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AbstractThis paper presents the formulation and numericalbuckling analysis of a circular cylindrical composite shell comprising

of fibers made of functionally graded material (FGM). The material

properties of the fibers vary through the shell length according to a

power-law distribution of the volume fraction. That is the fiber

material properties vary from the metal on the one end to the ceramicup to the middle of the shell then from the ceramic to the metal

towards the other end of the shell. Based on the first order shear

deformation theory (FSDT) the governing equations of the shells are

derived. Then to determine the buckling load of the composite shell

over simply supported edges these equations are solved using the

generalized differential quadrature method. The obtained results for

an isotropic shell are compared with those given in the literature.

Very good agreement is seen. Then, the effects of geometric

parameters and FG power index are also investigated on the

magnitude of the buckling load through number of examples. The

study of the obtained results shows that any decrease in the value of

FG power index will lead to a better buckling behavior of the

composite shell.

KeywordsBuckling analysis, Generalized differentialquadrature method, FG fiber with axial FG distribution, Functionally

graded cylindrical shell.

I. INTRODUCTIONN the last decade due to the increasing demands for high

heat-resisting, lightweight structures, the studies on

functionally graded structures, especially FG cylindrical

shells, have attracted much attention. FG materials were

reported first in Japan in 1984 [1]. FGMs are composite

materials in which the mechanical properties vary smoothly

and continuously from one surface to another. Buckling

behavior of the homogenous structures subjected tomechanical loads has been investigated by Brush and Almorth

[1]. Timoshenko [2] studied the exact solution of a thick

walled cylinder under inner and outer pressures. The cylinder

M. H. Kargarnovin, School of Mechanical Engineering, Sharif University

of Technology, Azadi Ave., Tehran 14588-89694, I.R. Iran (Corresponding

author to provide phone: +9821-66165510 ; fax: +9821-6600-0021 ; e-mail:

[email protected]).

M. Shahsanami, Graduate Student, Department of Mechanical

Engineering,Arak Branch, Islamic Azad University, Arak 38135-567, Iran

(e-mail: [email protected]).

is supposed to be axisymmetric and isotropic. The solution is

applicable for simple and quick solution of pressure vessels.

Mirsky and Hermann [3] employed the first order shear

deformation theory for the analysis of an isotropic cylinder.

Tutuncu and Ozturk [4] presented the exact solution of FG

spherical and cylindrical pressure vessels. Jabbari [5] analyzed

the thermo elastic analysis of a FG cylinder under the thermaland mechanical loads. Wu [6] investigated the elastic stability

of a FG cylinder. They employed the shell Donnells theory to

derive the strain-deformation relations. Shao [7] investigated

the thermo elastic analysis of a thick walled cylinder under

mechanical and thermal loads. Li and Batrab [8] studied

buckling behaviors of an axial compressed three-layer circular

cylindrical shell with the middle layer made of FGMs.

Najafizadeh [9] studied linear buckling behaviors of axially

compressed stiffened FG cylindrical shells employing Donnell

shell theory and a three-dimensional finite element code.

Due to the complexity of the problem, it is difficult to

obtain the exact solution. In present work, the generalized

differential quadrature method (GDQM) approach is used tosolve the governing equations of the FG cylindrical shell with

lengthwise material distribution. Since Bert [10] first used the

method to solve problems in structural mechanics, the method

has been applied successfully to a variety of problems [11].

Lam [12] successfully used the generalized differential

quadrature method to research on the instability of conical

shells, free vibration truncated conical panels [13], and free

vibration of rotating composite laminated conical shell

[14,15]. Better convergence behavior is observed by GDQM

compared with its peer numerical competent techniques such

as finite element method, finite difference method, boundary

element method and meshless technique.

II.GOVERNINGEQUATIONSConsider a FG cylindrical shell of mean radius R,

thickness h, and length l, refer to cylindrical coordinates

(x,,z). The shell properties are assumed to vary only through

the length direction according to power-law form, which is

given by:

Buckling Analysis of a Composite Cylindrical

Shell with Fibers Material Properties Changing

Lengthwise Using First-Order Shear

Deformation Theory

M. H. Kargarnovin, M. Shahsanami

I

International Conference on Mechanical, Automotive and Materials Engineering (ICMAME'2012) Jan. 7-8, 2012 Dubai

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( )

( )

20

2( )

22

2

k

c m m

k

c m m

x lP P P xl

P x

x lP P P x ll

+

=

+

(1)

Here subscript m and c are the metal and ceramic

constituents respectively and P denotes a material property ofFG cylindrical shell which may be substituted with the

modulus of elasticity E or mass density . k is power law

index that takes values greater than or equal to zero and l

indicates length of the shell.

The displacement field is based on first-order shear

deformation theory which is given by [16]:

0 1

0

0 1

( (( , , ) , , ) , , )

( , , ) ( , , ) ( , , )

( , , ) ( , )

u x z u x z zu x z

v x z v x z zv x z

w x z w x

= +

= +

=

(2)

Where ( , , )u x z , ( , , )v x z and ( , , )w x z are

displacement components along the x, , z direction

respectively.0

( , , )u x z ,0

( , , )v x z and0

( , , )w x z are

the middle surface displacements and1

( , , )u x z and

1( , , )v x z describe the rotations about the and x axes,

respectively.

According to displacement fields (2), the strain-

displacement relations of FG cylindrical shells are expressed

as:

0, 1,

2

0,

1

2x xxxx u zuw + +=

(3)

0, 0 1,

2

0,2( )

1

2

1v w vw

r

z

r r += + +

(4)

0, 0, 1, 1,0, 0,( )

11 1x xxx u z uw w

rv v

r r ++ + += (5)

1 0,xxz wu += (6)

1 0,

1z w

rv

+=

(7)

The stress-strain relations are given by Hooks law andafter substituting strain-displacement fields (3)-(7), the stress-

displacement relations are expressed as

0 , 1, 0 , 02

1,

2

0,

2

0,

( )[ (

1

1)]

2

1

2x x xx

E xu zu v w

zvr

wr

w

= + + + +

+ +

(8)

2

0, 0 1, 0 ,2 2

2

0, 1, 0 ,

( ) 1[ ( )

1 2

1( )]

2

1

x x x

E xv w zv w

r

u zu w

r

= + + +

+ + +

(9)

0 , 0 , 0 , 0 , 1, 1,

( ) 1[ ( )]

2(1 )

1 1x x x x

E xu w w z u

rv v

r r

= + + + ++

(10)

1 0,

( )[ ]

2(1 )xz x

E xwu

=

++

(11)

1 0,

( ) 1[ ]

2(1 )z

E xw

rv

=

+

+

(12)

The stress resultanti

N ,i

M ,i

Q are defined by:

2

2

2

2

( , ) (1, ) , ,

, ,

h

hi i i

h

hi iz

N M z dz i x

Q dz i x

= =

= =

(13)

Using the minimum potential energy criterion, the

equilibrium equations of FG cylindrical shells are derived as

follows:

, ,

10

x x xN N

r

+ = (14)

, ,

10

x xN N

r

+ = (15)

, , 0 , 0 ,

0,2

1 1 2

1

x x x xx x xN Q Q N w N wr r r

N w Pr

+ + + +

+ =

(16)

, ,

10

x x x xQ M M

r

+ + = (17)

, ,

10

x xQ M M

r

+ + = (18)

The stability equations of FG cylindrical shell may be

derived by the variational approach. If V is the total potential

energy of the shell, the first variationV is associated with thestate of equilibrium. The stability of the original configuration

of the shell in the neighbourhood of the equilibrium state [1]

can be determined by the sign of second variation V.

However the condition of V=0 is used to derive the

stability equations of many practical problems on the buckling

of shells. Thus the stability equations are represented by the

Euler equations for the integrand in the second variation

expression as:


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1, 1,

10

x x xN N

r

+ = (19)

1, 1,

10

x xN N

r

+ = (20)

1 1, 1, 0 1, 0 1,

0 1,2

1 1 2

1 0

x x x xx x xN Q Q N w N w

r r r

N wr

+ + + +

+ =

(21)

1 1, 1,

10

x x x xQ M M

r

+ + = (22)

1 1, 1,

10

x xQ M M

r

+ + = (23)

The terms with the subscript 0 are related to the state of

equilibrium and terms with subscript 1 are those

characterizing the state of stability. By substituting equation

(3)-(7) and (13) into (19)-(23), the stability equations can be

derived in terms of displacement components. To determinethe critical buckling loads, the prebuckling mechanical forces

should be found from the equilibrium equations and then

substituted into the stability equations for the buckling

analysis. Under a uniformly distributed axial compressive load

P, the prebuckling mechanical forces are given by:

0 0 0, 0, 0

2x x

PN N N

r

= = = (24)

III.2 B

SOLUTIONPROCEDURE

According to generalized differential quadrature method

(GDQ), the nth-order derivative of the solution function f(x)at grid point i in one dimension can be written as [17]:

( )

1

( )( )

1, 2, ..., 1, 2, ..., 1

n n

ni

ik kn

k

f xC f x

x

i N n N

=

=

= =

(25)

(1)

(1)

(1)

( )1, 2,..., ,

( ) ( ),i

ij

i j j

M xC j N i j

x x M x= =

(26)

(1)

1,

( ) ( )

N

i i j

j i j

M x x x=

= (27)

( 1)

( ) ( 1) (1)

( )( )

, 1, 2, ..., , , 2, 3, ..., 1

n

ijn n

ij ii ij

i j

CC n C C

x x

i j N i j n N

=

= =

(28)

( ) ( )

1,

1, 2, ..., , 1, 2, ..., 1

N

n n

ii ij

j i j

C C i N n N =

= = = (29)

Where( )n

ijC is the weighting coefficients of nth-order

derivative, and N is the total number of grid points. It has been

shown that the Chebyshev-Gauss-Lobatto (C-G-L) grid results

in the most convergence and stability among the other grid

distributions. Therefore this study applies C-G-L grid points,

whose positions in one-dimensional form are given by [17]:

1 1(1 cos( )) 1, 2,...,

2 1i

ix i N

N

= =

(30)

The displacement fields u, v, w for buckling of a circular

cylindrical shell are expressed as unknown functions along the

axial direction and known trigonometric functions along the

circumferential directions as follows:

0

0

0

1 1

1 1

( , ) ( ) cos

( , ) ( ) sin

( , ) ( ) cos

( , ) ( ) cos

( , ) ( ) sin

o

o

u x U x n

v x V x n

w x W x n

u x U x n

v x V x n

=

=

=

=

=

(31)

By writing the stability equations (19)-(23) in terms of

displacement relations (31) following equations are obtained:

(1)

0 0 0

1

( 2 ) (1) (1)

0 0 0

1 1 1

2

(1)

0 02

1

( )[ ( )] ( )

[ ( )]

( )(1 )[ ] 02

N

i

ik k i i i

k

N N N

ik k ik k ik k

k k k

N

i i ik k

k

K xh C U nV W K x

x r

h C U n C V C W r

h n nK x U C V

r r

=

= = =

=

+ + +

+ +

+ =

(32)

(1)

0 0

1

( 2 ) (1)

0 0

1 1

(1)

0 0 0

1

( ) (1 )( )

( )(1 )( )2

1( ) [ ( )] 0

N

i

ik k i

k

N N

i ik k ik k

k k

N

i ik k i i

k

K x nh C V U x r

h nK x C V C U

r

nK x h n C U nV W

r r

=

= =

=

+

+ =

(33)

(1)

0 0 0

1

(1) (1)

1 0 1

1 1

2

(2 )

0 1 0

1

(2 )

0

1

1 1( ) [ ( )]

( )( ) ( ) (

1) ( )(1 ) [ ]

02

N

i ik k i i

k

N N

i

i ik k i ik k

k k

N

ik k i i i

k

N

ik k

k

K x h C U nV W r r

K xh U C W K x h C U

x

nC W K x h nV W

r r

PC W

r

=

= =

=

=

+ + +

+ +

+ +

=

(34)


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(35)

(1)

1 0

1

3

(1)

1 1

1

3

( 2 ) (1)

1 1

1 1

3 2

(1)

1 1

1

( )1( )(1 ) ( )

2

( )12

( ) ( )12

1( )(1 ) ( ) 0

2 12

N

i

i i ik k

k

N

ik k i

k

N N

i ik k ik k

k k

N

i ik k i

k

K xK x h U C W

x

h nC U V

r

h nK x C U C V

r

h nK x n C V U

r r

=

=

= =

=

+ +

+

+ +

+ =

(36)

1 0

3

(1)

1 1

1

3

( 2 ) (1)

1 1

1 1

3 2

(1)

1 1

1

( )1 1( )(1 ) ( ) (1 )

2 2

( )]12

1( )(1 ) ( )

2 12

( ) ( ) 012

i

i i

N

ik k i

k

N N

i ik k ik k

k k

N

i ik i i

k

K xnK x h V W

r x

h nC V U

r

h nK x C V C U

r

h n

K x n C U V r r

=

= =

=

+

+

+ =

Where2

( )( )

1

E xK x

=

The boundary conditions for a simply supported shell are

given as:

0 10 0,

x xv w N M v x L= = = = = = (37)

In order to carry out the analyses, domain and boundary

degrees of freedom are separated, and in vector forms they are

denoted as (d) and (b), respectively. Based on this definition,

the matrix form of the equilibrium equations and the related

boundary conditions take the following form:

(38)

Where{ }b

U and{ }b

U are as follows:

{ } {{ },{ },{ }} ,

{ } {{ },{ },{ }} .

T

d xd d zd

T

b xb b zb

U U U U

U U U U

=

= (39)

In relations (38) and (39), subscripts b and d correspond

to the displacement vectors at the boundaries and domain of

the shell, respectively. Eliminating the boundary degrees of

freedom, this equation becomes:

[ ]{ } {0}d

A U = (40)

where

1[ ] [ ] [ ][ ] [ ]

dd db b b bd A A A A A

= (41)

By setting the determinant of [A] equal to zero to obtain the

non-zero solution, the value of P can be found. The critical

buckling load can be obtained by minimizing P with respect to

n, the number of circumferential buckling waves.

IV. 3BRESULTSANDDISCUSSIONBy using the first order shear deformation theory and the

adjacent equilibrium criterion method, numerical results based

on generalized differential quadrature method have been

obtained for axial compressive loading. Alumina and

aluminum are used as ceramic and metal materials of the FG

cylindrical shell, respectively. The Youngs modulus for

alumina and aluminum are considered as 380 GPa and 70

GPa, respectively. The shell thickness is set to be 0.001m and

Poissons ratio is assumed to be 0.3. For the given values of

the power law index k, thickness ratio R/h, and aspect ratio

L/R, the values of circumferential wave number to give thesmallest value of buckling load, are obtained by optimization

program. Results are verified with substituting power law

index k, equal to zero to obtain an isotropic shell. Comparison

of the results in this situation investigates accuracy of the

present method. Verification of the results for isotropic shell,

consist of alumina, with literature [18] is listed in table 1 and

values of buckling load for FG cylindrical shell with respect to

thickness ratio R/h, aspect ratio L/R and power law index and

grid distribution with N=13 are listed in table 2. The number

in parentheses indicates the circumferential wave number (n).

TABLE I

COMPARISON OF CRITICAL BUCKLING LOADS (MN) FOR

ISOTROPIC CYLINDRICAL SHELL

L/r r/h

Alumina

crP DQM

crP Ref

% of

error

0.5

5 1.5747(n=1) 1.598 1.52

10 1.3867(n=1) 1.403 1.21

30 1.5293(n=5) 1.566 2.36

100 1.4449(n=9) 1.443 0.06

300 1.4445(n=13) 1.443 0.06

1

5 1.4597(n=2) 1.472 0.88

10 1.4120(n=3) 1.403 0.64

30 1.4655(n=5) 1.435 2.09100 1.4375(n=7) 1.443 0.41

300 1.4506(n=9) 1.443 0.48

[ ] [ ] { }0

[ ] [ ] { }

bb bd b

db dd d

A A U

A A U=


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V.CONCLUSIONIn this paper equilibrium and stability equations of simply

supported composite cylindrical shell with fibers material

properties changing lengthwise based on first order sheardeformation theory, are obtained. Buckling analysis of the

shell subjected to axial compression is also investigated and

critical buckling load using generalized differential quadrature

method with different geometrical parameters and power law

indices is obtained. Some of the observations made based on

the results are as follows:

The critical buckling loads of FG cylindrical shells areless than the isotropic cylindrical shell.

The critical buckling loads of FG cylindrical shellswith lengthwise material distribution are less than the

ones with material distribution along their thickness.

The rate of change in critical loads reducessignificantly with the increase in the power law

index.

The value of circumferential wave number in whichshell buckles, varies with respect to the thickness

ratio and aspect ratio.

Critical buckling load decreases with increasing thelength to radius ratio.

VI. .REFERENCES[1] D.O. Brush, and B.O. Almorth, Buckling of bars, plates and shells,

McGraw-Hill, NewYork, 1975.

[2] S. P. Timoshenko, Strength of Materials, part II, 3rd ed., New York,Van Nostrand Reinhold Co., 1976.

[3] I. Mirsky, G. Hermann, Axially Motions of Thick CylindricalShells,Journal of Applied Mechanics, vol 25, pp. 97-102, 1958.[4] N. Tutuncu, M. Ozturk, Exact solution for Stresses in Functionally

Garded Pressure Vessels, Journal of Composites, Part B

(Engineering), vol 32, pp. 686, 2001.

[5] M. Jabbari, S. Sohrabpour, M. R. Eslami, Mechanical and ThermalStresses in a Functionally Graded Hollow Cylinder due to Radially

Symmetric Loads,International Journal of Pressure Vessels Piping,

vol. 79, pp. 493-497, 2002.

[6] Wu. Lanhe, J. Zhiqing, L. Jun, Thermoelastic Stability ofFunctionally Graded Cylindrical Shells, Journal of Composite

Structure, vol. 70, pp. 60-68, 2005.

[7] ZS. Shao, Mechanical and Thermal Stresses of a FunctionallyGraded Circular Hollow Cylinder with Finite Length, International

Journal of Pressure Vessels Piping, vol. 82, pp. 155-163, 2005.

[8] SL. Li, RC. Batrab, Buckling of Axially Compressed ThinCylindrical Shells with Functionally Graded Middle Layer,Journal

of Thin-Walled Structures, vol. 43, pp. 307-324, 2006.

[9] M. M. Najafizadeh, A. Hasani, P. Khazaeinejad, Mechanical stabilityof Functionally Graded Stiffened Cylindrical Shells, Journal of

Applied Mechanical Modelling, vol. 33, pp. 1151-1157, 2009.

[10] C. W. Bert, S. K. Jang, A.G. Striz, Two new Approximate Methodsfor Analysing Free Vibration of Structural Components, Journal of

AIAA, vol. 26, pp. 612-618, 1988.

[11] C. W. Bert, M. Malik, Differential Quadrature in ComputationalMechanics: a review, Journal of Applied Mechanics, rev 49, pp. 1-

27, 1996.

[12] T.Y. Ng, H. Li, K. Y. Lam, C. T. Loy, Parametric Instability ofConical Shells by the Generalized Differential Quadrature Method,

International Journal of Numerical Method Engineering, vol. 44, pp.

819-837, 1999.

[13] K. Y. Lam, H. Li, T. Y. NG, C. F. Chua, Generalized DifferentialQuadrature Method for the Free Vibration of Truncated Conical

Panels, Journal of Sound and Vibration, vol. 251, pp. 329-348,2002.

[14] T.Y. Ng, H. Li, K. Y. Lam, Generalized Differential Quadrature forFree Vibration of Rotating Composite Laminated Conical Shell with

Various Boundary Conditions,International Journal of Mechanical

Science, vol. 45, pp. 567-587, 2003.

[15] T.Y. Ng, H. Li, K. Y. Lam, C. F. Chua, Frequency Analysis ofRotating Conical Panels: A Generalized Differential Quadrature

Approach, Journal of Applied Mechanics, vol. 70, pp. 601-605,

2003.

[16] J. N. Reddy, Mechanics of Laminated Composite Plates and Shells:Theory and Analysis, 2nd edition, CRC Press, Boca Raton, FL,

2004.

[17] C. Shu, Differential Quadrature and Its Application in Engineering,Springer, Berlin, 2000.

[18] P. Khazaeinejad, M. M. Najafizadeh, Mechanical Buckling ofCylindrical Shells with Varying Material Properties, Journal of

Mechanical Engineering Science, vol. 224, pp. 1551-1557, 2010.

TABLE II

COMPARISON OF CRITICAL BUCKLING LOADS (MN) FOR SIMPLY SUPPORTED FG CYLINDRICAL SHELL WITH

DIFFERENT MATERIAL DISTRIBUTION

k=10k=5k=2k=1r/hL/r

0.3625(n=1)0.4510(n=1)0.6134(n=1)0.79149(n=1)5

0.5

0.3403(n=2)0.4278(n=2)0.6272(n=2)0.86166(n=2)10

0.3051(n=5)0.3586(n=5)0.5132(n=5)0.7341(n=5)30

0.2818(n=8)0.3174(n=8)0.4136(n=8)0.6069(n=8)100 0.2797(n=15)0.2919(n=15)0.3482(n=14)0.5047(n=14)300

0.2912(n=2)0.3449(n=2)0.4736(n=2)0.6609(n=2)5

1

0.2760(n=1)0.3398(n=3)0.4688(n=3)0.6749(n=3)10

0.2836(n=4)0.3116(n=4)0.3942(n=4)0.5775(n=4)30

0.2736(n=9)0.2829(n=8)0.3314(n=8)0.4756(n=8)100

0.2699(n=15)0.2720(n=15)0.2980(n=14)0.4106(n=14)300


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