DSpace at VNU: Nonlinear dynamical analyses of eccentrically stiffened functionally graded toroidal shell segments surrounded by elastic foundation in thermal environment

Composites Part B 95 (2016) 355e373

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Composites Part B
journal homepage: www.elsevier.com/locate/compositesb

Nonlinear dynamical analyses of eccentrically stiffened functionally
graded toroidal shell segments surrounded by elastic foundation in
thermal environment
Dao Huy Bich a, Dinh Gia Ninh b, *, Bui Huy Kien c, David Hui d
a

Vietnam National University, Hanoi, Viet Nam
School of Mechanical Engineering, Hanoi University of Science and Technology, Hanoi, Viet Nam
Faculty of Mechanical Engineering, Hanoi University of Industry, Hanoi, Viet Nam
d
Department of Mechanical Engineering, University of New Orleans, Louisiana, USA
b
c

a r t i c l e i n f o

a b s t r a c t

Article history:
Received 19 February 2016
Received in revised form
31 March 2016
Accepted 1 April 2016
Available online 9 April 2016

In this study, the nonlinear vibration and dynamic buckling of eccentrically stiffened functionally graded
toroidal shell segments surrounded by an elastic medium in thermal environment are presented. The
governing equations of motion of eccentrically stiffened functionally graded toroidal shell segments are
derived based on the classical shell theory with the geometrical nonlinear in von Karman-Donnell sense
and the smeared stiffeners technique. Furthermore, the dynamical characteristics of shells as natural
frequencies, nonlinear frequencyeamplitude relation, nonlinear dynamic responses and the nonlinear
dynamic critical buckling loads evaluated by Budiansky-Roth criterion are considered. The effects of
characteristics of functionally graded materials, geometrical ratios, elastic foundation, pre-loaded axial
compression and temperature on the dynamical behavior of shells are investigated.
© 2016 Elsevier Ltd. All rights reserved.

Keywords:
Toroidal shell segments
A. Discontinuous reinforcement
B. Thermomechanical
B. Vibration
C. Analytical modelling

1. Introduction
In the past, the dynamic problems of laminated composite
structures studied by many authors. Hui [1,2] presented the effects
of shear loads on vibration and buckling of typical antisymmetric
cross-ply thin cylindrical panels under combined loads; effects of
geometrical imperfections on the large amplitude vibration of
shallow spherical shells as well as effects of structural damping.
Dynamic fracture and delamination of unidirectional graphite/
epoxy composites for end-notched flexure center-noted flexure
pure mode II loading configurations using a modified split Hopkinson pressure bar were investigated by Nwosu et al. [3].
Functionally graded materials (FGMs) were invented by Japanese scientists in 1984 [4]. This composite material is a mixture of

ceramic and metallic constituent materials by continuously
changing in the volume fractions of their components. Mechanical

* Corresponding author. Tel.: þ84 988 287 789.
E-mail
addresses:
,
(D.G. Ninh).
/>1359-8368/© 2016 Elsevier Ltd. All rights reserved.



and physical behavior of FGMs are better than fiber reinforced
laminated composite materials with advantages of no stress concentration, high toughness, oxidation resistance and heatresistance so FGMs are applied to heat-resistant, lightweight
structures in aerospace, mechanical, and medical industry and so
forth. Therefore, the nonlinear vibration and dynamic buckling
problems of FGM structures have been attracted a vast amount
attention of researchers.
Pradhan et al. [5] studied vibration of FGM cylindrical shells
under various boundary conditions with the strain-displacement
relations form Love's shell theory. Based on the Rayleigh method,
the governing equations were derived and the natural frequencies
were investigated depending on the constituent volume fractions
and boundary condition. The general formulation for free, steadystate and transient vibration analyses of FGM shells of revolution
subjected to arbitrary boundary conditions was presented by Qu
et al. [6]. The formulation was derived by means of a modified
variational principle in conjunction with a multi-segment partitioning procedure on the basis of the first order shear deformation
shell theory. G. G. Sheng and X. Wang [7] investigated the nonlinear
vibrations control of FGM laminated cylindrical shell based on

356

D.H. Bich et al. / Composites Part B 95 (2016) 355e373

Hamilton's principle, von Karman nonlinear theory and constantgain negative velocity feedback approach. The thin piezoelectric
layers embedded on inner and outer surfaces of the smart FG
laminated cylindrical shell were acted as distributed sensor and
actuator, which is used to control nonlinear vibration of the smart
FG laminated cylindrical shell. The free vibration analysis of functionally graded cylindrical panels with cut-out and under temperature condition using the three-dimensional Chebyshev-Ritz
method was noticed by Malekzadeh et al. [8]. Chen and Babcock [9]
gave the large amplitude vibration of a thin-walled cylindrical shell
using the perturbation method the steady-state forced vibration
problem. Furthermore, the simply-supported boundary conditions
and the circumferential periodicity condition were satisfied. The
unified solution for the vibration analysis of functionally graded
material (FGM) doubly-curved shells of revolution with arbitrary
boundary conditions was given by Jin et al. [10]. The solution was
derived by means of the modified Fourier series method on the
basis of the first order shear deformation shell theory considering
the effects of the deepness terms. Kim [11] performed free vibration
characteristics of FGM cylindrical shells partially resting on elastic
foundation with an oblique edge using an analytical method. The
motion of shell was represented based on the first order shear
deformation theory to account for rotary inertia and transverse
shear strains. The nonlinear dynamic buckling and pre-buckling
deformation of FGM truncated conical shells under axial
compressive load varying as a linear function of time using the
Superposition principle, Galerkin and Runge-Kutta methods were
studied by Deniz and Sofiyev [12,13]. Shen and Yang [14] carried out

the free vibration and dynamic instability of functionally graded
cylindrical panels subjected to combined static and periodic axial
forces and in thermal environment with theoretical formulations
based on Reddy's higher order shear deformation shell theory
taking into account rotary inertia and the parabolic distribution of
the transverse shear strains through the panel thickness. The
characteristics of free vibration and nonlinear responses were
investigated using the governing equations of motion of eccentrically stiffened functionally graded cylindrical panels with
geometrically imperfections based on the classical shell theory
with the geometrical nonlinearity in von Karman-Donnell sense
and smeared stiffeners technique by Bich et al. [15]. Moreover, Bich
and Nguyen [16] presented the study of the nonlinear vibration of a
functionally graded cylindrical shell subjected to axial and transverse mechanical loads based on improved Donnell equations. The
dynamic behavior of moderately thick functionally graded conical,
cylindrical shells and annular plates with a four-parameter power
law distribution based on the First order Shear Deformation Theory
were focused by Tornabene [17]. The materials were assumed to be
isotropic and inhomogeneous though the thickness direction.
Sofiyev [18e20] studied the dynamic behavior of FGM structures
such as: the vibration of FGM conical shells under a compressive
axial load using the shear deformation theory using Donnell shell
theory; the parametric vibration of shear deformable functionally
graded truncated conical shells subjected to static and time
dependent periodic uniform lateral pressures based on the first
order shear deformation theory; the theoretical approach to solve
vibration problems of FGM truncated conical shells under mixed
boundary conditions using means of the Airy stress function
method to derive the fundamental relations, motion and strain
compatibility equations. The vibration of thin cylindrical shells
extracted from Ref. Függe's three equations of motion was investigated by Warburton [21]. This solution required the assumption of

a natural frequency and the determination of the corresponding
shell length for the prescribed end conditions. Based on straindisplacement relations from the Love's shell theory and the
eigenvalue governing equation using Rayleigh-Ritz method, Loy

et al. [22] gave the study at vibration filed of functionally cylindrical
shells. The influences of shear stresses and rotary inertia on the
vibration of FG coated sandwich cylindrical shells resting on Pasternak elastic foundation based on the modification of Donnell type
equations of motion were examined by Sofiyev et al. [23]. The basic
equations were reduced to an algebraic equation of the sixth order
and numerically solving this algebraic equation gave the dimensionless fundamental frequency.
Noda [24], Praveen et al. [25] first discovered the heat-resistant
FGM structures and studied material properties dependent on
temperature in thermo elastic analyses. Heydarpour and Malekzadeh [26] pointed out the free vibration analysis of rotating
functionally graded cylindrical shells in temperature environment
with the equations of motion and related boundary conditions
derived to Hamilton's principle. The initial thermo-mechanical
stresses were obtained by solving the thermo elastic equilibrium
equations. Sheng and Wang [27] researched the nonlinear response
of functionally graded cylindrical shells under mechanical and
thermal loads using con Karman nonlinear theory. The coupled
nonlinear partial differential equations are discretized based on a
series expansion of linear modes and a multiterm Galerkin's
method. Furthermore, Shen [28] took into account the nonlinear
vibration of shear deformable FGM cylindrical shells of finite length
embedded in a large outer elastic medium and in thermal environments. The motion equations were based on a higher order
shear deformation shell theory that included shellefoundation
interaction. The large amplitude vibration behavior of a shear
deformable FGM cylindrical panel resting on elastic foundations in
thermal environments based on a higher order shear deformation
shell theory was investigated by Shen and Wang [29]. The thermal

effects are also included and the material properties of FGMs are
assumed to be temperature-dependent. The equations of motion
are solved by a two step perturbation technique to determine the
nonlinear frequencies of the FGM cylindrical panel.
Toroidal shell segment has been used in such applications as
satellite support structures, rocket fuel tanks, fusion reactor vessels, diver's oxygen tanks and underwater toroidal pressure hull.
Today, FGMs have received mentionable attention in structural
applications. The smooth and continuous change in material
properties enables FGMs to avoid interface problems and unexpected thermal stress concentrations. Some components of the
above-mentioned structures may be made of FGM. Stein and
McElman [30] carried out the homogenous and isotropic toroidal
shell segments about the buckling problem. Moreover, the initial
post-buckling behavior of toroidal shell segments subjected to
several loading conditions based on the basic of Koiter's general
theory was performed by Hutchinson [31]. Parnell [32] gave a
simple technique for the analysis of shells of revolution applied to
toroidal shell segments. Recently, there have had some new
publications about toroidal shell segment structure. Bich et al. [33]
has studied the buckling of eccentrically stiffened functionally
graded toroidal shell segment under axial compression, lateral
pressure and hydrostatic pressure based on the classical thin shell
theory, the smeared stiffeners technique and the adjacent equilibrium criterion. Furthermore, the nonlinear buckling and postbuckling of ES-FGM toroidal shell segments surrounded by an
elastic medium under torsional load based on the analytical
approach are investigated by Ninh et al. [34,35]. Bich et al. [36,37]
studied the post-buckling of FGM and S-FGM toroidal shell
segment under external pressure loads by an analytical approach
using the Galerkin method.
To the best of the authors' knowledge, there has not been any
study to the nonlinear dynamical analysis of eccentrically stiffened
FGM toroidal shell segments surrounded by an elastic foundation

including temperature effects.


D.H. Bich et al. / Composites Part B 95 (2016) 355e373

In the present paper, the dynamic buckling behavior and
nonlinear vibration of eccentrically stiffened FGM toroidal shell
segments on elastic medium in thermal environment are investigated. Based on the classical shell theory with the nonlinear straindisplacement relation of large deflection, the Galerkin method,
Volmir's assumption and the numerical method using fourth-order
Runge-Kutta are performed for dynamic analysis of shells to give
expression of natural frequencies and nonlinear dynamic
responses.

357

and the stiffeners and easier to manufacture, the homogeneous
stiffeners can be used. Because the pure ceramic ones are brittleness the metal stiffeners are used and arranged at metal-rich side of
the shell. With the law indicated in (1) the outer surface of the shell
is metal-rich and the external metal stiffeners are arranged at this
side.
The strains across the shell thickness at a distance z from the
mid-surface are:

ε1 ¼ ε01 À zc1 ; ε2 ¼ ε02 À zc2 ; g12 ¼ g012 À 2zc12 ;

2. Governing equations
2.1. Functionally graded material (FGM)
Suppose that the material composition of the shell varies
smoothly along the thickness is such a way inner surface is ceramicrich and the outer surface is metal-rich by a simple power law in
terms of the volume fractions of the constituents.

We denote Vm and Vc being volume e fractions of metal and
ceramic phases respectively, which are related by Vm þ Vc ¼ 1 and

k
, where h is the thickness of thin
Vc is expressed as Vc ðzÞ ¼ 2zþh
2h
e walled structure, k is the volume e fraction exponent (k ! 0); z is
the thickness coordinate and varies from Àh/2 to h/2; the subscripts
m and c refer to the metal and ceramic constituents respectively.
According to the mentioned law, the Young modulus E(z), the mass
density r(z) and the thermal expansion coefficient a(z) can be
expressed in the form.


2z þ h k
EðzÞ ¼ Em Vm þ Ec Vc ¼ Em þ ðEc À Em Þ
;
2h


2z þ h k
aðzÞ ¼ am Vm þ ac Vc ¼ am þ ðac À am Þ
;
2h


2z þ h k
rðzÞ ¼ rm Vm þ rc Vc ¼ rm þ ðrc À rm Þ
:

2h

where ε01 and ε02 are normal strains, g012 is the shear strain at the
middle surface of the shell and cij are the curvatures.
According to the classical shell theory the strains at the middle
surface and curvatures are related to the displacement components
u, v, w in the x1, x2, z coordinate directions as [38]:

ε01 ¼





vu w 1 vw 2 0
vv
w 1 vw 2
À þ
; ε2 ¼
À þ
;
vx1 R 2 vx1
vx2 a 2 vx2

g012 ¼

The constitutive stressestrain equations by Hooke law for the
shell material are given

ssh

2 ¼
(1)

vu
vv
vw vw
v2 w
v2 w
v2 w
þ
þ
; c ¼ 2 ; c2 ¼ 2 ; c12 ¼
vx2 vx1 vx1 vx2 1
vx
vx1
vx2
1 vx2
(3)

ssh
1 ¼



(2)

ssh
12

EðzÞ

1 À n2
EðzÞ
2

ðε1 þ nε2 Þ À

EðzÞaðzÞ
DT; DT ¼ T À T0 ;
1Àn

ðε2 þ nε1 Þ À

EðzÞaðzÞ
DT;
1Àn

1Àn
EðzÞ
¼
g
2ð1 þ nÞ 12

(4)

and for metal stiffeners.

the Poisson's ratio n is assumed to be constant.

st
sst

1 ¼ Em ε1 À Em am DT; s2 ¼ Em ε2 À Em am DT:

2.2. Constitutive relations and governing equations

Integrating the stressestrain equations and their moments
through the thickness of the shell; and using the smeared stiffeners
technique, the expressions for force and moment resultants of a
FGM toroidal shell segment are obtained [38]:

Consider a functionally graded toroidal shell segment of thickness h, length L, which is formed by rotation of a plane circular arc
of radius R about an axis in the plane of the curve as shown in Fig. 1.
The geometry and coordinate system of a stiffened FGM toroidal
shell segments are depicted in Fig. 2. For the middle surface of a
toroidal shell segment, from the figures we have:

r ¼ a À Rð1 À sin4Þ;
where a is the equator radius and f is the angle between the axis of
revolution and the normal to the shell surface. For a sufficiently
shallow toroidal shell in the region of the equator of the torus, the
angle f is approximately equal to p/2, thus sinf z 1; cosf z 0 and
r ¼ a [30]. The form of governing equation is simplified by putting:

dx1 ¼ Rd4; dx2 ¼ adq
The radius of arc R is positive with convex toroidal shell segment
and negative with concave toroidal shell segment. The shell is
surrounded by an elastic foundation with Winkler foundation
modulus K1(N/m3) and the shear layer foundation stiffness of
Pasternak model K2(N/m).
Suppose the FGM toroidal shell segment is reinforced by string
and ring stiffeners. In order to provide continuity within the shell



Em A1 0
ε1 þ A12 ε02 À ðB11 þ C1 Þc1 À B12 c2 À Fa À F*a ;
N1 ¼ A11 þ
s1


Em A2 0
N2 ¼ A12 ε01 þ A22 þ
ε2 À B12 c1 À ðB22 þ C2 Þc2 À Fa À F**
a ;
s2
N12 ¼ A66 g012 À 2B66 c12 ;
(5)


Em I1
c1 À D12 c2 À Fm À F*m ;
M1 ¼ ðB11 þ C1 Þε01 þ B12 ε02 À D11 þ
s1


Em I2
M2 ¼ B12 ε01 þ ðB22 þ C2 Þε02 À D12 c1 À D22 þ
c2 À Fm À F**
m;
s2
M12 ¼ B66 g012 À 2D66 c12 ;

(6)
where Aij,Bij,Dij (i, j ¼ 1, 2, 6) are extensional, coupling and bending
stiffnesses of the shell without stiffeners.


358

D.H. Bich et al. / Composites Part B 95 (2016) 355e373

Fig. 1. Configuration of toroidal shell segments.

A11 ¼ A22 ¼
B11 ¼ B22 ¼
D11 ¼ D22 ¼

E1

;
2

A12 ¼

;
2

B12 ¼

1Àn
E2
1Àn

E3

1 À n2

;

E1 n

;
2

A66 ¼

E1
;
2ð1 þ nÞ

;
2

B66 ¼

E2
;
2ð1 þ nÞ

1Àn
E2 n
1Àn

D12 ¼

E3 n
1 À n2

;

D66

E3
;
¼
2ð1 þ nÞ

where


ðEc À Em Þkh2
Ec À Em
h; E2 ¼
;
kþ1
2ðk þ 1Þðk þ 2Þ



Em
1
1
1

À
þ
h3 ;
E3 ¼
þ ðEc À Em Þ
k þ 3 k þ 2 4k þ 4
12


(7)

E1 ¼

Em þ

(8)


D.H. Bich et al. / Composites Part B 95 (2016) 355e373

359

vN1 vN12
v2 u
þ
¼ r1 2 ;
vx1
vx2
vt
vN12 vN2

v2 v
þ
¼ r1 2 ;
vx1
vx2
vt
v2 M1
vx21

þ2

þN2
þK2

v2 M12 v2 M2
v2 w
v2 w
þ
þ
N
þ
2N
1
12
vx1 vx2
vx1 vx2
vx21
vx22

v2 w

vx22

À ph

v2 w
vx21

þ

v2 w

vx21
!
v2 w
vx22

þ

(11)

N1 N2
þ
þ q À K1 w
R
a

¼ r1

v2 w
vt

2

þ 2 r1 ε

vw
;
vt

where K1 (N/m3) is linear stiffness of foundation, K2 (N/m) is the
shear modulus of the sub-grade, ε is damping coefficient and



r1 ¼ rm þ
Fig. 2. Geometry and coordinate system of a stiffened FGM toroidal shell segments (a)
stringer stiffeners; (b) ring stiffeners.

Em A1 z1
Em A2 z1
C1 ¼ À
; C2 ¼ À
:
s1
s2

(9)

1
Fa ¼

1Àn

F**
a ¼

Fm ¼

F**
m ¼

d2
s2

Àh=2
Àh=2
Z

d
EðzÞaðzÞDTdz; F*a ¼ 1
s1

Àh=2
Z

Em am DTdz;

Àh=2Àh1

Em am DTdz;

Zh=2

EðzÞaðzÞDTzdz; F*m ¼

Àh=2
Àh=2
Z

d1
s1

Àh=2
Z

Em am DTzdz;

Àh=2Àh1

Em am DTzdz:

v2 u
vt 2

;

v2 v

;
vt 2
H31 ðuÞ þ H32 ðvÞ þ H33 ðwÞ þ P3 ðwÞ þ Q3 ðu; wÞ


v2 w
1
Fa þ F*a
þR3 ðv; wÞ À ph 2 þ q À
R
vx1

(13)

where the linear operators Hij( )(i,j ¼ 1, 2, 3) and the nonlinear
operators Pi( )(i ¼ 1, 2, 3), Q3 and R3 are demonstrated in Appendix
A:
Eq. (13) are the nonlinear governing equations used to investigate the nonlinear dynamical responses of eccentrically stiffened
functionally graded toroidal shell segments surrounded by elastic
foundation in thermal environment.

Àh=2Àh2

(10)
If DT ¼ const

F*a ¼

H11 ðuÞ þ H12 ðvÞ þ H13 ðwÞ þ P1 ðwÞ ¼ r1


1
v2 w
vw

;
¼ r1 2 þ 2 r1 ε
À Fa þ F**
a
a
vt
vt

Àh=2Àh2

1
1Àn
d2
s2

Zh=2

(12)

By substituting Eq. (3) into Eqs. (5) and (6) and then into Eq. (11),
the term of displacement components are expressed as follows:

H21 ðuÞ þ H22 ðvÞ þ H23 ðwÞ þ P2 ðwÞ ¼ r1

and

Fa ¼





ðrc À rm Þ
A
A
h þ rm 1 þ 2
kþ1
s1
s2

1
Em acm þ Ecm am Ecm acm
þ
PhDT; P ¼ Em am þ
1Àn
kþ1
2k þ 1
d1 h1
d2 h2
Em am DT; F**
Em am DT:
a ¼
s1
s2

where Ecm ¼ Ec À Em; acm ¼ ac À am.
The spacings of the stringer and ring stiffeners are denoted by s1
and s2 respectively. The quantities A1, A2 are the cross section areas
of the stiffeners and I1, I2, z1, z2 are the second moments of cross
section areas and eccentricities of the stiffeners with respect to the
middle surface of the shell respectively.

The nonlinear equilibrium equations of a toroidal shell segment
under a lateral pressure q, an axial compression p and surrounded
by an elastic foundation based on the classical shell theory are given
by Ref. [38]:

3. Nonlinear analysis
In the present paper, the simply-supported boundary conditions
are considered

w ¼ 0; v ¼ 0; M1 ¼ 0: at x1 ¼ 0 and x1 ¼ L

(14)

The approximate solutions of the system of Eq. (13) satisfying
the conditions Eq. (14) can be expressed as:

mpx1
nx
mpx1
nx
sin 2 ; v ¼ VðtÞsin
cos 2 ;
L
2a
L
2a
mpx1
nx2
sin
;

w ¼ WðtÞsin
L
2a
u ¼ UðtÞcos

(15)

where U, V, W are the time depending amplitudes of vibration, m
and n are numbers of half wave in axial direction and wave in
circumferential direction, respectively.
Substituting Eq. (15) into Eq. (13) and then applying the Galerkin
method leads to:


360

D.H. Bich et al. / Composites Part B 95 (2016) 355e373

h11 U þ h12 V þ h13 W þ n1 W 2 ¼ r1
h21 U þ h22 V þ h23 W þ n2 W 2 ¼ r1
h31 U þ h32 V þ h33 W þ p

hp2 m2

d2 U
dt

2

d2 V

dt 2

Solving Eq. (20) leads to three angular frequencies of the
toroidal shell in the axial, circumferential and radial directions, and
the smallest one is being considered.
On the other hand, the fundamental frequencies of the shell can
be approximately determined by explicit expression in Eq. (18).

;
;

W þ n3 W 2 þ n4 W 3

L2

 1

4d d
1
þn5 UW þ n6 VW þ 1 22 q À
Fa þ F*a À Fa þ F**
a
R
a
mnp

¼ r1

d2 W
dt 2

þ 2r1 ε

dW
;
dt
(16)

umn ¼

rffiffiffiffiffi
a1

(21)

r1

Solving Eq. (20) leads to exact solution but implicit expression
while Eq. (21) performs approximate frequencies but explicit
expression and simpler.
3.2. Frequency-amplitude curve

where hij; ni are given in Appendix B.
Otherwise, the Volmir's assumption [39] can be used in the
dynamic analysis. Taking the inertia forces r1(d2U/dt2) / 0 and
r1(d2V/dt2) / 0 into consideration because of u << w,v << w, Eq.
(16) can be rewritten as follows:

Consider nonlinear vibration of a toroidal shell segment under a
uniformly distributed transverse load q ¼ QsinUt including thermal

effects. Assuming pre-loaded compression p, Eq. (18) takes the
form.

h11 U þ h12 V þ h13 W þ n1 W 2 ¼ 0;

r1

h21 U þ h22 V þ h23 W þ n2 W 2 ¼ 0;
h31 U þ h32 V þ h33 W þ p

hp2 m2

W þ n3 W 2 þ n4 W 3

L2

 1

4d d
1
þn5 UW þ n6 VW þ 1 22 q À
Fa þ F*a À Fa þ F**
a
R
a
mnp

¼ r1

d2 W

dt 2

þ 2r1 ε

dW
:
dt
(17)

Solving the first and the second obtained equations with respect
to U and V and then substituting the results into the third equation
yields.

r1

d2 W
dW
þ a1 W À a2 W 2 þ a3 W 3
þ 2r1 ε
dt
dt 2

 1

4d d
1
¼ 1 22 q À
Fa þ F*a À Fa þ F**
a
R

a
mnp

(22)

Eq. (22) can be rewritten as



d2 W
dW
þ u2mn W À HW 2 þ KW 3 À F sin Ut þ G ¼ 0;
þ 2ε
2
dt
dt
(23)
where umn ¼

qffiffiffiffi
a1

r1 is the fundamental frequency of linear vibration
d1 d2 Q ,
of the toroidal shell segment and H ¼ a2/a1, K ¼ a3/a1, F ¼ r4mn
p2
1



d1 d2
G ¼ r4mn
p2
1

1
R ð Fa


þ F*a Þ þ 1a ðFa þ F**
a Þ .

For seeking amplitudeefrequency relation of nonlinear vibration we substitute

(18)

where

W ¼ A sin Ut;

(24)

into Eq. (23) to give.

a1 ¼ Àh33 À
Àp

d2 W
dW
þ a1 W À a2 W 2 þ a3 W 3

þ 2r1 ε
dt
dt 2

 1

4d d
1
:
¼ 1 22 Q sin Ut À
Fa þ F*a À Fa þ F**
a
R
a
mnp



Y ¼ A u2mn À U2 sin Ut þ 2εAU cos Ut À u2mn HA2 sin2 Ut

h31 ðh12 h23 À h22 h13 Þ h32 ðh21 h13 À h11 h23 Þ
À
h11 h22 À h12 h21
h11 h22 À h12 h21

hp2 m2

þ K u2mn A3 sin3 Ut À F sin Ut þ G ¼ 0

;

L2
h ðh n À h22 n1 Þ h32 ðh12 n1 À h11 n2 Þ
a2 ¼ n3 þ 31 12 2
þ
h11 h22 À h12 h21
h11 h22 À h12 h21

(25)

pZ=2U

n ðh h À h22 h13 Þ n6 ðh21 h13 À h11 h23 Þ
þ 5 12 23
þ
;
h11 h22 À h12 h21
h11 h22 À h12 h21
a3 ¼ Àn4 À n5

the frequencyeamplitude relation of nonlinear vibration is
obtained.

Taking linear parts of the set of Eq. (16) and putting p ¼ 0, q ¼ 0,
the natural frequencies of the shell can be directly calculated by
solving determinant.

h12
h22 þ r1 u2
h32

h13



¼0
h23


2

h33 þ r1 u

Y sin Utdt ¼ 0;

0

ðh12 n2 À h22 n1 Þ n6 ðh21 n1 À h11 n2 Þ
À
:
h11 h22 À h12 h21
h11 h22 À h12 h21

3.1. Natural frequencies

h11 þ r1 u2




h21




h31

Integrating over a quarter of vibration period

(19)

U2 À

p



U ¼ u2mn 1 À

8
3K 2
A
HA þ

3p
4


À

F 4G
þ
A Ap

(26)

By denoting a2 ¼ U2 =u2mn Eq. (26) is rewritten as

a2 À
(20)

4ε

4ε

pumn

a¼1À

8
3K 2
F
4G
A À

HA þ
þ
3p
4
Au2mn Apu2mn

(27)

For the nonlinear vibration of the shell without damping (ε ¼ 0),
this relation has of the form.


D.H. Bich et al. / Composites Part B 95 (2016) 355e373

a2 ¼ 1 À

8
3K 2
F
4G
A À
HA þ
þ
3p
4
Au2mn Apu2mn

(28)

If F ¼ 0, G ¼ 0 i.e. no force excitation and thermal effect acting on

the shell, the frequencyeamplitude relation of the free nonlinear
vibration without damping is obtained.



u2NL ¼ u2mn 1 À

8
3K 2
A
HA þ
3p
4


(29)

4.1. Validation

3.3. Nonlinear vibration responses
Consider an eccentrically stiffened functionally graded toroidal
shell segment acted on by a uniformly distributed transverse load
q(t) ¼ QsinUt and a pre-loaded axial compression p, the set of
motion Eq. (16) has of the form.

h21 U þ h22 V þ h23 W þ n2 W 2 ¼ r1

d2 U
dt 2
d2 V

dt 2

;
;

hp2 m2

W þ n3 W 2 þ n4 W 3
L2


4d d
1
þn5 UW þ n6 VW þ 1 22 Q sin Ut À
Fa þ F*a
R
mnp



2
1
d W
dW
;
¼ r1 2 þ 2 r1 ε
À Fa þ F**
a
a
dt

dt

h31 U þ h32 V þ h33 W þ p

(30)

And the motion Eq. (18) by the use of Volmir's assumption
becomes

r1

This criterion is based on that, for large value of loading speed
the amplitude-time curve of obtained displacement response increases sharply depending on time and this curve obtains a
maximum by passing from the slope point and at the corresponding time t ¼ tcr the stability loss occurs. Here t ¼ tcr is called
critical time and the load corresponding to this critical time is called
dynamic critical buckling load pcr ¼ c1tcr (in case 1) or qcr ¼ c2tcr (in
case 2), respectively.
4. Results and discussion

where uNL is the nonlinear vibration frequency.

h11 U þ h12 V þ h13 W þ n1 W 2 ¼ r1

361

d2 W
dW
þ a1 W À a2 W 2 þ a3 W 3
þ 2r1 ε
dt

dt 2

 1

4d d
1
¼ 1 22 Q sin Ut À
Fa þ F*a À Fa þ F**
a
R
a
mnp

(31)

Using the fourth-order Runge-Kutta method into Eq. (30) or
Eq. (31) combined with initial conditions, the nonlinear vibration
responses of ES-FGM toroidal shell segment can be investigated.
3.4. Nonlinear dynamic buckling analysis
Investigation of the nonlinear dynamic buckling of ES-FGM
toroidal shell segment can be performed in two cases:
Case 1: the shell under axial compression varying as linear
function of time p ¼ c1t (c is a loading speed) and.
Case 2: the shell under lateral pressure varying as linear function of time q ¼ c2t.
By solving Eq. (18) for each case respectively, the dynamic
critical time tcr can be obtained according to Budiansky-Roth criterion [40].

Up to now there has been no publication about nonlinear
vibration and dynamic buckling of FGM toroidal shell segment, that
is reason to compare the results in this paper with cylindrical shells

(i.e. a toroidal shell segment with R / ∞).
Firstly, the results of natural frequencies in present will be
compared with results for the un-stiffened isotropic cylindrical
shell studied by Bhimaraddi [41], Lam and Loy [42], Li [43] and Shen
[28] and can be seen in Table 1.
As can be seen in Table 1 that good agreements are obtained in
this comparison. Moreover the frequencies calculated by Eq. (20)
(the full order equation ODE) and Eq. (21) (the Volmir's assumption) are quite close to each other.
Secondly, the natural frequencies of FGM cylindrical shell
illustrated in Table 2 are computed and compared with the results
of Loy et al. [22] using RayleigheRitz method and Shen [28] with
two kinds of micromechanics models: Voigt model and MorieTanaka model based on a higher order shear deformation shell
theory. A FGM cylindrical shell is made of stainless steel and nickel
material in initial temperature T0 ¼ 300 K is considered with the
following material properties.
ENi ¼ 205.09 GPa; yNi ¼ 0.31; rNi ¼ 8900 kg/m3;
ESS ¼ 207.7877 GPa; y ¼ 0.32; rNi ¼ 8166 kg/m3.
As can be seen, a very good agreement is obtained in the comparison with the result of Ref [22], but there are a few differences
with those of Ref [28] because the author uses above-mentioned
other theories.
In the following sections, the materials consist of Aluminum and
Alumina with Em ¼ 70 Â 109 N/m2;rm ¼ 2702 kg/m3;am ¼ 23 Â
10À6  CÀ1;Ec ¼ 380 Â 109 N/m2;rc ¼ 3800 kg;ac ¼ 5.4 Â 10À6  CÀ1
and Poisson's ratio is chosen to be 0.3. The elastic foundation parameters are taken as K1 ¼ 2.5 Â 108 N/m3, K2 ¼ 5 Â 105 N/m with
Pasternak foundation. The parameters n1 ¼ 50 and n2 ¼ 50 are the
number of stringer and ring stiffeners, respectively.
4.2. The fundamental frequencies
The natural frequencies of ES-FGM toroidal shell segment in
three cases using Eq. (20) are illustrated in Table 3. The datum of
problem: h ¼ 0.01 m; a ¼ 200 h; R ¼ 500 h; L ¼ 2a; d1 ¼ d2 ¼ h/2;

h1 ¼ h2 ¼ h/2; n1 ¼ n2 ¼ 50; k ¼ 1. It can be observed that the
natural frequencies of the shell on Pasternak foundation are the
highest while the natural frequencies with pre-loaded axial
compression (p ¼ 1 GPa) are the lowest. When the shell is subjected

Table 1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Comparison of dimensionless frequencies u ¼ Uðh=pÞ 2ð1 þ nÞr=E for an isotropic cylindrical shell (a/L ¼ 2; h/a ¼ 0.06, E ¼ 210 GPa, y ¼ 0.3, r ¼ 7850 kg/m3).
(m, n)

Bhimaraddi [41]

Lam and Loy [42]

Li [43]

Shen [28]

Present (Eq. (21))

Present (Eq. (20))

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

0.03692
0.03612
0.03566

0.03632

0.03748
0.03671
0.03635
0.03720

0.03739
0.03666
0.03634
0.03723

0.03712
0.03648
0.03620
0.03700

0.03781
0.03761
0.03733
0.03705

0.03755
0.03751
0.03717
0.03684

1)
2)
3)

4)


362

D.H. Bich et al. / Composites Part B 95 (2016) 355e373

Table 2
Comparisons of natural frequencies f ¼ 2Up(Hz) for FGM cylindrical shells (L/a ¼ 20, a/h ¼ 20, h ¼ 0.05 m, T ¼ 300 K).
(m, n)

SUS304/Ni
(1, 7)

(1, 8)

Ni/SUS304
(1, 7)

(1, 8)

Source

k
0.0

0.5

1.0

2.0

5.0

15.0

Loy et al. [22]
Shen [28]
Present (Eq. (21))
Present (Eq. (20))
Loy et al. [22]
Shen [28]
Present (Eq. (21))
Present (Eq. (20))

580.78
585.788
601.267
578.011
763.98
759.914
784.929
761.367

570.25
575.266
591.025
568.187
750.12
746.278

771.557
748.427

565.46
570.471
586.279
563.630
743.82
740.065
765.362
742.425

560.93
565.925
581.741
559.267
737.86
734.176
759.437
736.677

556.45
561.399
577.242
554.934
731.97
728.311
753.565
730.971

553.37
558.274
574.228
552.027
737.92
724.262
749.631
727.140

Loy et al. [22]
Shen [28]
Present (Eq. (21))
Present (Eq. (20))
Loy et al. [22]
Shen [28]
Present (Eq. (21))
Present (Eq. (20))

551.22
556.073
572.189
550.057
725.08
721.406
746.968
724.546

560.94
565.780
581.633

559.113
737.87
733.988
759.297
736.474

565.63
570.478
586.279
563.574
744.04
740.074
765.362
742.349

570.25
575.119
590.913
568.028
750.13
746.088
771.411
748.217

575.03
579.936
595.704
572.642
756.41
752.327

777.666
754.294

578.40
583.355
599.008
575.829
760.84
756.757
781.979
758.492

Table 3
The fundamental frequencies (sÀ1) using Eq. (20) in various cases of ES-FGM toroidal shell segment.
Cases

u1 (1, 1)a

u2 (1, 2)

u3 (1, 3)

u4 (2, 1)

u5 (2, 2)

Natural frequencies
Natural frequencies of shell on Pasternak foundation
Natural frequencies of shell with pre-loaded axial compression (p ¼ 1 GPa)

3346.998
3858.938
3329.860

2940.068
3657.946
2918.333

2612.503
3518.059
2585.179

3986.191
4802.866
3896.451

3803.026
4643.698
3709.798

a

The numbers in brackets indicate the vibration buckling mode (m, n).

to pre-loaded axial compression, the natural frequencies will
lessen.
4.3. Frequencyeamplitude curve
The frequencyeamplitude curve of nonlinear free vibration of
the shell and the effects of pre-loaded axial compression, elastic
foundation are indicated in Fig. 3. As can be seen, the lowest frequency will increase when the shell is on elastic foundation.

Whereas, it will decrease when the shell bears the pre-loaded axial
compression without elastic foundation.
The effect of amplitude of external force on the frequencyeamplitude curve in case of forced vibration is investigated in

Fig. 4. The line 1 is corresponding to the free vibration case
(F ¼ 0, p ¼ 0) of the shell without elastic foundation. The lines 2
and 3 correspond to the forced vibration cases of the shell
without elastic foundation under excited forces with F ¼ 5 Â 105
and F ¼ 8 Â 105, respectively. Finally, the lines 4 and 5 correspond to the free vibration and the forced vibration cases of the
shell on Pasternak foundation, respectively. It can be seen, the
frequencyeamplitude curve trend further from the curve of the
free vibration case when the amplitude of external force increases. The frequencyeamplitude curves of the shell on elastic
foundation move ahead in the increasing frequency direction in
comparison with those curves of the shell without elastic
foundation.

Fig. 3. Effects of elastic foundation and pre-loaded axial compression on frequencyeamplitude curve of ES-FGM toroidal shell segment in case of free vibration and no damping.


D.H. Bich et al. / Composites Part B 95 (2016) 355e373

363

Fig. 4. The frequencyeamplitude curve in case of forced vibration.

4.4. Nonlinear vibration responses
The comparison of the nonlinear response of the shell calculated
by the approximate Eq. (31) (Volmir's assumption) and the full
order system Eq. (30) is illustrated in Fig. 5.
From Tables 1 and 2 and Fig. 5, we conclude that the Volmir's

assumption can be used to investigate nonlinear dynamical analysis
with an acceptable accuracy.
In the next sections we will use the full order system Eq. (30) to
investigate nonlinear vibration responses and the approximate Eq.
(31) (Volmir's assumption) to analyze nonlinear dynamical buckling of the shell. As following we consider the effects of the characteristics of functionally graded materials, the pre-loaded axial
compression, the dimensional ratios, the elastic foundation and
thermal loads on the nonlinear dynamic responses of the ES-FGM
toroidal shell segments.

4.4.1. The effect of R/h ratio
Figs. 6 and 7 describe the effect of R/h ratio on nonlinear vibration of convex and concave stiffened FGM toroidal shell
segment, respectively. As can be seen, when R/h ratio increase, the
amplitudes of nonlinear vibration of both stiffened FGM also increase and the frequency does not modify much. Furthermore, the
amplitudes of nonlinear vibration of convex ES-FGM shell are
smaller than ones of concave ES-FGM shell.
4.4.2. The effect of L/R ratio
From Figs. 8 and 9, it can be seen that when increasing L/R ratio,
the amplitudes of nonlinear vibration of convex ES-FGM shell also
go up while those of concave ES-FGM shell decrease. It means that
the amplitudes of the more convex shells are greater than that of the
less convex ones whereas this feature of concave shells is completely
on the contrary. On the other hand, the amplitudes of nonlinear
vibration of convex shell are lower than ones of concave shell.

Fig. 5. The comparison of the nonlinear dynamical response calculated by Eq. (31) (Volmir's assumption) and Eq. (30) (the full order equation system) (k ¼ 1).


364

D.H. Bich et al. / Composites Part B 95 (2016) 355e373

Fig. 6. Effect of R/h ratio on nonlinear vibration response of convex ES-FGM toroidal shell segment on elastic medium (k ¼ 1).

Fig. 7. Effect of R/h ratio on nonlinear vibration response of concave ES-FGM toroidal shell segment on elastic medium (k ¼ 1).

Fig. 8. Effect of L/R ratio on nonlinear vibration response of convex ES-FGM toroidal shell segment on elastic medium (k ¼ 1).


D.H. Bich et al. / Composites Part B 95 (2016) 355e373

Fig. 9. Effect of L/R ratio on nonlinear vibration response of concave ES-FGM toroidal shell segment on elastic medium (k ¼ 1).

Fig. 10. Effect of L/a on nonlinear vibration response of convex ES-FGM toroidal shell segment on elastic medium (k ¼ 1).

Fig. 11. Effect of L/a on nonlinear vibration response of concave ES-FGM toroidal shell segment on elastic medium (k ¼ 1).

365


366

D.H. Bich et al. / Composites Part B 95 (2016) 355e373

4.4.3. The effect of L/a ratio
As can be observed in Figs. 10 and 11, the influence of ratio L/a on
the nonlinear response of the shell is similar as one of ratio L/R. In
addition, the amplitude of nonlinear vibration response of ES-FGM
concave shell is unequal.
4.4.4. The effects of volume-fraction k
The effect of volume-fraction k is shown in Fig. 12. The amplitudes of nonlinear vibration of ES-FGM toroidal shell segment increase when the value of volume fraction index increases.

Obviously this property appropriates to the real characteristic
of material, because the higher value of k corresponds to a
metal-richer shell which has less stiffness than a ceramic-richer
one.
4.4.5. The effect of elastic foundation
Based on Fig. 13, the amplitudes of nonlinear vibration of shell
without elastic foundation are the highest and those on Pasternak foundation are lowest. Furthermore, the amplitudes of
nonlinear vibration of shell without elastic foundation in this
example are about 5 times more than ones on Pasternak
foundation.

4.4.6. Effect of pre-loaded axial compression
Fig. 14 depicts the effect of pre-loaded axial compression on the
nonlinear responses of ES-FGM shell. Moreover, the amplitude of
nonlinear vibration of the shell increases when the value of axial
compression load increases. The pre-loaded axial compression
makes the load bearing capacity of the shell reducing under
dynamical loads.
4.4.7. Effect of temperature
Based on Figs. 15 and 16, we can see that the temperature increases, the stiffness of the structure will reduce and the amplitude
will rise. Therefore, the load bearing capacity of the structure will
mitigate.
The shell is preheated, the temperature field makes the shell to be
deflected outward (negative deflection) before it is affected by lateral
load. It means that the amplitude of structure will be negative.
Moreover, the amplitude of nonlinear vibration of convex shell
is about 8 times higher than one of concave shell. This can be
explained as the following: The geometry of the shell is concave,
thus when temperature field affects, it is expanded and has trend as
cylindrical shell. In contrarily, the shape of convex shell is convex

thus it is expanded so much.

Fig. 12. Effect of volume-fraction k on nonlinear vibration response of ES-FGM toroidal shell segment on elastic medium.

Fig. 13. Effect of elastic foundation on nonlinear vibration response of ES-FGM toroidal shell segment.


D.H. Bich et al. / Composites Part B 95 (2016) 355e373

Fig. 14. Effect of pre-loaded axial compression on nonlinear vibration response of ES-FGM toroidal shell segment.

Fig. 15. Effect of thermal environment on nonlinear vibration response of ES-FGM convex toroidal shell segment.

Fig. 16. Effect of thermal environment on nonlinear vibration response of ES-FGM concave toroidal shell segment.

367