DSpace at VNU: Nonlinear static and dynamic buckling analysis of functionally graded shallow spherical shells including temperature effects

Composite Structures 94 (2012) 2952–2960

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

Nonlinear static and dynamic buckling analysis of functionally graded
shallow spherical shells including temperature effects
Dao Huy Bich, Dao Van Dung, Le Kha Hoa ⇑
Vietnam National University, Hanoi, Viet Nam

a r t i c l e

i n f o

Article history:
Available online 24 April 2012
Keywords:
Functionally graded materials
Static and dynamic buckling
Shallow spherical shells

a b s t r a c t
This paper presents an analytical approach to investigate the nonlinear static and dynamic unsymmetrical responses of functionally graded shallow spherical shells under external pressure incorporating the
effects of temperature. Governing equations for thin FGM spherical shells are derived by using the
classical shell theory taking into account von Karman–Donnell geometrical nonlinearity. Approximate
solutions are assumed and Galerkin procedure is applied to determine explicit expressions of static critical buckling loads of the shells. For the dynamical response, motion equations are numerically solved by
using Runge–Kutta method and the criterion suggested by Budiansky–Roth. A detailed analysis is carried
out to show the effects of material and geometrical parameters, boundary conditions and temperature on
the stability and dynamical characteristics of FGM shallow spherical shells.

Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction
Structures in the form of spherical shells are used widely in many
engineering applications. Most of these shells are subjected to static
and impulsive loads which is cause of instability and strength reduction of the structures. As a result, the investigation on the nonlinear
static and dynamical buckling of spherical shells is necessary and
has attracted attention of many researchers. Budiansky and Roth
[1] studied axisymmetrical dynamic buckling of clamped shallow
isotropic spherical shells. Their well-known results have received
considerable attention in the literature. Huang [2] considered the
unsymmetrical buckling of thin shallow spherical shells under
external pressure. He pointed out that unsymmetrical deformation
may be the source of discrepancy in critical pressures between axisymmetrical buckling theory and experiment. The static buckling
behavior of shallow spherical caps under an uniform pressure loads
was analyzed by Tillman [3]. Results on the dynamic buckling of
clamped shallow spherical shells subjected to axisymmetric and
nearly axisymmetric step-pressure loads using a digital computer
program were given by Ball and Burt [4]. Kao and Perrone [5] reported the dynamic buckling of isotropic axisymmetrical spherical
caps with initial imperfection. Two types of loading are considered,
in this paper, namely, step loading with infinite duration and right
triangular pulse. Wunderlich and Albertin [6] also studied on the
static buckling behavior of isotropic imperfect spherical shells.
New design rules in their work for these shells were developed,
which take into account relevant details like boundary conditions,
⇑ Corresponding author.
E-mail address: (L.K. Hoa).
0263-8223/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved.
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material properties and imperfections. The nonlinear static and dynamic response of spherical shells has been analyzed by Nath and

Alwar [7] using Chebyshev series expansion. Based on an assumed
two-term mode shape for the lateral displacement, Ganapathi and
Varadan [8] investigated the problem of dynamic buckling of orthotropic shallow spherical shells under instantaneously applied uniform step-pressure load of infinite duration. The same authors
analyzed the dynamical buckling of laminated anisotropic spherical
caps using the finite element method [9]. Static and dynamic snapthrough buckling of orthotropic spherical caps based on the classical
thin shell theory and Reissener’s shallow shell assumptions have
been considered by Chao and Lin [10] using finite difference method.
There were several investigations on the buckling of spherical shells
under mechanical or thermal loading taking into account initial
imperfection such as studies by Eslami et al. [11] and Shahsiah and
Eslami [12].
Functionally graded materials (FGMs) which are microscopically
composites and composed of ceramic and metal constituents or
combination of metals have received much interest in recent years.
Due to essential characteristics such as high stiffness, excellent temperature resistance capacity, functionally graded materials find
wide applications in many industries, especially in temperature
shielding structures and nuclear plants. Shahsiah et al. [13]
presented an analytical approach to study the instability of FGM
shallow spherical shells under three types of thermal loading
including uniform temperature rise, linear radial temperature, and
nonlinear radial temperature. Prakash et al. [14] gave results on
the nonlinear axisymmetric dynamic buckling behavior of clamped
FGM spherical caps. Also, the dynamic stability characteristics of
FGM shallow spherical shells have been considered by Ganapathi


2953

D.H. Bich et al. / Composite Structures 94 (2012) 2952–2960

[15] using the finite element method. In his study, the geometric
nonlinearity is assumed only in the meridional direction in strain–
displacement relations. Bich [16] studied the nonlinear buckling
of FGM shallow spherical shells using an analytical approach and
the geometrical nonlinearity was considered in all strain–displacement relations. By using Galerkin procedure and Runge–Kutta
method, Bich and Hoa [17] analyzed the nonlinear vibration of
FGM shallow spherical shells subjected to harmonic uniform external pressures. Recently, Bich and Tung [18] reported an analytical
investigation on the nonlinear axisymmetrical response of FGM
shallow spherical shells under uniform external pressure taking
the effects of temperature conditions into consideration. Shahsiah
et al. [19] used an analytical approach to investigate thermal linear
instability of FGM deep spherical shells under three types of thermal
loads using the first order shell theory basing Sander nonlinear kinematic relations. To best of authors’ knowledge, there is no analytical
investigation on the nonlinear dynamic stability of FGM shallow
spherical shells.
In the present paper, the nonlinear static and dynamical buckling behavior of clamped FGM shallow spherical shells under uniform external pressure and thermal loads are considered by using
an analytical approach. Governing equations for thin shallow
spherical shells are derived by using the classical shell theory taking into account von Karman–Donnell nonlinear terms. Material
properties are assumed to be temperature independent and
graded in the thickness direction according to a simple power
law function. Approximate one-term solutions of deflection and
stress function that satisfy the boundary conditions are assumed
and Galerkin procedure is used to obtain explicit expressions of
static critical buckling loads. For dynamical analysis, motion equation is solved numerically by applying Runge–Kutta method and
the criterion suggested by Budiansky–Roth. The effects of material
and geometrical properties, temperature and boundary conditions
on the response of FGM spherical shells are analyzed and
discussed.

2. Theoretical formulations


k

k
2z þ h
2z þ h
; qðzÞ ¼ qm þ ðqc À qm Þ
;
2h
2h

k

k
2z þ h
2z þ h
aðzÞ ¼ am þ ðac À am Þ
; KðzÞ ¼ K m þ ðK c À K m Þ
;
2h
2h
EðzÞ ¼ Em þ ðEc À Em Þ

mðzÞ ¼ m ¼ const;
ð1Þ
where k P 0 is volume fraction index and E, m, q, a, K are Young’s
modulus, Poisson’s ratio, mass density, coefficient of thermal
expansion, coefficient of thermal conduction, respectively, and subscripts m and c stand for the metal and ceramic constituents,
respectively.

2.2. Governing equations
It is convenient to introduce an additional variable r defined by
the relation r = Rsin u, where r is the radius of the parallel circle
with the base of shell. If the rise H of the shell is much smaller than
the base radius r0 we can take cos u % 1 and Rdu = dr, such that
points of the middle surface may be referred to coordinates r and h.
The strain components on the middle surface of shell based
upon the von Karman assumption are of the form

e0r ¼ u;r À w=R þ w2;r =2; e0h ¼ ðv;h þ uÞ=r À w=R þ w2;h =ð2r2 Þ;
c0rh ¼ rðv=rÞ;r þ u;h =r þ w;r w;h =r;

ð2Þ

vr ¼ w;rr ; vh ¼ w;hh =ðr2 Þ þ w;r =r; vrh ¼ w;rh =r À w;h =ðr2 Þ;

ð3Þ

where u, v and w are the displacements of the middle surface points
along meridional, circumferential and radial directions, respectively, and vr, vh, vrh are the change of curvatures and twist,
respectively.
Using Eqs. (2) and (3), the geometrical compatibility equation is
written as

Á
1 0
1
1
1À
1

e À e0 þ ðr2 e0h;r Þ;r À 2 rc0rh ;rh ¼ À r2 w þ v2rh À vr vh ;
r2 r;hh r r;r r2
r
R
2

ð4Þ

2

@
1 @
1 @
where r ¼ @r
is a Laplace’s operator.
2 þ r @r þ r 2
@h2
The strains across the shell thickness at a distance z from the
mid-plane are given by

2.1. Functionally graded shallow spherical shells
Consider a clamped FGM shallow spherical shell of thickness h,
base radius r0, curvature radius R, rise H as shown in Fig. 1. It is
defined in coordinate system (u, h, z), where u and h are in the
meridional and circumferential directions of the shell, respectively,
and z is perpendicular to the middle surface positive inward. Assume that the shell is made from a mixture of ceramic and metal
constituents and the effective material properties vary continuously along the thickness by the power law distribution.

er ¼ e0r À zvr ; eh ¼ e0h À zvh ; crh ¼ c0rh À 2zvrh :

ð5Þ

The stress–strain relationships including temperature effect for
an FGM spherical shell are defined by the Hooke law

EðzÞ
½ðer ; eh Þ þ mðeh ; er Þ À ð1 þ mÞa Á DT Á ð1; 1ފ;
1 À m2
EðzÞ
¼
c ;
2ð1 þ mÞ rh

ðrr ; rh Þ ¼

rrh

ð6Þ

where DT is temperature change from stress free initial state.
The force and moment resultants of an FGM shallow spherical
are expressed in terms of the stress components through the thickness as

H

r

ϕ ,u

fðNr ; Nh ; Nrh Þ;

r0

R

Fig. 1. Geometry and coordinate system of a spherical cap.

Z

h=2

frr ; rh ; rrh gð1; zÞdz:

ð7Þ

Àh=2

θ ,v
z, w

ðM r ; Mh ; M rh Þg ¼

Introduction of Eqs. (1), (5) and (6) into Eq. (7) gives the
constitutive relations

Á
E1 À 0
E
/
er þ me0h À 2 2 ðvr þ mvh Þ À m ;

2
1Àm
1Àm
1Àm
Á
E1 À 0
E
/
Nh ¼
e þ me0r À 2 2 ðvh þ mvr Þ À m ;
1 À m2 h
1Àm
1Àm
E1
E
c0 À 2 v ;
Nrh ¼
2ð1 þ mÞ rh 1 þ m rh

Nr ¼

ð8Þ


2954

D.H. Bich et al. / Composite Structures 94 (2012) 2952–2960

Á
E2 À 0

E
/
er þ me0h À 3 2 ðvr þ mvh Þ À b ;
2
1Àm
1Àm
1Àm
Á
E2 À 0
E
/
Mh ¼
e þ me0r À 3 2 ðvh þ mvr Þ À b ;
1 À m2 h
1Àm
1Àm
E2
E
c0 À 3 v ;
M rh ¼
2ð1 þ mÞ rh 1 þ m rh

3. Boundary conditions and solution of problem

Mr ¼

ð9Þ

The FGM shallow spherical shell is assumed to be clamped at its
base edge and subjected to external pressure uniformly distributed

on the outer surface of shell. Depending on the in-plane behavior at
the edge, two cases of boundary conditions will be considered.



Ecm h
1
1
2
;
; E2 ¼ Ecm h
À
E1 ¼ Em h þ
kþ1
k þ 2 2k þ 2


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


k #"

k #
Z h=2 "
2z þ h
2z þ h
ð/m ;/b Þ ¼
Em þ Ecm
am þ acm
DTð1;zÞdz;
2h
2h
Àh=2

Case (i). The base edge of shell is clamped and freely movable
(FM) in the meridional direction. The associated boundary conditions are

where

Ecm ¼ Ec À Em ; acm ¼ ac À am :
ð10Þ
The nonlinear equations of motion of perfect shallow spherical
shell according to the classical shell theory are [20]

1
1
1
ðrNr Þ;r þ Nrh;h À Nh ¼ q1 u;tt ;
r

r
r
1
1
1
ðrNrh Þ;r þ Nh;h þ Nrh ¼ q1 v ;tt ;
r
r
r


!
1
1
1
1
ðrM r Þ;rr þ 2 Mrh;rh þ M rh;h þ Mh;hh À M h;r þ ðNr þ N h Þ
r
r
r
R


1
1
1
Nrh w;r þ Nh w;h
þ q ¼ q1 w;tt ; ð11Þ
þ ðrNr w;r þ Nrh w;h Þ;r þ
r

r
r
;h
where q is an uniform external pressure acting on the shell outer
surface positive inward, and

q1 ¼ qm h þ

qcm h
;
kþ1

qcm ¼ qc À qm :

ð12Þ

By taking the inertia forces q1u,tt ? 0 andq1v,tt ? 0 into consideration because of u ( w, v ( w [21], two first of Eq. (11) are satisfied by introducing the stress function f

1
1
Nr ¼ f;r þ 2 f;hh ;
r
r

Nh ¼ f;rr ;

1
1
Nrh ¼ 2 f;h À f;rh ;
r

r

ð13Þ

and substituting relations (2), (3), (9) and (13) into third of Eq. (11)
gives





1
1
1
1
1
q1 w;tt þ Dr4 w À r2 f À f;r þ 2 f;hh w;rr À w;r þ 2 w;hh f;rr
R
r
r
r
r



1
1
1
1
ð14Þ

w;rh À 2 w;h À q ¼ 0;
þ 2 f;rh À 2 f;h
r
r
r
r
E E ÀE2

1 3
2
where D ¼ ð1À
m2 ÞE1 .
Eq. (14) includes two unknown functions w and f and to find a
second equation relating two these functions the geometrical compatibility Eq. (4) is used. For this aim, from Eq. (8) strain components can be expressed through force resultants as

Ã
1Â
ðNr ; Nh Þ À mðNh ; Nr Þ þ E2 ðvr ; vh Þ þ /m ð1; 1Þ ;
E1
Ã
2Â
c0rh ¼
ð1 þ mÞNrh þ E2 vrh :
E1
À

Nr ¼ 0;

Nrh ¼ 0 at r ¼ r 0 ;

ð17Þ

Case (ii). The base edge of shell is clamped and immovable (IM).
For this case, the boundary conditions are

u ¼ 0;

w ¼ w;r ¼ 0;

Nr ¼ N0 ;

Nrh ¼ 0 at r ¼ r 0 ;

ð18Þ

where N0 is fictitious compressive edge load at immovable edge.
The mentioned boundary conditions can be satisfied, when the
deflection w is represented by a single term of a Fourier series. This
approximate solution is acceptable in the vicinity of the buckling
load [22,23].

w¼W

16r2 ðr 0 À rÞ2
sin nh;
r40

ð19Þ

where W = W(t) is a time dependent total amplitude of deflection of

shell.
Regularly, the stress function f should be determined by the
substitution of deflection function w into compatibility equation
(16) and solving the resulting equation. However, such a procedure
is very complicated in mathematical treatment because obtained
equation is a variable coefficient partial differential equation.
Accordingly, integration to obtain exact stress function f(r, h) is extremely complex. Therefore, the stress function f satisfying boundary conditions (18) is chosen in the same form of deflection w as
mentioned in Refs. [24,25]

f ¼F

16r2 ðr 0 À rÞ2
r2
sin nh þ N0 :
2
r40

ð20Þ

Substituting Eqs. (19) and (20) into Eqs. (14) and (16) and
applying Galerkin procedure for the resulting equations yield

"
#
2E1
Wr 20
32n 2
2
F¼
W ;

ð3 þ 2n Þ À
7ð38 À 20n2 þ 12n4 Þ R
p
7Dð38 À 20n2 þ 12n4 Þ
ð3 þ 2n2 Þ
32nWF
Wþ
FÀ
4
pr40
4r 0
2Rr20


3 þ 2n2
7
7q
þ N0
WÀ
¼
:
4pRn
8pn
2r 20

ð21Þ

q1 W ;tt þ

ð22Þ

Eliminating F from Eqs. (21) and (22) leads to

Á

e0r ; e0h ¼

ð15Þ

q1 W ;tt

Substituting these equations into Eq. (4), taking into account
relations (3) and (13) leads to


2


1 4 1 2
1
1
1
1
r f þ r w À w;rh À 2 w;h þ w;rr 2 w;hh þ w;r ¼ 0:
E1
R
r
r
r
r

w ¼ w;r ¼ 0;

ð16Þ

Eqs. (14) and (16) are governing equations used to investigate
the nonlinear static and dynamic buckling of FGM shallow spherical shells.

À

"
#
7Dð38 À 20n2 þ 12n4 Þ
ð3 þ 2n2 Þ2 E1
W
þ
þ 2
4r40
7R ð38 À 20n2 þ 12n4 Þ
96E1 ð3 þ 2n2 Þn

W2 þ

7p
7p
À
þ


3 þ 2n2

7
7q
þ N0
WÀ
¼
:
4pRn
8pn
2r 20
Rr20 ð38

20n2

12n4 Þ

2048E1 n2
W3
À 20n2 þ 12n4 Þ

2 r 4 ð38
0

ð23Þ

Based on this equation, the mechanical and thermal stability
analysis of shells are considered below.


D.H. Bich et al. / Composite Structures 94 (2012) 2952–2960

4. Mechanical stability analysis
Consider an FGM shallow spherical shell being clamped and
freely movable at the edge r = r0 (case (i)). The outer surface of shell
is subjected to uniform external pressure q and without the effects
of temperature. In this case N0 = 0 and Eq. (23) reduces to

"
#
7Dð38 À 20n2 þ 12n4 Þ
ð3 þ 2n2 Þ2 E1
W
q1 W ;tt þ
þ
4r 40
7R2 ð38 À 20n2 þ 12n4 Þ
96E1 ð3 þ 2n2 Þn

À

Rr 20 ð38

7p
7q
¼
:
8pn

À

20n2

12n4 Þ

þ

W2 þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2  2ffi
1 49 Ã 38 À 20n2 þ 12n4
1 h
;
À
D
X1 ¼ 1 À
3 6
3 þ 2n2
n4 R
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2  2ffi
1 49 Ã 38 À 20n2 þ 12n4
1 h
X2 ¼ 1 þ
:
À
D
3 6
3 þ 2n2

n4 R

ð24Þ

By putting
2

EÃ2 ¼ E2 =h ;

n ¼ r 0 =R;

Ã

3

EÃ3 ¼ E3 =h ;

2

W Ã ¼ W=h;

Ã

q ¼ q1 =h:

D ¼ D=ðE1 h Þ;

ð25Þ

Eq. (24) is rewritten as

("
 4
EÃ1
7DÃ ð38 À 20n2 þ 12n4 Þ2 h
R
ð38 À 20n2 þ 12n4 Þ
4n4
#



3
2
2 2
2
ð3 þ 2n Þ h
96ð3 þ 2n Þn h
WÃ À
W Ã2
þ
R
R
7
7pn2
)
 4
2048n2 h
7q
Ã3

¼
þ
W
:
ð26Þ
8p n
7p2 n4 R

qà h2 W Ã;tt þ

Note that quantities qupper and qlower in Eq. (28) depend on the
buckling mode n and the minimum values of buckling loads i.e.
critical upper and lower buckling loads may be obtained by minimization of these loads with respect to n.

An FGM shallow spherical shells subjected to external pressure
loads varying as linear functions of time, q = st(s – a loading speed),
is considered. The aim of the problem is to determine the critical
dynamic buckling loads. In this case of load, Eq. (26) is rewritten as
("
 4
EÃ1
7DÃ ð38 À 20n2 þ 12n4 Þ2 h
qà h2 W Ã;tt þ
4
2
4
R
ð38 À 20n þ 12n Þ
4n
)

 2 #
 3
 4
2 2
2
2
ð3 þ 2n Þ h
96ð3 þ 2n Þn h
2048n h
Ã2
Ã3
W
þ
W
WÃ À
þ
R
R
7
7p2 n4 R
7pn2

À

7st
¼ 0:
8pn

4.1. Static buckling and postbuckling analysis

32ð38 À 20n2 þ 12n4 Þ

"
DÃ



38 À 20n2 þ 12n4
3 þ 2n2
#

5. Thermomechanical stability analysis

2  3
h X1
R n2

 

2n2 h 
2X1 À 3X21 þ X31 ;
49 R
" 
  3
2
4 2
p2 EÃ1 ð3 þ 2n2 Þ3
h X2
à 38 À 20n þ 12n
qlower ¼

D
R n2
32ð38 À 20n2 þ 12n4 Þ
3 þ 2n2
#
 

2n2 h 
2
3
þ
2X2 À 3X2 þ X2 ;
49 R

ð32Þ

ð27Þ

Eq. (27) may be used to find static critical buckling load and
trace postbuckling load–deflection curves of FGM spherical shells.
It is evident that q(W⁄) curves originate from the coordinate origin.
Eq. (27) indicates that there is no bifurcation-type buckling for
pressure loaded spherical shells and extremum-type buckling only
occurs under definite conditions.
The extremum buckling load of the shell can be found from Eq.
(27) using the condition dq/dW⁄ = 0 which give

p2 EÃ1 ð3 þ 2n2 Þ3

Ã

dW =dtjt¼0 ¼ 0:

The well-known criterion suggested by Budiansky and Roth [1]
is employed herein. According to this criterion, for large values of
loading speed, the average deflection–time curve (W⁄ À t) of obtained displacement response increases sharply depending on time
and this curve obtains a maximum by passing from the slope point,
and at the corresponding time t = tcr the stability loss occurs. The
value t = tcr is called critical time and the load corresponding to this
critical time is called dynamic critical buckling load. To obtain displacement responses, Eq. (31) in conjunction with initial conditions (32) will be solved by using the Runge–Kutta method.

Omitting the term of inertia force in Eq. (26) yields

("
 4
8pnEÃ1
7DÃ ð38 À 20n2 þ 12n4 Þ2 h
q¼
R
7ð38 À 20n2 þ 12n4 Þ
4n4
#




2
3
2
ð3 þ 2n2 Þ h

96ð3 þ 2n2 Þn h
WÃ À
W Ã2
þ
R
R
7
7pn2
)
 4
2048n2 h
Ã3
þ
W
:
7p2 n4 R

ð31Þ

In the present study, initial conditions are assumed as

W Ã jt¼0 ¼ W Ã0 ;

A clamped FGM shallow spherical shell with immovable edge
(case (ii)) subjected simultaneously to uniform external pressure
q and thermal load is considered. The condition expressing the
immovability of the boundary edge, i.e. u = 0 at r = r0, is fulfilled
on the average sense as

Z pZ

0

r0
0

@u
rdrdh ¼ 0:
@r

ð33Þ

From Eqs. (2) and (15) one can obtain the expression of @u/@r
and then substituting the result into Eq. (33) gives

þ

where

ð30Þ

4.2. Dynamic buckling analysis

EÃ1 ¼ E1 =h;

qupper ¼

ð29Þ

Providing is


2  2
1 49 Ã 38 À 20n2 þ 12n4
1 h
À
D
> 0:
3 6
3 þ 2n2
n4 R

2048E1 n2
W3
7p2 r 40 ð38 À 20n2 þ 12n4 Þ

2955

N0 ¼
ð28Þ

16Fn
16E1 W
128E1 W 2
/m
À
À
þ
:
2
3pr0 ð1 À mÞ 15pRnð1 À mÞ 105r 20 ð1 À mÞ ð1 À mÞ

ð34Þ

Introduction F from Eq. (21) into Eq. (34) leads to

16E1 ð266 À 170n2 þ 64n4 Þ
W
105pRnð1 À mÞð38 À 20n2 þ 12n4 Þ
 2
Ã
128E1 p ð38 À 20n2 þ 12n4 Þ À 40n2
/m
W2 À
:
þ
ð1 À mÞ
105p2 r20 ð1 À mÞð38 À 20n2 þ 12n4 Þ

N0 ¼ À

ð35Þ


2956

D.H. Bich et al. / Composite Structures 94 (2012) 2952–2960

5.2. Dynamic stability analysis

Substituting this relation N0 into Eq. (23) yields

("
 4
7DÃ ð38 À 20n2 þ 12n4 Þ2 h
q
R
4n4
!  2 #
2
ð3 þ 2n2 Þ 28ð266 À 170n2 þ 64n4 Þ
h
WÃ
þ
þ
105p2 n2 ð1 À mÞ
R
7
"
96ð3 þ 2n2 Þn 8ð3 þ 2n2 Þð266 À 170n2 þ 64n4 Þ
À
þ
7pn2
105pnð1 À mÞn2
 2
Ã# 3
224 p ð38 À 20n2 þ 12n4 Þ À 40n2
h
W Ã2
þ
R

105p3 nð1 À mÞn2
)
"
Â
Ã# 4
2048n2 64ð3 þ 2n2 Þ p2 ð38 À 20n2 þ 12n4 Þ À 40n2
h
Ã3
þ
þ
W
R
7p 2 n 4
105p2 n4 ð1 À mÞ
"


 #
2
2
/m
ð3 þ 2n Þ h
7 h
7q
WÃ À
À
:
ð36Þ
¼
R

4pn R
8p n
ð1 À mÞh
2n2
EÃ1
à 2
h W Ã;tt þ
ð38 À 20n2 þ 12n4 Þ

Eq. (36) is employed to investigate static and dynamic unsymmetric responses of FGM shallow spherical shells under combined
mechanical and thermal loads.
5.1. Static stability analysis
Environment temperature is uniformly raised from initial value
Ti, at which the shell is thermal stress free, to final one Tf and temperature change DT = Tf À Ti is independent to thickness variable.
The thermal parameter /m can be expressed in terms of DT due
to Eq. (10) as

/m ¼ Um0 DTh;

ð37Þ

where

Um0 ¼ Em am þ

Em acm þ Ecm am Ecm acm
þ
:
kþ1
2k þ 1

ð38Þ

Substituting /m from Eq. (37) into Eq. (36) and neglecting the
2
inertia force, i.e. qà h W Ã;tt ¼ 0, yields
("

 4
7DÃ ð38 À 20n2 þ 12n4 Þ2 h
4
R
4n
!  #
2
2 2
2
4
ð3 þ 2n Þ 28ð266 À 170n þ 64n Þ
h
þ
þ
WÃ
105p2 n2 ð1 À mÞ
R
7
"
96ð3 þ 2n2 Þn 8ð3 þ 2n2 Þð266 À 170n2 þ 64n4 Þ
À
þ

7 p n2
105pnð1 À mÞn2
 2
à #  3
224 p ð38 À 20n2 þ 12n4 Þ À 40n2
h
þ
W Ã2
R
105p3 nð1 À mÞn2
)
"
Â
Ã# 4
2048n2 64ð3 þ 2n2 Þ p2 ð38 À 20n2 þ 12n4 Þ À 40n2
h
Ã3
þ
þ
W
R
105p2 n4 ð1 À mÞ
7p2 n4
"


 #
2
2
8pnUm0 ð3 þ 2n Þ h

7 h
À
WÃ À
DT:
ð39Þ
7ð1 À mÞ
R
4pn R
2n2

8pnEÃ1
q¼
7ð38 À 20n2 þ 12n4 Þ

Eq. (39) is the explicit expression of external pressure-average
deflection curve incorporating the effects of temperature. These
expressions can be used to consider the nonlinear unsymmetric response of immovable clamped FGM spherical shell subjected to
external pressure and exposed to temperature conditions. The static critical buckling loads in this case can be obtained from Eq. (39)
by using condition dq/dW⁄ = 0.
Inversely, the temperature difference DT may be obtained in
terms of q, W⁄ as well as material and geometric properties due
to these expressions.

Similarly, suppose external pressure depending on time with
the law q = st, the motion Eq. (36) in conjunction with Eq. (37)
becomes

("
 4
7DÃ ð38 À 20n2 þ 12n4 Þ2 h

R
4n4
! 2 #
2 2
2
4
ð3 þ 2n Þ 28ð266 À 170n þ 64n Þ
h
WÃ
þ
þ
105p2 n2 ð1 À mÞ
R
7
"
96ð3 þ 2n2 Þn 8ð3 þ 2n2 Þð266 À 170n2 þ 64n4 Þ
À
þ
7pn2
105pnð1 À mÞn2
 2
Ã# 3
224 p ð38 À 20n2 þ 12n4 Þ À 40n2
h
W Ã2
þ
R
105p3 nð1 À mÞn2
"
)

Â
Ã# 4
2048n2 64ð3 þ 2n2 Þ p2 ð38 À 20n2 þ 12n4 Þ À 40n2
h
Ã3
þ
þ
W
R
7p2 n4
105p2 n4 ð1 À mÞ
"
#
 
 
Um0 ð3 þ 2n2 Þ h 2 Ã
7 h
7st
À
¼ 0:
ð40Þ
W
À
Á DT À
2
R
4
R
8
ð1 À mÞ

p
n
pn
2n

qà h2 W Ã;tt þ

EÃ1
ð38 À 20n2 þ 12n4 Þ

Eq. (40) is the governing equation to investigate dynamic
behavior of FGM shallow spherical shells under uniform external
pressure including temperature effects.
The analytical solution of Eq. (40) is very complicated, so this
equation may be solved approximately by applying Runge–Kutta
method and the Budiansky–Roth criterion to obtain critical dynamic buckling loads.
6. Numerical results and discussion
In the following discussions, the FGM spherical shell is made of
silicon nitride (Si3N4) and steel (SUS 304). The Young’s modulus,
mass densities and the coefficients of thermal expansion for Silicon
nitride are Ec = 348.43 GPa, qc = 2370 kg/m3, ac = 5.8723 Â 10À6
1/°C and for steel are Em = 201.04 GPa, qm = 8166 kg/m3, am =
12.33 Â 10À6 1/°C, respectively. The Poisson’s ratio is supposed to
be m = 0.3 for both constituent materials.
6.1. Validation of the proposed formulation
To validate the proposed formulation in static and dynamic stability analysis of FGM shallow spherical shell, the present results
are compared with results obtained by Ganapathi [15]. The nondimensional dynamic pressure Pcr and geometrical parameter k are
defined

Pcr ¼

1
qr4
½3ð1 À m2 ފ1=2 ðh=HÞ2 04 ;
8
Ec h
k ¼ 2½3ð1 À m2 ފ1=4 ðH=hÞ1=2 :


pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
where H ¼ R 1 À 1 À n2 is the central shell rise and the length of
response calculation time is introduced.
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R h=2
s ¼ ðh=r0 Þ2 Eef =½12ð1 À m2 Þqef h2 Š t; where Eef ¼ 12 Àh=2
EðzÞdz as
in Ref [15]. Using Eq. (36) with /m = 0, the dynamic buckling study
is conducted for step loading of infinite duration.
Nonlinear dynamic response history with time for the FGM
spherical shell parameter k = 6, r0/h = 400 and k = 1 considering different externally applied pressure loads is obtained and illustrated
in Fig. 2, that is similar with the result in the mentioned paper. As
can be seen, there is a sudden jump in the value of the average
deflection when the external pressure reaches the critical value


2957

D.H. Bich et al. / Composite Structures 94 (2012) 2952–2960

4

30
P

q

3.5
H

3
τ

Applied pressure
load

r0

3

t
Applied presume
load

20

2

15

1.5

W*

W*

2.5

2

3

1: P=0.565
2: P=0.566
3: Pcr =0.567

0
0

0.2

0

0.4

0.6

0.8

1

1.2

-5

1.4

Non dimensional time (τ)

0

0.5

1

1.5

2.5

3

Fig. 4. Effect of index k on dynamic response (dynamic, FM).

Pcr = 0.567.This result is in good agreement with the one of Ref.
[15] Pcr = 0.6063 obtained by using the finite element method.

5

x 10

4

ξ =0.2, k=1, n=1

6.2. Results for movable clamped FGM shallow spherical shell
3

1: R/h=1000
2: R/h=1200
3: R/h=1500

2

q (Pa)

To illustrate the proposed formulation a FGM shallow spherical
shell of geometric ratios R/h = 1000, n = r0/R = 0.2 and volume fraction index k = 1 under uniform external pressure is considered.
After calculation of the buckling load according to Eq. (28) with
various shape modes n, it can see that the smallest buckling load,
i.e the static critical buckling load, corresponds to the shape mode
n = 1 and receives the value qcr = 2.2167e+005 Pa.
The effect of material and geometric parameters on the nonlinear unsymmetrical static and dynamic response of the FGM shallow spherical shells with movable clamped edge under uniform
external pressure are considered in Figs. 3–6.
Fig. 3 shows the effects of volume fraction index k(=0, 1 and 5)
on the nonlinear unsymmetrical static response of FGM spherical
shells. As can be seen, the load- average deflection curves become
lower when k increases. The increase in the extremum-type buckling load and load carrying capacity of the shell when k reduces is
presented by a bigger difference between upper and lower buckling loads. Whereas Fig. 4 demonstrates these effects on the nonlinear dynamic response of FGM spherical shells.

1
3

2

0

1

-1

-2

0

25

q

t
Applied pressure
load

W*

25

ξ =0.2, k=1, n=1

20

3

1: R/h=1000
2: R/h=1200
3: R/h=1500

2
15

1

20

40

30

3

15

45

x 10

1

10

Fig. 5. Effect of R/h on load–average deflection curve (static, FM).

35

3

5

W*

5

2

2

time (s)

Fig. 2. Dynamic response W⁄–s with k = 6, k = 1, r0/h = 400.

q (Pa)

1

5

0.5

2

1

10

0
-1

R/h=1000
ξ =0.2

5

n=1

0

0

2

0

0.5

1

1.5

2

2.5

time (s)

1: k=0
2: k=1
3: k=5

-2
-3

2

1: k=0
2: k=1
3: k=5

10

1

1

-0.5

R/h=1000
ξ = 0.2
n=1

25

Fig. 6. Effect of R/h on dynamic response (dynamic, FM).

4

6

8

10

12

14

16

18

20

W*
Fig. 3. Effect of index k on load–average deflection curve (static, FM).

It is observed that a sudden jump in the value of the average
deflection occurs earlier when k increases, i.e. the corresponding
dynamic buckling load is smaller. This is expected because the


2958

D.H. Bich et al. / Composite Structures 94 (2012) 2952–2960

Table 1
Comparison between critical loads with the change of index k in case FM.

18

q

16

0

1

5

10

Pcr (static)
Pcr (dynamic)
%

2.8112e+005
2.9995e+005
6.7

2.2167e+005
2.3796e+005
7.3

1.8200e+005

1.9767e+005
8.6

1.7299e+005
1.8826e+005
8.8

14

t
Applied pressure
load

12

W*

k

higher value k corresponds to a metal-richer shell which usually
has less stiffness than a ceramic-richer shell.
Figs. 5 and 6 consider the effects of curvature radius-to-thickness ratio R/h(=1000, 1200 and 1500) on the nonlinear static and
dynamic characteristics respectively of the externally pressurized
FGM spherical shells.
As can be observed, the load bearing capability of the spherical
shell is considerably enhanced as R/h ratio decreases. Furthermore,
the increase of R/h ratio is accompanied by a drop of nonlinear
load–deflection curves and more severe snap-through static response and early occurrence of a jump of dynamic response.
Tables 1 and 2 demonstrate the comparison between nonlinear
critical static and dynamic buckling loads with the change of

power index k and R/h ratio, respectively. Clearly, the dynamic critical load is greater than the static critical load.

10

ξ=0.2, R/h=1000, n=1
1: k=0
2: k=1
3: k=5

8
6
4
2
0

0

1

2

3

ξ=0.2, k=1, n=1
1: R/h=1000
2: R/h=1200
3: R/h=1500
4: R/h=2000

1

4

q (Pa)

2

3
3

2
4

1
0

2

4

6

8

1200

1500

2000

Pcr (static)
Pcr (dynamic)
%

2.2167e+005
2.3796e+005
7.3

1.8481e+005
1.9565e+005
5.9

1.4789e+005
1.5505e+005
4.8

1.1093e+005
1.1579e+005
4.4

5

x 10

q (Pa)

W*

R/h=1000, ξ=0.2, n=1
1: k=0

2: k=1
3: k=5
4: k=10

18

20

q
15

0

2

2

3

4

1

Applied pressure
load

5

3
4

3
t

10

3

0

1

2

5

6

time (s)

1
0

16

ξ=0.2, k=1, n=1
1: R/h=1000
2: R/h=1200
3: R/h=1500

20

2

14

30
25

4

12

Fig. 9. Effect of ratio R/h on load–average deflection curve (static, IM).

1000

5

10

W*

R/h

1

7

5

Table 2
Comparison between critical loads with the change of ratio R/h in case FM.

6

6

x 10

6

-1
0

7

5

Fig. 8. Effect of index k on dynamic response (dynamic, IM).

6.3. Results for immovable clamped FGM spherical shell

8

4

time (s)

5

Similarly the effects of volume fraction index k and curvature
radius-to-thickness ratio R/h on the nonlinear static and dynamic
response of the FGM shallow spherical shell with immovable
clamped edge subjected to external pressure are illustrated in Figs.
7–10, respectively.
Effects of power index k and R/h ratio on static and dynamic
critical loads in IM case are given in Tables 3 and 4.
As can be seen, the trend of nonlinear static and dynamic responses of FGM spherical shells in IM case are very similar with re-

1

2

3

Fig. 10. Effect of ratio R/h on dynamic response (dynamic, IM).

0

5

10

15

W*
Fig. 7. Effect of index k on load–average deflection curve (static, IM).

sponses in FM case. The dynamic critical load is greater than the

static critical load, but the difference of these loads in IM case is
smaller than in FM case.


2959

D.H. Bich et al. / Composite Structures 94 (2012) 2952–2960
Table 3
Comparison between the nonlinear static and dynamic buckling loads vs k in case IM.
k

0

1

5

10

Pcr (static)
Pcr (dynamic)
%

6.2836e+005
6.4303e+005
2.3

4.9545e+005
5.0813e+005
2.6

4.0688e+005
4.1900e+005
3.0

3.8674e+005
3.9854e+005
3.1

5

x 10

6
5
4

q (Pa)

ξ=0.2

2

k=1
n=1

FM
IM

3

3

Table 4
Comparison between the nonlinear static and dynamic buckling loads vs R/h s in case
IM.

1

1

2

2

3

1
0

R/h

1000

1200

1500

2000

Pcr (static)
Pcr (dynamic)
%

4.9545e+005
5.0813e+005
2.6

4.1277e+005
4.2118e+005
2.0

3.3015e+005
3.3550e+005
1.6

2.4757e+005
2.5130e+005
1.5

(1): R/h=1000
(2): R/h=1200
(3): R/h=1500

-1
-2
0

5

10

15

20

25

W*
5

Fig. 12. Effect of in-plane restraint on nonlinear static response with the change of
R/h ratio.

x 10

7
6

1

5

R/h=1000

FM

ξ=0.2, n=1

IM

30

2

4

1: k=0

3

q (Pa)

1
3

IM

25

3: k=5

2

2

q

FM

2: k=1

t
20

3

R/h=1000
ξ=0.2, k=1
n=1

W*

1
0

Applied pressure
load

15

-1
10

-2
-3

0

5

10

15

5

W*
0

Fig. 11. Effect of in-plane restraint on nonlinear static response with the change of
index k.

0

1

2

3

4

5

6

time (s)
6.4. Effect of FM and IM boundary conditions
Graph of nonlinear static responses of clamped FGM spherical
shells with different boundary conditions are plotted in Figs. 11

and 12. As can be observed, the spherical shells with immovable
clamped edge have a comparatively higher capability of carrying
external pressure than shells with movable clamped edge. However,
their response is unstable. That means the IM shells experience a
snap-through with much higher intensity than their movable
clamped counterparts. Furthermore, these figures also show that
the effect of k index and R/h ratio on the critical buckling pressure
of shells is very strong.
Fig. 13 shows the comparison of the dynamic response of FGM
spherical with FM and IM boundary conditions. It also can see that
the dynamic critical buckling load of clamped FM shell is smaller
than the one of clamped IM shell.
Table 5 shows the effects of k index and R/h ratio on static
critical load in FM case and IM case. Once again it is indicated that
critical loads in IM case are greater about twice times than the ones
in FM case.
6.5. Effect of environment temperature
The effect of environment temperature on the thermomechanical behavior of FGM shallow spherical shells with immovable
clamped edge is considered in this subsection. The shells are

Fig. 13. Effect of in-plane restraint on nonlinear dynamic response.

exposed to temperature field prior to applying external pressure.
Figs. 14 and 15 analyze the nonlinear unsymmetrical static and
dynamic responses of FGM spherical shells for various values of
uniformly raised temperature DT(=0, 50, 100 and 150 °C).
As shown in Fig. 14, the temperature field makes shell to be deflected outward (negative deflection) prior to mechanical load acting on it. When the shell is subjected uniform of external pressure,
its outward deflection is reduced and when external pressure exceeds bifurcation point of load, an inward deflection occurs. Similar
behavior occur for dynamic clamped FGM shells, too. It is illustrated in Fig. 15.
Table 5

Effects of k and R/h on static critical load in FM and IM cases.
R/h

1000

1200

1500

2000

k=0

2.8112e+005FM
6.2836e+005IM

2.3438e+005
5.2350e+005

1.8756e+005
4.1871e+005

1.4068e+005
3.1398e+005

k=1

2.2167e+005
4.9545e+005

1.8481e+005
4.1277e+005

1.4789e+005
3.3015e+005

1.1093e+005
2.4757e+005

k=5

1.8200e+005
4.0688e+005

1.5175e+005
3.3897e+005

1.2144e+005
2.7112e+005

9.1091e+004
2.0330e+005

k = 10

1.7299e+005
3.8674e+005

1.4424e+005
3.2220e+005

1.1543e+005
2.5770e+005

8.6582e+004
1.9324e+005


2960

D.H. Bich et al. / Composite Structures 94 (2012) 2952–2960

FGM shallow spherical shells under uniform external pressure with
and without including the effects of temperature.
Approximate analytical one-term deflection mode for two types
boundary conditions is given and by applying Galerkin procedure
explicit expressions of static critical buckling loads and postbuckling load–deflection curves are determined.
For the nonlinear dynamic buckling analysis, the nonlinear
equation of motion of the shell is solved by using Runge–Kutta
method. The dynamic critical buckling loads are found according
to Budiansky–Roth criterion. The nonlinear unsymmetric response
of the shells is analyzed and the results are illustrated in graphic
form and numerical tables. The results indicate that the nonlinear
response of FGM shallow spherical shells is complex and greatly
influenced by the type of loading (static or dynamic), the material
and geometric parameters, the in-plane restraint and the preexistent temperature condition.

5

x 10

14

4

R/h=1000, ξ=0.2
k=1, n=1
1: ΔT=0
2: ΔT=50
3: ΔT=100
4: ΔT=150

q (Pa)

12
10

3

8

2

6

1

4
2
0

-2
-4

0

2

4

6

8

10

12

14

16

18

W*

Acknowledgement

Fig. 14. Effect of temperature on nonlinear static response.

This paper was supported by the National Foundation for

Science and Technology Development of Vietnam – NAFOSTED.
The authors are grateful for this financial support.

25
1: ΔT=0
2: ΔT=50
3: ΔT=100
4: ΔT=150

20

W*

15

R/h=1000, ξ=0.2, k=1, n=1

4

3

2

1

References

q

10

t
Applied pressure
load

5

0

-5

0

2

4

6

8

10

12

14

time (s)
Fig. 15. Effect of temperature on nonlinear dynamic response.

Table 6
Comparison of static and dynamic critical loads.

DT

0

50

100

150

Pcr (static)
Pcr (dynamic)
%

4.9545e+005
5.0813e+005
2.6

7.1219e+005
7.2917e+005
2.4

9.4978e+005
9.7132e+005
2.3

12.0582e+005

12.3231e+005
2.2

The enhancement of temperature difference is accompanied by
the increase of bifurcation points, and the intensity of snapthrough behavior of the spherical shells (in static analysis) and
the strengtheneth of load bearing capability of the spherical shells
under dynamic loading (in dynamic analysis).
Comparison between nonlinear static and dynamic critical
buckling loads with effect of temperature is given in Table 6. It
can see that in this case the dynamic critical buckling load also is
greater than static one.
7. Concluding remarks
This paper presents an analytical approach to investigate the
nonlinear unsymmetrical static and dynamic responses of clamped

[1] Budiansky B, Roth RS. Axisymmetric dynamic buckling of clamped shallow
spherical shells. NASA TND 1962;510:597–609.
[2] Huang NC. Unsymmetrical buckling of thin shallow spherical shells. Trans
ASME J Appl Mech 1964;31:447–57.
[3] Tillman SC. On the buckling behavior of shallow spherical caps under a
uniform pressure load. Int J Solids Struct 1970;6:37–52.
[4] Ball RE, Burt JA. Dynamic buckling of shallow spherical shells. ASME J Appl
Mech 1973;41:411–6.
[5] Kao R, Perrone N. Dynamic buckling of axisymmetric spherical caps with initial
imperfection. Comput Struct 1978;9:463–73.
[6] Wunderlich W, Albertin U. Buckling behavior of imperfect spherical shells. Int J
Nonlinear Mech 2002;37:589–604.
[7] Nath N, Alwar RS. Nonlinear static and dynamic response of spherical shells.
Int J Nonlinear Mech 1978;13:157–70.
[8] Ganapathi M, Varadan TK. Dynamic buckling of orthotropic shallow spherical

shells. Comput Struct 1982;15:517–20.
[9] Ganapathi M, Varadan TK. Dynamic buckling of laminated anisotropic
spherical caps. J Appl Mech 1995;62:13–9.
[10] Chao CC, Lin IS. Static and dynamic snap-through of orthotropic spherical caps.
Compos Struct 1990;14:281–301.
[11] Eslami MR, Ghorbani HR, Shakeri M. Thermoelastic buckling of thin spherical
shells. J Therm Stresses 2001;24(12):1177–98.
[12] Shahsiah R, Eslami MR. Thermal and mechanical instability of an imperfect
shallow spherical cap. J Therm Stresses 2003;26(7):723–37.
[13] Shahsiah R, Eslami MR, Naj R. Thermal instability of functionally graded
shallow spherical shells. J Therm Stresses 2006;29(8):771–90.
[14] Prakash T, Sundararajan N, Ganapathi M. On the nonlinear axisymmetric
dynamic buckling behavior of clamped functionally graded spherical caps. J
Sound Vib 2007;299:36–43.
[15] Ganapathi M. Dynamic stability characteristics of functionally graded
materials shallow spherical shells. Compos Struct 2007;79:338–43.
[16] Bich DH. Nonlinear buckling analysis of FGM shallow spherical shells. Vietnam
J Mech 2009;31:17–30.
[17] Bich DH, Hoa LK. Nonlinear vibration of functionally graded shallow spherical
shells. Vietnam J Mech 2010;32(4):199–210.
[18] Bich DH, Tung HV. Nonlinear axisymmetric response of functionally graded
shallow spherical shells under uniform external pressure including
temperature effects. Int J Nonlinear Mech 2011;46:1195–204.
[19] Shahsiah R, Eslami MR, Sabzikar Boroujerdy M. Thermal instability of
functionally
graded
deep
spherical
shell.
Arch

Appl
Mech
2011;81(10):1455–71.
[20] Brush DO, Almroth BO. Buckling of bar, plates and shells. NewYork:
McGrawHill; 1975.
[21] Volmir AS. Nonlinear dynamic of plates and shells. science ed. Moscow; 1972
[in Russian].
[22] Mushtari XM, Galimov KZ. Nonlinear theory of elastic shells. Kazan; 1957 [in
Russian].
[23] Birman V. Theory and comparison of the effect of composite and shape
memory along stiffness on stability of composite shells and plates. Int J Mech
Sci 1997;39(10):1139–49.
[24] Oghibalov PM. Dynamics and stability of shells. Moscow; 1963 [in Russian].
[25] Duc ND, Tung HV. Nonlinear analysis of stability for functionally graded
cylindrical panels under axial compression. Comput Mater Sci 2010;49:313–6.