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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:
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
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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
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Science, vol. 45, pp. 567-587, 2003.
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[16] J. N. Reddy, Mechanics of Laminated Composite Plates and Shells:Theory and Analysis, 2nd edition, CRC Press, Boca Raton, FL,
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[17] C. Shu, Differential Quadrature and Its Application in Engineering,Springer, Berlin, 2000.
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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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