DSpace at VNU: Nonlinear static and dynamic buckling analysis of imperfect eccentrically stiffened functionally graded circular cylindrical thin shells under axial compression

International Journal of Mechanical Sciences 74 (2013) 190–200

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International Journal of Mechanical Sciences
journal homepage: www.elsevier.com/locate/ijmecsci

Nonlinear static and dynamic buckling analysis of imperfect
eccentrically stiffened functionally graded circular cylindrical thin
shells under axial compression
Dao Huy Bich a, Dao Van Dung a, Vu Hoai Nam b,n, Nguyen Thi Phuong b
a
b

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

art ic l e i nf o

a b s t r a c t

Article history:
Received 24 November 2012
Received in revised form
20 May 2013
Accepted 3 June 2013
Available online 12 June 2013

An analytical approach is presented to investigate the nonlinear static and dynamic buckling of imperfect
eccentrically stiffened functionally graded thin circular cylindrical shells subjected to axial compression.
Based on the classical thin shell theory with the geometrical nonlinearity in von Karman–Donnell sense,

initial geometrical imperfection and the smeared stiffeners technique, the governing equations of motion
of eccentrically stiffened functionally graded circular cylindrical shells are derived. The functionally
graded cylindrical shells with simply supported edges are reinforced by ring and stringer stiffeners
system on internal and (or) external surface. The resulting equations are solved by the Galerkin
procedure to obtain the explicit expression of static critical buckling load, post-buckling load–deflection
curve and nonlinear dynamic motion equation. The nonlinear dynamic responses are found by using
fourth-order Runge–Kutta method. The dynamic critical buckling loads of shells under step loading of
infinite duration are found corresponding to the load value of sudden jump in the average deflection and
those of shells under linear-time compression are investigated according to Budiansky–Roth criterion.
The obtained results show the effects of stiffeners and input factors on the static and dynamic buckling
behavior of these structures.
& 2013 Elsevier Ltd. All rights reserved.

Keywords:
Static and dynamic buckling analysis
Stiffener
Functionally graded material
Stiffened circular cylindrical shell
Critical buckling load

1. Introduction
Functionally graded (FGM) plate and shell structures have became
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 scientists in the world.
In static analysis of FGM cylindrical shells, many studies have been
focused on the buckling and post-buckling of shells under mechanic
and thermal loading. Shen [1] presented the nonlinear post-buckling
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 Karman–Donnell sense. By
using higher order shear deformation theory; this author [2] continued to investigate the post-buckling of FGM hybrid cylindrical shells in
thermal environments under axial loading. Bahtui and Eslami [3]
investigated the coupled thermo-elasticity of FGM cylindrical shells.
Huang and Han [4–7] studied the buckling and post-buckling of unstiffened FGM cylindrical shells under axial compression, radial
pressure and combined axial compression and radial pressure based
on the Donnell shell theory and the nonlinear strain–displacement

n

Corresponding author. Tel.: +84 98 3843 387.
E-mail address: (V.H. Nam).

0020-7403/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.
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relations of large deformation. The post-buckling of shear deformable
FGM cylindrical shells surrounded by an elastic medium was studied
by Shen [8]. Sofiyev [9] analyzed the buckling of FGM circular shells
under combined loads and resting on the Pasternak type elastic
foundation. Zozulya and Zhang [10] studied the behavior of functionally graded axisymmetric cylindrical shells based on the high order
theory.
For dynamic analysis of FGM cylindrical shells, Ng et al. [11] and
Darabi et al. [12] presented respectively linear and nonlinear parametric resonance analyses for un-stiffened FGM cylindrical shells.
Three-dimensional vibration analysis of fluid-filled orthotropic FGM
cylindrical shells was investigated by Chen et al. [13]. Sofiyev and
Schnack [14] and Sofiyev [15] obtained critical parameters for unstiffened 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. Shariyat [16,17] investigated the nonlinear dynamic buckling
problems of axially and laterally preloaded FGM cylindrical shells

under transient thermal shocks and dynamic buckling analysis for unstiffened FGM cylindrical shells under complex combinations of
thermo–electro-mechanical loads. Geometrical imperfection effects
were also included in his research. Li et al. [18] studied the free
vibration of three-layer circular cylindrical shells with functionally
graded middle layer. Huang and Han [19] presented the nonlinear


D. Huy Bich et al. / International Journal of Mechanical Sciences 74 (2013) 190–200

Nomenclature
h
thickness of the shell
m
number of half waves in axial direction
n
number of wave in circumferential direction
k
volume-fraction index
z
coordinate in thickness direction
EðzÞ; Em ; Ec Young's modulus of shell, metal, ceramic,
respectively
ρðzÞ; ρm ; ρc mass density of shell, metal, ceramic, respectively
L
length of the shell
R
radius of the shell
Es ; Er
Young's modulus of stringer and ring stiffeners,
respectively

Aij ; Bij ; Dij extensional, coupling and bending stiffness of the unstiffened shell, respectively
Cs; Cr
coupling parameters
ss ; sr
spacing of the stringer and ring stiffeners, respectively

dynamic buckling problems of un-stiffened functionally graded cylindrical shells subjected to time-dependent axial load by using the
Budiansky–Roth dynamic buckling criterion [20]. Various effects of
the inhomogeneous parameter, loading speed, dimension parameters;
environmental temperature rise and initial geometrical imperfection
on nonlinear dynamic buckling were discussed. Shariyat [21] analyzed
the nonlinear transient stress and wave propagation analyses of the
FGM thick cylinders, employing a unified generalized thermoelasticity theory.
Recently, idea of eccentrically stiffened FGM structures has been
proposed by Najafizadeh et al. [22] and Bich et al. [23,24]. Najafizadeh
et al. [22] have studied linear static buckling of FGM axially loaded
cylindrical shell reinforced by ring and stringer FGM stiffeners. In order
to provide material continuity and easily to manufacture, the FGM
shells are reinforced by an eccentrically homogeneous stiffener
system; Bich et al. [23] have investigated the nonlinear static postbuckling of functionally graded plates and shallow shells and nonlinear dynamic buckling of functionally graded cylindrical panels [24].
Literature on the nonlinear static and dynamic analysis of
imperfect FGM stiffened circular cylindrical shells is still very
limited. In this paper, the mentioned just problem is investigated
by analytical approach. The nonlinear dynamic equations of
eccentrically stiffened FGM circular cylindrical shells are derived
based on the classical shell theory with the nonlinear strain–
displacement relation of large deflection and the smeared stiffeners technique. By using the Galerkin method, the closed-form
expression to determine the static critical buckling load and load–
deflection curves are obtained. The nonlinear dynamic responses
are found by using fourth-order Runge–Kutta method. The

dynamic buckling loads of shells under step loading of infinite
duration are found corresponding to the load value of sudden
jump in the average deflection and those of shells under lineartime compression are investigated according to Budiansky–Roth
criterion. The results show that the stiffener, volume-fractions
index, initial imperfection and geometrical parameters influence
strongly to the static and dynamic buckling of shells.
2. Eccentrically stiffened FGM (ES-FGM) circular cylindrical
shells
2.1. Functionally graded material
In this paper, functionally graded material is assumed to be
made from a mixture of ceramic and metal with the volume-

191

As ; Ar
Is ; Ir

cross-section areas of stiffeners
moments of inertia of stiffener cross sections relative
to the shell middle surface
zs ; zr
eccentricities of stiffeners with respect to the middle
surface of shell
ds ; dr
width of the stringer and ring stiffeners, respectively
hs ; hr
height of the stringer and ring stiffeners, respectively
f ¼ f ðtÞ time dependent total amplitude
f0
known imperfect amplitude

r0
compressive load per unit length
r 0 ¼ r 0 =h compressive stress
r sbu
static buckling stress
r scr
static critical buckling stress
r scr
static critical buckling loads per unit length
t; t cr
time and dynamic critical time
c
loading speed
r dcr
dynamic critical buckling stress
τcr
dynamic coefficient

fractions given by a power law
V m þ V c ¼ 1;
V c ¼ V c ðzÞ ¼


k
2z þ h
;
2h

where h is the thickness of shell; k≥0 is the volume-fraction index;
z is the thickness coordinate and varies from −h=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

k
2z þ h
EðzÞ ¼ Em V m þ Ec V c ¼ Em þ ðEc −Em Þ
;
2h

k
2z þ h
;
ð1Þ
ρðzÞ ¼ ρm V m þ ρc V c ¼ ρm þ ðρc −ρm Þ
2h
Poisson’s ratio ν is assumed to be constant.
2.2. Constitutive relations and governing equations
Consider a functionally graded thin circular cylindrical shell
with length L, mean radius R. This shell is assumed to be reinforced
by closely spaced [22,25,29] homogeneous ring and stringer
stiffener systems (see Fig. 1). Stiffener is pure-ceramic if it is
located at ceramic-rich side and is pure-metal if is located at
metal-rich side, such FGM stiffened circular cylindrical shells
provide continuity within shell and stiffeners and can be easier
manufactured. The origin of the coordinate 0 locates on the
middle plane and at the left end of the shell, x; y (y ¼ Rθ) and z
axes are in the axial, circumferential, and inward radial directions,
respectively.
According to the von Karman nonlinear strain–displacement

relations [25], the strain components at the middle plane of
imperfect circular cylindrical shells are of the form
 
∂u 1 ∂w 2 ∂w ∂w0
ε0x ¼
þ
;
þ
∂x 2 ∂x
∂x ∂x
 2
∂v w 1 ∂w
∂w ∂w0
− þ
ε0y ¼
;
þ
∂y R 2 ∂y
∂y ∂y
∂u ∂v ∂w ∂w ∂w ∂w0 ∂w ∂w0
þ þ
þ
þ
;
γ 0xy ¼
∂y ∂x ∂x ∂y
∂y ∂x
∂x ∂y
χx ¼

∂2 w
;
∂x2

χy ¼

∂2 w
;
∂y2

χ xy ¼

∂2 w
;
∂x∂y

ð2Þ


192

D. Huy Bich et al. / International Journal of Mechanical Sciences 74 (2013) 190–200

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

where u ¼ uðx; yÞ, v ¼ vðx; yÞ and w ¼ wðx; yÞ are displacements
along x; y and z axes respectively and w0 ¼ w0 ðx; yÞ denotes initial
imperfection of shell, which is very small compared with the shell
dimensions, but may be compared with the shell wall thickness.
The strains across the shell thickness at a distance z from the

mid-surface are given by



Er I r
M y ¼ B12 ε0x þ ðB22 þ C r Þε0y −D12 χ x − D22 þ
χy;
sr
M xy ¼ B66 γ 0xy −2D66 χ xy ;

where Aij ; Bij ; Dij ði; j ¼ 1; 2; 6Þ are extensional, coupling and
bending stiffness of the un-stiffened FGM cylindrical shell
E1
E1 ν
E1
;
; A12 ¼
; A66 ¼
2ð1 þ νÞ
1−ν2
1−ν2
E2
E2 ν
E2
B11 ¼ B22 ¼
;
; B12 ¼
; B66 ¼
2ð1 þ νÞ
1−ν2

1−ν2
E3
E3 ν
E3
;
D11 ¼ D22 ¼
; D12 ¼
; D66 ¼
2ð1 þ νÞ
1−ν2
1−ν2

A11 ¼ A22 ¼

εx ¼ ε0x −zχ x ;
εy ¼ ε0y −zχ y ;
γ xy ¼ γ 0xy −2zχ xy :

ð3Þ

From Eq. (2) the strains must be relative in the deformation
compatibility equation
 2
2
2
2
∂2 ε0x ∂ ε0y ∂ γ 0xy
1 ∂2 w
∂ w ∂2 w0
þ

¼
−
þ
−
þ
R ∂x2
∂x∂y ∂x∂y
∂y2
∂x2 ∂x∂y
 2
 2

2
2
∂ w ∂ w0
∂ w ∂ w0
−
þ
þ
:
∂x2
∂y2
∂x2
∂y2

ð4Þ

EðzÞ
ðεx þ νεy Þ;
1−ν2

ssh
y ¼

EðzÞ
ðεy þ νεx Þ;
1−ν2

ssh
xy ¼

EðzÞ
γ ;
2ð1 þ νÞ xy



2
Ec −Em
ðEc −Em Þkh
h; E2 ¼
;
E1 ¼ E m þ
kþ1
2ðk þ 1Þðk þ 2Þ



Em
1

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

sry ¼ Er εy ;

ð5Þ

ð6Þ

where Es ; Er is Young's modulus of stringer and ring stiffeners,
respectively.
The force and moment of an un-stiffened FGM circular cylindrical shell can be determine by
Z h=2
fðN x ; N y ; N xy Þ; ðM x ; M y ; M xy Þg ¼
fsx ; sy ; sxy gð1; zÞ dz:
ð7Þ
−h=2

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 shell, we obtain the expressions for force

and moment resultants of an ES-FGM circular cylindrical shell


Es As 0
N x ¼ A11 þ
εx þ A12 ε0y −ðB11 þ C s Þχ x −B12 χ y ;
ss


Er Ar 0
εy −B12 χ x −ðB22 þ C r Þχ y ;
Ny ¼ A12 ε0x þ A22 þ
sr
Nxy ¼ A66 γ 0xy −2B66 χ xy ;


Es I s
χ x −D12 χ y ;
M x ¼ ðB11 þ C s Þε0x þ B12 ε0y − D11 þ
ss

3

ds hs
dr h r
þ As z2s ; I r ¼
þ Ar z2r ;
12
12
Es As zs

E r Ar z r
Cs ¼ 7
; Cr ¼ 7
;
ss
sr
hs þ h
hr þ h
; zr ¼
;
zs ¼
2
2
Is ¼

and for stiffeners
ssx ¼ Es εx ;

ð10Þ

with

3

Hooke's stress–strain relation is applied for the shell
ssh
x ¼

ð9Þ

ð8Þ

ð11Þ

where the coupling parameters C s and C r are negative for outside
stiffeners and positive for inside ones. The spacing of the stringer
and ring stiffeners is denoted by ss and sr respectively.
The quantities As , Ar are the cross-section areas of stiffeners and
I s , I r , zs , zr are the second moments of cross section areas and the
eccentricities of stiffeners with respect to the middle surface of
shell, respectively. The width and thickness of the stringer and ring
stiffeners are denoted by ds ; hs and dr ; hr respectively. The Young
modulus of stiffeners Es , Er take the value Em if the full metal
stiffeners are put at the metal-rich side of the shell and conversely,
Ec if the full ceramic ones are put at the ceramic-rich side.
From the constitutive relations (8), one can write inversely
ε0x ¼ An22 N x −An12 N y þ Bn11 χ x þ Bn12 χ y ;
ε0y ¼ An11 N y −An12 N x þ Bn21 χ x þ Bn22 χ y ;
γ 0xy ¼ An66 þ 2Bn66 χ xy ;
in which




1
Es As
A12
1
Er Ar
A11 þ

A22 þ
An11 ¼
; An22 ¼
; An12 ¼
;
Δ
Δ
ss
Δ
sr



1
Es As
Er Ar
A22 þ
−A212 :
; Δ ¼ A11 þ
An66 ¼
A66
ss
sr

ð12Þ


D. Huy Bich et al. / International Journal of Mechanical Sciences 74 (2013) 190–200

Bn11 ¼ An22 ðB11 þ C s Þ−An12 B12 ;

n

n

n

B12 ¼ A22 B12 −A12 ðB22 þ C r Þ;
B66
Bn66 ¼
;
A66

Bn22 ¼ An11 ðB22 þ C r Þ−An12 B12 ;
n

n

þ2

n

B21 ¼ A11 B12 −A12 ðB11 þ C s Þ;

193


 2  2

∂2 φ ∂2 w ∂2 w0
∂ φ ∂ w ∂2 w0

þ
− 2
þ
¼ 0:
2
2
∂x∂y ∂x∂y ∂x∂y
∂x
∂y
∂y

ð20Þ

ð13Þ

Eqs. (19) and (20) are a nonlinear equation system in terms of
two dependent unknowns w and φ. They are used to investigate
the dynamic characteristics of imperfect ES-FGM circular cylindrical shells.

ð14Þ

3. Nonlinear static and dynamic buckling analysis

Substituting Eq. (12) into Eq. (9) leads to
M x ¼ Bn11 Nx þ Bn21 N y −Dn11 χ x −Dn12 χ y ;
M y ¼ Bn12 Nx þ Bn22 N y −Dn21 χ x −Dn22 χ y ;
M xy ¼ Bn66 Nxy −2Dn66 χ xy ;
in which

Suppose that an imperfect ES-FGM cylindrical shell is simply

supported and subjected to axial compressive load r 0 ¼ r 0 h where
r 0 is the average axial stress on the shell's end sections, positive
when the shells subjected to axial compression (in N/m2). Thus,
the boundary conditions considered in the current study are

Es I s
−ðB11 þ C s ÞBn11 −B12 Bn21 ;
ss
Er I r
−B12 Bn12 −ðB22 þ C r ÞBn22 ;
Dn22 ¼ D22 þ
sr
Dn12 ¼ D12 −ðB11 þ C s ÞBn12 −B12 Bn22 ;

Dn11 ¼ D11 þ

w ¼ 0;

Dn21 ¼ D12 −B12 Bn11 −ðB22 þ C r ÞBn21 ;
n

n

D66 ¼ D66 −B66 B66 :

ð15Þ

The nonlinear equations of motion of a cylindrical thin shell based
on the classical shell theory and the assumption [12,14,26]
u o o w and v o o w, ρ1 ð∂2 u=∂t 2 Þ-0, ρ1 ð∂2 v=∂t 2 Þ-0 are given by

[4,14]
∂Nxy ∂N y
∂N x ∂N xy
þ
¼ 0;
þ
¼ 0;
∂x
∂y
∂x
∂y
 2

2
2
2
∂ M xy ∂ M y
∂ Mx
∂ w ∂2 w0
þ
þ2
þ Nx
þ
2
2
2
2
∂x∂y
∂x
∂x

∂y
∂x
 2

 2

∂ w ∂2 w0
∂ w ∂ 2 w0
1
∂2 w
þ
þ Ny
þ
þ N y ¼ ρ1 2 ;
þ2Nxy
2
2
∂x∂y ∂x∂y
R
∂y
∂y
∂t

h=2

þ ρr

Ar
;
sr

w ¼ f ðtÞ sin

ð17Þ

ρs ¼ ρm ; ρr ¼ ρm for metal stiffeners;
ρs ¼ ρc ; ρr ¼ ρc for ceramic stiffeners:
Considering the first two of Eq. (16), a stress function φ may be
defined as
∂ φ
;
∂y2

Ny ¼

∂ φ
;
∂x2

N xy ¼ −

∂ φ
:
∂x∂y
2

ð18Þ

Substituting Eq. (11) into the compatibility Eqs. (4) and (14) into
the third of Eq. (16), taking into account Eqs. (2) and (18)

neglecting small terms of higher second order with respect to
w0 , yields
∂4 φ
∂4 φ
∂4 φ
∂4 w
þ ðAn66 −2An12 Þ 2 2 þ An22 4 þ Bn21 4
4
∂x
∂x ∂y
∂y
∂x
4
∂4 w
1 ∂2 w
n
n
n
n ∂ w
þðB11 þ B22 −2B66 Þ 2 2 þ B12 4 þ
R ∂x2
∂x ∂y
∂y
"
#
2 2
2
2
2
2

∂ w
∂ w∂ w
∂ w ∂ w0
−
− 2
−2
∂x∂y
∂x∂y ∂x∂y
∂x ∂y2

ρ1

∂2 w ∂2 w0 ∂2 w ∂2 w0
þ 2
¼ 0;
∂x2 ∂y2
∂y ∂x2

ð22Þ

mπx
ny
sin
;
L
R

ð23Þ

where f 0 is the known imperfect amplitude.

Parameters

Present

R/h ¼500, L/R ¼2, c ¼ 100 MPa/s
k ¼0.2
194.94(2,11)
k ¼1.0
169.94(2,11)
k ¼5.0
149.98(2,11)
R/h ¼500, L/R ¼2, k ¼ 0.5
c¼ 100 MPa/s
181.68(2,11)
c¼ 50 MPa/s
179.38(2,11)
c¼ 10 MPa/s
177.02(2,11)
L/R ¼2, k¼ 0.2, c ¼ 100 MPa/s
R/h ¼ 800
124.67(2,12)
R/h ¼ 600
162.18(3,14)
R/h ¼ 400
239.56(5,15)

Huang and Han [19]
r dcr ðm; nÞ

τcr ¼

1.030
1.034
1.041

194.94(2,11)
169.94(2,11)
150.25(2,11)

1.030
1.034
1.040

1.032
1.019
1.006

181.67(2,11)
179.37(2,11)
177.97(1,8)

1.032
1.019
1.009

1.049
1.026
1.013

124.91(2,12)
162.25(3,14)
239.18(5,15)

1.051
1.027
1.011

τcr ¼

r dcr
r scr

r dcr
r scr

Table 2

An11

þ

ð21Þ

L:

mπx
ny
sin
;

L
R

r dcr ðm; nÞ

2

at x ¼ 0;

where f ðtÞ is the time dependent total amplitude, m is the number
of half waves in axial direction and n is the number of wave in
circumferential direction.
The initial-imperfection w0 is assumed to be the same form of
the deflection as

with

2

Nxy ¼ 0;

Table 1
Comparisons of dynamic critical buckling stress r dcr (MPa) and dynamic coefficient
τcr ¼ r dcr =r scr of perfect un-stiffened FGM cylindrical shells under linear-time
compression.



As
Ar

ρ −ρ
As
ρðzÞdz þ ρs þ ρr
¼ ρm þ c m h þ ρs
ss
sr
kþ1
ss
−h=2

Nx ¼

N x ¼ −r 0 h;

The deflection of shell is satisfying the mentioned condition (21) is
represented by

w0 ¼ f 0 sin
ð16Þ

where
Z
ρ1 ¼

M x ¼ 0;

Comparisons of static critical buckling load per unit length r scr ¼ r scr h (Â106 N/m)
of perfect stiffened homogeneous cylindrical shells under axial compression.
Present

Brush and Almroth [25]

Difference (%)

50 rings, 50 stringers, L ¼ 1 m, R¼ 0.5 m, E ¼ 7 Â 1010 N=m2 , υ ¼ 0:3,
dr ¼ ds ¼ 0:0025 m, hr ¼ hs ¼ 0:01 m

ð19Þ

∂2 w
∂4 w
∂4 w
∂4 w
∂4 φ
þ Dn11 4 þ ðDn12 þ Dn21 þ 4Dn66 Þ 2 2 þ Dn22 4 −Bn21 4
2
∂x
∂x ∂y
∂y
∂x
∂t


4
∂4 φ
1 ∂2 φ ∂2 φ ∂2 w ∂ 2 w0
n
n
n
n ∂ φ

−ðB11 þ B22 −2B66 Þ 2 2 −B12 4 −
−
þ
R ∂x2 ∂y2 ∂x2
∂x ∂y
∂y
∂x2

Internal stiffeners
R/h ¼ 100
3.0725(6,7)
R/h ¼ 200
1.4147(6,7)
R/h ¼ 500
0.6924(5,6)
External stiffeners
R/h ¼ 100
3,9529(9,3)
R/h ¼ 200
2.1410(9,4)
R/h ¼ 500
1.2764(6,6)

3.0906(6,7)
1.4328(6,7)
0.7057(5,6)

0.59
1.28
1.92

3.9551(9,2)
2.1469(9,4)
1.2897(6,6)

0.06
0.28
1.04


194

D. Huy Bich et al. / International Journal of Mechanical Sciences 74 (2013) 190–200

Substituting Eqs. (22) and (23) into Eq. (19) and solving
obtained equation for unknown φ lead to
φ ¼ φ1 cos

2mπx
2ny
mπx
ny
y2
þ φ2 cos
−φ3 sin
sin
−r 0 h ;
L
R
L

R
2

ð24Þ

þ Gf ðf þ f 0 Þðf þ 2f 0 Þ−L2 m2 π 2 hr 0 ðf þ f 0 Þ ¼ 0;

where denote
φ1 ¼

φ3 ¼

A ¼ An11 m4 π 4 þ ðAn66 −2An12 Þm2 n2 π 2 λ2 þ An22 n4 λ4 ;

m2 π 2
f ðf þ 2f 0 Þ;
32n2 λ2 An22
h
i
2
Bn21 m4 π 4 þ ðBn11 þ Bn22 −2Bn66 Þm2 n2 π 2 λ2 þ Bn12 n4 λ4 − LR m2 π 2

f ¼ f ðtÞ;

An11 m4 π 4 þ ðAn66 −2An12 Þm2 n2 π 2 λ2 þ An22 n4 λ4
λ¼

L
:
R

ð26Þ

where

n2 λ2
f ðf þ 2f 0 Þ;
32m2 π 2 An11

φ2 ¼

Galerkin method to the resulting equation yield
!
B2
4€
ρ1 L f þ D þ
f
A

L2 2 2
m π ;
R
D ¼ Dn11 m4 π 4 þ ðDn12 þ Dn21 þ 4Dn66 Þm2 n2 π 2 λ2 þ Dn22 n4 λ4 ;
!
n4 λ 4
m4 π 4
þ
G¼
:
ð27Þ

16An11 16An22
B ¼ Bn21 m4 π 4 þ ðBn11 þ Bn22 −2Bn66 Þm2 n2 π 2 λ2 þ Bn12 n4 λ4 −

f;

ð25Þ

Substituting the expressions (22)–(24) into Eq. (20) and applying

Fig. 2. Effect of k on the static post-buckling of un-stiffened shells.

Fig. 3. Effect of k on the static post-buckling of external ring and stringer stiffened
shells.

Fig. 4. Effect of k on the static post-buckling of internal ring and stringer stiffened
shells.

Introduce parameters
D¼

D
3

h

;

B¼

B

;
h

A ¼ Ah;

G¼

G
;
h

ξ¼

f
;
h

ξ0 ¼

f0
;
h

ð28Þ

Fig. 5. Dynamic responses of un-stiffened shell under step loading of infinite
duration.

Fig. 6. Dynamic response of external rings and stringers stiffened shell under step
loading of infinite duration.

Fig. 7. Dynamic response of internal rings and stringers stiffened shell under step
loading of infinite duration.


D. Huy Bich et al. / International Journal of Mechanical Sciences 74 (2013) 190–200

195

3.2. Dynamic buckling analysis
For dynamic buckling analysis, this paper investigates two
cases as following.

Fig. 8. Effect of k on the dynamic responses of un-stiffened shells under linear-time
compression.

Case 1. Consider a cylindrical shell subjected to the axial compression linearly varying on time r 0 ¼ ct in which c is a loading
speed. By using the Runge–Kutta method, the responses of ES-FGM
cylindrical shells can be determined from Eq. (29). The dynamic
critical time t cr can be obtained according to Budiansky–Roth
criterion [20]: for large value of loading speed, the amplitude–
time curve of obtained displacement response increases sharply
and this curve obtain a maximum by passing from the slope point
and at the corresponding time t ¼ t cr the stability loss occurs. Here,
t cr is called critical time and the corresponding dynamic critical
buckling stress r dcr ¼ ct cr and dynamic coefficient τcr ¼ r dcr =r scr .
Case 2. Assume that a shell is conducted for step loading of
infinite duration r 0 ¼ const; ∀t. The dynamic critical load is
found based on the criterion mentioned in [27]: the load corresponding to a sudden jump in the maximum average deflection in
the time history of the shell is taken as the critical buckling

step load.
4. Numerical results and discussions

Fig. 9. Effect of k on the dynamic responses of external ring and stringer stiffened
shells under linear-time compression.

the non-dimension form of Eq. (26) is written as
!
2
ρ1 L4 €
B
ξ
ξþ Dþ
3
A
h
 2
L
m2 π 2 ðξ þ ξ0 Þr 0 ¼ 0:
þ Gξðξ þ ξ0 Þðξ þ 2ξ0 Þ−
h

ð29Þ

3.1. Static buckling and post-buckling analysis
Omitting the term of inertia, Eq. (29) leads to
!
2
2
2

h
B
ξ
h
þ 2
Dþ
Gξðξ þ 2ξ0 Þ:
r0 ¼ 2
A ðξ þ ξ0 Þ L m2 π 2
L m2 π 2
Putting ξ0 ¼ 0 in Eq. (30), yields
!
2
2
2
h
B
h
þ 2
Dþ
Gξ2 :
r0 ¼ 2
2
2
A
L m π
L m2 π 2

ð30Þ

To validate the present formulation, two comparisons on
critical load are carried out with results from open literatures.
First, the dynamic buckling of perfect un-stiffened FGM cylindrical shells under linear-time compression is given in Table 1,
which was also analyzed by Huang and Han [19] using the energy
method and classical shell theory. As can be seen, the good
agreements are observed.
Second, the present static buckling load (Table 2) of stiffened
homogeneous cylindrical shells under axial compression is compared with the results in the monograph of Brush and Almroth
[25] (based on equations in page 180) where the smeared stiffeners technique, equilibrium path and classical shell theory are used.
This comparison once again also shows that the good agreements
are obtained.
To illustrate the proposed approach of eccentrically stiffened
FGM cylindrical shells, the stiffened and un-stiffened FGM cylindrical shells are considered with R ¼ 0:5 m, L ¼ 0:75 m, R=h ¼ 250.
The combination of materials consists of aluminum Em ¼ 7Â
1010 N/m2, ρm ¼ 2702 kg/m3 and alumina Ec ¼ 38 Â 1010 N/m2,
ρc ¼ 3800 kg/m3. The compressive stress of dynamic analysis is
taken to be r 0 ¼ 1010 t. Poisson's ratio ν is chosen to be 0.3 for
simplicity. The height of stiffeners is equal to 0:01 m, its width
0:0025 m. The material properties are Es ¼ Ec and Er ¼ Ec , ρs ¼ ρc
and ρr ¼ ρc with internal stringer stiffeners and internal ring

ð31Þ

From Eq. (31), by taking ξ ¼ 0 the buckling stress of perfect ESFGM cylindrical shells can be determined as
!
2
2
h
B
r sbu ¼ 2

Dþ
:
ð32Þ
A
L m2 π 2
The static critical buckling stress of perfect ES-FGM cylindrical
shells are determined by condition r scr ¼ minr sbu vs. ðm; nÞ and the
static post-buckling curves of perfect and imperfect shells may be
traced by using Eqs. (30) and (31) with the same buckling mode
shape of critical buckling stress for evaluate static behavior of
these structures.

Fig. 10. Effect of k on the dynamic responses of internal ring and stringer stiffened
shells under linear-time compression.


196

D. Huy Bich et al. / International Journal of Mechanical Sciences 74 (2013) 190–200

Table 3
Effect of k on critical static and dynamic buckling stress r 0 ( Â 108 N/m2).
0.2

k
Un-stiffened
Static
Dynamic r 0 ¼ const
Dynamic r 0 ¼ ct
τcr

External rings and stringers
Static
Dynamic r 0 ¼ const
Dynamic r 0 ¼ ct
τcr
Internal rings and stringers
Static
Dynamic r 0 ¼ const
Dynamic r 0 ¼ ct
τcr

1

7.743(12,10)
7.743(12,10)
8.002(12,10)
1.033

5

4.998(7,15)
4.998(7,15)
5.310(7,15)
1.062

10

2.985(13,6)
2.986(13,6)
3.185(13,6)

1.067

2.560(12,8)
2.560(12,8)
2.770(12,8)
1.082

17.814(5,10)
17.815(5,10)
18.315(5,10)
1.028

14.658(4,9)
14.658(4,9)
15.245(4,9)
1.040

10.926(4,8)
10.927(4,8)
11.570(4,8)
1.059

9.815(4,8)
9.816(4,8)
10.361(4,8)
1.056

26.660(3,7)
26.660(3,7)
27.466(3,7)

1.030

20.350(3,7)
20.350(3,7)
21.081(3,7)
1.036

13.181(3,6)
13.182(3,6)
13.912(3,6)
1.055

11.480(3,6)
11.480(3,6)
12.455(3,6)
1.085

Table 4
Effects of number, type and position of stiffeners on critical static and dynamic buckling stress r 0 ( Â 108 N/m2).
15 rings and 63 stringers
Static

Un-stiffened
ER
IR
ES
IS
IR and IS
ER and ES
IR and ES

ER and IS

4.998(7,15)
5.201(15,1)
5.184(13,8)
5.603(1,8)
5.222(1,8)
20.350(3,7)
14.658(4,9)
12.345(6,8)
16.448(2,8)

20 rings and 84 stringers
Dynamic

Static

r 0 ¼ const

r 0 ¼ ct

4.998(7,15)
5.201(15,1)
5.184(13,8)
5.604(1,8)
5.223(1,8)
20.350(3,7)
14.658(4,9)
12.345(6,8)
16.449(2,8)

5.310(7,15)
5.397(15,1)
5.378(14,8)
6.903(1,8)
6.785(1,8)
21.081(3,7)
15.245(4,9)
12.917(6,8)
17.305(2,8)

4.998(7,15)
5.261(15,1)
5.238(14,8)
5.779(1,8)
5.273(1,8)
22.351(3,6)
16.850(4,9)
13.099(5,7)
18.805(2,7)

Dynamic
r 0 ¼ const

r 0 ¼ ct

4.998(7,15)
5.261(15,1)
5.239(14,8)
5.779(1,8)

5.273(1,8)
22.352(3,6)
16.850(4,9)
13.099(5,7)
18.806(2,7)

5.310(7,15)
5.441(16,1)
5.447(14,8)
7.273(1,8)
6.670(1,8)
23.108(3,6)
17.414(4,9)
13.754(5,7)
19.727(2,7)

ER, external rings; IR, internal rings; ES, external stringers; IS, internal stringers.

Fig. 11. Effect of external ring and external stringer stiffeners on the static postbuckling curves.

Fig. 12. Effect of internal ring and internal stringer stiffeners on the static postbuckling curves.

stiffeners; Es ¼ Em , Er ¼ Em , ρs ¼ ρm and ρr ¼ ρm with external
stringer stiffeners and external ring stiffeners, respectively. The
stiffener system includes 15 ring stiffeners and 63 stringer stiffeners distributed regularly in the axial and circumferential directions, respectively.
In Figs. 2–4, the static post-buckling curves of un-stiffened and
stiffened shells are traced by Eqs. (30) and (31) of perfect (ξ0 ¼ 0)
and imperfect (ξ0 ¼ 0:1) cases versus three different values of
volume fraction index k (¼0.2, 1, 5). As can be seen, the postbuckling curves are lower with increasing values of k. Furthermore, the post-buckling curves of imperfect shells are lower than
those of perfect shells when deflection is small and post-buckling

curves of imperfect shells is higher than that of perfect shells
when the deflection is sufficiently large.

By using the fourth-order Runge–Kutta method, Eq. (29) is
solved to obtain the dynamic responses of perfect (ξ0 ¼ 0) shells
under step loading of infinite duration. Dynamic responses of unstiffened and stiffened shells are presented in Figs. 5–7. As can be
seen, there is a sudden jump in the value of the average deflection
when the axial compression reaches the critical value. In addition,
the dynamic critical step load corresponding to internal ring and
stinger stiffened shell is biggest. This value is bigger than about
1.4 times in comparison with the external ring and stinger
stiffened shell.
Figs. 8–10 show the effect of k on the dynamic responses of
perfect and imperfect un-stiffened and stiffened shells under
linear-time compression. These figures also show that there is no
definite point of instability as in static analysis. Rather, there is a


D. Huy Bich et al. / International Journal of Mechanical Sciences 74 (2013) 190–200

Fig. 13. Effect of internal stiffeners and external stiffeners on the static postbuckling curves.

Fig. 14. Effect of position of stiffeners on the static post-buckling of stiffened shells.

197

Fig. 17. Effect of internal stiffeners and external stiffeners on the dynamic
responses of shells under linear-time compression.

Fig. 18. Effect of stiffeners position on the dynamic responses of shells under

linear-time compression.

Table 5
Effect of R/h on critical static and dynamic buckling load per unit length r 0 ( Â 106
N/m).
R/h

Fig. 15. Effect of external ring and external stringer stiffeners on the dynamic
responses of shells under linear-time compression.

100

Un-stiffened
Static
6.247(6,9)
Dynamic r 0 ¼ const
6.247(6,9)
Dynamic r 0 ¼ ct
6.457(6,9)
External rings and stringers
Static
8.341(4,8)
Dynamic r 0 ¼ const
8.341(4,8)
Dynamic r 0 ¼ ct
8.607(4,8)
Internal rings and stringers
Static
9.964(3,7)
Dynamic r 0 ¼ const

9.964(3,7)
Dynamic r 0 ¼ ct
10.288(3,7)

250

500

1000

0.999(7,15)
0.999(7,15)
1.062(7,15)

0.250(9,21)
0.250(9,21)
0.277(9,21)

0.062(25,20)
0.062(25,20)
0.069(25,20)

2.932(4,9)
2.932(4,9)
3.049(4,9)

1.859(4,9)
1.859(4,9)
1.920(4,9)

1.292(4,8)
1.293(4,8)
1.326(4,8)

4.070(3,7)
4.070(3,7)
4.216(3,7)

2.380(3,6)
2.381(3,6)
2.457(3,6)

1.480(3,6)
1.480(3,6)
1.528(3,6)

Fig. 16. Effect of internal ring and internal stringer stiffeners on the dynamic
responses of shells under linear-time compression.

Fig. 19. Effect of R/h on the static post-buckling of un-stiffened shells.

region of instability where the slope of ξ vs. t curve increases
rapidly (in perfect shell cases). According to the Budiansky–Roth
criterion [20], the critical time t cr can be taken as an intermediate

value of this region. Therefore, one can choose the inflexion point
2
of curve i.e. d ξ=dt 2 jt ¼ t cr ¼ 0 as Huang and Han [19]. This region is
clearly recognized with perfect shells but it is very difficult to

198

D. Huy Bich et al. / International Journal of Mechanical Sciences 74 (2013) 190–200

Fig. 20. Effect of R/h on the static post-buckling of external stiffened shells.

Fig. 24. Effect of R/h on the dynamic responses of internal ring and stringer
stiffened shells under linear-time compression.

Fig. 21. Effect of R/h on the static post-buckling of internal stiffened shells.
Fig. 25. Effect of loading speed on the dynamic responses of un-stiffened shells.

Fig. 22. Effect of R/h on the dynamic responses of un-stiffened shells under lineartime compression.
Fig. 26. Effect of loading speed on the dynamic responses of internal stiffened
shells.

Fig. 23. Effect of R/h on the dynamic responses of internal ring and stringer
stiffened shells under linear-time compression.

define that with imperfect shells. Therefore, critical dynamic
buckling compressions of imperfect un-stiffened and stiffened
cylindrical shells cannot accurately predict by Budiansky–Roth
criterion (like a remark given by Huang and Han [19] for FGM

un-stiffened shells). This figure also shows that a sudden jump in
the value of deflection occurs earlier when k increases and it
corresponds a smaller dynamic buckling compression.
In the next figures, the dynamic response is traced by relation
of deflection ratio ξ versus excited load r 0 (where r 0 ¼ ct).

Table 3 shows the critical static and dynamic buckling stresses
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
input parameters, the effectiveness of stiffeners is obviously
proven; the critical buckling stress of stiffened shell is greater
than one of un-stiffened shell. Table 3 also shows that the dynamic
critical stress decreases with the increase of the volume fraction
index k and the buckling modes ðm; nÞ seem smaller with stiffened
shells. The critical parameter τcr is larger than 1, it denotes that the
dynamic critical buckling stress of linear-time compression case is
larger than static buckling stress The largest value of and τcr is
equal to 1.085 for the internal rings and stringers stiffened shell
with k ¼10 and the smallest τcr ¼ 1:028 corresponds to external


D. Huy Bich et al. / International Journal of Mechanical Sciences 74 (2013) 190–200

rings and stringers stiffened shell with k ¼0.2. In addition, when
the shell subjected to the step loading of infinite duration, it seems
that the dynamic critical buckling compression is approximately
equal to the static critical buckling compression.
Effect of the stiffener number, type and position of stiffeners on
the nonlinear critical buckling stress is given in Table 4. Clearly, the
ring or stringer stiffeners lightly influence to the critical buckling
stress of shells. But, the combination of ring and stringer stiffeners
has a considerable effect on the stability of shell. Especially, the
critical buckling stress of internal rings and stringers stiffened
shell is greatest and the critical buckling stress of internal rings
stiffened shell is smallest. When the number of stiffeners
increases, it is evident that critical buckling stresses increase.

Figs. 11–14 show the effect of type and position of stiffeners on
the static post-buckling of stiffened and un-stiffened shells (k ¼1,
15 rings and 63 stringers). According to the critical buckling
values, the post-buckling of un-stiffened shells is lower than one
of stiffened shells. For stiffened shells, the post-buckling of
internal rings stiffened shell is the lowest and one of internal
rings and stringers stiffened shell is the highest.
Figs. 15–18 show the effect of type and position of stiffeners on
the dynamic response of stiffened and un-stiffened shells under
linear-time compression (k¼ 1, 15 rings and 63 stringers). For one
type of stiffeners shells (Figs. 15 and 16), it seem that the
amplitude responses of perfect rings stiffened shells are smallest
and those of perfect stringers stiffened shells are the biggest.
In the results considered, the slope of instability region of stringer
stiffened shells is smaller.
The effects of R/h on the behavior of buckling loads per unit
length (r 0 ¼ r 0 h) are illustrated in Table 5. Clearly, the critical
buckling load decreases when the R/h ratio increases, the stiffeners
are more effective with thinner shells. When R/h ratio increases,
the critical buckling loads of un-stiffened shells strongly decrease
(about 100 times for variation of R/h from 100 to 1000) but lightly
with stiffened shells (about 6 times for variation of R/h from 100
to 1000).
Figs. 19–21 show the static post-buckling of un-stiffened and
stiffened shells. The post-buckling curves of shells are much
higher when R/h ratio decreases.
The dynamic responses of un-stiffened and stiffened shells
under linear-time compression are presented in Figs. 22–24.
As can be observed, maximal amplitude responses of instability
region increase when R/h ratio increases. These figures also show

that the slope of instability region of thinner shells is greater.
Effects of the loading speed on the dynamic responses of unstiffened and internal stiffened shells under linear-time compression are shown in Figs. 25 and 26. Three values of loading speed
are used, i.e. c ¼1010 , c ¼2 Â 1010 , c ¼5 Â 1010 . Clearly, the critical
dynamic buckling loads and amplitude response increase when
the loading speed increases. It mean that rapidly compressed
cylindrical shell will buckle at a higher critical stress than a very
slowly compressed cylindrical shell.

5. Conclusions
A formulation of governing equations of eccentrically stiffened
functionally graded circular cylindrical thin shells based on the
classical shell theory and the smeared stiffeners technique with
von Karman–Donnell nonlinear terms is presented in this paper.
By using the Galerkin method the explicit expressions of static
buckling compression, post-buckling load–deflection curve and
the nonlinear dynamic equation of ES-FGM circular cylindrical
shells are obtained, the later is solved by using the Runge–Kutta
method and the criteria for determining critical dynamic compressions are applied.

199

Some conclusions can be obtained from the present analysis:
(i) Stiffeners enhance the static and dynamic stability and loadcarrying capacity of FGM circular cylindrical shells.
(ii) Combination of ring and stringer stiffeners has a large effect
on the stability of shell. The critical buckling compression of
internal rings and stringers stiffened shell is greatest.
(iii) In dynamic linear-time load case, there is no definite point of
instability as in static analysis. Rather, there is a region of
instability where the slope of ξ vs. t curve increases rapidly in
perfect shell cases.

(iv) For imperfect FGM shell, it is difficult to accurately predict the
critical buckling compression.
(v) The dynamic critical buckling compressions of linear-time
compression case is larger than static critical buckling compressions (τcr is about 1.028–1.085) and the dynamic critical
buckling compression of step loading of infinite duration
is approximately equal to the static critical buckling
compression.
(vi) Initial geometrical imperfection, radius-to-thickness ratio,
position, type and number of stiffeners significantly influence
on the static and dynamic behavior of cylindrical shell.
Major purpose of this study is to analyze the global buckling
and post-buckling behavior of FGM cylindrical shells reinforced by
closely spaced stiffener system. For local buckling analysis, the
approach of Stamatelos et al. [28] may be used.

Acknowledgments
This research is funded by Vietnam National Foundation for
Science and Technology Development (NAFOSTED) under grant
number 107.01-2012.02.
References
[1] Shen HS. Postbuckling analysis of pressure-loaded functionally graded cylindrical shells in thermal environments. Eng Struct 2003;25(4):487–97.
[2] Shen HS. Postbuckling of axially-loaded FGM hybrid cylindrical shells in
thermal environments. Compos Sci Technol 2005;65(11–12):1675–90.
[3] Bahtui A, Eslami MR. Coupled thermoelasticity of functionally graded cylindrical shells. Mech Res Commun 2007;34(1):1–18.
[4] Huang H, Han Q. Buckling of imperfect functionally graded cylindrical shells
under axial compression. Eur J Mech—A/Solids 2008;27(6):1026–36.
[5] Huang H, Han Q. Nonlinear elastic buckling and postbuckling of axially
compressed functionally graded cylindrical shells. Int J Mech Sci 2009;51
(7):500–7.
[6] Huang H, Han Q. Nonlinear buckling and postbuckling of heated functionally

graded cylindrical shells under combined axial compression and radial
pressure. Int J Non-Linear Mech 2009;44(2):209–18.
[7] Huang H, Han Q. Research on nonlinear postbuckling of FGM cylindrical shells
under radial loads. Compos Struct 2010;92(6):1352–7.
[8] Shen HS. Postbuckling of shear deformable FGM cylindrical shells surrounded
by an elastic medium. Int J Mech Sci 2009;51(5):372–83.
[9] Sofiyev AH. Buckling analysis of FGM circular shells under combined loads and
resting on the Pasternak type elastic foundation. Mech Res Commun 2010;37
(6):539–44.
[10] Zozulya VV, Zhang Ch. A high order theory for functionally graded axisymmetric cylindrical shells. Int J Mech Sci 2012;60(1):12–22.
[11] Ng TY, Lam KY, Liew KM, Reddy JN. Dynamic stability analysis of functionally
graded cylindrical shells under periodic axial loading. Int J Solids Struct
2001;38(8):1295–309.
[12] Darabi M, Darvizeh M, Darvizeh A. Non-linear analysis of dynamic stability for
functionally graded cylindrical shells under periodic axial loading. Compos
Struct 2008;83(2):201–11.
[13] Chen WQ, Bian ZG, Ding HJ. Three-dimensional vibration analysis of fluidfilled orthotropic FGM cylindrical shells. Int J Mech Sci 2004;46(1):159–71.
[14] Sofiyev AH, Schnack E. The stability of functionally graded cylindrical shells
under linearly increasing dynamic torsional loading. Eng Struct 2004;26
(10):1321–31.
[15] Sofiyev AH. The stability of compositionally graded ceramic–metal cylindrical
shells under aperiodic axial impulsive loading. Compos Struct 2005;69
(2):247–57.


200

D. Huy Bich et al. / International Journal of Mechanical Sciences 74 (2013) 190–200

[16] Shariyat M. Dynamic thermal buckling of suddenly heated temperaturedependent FGM cylindrical shells, under combined axial compression and

external pressure. Int J Solids Struct 2008;45(9):2598–612.
[17] Shariyat M. Dynamic buckling of suddenly loaded imperfect hybrid FGM
cylindrical shells with temperature-dependent material properties under
thermo-electro-mechanical loads. Int J Mech Sci 2008;50(12):1561–71.
[18] Li SR, Fu XH, Batra RC. Free vibration of three-layer circular cylindrical shells
with functionally graded middle layer. Mech Res Commun 2010;37(6):577–80.
[19] 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.
[20] Budiansky B, Roth RS. Axisymmetric dynamic buckling of clamped shallow
spherical shells. NASA technical note; 1962; D_510:597–609.
[21] Shariyat M. Nonlinear transient stress and wave propagation analyses of the
FGM thick cylinders, employing a unified generalized thermoelasticity theory.
Int J Mech Sci 2012;65(1):24–37.
[22] Najafizadeh MM, Hasani A, Khazaeinejad P. Mechanical stability of functionally graded stiffened cylindrical shells. Appl Math Model 2009;54(2):1151–7.

[23] Bich DH, Nam VH, Phuong NT. Nonlinear postbuckling of eccentrically
stiffened functionally graded plates and shallow shells. Vietnam J Mech
2011;33(3):132–47.
[24] Bich DH, Dung DV, Nam VH. Nonlinear dynamical analysis of eccentrically
stiffened functionally graded cylindrical panels. Compos Struct 2012;94
(8):2465–73.
[25] Brush DO, Almroth BO. Buckling of bars, plates and shells; 1975. Mc Graw-Hill.
[26] Volmir AS. Non-linear dynamics of plates and shells. Science Edition M; 1972
[in Russian].
[27] Ganapathi M. Dynamic stability characteristics of functionally graded materials shallow spherical shells. Compos struct 2007;79:338–43.
[28] Stamatelos DG, Labeas GN, Tserpes KI. Analytical calculation of local buckling
and post-buckling behavior of isotropic and orthotropic stiffened panels. ThinWalled Struct 2011;49:422–30.
[29] Reddy JN, Starnes JH. General buckling of stiffened circular cylindrical shells
according to a Layerwise theory. Comput & Struct 1993;49:605–16.