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Journal of />Vibration and Control

Nonlinear dynamic analysis of imperfect functionally graded material double curved thin shallow shells
with temperature-dependent properties on elastic foundation
Nguyen Dinh Duc and Tran Quoc Quan
Journal of Vibration and Control published online 31 July 2013
DOI: 10.1177/1077546313494114
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Article

Nonlinear dynamic analysis of imperfect
functionally graded material double
curved thin shallow shells with
temperature-dependent properties
on elastic foundation

Journal of Vibration and Control
0(0) 1–23
! The Author(s) 2013
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DOI: 10.1177/1077546313494114
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Nguyen Dinh Duc and Tran Quoc Quan

Abstract
This paper presents an analytical investigation on the nonlinear dynamic analysis of functionally graded double curved thin
shallow shells using a simple power-law distribution (P-FGM) with temperature-dependent properties on an elastic
foundation and subjected to mechanical load and temperature. The formulations are based on the classical shell
theory, taking into account geometrical nonlinearity, initial geometrical imperfection, temperature-dependent properties
and unlike other publications, Poisson ratio is assumed to be varied smoothly along the thickness  ¼ ðzÞ. The nonlinear
equations are solved by the Bubnov-Galerkin and Runge-Kutta methods. The obtained results show the effects of
temperature, material and geometrical properties, imperfection and elastic foundation on the nonlinear vibration and
nonlinear dynamical response of double curved FGM shallow shells. Some results were compared with those of other
authors.

Keywords
Elastic foundation, FGM double curved thin shallow shells, imperfection, nonlinear dynamic analysis, temperaturedependent properties

1. Introduction
Functionally graded materials (FGMs), which are
microscopically composites and made from a mixture
of metal and ceramic constituents, have received considerable attention in recent years due to their high performance heat resistance capacity and their excellent
characteristics in comparison with conventional composites. By continuously and gradually varying the
volume fraction of constituent materials through a specific direction, FGMs are capable of withstanding ultrahigh temperature environments and extremely large
thermal gradients. Therefore, these novel materials
are chosen for use in temperature shielding structure
components of aircraft aerospace vehicles, nuclear
plants and engineering structures in various industries.
As a result, in recent years important studies have been
undertaken about the stability and vibration of FGM
plates and shells.

The research on FGM shells and plates under
dynamic load is attractive to many researchers in different parts of the world. Firstly we have to mention the
research group of Reddy et al. The vibration of functionally graded cylindrical shells has been investigated
by Loy et al. (1999). Lam and Li Hua (1999) has taken
into account the influence of boundary conditions on
the frequency characteristics of a rotating truncated
circular conical shell. Pradhan et al. (2000) studied
vibration characteristics of FGM cylindrical shells
under various boundary conditions. Ng et al. (2001)

Vietnam National University, Hanoi, Vietnam
Received: 22 March 2013; accepted: 19 May 2013
Corresponding author:
Nguyen Dinh Duc, Vietnam National University, Hanoi, 144 Xuan Thuy,
Cau Giay, Hanoi, Vietnam.
Email:

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studied the dynamic stability analysis of functionally
graded cylindrical shells under periodic axial loading.
The group of Ng et al. (2003) also published results on
generalized differential quadrate for free vibration of
rotating composite laminated conical shell with various
boundary conditions. In the same year, Yang and Shen
(2003) published the nonlinear analysis of FGM plates
under transverse and in-plane loads. Zhao et al. (2004)
studied the free vibration of a two-sided simply supported laminated cylindrical panel via the mesh-free
kp-Ritz method. With regard to vibration of FGM
plates, Vel and Batra (2004) gave a three-dimensional
exact solution for the vibration of FGM rectangular
plates. Sofiyev and Schnack (2004) investigated the stability of functionally graded cylindrical shells under linearly increasing dynamic tensional loading and

obtained the result for the stability of functionally
graded truncated conical shells subjected to a periodic
impulsive loading. They also published the result of the
stability of functionally graded ceramic–metal cylindrical shells under a periodic axial impulsive loading
in 2005. Ferreira et al. (2006) received natural frequencies of FGM plates by a meshless method. Zhao et al.
(2006) used the element-free kp-Ritz method for free
vibration analysis of conical shell panels. Liew et al.
(2006a, 2006b) studied the nonlinear vibration of a
coating-FGM-substrate cylindrical panel subjected to
a temperature gradient and dynamic stability of rotating cylindrical shells subjected to periodic axial loads.
Woo et al. (2006) investigated the nonlinear free vibration behavior of functionally graded plates. Ravikiran
Kadoli and Ganesan (2006) studied the buckling and
free vibration analysis of functionally graded cylindrical shells subjected to a temperature-specified
boundary condition. Wu et al. (2006) published their
results of nonlinear static and dynamic analysis of functionally graded plates. Sofiyev (2007) has considered
the buckling of functionally graded truncated conical
shells under dynamic axial loading. Prakash et al.
(2007) studied the nonlinear axisymmetric dynamic
buckling behavior of clamped functionally graded
spherical caps. Darabi et al. (2008) obtained the nonlinear analysis of dynamic stability for functionally
graded cylindrical shells under periodic axial loading.
Natural frequencies and buckling stresses of FGM
plates were analyzed by Hiroyuki Matsunaga (2008)
using 2-D higher-order deformation theory. Shariyat
(2008a, 2008b) also obtained the dynamic thermal
buckling of suddenly heated temperature- dependent
FGM cylindrical shells under combined axial compression and external pressure and dynamic buckling of
suddenly loaded imperfect hybrid cylindrical FGM
with temperature-dependent material properties under
thermo-electro-mechanical loads. Allahverdizadeh

et al. (2008) studied nonlinear free and forced vibration

analysis of thin circular functionally graded plates.
Sofiyev (2009) investigated the vibration and stability
behavior of freely supported FGM conical shells subjected to external pressure. Shen (2009) published a
valuable book, Functionally Graded Materials, Non
linear Analysis of Plates and Shells, in which the results
about nonlinear vibration of shear deformable FGM
plates are presented. Zhang and Li (2010) published
the dynamic buckling of FGM truncated conical
shells subjected to non-uniform normal impact load.
Ibrahim and Tawfik (2010) studied FGM plates subject
to aerodynamic and thermal loads. Fakhari and Ohadi
(2011) investigated nonlinear vibration control of functionally graded plate with piezoelectric layers in thermal environment. Ruan et al. (2012) analyzed dynamic
stability of functionally graded materials’ skew plates
subjected to uniformly distributed tangential follower
forces. Najafov et al. (2012) studied vibration and stability of axially compressed truncated conical shells
with a functionally graded middle layer surrounded
by elastic medium. Bich et al. (2012, 2013) investigated
nonlinear dynamical analysis of eccentrically stiffened
functionally graded cylindrical panels and shallow
shells using the classical shell theory. Recently, Duc
(2013) investigated the nonlinear dynamic response of
imperfect FGM double curved shallow shells eccentrically stiffened on an elastic foundation.
However, in practice, the FGM structure is usually
exposed to high-temperature environments, where significant changes in material properties are unavoidable.
Therefore, the temperature dependence of their properties should be considered for an accurate and reliable
prediction of deformation behavior of the composites.
Duc and Tung (2010) investigated mechanical and
thermal post-buckling of FGM plates with temperature-dependent properties using first order shear

deformation theory. Huang and Shen (2004) studied
nonlinear vibration and dynamic response of FGM
plates in a thermal environment with temperaturedependent material properties – volume fraction follows
a simple power law for P-FGM plate. Shariyat investigated vibration and dynamic buckling control of imperfect hybrid FGM plate subjected to thermo-electromechanical conditions (2009) and dynamic buckling of
suddenly load imperfect hybrid FGM cylindrical shells
(2008) with temperature-dependent material properties.
Kim (2005) studied temperature dependent vibration
analysis of functionally graded rectangular plates by
the finite element method. It is evident from the literature that investigations considering the temperature
dependence of material properties for FGM shells are
few in number. It should be noted that all the publications mentioned above (Huang and Shen, 2004; Kim,
2005; Shariyat, 2009) use displacement functions and
the volume fraction follows a simple power law.

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Duc and Quan

3

This paper presents a dynamic nonlinear response of
double curved FGM thin shallow shells with temperature-dependent properties subjected to mechanical load
and temperature on an elastic foundation. The formulations are based on the classical shell theory taking
into account geometrical nonlinearity, initial geometrical with Pasternak type elastic foundation. The
obtained results show the effects of material, geometrical properties, elastic foundation and imperfection on
the dynamical response of FGM shallow shells.

ðz, TÞ and Kðz, TÞ are obtained by substituting equation (2.1) into (2.2) as

2. Double curved FGM shallow shell
on elastic foundation

where

2z þ h
Vm ðzÞ ¼
2h

N
, Vc ðzÞ ¼ 1 À Vm ðzÞ

ð1Þ

where N is volume fraction index (0 N 5 1).
Effective properties Preff of the FGM panel are determined by linear rule of mixture as
PrðzÞ ¼ Pr Vm ðzÞ þ Pr Vc ðzÞ
eff

m

c

¼ ½Ec ðT Þ, c ðT Þ, ðT Þ, c ðT Þ, Kc ðT ފ
þ ½Emc ðT Þ, mc ðT Þ, mc ðT Þ, mc ðT Þ, Kmc ðT ފ



2z þ h N
2h
ð3Þ

Emc ðT Þ ¼ Em ðT Þ À Ec ðT Þ,

Consider an FGM double curved thin shallow shell of
radii of curvature Rx ,Ry length of edges a, b and uniform thickness h. A coordinate system ðx, y, zÞ is established in which ðx, yÞ plane on the middle surface of the
shell and z is the thickness direction ðÀh=2 z h=2Þ,
as shown in Figure 1.
For the P-FGM shell, the volume fractions of constituents are assumed to vary through the thickness
according to the following power law distribution:


½Eðz, TÞ, vðz, TÞ, ðz, TÞ, ðz, TÞ, Kðz, Tފ

ð2Þ

where Pr denotes a temperature independent material
property, and subscripts m and c stand for the metal
and ceramic constituents, respectively. Specific expressions of modulus of elasticity Eðz, TÞ, ðz, TÞ, ðz, TÞ,

mc ðT Þ ¼ m ðT Þ À c ðT Þ,

mc ðT Þ ¼ m ðT Þ À c ðT Þ,

ð4Þ

ðmc ðT Þ ¼ m ðT Þ À c ðT Þ,
Kmc ðT Þ ¼ Km ðT Þ À Kc ðT Þ
The values with subscripts m and c belong to metal
and ceramic respectively, and unlike other publications,
the Poisson ratio is assumed to be varied smoothly
along the thickness  ¼ ðzÞ. It is evident from equations (2.3) and (2.4) that the upper surface of the panel
(z ¼ Àh=2) is ceramic-rich, while the lower surface
(z ¼ h=2) is metal-rich, and the percentage of ceramic
constituent in the panel is enhanced when N increases.
A material property Pr, such as the elastic modulus E,
Poisson ratio , the mass density , the thermal expansion coefficient  and coefficient of thermal conduction
K can be expressed as a nonlinear function of
temperature:
À
Á
Pr ¼ P0 PÀ1 TÀ1 þ 1 þ P1 T1 þ P2 T2 þ P3 T3

ð5Þ

in which T ¼ T0 þ ÁTðzÞ and T0 ¼ 300K (room temperature); P0, P-1, P1, P2 and P3 are coefficients characteristic of the constituent materials.
The shell–foundation interaction is represented by
the Pasternak model as
qe ¼ k1 w À k2 r2 w

ð6Þ

where r2 ¼ @2 =@x2 þ @2 =@y2 , w is the deflection of the
panel, k1 is Winkler foundation modulus and k2 is
the shear layer foundation stiffness of the Pasternak
model.

Figure 1. Geometry and coordinate system of a P-functionally
graded material double curved shallow shell on elastic
foundation.

3. Theoretical formulation
In this study, the classical shell theory is used to establish governing equations and determine the nonlinear

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response of FGM thin shallow double curved shells
(Brush and Almroth, 1975; Reddy, 2004):

0

1

0

1

0

1
ð7Þ

where
"0x

1

0

u, x À w=Rx þ w2,x =2

1 0

B
C B
C
B "0 C B
C
B y C ¼ B v, y À w=Ry þ w2,y =2 C,

@
A @
A
0
xy
u, y þ v, x þ w, x w, y

kx

1

0

Àwx, x

1

B
C B
C
Bk C B
C
B y C ¼ B Àwy, y C
@
A @
A
kxy
Àw, xy
ð8Þ

In which u, v are the displacement components along
the x, y directions, respectively.
Hooke law for an FGM shell is defined as
ðx , y Þ ¼
xy ¼

Z

EðzÞ
zj dz
2
Àh=2 1 À ðzÞ
Z h=2
EðzÞðzÞ j
z dz
I2j ¼
2
Àh=2 1 À ðzÞ
Z h=2
EðzÞ
1
zj dz ¼ ðI1j À I2j Þ
I3j ¼
2
Àh=2 2½ð1 þ ðzÞ Š
Z h=2
EðzÞðzÞ
ÁTðzÞð1, zÞdz
ðÈ1 , È2 Þ ¼ À
Àh=2 1 À ðzÞ

E
xy
2ð1 þ Þ

ð12Þ

The nonlinear equilibrium equations of a perfect
FGM double curved shallow shell based on the classical
shell theory are (Nayfeh and Pai, 2004):
@2 u
@2 t
@2 v
¼ 1 2 Mx, xx
@ t

Nx, x þ Nxy, y ¼ 1
Nxy, x þ Ny, y

þ 2Mxy, xy þ My, yy þ

Nx Ny
þ
Rx Ry

ð13Þ

þ Nx w, xx þ 2Nxy w, xy þ Ny w, yy

Ã

E Â
ð"x , "y Þ þ vð"y , "x Þ À ð1 À vÞÁTð1, 1Þ
2
1Àv

þ q À k1 w þ k2 r2 w ¼ 1

@2 w
@2 t

where
ð9Þ

where ÁT is the temperature rise from a stress-free initial state. The force and moment resultants of the FGM
shallow shell are determined by
Z

h=2

I1j ¼

"0x
kx
"x
B
C B
C
B
C
B

C B 0 C
B
C
B "y C ¼ B "y C þ zB ky C
@
A @
A
@
A
0
2k
xy
xy
xy

0

where Iij ði ¼ 1, 2, 3; j ¼ 0, 1, 2Þ:

h=2

ðNi , Mi Þ ¼

i ð1, zÞdz i ¼ x, y, xy

ð10Þ



c À m

1 ¼
ðzÞdz ¼ m þ
h
Nþ1
Àh=2
Z

h=2

ð14Þ

With Volmir’s assumption (Volmir, 1972):
2
u ( w, v ( w, in (13) the inertia 1 @@t2u ! 0 and
2
1 @@t2v ! 0, the nonlinear motion equations for perfect
FGM shells can be written in terms of deflection w and
force resultants as

Àh=2

Nx, x þ Nxy, y ¼ 0
Substitution of equations (7) and (9) into equation
(10) and the result into equation (10) gives the constitutive relations as
À

Á
Nx , Ny , Mx , My ¼ ðI10 , I20 , I11 , I21 Þ"0x

þ ðI21 , I11 , I22 , I12 Þky

þ ðÈ1 , È1 , È2 , È2 ÞÁT
À
Á
0
þ 2ðI31 , I32 Þkxy
Nxy , Mxy ¼ ðI30 , I31 Þ
xy

Mx, xx þ 2Mxy, xy þ My, yy þ

Nx Ny
þ
Rx Ry

ð15Þ

þ Nx w, xx þ 2Nxy w, xy þ Ny w, yy
þ q À k1 w þ k2 r2 w ¼ 1

þ ðI20 , I10 , I21 , I11 Þ"0y
þ ðI11 , I21 , I12 , I22 Þkx

Nxy, x þ Ny, y ¼ 0

@2 w
@2 t

Calculated from equation (11)
ð11Þ
"0x ¼ D0 ðI10 Nx À I20 Ny þ D1 w, xx þ D2 w, yy À D3 È1 ÁTÞ

"0y ¼ D0 ðI10 Ny À I20 Nx þ D1 w, yy þ D2 w, xx À D3 È1 ÁTÞ
1
ðNxy þ 2I31 w, xy Þ
0xy ¼
I30
ð16Þ

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where

From the constitutive relations (16) in conjunction
with equation (20):
D0 ¼

1

, D1 ¼ I10 I11 À I20 I21
I210 À I220

D2 ¼ I10 I21 À I20 I11 , D3 ¼ I10 À I20

ð17Þ

Once again substituting equation (16) into
the expression of Mij in (11), then Mij into the equation (15) leads to
Nx, x þ Nxy, y ¼ 0
Nxy, x þ Ny, y ¼ 0
P1 r4 f þ P2 r4 w þ þNx w, xx
þ 2Nxy w, xy þ Ny w, yy
þ

P3 ¼

where
P1 ¼ D0 D2 , P2 ¼ D0 ðI11 D1 þ I21 D2 Þ À I12

ð19Þ

f(x,y) is stress function defined by
Nx ¼ f, yy , Ny ¼ f, xx , Nxy ¼ Àf, xy

ð20Þ

For imperfect FGM shells, equation (18) is modified
into form as



P1 r4 f þ P2 r4 w þ f, yy w, xx þ wÃ, xx
À 2f, xy w, xy þ wÃ, xy
þ

ð24Þ
where

Nx Ny
þ
þ q À k1 w
Rx Ry



Setting equation (23) into equation (22) gives the
compatibility equation of an imperfect FGM double
curved shell as
0 2
1
w, xy À w, xx w, yy þ 2w, xy wÃ, xy
B
C
B Àw, xx wÃ, yy À w, yy wÃ, xx
C
r4 f þ P3 r4 w À P4 B
C ¼ 0:
@
A
w, yy w, xx

À
À
Rx
Ry

ð18Þ

@2 w
þ k2 r2 w ¼ 1 2
@ t



"0x ¼ D0 ðI10 f, yy À I20 f, xx þ D1 w, xx þ D2 w, yy À D3 È1 ÁTÞ
"0y ¼ D0 ðI10 f, xx À I20 f, yy þ D1 w, yy þ D2 w, xx À D3 È1 ÁTÞ
1
ðÀf, xy þ 2I31 w, xy Þ
0xy ¼
I30
ð23Þ

D2
,
I10

P4 ¼

1
D0 I10

ð25Þ

Equations (21) and (24) are nonlinear equations in
terms of variables w and f and are used to investigate
the dynamic response of thick double curved shallow
FGM shells on elastic foundations subjected to mechanical, thermal and thermo-mechanical loads.
Depending on the in-plane restraint at the edges,
three cases of boundary conditions, labeled as Cases
1, 2 and 3 may be considered (Reddy, 2004; Duc and
Tung, 2010):
Case 1: Four edges of the shallow shell are simply supported and freely movable (FM). The associated
boundary conditions are
w ¼ Nxy ¼ Mx ¼ 0, Nx ¼ Nx0 at x ¼ 0, a



þ f, xx w, yy þ wÃ, yy

ð21Þ

f, yy f, xx
@2 w
þ
þ q À k1 w þ k2 r2 w ¼ 1 2
@ t
Rx
Ry
Ã

in which w ðx, yÞ is a known function representing initial small imperfection of the shell. Following Volmir’s

approach, the geometrical compatibility equation for
an imperfect double curved shallow shell is written as

ð26Þ
w ¼ Nxy ¼ My ¼ 0, Ny ¼ Ny0 at y ¼ 0, b:
Case 2: Four edges of the shallow shell are simply supported and immovable (IM). In this case, boundary
conditions are
w ¼ u ¼ Mx ¼ 0, Nx ¼ Nx0 at x ¼ 0, a
w ¼ v ¼ My ¼ 0, Ny ¼ Ny0 at y ¼ 0, b

ð27Þ

Case 3: All edges are simply supported. Two edges
x ¼ 0, a are freely movable, whereas the remaining
two edges y ¼ 0, b are immovable. For this case, the
boundary conditions are defined as

0
2
"0x, yy þ "0y, xx À
xy,
xy ¼ w, xy À w, xx w, yy

þ 2w, xy wÃ, xy À w, xx wÃ, yy À w, yy wÃ, xx À

w, yy w, xx
À
:
Rx
Ry

ð22Þ

w ¼ Nxy ¼ Mx ¼ 0, Nx ¼ Nx0 at x ¼ 0, a
w ¼ v ¼ My ¼ 0, Ny ¼ Ny0 at y ¼ 0, b

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where Nx0 , Ny0 are in-plane compressive loads at movable edges (i.e. Case 1 and the first of Case 3) or are
fictitious compressive edge loads at immovable edges
(i.e. Case 2 and the second of Case 3). In the present
study, the edges of curved shallow shells are assumed to
be simply supported and immovable (Case 2).
Taking into account temperature-dependent material properties, the mentioned conditions (27) can
be satisfied if the deflectionw, wà and the stress function

f are written in the form similar to (Duc and Tung,
2010):
mx
ny
sin
w ¼ ðtÞ sin
a
b
mx
ny
Ã
w ¼ 0 sin
sin
a
b
2mx
2ny
f ¼ A1 cos
þ A2 cos
a
b
mx
ny
þ A3 sin
sin
a
b
mx
ny 1
1

þ A4 cos
cos
þ Nx0 y2 þ Ny0 x2
a
b
2
2

A simply supported FGM double curved shallow
shell on elastic foundations with all immovable edges
is considered. The FGM shell is subjected to uniform
external pressure q and simultaneously exposed to temperature environments or subjected to temperature gradient which varies through the thickness of the shell.
The in-plane condition on immovability at all edges, i.e.
u ¼ 0 at x ¼ 0, a and v ¼ 0 at y ¼ 0, b, is fulfilled in an
average sense as (Shen, 2004; Kim, 2005; Duc and
Tung, 2010)
Z bZ
0

@u
dx dy ¼ 0,
@x

Z aZ
0

0

b

@v
dydx ¼ 0
@y

ð31Þ

From equations (7) and (16) the following expressions, in which equation (20) and imperfections have
been included, can be obtained:
@u
¼ D0 ðI10 f, yy À I20 f, xx þ D1 w, xx þ D2 w, yy À D3 È1 Þ
@x
1
w
À w2,x À w, x wÃ, x þ
2
Rx
@v
¼ D0 ðI10 f, xx À I20 fyy þ D1 w, yy þ D2 w, xx À D3 È1 Þ
@y
1
w
À w2,y À w, y wÃ, y þ
2
Ry

ð29Þ

m ¼ m=a, n ¼ n=b, m, n ¼ 1, 2, . . . , are natural
numbers representing the number of half waves in the
x and y directions respectively;  is the deflection amplitude; 0 ¼ const, varying between 0 and 1, represents

the size of the imperfections. We should note that the
choice of f in (29) is different from the form used in Duc
and Tung, 2010 and Duc, 2013.
The coefficients Ai ði ¼ 1 Ä 3Þ are determined by
substitution of equation (29) into equation (24) as
P4 2n
P4 2m
ð þ 20 Þ, A2 ¼
ð þ 20 Þ,
2
32m
322n
 2

P4
n 2m
A3 ¼ À
þ
 À P3 , A4 ¼ 0:
Á2
Rx Ry
2m þ 2n

0

a

ð32Þ
Substitution of equation (29) into equation (32) and
then the result into equation (31) gives fictitious edge

compressive loads as
&


4
4
I10 I20
2
2
ðI

þ
I

Þ
À
þ
Nx0 ¼ È1 þ
21 n
11 m
mn2
mn2 Rx Ry
"
#
 2

P4
n 2m
n o
þ

þ4 À

À P3
Á2
mb2
Rx Ry
2 þ 2

A1 ¼

ð30Þ

m

n

1
þ ðI10 2m þ I20 2n Þð þ 20 Þ,
8
ð33Þ

Table 1. Material properties of the constituent materials of the considered functionally graded material shells.
Material

Property

P0

PÀ1

P1

P2

P3

Si3N4 (Ceramic)

E (Pa)
 (kg/m3)
 ðKÀ1 Þ
k ðW=mKÞ

E (Pa)
 (kg/m3)
 ðKÀ1 Þ
k ðW=mKÞ


348.43e9
2370
5.8723eÀ6
13.723
0.24
201.04e9
8166
12.330eÀ6
15.379
0.3177

0
0
0
0
0
0
0
0
0
0

À3.70eÀ4
0
9.095eÀ4
0
0
3.079eÀ4
0
8.086eÀ4
0
0

2.160eÀ7
0
0
0
0
À6.534eÀ7
0
0

0
0

À8.946eÀ11
0
0
0
0
0
0
0
0
0

SUS304 (Metal)

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&


4
4
I10 I20
2
2
Ny0 ¼ È1 þ
ðI

þ
I

Þ
À
þ
11 n
21 m
mn2
mn2 Ry Rx
"
# '
 2

P4
n 2m
m

þ
þ4 À
À P3

Á2
2
2
2
na
R
R
x
y
m þ n
1
þ ðI20 2m þ I10 2n Þð þ 20 Þ:
8
Specific expressions of parameter È1 in two cases of
thermal loading will be determined.
Subsequently, substitution of equation (29) into
equation (21) and applying the Galerkin procedure
for the resulting equation yields
 2

À
Á2
ab
n 2m
½ðP3 þ P1 P4 Þ
þ

þ ðP2 À P1 P3 Þ 2m þ 2n
4
Rx Ry
 2
2
À
Á
P4
n 2m
ÀÀ
þ
Àk2 2m þ 2n À k1 Š
Á2
Rx Ry
2m þ 2n
"
#
 2

8m n
P4
n 2m
þ
þ
À P3 ð þ 0 Þ
À
Á2
3
Rx Ry
2m þ 2n

 2

!
P4
m 2n
4
þ
þ
À P1 P4 m n ð þ 20 Þ
3
6m n Ry Rx
Á
P4 ab À 4
m þ 4n ð þ 0 Þð þ 20 Þ
À
64
Á
ab À
Nx0 2m þ Ny0 2n ð þ 0 Þ
À
4


4
Nx0 Ny0
4q
ab @2 
þ
¼ 1
þ

þ
m n Rx
m n
4 @2 t
Ry
ð34Þ
Where m, n are odd numbers. This is a basic equation governing the nonlinear dynamic response for thin
imperfect FGM double curved shallow shells under
mechanical, thermal and thermo-mechanical loading
conditions. In what follows, some thermal loading conditions will be considered.
Introducing Nx0 , Ny0 at (33) into equation (34) gives
€ þ m1 ðtÞ þ m2 2 ðtÞ þ m3 3 ðtÞ
ðtÞ
2

þ m4 ðtÞ0 þ m5  ðtÞ0 þ
þ m 7 0 ¼ m 8 q þ m 9

m6 ðtÞ20

ð35Þ

and specific expressions of coefficients mi ði ¼ 1 Ä 8Þ
are given in Appendix A and ðtÞ -deflection of

middle point of the plate ðtÞ ¼ wx¼a=2
y¼b=2 .
For linear free vibration for FGM plate equation
(35) gets the form:
::

ðtÞ þ m1 ðtÞ ¼ 0

ð36Þ

The fundamental frequency of natural vibration of
the FGM plate can be determined:
!L ¼

pffiffiffiffiffiffi
m1

ð37Þ

Equation (35) – for obtaining the nonlinear dynamic
response the
initial conditions are assumed as
:
ð0Þ ¼ 0 , ð0Þ ¼ 0. The applied loads are varying as
function of time. The nonlinear dynamic response of
the FGM shell acted on by the harmonic uniformly
excited transverse load qðtÞ ¼ Q0 sin t are obtained
by solving equation (35) combined with the initial conditions and with the use of the Runge-Kutta method.

4. Numerical results and discussion
Here, several numerical examples will be presented for
perfect and imperfect simply supported midplanesymmetric of the FGM shell. The typical values of
the coefficients of the materials mentioned in equation
(5) are listed in Table 1 (Reddy and Chin, 1998).

Table 2. Frequency of natural vibration (rad/s) of spherical
shallow shells with Rx ¼ Ry ¼ 5 ðmÞ and N ¼ 1.
T-ID
T-D
ðm, nÞ

Present

Bich et al.
(2012)

Present

(1,1)
(1,2) and (2,1)
(2,2)
(1,3) and (3,1)
(2,3) and (3,2)

3.4582e3
7.3694e3
11.1643e3
15.9583e3
18.3469e3

2.5604e3
6.7431e3
9.3093e3
14.5762e3
16.0638e3

2.4229e3
6.0439e3
9.5415e3
14.3293e3
16.1738e3

T-D: temperature dependent; T-ID: temperature independent.

Table 3. A comparison
the fundamental natural
ffi
pffiffiffiffiffiffiffiffiffiffiamong
frequency É ¼ !L h c =Ec of the square Al/Al2O3 functionally
graded material plates.
pffiffiffiffiffiffiffiffiffiffiffi
É ¼ !L h c =Ec
a=h

N

Matsunaga (2008)

Shariyat (2009)

Present

5

0

0.5
1
0
0.5
1

0.2121
0.1819
0.1640
0.05777
0.04917
0.04426

0.2083
0.1762
0.1594
0.05682
0.04876
0.04369

0.2287
0.1575
0.1380
0.4865
0.0386
0.0297

10

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À
ÁÂ À
Á Ã1=2
Table 4. Natural frequency parameter !L a2 =h 0 1 À 2 E0
for Si3N4/SUS304 double curved shallow shells in thermal
environments.
T-ID
(m,n)
Tm ¼ 300 K

Tm ¼ 300 K

Tc ¼ 400 K

Tc ¼ 600 K

(1,1)
(1,2)
(2,2)
(1,3)
(2,3)
(1,1)
(1,2)
(2,2)
(1,3)
(2,3)

and (2,1)
and (3,1)
and (3,2)
and (2,1)
and (3,1)
and (3,2)

T-D

Shen (2004)

Present

Present

Shen (2004)

7.514

17.694
26.717
32.242
39.908
7.305
17.486
26.506
31.970
39.692

7.0683
18.9712
29.8519
34.7710
40.6462
7.0683
18.9712
30.8519
36.7710
44.6462

6.7888
16.1335
25.4717
30.6960
40.0313
6.7574
15.5673
28.4734
33.0257

43.9825

7.474
17.607
26.590
32.088
39.721
7.171
17.213
26.109
31.557
39.114

T-D: temperature dependent; T-ID: temperature independent.

Figure 2. Effect of b=a on nonlinear dynamic response of the functionally graded material shell (temperature-independent
ÁT ¼ 300 ðKÞ). b=h ¼ 30, m ¼ n ¼ 1, N ¼ 1, Rx ¼ Ry ¼ 0:6 ðmÞ, Q0 ¼ 75000 ðN=m2 Þ,  ¼ 3000 ðrad=sÞ, 0 ¼ 0: k1 ¼ k2 ¼ 0
ðÀÀ : b=a ¼ 1; À À À : b=a ¼ 2Þ.

The metal-rich surface temperature Tm is maintained
at a stress-free initial value while ceramic-rich surface
temperature Tc is elevated and nonlinear steady temperature conduction is governed by one-dimensional

Fourier equation
!
d
dT
KðzÞ
¼ 0, Tðz ¼ Àh=2Þ ¼ Tc ,
dz

dz
Tðz ¼ h=2Þ ¼ Tm

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Figure 3. Effect of b=a on nonlinear dynamic response of the functionally graded material shell (temperature-dependent,
ÁT ¼ ÁT ðzÞ, Tc ¼ 500 ðKÞ, Tm ¼ 300 ðKÞ). b=h ¼ 30, m ¼ n ¼ 1, N ¼ 1, Rx ¼ Ry ¼ 0:6 ðmÞ, Q0 ¼ 75000 ðN=m2 Þ,
 ¼ 3000 ðrad=sÞ, 0 ¼ 0: k1 ¼ k2 ¼ 0: ðÀÀ : b=a ¼ 1; À À À : b=a ¼ 2Þ.

Figure 4. Effect of b=h on nonlinear dynamic response of the functionally graded material shell (temperature-independent,
ÁT ¼ 300 ðKÞ). b=a ¼ 2, N ¼ 1, m ¼ n ¼ 1, Rx ¼ Ry ¼ 0:6 ðmÞ, Q0 ¼ 75000 ðN=m2 Þ,  ¼ 3000 ðrad=sÞ, 0 ¼ 0: k1 ¼ k2 ¼ 0.

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Figure 5. Effect of b=h on nonlinear dynamic response of the functionally graded material shell (temperature-dependent,
ÁT ¼ ÁT ðzÞ, Tc ¼ 500 ðKÞ, Tm ¼ 300 ðKÞ). b=a ¼ 2, N ¼ 1, m ¼ n ¼ 1, Rx ¼ Ry ¼ 0:6 ðmÞ, Q0 ¼ 75000 ðN=m2 Þ,
 ¼ 3000 ðrad=sÞ, 0 ¼ 0: k1 ¼ k2 ¼ 0. ðÀÀÀ : b=h ¼ 20; À À À : b=h ¼ 30Þ.

Figure 6. Effect of imperfection 0 on nonlinear dynamic response of functionally graded material shell (temperature-independent,
ÁT ¼ 300 ðKÞ). b=a ¼ 1, b=h ¼ 30, N ¼ 1, m ¼ n ¼ 1, Rx ¼ Ry ¼ 0:6 ðmÞ, k1 ¼ k2 ¼ 0.

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Figure 7. Effect of imperfection 0 on nonlinear dynamic response of functionally graded material shell (temperature-dependent,
ÁT ¼ ÁTðzÞ), b=a ¼ 1, b=h ¼ 30, N ¼ 1, m ¼ n ¼ 1, Rx ¼ Ry ¼ 0:6 ðmÞ, k1 ¼ k2 ¼ 0: ðÀÀÀ : 0 ¼ 0 . . . : 0 ¼ 0:001;
À À À : 0 ¼ 0:003Þ.

Figure 8. Deflection-velocity relation (d=dtÀ ) with temperature-independent, ÁT ¼ 300 ðKÞ of the functionally graded material
shallow spherical shell.

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Figure 9. Deflection-velocity (d=dtÀ) with temperature-dependent, ÁT ¼ ÁT ðzÞ, Tc ¼ 500 ðKÞ, Tm ¼ 300 ðKÞ of the functionally
graded material shallow spherical shell.

Figure 10. Effect of  on nonlinear dynamic response of the functionally graded material shell (temperature-independent,
ÁT ¼ 300 ðKÞ. b=a ¼ 1, b=h ¼ 30, N ¼ 1, m ¼ n ¼ 1, Rx ¼ Ry ¼ 0:6 ðmÞ, Q0 ¼ 75000 ðN=m2 Þ, 0 ¼ 0: k1 ¼ k2 ¼ 0.

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Figure 11. Effect of  on nonlinear dynamic of the functionally graded material shell (T À D, ÁT ¼ ÁTðzÞ, Tc ¼ 500 ðKÞ,
Tm ¼ 300 ðKÞ). b=a ¼ 1, b=h ¼ 30, N ¼ 1, m ¼ n ¼ 1, Rx ¼ Ry ¼ 0:6 ðmÞ, Q0 ¼ 75000 ðN=m2 Þ, 0 ¼ 0: k1 ¼ k2 ¼ 0.
ðÀÀÀ :  ¼ 2500ðrad=sÞ; À À À :  ¼ 3000ðrad=sÞÞ.

Figure 12. Effect of amplitude Q0 on dynamic response of the functionally graded material shell (temperature-independent,
ÁT ¼ 300 ðKÞ). b=a ¼ 1, b=h ¼ 30, N ¼ 1, m ¼ n ¼ 1, Rx ¼ Ry ¼ 0:6 ðmÞ,  ¼ 3000 ðrad=sÞ, 0 ¼ 0: k1 ¼ k2 ¼ 0.

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Figure 13. Effect of amplitude Q0 on dynamic response of the functionally graded material shell
(T À D, ÁT ¼ ÁT ðzÞ, Tc ¼ 500 ðKÞ, Tm ¼ À300 ðKÞ).
Á b=a ¼ 1, b=h ¼ 30, N À¼ 1, m Á¼ n ¼ 1, Rx ¼ Ry ¼ 0:6 ðmÞ,  ¼ 3000 ðrad=sÞ,
0 ¼ 0: k1 ¼ k2 ¼ 0. ðÀÀÀ : Q0 ¼ 75000 N=m2 ; À À À : Q0 ¼ 95000 N=m2 Þ.

Figure 14. Effect of elastic foundations k1 , k2 on nonlinear dynamic response of the functionally graded material shell (temperatureindependent, ÁT ¼ 300 ðKÞ). b=a ¼ 1, b=h ¼ 30, N ¼ 1, m ¼ n ¼ 1, Rx ¼ Ry ¼ 0:6 ðmÞ, Q0 ¼ 75000 ðN=m2 Þ,  ¼ 3000 ðrad=sÞ,
0 ¼ 0.

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Figure 15. Effect of elastic foundations k1 , k2 (temperature-dependent, ÁT ¼ ÁT ðzÞ, Tc ¼ 500 ðKÞ, Tm ¼ 300 ðKÞ) on nonlinear
dynamic response of the functionally graded material shell. b=a ¼ 1, b=h ¼ 30, N ¼ 1, m ¼ n ¼ 1, Rx ¼ Ry ¼ 0:6 ðmÞ,
Q0 ¼ 75000 ðN=m2 Þ,  ¼ 3000 ðrad=sÞ, 0 ¼ 0. ðÀÀ: k1 ¼ 20, k2 ¼ 100; À À À : k1 ¼ 50, k2 ¼ 200Þ.

Figure 16. Effect of temperature (temperature-independent) on nonlinear dynamic response of the shell. b=a ¼ 1, b=h ¼ 30, N ¼ 1,
m ¼ n ¼ 1, Rx ¼ Ry ¼ 0:6 ðmÞ, Q0 ¼ 75000 ðN=m2 Þ,  ¼ 3000 ðrad=sÞ, 0 ¼ 0.

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Figure 17. Effect of temperature (temperature-dependent) on nonlinear dynamic response of the shell. b=a ¼ 1, b=h ¼ 30, N ¼ 1,
m ¼ n ¼ 1, Rx ¼ Ry ¼ 0:6 ðmÞ, Q0 ¼ 75000 ðN=m2 Þ,  ¼ 3000 ðrad=sÞ, 0 ¼ 0. ——: Tc ¼ 400 ðKÞ, Tm ¼ 300 ðKÞ;
À À À : Tc ¼ 500 ðKÞ, Tm ¼ 300 ðKÞ.

Figure 18. The frequency-amplitude relation of nonlinear free vibration !nl - (temperature-dependent, ÁT ¼ 300 ðKÞ).

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Figure 19. The frequency-amplitude relation of nonlinear free vibration !nl - (temperature-independent, ÁT ¼ ÁT ðzÞ,
Tc ¼ 500 ðKÞ, Tm ¼ 300 ðKÞ).

Figure 20. Effect of Poisson ratio on dynamic response of the functionally graded material spherical shallow shell.
b=a ¼ 1, b=h ¼ 30, N ¼ 1, m ¼ n ¼ 1, Rx ¼ Ry ¼ 0:6 ðmÞ, Q0 ¼ 75000 ðN=m2 Þ,  ¼ 3000 ðrad=sÞ, 0 ¼ 0, Tc ¼ 500 ðKÞ, Tm ¼ 300 ðKÞ.
——:  ¼ ðzÞ; À À À :  ¼ constÞ.

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Figure 21. Nonlinear dynamic responses of functionally graded material spherical shallow shells (without the temperature gradient,
1: Bich (2012); 2: Present).

Using KðzÞ in equation (3), the solution of equation
(38) may be found in terms of polynomial series, and
the first eight terms of this series gives the following
approximation (Shen, 2004; Kim, 2005; Duc and
Tung, 2010):

2

P5

ðÀKmc =Kc Þj
j¼0
jNþ1

H¼

P5

4

j¼0

r

mc þEmc c
þ Ecðjþ1ÞNþ2

Emc mc
þ ð jþ2ÞNþ2

3
5

ðÀKmc =Kc Þj
jNþ1

ð43Þ

P5 ðÀrN Kmc =Kc Þj
j¼0

jNþ1
ðÀKmc =Kc Þj
j¼0
jNþ1

TðzÞ ¼ Tm þ ÁT À ÁT P5

ð39Þ

where r ¼ ð2z þ hÞ=2h and ÁT ¼ Tc À Tm is defined as
the temperature change between two surfaces of the
FGM shallow shell.
Introduction of equation (39) into equation (12)
gives the thermal parameter as
È1 ¼ ðL À HÞhÁT

The equation of nonlinear free vibration of perfect
FGM shells can be obtained from (35)
::

ðtÞ þ m1 ðtÞ þ m2 2 ðtÞ þ m3 3 ðtÞ ¼ 0

!NL
Emc mc
P
þ 2 À ðEc mc þ Emc c Þ
ðN þ 1Þmc mc



Z1
1
d
Â
þ
N
1 À c À mc
0 1 À c À mc 

P ¼ mc ðEc mc þ c Emc Þ À Emc mc þ Emc mc c

ð41Þ

ð42Þ

ð44Þ

Seeking solution as ðtÞ ¼ cosð!tÞ and applying a
procedure such as Galerkin’s method to equation (44),
the frequency-amplitude relation of nonlinear free
vibration is obtained

ð40Þ

where
L¼

Ec c
jNþ2


1
3m3 2 2
¼ !L 1 þ 2 
4!L

ð45Þ

where !NL is the nonlinear vibration frequency and  is
the amplitude of nonlinear vibration.
Table 2 illustrates the numerical calculation of the
fundamental frequency of natural vibration (37)
applied for the spherical shallow shells in the temperature dependent case (T-D) and the temperature independent one (T-ID). We can conclude that the
temperature has a significant effect on the frequency

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Figure 22. Nonlinear response of functionally graded material cylindrical panel. b ¼ a ¼ 1:5 ðmÞ, h ¼ 0:008 ðmÞ, N ¼ 5, Rx ¼ 1,
Ry ¼ 3 ðmÞ, k1 ¼ k2 ¼ 0, ÁT ¼ 0. 1. Bich (2012); 2. Present.

of the FGM shells. In the case of T-ID, we have made a
comparison with Bich et al. (2012) and we have found
good agreement between the two calculations. pffiffiffiffiffiffiffiffiffiffiffiffi
Table 3 presents the calculation of É ¼ !L h c =Ec
for Al/Al2O3 FGM (i.e.1=Rx ¼ 1=Ry ¼ 0) in the
case of T-D without the plastic foundation
(k1 ¼ k2 ¼ 0, b=a ¼ 1, b=h ¼ 30, N ¼ 1, Tc ¼ 500 ðKÞ,
Tm ¼ 300 ðKÞ). We can see that there is great agreement
between our results and those reported in Matsunaga
(2008) and Shariyat (2009).
illustrates
the
calculation
of
ÀTableÁÂ À4
Á Ã1=2
!L a2 =h 0 1 À 2 E0
for Si3N4/SUS304 FGM
plate in a thermal environment for two cases (T-ID
and T-D) with b=a ¼ 1, b=h ¼ 30, 1=Rx ¼ 1=Ry ¼ 0,
k1 ¼ k2 ¼ 0, N ¼ 1. This finding agrees greatly with
the one captured by Shen (2004) for FGM plates.
Figures 2 and 3 describe the effect of a geometric
parameter b=a on the nonlinear dynamic response of
the double curved FGM shallow shells in the case of
T-ID (Figure 2) and T-D (Figure 3). Obviously, the

larger the ratio b=a, the stronger the effect on the nonlinear dynamic response of the shell.
Figures 4 and 5 illustrate the effect of the geometric
parameter b=h (where h is the height of the shell) on the
nonlinear dynamic response of the double curved FGM
shallow shells for the T-ID and T-D cases, respectively.
It is clear that the larger the ratio b=h, the stronger the

effect on the nonlinear dynamic response of the FGM
shallow shells.
Figures 6 and 7 illustrate the effect of the imperfect
coefficient 0 on the nonlinear dynamic response of
FGM shallow shells under the conditions:
b=a ¼ 1, b=h ¼ 30, N ¼ 1, k1 ¼ k2 ¼ 0.
Also,
the
imperfect coefficient has a significant effect on the
dynamic response of the FGM shell.
Figures 8 and 9 describe the deflection-velocity
relation of the spherical FGM shallow shell in the
T-ID case (Figure 8) and in the T-D case (Figure 9)
under the conditions: b=a ¼ 1, b=h ¼ 30, N ¼ 1,
Rx ¼ Ry ¼ 0:6 ðmÞ,
Q0 ¼ 75000 ðN=m2 Þ,  ¼ 3000
ðrad=sÞ, 0 ¼ 0, k1 ¼ k2 ¼ 0. Obviously, the temperature has a significant effect on the deflection-velocity
relation of the shell.
Figures 10 and 11 illustrate the effect of the frequency  on the nonlinear dynamic response of the
double curved FGM shallow shell in the T-ID case
(Figure 10) and in the T-D case (Figure 11), respectively. At a frequency , its effect on the dynamic
response of the shell in the T-D case is stronger and
faster than its effect in the T-ID case.

Figures 12 and 13 show the effect of the dynamical
load (i.e. an amplitude Q0 ) on the nonlinear dynamic
response of the double curved FGM shallow shell. As a
result, the larger the amplitude of the dynamic loads are

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Journal of Vibration and Control 0(0)

Figure 23. Nonlinear dynamic response of the functionally graded material spherical shallow shell. b ¼ a ¼ 2 ðmÞ, h ¼ 0:01 ðmÞ,
Rx ¼ Ry ¼ 5 ðmÞ.

and the higher the temperature is, the stronger its
effects on dynamic of the FGM shell.
Figures 14 and 15 illustrate the effect of the elastic
coefficient k1 , k2 on the nonlinear dynamic response of
the double curved FGM shallow shells under the conditions: b=a ¼ 1, b=h ¼ 30, N ¼ 1, Q0 ¼ 75000 ðN=m2 Þ,
 ¼ 3000 ðrad=sÞ, 0 ¼ 0: We conclude that these coefficients have a strong effect on the nonlinear dynamic

response of the FGM shells. Compared to the case corresponding to the coefficient k1 , the Pasternak type
elastic foundation with the coefficient k2 has a stronger
effect.
Figures 16 and 17 show the effect of the temperature
on the nonlinear dynamic response of the double
curved FGM shallow shell. Obviously, the temperature
has a strong effect on the dynamic response of the shell.
Figures 18 and 19 illustrate the frequency-amplitude
relation of nonlinear free vibration !nl - in the case of
T-ID (Figure 18) and T-D (Figure 19). The higher the
temperature, the stronger the amplitude of nonlinear
free vibration of the FGM shell.
Unlike other publications, the Poisson ratio is
assumed to be varied smoothly along the thickness
 ¼ ðzÞin this framework. The analytical calculation
and transformation are, therefore, more complex.
However, Figure 20 shows us that there is no significant
change in the dynamic response of the FGM spherical
shallow shell in case of  ¼ ðzÞ and  ¼ const.

In the case of 1=Rx ¼ 1=Ry , the shell is a spherical
shallow shell. Figure 21 shows the nonlinear dynamic
response of the FGM spherical shallow shell (without
the temperature gradient ÁT ¼ 0) has a good agreement with the recent report by Bich et al. (2012).
In particular, Rx ¼ 1 or 1=Rx ¼ 0 makes the FGM
shells in the form of the cylindrical shallow panel.
Figure 22 shows that our findings are in good agreement with the calculation in Bich (2012) applied for the
same system.
Figure 23 compares the nonlinear dynamic response
of FGM spherical shallow shells in the presence of the

temperature gradient (i.e. ÁT ¼ 300K) and in the
absence of temperature gradient (i.e. ÁT ¼ 0) reported
in recent work Duc (2013). This result shows us the
significant effect of the temperature on the dynamic
response between the bending and time.

5. Conclusion
This paper presents an analytical investigation on the
nonlinear dynamic response of double curved P-FGM
thin shallow shells with temperature-dependent properties on an elastic foundation and subjected to mechanical load and temperature. The formulations are based
on the classical shell theory taking into account geometrical nonlinearity, initial geometrical imperfection
with Pasternak type elastic foundation. The nonlinear

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Duc and Quan

21

equations are solved by Bubnov-Galerkin and the

Runge-Kutta methods. The obtained results show the
effects of material and geometrical properties, imperfection, elastic foundation and temperature on the dynamical response of P-FGM double curved thin shallow
shells. Therefore, when we change these factors, we
can control the dynamic response and vibration of the
FGM shell actively. Some results were compared with
those of other authors.
Funding
This work was supported by the grant in mechanics of the
National Foundation for Science and Technology
Development of Vietnam – NAFOSTED. The authors are
grateful for this support.

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Appendix A

 2

 2
2
!
À
Á2
À
Á
À1


2
P4
n 2m
þ
Àk2 2m þ 2n À k1
ðP3 þ P1 P4 Þ n þ m þ ðP2 À P1 P3 Þ 2m þ 2n À À
Á2
1
Rx Ry
Rx Ry
2m þ 2n
&
2
2
È1 ðm þ n Þ
16
4
þ
À
ðI21 2n þ I11 2m Þ
2
1
mn 1 mn2 Rx
"
#


 2


'
4
I10 I20
P4
n 2m
n
À
þ
þ
þ4 À
À P3
Á2
mn2 Rx Rx Ry
mb2 Rx
Rx Ry
2m þ 2n
(
"
#
)
 2



P4
n 2m
m
16
4
4

I10 I20
2
2
þ
ðI11 n þ I21 m Þ À
þ
À
þ4 À
À P3
Á2
na2 Ry
mn2 1 mn2 Ry
mn2 Ry Ry Rx
Rx Ry
2m þ 2n

m1 ¼

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23

"
#
 2

 2

!
À32mn2
P4
n 2m
4
P4
m 2n
4
P
m2 ¼
þ
þ
P


À
À
P
À
À

Á2
3
1 4 m n
ab1 6m n Ry Rx
3
3a2 b2 1
Rx Ry
2m þ 2n
!
2
ðI10 2m þ I20 2n Þ ðI20 2m þ I10 2n Þ
À
þ
mn2 1
Rx
Ry
"
#
&


 2

'
m2 2
4
4
I10 I20
P4
n 2m

n
2
2
þ 2
ðI

þ
I

ÞÀ
þ
þ
þ
4
À
P
À
Á2
21 n
11 m
3
mn2 Rx Rx Ry
mb2 Rx
a 1 mn2 Rx
Rx Ry
2m þ 2n
(
"
#
)



 2

2
n2 2
4
4
I
I
P


m
10
20
4
n
ðI11 2n þ I21 2m Þ À
þ
þ m À P3
þ 2
þ4 À
Á2
mn2 Ry Ry Rx
na2 Ry
b 1 mn2 Ry
Rx Ry
2m þ 2n
Á

P4 À 4
1
m3 ¼
m þ 4n þ
ðI10 4m þ 2I20 2m 2n þ I10 4n Þ
81
161
"
#
 2

 2

!
À32mn2
P4
n 2m
8
P4
m 2n
4
þ
þ
m4 ¼
À P3 À
À P1 P4 m n
À
Á2
ab1 6m n Ry Rx
3

3a2 b2 1
Rx Ry
2m þ 2n
"
#
&




'
m2 2
4
4
I10 I20
P4
2n 2m
n
2
2
ðI

þ
I

ÞÀ
þ
þ
þ 2
þ

4
À
P
À
Á2
21 n
11 m
3
mn2 Rx Rx Ry
mb2 Rx
a 1 mn2 Rx
Rx Ry
2m þ 2n
(
"
#
)


 2

n2 2
4
4
I10 I20
P4
n 2m
m
2
2

ðI11 n þ I21 m Þ À
þ
þ
þ 2
þ4 À
À P3
Á2
mn2 Ry Ry Rx
na2 Ry
b 1 mn2 Ry
Rx Ry
2m þ 2n
!
4
ðI10 2m þ I20 2n Þ ðI20 2m þ I10 2n Þ
À
þ
mn2 1
Rx
Ry
Á
3P4 À 4
1
m5 ¼
 þ 4n þ
ðI10 4m þ 2I20 2m 2n þ I10 4n Þ
81
161 m
Á
Á

P4 À 4
1 À
m6 ¼
m þ 4n þ
I10 4m þ 2I20 2n 2m þ I10 4n
41
81
Á
È1 À 2
m7 ¼
 þ 2n
1 m
16
m8 ¼
1 mn2


4È1 1
1
m9 ¼
þ
m n Rx Ry

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