DSpace at VNU: Nonlinear dynamic analysis of eccentrically stiffened imperfect functionally graded doubly curved thin shallow shells

Composite Structures 96 (2013) 384–395

Contents lists available at SciVerse ScienceDirect

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

Nonlinear dynamic analysis of eccentrically stiffened imperfect functionally
graded doubly curved thin shallow shells
Dao Huy Bich a, Dao Van Dung a, Vu Hoai Nam b,⇑
a
b

Vietnam National University, Ha Noi, Viet Nam
University of Transport Technology, Ha Noi, Viet Nam

a r t i c l e

i n f o

Article history:
Available online 29 October 2012
Keywords:
Functionally graded material
Dynamic analysis
Critical dynamic buckling load
Vibration
Shallow shells
Stiffeners

a b s t r a c t

This paper presents a semi-analytical approach to investigate the nonlinear dynamic of imperfect eccentrically stiffened functionally graded shallow shells taking into account the damping subjected to
mechanical loads. The functionally graded shallow shells are simply supported at edges and are reinforced by transversal and longitudinal stiffeners on internal or external surface. The formulation is based
on the classical thin shell theory with the geometrical nonlinearity in von Karman–Donnell sense and the
smeared stiffeners technique. By Galerkin method, the equations of motion of eccentrically stiffened
imperfect functionally graded shallow shells are derived. Dynamic responses are obtained by solving
the equation of motion by the Runge–Kutta method. The nonlinear critical dynamic buckling loads are
found according to the Budiansky–Roth criterion. Results of dynamic analysis show the effect of stiffeners, damping, pre-loaded compressions, material and geometric parameters on the dynamical behavior of
these structures.
Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction
Eccentrically stiffened shallow shell is a very important structure in engineering design of aircraft, missile and aerospace industries. There are many researches on the static and dynamic
behavior of this structure with different materials.
Studies on the dynamics first were carried out with eccentrically stiffened shallow shells made of homogeneous material.
Khalil et al. [1] presented a finite strip formulation for the nonlinear analysis of stiffened plate structures subjected to transient
pressure loadings using an explicit central difference/diagonal
mass matrix time stepping method. Shen and Dade [2] investigated
dynamic analysis of stiffened plates and shells using spline gauss
collocation method. The free vibration of stiffened shallow shells
was studied by Nayak and Bandyopadhyay [3]. By using the finite
element method, the stiffened shell element was obtained by the
appropriate combinations of the eight-/nine-node doubly curved
isoparametric thin shallow shell element with the three-node
curved isoparametric beam element. Sheikh and Mukhopadhyay
[4] applied the spline finite strip method to investigate linear
and nonlinear transient vibration analysis of plates and stiffened
plates. The von Karman’s large deflection plate theory has been
used and the formulation was done in total Lagrange coordinate
system. Dynamic instability analysis of stiffened shell panels subjected to uniform in-plane harmonic edge loading and partial edge
⇑ Corresponding author.

E-mail address: (V.H. Nam).
0263-8223/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved.
/>
loading along the edges was studied by Patel et al. [5–8]. In these
studies, the new formulation for the beam element requires five
degrees of freedom per node as that of shell element.
For dynamic analysis of eccentrically stiffened laminated composite plates and shallow shells, Satish Kumar and Mukhopadhyay
[9] studied the transient response analysis of laminated stiffened
plates using the first order shear deformation theory. Parametric
study on the dynamic instability behavior of laminated composite
stiffened plate was studied by Patel et al. [10]. The same authors
[11] investigated the dynamic instability of laminated composite
stiffened shell panels subjected to in-plane harmonic edge loading.
By using the commercial ANSYS finite element software, Less and
Abramovich [12] studied the dynamic buckling of a laminated
composite stringer stiffened cylindrical panel. Bich et al. [13] presented an analytical approach to investigate the nonlinear dynamic
of imperfect reinforced laminated composite plates and shallow
shells using the classical thin shell theory with the geometrical
nonlinearity in von Karman–Donnell sense and the smeared stiffeners technique.
For functionally graded materials (FGM), many researches focused on the dynamical analysis of un-stiffened shallow shells.
Liew et al. [14] presented the nonlinear vibration analysis of the
coating FGM substrate cylindrical panel subjected to a temperature
gradient arising from steady heat conduction through the panel
thickness. Matsunaga et al. [15] investigated free vibrations and
stability of FGM doubly curved shallow shells according to a 2-D
higher order deformation theory. Chorfi and Houmat [16] investigated nonlinear free vibrations of FGM doubly curved shallow


385

D.H. Bich et al. / Composite Structures 96 (2013) 384–395

shells with an elliptical plan-form. Nonlinear vibrations of functionally graded doubly curved shallow shells under a concentrated
force were studied by Alijani et al. [17]. Nonlinear dynamical analysis of imperfect functionally graded material shallow shells subjected to axial compressive load and transverse load was studied
by Bich and Long [18], Dung and Nam [19]. The motion, stability
and compatibility equations of these structures were derived using
the classical shell theory. The nonlinear transient responses of
cylindrical and doubly-curved shallow shells subjected to excited
external forces were obtained and the dynamic critical buckling
loads were evaluated based on the displacement responses using
Budiansky–Roth dynamic buckling criterion.
Recently, some authors have studied static and dynamical
behaviors of some kind of shells. Najafizadeh et al. [20] have studied static buckling behaviors of FGM cylindrical shell. Bich et al.
[21] have studied the nonlinear static post-buckling of eccentrically stiffened imperfect functionally graded plates and shallow
shells. The nonlinear dynamical analysis of imperfect eccentrically
stiffened FGM cylindrical panels based on the classical theory with
the von Karman–Donnell geometrical nonlinearity are investigated
by Bich et al. [22].
Following the idea of work [22], in this paper, dynamic governing equations taking into account the effect of damping, nonlinear
vibration and dynamic critical buckling loads of eccentrically stiffened imperfect FGM doubly curved thin shallow shells are established. Effects of stiffeners, material, geometric parameters and
damping on the dynamic behavior of structure are considered.

V c ðzÞ ¼


k
2z þ h
;
2h

V m ðzÞ ¼ 1 À V c ðzÞ;

ð1Þ

where k is the volume fraction exponent (k P 0), z is the thickness
coordinate and varies from Àh/2 to h/2; the subscripts m and c refer
to the metal and ceramic constituents respectively. Effective properties Preff of FGM shell are determined by linear rule of mixture as

Preff ¼ Prm ðzÞV m ðzÞ þ Prc ðzÞV c ðzÞ:

ð2Þ

According to Eqs. (1) and (2), the modulus of elasticity E and the
mass density q can be expressed in the form


k
2z þ h
;
2h

k
2z þ h
qðzÞ ¼ qm V m þ qc V c ¼ qm þ ðqc À qm Þ
:
2h
EðzÞ ¼ Em V m þ Ec V c ¼ Em þ ðEc À Em Þ

ð3Þ

Poisson’s ratio m and linear damping coefficient e are assumed to
be constants.
Assume that the shell is reinforced by eccentrically longitudinal
and transversal homogeneous stiffeners with the elastic modulus
E0 and the mass density q0 of stiffeners. In order to provide the
continuity between the shell and stiffeners, the full metal stiffeners
are put at the metal-rich side of the shell thus E0 and q0 take the
value E0 = Em, q0 = qm and conversely the full ceramic ones at the
ceramic-rich side, so that E0 = Ec, q0 = qc.
3. Theoretical formulation

2. Eccentrically stiffened FGM shallow shells (ES-FGM shallow
shells)
Consider a doubly curved functionally graded shallow thin shell
(see Fig. 1) of thickness h and in-plane edges a and b. The shallow
shell is assumed to have a relative small rise as compared with its
span. Let the (x1,x2) plane of the Cartesian coordinates overlaps the
rectangular plane area of the shell. Note that the middle surface of
the shell generally is defined in terms of curvilinear coordinates,
but for the shallow shell, so the Cartesian coordinates can replace
the curvilinear coordinates on the middle surface.
The volume fractions of constituents are assumed to vary
through the thickness according to the following power law
distribution

The strains at the middle surface and curvatures relating to the
displacement components u, v, w based on the classical shell theory and von Karman–Donnell geometrical nonlinearity assumption
are of the form [24].

 2

@u
1 @w
À k1 w þ
;
@x1
2 @x1
 2
@v
1 @w
¼
À k2 w þ
;
2 @x2
@x2

e01 ¼

v1 ¼

@2w
;
@x21

e02

v2 ¼

@2w
;
@x22

c012 ¼

@u
@ v @w @w
þ
þ
;
@x2 @x1 @x1 @x2

v12 ¼

ð4Þ

@2w
;
@x1 @x2

in which k1 ¼ R11 ; k2 ¼ R12 and R1, R2 are curvatures and radii of curvatures of the shell respectively.

Fig. 1. Configuration of an eccentrically stiffened shallow shell.


386

D.H. Bich et al. / Composite Structures 96 (2013) 384–395

The strain components across the shell thickness at a distance z
from the middle surface are given by

e1 ¼ e01 À zv1 ; e2 ¼ e02 À zv2 ; c12 ¼ c012 À 2zv12 :

ð5Þ

From Eq. (4), the strains must be relative in the deformation
compatibility equation

@ 2 e01 @ 2 e02
@ 2 c012
þ 2 À
¼
@x22
@x1 @x1 @x2

@2w
@x1 @x2

!2
À

@2w @2w
@2w
@2w
À k1 2 À k2 2 :
@x21 @x22
@x2
@x1
ð6Þ

Hooke’s stress–strain relation is applied for the shell

rsh
1 ¼
r

sh
2

EðzÞ
ðe1 þ me2 Þ;
1 À m2

EðzÞ
¼
ðe2 þ me1 Þ;
1 À m2

ssh
12 ¼

ð7Þ

EðzÞ
c ;
2ð1 þ mÞ 12

r ¼ E0 e1 ;
r ¼ E0 e2 :

ð8Þ

Taking into account the contribution of stiffeners by the
smeared stiffeners technique and omitting the twist of stiffeners
because these torsion constants are smaller more than the moments of inertia [24] and integrating the stress–strain equations
and their moments through the thickness of the panel, lead to
the expressions for force and moment resultants of an ES-FGM
shallow shells as [22]



E 0 A1 0
N 1 ¼ A11 þ
e1 þ A12 e02 À ðB11 þ C 1 Þv1 À B12 v2 ;
s1


E 0 A2 0
e2 À B12 v1 À ðB22 þ C 2 Þv2 ;
N 2 ¼ A12 e01 þ A22 þ
s2





1
E0 A1
1
E 0 A2
; AÃ22 ¼

;
A11 þ
A22 þ
D
s1
D
s2
A12
1
AÃ12 ¼
; AÃ66 ¼
;
A66
D



E 0 A1
E0 A2
A22 þ
À A212 ;
D ¼ A11 þ
s1
s2
BÃ22 ¼ AÃ11 ðB22 þ C 2 Þ À AÃ12 B12 ;
BÃ12 ¼ AÃ22 B12 À AÃ12 ðB22 þ C 2 Þ;
BÃ21 ¼ AÃ11 B12 À AÃ12 ðB11 þ C 1 Þ;

B66
:

A66

M 1 ¼ BÃ11 N1 þ BÃ21 N2 À DÃ11 v1 À DÃ12 v2 ;
ð15Þ

where

E1
E1 m
E1
; A12 ¼
; A66 ¼
;
1 À m2
1 À m2
2ð1 þ mÞ
E2
E2 m
E2
; B12 ¼
; B66 ¼
B11 ¼ B22 ¼
;
1 À m2
1 À m2
2ð1 þ mÞ
E3
E3 m
E3
; D12 ¼

; D66 ¼
D11 ¼ D22 ¼
;
1 À m2
1 À m2
2ð1 þ mÞ

E0 I1
À ðB11 þ C 1 ÞBÃ11 À B12 BÃ21 ;
s1

DÃ22 ¼ D22 þ

E0 I2
À B12 BÃ12 À ðB22 þ C 2 ÞBÃ22 ;
s2

DÃ12 ¼ D12 À ðB11 þ C 1 ÞBÃ12 À B12 BÃ22 ;

ð11Þ

DÃ21 ¼ D12 À B12 BÃ11 À ðB22 þ C 2 ÞBÃ21 ;

Based on the classical shell theory and the Volmir’s assumption
2
2
[23] u ( w and v ( w; q1 @@t2u ! 0 and q1 @@t2v ! 0, the nonlinear
motion equations of a shallow thin shell with damping force is
written in the form

@N1 @N12
þ
¼ 0;
@x1
@x2

with

@N12 @N2
þ
¼ 0;
@x1
@x2

E1 ¼

ð12Þ

@ 2 M1
@ 2 M 12 @ 2 M 2
@2w
@2w
@2w
þ2
þ
þ N1 2 þ 2N12
þ N2 2
2
2
@x1 @x2

@x1 @x2
@x1
@x2
@x1
@x2

3

d2 h2
þ A2 z22 :
12

ð16Þ

DÃ66 ¼ D66 À B66 BÃ66 :

A11 ¼ A22 ¼



2
Ec À Em
ðEc À Em Þkh
h; E2 ¼
Em þ
;
kþ1
2ðk þ 1Þðk þ 2Þ

!

Em
1
1
1
3
À
þ
h ;
þ ðEc À Em Þ
E3 ¼
k þ 3 k þ 2 4k þ 4
12

DÃ11 ¼ D11 þ
ð10Þ

where Aij, Bij, Dij (i, j = 1, 2, 6) are extensional, coupling and bending
stiffness of the un-stiffened shell.

I2 ¼

BÃ66 ¼

M 12 ¼ BÃ66 N12 À 2DÃ66 v12 ;

M 12 ¼ B66 c012 À 2D66 v12 ;

3

ð14Þ

BÃ11 ¼ AÃ22 ðB11 þ C 1 Þ À AÃ12 B12 ;

M 2 ¼ BÃ12 N1 þ BÃ22 N2 À DÃ21 v1 À DÃ22 v2 ;

c À 2B66 v12 ;



E0 I 1
M 1 ¼ ðB11 þ C 1 Þe01 þ B12 e02 À D11 þ
v1 À D12 v2 ;
s1


E0 I2
M 2 ¼ B12 e01 þ ðB22 þ C 2 Þe02 À D12 v1 À D22 þ
v2 ;
s2

and

ð13Þ

Substituting Eq. (13) into Eq. (10) yields

ð9Þ

A66 012

d 1 h1
þ A1 z21 ;
12

e01 ¼ AÃ22 N1 À AÃ12 N2 þ BÃ11 v1 þ BÃ12 v2 ;
e02 ¼ AÃ11 N2 À AÃ12 N1 þ BÃ21 v1 þ BÃ22 v2 ;
c012 ¼ AÃ66 þ 2BÃ66 v12 ;

AÃ11 ¼

st
1
st
2

I1 ¼

where the coupling parameters C1, C2 are negative for outside stiffeners and positive for inside ones, s1 and s2 are the spacing of the
longitudinal and transversal stiffeners, A1, A2 are the cross-section
areas of stiffeners, I1,I2 are the second moments of cross-section
areas, z1, z2 are the eccentricities of stiffeners with respect to the
middle surface of shell, and the width and thickness of longitudinal
and transversal stiffeners are denoted by d1, h1 and d2, h2,
respectively.
For later use, the strain-force resultant reverse relations are obtained from Eq. (9)

where

and for stiffeners

N 12 ¼

E 0 A1 z 1
E 0 A2 z 2
; C2 ¼ Æ
;
s1
s2
h1 þ h
h2 þ h
z1 ¼
; z2 ¼
;
2
2

C1 ¼ Æ

þ k1 N1 þ k2 N 2 þ q0 ¼ q1

@2w
@w
;
þ 2q1 e
@t
@t 2

where e is damping coefficient and

ð17Þ

387

D.H. Bich et al. / Composite Structures 96 (2013) 384–395



A1 A2
þ
s1 s2
Àh=2




qc À qm
A1 A2
:
þ
¼ qm þ
h þ q0
kþ1
s1 s2
Z

q1 ¼

h=2

The couple of nonlinear Eqs. (19) and (20) or Eqs. (21) and (22)
in terms of two dependent unknowns w and u are used to investigate the nonlinear vibration and dynamic stability of ES-FGM
shells.

qðzÞdz þ q0

The first two of Eq. (16) are satisfied automatically by introducing a stress function u as
2

2

@ u
;
@x22

N1 ¼

N2 ¼

@ u
;
@x21

2

N12 ¼ À

@ u
:
@x1 @x2

ð18Þ

Substituting Eq. (13) into the compatibility Eqs. (6) and (15)
into the third of Eq. (17), taking into account expressions (4) and
(18), yields

AÃ11

Á @4u
@4u À Ã
@4u
@4w
þ A66 À 2AÃ12
þ AÃ22 4 þ BÃ21 4
4
2
2
@x1
@x1 @x2
@x2
@x1

þ
¼

À

BÃ11

þ

BÃ22

@2w
@x1 @x2

À

2BÃ66

!2
À

An imperfect ES-FGM shallow thin shell considered in this paper is assumed to be simply supported and subjected to uniformly
distributed pressure of intensity q0 and axial compression of intensities r0 and p0 respectively at its cross-section. Thus the boundary
conditions are

w ¼ 0;

M 1 ¼ 0;

N1 ¼ Àr 0 h;

N12 ¼ 0;

at

x1 ¼ 0;

a;

w ¼ 0;

M 2 ¼ 0;

N2 ¼ Àp0 h;

N12 ¼ 0;

at

x2 ¼ 0;

b:

Á @4w
@4w
@2w
@2w
þ BÃ12 4 þ k1 2 þ k2 2
2
2
@x1 @x2
@x2
@x2
@x1

@2w @2w
;

@x21 @x22

ð23Þ

ð19Þ

The mentioned conditions (23) can be satisfied identically if the
buckling mode shape is chosen by

w ¼ f ðtÞ sin

Á @4w
@2w
@w
@4w À
q1 2 þ 2q1 e þ DÃ11 4 þ DÃ12 þ DÃ21 þ 4DÃ66 2 2
@t
@x1
@x1 @x2
@t
4
4
4
À
Á
@
w
@
u
@

u
@4u
þ DÃ22 4 À BÃ21 4 À BÃ11 þ BÃ22 À 2BÃ66
À BÃ12 4
@x1
@x21 @x22
@x2
@x2
À k1

4. Nonlinear dynamic analysis

@2u
@2u @2u @2w
@2u
@2w
@2u @2w
À k2 2 À 2
þ2
À 2
¼ q0 :
2
2
@x1 @x2 @x1 @x2 @x1 @x22
@x2
@x1
@x2 @x1
ð20Þ

mpx1

npx2
sin
;
a
b

ð24Þ

where f(t) is time dependent total amplitude and m, n are numbers
of haft waves in x1 and x2 directions respectively.
The initial imperfection w0 is assumed to have the same form of
the shell deflection w, i.e.

w0 ¼ f0 sin

mpx1
npx2
sin
;
a
b

ð25Þ

For an initial imperfection shell: The initial imperfection of the
shell considered here can be seen as a small deviation of the shell
middle surface from the perfect shape, also seen as an initial
deflection which is very small compared with the shell dimensions,
but may be compared with the shell wall thickness. Let
w0 = w0(x1, x2) denote a known small imperfection, proceeding

from the motion Eqs. (19) and (20) of a perfect FGM shallow shell
and following to the Volmir’s approach [23] for an imperfection
shell, we can formulate the system of motion equations for an
imperfect eccentrically stiffened functionally graded shallow shell
(Imperfect ES-FGM shallow shell) as

where f0 is the known initial amplitude.
Substituting Eqs. (24) and (25) into Eq. (21) and solving the
resulting equation for unknown u yield

Á @4u
@4u À
@4u
@ 4 ðw À w0 Þ
AÃ11 4 þ AÃ66 À 2AÃ12
þ AÃ22 4 þ BÃ21
2
2
@x1
@x1 @x2
@x2
@x41
À
Á @ 4 ðw À w0 Þ
@ 4 ðw À w0 Þ
þ BÃ11 þ BÃ22 À 2BÃ66
þ BÃ12
2
2
@x1 @x2

@x42
2
3
!2
@ 2 ðw À w0 Þ
@ 2 ðw À w0 Þ 4 @ 2 w
@ 2 w @ 2 w5
þ k1
þ k2
À
À 2
@x1 @x2
@x22
@x21
@x1 @x22
2
3
!2
@ 2 w0
@ 2 w0 @ 2 w0 5
¼ 0;
ð21Þ
À
þ4
@x1 @x2
@x21 @x22

u1 ¼

2mpx1

2npx2
mpx1
npx2
þ u2 cos
À u3 sin
sin
a
b
a
b
x22
x21
À r 0 h À p0 h ;
2
2

u ¼ u1 cos

where
n 2 k2 f 2
m2 f 2
;
à ; u2 ¼
32m2 A11
32n2 k2 AÃ22
h

i
À
Á

2
BÃ21 m4 þ BÃ11 þ BÃ22 À 2BÃ66 m2 n2 k2 þ BÃ12 n4 k4 À pa 2 k1 n2 k2 þ k2 m2
u3 ¼
:
À Ã
Ã
à Á
Ã
A11 m4 þ A66 À 2A12 m2 n2 k2 þ A22 n4 k4
ð27Þ

Substituting the expressions (24)–(26) into Eq. (22) and applying the Galerkin’s method to the obtained equation

Z
0

U  q1

À

2

2

where w is a total deflection of shell.

0

U sin

mpx1
npx2
sin
dx1 dx2 ¼ 0
a
b

p

2

@ u@ w
@ u
@ w
@ u@ w
þ2
À
À q0 ¼ 0;
@x1 @x2 @x1 @x2 @x21 @x22
@x22 @x21

a

À
Á
À
Á
Á
a2 h À
þ H f 2 À f02 þ K f 2 À f02 f À 2 r 0 m2 þ p0 n2 k2 f

4
À
Á @4u
@2u
@2u
à @ u
À BÃ11 þ BÃ22 À 2BÃ66
À
B
À
k
À
k
1
2
12
@x21 @x22
@x42
@x22
@x21
2

Z

!
B2
8mnk2
B
ðf À f0 Þ þ

M€f þ 2Mef_ þ D þ
d1 d2 ðf À f0 Þf
A
A
3p2

@ 4 ðw À w0 Þ
@ 4 ðw À w0 Þ
@4u
þ DÃ22
À BÃ21 4
2
2
4
@x1 @x2
@x2
@x1

2

b

lead to

Á
@2w
@w
@ 4 ðw À w0 Þ À Ã
þ DÃ11
þ 2q1 e

þ D12 þ DÃ21 þ 4DÃ66
2
4
@t
@x1
@t

2

ð26Þ

ð22Þ

4a4 h
4a4
d1 d2 ðk1 r 0 þ k2 p0 Þ À q0
d1 d2 ¼ 0;
þ
6
mnp
mnp6
where the coefficients are given by

ð28Þ


388

D.H. Bich et al. / Composite Structures 96 (2013) 384–395

M¼

a4

p4

À

Á

q1 ; A ¼ AÃ11 m4 þ AÃ66 À 2AÃ12 m2 n2 k2 þ AÃ22 n4 k4 ;

À
Á
Á
a2 À
B ¼ BÃ21 m4 þ BÃ11 þ BÃ22 À 2BÃ66 m2 n2 k2 þ BÃ12 n4 k4 À 2 k1 n2 k2 þ k2 m2 ;
p
À
Á
D ¼ DÃ11 m4 þ DÃ12 þ DÃ21 þ 4DÃ66 m2 n2 k2 þ DÃ22 n4 k4 ;
!


2mnk2 BÃ21 BÃ12
a2
k2 n2 k2 k1 m2
d
H¼
þ

d
À
þ
ð29Þ
d1 d2 ;
1 2
3p2 AÃ11 AÃ22
6p4 mn
AÃ11
AÃ22
!
1 m 4 n 4 k4
a
K¼
þ
;k ¼ ;d1 ¼ ðÀ1Þm À 1; d2 ¼ ðÀ1Þn À 1:
16 AÃ22 AÃ11
b
The governing Eq. (28) is a basic equation for dynamic analysis
of imperfect ES-FGM doubly-curved shallow thin shells in general.
Based on this equation, the nonlinear vibration of perfect and
imperfect FGM shallow shells can be investigated and the dynamic
buckling analysis of shells under various loadings can be
performed.
Particularly for a spherical panel k1 = k2 = 1/R, for a cylindrical
panel k1 = 0, k2 = 1/R, and for a plate k1 = k2 = 0 are taken in Eqs.
(28) and (29).
4.1. Nonlinear vibration
Consider an imperfect ES-FGM shallow shell under uniformly
lateral pressure q0 = Q sin Xt and pre-loaded compressions r0, p0,

Eq. (28) becomes

M€f þ 2Mef_ þ D þ

!
B2
8mnk2 B
d1 d2 ðf À f0 Þf
ðf À f0 Þ þ
A
3p2 A

p

ð30Þ

By using this equation, the fundamental frequencies of natural
vibration of ES-FGM shell and FGM shell without stiffeners, and
frequency-amplitude relation of nonlinear vibration and nonlinear
response of ES-FGM shell are taken into consideration. The nonlinear dynamical responses of ES-FGM shells can be obtained by solving Eq. (30) by the fourth order Runge–Kutta iteration method with
the time step Dt and with initial conditions to be assumed as
f ð0Þ ¼ 0; f_ ð0Þ ¼ 0.
If the vibration is free and linear, and without damping, Eq. (30)
reduces to

M€f þ D þ

!
B2
ðf À f0 Þ ¼ 0:

A

ð31Þ

The fundamental frequencies of natural vibration of imperfect
ES-FGM shallow shells can be determined by

xmn

ð32Þ

If the shell is perfect ES-FGM and the vibration is nonlinear
forced vibration without pre-loaded compressions r0 = p0 = 0, Eq.
(30) can be rewritten a

€f þ 2ef_ þ x2 Àf þ H1 f 2 þ H2 f 3 Á ¼ F sinðXtÞ;
mn

ð33Þ

in which

H1 ¼

8mnk2 B
d d
3p2 A 1 2

M x2mn

þH

4e

p



X ¼ x2mn 1 þ


8H1
3H
F
g þ 2 g2 À ;
3p
4
g

ð35Þ

where g is the amplitude of nonlinear vibration.
By introducing the non-dimension frequency parameter n ¼ xXmn ,
Eq. (35) becomes

n2 À



8H1

3H
F
n¼ 1þ
g þ 2 g2 À 2 :
pxmn
3p
4
gxmn
4e

ð36Þ

In the case of the nonlinear forced vibration without damping, Eq.
(36) leads to

n2 ¼ 1 þ

8H1
3H
F
g þ 2 g2 À 2 :
3p
4
gxmn

ð37Þ

If F = 0 and without damping, the frequency–amplitude relation of
the nonlinear free vibration, from Eq. (37), is as

n2 ¼ 1 þ

8H1
3H
g þ 2 g2 :
3p
4

ð38Þ

4.2. Nonlinear dynamic buckling
Investigate the nonlinear dynamic buckling analysis of imperfect ES-FGM doubly-curved panels under lateral pressure varying
as linear function of time q0 = ct (c is a loading speed) and preloaded compressions r0 = const, p0 = const, Eq. (28) becomes

À
Á
À
Á
Á
a2 h À
þ H f 2 À f02 þ K f 2 À f02 f À 2 r 0 m2 þ p0 n2 k2 f

p

4a4 h
4a4
þ
d1 d2 ðk1 r 0 þ k2 p0 Þ ¼ 6
d d ct:
6

mnp
p mn 1 2

ð39Þ

By solving Eq. (39), the dynamic critical time tcr can be obtained
according to Budiansky–Roth criterion [25]. This criterion is based
on that, for large value of loading speed, the amplitude-time curve
of obtained displacement response increases sharply depending on
time and this curve obtain a maximum by passing from the slope
point and at the corresponding time t = tcr the stability loss occurs.
Here t = tcr is called critical time and the load corresponding to this
critical time is called dynamic critical buckling load qdcr = ctcr.
For static buckling analysis of ES-FGM shallow shell, the explicit
expressions of the upper and lower buckling load were obtained in
[21].
5. Numerical results and discussions
5.1. Validation of the present formulation

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
!
u
u1
B2
t
:
Dþ
¼
M
A



X2 À

!
B2
8mnk2 B
_
€
d1 d2 ðf À f0 Þf
Mf þ 2M ef þ D þ
ðf À f0 Þ þ
A
3p2 A

À
Á
À
Á
Á
a2 h À
þ H f 2 À f02 þ þK f 2 À f02 f À 2 r0 m2 þ p0 n2 k2 f
4a4 h
4a4
þ
d1 d2 ðk1 r0 þ k2 p0 Þ ¼ 6
d d Q sin Xt:
6
mnp

p mn 1 2

Seeking solution as f(t) = g sin (Xt) and applying procedure like
Galerkin method to Eq. (33), the frequency–amplitude relation of
nonlinear forced vibration is obtained


;

H2 ¼

K
M x2mn

;

F¼

4a4 d1 d2 Q
:
Mp6 mn

ð34Þ

To validate the present study, the fundamental frequency
parameter of unstiffened FGM shallow shells is compared with
other studies.
Table 1 shows the present results in comparison with those presented by Matsunaga [15] based on the two-dimensional (2D)
higher-order theory, Chorfi and Houmat [16] accorded to the
first-order shear deformation theory and Alijani et al. [17] used

Donnell’s nonlinear shallow shell theory. In p
this
comparison, the
ffiffiffiffiffiffiffiffiffiffiffiffi
~ ¼ xmn h qc =Ec of the perfect
fundamental frequency parameter x
un-stiffened FGM shallow shell (a/b = 1, h/a = 0.1) with simply supported edges. The material properties are Aluminum and Alumina,
i.e. Em = 70 Â 109 N/m2; qm = 2702 kg/m3 and Ec = 380 Â 109 N/m2;
qc = 3800 kg/m3 respectively. The Poisson’s ratio is chosen to be


389

D.H. Bich et al. / Composite Structures 96 (2013) 384–395
Table 1
~ with results reported by Matsunaga [15], Chorfi and Houmat [16]
Comparison of x
and Alijani et al. [17].
b/R2

a/R1

k

Present

Ref. [15]

Ref. [16]

Ref. [17]

0
0.5
1
4
10

0.0597
0.0506
0.0456
0.0396
0.0381

0.0588
0.0492
0.0430
0.0381
0.0364

0.0577
0.0490
0.0442
0.0383
0.0366

0.0597
0.0506
0.0456
0.0396

0.0380

FGM spherical panel
0.5
0.5
0
0.5
1
4
10

0.0779
0.0676
0.0617
0.0520
0.0482

0.0751
0.0657
0.0601
0.0503
0.0464

0.0762
0.0664
0.0607
0.0509
0.0471

0.0779

0.0676
0.0617
0.0519
0.0482

FGM cylindrical panel
0
0.5
0
0.5
1
4
10

0.0648
0.0553
0.0501
0.0430
0.0409

0.0622
0.0535
0.0485
0.0413
0.0390

0.0629
0.0540
0.0490
0.0419

0.0395

0.0648
0.0553
0.0501
0.0430
0.0408

FGM hyperbolic paraboloidal panel
À0.5
0.5
0
0.0597
0.5
0.0506
1
0.0456
4
0.0396
10
0.0381

0.0563
0.0479
0.0432
0.0372
0.0355

0.0580
0.0493

0.0445
0.0385
0.0368

0.0597
0.0506
0.0456
0.0396
0.0380

FGM plate
0
0

0.3. As can be seen, a very good agreement is obtained in the comparison with the result of Ref. [17], but there are a little differences
with those of Refs. [15,16] because they use above mentioned
other theories.
5.2. Nonlinear vibration results
To illustrate the proposed approach of eccentrically stiffened
FGM shallow shells, the stiffened and un-stiffened FGM shallow
shells are considered with in-plane edges a = b = 0.8 m;
h = 0.01 m;. The shells are simply supported at all its edges. The
combination of materials consists of Aluminum Em = 70 Â 109 N/
m2, qm = 2702 kg/m3 and Alumina Ec = 380 Â 109 N/m2 qc = 3800 kg/m3. The Poisson’s ratio is chosen to be 0.3 for simplicity. Material of stiffeners has elastic modulus Es1 = Es1 = 380 Â 109 N/m2,
qs1 = qs2 = 3800 kg/m3. The height of stiffeners is equal to 50 mm,
its width 2.5 mm, the spacing of stiffeners s1 = s2 = 0.1 m.
The obtained results in Table 2 show that effects of stiffeners on
the fundamental frequencies of natural vibration are considerable.
Obviously the natural fundamental frequencies of un-stiffened and
stiffened FGM spherical panels observed to be dependent on the

constituent volume fractions, they decrease when increasing the
power index k, furthermore with greater value k the effect of stiffeners is observed to be stronger. This is completely reasonable due
to the lower value of the elasticity modulus of the metal constituent. The natural frequencies of stiffened spherical panels are greater than one of un-stiffened panels.
Table 3 shows high frequencies of natural vibration of spherical
panels with R1 = R2 = 5 m, k = 1. Clearly, all modes of stiffened
spherical panel are greater than ones of un-stiffened panel. Especially, the difference is larger with higher modes.
Table 4 shows the fundamental frequencies of doubly curved
FGM shallow shells. In this case, with k = 1, It seem that the natural
frequencies of un-stiffened panels depend to the value jk1 + k2j, i.e.
they have the same value for all panels of same jk1 + k2j, the natural
frequencies of un-stiffened panels increases when this value increases, when k1 + k2 = 0 (R1 = 5, R2 = À5 or R1 = À5, R2 = 5) the nat-

Table 2
Fundamental frequencies of natural vibration (rad/s) of spherical panels.
R1 = R2

k

Un-stiffened

Stiffened

3

0.2
1
5
10

3291.88

2863.12
2250.15
2060.96

3446.03
3112.45
2687.97
2579.30

5

0.2
1
5
10

2093.17
1809.95
1436.76
1325.01

2724.96
2560.43
2376.61
2339.12

10

0.2
1

5
10

1287.67
1095.27
893.94
839.28

2412.76
2348.57
2294.78
2292.47

1 (plate)

0.2
1
5
10

866.22
712.41
614.33
594.95

2388.11
2365.91
2360.03
2359.30

Table 3
Frequencies of natural vibration (rad/s) of spherical panels with R1 = R2 = 5 m, k = 1.

x1(m = 1, n = 1)
x2(m = 1, n = 2) and (m = 2, n = 1)
x3(m = 2, n = 2)
x4(m = 1, n = 3) and (m = 3, n = 1)
x5(m = 2, n = 3) and (m = 3, n = 2)

Un-stiffened

Stiffened

1809.95
2437.29
3299.81
3931.47
4920.50

2560.43
6743.12
9309.30
14576.20
16063.76

ural frequencies of un-stiffened panels are equal to the natural
frequencies of un-stiffened plates (see Tables 2 and 4), but as can
be seen, this phenomenon is not observed for stiffened panels.
For stiffened panels of the same value jk1 + k2j have different natural frequencies. Except in special cases when k1 + k2 = 0, the natural
frequencies of stiffened panels have the same value with stiffened

respective plate.
The frequency–amplitude curves of nonlinear vibration of FGM
spherical panels without damping are presented in Fig. 2. This figure shows that the frequency–amplitude curves of forced vibration
are asymptotic with the frequency–amplitude curve of free vibration and the extreme point of stiffened panel is greater than one
of un-stiffened panel. In forced vibration, a value of n corresponds
to the maximum five distinguished values of g.
Fig. 3 shows the effect of excitation force Q on the frequency–
amplitude curves of nonlinear vibration of spherical panels. As
can be seen, when the excitation force decreases, the curves of
forced vibration are closer to the curve of free vibration.
Figs. 4 and 5 show the effect of volume-fraction index and radius of panels on the frequency–amplitude curve of ES-FGM spherical panel without damping. Clearly, the extreme points of
frequency–amplitude curve decrease when the volume-fraction index or the radius decreases.
Four cases of Gauss curvature of stiffened and un-stiffened panels are taken into consideration: k1k2 > 0 and k1 + k2 > 0 when
R1 = 3 m, R2 = 10 m, k1k2 > 0 and k1 + k2 < 0 when R1 = À3 m,
R2 = À10 m, k1k2 < 0 and k1 + k2 > 0 when R1 = 3 m, R2 = À10 m
and k1k2 < 0 and k1 + k2 < 0 when R1 = À3 m, R2 = 10 m (Figs. 6–9).
It seems that the frequency–amplitude curve depends on the value
k1 + k2. The minimal point of frequency–amplitude curve increases
when this value decreases. Especially, when k1 + k2 < 0, the frequency–amplitude curve of free vibration does not exist extreme
points (see Figs. 7 and 9).


390

D.H. Bich et al. / Composite Structures 96 (2013) 384–395

Table 4
Fundamental frequencies of natural vibration (rad/s) of doubly curved shallow shells.
jR1j

jR2j

k

R1 > 0, R2 > 0

R1 > 0, R2 < 0

R1 < 0, R2 > 0

R1 < 0, R2 < 0

Unstiffened

Stiffened

Unstiffened

Stiffened

Unstiffened

Stiffened

Unstiffened

Stiffened

3

5

0.2
1
5
10

2684.30
2330.04
1837.47
1686.97

3052.89
2805.76
2506.85
2436.09

1074.14
902.84
751.55
713.94

2370.00
2327.75
2300.22
2304.56

1074.14
902.84
751.55

713.94

2539.60
2505.03
2480.60
2484.22

2684.30
2330.04
1837.47
1686.97

3556.80
3361.46
3119.90
3062.03

3

10

0.2
1
5
10

2238.69
1938.18
1535.30
1413.86

2799.50
2614.99
2403.89
2358.66

1409.22
1203.97
975.40
911.55

2446.54
2368.80
2298.28
2291.62

1409.22
1203.97
975.40
911.55

2728.14
2666.60
2606.00
2599.46

2238.69
1938.18
1535.30
1413.86

3247.06
3102.05
2929.40
2891.28

5

5

0.2
1
5
10

2093.17
1809.95
1436.76
1325.01

2724.96
2560.43
2376.61
2339.12

866.22
712.41
614.33
594.95

2388.11
2365.91
2360.03
2359.30

866.22
712.41
614.33
594.95

2388.11
2365.91
2360.03
2359.30

2093.17
1809.95
1436.76
1325.01

3150.15
3020.95
2869.74
2837.71

Fig. 2. The frequency–amplitude curve of nonlinear vibration of FGM spherical
panels (R = 5 m, k = 1, Q = 105 N/m2).

Fig. 3. Effect of excitation force Q on the frequency–amplitude curve of stiffened
spherical panel (R = 5 m, k = 1).

Fig. 10 investigate the frequency–amplitude curve of nonlinear
vibration of stiffened plate and shallow shells with jR1j = jR2j, the
phenomenon is similar with Figs. 6–9, however when k1 + k2 = 0
with R1 = À5, R2 = 5 and R1 = 5, R2 = À5 the frequency–amplitude
curves coincide to the frequency–amplitude curve of plate.
Consider the un-stiffened and stiffened perfect FGM spherical
panel without damping under the uniformly harmonic load q0
(t) = Q sin (Xt), nonlinear responses are obtained solving the Eq.
(30) by fourth order Runge–Kutta method with the time step Dt.

Fig. 4. Effect of index k on the frequency–amplitude curves of stiffened spherical
panels (R = 5 m, Q = 105 N/m2).

Nonlinear responses of un-stiffened FGM spherical panel with
difference time steps are presented in Fig. 11. As can be observed,
difference of nonlinear responses of time steps Dt = 2 Â 10À4 and
Dt = 10À4 is very small. Therefore, the next results are calculated
with time step Dt = 10À4 to ensure the accuracy of this method.
Nonlinear responses of stiffened and un-stiffened functionally
graded spherical panel with k = 1, R = 5 are presented in Figs. 12
and 13. Natural frequencies of un-stiffened and stiffened spherical
panel are 1809.95 rad/s and 2560.43 rad/s, respectively (see
Table 2). The excitation frequencies are much smaller (Fig. 12,
q0(t) = 105 sin (100t)) and much greater (Fig. 13, q0(t) = 105 sin
(104t)) than natural frequencies. These results show that the
stiffeners strongly decrease vibration amplitude of the shell when
excitation frequencies are much smaller or much greater than
natural frequencies.
When the excitation frequencies are near to natural frequencies, the interesting phenomenon is observed like the harmonic

beat phenomenon of a linear vibration (Figs. 14 and 15). The excitation frequencies are 2510 rad/s and 2530 rad/s which are very
near to natural frequencies 2348.57 rad/s of stiffened spherical panel. These results show that the amplitude of beats of stiffened
panels increases rapidly when the excitation frequency approaches
the natural frequencies. The maximal amplitude of harmonic beat
increases and the response time of beat decreases when the excitation force increases as shown in Fig. 15.
The deflection–velocity relation has the closed curve form as in
Fig. 16. Deflection f and velocity f_ are equal to 0 at initial time and
final time of beat and the contour of this relation corresponds to
the period which has the greatest amplitude of beat.


D.H. Bich et al. / Composite Structures 96 (2013) 384–395

Fig. 5. Effect of radius R on the frequency–amplitude curves of stiffened spherical
panels (k = 1, Q = 105 N/m2).

Fig. 6. The frequency–amplitude curve of nonlinear vibration of un-stiffened
shallow shells (k = 1, Q = 105 N/m2).

Fig. 7. The frequency–amplitude curve of nonlinear vibration of un-stiffened
shallow shells (k = 1, Q = 105 N/m2).

Figs. 17 and 18 show the effect of pre-loaded compressions and
of known initial amplitude on the nonlinear responses of ES–FGM
spherical panel. In these investigations, pre-loaded compressions
and known initial amplitude slightly influence on the amplitude
of nonlinear vibration of panels.
Effect of damping on nonlinear responses is presented in Figs. 19
and 20 with linear damping coefficient e = 0.3. The damping influences very small to the nonlinear response in the first vibration
periods (Fig. 19) however it strongly decreases amplitude at the

next far periods (Fig. 20).

391

Fig. 8. The frequency–amplitude curve of nonlinear vibration of stiffened shallow
shells (k = 1, Q = 105 N/m2).

Fig. 9. The frequency–amplitude curve of nonlinear vibration of stiffened shallow
shells (k = 1, Q = 105 N/m2).

Fig. 10. The frequency–amplitude curve of nonlinear vibration of stiffened plate
and shallow shells (k = 1, Q = 105 N/m2).

5.3. Nonlinear dynamic buckling results
To evaluate the effectiveness of stiffener in the nonlinear
dynamic buckling problem, we consider an imperfect ES-FGM
cylindrical panel and spherical panel under lateral pressure and
pre-loaded compression. Materials of shells and stiffeners used in
this section are the same in the previous section.
The effect of stiffeners to the critical buckling of perfect FGM
shallow shells under only lateral pressure is investigated for two
cases of cylindrical and spherical panel under uniformly lateral


392

D.H. Bich et al. / Composite Structures 96 (2013) 384–395

Fig. 11. Nonlinear responses of un-stiffened FGM spherical panel with difference
time steps (R = 5 m, k = 1, q(t) = 105 sin 100t).

Fig. 12. Nonlinear
q(t) = 105 sin 100t).

responses

Fig. 13. Nonlinear
q(t) = 105 sin 104t).

responses

of

FGM

spherical

panel

(R = 5 m,

Fig. 14. Harmonic beat phenomenon of stiffened spherical panel (R = 5 m, k = 1,
q(t) = 105 sin Xt).

k = 1,
Fig. 15. Harmonic beat phenomenon of stiffened spherical panel (R = 5 m, k = 1,
q(t) = Q sin 2530t).

of

FGM

spherical

panel

(R = 5 m,

k = 1,

pressure varying on time as q0 = 105t (N/m2) (see Figs. 21 and 22).
The critical buckling load corresponds to the buckling mode shape
m = 1, n = 1 in all cases. These figures also show that there is no definite point of instability as in static analysis. Rather, there is a region of instability where the slope of t vs. f/h curve increases
rapidly. According to the Budiansky–Roth criterion, the critical
time tcr can be taken as an intermediate value of this region. There
2 
¼ 0 as in
fore one can choose the inflexion point of curve i.e. ddt2f 
t¼t cr

Ref. [26]. As can be seen, the dynamic buckling loads of stiffened
panels are greater than one of un-stiffened panels.

Fig. 16. Deflection–velocity relation of stiffened spherical panel (R = 5 m, k = 1,
q(t) = 2 Â 105 sin Xt).

Effect of volume-fraction index k and loading speed c on critical
dynamic buckling of stiffened cylindrical panels and spherical panel are showed in Table 5. Clearly, the critical dynamic buckling
of panel decrease when the volume-fraction index increases or
the loading speed decreases. As can be also observed in Table 5,

the critical dynamic buckling load is greater than the static critical
load.
Table 6 shows the effect of thickness h on the critical dynamic
buckling load of cylindrical and spherical panels. The critical dynamic buckling loads of un-stiffened and stiffened panels increase
when the thickness of panels increases. In addition, Table 6 also


393

D.H. Bich et al. / Composite Structures 96 (2013) 384–395

Fig. 17. Effect of pre-loaded compressions on nonlinear responses of stiffened
spherical panel (R = 5 m, k = 1, q(t) = 105 sin 105t).

Fig. 20. Effect of damping on nonlinear responses of stiffened spherical panel
(R = 5 m, k = 1, q(t) = 5 Â 103 sin 2530t).

Fig. 18. Effect of known initial amplitude on nonlinear responses of stiffened
spherical panel (R = 5 m, k = 1, q(t) = 105 sin 100t).

Fig. 21. Effect of stiffeners on dynamic buckling of cylindrical panel under lateral
pressure (R = 6 m, f0 = 0, d1 = d2 = 3 Â 10À3 m, h1 = h2 = 3 Â 10À2 m, s1 = s2 = 0.25 m).

Fig. 19. Effect of damping on nonlinear responses of stiffened spherical panel
(R = 5 m, k = 1, q(t) = 5 Â 103 sin 2530t).

Fig. 22. Effect of stiffeners on dynamic buckling spherical panel under lateral
pressure
(R = 10 m,
f0 = 0,

d1 = d2 = 3 Â 10À3 m,
h1 = h2 = 4 Â 10À2 m,
s1 = s2 = 0.25 m).

shows that the effect of stiffeners decreases when the thickness
increases.
Table 7 gives the results on the effect of known initial amplitude
f0 on the nonlinear buckling of stiffened cylindrical and stiffened
spherical panels. Clearly, the known initial amplitude slightly
influences on the critical dynamic buckling loads of these structures when they only subjected to lateral pressure.
Fig. 23 shows the dynamic response of stiffened spherical panels under combination of lateral pressure varying on time q0 = 105t
(N/m2) and pre-loaded compressions r0 = const, p0 = const. As can
be observed, the pre-loaded compressions strongly influence on

Table 5
Effect of volume fraction index k, loading speed c on the critical dynamic buckling
load of ES-FGM shallow panels (Â105 N/m2).
k

0.2

Cylindrical panels
Static
Dynamic q0 = 105t
Dynamic q0 = 106t

2.3179
2.3423
2.4574

Spherical panels
Static
Dynamic q0 = 105t
Dynamic q0 = 106t

17.7013
17.7236
17.8372

1

5

10

1.6022
1.6272
1.7441

0.9001
0.9256
1.0378

0.7461
0.7709
0.8732

12.0989
12.1219
12.2372

6.5172
6.5421
6.6619

5.2513
5.2770
5.4012

R = 5 m, d1 = d2 = 3 Â 10À3 m, h1 = h2 = 35 Â 10À3 m, s1 = s2 = 0.25 m.


394

D.H. Bich et al. / Composite Structures 96 (2013) 384–395

Table 6
Effect of thickness h on the critical dynamic buckling load of ES-FGM shallow panels
(Â105 N/m2).
h

0.006

0.008

0.01

Cylindrical panels
Un-stiffened
Stiffened

Difference (%)

0.2262
0.3678
62.60

0.3084
0.4635
50.29

0.4024
0.5744
42.74

1.5663
1.6855
7.61

2.1071
2.2353
6.08

2.6695
2.8099
5.26

Spherical panels
Un-stiffened
Stiffened
Difference (%)

R = 10 m, d1 = d2 = 3 Â 10À3 m, h1 = h2 = 4 Â 10À3 m, s1 = s2 = 0.25 m, q0 = 105t, k = 1.

Table 7
Effect of known initial amplitude f0 on the critical dynamic buckling load of ES-FGM
shallow panels (Â105 N/m2).
f0

0

À4

10

5 Â 10

À4

10

À3

R = 5 m, d1 = d2 = 3 Â 10À3 m, h1 = h2 = 35 Â 10À3 m, s1 = s2 = 0.25 m, k = 1,
q0 = 105t
Cylindrical panels
Spherical panels

1.6272
12.1219

1.6241
12.1071

1.6103
12.0471

1.5884
11.9739

Fig. 23. Effect of pre-loaded compression on dynamic buckling stiffened spherical
panel (R = 10 m, f0 = 0, d1 = d2 = 3 Â 10À3 m, h1 = h2 = 3.5 Â 10À2 m, s1 = s2 = 0.25 m).

the critical dynamic buckling of stiffened spherical panel. The critical dynamic buckling of panel increases when the pre-loaded
compressions increase.
6. Conclusions
The formulation of the governing equations of eccentrically
stiffened functionally graded doubly curved imperfect shallow
shells based upon the classical shell theory and the smeared stiffeners technique with von Karman–Donnell nonlinear terms is
presented. By using the Galerkin method, the nonlinear dynamic
equation for analysis of dynamic and stability characteristics of
ES-FGM shallow shells is obtained. The effects of material parameters, stiffeners and initial geometrical imperfection to the static
and dynamic behavior of FGM shallow shells are investigated.
The results of considered cases show some special phenomena
as:
(i) Fundamental frequencies of natural vibration of un-stiffened
panels seem to depend only on the value jk1 + k2j, but this
phenomenon is not observed for stiffened panels. Except in

(ii)

(iii)
(iv)
(v)

special cases when k1 + k2 = 0, the natural frequencies of
stiffened panels have the same value with stiffened respective plate.
The frequency–amplitude curves of nonlinear vibration of
shallow shells which have k1 + k2 = 0 are coincide to the frequency–amplitude curve of respective plate.
The stiffener system strongly enhances the stability and
load-carrying capacity of FGM shells.
The pre-loaded compressions strongly influence on the critical dynamic buckling of stiffened spherical panel.
The damping influences very small to the nonlinear response
in the first vibration periods, however it strongly decreases
amplitude at the next far periods.

Acknowledgements
This research is funded by Vietnam National Foundation for
Science and Technology Development (NAFOSTED) under Grant
No. 107.01-2012.02. The authors are grateful for this financial
support.

References
[1] Khalil MR, Olson MD, Anderson DL. Nonlinear dynamic analysis of stiffened
plates. Comput Struct 1988;29(6):929–41.
[2] Shen PC, Dade C. Dynamic analysis of stiffened plates and shells using spline
gauss collocation method. Comput Struct 1990;36(4):623–9.
[3] Nayak AN, Bandyopadhyay JN. On the free vibration of stiffened shallow shells.
J Sound Vib 2002;255(2):357–82.
[4] Sheikh AH, Mukhopadhyay M. Linear and nonlinear transient vibration
analysis of stiffened plate structures. Finite Elem Anal Des

2002;38(6):477–502.
[5] Patel SN, Datta PK, Sheikh AH. Buckling and dynamic instability analysis of
stiffened shell panels. Thin-Wall Struct 2006;44(03):321–33.
[6] Patel SN, Datta PK, Sheikh AH. Dynamic instability analysis of stiffened shell
panels subjected to partial edge loading along the edges. Int J Mech Sci
2007;49(12):1309–24.
[7] Patel SN, Datta PK, Sheikh AH. Dynamic stability analysis of stiffened shell
panels with cutouts. J Appl Mech 2009;76(4):041004–0410013.
[8] Patel SN, Datta PK, Sheikh AH. Effect of harmonic in-plane edge loading on
dynamic stability of stiffened shell panels with cutouts. Int J Appl Mech
2010;02(04):759–85.
[9] Satish Kumar YV, Mukhopadhyay M. Transient response analysis of laminated
stiffened plates. Compos Struct 2002;58:97–107.
[10] Patel SN, Datta PK, Sheikh AH. Parametric study on the dynamic instability
behavior of laminated composite stiffened plate. J Eng Mech ASCE
2009;135(11):1331–41.
[11] Patel SN, Datta PK, Sheikh AH. Dynamic instability analysis of laminated
composite stiffened shell panels subjected to in-plane harmonic edge loading.
Struct Eng Mech 2006;22(4):483–510.
[12] Less H, Abramovich H. Dynamic buckling of a laminated composite stringerstiffened cylindrical panel. Compos Part B: Eng 2012;43(5):2348–58.
[13] Bich DH, Dung DV, Long VD. Dynamic buckling of imperfect reinforced
laminated composite plates and shallow shells. In: The international
conference on computational solid mechanics, Hochiminh City, Vietnam,
2008, pp. 15–25.
[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] Matsunaga H. Free vibration and stability of functionally graded shallow shells
according to a 2D higher order deformation theory. Compos Struct
2008;84:132–46.

[16] 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.
[17] 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.
[18] Bich DH, Long VD. Non-linear dynamical analysis of imperfect functionally
graded material shallow shells. Vietnam J Mech 2010;32(1):1–14.
[19] 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, pp. 130–41.
[20] Najafizadeh MM, Hasani A, Khazaeinejad P. Mechanical stability of
functionally graded stiffened cylindrical shells. Appl Math Model
2009;54(2):1151–7.


D.H. Bich et al. / Composite Structures 96 (2013) 384–395
[21] Bich DH, Nam VH, Phuong NT. Nonlinear post-buckling of eccentrically
stiffened functionally graded plates and shallow shells. Vietnam J Mech
2011;33(3):132–47.
[22] Bich DH, Dung DV, Nam VH. Nonlinear dynamical analysis of eccentrically
stiffened functionally graded cylindrical panels. Compos Struct
2012;94(8):2465–73.
[23] Volmir AS. Non-linear dynamics of plates and shells. Science Edition, M, 1972
(in Russian).

395

[24] Brush DD, Almroth BO. Buckling of bars, plates and shells. Mc. Graw-Hill;
1975.

[25] Budiansky B, Roth RS. Axisymmetric dynamic buckling of clamped shallow
spherical shells. NASA Technical Note D_510 1962;25:597–609.
[26] Huang H, Han Q. Nonlinear dynamic buckling of functionally graded
cylindrical shells subjected to a time-dependent axial load. Compos Struct
2010;92(2):593–8.