DSpace at VNU: Nonlinear dynamical analysis of eccentrically stiffened functionally graded cylindrical panels

Composite Structures 94 (2012) 2465–2473

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

Nonlinear dynamical analysis of eccentrically stiffened functionally graded
cylindrical panels
Dao Huy Bich a, Dao Van Dung a, Vu Hoai Nam b,⇑
a
b

Vietnam National University, Hanoi, Viet Nam
Faculty of Civil Engineering, University of Transport Technology, Ha Noi, Viet Nam

a r t i c l e

i n f o

Article history:
Available online 28 March 2012
Keywords:
Functionally graded material (FGM)
Dynamical analysis
Critical dynamic buckling load
Vibration
Cylindrical panel
Stiffeners

a b s t r a c t

Based on the classical shell theory with the geometrical nonlinearity in von Karman–Donnell sense and
the smeared stiffeners technique, the governing equations of motion of eccentrically stiffened functionally graded cylindrical panels with geometrically imperfections are derived in this paper. The characteristics of free vibration and nonlinear responses are investigated. The nonlinear dynamic buckling of
cylindrical panel acted on by axial loading is considered. The nonlinear dynamic critical buckling loads
are found according to the criterion suggested by Budiansky–Roth. Some numerical results are given
and compared with the ones of other authors.
Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction
Functionally graded materials (FGMs) are composite materials
which have mechanical properties varying continuously from one
surface to the other of structure. The concept of functionally
graded material was proposed in 1984 [1] and it is often used in
heat-resistance structure as elements in aerospace and nuclear
reactors [2]. Today, the application of this material is getting varied, so the problems of static and dynamic behaviors of structures
such as FGM plates and shells have been properly noticed.
For dynamical analysis of FGM shells, many studies have been
focused on the characters of vibration and behavior of buckling
of shells. Ng et al. [3] and Darabi et al. [4] presented respectively
linear and nonlinear parametric resonance analyses for FGM cylindrical shells. Loy et al. [5] and Pradhan et al. [6] studied the free
vibration characteristics of FGM cylindrical shells. By using
Galerkin technique together with Ritz type variational method,
Sofiyev [7] and Sofiyev and Schnack [8] obtained critical parameters for cylindrical thin shells under linearly increasing dynamic
torsional loading, and under a periodic axial impulsive loading.
By using a higher order shell theory and a finite element solving
method, Shariyat [9] investigated nonlinear dynamic buckling
problems of axially and laterally preloaded FGM cylindrical shells
under transient thermal shocks. Geometrical imperfection effects
were also included in his research. Using the similar method, he
also presented a dynamic buckling analysis for FGM cylindrical
⇑ Corresponding author.

E-mail address: (V.H. Nam).
0263-8223/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved.
/>
shells under complex combinations of thermo–electro-mechanical
loads [10]. Huang and Han [11] presented nonlinear dynamic buckling problems of functionally graded cylindrical shells subjected to
time-dependent axial load by using Budiansky–Roth dynamic
buckling criterion [12]. Various effects of the inhomogeneous
parameter, loading speed, dimension parameters; environmental
temperature rise and initial geometrical imperfection on nonlinear
dynamic buckling were discussed. Liew et al. [13] presented the
nonlinear vibration analysis for layered cylindrical panels containing FGMs and subjected to a temperature gradient arising from
steady heat conduction through the panel thickness.
Ganapathi [14] studied the dynamic stability behavior of a
clamped FGMs spherical shell structural element subjected to
external pressure load. He solved the governing equations employing the Newmark’s integration technique coupled with a modified
Newton–Raphson iteration scheme. Sofiyev [15–17] studied the
vibration and buckling of the FGM truncated conical shells under
dynamic axial loading. Based on first-order shear deformation theory, the dynamic thermal buckling behavior of functionally graded
spherical caps is studied by Prakash et al. [18]. Dynamic buckling of
functionally graded materials truncated conical shells subjected to
normal impact loads is discussed by Zang and Li [19].
For FGM shallow shells, Alijani et al. [20], Chorfi and Houmat
[21] and Matsunaga [22] investigated nonlinear forced vibrations
of FGM doubly curved shallow shells with a rectangular base. Nonlinear dynamical analysis of imperfect functionally graded material
shallow shells subjected to axial compressive load and transverse
load was studied by Bich and Long [23] and Dung and Nam [24].
The motion, stability and compatibility equations of these


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D.H. Bich et al. / Composite Structures 94 (2012) 2465–2473

structures were derived using the classical shell theory. The nonlinear transient responses of cylindrical and doubly-cuvred 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 [12].
However, there are very little researches on nonlinear dynamic
problems of imperfect eccentrically stiffened functionally graded
shells. Recently, Najafizadeh et al. [25] studied statical buckling
behaviors of FGM cylindrical shell. Bich et al. [26] have studied
the nonlinear statical postbuckling of eccentrically stiffened functionally graded plates and shallow shells.
Following the idea of Ref. [26], this paper establishes dynamics
governing equations and investigates nonlinear vibration and dynamic buckling of imperfect reinforced FGM cylindrical panel. It
shows the influences of stiffener, of volume-fraction index, of initial imperfection and of geometrical parameters to the dynamic
characteristics of panels.

According to the classical shell theory and geometrical nonlinearity in von Karman–Donnell sense, the strains at the middle surface and curvatures are related to the displacement components u,
v, w in the x1, x2, z coordinate directions as [28].

 2
@u 1 @w
@2w
þ
; v1 ¼ 2 ;
@x1 2 @x1
@x1
 2
@v 1
1 @w

@2w
¼
À wþ
; v2 ¼ 2 ;
2 @x2
@x2 R
@x2

e01 ¼
e02

c012 ¼

@u
@ v @w @w
þ
þ
;
@x2 @x1 @x1 @x2

where R is radius of the cylindrical shell.
The strains across the shell thickness at a distance z from the
mid-surface are

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

@ 2 e01 @ 2 e02
@ 2 c012
þ 2 À
¼

2
@x2
@x1 @x1 @x2

2.1. Functionally graded material
Functionally graded material in this paper, is assumed to be
made from a mixture of ceramic and metal with the volume-fractions given by a power law

V m þ V c ¼ 1;

where h is the thickness of panel; k P 0 is the volume-fraction
index; 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. According to the mentioned law, the Young modulus
and the mass density can be expressed in the form

EðzÞ ¼ Em V m þ Ec V c ¼ Em þ ðEc À Em Þ

ð1Þ

the Poissons’s ratio m is assumed to be constant.
2.2. Constitutive relations and governing equations.
Consider a functionally graded cylindrical thin panel in-plane
edges a and b. The panel is reinforced by eccentrically longitudinal
and transversal stiffeners. The cylindrical panel 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 panel. Note that the middle surface of the panel generally is defined in terms of curvilinear coordinates, but for the cylindrical panel, so the Cartesian coordinates can replace the curvilinear
coordinates on the middle surface (see Fig. 1).

s1
(i) Longitudinal Stiffeners

z1

À

@2w @2w 1 @2w
À
:
@x21 @x22 R @x21

ð4Þ

ð5aÞ

and for stiffeners
st
1
st
2

r ¼ E0 e1 ;
r ¼ E0 e2

ð5bÞ

where E0 is Young’s modulus of ring and stringer stiffeners
Taking into account the contribution of stiffeners by the
smeared stiffeners technique and omitting the twist of stiffeners

and integrating the stress–strain equations and their moments
through the thickness of the panel, we obtain the expressions for
force and moment resultants of an ES-FGM cylindrical panel



E0 A1 0
N1 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 ;
N2 ¼ A12 e01 þ A22 þ
s2

ð6Þ

N12 ¼ A66 c012 À 2B66 v12 ;


E0 I1
v1 À D12 v2 ;
M 1 ¼ ðB11 þ C 1 Þe01 þ B12 e02 À D11 þ
s1


E0 I 2
v2 ;

M 2 ¼ B12 e01 þ ðB22 þ C 2 Þe02 À D12 v1 À D22 þ
s2
M 12 ¼ B66 c012 À 2D66 v12 ;

a
x2

!2

EðzÞ
ðe1 þ me2 Þ;
1 À m2
EðzÞ
¼
ðe2 þ me1 Þ;
1 À m2
EðzÞ
¼
c ;
2ð1 þ mÞ 12

b
0
d1

@2w
@x1 @x2

rsh
1 ¼

ssh
12

z

ð3Þ

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

rsh
2


k
2z þ h
;
V c ¼ V c ðzÞ ¼
2h

h1

@2w
;
@x1 @x2

v12 ¼

From Eq. (3) the strains must be relative in the deformation
compatibility equation

2. Eccentrically stiffened FGM cylindrical panels (ES-FGM
cylindrical panels)


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

ð2Þ

x1

0
h2
z

d2

s2
(ii) Transversal Stiffeners

Fig. 1. Configuration of an eccentrically stiffened cylindrical panel.

z2

ð7Þ


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D.H. Bich et al. / Composite Structures 94 (2012) 2465–2473

where Aij, Bij, Dij (i, j = 1, 2, 6) are extensional, coupling and bending
stiffnesses of the panel without stiffeners.

Z

h=2

Z

h=2

EðzÞ
E1
EðzÞm
E1 m
A11 ¼ A22 ¼
dz ¼
; A12 ¼
dz ¼
;

2
2
1 À m2
1 À m2
Àh=2 1 À m
Àh=2 1 À m
Z h=2
EðzÞ
E1
A66 ¼
dz ¼
;
2ð1 þ mÞ
Àh=2 2ð1 þ mÞ
Z h=2
Z h=2
zEðzÞ
E2
zEðzÞm
E2 m
B11 ¼ B22 ¼
dz
¼
;
B
¼
dz ¼
;
12
2

2
2
1
À
m
1
À
m
1
À
m
1
À m2
Àh=2
Àh=2
Z h=2
zEðzÞ
E2
dz ¼
;
B66 ¼
2ð1
þ
m
Þ
2ð1
þ mÞ
Àh=2
Z h=2 2
Z h=2 2

z EðzÞ
E3
z EðzÞm
E3 m
D11 ¼ D22 ¼
dz
¼
;
D
¼
dz ¼
;
12
2
2
2
1
À
m
1
À
m
1
À
m
1
À m2
Àh=2
Àh=2
Z h=2 2

z EðzÞ
E3
D66 ¼
dz ¼
;
2ð1
þ
m
Þ
2ð1
þ mÞ
Àh=2
ð8Þ
with



2
Ec À Em
ðEc À Em Þkh
E1 ¼ Em þ
h; E2 ¼
;
kþ1 
2ðk þ 1Þðk þ 2Þ
!
Em
1
1
1

3
h ;
E3 ¼
À
þ
þ ðEc À Em Þ
k þ 3 k þ 2 4k þ 4
12
3
3
d1 h1
d2 h2
I1 ¼
þ A1 z21 ; I2 ¼
þ A2 z22 :
12
12

Substituting Eq. (10) into Eq. (7) yields

M 1 ¼ BÃ11 N1 þ BÃ21 N2 À DÃ11 v1 À DÃ12 v2 ;
M 2 ¼ BÃ12 N1 þ BÃ22 N2 À DÃ21 v1 À DÃ22 v2 ;
M 12 ¼ BÃ66 N12 À 2DÃ66 v12 ;
where

E0 I1
À ðB11 þ C 1 ÞBÃ11 À B12 BÃ21 ;
s1
E0 I2
¼ D22 þ

À B12 BÃ12 À ðB22 þ C 2 ÞBÃ22 ;
s2
¼ D12 À ðB11 þ C 1 ÞBÃ12 À B12 BÃ22 ;

DÃ11 ¼ D11 þ
DÃ22
DÃ12

DÃ66 ¼ D66 À B66 BÃ66 :
The nonlinear equations of motion of a cylindrical thin panel
based on the classical shell theory and the assumption (Refs.
2
2
[4,8,27]) u ( w and v ( w, q1 @@t2u ! 0, q1 @@t2v ! 0 are given by

@N1 @N12
þ
¼ 0;
@x1
@x2
ð9Þ

@N12 @N2
þ
¼ 0;
@x1
@x2
@ 2 M1
@ 2 M12 @ 2 M2
@2w

@2w
þ2
þ
þ N 1 2 þ 2N12
@x1 @x2
@x1 @x2
@x21
@x22
@x1
þ N2

E 0 A1 z 1
E0 A2 z2
C1 ¼
; C2 ¼
;
s1
s2
h1 þ h
h2 þ h
; z2 ¼
:
z1 ¼
2
2

ð10Þ

q1 ¼

þ q0

BÃ21 ¼ AÃ11 B12 À AÃ12 ðB11 þ C 1 Þ;
B66
BÃ66 ¼
:
A66

ð14Þ

h=2

qðzÞdz þ q0


A1 A2
;
þ
s1 s2


 

A1 A2
q À qm
¼ qm þ c
þ
h
s1 s2
kþ1

with

q0 = qm for metal stiffener,
q0 = qc for ceramic stiffener.
The first two of Eq. (14) are satisfied automatically by choosing
a stress function u as

N1 ¼

@2u
;
@x22

N2 ¼

@2u
;
@x21

N12 ¼ À

@2u
:
@x1 @x2

Á @4/
@4u À Ã
@4u
@4w

þ A66 À 2AÃ12
þ AÃ22 4 þ BÃ21 4
4
2
2
@x1 @x2
@x1
@x1
@x2

À
Á @4w
@4w 1 @2w
þ BÃ11 þ BÃ22 À 2BÃ66
þ BÃ12 4 þ
2
2
@x1 @x2
@x2 R @x21
!2
@2w
@2w @2w
¼
À 2
;
@x1 @x2
@x1 @x22
ð11Þ

ð15Þ

The substitution of Eq. (10) into the compatibility Eqs. (4) and
(12) into the third of Eq. (14), taking into account expressions (2)
and (15), yields a system of equations

AÃ11 ¼

BÃ22 ¼ AÃ11 ðB22 þ C 2 Þ À AÃ12 B12 ;
BÃ12 ¼ AÃ22 B12 À AÃ12 ðB22 þ C 2 Þ;

Z

Àh=2

AÃ11

where

BÃ11 ¼ AÃ22 ðB11 þ C 1 Þ À AÃ12 B12 ;

@2w 1
@2w
þ N2 þ q0 ¼ q1 2 ;
2
@x2 R
@t

where

In above relations (6), (7) and (9) the quantityE0 is the Young

modulus in the axial direction of the corresponding stiffener which
is assumed identical for both types of stiffeners, it takes the value
E0 = Em if the full metal stiffeners are put at the metal-rich side of
the panel and conversely E0 = Ec if the full ceramic ones at the ceramic-rich side. Such FGM stiffened cylindrical panels provide continuity within panel and stiffeners and can be easier manufactured.
The spacings of the longitudinal and transversal stiffeners are denoted by s1 and s2 respectively. The quantities A1, A2 are the
cross-section areas of stiffeners and I1, I2, z1, z2 are the second moments of cross section areas and the eccentricities of stiffeners
with respect to the middle surface of panel respectively.
The strain-force resultant relations reversely are obtained from
Eq. (6)





1
E 0 A1
1
E0 A2
; AÃ22 ¼
;
A11 þ
A22 þ
D
s1
D
s2
A12
1
AÃ12 ¼
; AÃ66 ¼

;
A66
D



E0 A1
E 0 A2
A22 þ
À A212 ;
D ¼ A11 þ
s1
s2

ð13Þ

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

and

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 ;

ð12Þ

ð16Þ

Á @4w
@2w

@4w À
@4w
þ DÃ11 4 þ DÃ12 þ DÃ21 þ 4DÃ66
þ DÃ22 4
2
2
2
@x1
@x1 @x2
@x2
@t
4
4
4
À
Á
@
u
@
u
@
u
1
@2u
Ã
À BÃ21 4 À BÃ11 þ BÃ22 À 2BÃ66
À
B
À
12

@x1
@x21 @x22
@x42 R @x21

q1

À

@2u @2w
@2u
@2w
@2u @2w
þ2
À 2
¼ q0 ;
2
2
@x1 @x2 @x1 @x2 @x1 @x22
@x2 @x1

ð17Þ


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D.H. Bich et al. / Composite Structures 94 (2012) 2465–2473

For initial imperfection ES-FGM panels: The initial imperfection of the
panel considered here can be seen as a small deviation of the panel
middle surface from the perfect shape, also seen as an initial deflection which is very small compared with the panel dimensions, but

may be compared with the panel wall thickness. Let
w0 = w0(x1, x2) denote a known small imperfection, proceeding from
the motion Eqs. (16) and (17) of a perfect FGM cylindrical panel and
following to the Volmir’s approach [27] for an imperfection panel
we can formulate the system of motion equations for an imperfect
eccentrically stiffened functionally graded cylindrical panel (imperfect ES-FGM cylindrical panel) as

AÃ11

Á @4u
@4u À Ã
@4u
@ 4 ðw À w0 Þ
þ A66 À 2AÃ12
þ AÃ22 4 þ BÃ21
4
2
2
@x41
@x1
@x1 @x2
@x2

1 @ 2 ðw À w0 Þ
¼ 0;
R
@x21

À r0 h

2mpx1
2npx2
mpx1
npx2
þ u2 cos
À u3 sin
sin
a
b
a
b

x22
; ð23Þ
2

where denote

u1 ¼

n2 k2 f 2
;
32m2 AÃ11
m2 f 2

u2 ¼

;
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 1R m2
u3 ¼
:
À
Á
AÃ11 m4 þ AÃ66 À 2AÃ12 m2 n2 k2 þ AÃ22 n4 k4
ð24Þ

Á @ 4 ðw À w0 Þ
@ 4 ðw À w0 Þ À Ã
þ B11 þ BÃ22 À 2BÃ66
@x42
@x21 @x22
2
3
2
3
!2
!2
2
2
2
2
2
2
@

w
@
w
@
w
@
w
@
w
@
w
0
0
0
5þ4
5
À4
À 2
À
@x1 @x2
@x1 @x2
@x21 @x22
@x1 @x22
þ BÃ12

þ

u ¼ u1 cos

Substituting the expressions (21)–(23) into Eq. (19) and applying Galerkin method to the resulting equation yield

!
À
Á
B2
8mnk2 B
d1 d2 ðf À f0 Þf þ H f 2 À f02
ðf À f0 Þ þ
M€f þ D þ
A
3p2 A
ð18Þ

À
Á
a2 h
4q a4
þ K f 2 À f02 f À 2 r0 m2 f À 0 6 d1 d2 ¼ 0;
p
mnp

ð25Þ

where denote

Á @ 4 ðw À w0 Þ
@2w
@ 4 ðw À w0 Þ À Ã
q1 2 þ DÃ11
þ D12 þ DÃ21 þ 4DÃ66

4
@x1
@x21 @x22
@t
þ DÃ22

2

2

2

2

2

2

a4

p4

q1 ;

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

Á @4u
@ 4 ðw À w0 Þ

@4u À
À BÃ21 4 À BÃ11 þ BÃ22 À 2BÃ66
@x1
@x21 @x22
@x42
4

À BÃ12

M¼

2

@ u 1@ u @ u@ w
@ u @ w
@ u@ w
À
À
þ2
À
¼ q0 ; ð19Þ
@x1 @x2 @x1 @x2 @x21 @x22
@x42 R @x21 @x22 @x21

where w is a total deflection of panel.
Hereafter, the couple of Eqs. (16) and (17) or of Eqs. (18) and
(19) are used to investigate the nonlinear vibration and dynamic
stability of panels. They are nonlinear equations in terms of two
dependent unknowns w and u.

À
Á
a2 1
B ¼ BÃ21 m4 þ BÃ11 þ BÃ22 À 2BÃ66 m2 n2 k2 þ BÃ12 n4 k4 À 2 m2 ;
p R
À
Á
D ¼ DÃ11 m4 þ DÃ12 þ DÃ21 þ 4DÃ66 m2 n2 k2 þ DÃ22 n4 k4 ;
"
#


2mnk2 BÃ21 BÃ12
a2 n2 k2 1
d1 d2 ;
À
þ
H¼
3p2
AÃ11 AÃ22
6p4 mn AÃ11 R
!
1 m4 n4 k4
a
; k¼ ;
þ Ã
K¼
16 AÃ22
b
A11

ð26Þ

d1 ¼ ðÀ1Þm À 1;
3. Nonlinear dynamic analysis

d2 ¼ ðÀ1Þn À 1:

3.1. Solution of the problem

The obtained Eq. (25) is the governing equation for dynamic
analysis of ES-FGM cylindrical panels in general. Based on this
equation the non-linear vibration of perfect and imperfect FGM
cylindrical panels can be investigated and the dynamic buckling
analysis of panels under various loading cases can be performed.
Particularly for a plate, R = 1 is taken in Eqs. (25) and (26).

Suppose that an imperfect ES-FGM cylindrical panel is simply
supported and subjected to uniformly distributed pressure of
intensity q0 and in plane compressive load of intensities r0 at its
cross-section (in Pa). Thus the boundary conditions considered in
the current study are

w ¼ 0;

M 1 ¼ 0;

N1 ¼ Àr 0 h;

w ¼ 0;

M 2 ¼ 0;

N2 ¼ 0;

N 12 ¼ 0;

N 12 ¼ 0;

at

at

x1 ¼ 0;

x2 ¼ 0;

b:

a;

3.2. Vibration analysis

ð20Þ

where a and b are the lengths of in-plane edges of the panel.
The mentioned conditions (20) can be satisfied identically if the
buckling mode shape is represented by

w ¼ f ðtÞ sin

mpx1
npx2
sin
;
a
b

ð21Þ

where f(t) is time dependent total amplitude and m, n are numbers
of haft wave in axial and circumferential directions, respectively.
The initial-imperfection w0 is assumed to have similar form of
the panel deflection w, i.e.

w0 ¼ f0 sin

mpx1
npx2
sin
;
a
b

ð22Þ

where f0 is the known initial amplitude.
Substituting Eqs. (21) and (22) into Eq. (18) and solving obtained equation for unknown u lead to

Consider an imperfect ES-FGM cylindrical panel acted on by an

uniformly distributed excited transverse load q0 = Q sin Xt and
r0 = 0, the non-linear Eq. (25) has of the form

!
À
Á
B2
8mnk2 B
€
ðf À f0 Þ þ
d1 d2 ðf À f0 Þf þ H f 2 À f02
Mf þ D þ
A
3p2 A
À
Á
4a4
þ K f 2 À f02 f ¼ 6
d d Q sin Xt:
p mn 1 2

ð27Þ

By using Eq. (27), three aspects are taken into consideration:
fundamental frequencies of natural vibration of ES-FGM panel
and FGM panel without stiffeners, frequency–amplitude relation
of non-linear free vibration and non-linear response of ES-FGM panel. The non-linear dynamical responses of ES-FGM panels can be
obtained by solving this equation combined with initial conditions
to be assumed as f ð0Þ ¼ 0; f_ ð0Þ ¼ 0 by using the Runge–Kutta iteration schema.

2469

D.H. Bich et al. / Composite Structures 94 (2012) 2465–2473

If the vibration is free and linear, Eq. (27) leads to
2

m2 a2 h

!

p2

B
M€f þ D þ
ðf À f0 Þ ¼ 0;
A

ð28Þ

from which the fundamental frequencies of natural vibration of
imperfect ES-FGM cylindrical panels can be determined by

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
!
u
u1
B2
t

:
Dþ
xL ¼
M
A

ð29Þ

rupper ¼

where

!
1
B2
;
Dþ
M
A

xNL ¼ xL 1 þ

8H2
3H
g þ 32 g2
3px2L
4xL

ð35Þ

p2

Dþ

m2 a2 h

!
B2
;
A

ð36Þ

p2

ð37Þ

Suppose axial load varying linearly on time r0 = ct(c (in Pa/s) is a
loading speed) and introduce parameters:

H3 ¼

K
:
M

ð31Þ

Seeking solution as f(t) = gcosxt and applying procedure like
Galerkin method to Eq. (30), the frequency–amplitude relation of

non-linear free vibration is obtained



8mnk2 B
2
d1 d2 þ H f þ Kf :
3p2 A

2
b ¼ 8mnk B d1 d2 þ H:
H
2
3p A

where denoting

!
1 8mnk2 B
d
þ
H
;
d
1 2
M
3p 2 A

þ

!
^2
B2 H
¼ 2 2
Dþ À
;
m a h
A 4K

€f þ H f þ H f 2 þ H f 3 ¼ 0;
1
2
3

H2 ¼

B2
A

and the lower static buckling load is found using the condition
dr0
¼ 0, it follows
df

rlower

H1 ¼ x2L ¼

Dþ

!

From Eq. (35), the upper static buckling load can be determined
by putting f = 0

The equation of non-linear free vibration of a perfect panel can
be obtained from (27)

ð30Þ

r0 ¼

!

12
;

ð32Þ

where xNL is the non-linear vibration frequency and g is the amplitude of non-linear vibration.

D¼

D
3

h
f
n¼ ;
h

;

B
;
h
f0
n0 ¼ ;
h
B¼

A ¼ Ah;

s¼

H¼

H
h

2

;

K¼

K
;
h

r0
ct
¼
;
r scr r scr

ð38Þ

where rscr = minrupper vs. (m, n).
The non-dimension form of Eq. (33) is written as
2

1 d n
þ
S1 ds2

"
Dþ

À
Á i
þk n2 À n20 n

3.3. Nonlinear dynamic buckling analysis

B2

!

A

p2

ðn À n0 Þ þ

À Á2
k4 hb rscr

À

À
Á
8mnk2 B
d1 d1 ðn À n0 ÞnþþH n2 À n20
3p2 A

m2
k

ns ¼ 0:

ð39Þ

where
Investigate the non-linear dynamic buckling of imperfect ESFGM cylindrical panels in some cases of active loads varying as linear function of time. The aim of considered problems is to seek the
critical dynamic buckling loads. They can be evaluated based in the
displacement responses obtained from the motion Eqs. (25) and
(26).
The criterion suggested by Budiansky and Roth [12] is employed
here as it is widely accepted. 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 obtains 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.
Consider an ES-FGM cylindrical panel subjected to axial load
r0(t). In this case q0 = 0, Eq. (25) gives

M€f þ D þ

!
À
Á
B2
8mnk2 B
ðf À f0 Þ þ
d1 d2 ðf À f0 Þf þ H f 2 À f02
A
3p 2 A

À
Á
m2 a2 h
r0 ðtÞ f ¼ 0
þ K f 2 À f02 f À
2

p

ð33Þ

Omitting the term of inertia and putting f0 = 0 in Eq. (33), yields
an equation for determining the static critical load of ES-FGM
cylindrical panels as

m2 a2 h

p2

r0 f ¼

!
!
B2
8mnk2 B
3
fþ
d1 d2 þ H f 2 þ Kf :
Dþ
A
3p 2 A

ð34Þ

Taking f – 0, i.e. considering the panel after the lost of stability
we obtain

S1 ¼

p2 r3scr h

:
2
b c2 q1

ð40Þ

Solving Eq. (39) by Runge–Kutta method and applying Budiansky–Roth criterion, the critical value sdcr, the dynamic critical time
tdcr ¼ sdcrcrscr and dynamical buckling load rdcr = ctdcr respectively are
obtained.
4. Numerical results and discussions
4.1. Validation of the present formulation
In this section, first of all, the
qffiffiffifficomparison on the fundamental
~ ¼ xL h qE c (xL is calculated from Eq. (29))
frequency parameter x
c
given by the present analysis with the results of Alijani et al.[20]
based on the Donnell’s nonlinear shallow-shell theory, Chorfi and
Houmat [21] based on the first-order shear deformation theory
and Matsunaga [22] based on the two-dimensional (2D) higher-order theory for the perfect unreinforced FGM cylindrical panel
Àa
Á
¼ 1; ha ¼ 0:1 with simply supported movable edges is suggested.
b
The material properties in Refs. [20–22] are aluminium and alumina, i.e. Em = 70.109 N/m2, qm = 2702 kg/m3 and Ec = 380.109
N/m2, qc = 3800 kg/m3 respectively. The Poisson’s ratio is chosen
to be 0.3. As can be observed in Table 1, a very good agreement
is obtained in this comparison study.
Next, the present frequency xL (in Eq. (29)) is compared with
the result of Szilard [29] and Troitsky [30] based on the classical

assumptions of small deformations and thin plates. Consider a simply supported homogeneous plate that is biaxial stiffened with
multiple stiffeners (see Fig. 2). As shown in Table 2, a good agreement can be witnessed.


2470

D.H. Bich et al. / Composite Structures 94 (2012) 2465–2473

Table 1
~ with results reported by Alijani et al. [20], Chorfi and Houmat [21]
Comparison of x
and Matsunaga [22].
a/R

k

Present

Ref. [20]

Ref. [21]

Ref. [22]

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
0.0430
0.0381
0.0364

FGM cylindrical panel
0.5
0
0.0648
0.5
0.0553
1
0.0501
4
0.0430
10
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
0.5
1
4
10

Table 3
The fundamental frequencies of natural vibration (rad/s) of FGM cylindrical panels.
R (m)

k

Unreinforced (m, n)

Reinforced (m, n)

0.2
1
5
10

1172.51
982.14
822.19
783.56

(1,
(1,
(1,
(1,

3)
3)
3)
3)

1571.27
1435.02
1266.54

1224.47

(1,
(1,
(1,
(1,

2)
2)
2)
2)

0.2
1
5
10

803.92
686.91
556.39
519.90

(1,
(1,
(1,
(1,

2)
2)
2)

2)

1192.51
1128.40
1011.97
924.63

(1,
(1,
(1,
(1,

2)
2)
1)
1)

0.2
1
5
10

622.96
524.39
435.45
413.06

(1,
(1,
(1,

(1,

2)
2)
2)
2)

930.82
812.67
647.97
599.93

(1,
(1,
(1,
(1,

1)
1)
1)
1)

0.2
1
5
10

515.55
438.11
353.51

325.48

(1,
(1,
(1,
(1,

1)
2)
1)
1)

551.26
494.97
427.52
411.30

(1,
(1,
(1,
(1,

1)
1)
1)
1)

0.2
1
5

10

197.11
162.11
139.79
135.38

(1,
(1,
(1,
(1,

1)
1)
1)
1)

376.11
364.17
361.77
364.92

(1,
(1,
(1,
(1,

1)
1)
1)

1)

1.5

3

5

10

0.6m

0.41 m

0.0127 m

0.02222 m

1 (plates)

E=211GPa
=0.3
=7830 kg/m3

0.00633 m
0.02222 m

0.0127 m
Fig. 2. Configuration of an eccentrically stiffened plate.

Table 2
Comparison of present frequency (Hz) with results reported by Szilard [29] and
Troitsky [30].
Mode

Present

Refs. [29,30]

1
2
3
4
5
10
15
20

357.81
708.34
1248.97
1417.82
1440.38
2888.10
4927.78
5679.99

381.9
719.9
1339.8

1356.1
1512.4
3066.7
5029.2
5589.8

4.2. Vibration results
To illustrate the proposed approach to eccentrically stiffened
FGM cylindrical panels, the panels considered here are cylindrical
panels and plates with in-plane edges a = b = 1.5 m; h = 0.008 m;
f0 = 0. The panels 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 reinforced stiffeners has elastic modulus E = 380.109 N/m2;
q = 3800 kg/m3. The height of stiffeners is equal to 30 mm, its
width 3 mm, the spacing of stiffeners s1 = s2 = 0.15 m, the eccentricities of stiffeners with respect to the middle surface of panel
z1 = z2 = 0.019 m.
4.2.1. Results of fundamental frequencies of natural vibration
The obtained results in Table 3 show that the effect of stiffeners
on fundamental frequencies of natural vibration xL (xL is

calculated from Eq. (29)) is considerable. Obviously the natural frequencies of unreinforced and reinforced FGM cylindrical 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 because the lower value is the elasticity modulus of the metal constituent.
4.2.2. Results of frequency–amplitude of non-linear free vibration
Fig. 3 shows the relation frequency–amplitude of non-linear
free vibration of reinforced and unreinforced panel (calculated
from Eq. (32)) with m = 1, n = 1. As expected the non-linear vibration frequencies of reinforced panels are greater than ones of unreinforced panels.
4.2.3. Non-linear response results

For obtaining the non-linear dynamical responses of FGM cylindrical panel acted on by the harmonic uniformly load
q0(t) = Qsin(Xt) with Q = 5 Â 103 N/m2, X = 975 rad/s and
X = 950 rad/s, the Eq. (27) is solved using Runge–Kutta method.
Fig. 4 shows non-linear responses of ES-FGM cylindrical panel.
In this case, exited frequencies are near to fundamental frequencies of natural vibration x = 1011.97 rad/s (see Table 3). From obtained results, the interesting phenomenon is observed like the
harmonic beat phenomenon of a linear vibration, in which the
amplitude of beats of reinforced panels increased rapidly when
the exited frequency approached the natural frequency.
When the exited frequencies X = 500 rad/s and X = 600 rad/s
are away from the natural frequencies of ES-FGM cylindrical panel.
The obtained non-linear dynamical responses are shown in Fig. 5.
It shows that, the harmonic beat phenomenon does not appear
as in the previous case. The amplitude of beats of reinforced panels
increased slowly when the exited frequency is close to the natural
frequency.
Fig. 6 shows the Influence of initial imperfection with amplitudes f0 = 0, f0 = 10À5 and f0 = 5 Â 10 À5 m on the non-linear responses of ES-FGM cylindrical panel. The initial imperfection f0
has a slight influence to the nonlinear response of panel.


2471

D.H. Bich et al. / Composite Structures 94 (2012) 2465–2473

1.5E+3

ω

8.0E-4

(rad/s)

NL

R=3m, k=5, q 0=5000sin(500t)

f(m)

R=10m, k=0.2

6.0E-4

1.2E+3

4.0E-4

Reinforced
9.0E+2

Unreinforced

2.0E-4
0.0E+0

6.0E+2

-2.0E-4
R=10m, k=5

3.0E+2

-4.0E-4

0.0E+0
0.0E+0

η (m)
3.0E-2

6.0E-2

9.0E-2

1.2E-1

-6.0E-4
0

1.5E-1

0.6

f(m)

Ω=975(rad/s)

f0 =1e-5

f 0 =5e-5

0.015

0.03

0.045

t(s)

0.06

Fig. 6. Influence of initial imperfection on non-linear responses.

Fig. 3. Frequency–amplitude relation.

1.2E-2

Perfect

Ω=950(rad/s)

ξ

Unreinforced Panel

0.5

8.0E-3

0.4

4.0E-3

R=3m, k=1
m=5, n=2

0.3

0.0E+0

m=6, n=1

0.2

m=4, n=3

-4.0E-3

m=5, n=1

0.1

-8.0E-3
R=3m, k=5, q 0=5000sin(Ωt)

-1.2E-2

0

0.2

t (s)

f(m)

Ω=500 (rad/s)

τ

9.50E-01

1.00E+00

1.05E+00

1.10E+00

0.4
Fig. 7. Effect of buckling mode shapes on load–deflection curve of unreinforced
panel.

Fig. 4. Nonlinear response of ES-FGM cylindrical panel.

9.0E-4

0
9.00E-01

1.2

Ω=600 (rad/s)

6.0E-4

1

3.0E-4

0.8

0.0E+0

0.6

-3.0E-4

0.4

-6.0E-4

0.2

ξ

Reinforced Panel

R=3m, k=1
m=2, n=2
m=3, n=2

R=3m, k=5, q 0=5000sin(Ωt)

-9.0E-4
0

0.025

0.05

0.075

t (s)

0.1

Fig. 5. Nonlinear response of FGM cylindrical panel.

4.3. Nonlinear dynamic buckling results
To evaluate the effectiveness of the reinforcement of stiffener in
the nonlinear dynamic buckling problem, we consider the case of
imperfect ES-FGM cylindrical panel subjected to an axial compressive load. The critical dynamic buckling loads is determined by
solving Eq. (39) and applying Budiansky–Roth criterion.
Materials and structures used in this section are the same in the
previous section.
Figs. 7 and 8 show the effect of buckling mode shapes on load –
deflection curve of reinforced and unreinforced FGM cylindrical
panel subjected to an axial compressive load with the power law index k = 1, R = 3 m and compressive load r0 = 1.5 Â 109 t. Clearly, the
smallest critical dynamic buckling load corresponds to the buckling
mode shape m = 5, n = 2 in the case of unreinforced panel and m = 2,
n = 2 in the case of reinforced panel. This figure also shows that
there is no definite point of instability as in static analysis. Rather,

0
9.00E-01

m=1, n=2
m=2, n=1
τ

1.10E+00

1.30E+00

1.50E+00

1.70E+00

Fig. 8. Effect of buckling mode shapes on load–deflection curve of reinforced panel.

there is a region of instability where the slope of n vs. s curve
increases rapidly. In this paper, the critical parameter scr is taken

2 
¼ 0.
as an intermediate value satisfying the condition dds2n
s¼scr

Table 4 shows the critical loads of two cases of reinforcement
and unreinforcement cylindrical panel. The results show that the
reinforcement by stiffeners has large effect in the dynamic stability
problems of cylindrical panels under axial compressive load. With
the same input parameters, effectiveness of reinforcement increases as the curvature radius or the power index increases. Table

4 also considers the effect of loading speed to the dynamic buckling
load; the results show that the dynamic buckling loads increases
when the loading speed increases.
Fig. 9 shows the influence of initial imperfection amplitude f0 on
the non-linear buckling of ES–FGM Cylindrical panel. Clearly, the
initial imperfection strongly influences on the critical dynamic
buckling loads of ES-FGM cylindrical panel subjected to an axial
compressive load.


2472

D.H. Bich et al. / Composite Structures 94 (2012) 2465–2473

Table 4
Nonlinear critical buckling loads of the cylindrical panels subjected to an axial compressive load (Â108 N/m2).
R (m)

k

Unreinforced
Static (m, n)

Reinforced
Dynamic (m, n)

Static (m, n)

c = 1.5 Â 109

c = 2 Â 109

Dynamic (m, n)
c = 1.5 Â 109

c = 2 Â 109

3
0.2
1
5
10

5.1667
3.3323
1.9971
1.7111

(5,
(5,
(5,
(5,

2)
2)
2)
1)

5.1945 (5,
3.3795 (5,

2.0700 (5,
1.7895 (5,

2)
2)
2)
1)

5.2125
3.4016
2.0960
1.8160

(5,
(5,
(5,
(5,

2)
2)
2)
1)

9.5082
7.1505
5.0807
4.6866

(2,
(2,

(2,
(2,

2)
2)
2)
2)

9.6285
7.2975
5.2560
4.8690

(2,
(2,
(2,
(2,

2)
2)
2)
2)

9.6778
7.3496
5.3082
4.9308

(2,
(2,

(2,
(2,

2)
2)
2)
2)

0.2
1
5
10

3.0985 (3,
2.0070 (3,
1.1942 (3,
1.0243 (4,

2)
2)
2)
1)

3.1800
2.1120
1.3109
1.1483

(4,
(4,

(4,
(4,

1)
1)
1)
1)

3.2097
2.1419
1.3497
1.1857

(4,
(4,
(4,
(4,

1)
1)
1)
1)

6.5923
5.2586
3.3071
2.7666

(2,
(2,

(1,
(1,

2)
2)
1)
1)

6.7395
5.4255
3.6690
3.1575

(2,
(2,
(1,
(1,

2)
2)
1)
1)

6.7942
5.4763
3.7804
3.2803

(2,
(2,

(1,
(1,

2)
2)
1)
1)

0.2
1
5
10

1.5636
1.0027
0.6067
0.5266

(3,
(3,
(3,
(3,

1)
1)
1)
1)

1.7100
1.1723

0.7968
0.7176

(3,
(3,
(3,
(3,

1)
1)
1)
1)

1.7615
1.2218
0.8488
0.7672

(3,
(3,
(3,
(3,

1)
1)
1)
1)

2.9007
2.1341

1.4396
1.3004

(1,
(1,
(1,
(1,

1)
1)
1)
1)

3.2760
2.5485
1.9020
1.7865

(1,
(1,
(1,
(1,

1)
1)
1)
1)

3.3957
2.6758

2.0288
1.9180

(1,
(1,
(1,
(1,

1)
1)
1)
1)

0.2
1
5
10

0.3204
0.1948
0.1285
0.1171

(1,
(1,
(1,
(1,

1)
1)

1)
1)

0.7958
0.6194
0.5138
0.4980

(2,
(2,
(2,
(2,

1)
1)
1)
1)

0.8613
0.6906
0.5905
0.5773

(2,
(2,
(2,
(2,

1)
1)

1)
1)

1.3503
1.1552
1.0309
1.0236

(1,
(1,
(1,
(1,

1)
1)
1)
1)

1.8405
1.6575
1.5315
1.5255

(1,
(1,
(1,
(1,

1)
1)

1)
1)

1.9772
1.7740
1.6686
1.6607

(1,
(1,
(1,
(1,

1)
1)
1)
1)

5

10

1 (plates)

References
1

ξ

Reinforced Panel

0.8
0.6
0.4
0.2

R=3m, k=1,
m=2, n=2.
ξ0 =1e-5/h
ξ =2e-5/h
0

ξ =3e-5/h
0

0
0.65

0.75

τ
0.85

0.95

1.05

Fig. 9. Influence of initial imperfection on critical dynamic buckling load of
reinforced panel.

5. Conclusions
A formulation of the governing equations of eccentrically reinforced functionally graded cylindrical panels based upon the classical shell theory and the smeared stiffeners technique with von
Karman–Donnell nonlinear terms has been presented.
By use of Galerkin method a nonlinear dynamic equation for
analysis of dynamic and stability characteristics of ES-FGM cylindrical panels is obtained.
Fundamental frequencies of unreinforced and reinforced FGM
panels are considered. Some results were compared with the ones
of other authors.
Nonlinear dynamic responses and critical dynamic loads of ESFGM cylindrical panels are investigated according to the criterion
Budiansky–Roth. They are significantly influenced by material
parameters, stiffeners and initial geometrical imperfection. Clearly,
stiffeners enhance the stability and load-carrying capacity of FGM
cylindrical panels.
Acknowledgements
This paper was supported by the National Foundation for
Science and Technology Development of Vietnam – NAFOSTED.
The authors are grateful for this financial support.

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