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International Journal of Basic Sciences & Applied Research. Vol., 6 (2), 110-122, 2017
Available online at http://www.isicenter.org
ISSN 2147-3749 ©2017
Buckling Analysis of Shallow Spherical FGM Shells Based on the First Order
Shear Deformation Theory
Farshid Karami, Mehdi Hosseini*
Mechanical Engineering Department, Faculty of Engineering, Malayer University, Malayer, Iran
*Corresponding Author Email: [email protected]
Abstract
In this paper, the buckling analysis of simply supported thin shallow spherical
shells made of functionally graded material (FGM) is studied using the first order
shear deformation theory (FSDT).The equilibrium equations are derived by
minimization of the total potential energy functional. The Galerkin method is
employed to solve the derived equations. Since the employed strain-displacement
relations are nonlinear, so the presented analysis is nonlinear with high accuracy.
The obtained results are compared and validated with the results existed in the
literature. The effects of some significant geometrical and mechanical parameters
on the buckling load are studied and discussed. It is observed that when the power
law index increases, the critical buckling hydrostatic pressure of the FGM shallow
spherical shell decreases.
Keywords: Buckling, FGM, First Order Shear Deformation Theory, Hydrostatic
Pressure, Spherical Shell.
Introduction
In recent years, with the increasing and rapid development of various industries, the advancements in industrial
machinery and the developments in high-powered engines of aerospace industry, turbines and reactors and other
machinery, the need for mechanically stronger materials with higher thermal resistance has been felt .In previous
years, pure ceramic materials were used in the aerospace industry for the coating of the components with high
performance. These materials were very good non-conductors but were not resistant enough to residual stress.
Residual stresses caused many problems such as cavities and cracks in these materials. Layered composites were
used later to overcome this problem. Thermal stresses also made them chip. Regarding these problems, producing a
composite material which has high thermal and mechanical resistance and is immune to chipping gained in
importance.
In 2000, Zang et al (2000) investigated dynamic non-linear axisymmetric buckling of laminated cylindrically
orthotropic composite shallow spherical shells including third-order transverse shear deformation. In 2001, Nie
Intl. J. Basic. Sci. Appl. Res. Vol., 6 (2), 110-122, 2017
111
(2001) formulated an asymptotic solution for non-linear buckling of elastically restrained imperfect shallow
spherical shells continuously supported on a non-linear elastic foundation. In 2003, Li et al established the large
deflection equation of a shallow spherical shell under uniformly distributed transverse loads is with consideration of
effects of transverse shear deformation on flexural deformation. In 2008, Gupta et al conducted experiments,
simulation as well as analysis of the collapse behavior of thin spherical shells under quasi-static loading. In 2008,
Sørensen and Jensen carried out an analysis of buckling-driven delamination of a layer in a spherical, layered shell.
In 2010, Hutchinson revealed that cylindrical shells under uniaxial compression and spherical shells under equi-
biaxial compression display the most extreme buckling sensitivity to imperfections. In 2012, Sato et al studied the
critical buckling characteristics of hydrostatically pressurized complete spherical shells filled with an elastic medium
are demonstrated. In 2014, Sabzikar and Boroujerdy studied the snap-through buckling of shallow clamped spherical
shells made of functionally graded material (FGM) and surface-bonded piezoelectric actuators under the thermo-
electro-mechanical loading.
In the present paper, the buckling analysis of thin shallow spherical shells is studied using the first order shear
deformation theory. The material of the shell is supposed to be FGM and the boundary conditions are assumed to be
simply supported. The equilibrium equations are derived using the minimum total potential energy principle. The
Galerkin method is used to solve the derived equations. The validity of the present results is confirmed through
comparison with the existed results in the literature. The effects of some important geometrical and mechanical
parameters on the buckling load are studied and discussed.
Analytical formulation
The considered FGM shell is made of metal and ceramics, so that its properties gradually vary from pure
ceramics in the outer surface to pure metal in the inner surface (Figure 1). Voigt notation is used for elastic modulus,
but Poisson's ratio is assumed to be constant. The mechanical properties are assumed to be (Mirzavand & Eslami,
2007):
Figure 1. A schematic view of the FGM shallow spherical shell.
( )
( )
c c m mE z E V E V
z
(1)
Where mV and
cV are the volume fraction of metal and ceramics and are defined as follows (Nie, 2001):
( / 1/ 2)
1
k
m
c m
V z h
V V
(2)
In Eqs. (1) And (2), z is measured from the middle surface of the shell. The positive direction of z is outwards
and varies from –h.2 to h/2. In the spherical shells φ, θ and z are meridional, circumferential and radial directions
respectively and create a vertical coordinate system. Subscripts m and c indicate metal and ceramic respectively.
The parameter k is the power law index for volume fraction for which values greater than or equal to zero can be
considered. Substituting Eq. (2) in Eq. (1) yields:
Intl. J. Basic. Sci. Appl. Res. Vol., 6 (2), 110-122, 2017
112
2( )
2
( )
k
m cm
z hE z E E
h
z
(3)
Where
cm c mE E E (4)
According to the first-order shell theory of Love and Kirchhoff, the normal and shear strains at a z distance
from the middle surface of the shell are (Wunderlich & Albertin, 2002):
2
m
m
m
zk
zk
zk
(5)
where i and ij are the normal and shear strains respectively, k and k are the middle surface bending
curvatures, k is the middle surface twisting curvature, and the subscript m refers to the middle surface. The strains
and bending of the middle surface relate through nonlinear kinematics relations to the displacement components u,
v, and w. Sanders nonlinear kinematics relations for thin spherical shell are reported in reference (Wunderlich &
Albertin, 2002). According to the theory of shallow spherical shells, the DMV nonlinear relations for the spherical
shell are in the form of equation (6):
2
,
2,
, ,
2
,,
2 2
, ,
2
2
cos sin
sin 2
sin cos
sin
cot
sin
cot
sin
m
m
m
u w
R
v u w
R
u v v
R
wk
R
wwk
R R
w wk
R
(6)
Where and are the rotations of the normal vector to the middle surface about the θ and φ axes
respectively and:
,
sin
w
R
w
R
(7)
Figure (2) shows the stresses in a segment of the spherical shell. According to the theory of shallow shells, the
effects of transverse shear stresses z and z are ignored. According to Hooke's Law, the stress-strain relations
for spherical shells are as follows (Hutchinson, 1967):
Intl. J. Basic. Sci. Appl. Res. Vol., 6 (2), 110-122, 2017
113
Figure 2. A view of the stresses in a segment of the spherical shell.
2
2
( )
1
( )
1
( )
E z
E z
G z
(8)
Substituting Eq. (5) and the ( )
( )2(1 )
E zG z
relation in Eq. (8) yields:
2
2
( )
1
( )
1
m
m m
E zzk zk
E zzk zk
(9)
( )
22(1 )
m
E zzk
According to the theory of shallow shells, the effects of transverse shear forceszQand zQ are ignored. The
force resultants ijN and moment resultants ijM are linked to strains through the following equations (Nie, 2003):
/ 2
/2
/2
/2
h
ij ijh
h
ij ijh
N dz
M zdz
(10)
Substituting Eq. (9) in Eq. (10) and calculating the integrals, the force and moment resultants form as functions
of strains and curvatures. Generally, the total potential energy V of a loaded object is the sum of the strain energy U
and the potential energy of the applied loads Ω (Nie, 2003):
V U (11)
Generally, the strain energy U for a three-dimensional object with a desired vertical coordinates x, y, and z is
as follows (Nie, 2003):
1
2x x y y z z xy xy yz yz zx zxU dxdydz
(12)
Regarding this point that the shallow spherical shells are under plane stress and plane strain simultaneously, the
strain energy U would be equal to:
Intl. J. Basic. Sci. Appl. Res. Vol., 6 (2), 110-122, 2017
114
1
2U ds ds dz (13)
ds And ds are the differential of arc length about the θ and φ axes respectively and are:
ds Ad
ds Bd
(14)
Where A and B are the Lame parameters whose values for a spherical shell are A = R and B = R sin φ.
Substituting these values in Eq. (14) yields:
sin
ds Rd
ds R d
(15)
Substituting the Love-Kirchhoff equations(5), stress-strains of the middle surface (8), and the differential of arc
length (15) in the strain energy Eq. (13) and then substituting in the resultant equation the mechanical properties
FGM equations(3) and kinematic relations DMV Eq. (6) and calculating the integral with respect to coordinate z,
yields:
, , , , , , , ,, , , , , , , , , , , , , ,U D u v w u u v v w w w w w d d (16)
Generally, the potential energy of applied loads Ω for a shell with custom vertical coordinates of x, y, and z
and which is under three-dimensional pressure of , ,z y xP P P on all its surface, is equal to a negative times of the work
done due to this pressure and is as follows:
x y z x yP u P v P w ds ds (17)
As the pressure is only applied to z direction and in a negative way, the potential energy due to external
hydrostatic pressure Ω is obtained through using Eqs. (15) and (17) as follows: 2 sinPwR d d (18)
Substituting the strain energy Eq. (16) and the potential energy due to hydrostatic pressure Eq. (18) in the total
potential energy Eq. (11) yields the total potential energy for the considered shell in this study as follows:
, , , , ,
,
, , , , , , , , ,
, , , ,
u v w u u v v wV F d d
w w w w
(19)
Where:
2 sinF D PwR
(20)
According to the minimum potential energy criterion for static problems, in order to derive the equilibrium
equations, the total potential energy V should be minimized. To do so, its changes δV must be set equal to zero.
According to the concepts of calculus of variations, for zero total potential energy change (δV = 0), the function of
Eq. (20) should apply to the corresponding Euler equations. Functional Euler equations like the Eq. (19) are as
follows (Nie, 2001):
Intl. J. Basic. Sci. Appl. Res. Vol., 6 (2), 110-122, 2017
115
, ,
, ,
2
2
, , ,
2 2
2
, ,
0
0
0
F F F
u u u
F F F
v v v
F F F F
w w w w
F F
w w
(21)
Substituting the equation (20) in the Euler equations (21) yields nonlinear equilibrium equations for shallow
spherical shells under mechanical load. The equilibrium equations for these shells in the form of the force and
moment resultants are:
, ,
, ,
, , ,
,
2
,
cos (sin ) 0
(sin ) cos 0
1(sin ) sin ( ) cos
sin
2( cot ) sin ( )
( ) sin
N N N
N N N
M R N N M
M R N N
R N N PR
(22)
According to the Adjacent-Equilibrium Criterion, for obtaining the equations of stability, displacement
variables are given a very small virtual development (Nie, 2001), in a way that the displacements of the middle
surface in pre-buckling or equilibrium state, due to the applied loads (no development is made in the applied loads),
and the displacements of the virtual middle surface are very small. The displacements of the virtual middle surface
result in corresponding changes in the force and moment resultants (Nie, 2001), in a way that the pre-buckling force
and moment resultants and the virtual force and moment resultants correspond to the displacements of the virtual
middle surface. Substituting the force and moment resultants and0
,1
,0
,1
in the equilibrium equations
(22) yields the removal of many of the expressions according to the equations (22) and since the applied mechanical
load is symmetrical, the pre-buckling rotation of the normal vector to the middle surface about φ axis (0
) is equal
to zero and the effect of the pre-buckling rotation of the normal vector to the middle surface about θ axis (0
) is
ignored. So the equations get simplified as follows:
1 1
1 1 1
1
1 0 1 0 1 1
1 1
1 1 0 1 0 1
, ,
, ,
,
,,
, ,
,
cos (sin ) 0
(sin ) cos 0
(sin ) sin ( ) cossin
2 cot
sin ( ) ( ) 0
N N N
N N N
MM R N N M
M M
R N N R N N
(23)
Relations (23) are stability equations for shallow spherical shells, as said before the subscripts «1, 0» represent
equilibrium and stability states respectively. In fact, the expressions with "0" subscript are obtained by solving the
equilibrium equations.
In order to calculate the pre-buckling force resultants, for simplicity's sake, the membrane solution of the
equilibrium equations is considered or in other words we ignore the moment resultants.
According to this point that 0 0
0 and through the removal of the moment resultants expressions,
membrane form of the equilibrium equations (22) is obtained:
Intl. J. Basic. Sci. Appl. Res. Vol., 6 (2), 110-122, 2017
116
0 0 0
0 0 0
0 0
, ,
, ,
2
cos (sin ) 0
(sin ) cos 0
sin ( ) sin
N N N
N N N
R N N PR
(24)
As the applied hydrostatic pressure is symmetric, it is found that:
00
N N
N
(25)
Noting that 0
Nand
0N
are equal, it can be obtained:
0 0 2
PRN N (26)
According to Eqs. (25) and (26) the first and the second variations of Eq. (24) are satisfied. Thus, the pre-
buckling force resultants for a shell under external hydrostatic pressure are as follows:
0 0
0
2
0
PRN N
N
(27)
Simply supported boundary conditions are considered, so:
1,φ 1 1,φφ 1u =v =w =w =0 at = L (28)
The one-term solution to the stability equations (23) which satisfies the assumed boundary conditions (26) is
considered as follows (Nie, 2001; Wunderlich & Albertin, 2002):
1 1
1 1
1 1
cos cos
sin sin
cos sin
L
u A n
v B n
w C n
(29)
Where / Lm is the angle between the perpendicular line to spherical shell and the passing line through
the border and m=1, 2, 3 … and n=1, 2, 3 … are the numbers of the meridional and circumferential waves
respectively and 1 1 1,C B Aوare constant coefficients.
Substituting the approximate solutions (29) in the stability equations (23) yields a system of equations as
follows:
11 1 12 1 13 1 1
21 1 22 1 23 1 2
31 1 32 1 33 1 3
R A R B R C E
R A R B R C E
R A R B R C E
(30)
Where ijR is a function of parameters and n; geometrical parameters h and R; applied load P; mechanical
properties FGM and independent variables φ and θ. Errors 3 1 2,E E Eوmade are due to the approximate nature of the
solutions. The Galerkin method is used to minimize these errors. The approximate solutions (30) are substituted in
stability equation (24) utilizing the Galerkin minimization technique, to yield:
Intl. J. Basic. Sci. Appl. Res. Vol., 6 (2), 110-122, 2017
117
11 1 12 1 13 1
21 1 22 1 23 1
31 1 32 1 33 1
0
0
0
a A a B a C
a A a B a C
a A a B a C
(31)
For this system of equations (19) to have a nontrivial solution (where the shell loses stability or buckling
occurs), the determinant of the coefficients matrix of these equations must be set equal to zero:
11 12 13
21 22 23
31 32 33
0
a a a
a a a
a a a
(32)
When equation (32) is solved, buckling load is obtained as a function of m and n. The critical buckling load is
the minimum load that is obtained through different values of m and n.
Results
In this section we first calculate those buckling modes in which the minimum critical buckling pressure of
steel spherical shells occurs. Then the classical buckling load of steel and FGM spherical shells for different
parameters such as dimensionless parameter /h R is obtained. Following that, the dimensionless critical pressure
versus dimensionless parameter /h R for isotropic (steel) shallow spherical shells with different values of L is
calculated and in the end the numerical modeling of the problem is examined using ABAQUS software and the
numerical results are compared with the analytical results and the error degree is estimated.
The component elements of FGM are a type of steel and ceramics with the elastic modulus incited in Table 1.
Table 1. The elastic modulus of a type of steel and ceramics.
steel ceramics
202.97 GPa 361.3 GPa
Poisson's ratio for steel and ceramics are assumed to be equal to 0.3. The boundary conditions of the shell are
considered as simply supported. Results shown in graphs are related to mechanical loading (external hydrostatic
pressure).
The classical hydrostatic buckling pressure for a shallow spherical shell is (Hutchinson, 1967):
2
2
2
3 1cl
E hP
R
(33)
Figure (3) compares the results of reference (Hutchinson, 1967) with the results of this study on the shallow
spherical shells with pure isotropic material (metal) where mE E . Figure (3) shows the dimensionless critical
hydrostatic pressure versus the dimensionless parameter /h R for different values of L . Close agreements between
the results of this study and those reported in reference (Hutchinson, 1967) can be seen in Figure (3) and based on
that it is found that the larger the values of L , better agreement between the values of the critical hydrostatic
pressure and the classical one are observed.
Intl. J. Basic. Sci. Appl. Res. Vol., 6 (2), 110-122, 2017
118
Figure 3. The dimensionless critical hydrostatic pressure versus the dimensionless parameter /h R for isotropic
(steel) shallow spherical shells with different values of L .
Figure (4) represents the critical hydrostatic pressure to the dimensionless parameter /h R for steel shallow
spherical shells with 30 , 20 , 10L L L .Based on this figure it is found that the classical buckling hydrostatic
pressure expression (1) has more reliability for larger values of L . The most precise expression is for a complete
spherical shell.
Figure (5) represents the dimensionless critical hydrostatic pressure versus the dimensionless parameter /h R
for the steel, the ceramic, and the FGM shallow spherical shells with 10L and different power law indices k.
The curves indicate that the critical hydrostatic pressure decreases as the power law index increases, in a way
that the dimensionless critical hydrostatic pressure is greater in FGM spherical shells than steel spherical shells but it
is lower than ceramic spherical shells. Moreover, the dimensionless critical hydrostatic pressure in FGM spherical
shells decreases with an increase in the power law index. In other words, the dimensionless critical hydrostatic
pressure has a negative relationship with k.
Figure 4. The dimensionless critical hydrostatic pressure versus the dimensionless parameter /h R for isotropic
(steel) shallow spherical shells with different values of L .
Intl. J. Basic. Sci. Appl. Res. Vol., 6 (2), 110-122, 2017
119
Figure 5. The dimensionless critical hydrostatic pressure versus the dimensionless parameter /h R for the steel, the
ceramic, and the FGM shallow spherical shells with different power law indices k.
Figure (6) represents the dimensionless critical hydrostatic pressure versus the dimensionless parameter /h R
for FGM shallow spherical shells with k=2 and different values of L . Based on Figure 6 it is found that in FGM
shallow spherical shells the larger the value of L , better agreement between the values of the critical hydrostatic
pressure and the classical one are observed.
Figure 6. The dimensionless critical hydrostatic pressure versus the dimensionless parameter /h R for FGM shallow
spherical shells with different values of L .
Figure (7) represents the dimensionless critical hydrostatic pressure versus the dimensionless parameter /h R
for FGM shallow spherical shells with different values of L . This figure indicates an increase in hydrostatic
pressure due to an increase in the amount of dimensionless parameter /h R . Based on figure 7, it is found that the
critical hydrostatic pressure of buckling in FGM spherical shells is less than ceramic spherical shells and is greater
than steel spherical shells and it decreases as the power law index k increases.
Intl. J. Basic. Sci. Appl. Res. Vol., 6 (2), 110-122, 2017
120
Figure 7. The dimensionless critical hydrostatic pressure versus the dimensionless parameter /h R for FGM
shallow spherical shells with different values of L .
Figure (8) represents the hydrostatic buckling pressure of FGM spherical shells versus the dimensionless parameter
/h R for values of 10 ,20 ,30L and k=2. As you can see, as the dimensionless parameter /h R increases, the
hydrostatic buckling pressure also increases.
Figure 8. The hydrostatic buckling pressure of FGM spherical shells versus the dimensionless parameter /h R .
Figure (9) represents the hydrostatic buckling pressure of FGM spherical shells versus the dimensionless
parameter /h R with different power law indices k. This figure shows that the hydrostatic buckling pressure of FGM
spherical shells decreases as the power law index k increases and also for certain values of the power law index k,
the hydrostatic pressure increases as the dimensionless parameter /h R increases.
Intl. J. Basic. Sci. Appl. Res. Vol., 6 (2), 110-122, 2017
121
Figure 9. The hydrostatic buckling pressure of FGM spherical shells versus the power law index k for different
values of the dimensionless parameter /h R .
Figure (10) represents the hydrostatic buckling pressure of FGM spherical shells versus the parameter cmE for
different values of L .In this figure, k=2 and the value of the dimensionless parameter /h R is assumed to be equal to
0.005. The hydrostatic buckling pressure for 10 ,20 ,30 ,45 ,60L is obtained.
Figure 10. The hydrostatic buckling pressure of FGM spherical shells versus the parameter cmE .
Based on figure (10), it is found that the hydrostatic buckling pressure of FGM spherical shells increases as the
parameter cmE increases and also the larger the values of L gets, the more the hydrostatic buckling pressure
decreases so that their diagrams get closer to each other. In figure (10) the value of 200 9mE e is assumed to be
constant but the value of cE varies and it starts from 250 9e and in each level 50 9e is added to its value until it reaches
to 450 9e .
Conclusions
In the present paper, the buckling analysis of simply supported thin shallow spherical shells made of
functionally graded material was studied using the first order shear deformation theory. The equilibrium equations
were derived using the minimum total potential energy principle. The derived equations were solved by Galerkin
method. The obtained results were compared and validated with the results existed in the literature. The effects of
some significant geometrical and mechanical parameters on the buckling load were studied and discussed. The
following results are drawn:
Intl. J. Basic. Sci. Appl. Res. Vol., 6 (2), 110-122, 2017
122
1. The critical mechanical buckling load of an FGM shallow spherical shell is less than the critical mechanical
buckling load of ceramic shallow spherical shell and is higher than that of a steel shallow spherical shell.
2. The stability of an FGM shallow spherical shell decreases with an increase in the hydrostatic pressure.
3. When the power law index k increases, the critical buckling hydrostatic pressure of an FGM shallow
spherical shell decreases.
4. The higher the value of h/R, the higher the stability of the FGM shallow spherical shell subjected to
mechanical loading.
Conflict of interest
The authors declare no conflict of interest
References
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