buckling analysis of shallow spherical fgm shells … · buckling analysis of shallow spherical fgm...

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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

mailto:[email protected]

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:


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):


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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:


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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):


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, ,

, ,

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:


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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:


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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.


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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 .


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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.


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 .


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.


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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.


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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:


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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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