DSpace at VNU: Nonlinear dynamic analysis of eccentrically stiffened functionally graded circular cylindrical thin shells under external pressure and surrounded by an elastic medium

European Journal of Mechanics A/Solids 46 (2014) 42e53

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European Journal of Mechanics A/Solids
journal homepage: www.elsevier.com/locate/ejmsol

Nonlinear dynamic analysis of eccentrically stiffened functionally
graded circular cylindrical thin shells under external pressure and
surrounded by an elastic medium
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

a b s t r a c t

Article history:
Received 17 July 2013
Accepted 9 February 2014
Available online 18 February 2014

A semi-analytical approach eccentrically stiffened functionally graded circular cylindrical shells surrounded by an elastic medium subjected to external pressure is presented The elastic medium is
assumed as two-parameter elastic foundation model proposed by Pasternak. Based on the classical thin
shell theory with the geometrical nonlinearity in von KarmaneDonnell sense, the smeared stiffeners
technique and Galerkin method, this paper deals the nonlinear dynamic problem. The approximate
three-term solution of deflection shape is chosen and the frequencyeamplitude relation of nonlinear

vibration is obtained in explicit form. The nonlinear dynamic responses are analyzed by using fourth
order RungeeKutta method and the nonlinear dynamic buckling behavior of stiffened functionally
graded shells is investigated according to BudianskyeRoth criterion. Results are given to evaluate effects
of stiffener, elastic foundation and input factors on the frequencyeamplitude curves, natural frequencies,
nonlinear responses and nonlinear dynamic buckling loads of functionally graded cylindrical shells.
Ó 2014 Elsevier Masson SAS. All rights reserved.

Keywords:
Functionally graded material
Nonlinear dynamic analysis
Stiffened circular cylindrical shell

1. Introduction
Functionally graded material (FGM) cylindrical shell has become
popular in engineering designs of coating of nuclear reactors and
space shuttle. The static and dynamic behavior of FGM cylindrical
shell attracts special attention of a lot of researchers in the world.
In static analysis of FGM cylindrical shells, many studies have
been focused on the buckling and postbuckling of shells under
mechanic and thermal loading. Shen (2003) presented the
nonlinear postbuckling of perfect and imperfect FGM cylindrical
thin shells in thermal environments under lateral pressure by using
the classical shell theory with the geometrical nonlinearity in von
KarmaneDonnell sense. By using higher order shear deformation
theory; this author (Shen, 2005) continued to investigate the
postbuckling of FGM hybrid cylindrical shells in thermal environments under axial loading. Huang and Han (2008, 2009a, 2009b,
2010a, 2010b) studied the buckling and postbuckling of unstiffened FGM cylindrical shells under torsion load, axial
compression, radial pressure, combined axial compression and

* Corresponding author. Tel.: þ84 983843387.

E-mail address: (V.H. Nam).
/>0997-7538/Ó 2014 Elsevier Masson SAS. All rights reserved.

radial pressure based on the Donnell shell theory and the nonlinear
strainedisplacement relations of large deformation. Shen (2009b)
investigated the torsional buckling and postbuckling of FGM cylindrical shells in thermal environments. The non-linear static
buckling of FGM conical shells which is more general than cylindrical shells, were studied by Sofiyev (2011a,b). Zozulya and Zhang
(2012) studied the behavior of functionally graded axisymmetric
cylindrical shells based on the high order theory.
For dynamic analysis of FGM cylindrical shells, Darabi et al.
(2008) presented respectively linear and nonlinear parametric
resonance analyses for un-stiffened FGM cylindrical shells. Sofiyev
and Schnack (2004) and Sofiyev (2005) obtained critical parameters for un-stiffened cylindrical thin shells under linearly increasing
dynamic torsional loading and under a periodic axial impulsive
loading by using the Galerkin technique together with Ritz type
variation method. Sheng and Wang (2008) presented the thermomechanical vibration analysis of FGM shell with flowing fluid.
Sofiyev (2003, 2004, 2009, 2012) and Deniz and Sofiyev (2013)
were investigated the vibration and dynamic instability of FGM
conical shells. Hong (2013) studied thermal vibration of magnetostrictive FGM cylindrical shells. Huang and Han (2010c) presented
the nonlinear dynamic buckling problems of un-stiffened functionally graded cylindrical shells subjected to time-dependent axial


D.V. Dung, V.H. Nam / European Journal of Mechanics A/Solids 46 (2014) 42e53

load by using the BudianskyeRoth dynamic buckling criterion
(Budiansky and Roth, 1962). Various effects of the inhomogeneous
parameter, loading speed, dimension parameters; environmental
temperature rise and initial geometrical imperfection on nonlinear
dynamic buckling were discussed.
For FGM cylindrical shell surrounded by an elastic foundation,

the postbuckling of shear deformable FGM cylindrical shells surrounded by an elastic medium was studied by Shen (2009a). Shen
et al. (2010) investigated postbuckling of internal pressure loaded
FGM cylindrical shells surrounded by an elastic medium.
Bagherizadeh et al. (2011) investigated mechanical buckling of FGM
cylindrical shells surrounded by Pasternak elastic foundation.
Sofiyev (2010) analyzed the buckling of FGM circular shells under
combined loads and resting on the Pasternak type elastic foundation. Torsional vibration and stability of functionally graded
orthotropic cylindrical shells on elastic foundations is presented by
Najafov et al. (2013). For the FGM conical shell e general case of
FGM cylindrical shells, mechanic behavior of shell on elastic foundation was studied by Sofiyev (2011c), Najafov and Sofiyev (2013),
Sofiyev and Kuruoglu (2013).
In practice, FGM plates and shells, as other composite structures,
usually reinforced by stiffeners system to provide the benefit of
added load carrying capability with a relatively small additional
weight. Thus study on nonlinear static and dynamic behavior of
theses structures are significant practical problem. However, up to
date, the investigation on this field has received comparatively
little attention. Recently, Najafizadeh et al. (2009) have studied
linear static buckling of FGM axially loaded cylindrical shell reinforced by ring and stringer FGM stiffeners. Bich et al. (2011, 2012,
2013) have investigated the nonlinear static and dynamic analysis
of FGM plates, cylindrical panels and shallow shells with eccentrically homogeneous stiffener system. Dung and Hoa (2013a,
2013b) presented an analytical study of nonlinear static buckling
and post-buckling analysis of eccentrically stiffened functionally
graded circular cylindrical shells under external pressure and
torsional load with FGM stiffeners and approximate three-term
solution of deflection taking into account the nonlinear buckling
shape.
The review of the literature signifies that there are very little
researches on the nonlinear dynamic analysis of FGM stiffened
shells surrounded by an elastic foundation by analytical approach.

In this paper, the dynamic behavior of eccentrically stiffened FGM
(ES-FGM) cylindrical circular shells reinforced by eccentrically ring
and stringer stiffener system on internal and (or) external surface of
shell under external pressure loads is investigated. The nonlinear
dynamic equations are derived by using the classical shell theory
with the nonlinear strainedisplacement relation of large deflection,
the smeared stiffeners technique and Galerkin method. The present
novelty is that an approximate three-term solution of deflection
including the pre-buckling shape, the linear buckling shape and the
nonlinear buckling shape are more correctly chosen and the frequencyeamplitude relation of nonlinear vibration is obtained in
explicit form. In addition, the nonlinear dynamic responses are
found by using fourth order RungeeKutta method and the dynamic
buckling loads of stiffened FGM shells are investigated according to
BudianskyeRoth criterion. The results show that the stiffener,
volume-fractions index and geometrical parameters strongly influence to the dynamic behavior of shells.
2. Formulation
2.1. FGM power law properties
Functionally graded material in this paper, is assumed to be
made from a mixture of ceramic and metal in two cases: inside

43

ceramic surface, outside metal surface and outside ceramic surface,
inside metal surface. The volume-fractions is assumed to be given
by a power law


Vin ¼ Vin ðzÞ ¼



2z þ h k
; Vou ¼ Vou ðzÞ ¼ 1 À Vin ðzÞ;
2h

(1)

where h is the thickness of shell; k ! 0 is the volume-fraction index;
z is the thickness coordinate and varies from Àh/2 to h/2; the
subscripts in and ou refer to the inside and outside material constituents, respectively.
For case of inside ceramic surface and outside metal surface
Vin ¼ Vc and Vou ¼ Vm, for the case of outside ceramic surface and
inside metal surface Vin ¼ Vm and Vou ¼ Vc. In which, Vc is volumefraction of ceramic and Vm is volume-fraction of metal.
Effective properties Preff of FGM shell are determined by linear
rule of mixture as

Preff ¼ Prou ðzÞVou ðzÞ þ Prin ðzÞVin ðzÞ:

(2)

According to the mentioned law, the Young’s modulus and the
mass density of shell can be expressed in the form

2z þ hk
;
2h
2z þ hk
¼ rou þ ðrin À rou Þ
;
2h

EðzÞ ¼ Eou Vou þ Ein Vin ¼ Eou þ ðEin À Eou Þ

rðzÞ ¼ rou Vou þ rin Vin

(3)

For case of inside ceramic surface and outside metal surface
Ein ¼ Ec, rin ¼ rc and Eou ¼ Em, rou ¼ rm, for the case of outside
ceramic surface and inside metal surface Ein ¼ Em, rin ¼ rm and
Eou ¼ Ec, rou ¼ rc. Ec, rc, Em, rm are the Young’s modulus and the mass
density of ceramic and metal, respectively.
2.2. Constitutive relations and governing equations
Consider a functionally graded cylindrical thin shell surrounded
by an elastic foundation with length L, mean radius R and reinforced by closely spaced (Najafizadeh et al., 2009; Brush and
Almroth, 1975; Reddy and Starnes, 1993) pure-metal ring and
stringer stiffener systems (see Fig. 1). The stiffener is located at
outside surface for outside metal surface case and at inside surface
for inside metal surface case. The origin of the coordinate O locates
on the middle surface and at the left end of the shell, x,y ¼ Rq and z
axes are in the axial, circumferential, and inward radial directions
respectively.
According to the von Karman nonlinear strainedisplacement
relations (Brush and Almroth, 1975), the strain components
at the middle surface of perfect circular cylindrical shells are the
form

 
vu 1 vw 2
þ
;

vx 2 vx
 
vv w 1 vw 2
À þ
ε0y ¼
;
vy R 2 vy
ε0x ¼

g0xy

vu vv vw vw
þ þ
;
¼
vy vx vx vy

cx ¼

(4)

v2 w
v2 w
v2 w
c
c
;
;
¼
;

¼
y
xy
vxvy
vx2
vy2

where ε0x and ε0y are normal strains, g0xy is the shear strain at the
middle surface of shell, cx, cy, cxy are the change of curvatures and
twist of shell, and u ¼ u(x,y), v ¼ v(x,y), w ¼ w(x,y) are displacements along x, y and z axes respectively.


44

D.V. Dung, V.H. Nam / European Journal of Mechanics A/Solids 46 (2014) 42e53

Fig. 1. Configuration of an eccentrically stiffened cylindrical shell surrounded by an elastic medium.

The strains across the shell thickness at a distance z from the
mid-surface are represented by

εx ¼

ε0x

À zcx ;

εy ¼

ε0y

À zcy ;

gxy ¼

g0xy

À 2zcxy :

(5)



Em Is
cx À D12 cy ;
Mx ¼ ðB11 þ Cs Þε0x þ B12 ε0y À D11 þ
ss


Em Ir
cy ;
My ¼ B12 ε0x þ ðB22 þ Cr Þε0y À D12 cx À D22 þ
sr

(10)

Mxy ¼ B66 g0xy À 2D66 cxy ;

The deformation compatibility equation is derived from Eq. (4)

v2 ε0x
vy2

þ

v2 ε0y
vx2

À

v2 g0xy
vxvy

2

1v w
¼ À
þ
R vx2

2

!2

v w
vxvy

À

2

2

v wv w
:
vx2 vy2

(6)

where Aij, Bij, Dij (i,j ¼ 1,2,6) are extensional, coupling and bending
stiffness of the un-stiffened FGM cylindrical shell, Nx, Ny are inplane normal force intensities, Nxy is in-plane shearing force intensity, Mx, My are bending moment intensities and Mxy is twisting
moment intensity.

The stressestrain relations for FGM shells are

Á
EðzÞ À
εx þ nεy ;
2
1Àn
Á
EðzÞ À
¼
εy þ nεx ;
1 À n2
EðzÞ
g ;
¼
2ð1 þ nÞ xy

ssh
x ¼
ssh
y
ssh
xy

(7)

sh
where the Poisson’s ratio n is assumed to be constant, ssh
x ; sy are
normal stress in x, y direction of un-stiffened shell, respectively, ssh
xy
is shearing stress in of un-stiffened shell.
The stressestrain relation is applied for homogenous stiffeners

sst
s ¼ Es εx ;
sst
r ¼ Er εy ;


Em As 0
εx þ A12 ε0y À ðB11 þ Cs Þcx À B12 cy ;
ss


Em Ar 0
Ny ¼ A12 ε0x þ A22 þ

εy À B12 cx À ðB22 þ Cr Þcy ;
sr


(8)

A11 þ

Nxy ¼ A66 g0xy À 2B66 cxy ;

E1
;
1 À n2

A12 ¼

E1 n
;
1 À n2

A66 ¼

E1
;
2ð1 þ nÞ

B11 ¼ B22 ¼

E2
;

1 À n2

B12 ¼

E2 n
;
1 À n2

B66 ¼

E2
;
2ð1 þ nÞ

D11 ¼ D22 ¼

E3
;
1 À n2

D12 ¼

E3 n
;
1 À n2

D66 ¼

E3
;

2ð1 þ nÞ
(11)

with


Ein À Eou
ðEin À Eou Þkh2
h; E2 ¼
;
kþ1
2ðk þ 1Þðk þ 2Þ
!

Eou
1
1
1
À
þ
h3 ;
E3 ¼
þ ðEin À Eou Þ
k þ 3 k þ 2 4k þ 4
12


st
where sst
s ; sr are normal stress of stringer and ring stiffeners,

respectively. Es, Er are Young’s modulus of stringer and ring stiffeners, respectively. In this paper, the stringer and ring are assumed
to be metal stiffeners, so Es ¼ Er h Em.
Taking into account the contribution of stiffeners by the
smeared stiffeners technique and omitting the twist of stiffeners
because the torsion constants are smaller more than the moment of
inertia (Brush and Almroth, 1975) and integrating the stressestrain
equations and their moments through the thickness of shell, the
expressions for force and moment resultants of an ES-FGM cylindrical shell are of the form

Nx ¼

A11 ¼ A22 ¼

(9)

E1 ¼

Eou þ

ds h3s
dr h3r
þ As z2s ; Ir ¼
þ Ar z2r ;
12
12
Em As zs
Em Ar zr
Cs ¼ Æ
; Cr ¼ Æ
;

ss
sr

Is ¼

zs ¼

hs þ h
;
2

zr ¼

(12)

hr þ h
;
2

where the coupling parameters Cs and Cr are negative for outside
stiffeners and positive for inside ones. The spacing of the longitudinal and transversal stiffeners is denoted by ss and sr,
respectively. The width and thickness of the stringer and ring
stiffeners are denoted by ds, hs and dr, hr, respectively. The
quantities As, Ar are the cross-section areas of stiffeners and Is, Ir,
zs, zr are the second moments of cross section areas and the eccentricities of stiffeners with respect to the middle surface of
shell respectively.


D.V. Dung, V.H. Nam / European Journal of Mechanics A/Solids 46 (2014) 42e53
=

h

From the constitutive relations (9), one can obtain inversely

ε0x ¼ A*22 Nx À A*12 Ny þ
ε0y ¼ A*11 Ny À A*12 Nx þ
g0xy ¼ A*66 þ 2B*66 cxy ;

B*11 cx
B*21 cx

þ
þ

r1 ¼

B*12 cy ;
B*22 cy ;

1



D


Em As
;
ss

D

;



¼

A*22 ¼

1



D

A22 þ


Em Ar
;
sr

rin À rou



kþ1

h þ rm

As
Ar
þ rm :
ss
sr

(18)

Nx ¼

v2 4
;
vy2

Ny ¼

v2 4
;
vx2

Nxy ¼ À

v2 4
:
vxvy

(19)

A*66 ¼

B*11 ¼ A*22 ðB11 þ Cs Þ À A*12 B12 ;

Substituting Eq. (13) into the compatibility Eq. (6) and Eq.
(15) into the third of Eq. (17), taking into account Eqs. (4) and (19),
yields

(14)
A*11

B*22 ¼ A*11 ðB22 þ Cr Þ À A*12 B12 ;
B*12 ¼ A*22 B12 À A*12 ðB22 þ Cr Þ;
B*21 ¼ A*11 B12 À A*12 ðB11 þ Cs Þ;
B*66 ¼

rou þ

As
Ar
þ rm
ss
sr

Considering the first two of Eq. (17), a stress function 4 may be
defined as

1
;
A66




E A
E A
D ¼ A11 þ m s A22 þ m r À A212 ;
ss
sr

A*12 ¼

A12

A11 þ

rðzÞdz þ rm

Àh=
2

(13)

in which

A*11 ¼

Z2

45

B66
:
A66

 v4 4
v4 4  *
v4 4
v4 w
þ A66 À 2A*12
þ A*22 4 þ B*21 4
4
2
2
vx
vx vy
vy
vx
 v4 w

4
v
w
1 v2 w
*
þ B*11 þ B*22 À 2B*66
þ
B
þ
12
vx2 vy2

vy4 R vx2
2
3
!2
v2 w
v2 w v2 w5
À4
À 2
¼ 0;
vxvy
vx vy2

(20)

Substituting Eq. (13) into Eq. (10) leads to

Mx ¼ B*11 Nx þ B*21 Ny À D*11 cx
My ¼ B*12 Nx þ B*22 Ny À D*21 cx
Mxy ¼ B*66 Nxy À 2D*66 cxy ;

À D*12 cy ;
À D*22 cy ;

r1
(15)

v4 4 1 v2 4 v2 4 v2 w
v2 4 v2 w v2 4 v2 w
À
À

À
þ
2
vxvy vxvy vx2 vy2
vy4 R vx2 vy2 vx2
!
v2 w v2 w
À q0 þ k1 w À k2
þ 2 ¼ 0:
vx2
vy
À B*12

in which

D*11 ¼ D11 þ
D*22

 v4 w
v2 w
vw
v4 w 
þ D*11 4 þ D*12 þ D*21 þ 4D*66
þ 2r1 ε
2
vt
vt
vx
vx2 vy2



4
4
v w
v 4
v4 4
þ D*22 4 À B*21 4 À B*11 þ B*22 À 2B*66
vy
vx
vx2 vy2

Em Is
À ðB11 þ Cs ÞB*11 À B12 B*21 ;
ss

Em Ir
¼ D22 þ
À B12 B*12 À ðB22 þ Cr ÞB*22 ;
sr

D*12 ¼ D12 À ðB11 þ Cs ÞB*12 À B12 B*22 ;

(21)
(16)

D*21 ¼ D12 À B12 B*11 À ðB22 þ Cr ÞB*21 ;

Eqs. (20) and (21) are a nonlinear equation system in terms of
two dependent unknowns w and 4. They are used to investigate the
dynamic characteristics of ES-FGM circular cylindrical shells.

D*66 ¼ D66 À B66 B*66 :

3. Dynamic Galerkin method approach

The nonlinear equations of motion of a thin circular cylindrical
shell based on the classical shell theory and the assumption (Darabi
et al., 2008; Sofiyev and Schnack, 2004; Volmir, 1972) u << w and
v << w, r1v2u/vt2 / 0, r1v2v/vt2 / 0 are given by

Suppose that an ES-FGM cylindrical shell is simply supported
and subjected to uniformly distributed pressure of intensity q0
(N/m2) surrounded by an elastic foundation. Thus the boundary
conditions are of the form

w ¼ 0;

vNx vNxy
þ
¼ 0;
vx
vy
vNxy vNy
þ
¼ 0;
vx
vy

Mx ¼ 0;

Nx ¼ 0;

Nxy ¼ 0;

at x ¼ 0; L:

(22)

The deflection of cylindrical shells in this case can be chosen by
Volmir (1972), Huang and Han (2010a)

v2 Mxy v2 My
v2 Mx
v2 w
v2 w
v2 w
þ Ny 2
þ
þ2
þ Nx 2 þ 2Nxy
2
2
vxvy
vxvy
vx
vy
vx
vy
!
1

v2 w v2 w
v2 w
vw
;
þ Ny þ q0 À k1 w þ k2
þ 2 ¼ r1 2 þ 2r1 ε
R
vt
vx2
vy
vt
(17)
where k1 is Winkler foundation modulus and k2 is the shear layer
foundation stiffness of Pasternak model, q0 is external pressure, t is
time (s), ε is damping coefficient and

w ¼ f0 þ f1 sin

mpx
ny
mpx
sin
þ f2 sin2
;
L
R
L

(23)

in which f0 ¼ f0(t) is time dependent pre-buckling uniform unknown amplitude, f1 ¼ f1(t) is time dependent linear unknown
amplitude, f2 ¼ f2(t) is time dependent nonlinear unknown
amplitude, sin(mpx/L)sin(ny/R) is linear buckling shape, sin2mpx/L
is nonlinear buckling shape in axial direction, m is number of half
waves and n is number of full wave in axial and circumferential
directions, respectively.


46

D.V. Dung, V.H. Nam / European Journal of Mechanics A/Solids 46 (2014) 42e53

As can be seen that the simply supported boundary condition at
x ¼ 0 and x ¼ L is fulfilled on the average sense.
Substituting Eq. (23) into Eq. (20) and solving obtained equation
for unknown 4 leads to

2mpx
2ny
mpx
ny
þ 42 cos
À 43 sin
sin
L
R
L
R
3mpx
ny

x2
sin
À s0y h ;
þ 44 sin
L
R
2

r1

4 ¼ 41 cos

(24)

where s0y is the average circumferential stress, and


41 ¼
42 ¼

2
n2 l

32A*11 m2 p2
m2 p2

f12 À

4lL À 16B*21 m2 p2
32A*11 m2 p2


f2 ;

f 2;
2 1

32A*22 n2 l

(25)

(
mp4
v2 f0
3 v2 f2
vf0 3
vf2
r
r
r
þ
þ
4B*21
þ
þ
2
ε
ε
1
1

1
2
2
4 vt
2
vt
vt
L
vt
)
!
2






2
2
2
2
1 mp
n l
1 B mp
n
À
þ
f12
R L

R
16A*11 m2 p2 2 A L

2  n 2  1 1 
1
2 mp
À
f 2f
þ m2 n2 p2 l
2
R
A G 1 2
L
(
mp4
mp4
þ 4D*11
À 4B*21
L
L
)
!


*
2
2
s0y h
1 mp 2 lL À 4B21 m p
À

À q0
f2 þ
R L
R
4A*11 m2 p2


mp2
3
þ k1 f2 þ f0 þ k2 f2
¼ 0:
4
L

(29)

In addition, the cylindrical shell must satisfy the circumferential
closed condition as (Huang and Han, 2010c; Volmir, 1972)

2

43 ¼
44 ¼

B
m2 n2 p2 l
f1 þ
f1 f2 ;
A
A

2
m2 n2 p2 l

G

2pR Z L
Z

f1 f2 ;

0

0

vv
dxdy ¼
vy

2pR Z L "
Z

ε0y
0

0

  #
w 1 vw 2
þ À
dxdy ¼ 0:

R 2 vy

(30)



2
4
A ¼ A*11 m4 p4 þ A*66 À 2A*12 m2 n2 p2 l þ A*22 n4 l ;


L2
2
4
B ¼ B*21 m4 p4 þ B*11 þ B*22 À 2B*66 m2 n2 p2 l þ B*12 n4 l À m2 p2 ;
R


2
4
*
4
4
*
*
*
2
2
2
*

4
D ¼ D11 m p þ D12 þ D21 þ 4D66 m n p l þ D22 n l ;


2
4
G ¼ 81A*11 m4 p4 þ 9 A*66 À 2A*12 m2 n2 p2 l þ A*22 n4 l ;

l ¼

(26)

L
:
R

Substituting the expressions ((23) and (24)) into Eq. (20) and
then applying Galerkin method in the ranges 0
x
L and
0 y 2pR leads to

r d2 f
1
d2 f
s0y h ¼ Rq0 À Rk1 ðf2 þ 2f0 Þ À Rr1 20 À R 1 22
2
2 dt
dt
df0

df2
À Rr1 ε
;
À 2Rr1 ε
dt
dt

Using Eqs. (13), (19), (23) and (24), this integral becomes

À2A*11 s0y h þ

1
1 n 2 2
ðf þ 2f0 Þ À
f ¼ 0:
R 2
4 R 1

(31)

Eliminating s0y from Eqs. (27)e(29) and the condition of closed
form (31), lead to

(27)

d2 f 0
df
þ 2ε 0
dt
dt 2

!

1 d2 f2
df
þ
þ 2ε 2
2 dt 2
dt

!
þ a11 ðf2 þ 2f0 Þ

(32)

À a12 f12 À a13 q0 þ a14 k1 ðf2 þ 2f0 Þ ¼ 0;


3
2
2


2 n2 p2 l2
n2 l lL À 4B*21 m2 p2
v2 f1
vf1
B2
m
B

2
4
2
2
2
5f1 f2 þ
þ Dþ
f þ4
þ m n p l À
L r1 2 þ 2L r1 ε
A
vt
A 1
A
vt
4A*11
!
4
h
i
m4 p4
n4 l
2
2
3
2 2 2
4
2
s
l

l
p
ð
¼ 0;
þ
þ
À
hn
L
f
þ
L
k
f
þ
L
k
f
nÞ
þ
ðm
Þ
f
0y
1
1
1
2
1
1

16A*22 16A*11
4

m4 n4 p4 l
m4 n4 p4 l
þ
A
G
4

4

!
f1 f22

(28)


D.V. Dung, V.H. Nam / European Journal of Mechanics A/Solids 46 (2014) 42e53

a21 f1

d2 f0
df
þ 2ε 0
dt
dt 2

!

d2 f1
df
þ 2ε 1
dt
dt 2

þ

!
þ

a21
2

f1

d2 f2
df
þ 2ε 2
dt
dt 2

!

d2 f1
df
þ 2ε 1
dt
dt 2

47

!
þ a22 f1 þ b21 f1 f0 þ b22 f1 f2 þ a25 f1 f22 þ b23 f13
À b24 k1 f1 f0 À b25 k1 f1 f2 þ b26 k2 f1 f2 þ a27 k1 f1

þ a22 f1 þ a23 f1 f2 þ a24 f13 þ a25 f1 f22 À a26 q0 f1 þ a27 k1 f1

þ a28 k2 f1 ¼ 0;

þ a28 k2 f1 ¼ 0;

(39)

(33)

d2 f2
df
þ 2ε 2
dt
dt 2

!
þ a31 f12 þ a32 f12 f2 þ a33 f2 þ a34 k1



3
f þ f0
42



d2 f2
df
þ 2ε 2
dt
dt 2

!
þ a31 f12 þ a32 f12 f2 þ a33 f2 þ a34 k1



3
f þ f0
42



þ a35 k2 f2 ¼ 0;

þ a35 k2 f2 ¼ 0;

(40)
(34)
where

where

b11 ¼ 2a11 ; b12 ¼
a11 ¼

1
2A*11 R2 r1

; a12 ¼

n2
8A*11 R3 r1

; a13 ¼

1

r1

; a14 ¼

1
;
2r1

(35)

À



1

2


À1
a31 þ a12 ;
2

b14 ¼ a11 À a33 ; b15
À



3
8

1
2

À
1
¼ 2a14 À a34 ;
2

b13 ¼ a32 ;
(41)

1
2

b16 ¼ a14 À a34 ; b17 ¼ a35 ;



2
Rn2 l
1
B2
a
;
D
þ
;
¼
22
A
L4 r1
L2

3
2
2 2
*
2 2
2
1 4m2 n2 p2 l
B 2 2 2 2 n l lL À 4B21 m p 5
þ m n p l À
;
¼ 4
A
A

L r1
4A*11
!
4
1
m4 p4
n4 l
¼ 4
þ
;
L r1 16A*22 16A*11
!
4
4
1
m4 n4 p4 l
m4 n4 p4 l
þ
;
¼ 4
A
G
L r1

a21 ¼
a23
a24
a25

a26 ¼

i
Rn2 l
1
1 h
2
ðlnÞ þ ðmpÞ2 ;
; a27 ¼ ; a28 ¼ 2
2
r
L r1
L r1
1
2

(36)

a31 ¼

1

r1

(
4B*21
þ2

a34

L

À

!
1mp2
R

L
)

2

b22 ¼ Àb14 a21 À a33

b23 ¼ b12 a21 À a31

a21

À

a21

À

a21

b24 ¼ a21 b15 þ
b26 ¼ a21 b17 À

2

2

2

a21
2

þ a23 ;

þ a24 ;


a34 ;

b25 ¼





3
8



a21 b16 þ a21 a34 ;

a35 :
(42)

Putting f ¼ wmax, from Eq. (23), it is obvious that the maximal
deflection of the shells

f ¼ f0 þ f1 þ f2 ;

(43)

locates at x ¼ iL/2m, y ¼ jpR/2n where i, j are odd integer numbers.
Note that f0 ¼ f0(t), f1 ¼ f1(t), f2 ¼ f2(t) and f ¼ f(t) in Eq. (43).
Eqs. (38)e(40) and (43) are used to analyze the effects of input
parameters on the load-maximum deflection curves of ES-FGM
shells.

2
n2 l

4A*11 m2 p2

Bmp2 n2
;
A L
R
mp2 n2 1


1
2
À
;
m2 n2 p2 l

r1
R
A G
L
(
mp4
mp4 1mp2 !
1
¼
À 4B*21
À
16D*11
r1
R L
L
L
)
*
2
lL À 4B21 m p2
Â
;
A*11 m2 p2
4
4 mp2
¼ ; a35 ¼
:
r1
r1 L

a32 ¼
a33

mp4

b21 ¼ Àb11 a21 ;

3.1. Nonlinear vibration analysis

(37)
Consider an ES-FGM cylindrical thin shell under uniformly
external pressure with the law q0 ¼ QsinUt, Eqs. (38)e(40) become

d2 f0
df
þ 2ε 0
dt
dt 2

!
þ b11 f0 À b12 f12 À b13 f12 f2 þ b14 f2
À a13 Q sinðUtÞ þ b15 k1 f0 þ b16 k1 f2
À b17 k2 f2 ¼ 0;

(44)

Simplifying Eqs. (32)e(34), leads to

d2 f0
df

þ 2ε 0
dt
dt 2

d2 f1
df
þ 2ε 1
dt
dt 2

!
þ b11 f0 À b12 f12 À b13 f12 f2 þ b14 f2 À a13 q0

!
þ a22 f1 þ b21 f1 f0 þ b22 f1 f2 þ a25 f1 f22 þ b23 f13
À b24 k1 f1 f0 À b25 k1 f1 f2 þ b26 k2 f1 f2 þ a27 k1 f1
þ a28 k2 f1 ¼ 0;

þ b15 k1 f0 þ b16 k1 f2 À b17 k2 f2 ¼ 0;
(38)

(45)


48

D.V. Dung, V.H. Nam / European Journal of Mechanics A/Solids 46 (2014) 42e53

d2 f2
df

þ 2ε 2
dt
dt 2

!
þ a31 f12 þ a32 f12 f2 þ a33 f2 þ a34 k1



3
f þ f0
42



þ a35 k2 f2 ¼ 0;
(46)
where Q is amplitude of excitation force and U is excitation
frequency.
By using these equations, the fundamental frequencies of natural vibration of ES-FGM shell and un-stiffened FGM shell, and
frequencyeamplitude relation of nonlinear vibration and nonlinear
response of ES-FGM shell are taken into account. The nonlinear
dynamic responses of ES-FGM shells can be obtained by solving
Eqs. (44)e(46) by the fourth order RungeeKutta iteration method.
If the uniform buckling shape and nonlinear buckling shape are
ignored, Eq. (33) reduces to

d2 f1
df
þ 2ε 1

dt
dt 2

!
þ ða22 þ a27 k1 þ a28 k2 Þf1 þ a24 f13

(47)

À a26 f1 Q sinðUtÞ ¼ 0:
For the free and linear vibration without damping, the Eq. (47)
becomes

d2 f1
þ ða22 þ a27 k1 þ a28 k2 Þf1 ¼ 0:
dt 2

(48)

The fundamental frequency of natural vibration of ES-FGM cylindrical shells can be determined by

umn ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a22 þ k1 a27 þ k2 a28 ;

(49)

where umn is fundamental frequency of natural vibration of shell.
Seeking solution as f1(t) ¼ hsin(Ut) and applying procedure like
Galerkin method to Eq. (47), the frequencyeamplitude relation of

nonlinear vibration is obtained

4

3

8

U2 À εU ¼ ða22 þ k1 a27 þ k2 a28 Þ þ a24 h2 À a26 Q :
p
4
3p

(50)

where h is the amplitude of nonlinear vibration of f1(t).
By introducing the non-dimension frequency parameter x ¼ U/
umn, Eq. (50) becomes

x2 À

4ε

pumn

x ¼ 1þ

3 a24 2
8 a26
h À

Q:
4 u2mn
3p u2mn

(51)

If Q ¼ 0, the frequencyeamplitude relation of nonlinear free
vibration is obtained

x2 À

4ε

pumn

x ¼ 1þ

3 a24 2
h :
4 u2mn

Fig. 2. Comparison of natural frequency of isotropic un-stiffened cylindrical shells
(m ¼ 1).

where qsbu
is the linear upper static buckling load of ES-FGM cy0
lindrical shells.
The linear static critical buckling loads of ES-FGM cylindrical
shells are determined by conditions qscr ¼ minqsbu
0 vs. (m,n).

3.2.2. Nonlinear dynamic buckling analysis of ES-FGM cylindrical
shells
Based on Eqs. (38)e(40), the nonlinear dynamic critical buckling
analysis of ES-FGM circular cylindrical shells is investigated in case
of lateral pressure varying as linear function of time q0 ¼ ct in which
c (N/m2 s) is a loading speed.
Eqs. (38)e(40) are the nonlinear second-order differential three
equation system. Therefore their analytical solution may be very
difficult to find mathematically. In this paper, this equation system
is solved by four order RungeeKutta method. The dynamic critical
time tcr can be obtained according to BudianskyeRoth criterion
(Budiansky and Roth, 1962). This criterion is based on that, for large
value of loading speed, the amplitudeetime 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 ¼ tcr the stability loss occurs. The load
corresponding to the dynamic critical time is called dynamic critical
buckling load.
4. Numerical results
4.1. Validation of the present approach
To validate the present formulation, the natural frequencies of
perfect stiffened isotropic cylindrical shells without elastic

(52)

3.2. Buckling analysis
3.2.1. Linear static buckling analysis of ES-FGM cylindrical shells
Omitting the uniform buckling shape and nonlinear buckling
shape and putting f_ 1 ¼ 0; €f 1 ¼ 0, and taking f1 s 0 Eq. (33)
becomes

a22 þ k1 a27 þ k2 a28 þ a24 f12 À a26 q0 ¼ 0:

(53)

By ignoring the nonlinear term of f1 in Eq. (53), leads to

qsbu
¼
0

a22 þ k1 a27 þ k2 a28
:
a26

(54)

Fig. 3. Comparison of natural frequency of isotropic external stiffened cylindrical
shells (m ¼ 1).


D.V. Dung, V.H. Nam / European Journal of Mechanics A/Solids 46 (2014) 42e53

49

Table 2
Comparison on the frequency parameter for un-stiffened cylindrical shell surrounded by a Winkler foundation (m ¼ 1).
n

Present

Sofiyev et al. (2009)

Paliwal et al. (1996)

Errors (%)
1
2
3
4

0.67480
0.36223
0.20670
0.13747

0.67921
0.36463
0.20804
0.13824

0.65
0.66
0.65
0.56

Errors (%)
0.67882
0.36394
0.20526

0.12745

0.60
0.47
0.70
7.29

Table 3
Effect of R/h ratio and volume-fraction index k on the fundamental frequency of
natural vibration (rad/s) of ES-FGM cylindrical shells surrounded by an elastic
foundation.
Fig. 4. Comparison of natural frequency of isotropic internal stiffened cylindrical shells
(m ¼ 1).

R/h

100

foundation are considered in Figs. 2e4, which were also analyzed
by Sewall and Naumann (1968) and Sewall et al. (1964). The static
buckling of stiffened isotropic cylindrical shells without elastic
foundation under external pressure was studied by Baruch and
Singer (1963), Reddy and Starnes (1993) and Shen (1998) (see
Table 1) and the natural frequencies of isotropic cylindrical shell
surrounded by an elastic foundation investigated by Sofiyev et al.
(2009) and Paliwal et al. (1996) (Table 2).
As can be seen, the good agreements are obtained in these
comparisons.

175

250

a

k

0.2
1
5
10
0.2
1
5
10
0.2
1
5
10

Inside ceramic surface

Outside ceramic surface

Un-stiffened

External stiffeners

Un-stiffened

Internal stiffeners

3284.18
2925.82
2495.50
2362.77
2814.44
2650.02
2331.08
2214.15
2732.24
2614.92
2301.47
2195.37

2187.64
2650.64
3023.68
3132.36
1806.34
2117.46
2376.06
2453.21
1686.54
1933.94
2132.33
2185.58

2448.54
2912.85

3232.89
3309.35
2282.83
2668.73
2814.52
2850.82
2246.27
2613.61
2747.12
2771.16

3111.37
2650.64
2212.02
2098.85
2439.80
2117.46
1823.20
1735.86
2167.31
1933.94
1695.41
1633.08

a

(6)
(6)
(6)
(6)

(7)
(7)
(7)
(6)
(7)
(7)
(7)
(7)

(6)
(5)
(5)
(5)
(6)
(6)
(5)
(5)
(6)
(6)
(5)
(5)

(6)
(6)
(6)
(6)
(7)
(7)
(7)
(7)

(7)
(7)
(7)
(7)

(5)
(5)
(6)
(6)
(5)
(6)
(6)
(6)
(5)
(5)
(6)
(6)

The numbers in the parenthesis denote the buckling modes (n), m ¼ 1.

4.2. Dynamic responses of ES-FGM cylindrical shell
In this section, the stiffened and un-stiffened FGM cylindrical
shells surrounded by an elastic foundation are considered with
R ¼ 0.5 m, L ¼ 0.75 m. The combination of materials consists of
Aluminum Em ¼ 7 Â 1010 N/m2, rm ¼ 2702 kg/m3 and Alumina
Ec ¼ 38 Â 1010 N/m2, rc ¼ 3800 kg/m3. The Poisson’s ratio n is chosen
to be 0.3 for simplicity. The height of stiffeners is equal to 0.01 m, its
width 0.0025 m. The stiffener system includes 15 ring stiffeners and
63 stringer stiffeners distributed regularly in the axial and
circumferential directions, respectively.

Table 3 shows the fundamental frequency of natural vibration of
ES-FGM cylindrical shells with foundation parameters
k1 ¼ 5 Â 105 N/m3, k2 ¼ 2.5 Â 104 N/m, and geometric parameters
hs ¼ hr ¼ 0.01 m, bs ¼ br ¼ 0.0025 m. Clearly, the natural frequency
of stiffened shells is greater than one of un-stiffened shells. The
natural frequency decreases when the proportion of metal increases. Table 3 also shows that the natural frequency of shell increases when the R/h ratio decreases. When k ¼ 1 (the proportions
of ceramic and metal of internal ceramic surface case is equal to
ones of external ceramic surface case), the natural frequency of two
case attain the same value.
Table 4 shows effects of foundation and stiffener on the umn of
cylindrical shells with input parameters k ¼ 1, R/h ¼ 250,

hs ¼ hr ¼ 0.01 m, bs ¼ br ¼ 0.0025 m and different parameters of
foundation. As can be found that parameters of foundation k1 and
k2 affect strongly to fundamental frequency of natural vibration of
shells. Especially, with the presence of the both two parameters of
foundation, the umn is biggest.
The effect of excitation force q0 on the h/hÀx frequencye
amplitude curves of nonlinear vibration of internal stiffened
FGM cylindrical shell is presented in Fig. 5. Two foundation
coefficients are considered as k1 ¼ 5 Â 105 N/m3,
k2 ¼ 2.5 Â 104 N/m and two values of Q are taken as
Q ¼ 105 N/m2 and Q ¼ 2 Â 104 N/m2. As can be observed, when
the excitation force decreases, the curves of forced vibration are
closer to the curve of free vibration.
Fig. 6 investigates effect of the both stiffeners and foundation on
the h/hÀx frequencyeamplitude curve of nonlinear free vibration
for parameters k1 ¼ 5 Â 105 N/m3, k2 ¼ 2.5 Â 104 N/m, R/h ¼ 250,
k ¼ 1 and modes m ¼ 1, n ¼ 8. The obtained results show that the
frequencyeamplitude curve of un-stiffened shell is lower than one

of stiffened shell with and without elastic foundation (EF).
Nonlinear responses of stiffened and un-stiffened functionally
graded cylindrical shell are illustrated in Fig. 7. Computations have
been carried out for the following data: k1 ¼ 5 Â 105 N/m3,
k2 ¼ 2.5 Â 104 N/m, R/h ¼ 250, k ¼ 1, m ¼ 1, n ¼ 5.

Table 1
Comparisons on the static buckling of internal stiffened isotropic cylindrical shells under external pressure (Psi) (m ¼ 1).
Present

Baruch and Singer (1963)

Reddy and Starnes (1993)

Errors (%)
Un-stiffened
Stringer stiffened
Ring stiffened
Orthogonal stiffened
a

103.327
104.494
379.694
387.192

(4)
(4)
(3)
(3)

102
103
370
377

The numbers in the parenthesis denote the buckling modes (n).

1.28
1.43
2.55
2.63

Shen (1998)

Errors (%)
93.5
94.7
357.5
365

9.51
9.37
5.85
5.73

Errors (%)
100.7
102.2
368.3

374.1

(4)a
(4)
(3)
(3)

2.54
2.20
3.00
3.38


50

D.V. Dung, V.H. Nam / European Journal of Mechanics A/Solids 46 (2014) 42e53

Table 4
Effect of foundation parameters k1, k2 on the fundamental frequency of natural vibration (rad/s) of ES-FGM cylindrical shells surrounded by an elastic foundation.
k1 N/m3

k2 N/m

Un-stiffened

0

0
104
2.5 Â 104

5 Â 104
0
104
2.5 Â 104
5 Â 104
0
104
2.5 Â 104
5 Â 104
0
104
2.5 Â 104
5 Â 104

1654.05
1776.68
1913.95
2117.62
1658.70
1781.01
1917.97
2121.24
1677.14
1798.19
1933.94
2135.70
1699.91
1819.45
1953.72
2153.62

105

5 Â 105

106

a

(8)a
(8)
(7)
(7)
(8)
(8)
(7)
(7)
(8)
(8)
(7)
(7)
(8)
(8)
(7)
(7)

External stiffeners

Internal stiffeners

2518.90
2553.49
2604.51
2687.40
2521.06
2555.62
2606.60
2689.42
2529.66
2564.10
2614.92
2697.48
2540.37
2574.67
2625.28
2707.53

2539.43
2566.15
2603.20
2663.79
2541.57
2568.27
2605.28
2665.83
2550.10
2576.71
2613.61
2673.96
2560.72

2587.23
2623.97
2684.10

(6)
(6)
(6)
(6)
(6)
(6)
(6)
(6)
(6)
(6)
(6)
(6)
(6)
(6)
(6)
(6)

(6)
(5)
(5)
(5)
(6)
(5)
(5)
(5)
(6)

(5)
(5)
(5)
(6)
(5)
(5)
(5)

Fig. 7. Nonlinear responses of un-stiffened and internal stiffened FGM cylindrical
shells.

The numbers in the parenthesis denote the buckling modes (n), m ¼ 1.

Fig. 5. The frequencyeamplitude curve of nonlinear vibration of internal stiffened
FGM cylindrical shell (R/h ¼ 250, k ¼ 1, m ¼ 1, n ¼ 5).

The excitation frequencies corresponding to q0 ¼
106sin(300t) N/m2 are much smaller than natural frequencies.
These results show that the stiffeners strongly decrease vibration
amplitude of the shell when excitation frequencies are far from
natural frequencies.
Considers an internal stiffened cylindrical shell with
k1 ¼ 5 Â 105 N/m3, k2 ¼ 2.5 Â 104 N/m and R/h ¼ 250, k ¼ 1,
U ¼ 300 rad/s, Q ¼ 106 N/m2, m ¼ 1, n ¼ 5. As can be seen that when

Fig. 6. The frequencyeamplitude curve of nonlinear vibration of un-stiffened and
internal stiffened FGM cylindrical shell.

Fig. 8. Deflection-velocity relation of internal stiffened cylindrical shell under
Q ¼ 106 N/m2.

the excitation force is small, the deflection-velocity relation with
has the closed curve form as in Fig. 8. But when the excitation force
increases (Q ¼ 1.5 Â 106 N/m2), the deflection-velocity curve becomes more disorderly as Fig. 9.
Fig. 10 shows that when the excitation frequencies are near to
natural frequencies, the interesting phenomenon is observed like
the harmonic beat phenomenon of a linear vibration. The excitation
frequency is 2600 rad/s which is near to natural frequencies
2613.61 rad/s of internal stiffened cylindrical shell. As can be seen,

Fig. 9. Deflection-velocity relation of internal stiffened cylindrical shell under
Q ¼ 1.5 Â 106 N/m2.


D.V. Dung, V.H. Nam / European Journal of Mechanics A/Solids 46 (2014) 42e53

Fig. 10. Nonlinear responses of internal stiffened FGM cylindrical shells
(k1 ¼ 5 Â 105 N/m3, k2 ¼ 2.5 Â 104 N/m, R/h ¼ 250, k ¼ 1, q0(t) ¼ 5 Â 105sin(2600t) N/
m2, m ¼ 1, n ¼ 5).

51

Fig. 12. Effect of damping on nonlinear responses of external stiffened cylindrical
shells in the far periods (k1 ¼ 5 Â 105 N/m3, k2 ¼ 2.5 Â 104 N/m, R/h ¼ 250, k ¼ 1,
q0(t) ¼ 5 Â 105sin(2600t) N/m2, m ¼ 1, n ¼ 5).

To investigate the nonlinear dynamic buckling approach of
eccentrically stiffened FGM cylindrical shells, the stiffened and unstiffened FGM cylindrical shells with and without elastic foundation are considered with R ¼ 0.5 m, L ¼ 0.75 m. The combination of
materials is the same with previous section. The height of stiffeners
is equal to 0.005 m, its width 0.002 m. The stiffener system includes

15 ring stiffeners and 63 stringer stiffeners distributed regularly in
the axial and circumferential directions, respectively.
Figs. 13e14 show the dynamic responses of un-stiffened and
stiffened shells under mechanic load. 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 f vs t curve increases rapidly. According to the BudianskyeRoth criterion
(Budiansky and Roth, 1962), the critical time tcr can be taken as an
intermediate value of this region. Therefore, one can choose the



inflexion point of curve i.e. d2 f =dt 2 
¼ 0 as Huang and Han
t¼tcr
(2010c).
Effect of elastic foundation and stiffener on the nonlinear critical
buckling loads is given in Table 5. The computation parameters are
assumed as k1 ¼ 5 Â 105 N/m3, k2 ¼ 2.5 Â 104 N/m, R/h ¼ 250, k ¼ 1,
c ¼ 106 N/m2 s. Clearly, elastic foundation considerably enhances
the critical buckling load of shell. It seems that, the stringer stiffeners lightly influence and the ring stiffeners strongly influence to
the critical buckling load of shells. Table 5 also shows that the
critical dynamic buckling loads are greater than the critical static
buckling loads of shells.
Table 6 shows the critical dynamic buckling loads of stiffened
and un-stiffened cylindrical shells vs. four different values of volume fraction index k ¼ (0.2,1,5,10). With the same value of foundation parameters k1 ¼ 5 Â 105 N/m3, k2 ¼ 2.5 Â 104 N/m and
loading speed. c ¼ 106 N/m2 s, it is found that the effectiveness of
stiffeners is obviously proven; the critical buckling load of stiffened
shell is greater than one of un-stiffened shell. Table 6 also shows
that the critical dynamic load decreases with the increase of the
proportion of metal. The results in Table 6 also show the effect of R/

h ratio on the critical dynamic buckling of FGM cylindrical shells. As
can be seen, the critical dynamic buckling of FGM cylindrical shell is
considerably decreased when the R/h ratio increases. It is reasonable because the critical buckling loads decrease with the thinner
shell.

Fig. 11. Effect of damping on nonlinear responses of external stiffened cylindrical
shells in the first periods (k1 ¼ 5 Â 105 N/m3, k2 ¼ 2.5 Â 104 N/m, R/h ¼ 250, k ¼ 1,
q0(t) ¼ 5 Â 105sin(2600t) N/m2, m ¼ 1, n ¼ 5).

Fig. 13. Effect of loading speed on the dynamic responses of internal stiffened shells
under external pressure (R/h ¼ 250, k ¼ 1, m ¼ 1, n ¼ 8, k1 ¼ 5 Â 105 N/m3,
k2 ¼ 2.5 Â 104 N/m).

the amplitude of beats increases rapidly when the excitation frequency approaches the natural frequencies.
Effect of damping on nonlinear responses is presented in Figs. 11
and 12 with linear damping coefficient ε ¼ 0.3. The damping influences very small to the nonlinear response in the first vibration
periods (Fig. 11) however, it strongly decreases amplitude at the
next far periods (Fig. 12).
4.3. Nonlinear dynamic buckling of ES-FGM shell


52

D.V. Dung, V.H. Nam / European Journal of Mechanics A/Solids 46 (2014) 42e53

stiffeners technique with von KarmaneDonnell nonlinear terms,
the governing equations are presented in this paper. Approximate
three-term solution of deflection taking into account the nonlinear
buckling shape is chosen. By using the Galerkin method, the
nonlinear dynamic equations of ES-FGM circular cylindrical shells

are obtained. The frequencyeamplitude relation of nonlinear vibration is obtained in explicit form. The nonlinear dynamic
response and critical dynamic buckling load are analyzed by using
the RungeeKutta method and the BudianskyeRoth criterion.
Some conclusions are obtained from this study

Fig. 14. Effect of loading speed on the dynamic responses of external stiffened shells
under external pressure (R/h ¼ 250, k ¼ 1, m ¼ 1, n ¼ 8, k1 ¼ 5 Â 105 N/m3,
k2 ¼ 2.5 Â 104 N/m).

Table 5
Effects of elastic foundation and stiffener on the critical buckling load of FGM cylindrical shells (Â105 N/m2).

Un-stiffened
External rings
Internal rings
External stringers
Internal stringers
External rings
and stringers
Internal rings
and stringers
a

Without elastic foundation

With elastic foundation

Static

Dynamic

Static

1.568
2.738
2.854
1.571
1.575
2.791

1.850
3.025
3.138
1.873
1.852
3.064

1.292
2.452
2.565
1.315
1.294
2.490

(9)a
(8)
(8)
(9)
(9)
(8)

2.568 (8)

(9)
(8)
(8)
(9)
(9)
(8)

2.869 (8)

Dynamic
(9)
(8)
(8)
(9)
(9)
(8)

3.141 (8)

2.114
3.303
3.424
2.129
2.115
3.358

(9)

(8)
(8)
(9)
(9)
(8)

3.438 (8)

The numbers in the parenthesis denote the buckling modes (n), m ¼ 1.

Effects of the loading speed on the dynamic responses of cylindrical shells are shown in Figs. 13 and 14. Three values of loading
speed are chosen c ¼ 106 N/m2s, c ¼ 2 Â 106 N/m2s and
c ¼ 5 Â 106 N/m2s. Clearly, the critical dynamic buckling loads and
maximal amplitude response increase when the loading speed
increases.
5. Conclusions
A semi-analytical approach of eccentrically stiffened functionally graded circular cylindrical thin shells subjected to time
dependent external pressure surrounded by an elastic foundation is
proposed. Base on the classical shell theory and the smeared
Table 6
Dynamic critical buckling of FGM cylindrical shells (Â105 N/m2).
R/h

100

175

250

a

k

0.2
1
5
10
0.2
1
5
10
0.2
1
5
10

Inside ceramic surface

Outside ceramic surface

Un-stiffened

External stiffeners

Un-stiffened

Internal stiffeners

23.694
15.977

11.262
10.220
7.287
5.476
4.252
3.905
4.082
3.358
2.843
2.675

9.068
14.207
20.414
22.615
2.822
4.082
5.596
6.115
1.600
2.114
2.725
2.948

10.875
16.113
22.158
24.296
4.221
5.584

7.019
7.486
2.815
3.438
4.053
4.226

22.118
14.207
9.313
8.393
6.627
4.082
2.885
2.652
2.903
2.114
1.636
1.522

(7)a
(7)
(7)
(7)
(7)
(8)
(8)
(8)
(9)
(9)

(9)
(9)

(7)
(7)
(7)
(7)
(8)
(8)
(7)
(7)
(8)
(8)
(8)
(8)

(7)
(7)
(7)
(7)
(8)
(8)
(8)
(8)
(9)
(9)
(9)
(9)

(7)

(7)
(7)
(7)
(7)
(8)
(8)
(8)
(8)
(8)
(8)
(8)

The numbers in the parenthesis denote the buckling modes (n), m ¼ 1.

i) Stiffeners and elastic foundation have the large effect on the
increase of natural frequencies of shells.
ii) When the excitation frequencies are near to natural frequencies, the interesting phenomenon is observed like the
harmonic beat phenomenon of a linear vibration.
iii) Damping lightly influences to the nonlinear response in the
first vibration periods, however it strongly decreases
amplitude at the next far periods.
iv) Stiffeners and elastic foundation enhance the dynamic stability and load-carrying capacity of FGM cylindrical shells.
v) Fundamental frequency of natural vibration corresponding
to the presence of the both foundation parameters k1 and k2
is biggest.
vi) Radius-to-thickness ratio, elastic foundation and position of
stiffeners significantly influence on the dynamic behavior of
cylindrical shell.
vii) Stringer stiffeners lightly influence and the ring stiffeners
strongly influence on the critical buckling load of shells for

FGM cylindrical shells.
Acknowledgments
This research is funded by Vietnam National Foundation for
Science and Technology Development (NAFOSTED) under grant
number 107.02-2013.02.
References
Bagherizadeh, E., Kiani, Y., Eslami, M.R., 2011. Mechanical buckling of functionally
graded material cylindrical shells surrounded by Pasternak elastic foundation.
Compos. Struct. 93, 3063e3071.
Baruch, M., Singer, J., 1963. Effect of eccentricity of stiffeners on the general instability of stiffened cylindrical shells under hydro-static pressure. J. Mech. Eng.
Sci. 5, 23e27.
Bich, D.H., Nam, V.H., Phuong, N.T., 2011. Nonlinear postbuckling of eccentrically
stiffened functionally graded plates and shallow shells. Vietnam J. Mech. 33,
132e147.
Bich, D.H., Dung, D.V., Nam, V.H., 2012. Nonlinear dynamical analysis of eccentrically stiffened functionally graded cylindrical panels. Compos. Struct. 94, 2465e
2473.
Bich, D.H., Dung, D.V., Nam, V.H., 2013. Nonlinear dynamic analysis of eccentrically
stiffened imperfect functionally graded doubly curved thin shallow shells.
Compos. Struct. 96, 384e395.
Brush, D.O., Almroth, B.O., 1975. Buckling of Bars, Plates and Shells. Mc Graw-Hill,
New York.
Budiansky, B., Roth, R.S., 1962. Axisymmetric Dynamic Buckling of Clamped Shallow
Spherical Shells. NASA technical note D_510.
Darabi, M., Darvizeh, M., Darvizeh, A., 2008. Non-linear analysis of dynamic stability
for functionally graded cylindrical shells under periodic axial loading. Compos.
Struct. 83, 201e211.
Deniz, A., Sofiyev, A.H., 2013. The nonlinear dynamic buckling response of functionally graded truncated conical shells. J. Sound Vib. 332, 978e992.
Dung, D.V., Hoa, L.K., 2013a. Nonlinear buckling and post-buckling analysis of
eccentrically stiffened functionally graded circular cylindrical shells under
external pressure. Thin-Walled Struct. 63, 117e124.

Dung, D.V., Hoa, L.K., 2013b. Research on nonlinear torsional buckling and postbuckling of eccentrically stiffened functionally graded thin circular cylindrical
shells. Compos. B 51, 300e309.
Hong, C.C., 2013. Thermal vibration of magnetostrictive functionally graded material shells. Eur. J. Mech. A/Solids 40, 114e122.
Huang, H., Han, Q., 2008. Buckling of imperfect functionally graded cylindrical
shells under axial compression. Eur. J. Mech. A/Solids 27, 1026e1036.


D.V. Dung, V.H. Nam / European Journal of Mechanics A/Solids 46 (2014) 42e53
Huang, H., Han, Q., 2009a. Nonlinear elastic buckling and postbuckling of axially
compressed functionally graded cylindrical shells. Int. J. Mech. Sci. 51, 500e507.
Huang, H., Han, Q., 2009b. Nonlinear buckling and postbuckling of heated functionally graded cylindrical shells under combined axial compression and radial
pressure. Int. J. Non-Linear Mech. 44, 209e218.
Huang, H., Han, Q., 2010a. Research on nonlinear postbuckling of FGM cylindrical
shells under radial loads. Compos. Struct. 92, 1352e1357.
Huang, H., Han, Q., 2010b. Nonlinear buckling of torsion-loaded functionally graded
cylindrical shells in thermal environment. Eur. J. Mech. A/Solids 29, 42e48.
Huang, H., Han, Q., 2010c. Nonlinear dynamic buckling of functionally graded cylindrical shells subjected to a time-dependent axial load. Compos. Struct. 92,
593e598.
Najafizadeh, M.M., Hasani, A., Khazaeinejad, P., 2009. Mechanical stability of functionally graded stiffened cylindrical shells. Appl. Math. Model 54, 1151e1157.
Najafov, A.M., Sofiyev, A.H., Kuruoglu, N., 2013. Torsional vibration and stability of
functionally graded orthotropic cylindrical shells on elastic foundations. Meccanica 48, 829e840.
Najafov, A.M., Sofiyev, A.H., 2013. The non-linear dynamics of FGM truncated
conical shells surrounded by an elastic medium. Int. J. Mech. Sci. 66, 33e44.
Paliwal, D.N., Pandey, R.K., Nath, T., 1996. Free vibration of circular cylindrical shell
on Winkler and Pasternak foundation. Int. J. Pres. Ves. Pip. 69, 79e89.
Reddy, J.N., Starnes, J.H., 1993. General buckling of stiffened circular cylindrical
shells according to a layerwise theory. Comput. Struct. 49, 605e616.
Sewall, J.L., Clary, R.R., Leadbetter, S.A., 1964. An Experimental and Analytical Vibration Study of a Ring-stiffened Cylindrical Shell Structure with Various Support Conditions. NASA TN D-2398.
Sewall, J.L., Naumann, E.C., 1968. An Experimental and Analytical Vibration Study of
Thin Cylindrical Shells with and Without Longitudinal Stiffeners. NASA technical note D-4705.

Shen, H.S., 1998. Post-buckling analysis of imperfect stiffened laminated cylindrical
shells under combined external pressure and thermal loading. Int. J. Mech. 40,
339e355.
Shen, H.S., 2003. Postbuckling analysis of pressure-loaded functionally graded cylindrical shells in thermal environments. Eng. Struct. 25, 487e497.
Shen, H.S., 2005. Postbuckling of axially-loaded FGM hybrid cylindrical shells in
thermal environments. Compos. Sci. Technol. 65, 1675e1690.
Shen, H.S., 2009a. Postbuckling of shear deformable FGM cylindrical shells surrounded by an elastic medium. Int. J. Mech. Sci. 51, 372e383.

53

Shen, H.S., 2009b. Torsional buckling and postbuckling of FGM cylindrical shells in
thermal environments. Int. J. Non-Linear Mech. 44, 644e657.
Shen, H.S., Yang, J., Kitipornchai, S., 2010. Postbuckling of internal pressure loaded
FGM cylindrical shells surrounded by an elastic medium. Eur. J. Mech. A/Solids
29, 448e460.
Sheng, G.G., Wang, X., 2008. Thermomechanical vibration analysis of a functionally
graded shell with flowing fluid. Eur. J. Mech. A/Solids 27, 1075e1087.
Sofiyev, A.H., 2003. Dynamic buckling of functionally graded cylindrical shells under non-periodic impulsive loading. Acta Mech. 165, 151e163.
Sofiyev, A.H., 2004. The stability of functionally graded truncated conical shells
subjected to aperiodic impulsive loading. Int. J. Solids Struct. 41, 3411e3424.
Sofiyev, A.H., Schnack, E., 2004. The stability of functionally graded cylindrical shells
under linearly increasing dynamic torsional loading. Eng. Struct. 26, 1321e1331.
Sofiyev, A.H., 2005. The stability of compositionally graded ceramic-metal cylindrical shells under aperiodic axial impulsive loading. Compos. Struct. 69, 247e
257.
Sofiyev, A.H., 2009. The vibration and stability behavior of freely supported FGM
conical shells subjected to external pressure. Compos. Struct. 89, 356e366.
Sofiyev, A.H., Avcar, M., Ozyigit, P., Adigozel, S., 2009. The free vibration of non
homogeneous truncated conical shells on a Winkler foundation. Int. J. Eng.
Appl. Sci. 1, 34e41.
Sofiyev, A.H., 2010. Buckling analysis of FGM circular shells under combined loads

and resting on the Pasternak type elastic foundation. Mech. Res. Commun. 37,
539e544.
Sofiyev, A.H., 2011a. Non-linear buckling behavior of FGM truncated conical shells
subjected to axial load. Int. J. Non-Linear Mech. 46, 711e719.
Sofiyev, A.H., 2011b. Influence of the initial imperfection on the non-linear buckling
response of FGM truncated conical shells. Int. J. Mech. Sci. 53, 753e761.
Sofiyev, A.H., 2011c. Thermal buckling of FGM shells resting on a two-parameter
elastic foundation. Thin-Walled Struct. 49, 1304e1311.
Sofiyev, A.H., 2012. The non-linear vibration of FGM truncated conical shells.
Compos. Struct. 94, 2237e2245.
Sofiyev, A.H., Kuruoglu, N., 2013. Non-linear buckling of an FGM truncated conical
shell surrounded by an elastic medium. Int. J. Pres. Ves. Pip. 107, 38e49.
Volmir, A.S., 1972. Non-linear Dynamics of Plates and Shells. Science Edition M. (in
Russian).
Zozulya, V.V., Zhang, Ch, 2012. A high order theory for functionally graded
axisymmetric cylindrical shells. Int. J. Mech. Sci. 60, 12e22.