W
Þ
þ
0
3a4
P1
3a2
Ry
Rx P 1

À WðW 2 À W 20 Þ



512m2 n2 k 1
16h r0 p0
16q
@2W
À
þ
¼q 2
þ
4
2
9a

P1 mnp Rx Ry
mnp2
@t

ð28Þ
where:

4. Nonlinear dynamic analysis

P1 ¼ A11 m4 þ ðA66 À 2A12 Þm2 n2 k2 þ A22 n4 k4
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.

P2 ¼ B21 m4 þ ðB11 þ B22 À 2B66 Þm2 n2 k2 þ B12 n4 k4

ð29Þ

P3 ¼ D11 m4 þ ðD12 þ D21 þ 4D66 Þm2 n2 k2 þ D22 n4 k4

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.


92

N.D. Duc / Composite Structures 99 (2013) 88–96

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



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 À W 0 Þ


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

ð30Þ

where

Q 1 ¼ k1 þ k2

p

2

a2

ðm2 þ k2 n2 Þ þ

p P2
4

a4 P1

À

p

2

a2

2

2 2

m
n k

þ
Ry
Rx

!

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

P2
P1

!
!2
n2 k2 P 2 p4
m2 n2 k2
1
Q2 ¼ À 2
þ
þ P3 þ
þ
P1
a
Ry

Rx P 1 a4
Ry
Rx
!
1 m2 n2 k2 16mnk2 16mnp2 k2 P 2
Q3 ¼ 2
þ
À
a
Ry
Rx
3P1
a4
P1

p2 m2

Q4 ¼

32mnp2 k2 P2
3a4
P1

p2 h

p4 P2 p2 m2 P2
ðm2 þ k2 n2 Þ À 4 þ 2
a2
a2
a P1

a RP 1
!
p2 m2 P2 p4
m4
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
a2 3P1 R
a4
P1

W

ð31Þ

512m2 n2 k 1
9a4
P1

q

W

p2 h
a2

p4 P2 p2 m2 P2
ðm2 þ k2 n2 Þ þ 4 À 2
a P1
a RP 1
!
2 2
4
4
p m P2 p
m
þ 4 P3 þ 2
À 2
a RP 1
a
R P1
"
2
3
16mnp2 k2 P2 32mnp2 k2 P2
2 1 16m nk
À
À
ÀW

2
a
3P1 R
a4
P1
3a4
P1
#
2
32mnk m2 1
512m2 n2 k 1
þ W3
ð37Þ
þ
9a4
P1
3a2
R P1

m2 r 0 ¼W k1 þ k2

ð32Þ

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

H2 ¼
H3 ¼

1

q
q

p2 h
ð34Þ

a2

m2 r0 ¼ k1 þ k2

q

Seeking solution as W (t) = scosxt and applying procedure like
Galerkin method to Eq. (33), the frequency–amplitude relation of
nonlinear free vibration is obtained

xNL

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

p

4

p2
a2

ðm2 þ k2 n2 Þ þ

p4 P2
a4 P

1

À

p2 m2 P2
a2 RP

À

p2 m 2 P 2

1

a2 RP 1

4

m
P3 þ 2

R P1
"
1 16m3 nk2 16mnp2 k2 P2 32mnp2 k2 P2
ÀW 2
À
À
a
3P1 R
a4
P1
3a4
P1
#
2
2
2 2
32mnk m 1
512m n k 1
þ W2
þ
9a4
P1
3a2
R P1
þ

Q6


1

8H2
3H3 2 2
¼ xL 1 þ
s
þ
s
3px2L
4x2L

a2

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

ðQ 1 þ Q 2 Þ

Q4 À Q3 À Q5

p2

ð33Þ

where denoting

H1 ¼ x2L ¼

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

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

!

À W2

16Q 0
Q7 ¼
mnp2

xL ¼

p2

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

2

 512m2 n2 k 1
@ W
2
2
À W W À W0
¼q 2
9a4
P1
@t

!
32mnk2 m2 n2 k2 1
Q5 ¼
þ
3a2
Ry
Rx P 1
Q6 ¼

m2 r0 À k1 À k2

a4

ð38Þ

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

p4 P2 p2 m2 P2

ðm2 þ k2 n2 Þ þ 4 À 2
p
a P1
a RP 1
!
p2 m2 P2 p4
m4
þ 4 P3 þ 2
À 2
a RP 1
a
R P1

rupper ¼

a2

m2 h

2

k1 þ k2

p2
a2

And the lower buckling load is found using the condition
follows

ð39Þ

dr0
dW

¼ 0, it


93

N.D. Duc / Composite Structures 99 (2013) 88–96

rlower ¼

a2

4

p

2

2

2

2

4

9a P 1
k1 þ k2 pa2 ðm2 þ k2 n2 Þ þ pa4 PP2 À pa2mRPP2 À pa2mRPP2 þ pa4 P 3 þ Rm2 P À 1024m

2 n2 k
2

2 hm2

1

1

4

4

1

h

1

1 16m3 nk2
a2 3P1 R

2 k2

À 16mna4p

P2
P1

2 k2

p
À 32mn
3a4

P2
P1

2

m2 1
R P1

þ 32mnk
3a2

i2

þ4



512m2 n2 k 1
P1
9a4

2 h

1 16m3 nk2
a2 3P 1 R

2 k2

À 16mna4p

P2
P1

2 k2

p
À 32mn
3a4

P2
P1

2

þ 32mnk
3a2

m2 1
R P1

i!

ð40Þ
4.2.2. Imperfect eccentrically stiffened shallow FGM cylindrical panel
subjected to transverse load

The Eq. (28) in this case Rx ? 1, Ry = R, p0 = r0 = 0 can be rewritten as:

"
W Àk1 À k2

p

2

a2

p2 m2 P2

þ ðW À W 0 Þ


2

þ W À

ðm2 þ k2 n2 Þ À

W 20

À

a4 P1

þ

p4

m4
P3 À
a4
RP 1

a2 RP 1
"
 1 16m3 nk2
a2

p P2

2 2 2

4

3RP 1

p n k P2

Em ¼ 70 Â 10 N=m2 ;

!
#

4

ð41Þ

W Àk1 À k2

a2

ðm þ k n Þ À

2

p4 P 2
a4

P1

R

#

ωNL (rad/s)

5.7415

5.7414

0.05

P1

5. Numerical results and discussions
The eccentrically stiffened FGM shells considered here are shallow shell with in-plane edges:

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

xL (rad/s)
Reinforced

Unreinforced
5

56.130 Â 10
39.034 Â 105
31.982 Â 105
24.047 Â 105

5

55.667 Â 10
38.515 Â 105
31.441 Â 105
23.477 Â 105

0.1

τ

0.15

Fig. 3. Frequency–amplitude relation.

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

xL (rad/s)

K1, K2

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

Reinforced

Unreinforced

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

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)

0
1
2
5

0

!

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.

N

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

5.7416

5.7413

!2 3
p2 m2 þ n k P2 p4
m2 þ n2 k2
15
4
À P3 À
þ ðW À W 0 Þ 2
P1
a
R
P1 a4
R
!
"
#

 1 m2 þ n2 k2 16mnk2 16mnp2 k2 P
2
2
2
À
þ W À W0
a2
R

3P1
a4
P1
!
2
2
2
32mnk m2 þ n2 k2 1
2 32mnp k P 2
þ WðW À W 0 Þ
ÀW
P1
3a4
P1
3a2
R
2

 512m2 n2 k 1
16q
@ W
À W W 2 À W 20
þ
¼q 2
9a4
P 1 mnp2
@t
ð42Þ
2 2

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

p2 m2 þ n2 k2 P2
a2

Reinforced, Rx=R y=3(m), N=5

5.7417

!

þ

x 10

5.7419

5.7418

4.2.3. Imperfect eccentrically stiffened FGM shallow spherical panel
under transverse load
The Eq. (28) in this case Rx = Ry = R, p0 = r0 = 0 can be rewritten
as:
2 2

m ¼ 0:3

z1 ¼ z2 ¼ 0:0225ðmÞ;

The 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 2ðmÞ; h ¼ 0:01ðmÞ; K 1 ¼
200; K 2 ¼ 10; Rx ¼ Ry ¼ 3ðmÞ; W 0 ¼ 1e À 5.

The static critical load can be determined by the equation to be
€ ¼ 0; W 0 ¼ 0 and using condireduced from Eq. (41) by putting W
dq
tion dW
¼ 0.

2

ð43Þ

qm ¼ 2702 kg=m ; qc ¼ 3800 kg=m3 ;

32mnp2 k2 P 2
32m3 nk2 1
þ WðW À W 0 Þ
ÀW
4
3a
P1
3Ra2 P1

 512m2 n2 k 1
16q
@2W
À W W 2 À W 20
þ
¼q 2

4
2
9a
P 1 mnp
@t

p2

Ec ¼ 380 Â 109 N=m2 ;

3

2

"

h ¼ 0:01m;
9

s1 ¼ s2 ¼ 0:4;

#

a2 RP 1

16mnp2 k2 P2
À
a4
P1

a ¼ b ¼ 2m;


94

N.D. Duc / Composite Structures 99 (2013) 88–96

Fig. 5. Deflection–velocity relation of the eccentrically stiffened shallow spherical
FGM shell.
Fig. 4. Effect of eccentrically stiffeners on nonlinear dynamic response of the
shallow spherical FGM shell.

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. 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 q
the
ffiffiffiffi comparison on the fundamental frequency
parameter - ¼ xL h qEcc (In the Table 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 2ðmÞ; h ¼ 0:01ðmÞ; K 1 ¼ 200; K 2 ¼ 10; Rx ¼
Ry ¼ 3ðmÞ; 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. 6. Influence of elastic foundations on nonlinear dynamic response of the
eccentrically stiffened shallow spherical FGM shell.

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


N.D. Duc / Composite Structures 99 (2013) 88–96

95

Fig. 9. Effect of Rx on nonlinear dynamic response.
Fig. 7. Effect of volume metal-ceramic on nonlinear response of the eccentrically
stiffened shallow spherical FGM shell.

Fig. 8. Effect of dynamic loads on nonlinear response.

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. 10. Influence of initial imperfection on nonlinear dynamic response of the
eccentrically stiffened spherical 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


96

N.D. Duc / Composite Structures 99 (2013) 88–96

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

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
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 Vietnam-NAFOSTED. The author is grateful for this financial
support.
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