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Dynamic Stability of FGM Cylindrical Shells

F. Ebrahimi*

*Department of Mechanical Eng.., University of Tehran, North

Karegar Avenue, Tehran, Irantel/fax: +98 21 88 005 67

e-mail:[email protected]

Submitted: 23/03/2010Accepted: 16/04/2010

Appeared: 29/04/2010

HyperSciences.Publisher

AbstractIn this study, a formulation for the stability of cylindrical shells made of functionally graded

material (FGM) subjected to combined static and periodic axial loadings are presented. The properties are

temperature dependent and graded in the thickness direction according to a volume fraction power law

distribution. The analysis is based on two different methods of first-order shear deformation theory(FSDT) considering the transverse shear strains and the rotary inertias and the classical shell theory

(CST). The results obtained show that the effect of transverse shear and rotary inertias on dynamic

stability of functionally graded cylindrical shells is dependent on the material composition, the

temperature environment, the amplitude of static load, the deformation mode, and the shell geometry

parameters.

Keywords:FGM, CST, Dynamic Stability, Cylindrical Shell.

1. INTRODUCTION

Functionally graded materials are microscopically

inhomogeneous materials. By choosing specific

manufacturing processing, the property of the producedFGMs may vary from points to points or from layer to layers.

Most importantly, certain properties (usually the desired

ones) of the specifically produced FGMs are superior to those

of corresponding homogeneous materials. Recent advance in

the manufacturing technology has made FGMs morepreferential from both functionality and economy points of

views. As a result, the usage of FGMs has been significantly

broadened. For example, early successful applications of

FGMs were reported as the high-temperature materials in

nuclear reactors and chemical plants in Japan (Koizumi andNiino (1995)). Now, FGMs are also considered as potential

structural materials for the future high-speed spacecrafts.

Formulation and theoretical analysis of the FGM plates andshells were presented by Reddy (2004), Reddy and Chin

(1998), Arciniega et al. (2007) and Praveen et al. (1999).

Shell structure made up of this composite material (FGM) isalso one of the basic structural elements used in many

engineering structures.

Despite the evident importance in practical applications,investigations on the static and dynamic characteristics of

FGM shell structures are still limited in number.

Among those available, Loy et al. (1999) investigated the free

vibration of simply supported FGM cylindrical shells, which

was later extended by Pradhan et al. (2000) to cylindricalshells under various end supporting conditions. Gong et al.

(1999) presented elastic response analysis of simplysupported FGM cylindrical shells under low-velocity impact.

Ng et al. (2001) studied dynamic instability of simply

supported FGM cylindrical shells, a normal-mode expansion

and Bolotin method were used to determine the boundaries of

the unstable regions. In all the above studies, theoreticalformulations were all based on classical shell theory, i.e.,neglecting the effect of transverse shear strains. Using Loves

shell theory and the Galerkin method, Sofiyev (2005)

presented an analytic solution for the stability behavior of

cylindrical shells made of compositionally (or functionally)

graded ceramicmetal materials under the axial compressive

loads.

This paper studies the effect of transverse shear and rotaryinertias on dynamic stability of the functionally graded

cylindrical shells subjected to combined static and periodic

axial forces, through a comparison of results obtained by

using two different methods such as the first-order sheardeformation theory (FSDT) considering the transverse shear

strains and the rotary inertias and the classical shell theory

(CST). Material properties of functionally graded cylindrical

shells are considered as temperature dependent and graded in

the thickness direction according to a power-law distribution

in terms of the volume fractions of the constituents. Theresults obtained show that the effect of transverse shear and

rotary inertias on dynamic stability of functionally graded

cylindrical shells subjected to combined static and periodic

axial forces is dependent on the material composition, the

temperature environment, the amplitude of static load, the

deformation mode, and the shell geometry parameters. It is

found that the effect of transverse shear and rotary inertias ondynamic stability of functionally graded cylindrical shells

Journal of Advanced Research in Mechanical Engineering (Vol.1-2010/Iss.2)Ebrahimi / Dynamic Stability of FGM Cylindrical Shells / pp. 111-119

111Copyright 2010 HyperSciences_Publisher. All rights reserved www.hypersciences.org

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subjected to combined static and periodic axial forces is not

neglected in some cases. The new features of the effect of

transverse shear and rotary inertias on dynamic stability of

functionally graded cylindrical shells and some meaningful

results in this paper are helpful for the application and the

design of nuclear reactors, space planes and chemical plants,

in which functionally graded cylindrical shells act as basicelements.

2. THEORETICAL FORMULATIONS

A functionally graded materials cylindrical shell with meanradius of R, thickness h, and the length L is shown in Fig. 1.

The displacement components in the x, and z direction aredenoted by u, vand wrespectively. The pulsating axial load

is given by

PtNNN doa cos+= (1)

where P is the frequency of excitation in radians per unittime. The material properties of FGM cylindrical shells with

both temperature dependent and position dependent are

accurately modeled, by using a simple rule of mixtures for

the stiffness parameters coupled with the temperature

dependent properties of the constituents. The volume fraction

is described by a spatial function as follows

( )+= 0,)21()( hzzV (2)

where expresses the volume fraction exponent. The

combination of these functions gives rise to the effectiveproperties of functionally graded materials. An FGMcylindrical shell that is metal rich at the inner surface and

ceramic rich at the outer surface is considered as Type A. The

corresponding effective material properties are expressed as

( )( )zVTFzVTFzTF mceff += 1)()()(),( (3)

where effF is the effective material property of the FGM

cylindrical shell, including the effective elastic modulus,

effective mass density and effective Poissons ratio. cF and

mF are the temperature dependent properties of the ceramic

and metal, respectively . On the other hand, an FGMcylindrical shell that is ceramic rich at the inner surface and

metal rich at the outer surface is defined as Type B, whose

effective material properties are given by

( )( )zVTFzVTFzTF cmeff += 1)()()(),( (4)

Based on the first-order shear deformation theory (FSDT),

the equations of motion for an FGM cylindrical shell underaxially dynamic load are as follows (Timoshenko and

Woinowsky-Krieger 1959)

+=

+

121

61 1

IuIN

Rx

N(5)

+=

++

+

2212

2

2

16 11

IvIx

vNQ

R

N

Rx

Na

(6)

wIx

wN

R

NQ

Rx

Qa

=

+

+

12

2

221 1

(7)

+=

+

1321

61 1

IuIQM

Rx

M(8)

+=

+

2322

26 1

IvIQM

Rx

M(9)

where 1 and 2 are the rotations of a normal to the

reference surface, )3,2,1( =iIi is the mass inertia termsdefined as

( )

= 2

2

2

321 ),,1)((,,h

h zzzIII (10)

and )(z is the effective mass density of functionallygraded materials. The stress resultants of FGM cylindrical

shells are given by

Fig. 1. Coordinate system of the FGM cylindrical shell


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=

6

2

1

6

2

1

6666

22212221

12111211

6666

22212221

12111211

6

2

1

6

2

1

0000

00

00

0000

00

00

DB

DDBB

DDBB

BA

BBAA

BBAA

M

M

M

N

N

N

=

4

5

55

44

2

1

0

0

C

C

Q

Q(11)

where ijA , ijB , ijD and ijC are ,respectively, the

extensional, coupling, bending, and shear stiffness, which are

given by

( ) )6,2,1(,),,1(,,2

2

2 == idzzzQDBAh

h ijijijij

222111 eff

effEQQ

== ,

( )eff

effEQ

+=

1244

)1(22112

eff

effeff

A

EQQ

== , AQQ 4455=

RzA += 1 ,)1(2

66

eff

eff

A

EQ

+=

==2

2

55552

2

4444 ,h

h

h

h dzQCdzQC (12)

where effE and eff are the effective elastic modulus and

effective Poissons ratio of FGM cylindrical shells,

respectively. is the shear correction factor introduced byReddy (2004) and is equal to 5/6 . The strains are expressed

as

x

u

=1 ,

+

= w

v

R

12 ,

+

=

u

Rx

v 16 ,

+= w

R

124 ,

x

w

+= 15 ,

x

= 11

,

= 22

1

R ,

+

= 126

1

Rx

(13)

Utilizing Eqs.(5)-(9), (11) and (13), the equations of motion

can be expressed in terms of generalized displacement

),,,,( 21wvu as follows

+= 121211 ),,,,( IuIwvuL

(14)

+=

+ 2212

2

212 ),,,,( IvIx

vNwvuL a (15)

wIx

wNwvuL a =

+ 12

2

213 ),,,,( (16)

+= 132214 ),,,,( IuIwvuL (17)

+= 232215 ),,,,( IvIwvuL (18)

By neglecting terms 2I and 3I involving in equations

(5)-(9) and setting

,1x

w

= ,

12

=

w

R(19)

the equations of motion based on a classical shell theory(CST) can be easily obtained.

Here, the two ends of FGM cylindrical shells are consideredas simply supported, so that a solution for the motion

equations (14)-(18) can be described by

nxeAu mti

mnmn coscos=

nxeBv mti

mnmn sinsin=

nxeCw mti

mnmn cossin=

nxeH mti

mnmn coscos1 =

nxeK mti

mnmn sinsin2 = (20)

Fig. 2. Comparison of CST and FSDT unstable regions for a

simply supported FGM cylindrical shell under

combined static axial compressive loading and periodic

axial loading ( m=1,2 , n=1,2,crNN 5.00= , L/R=1.0,

T=300K,=1.0 , h/R=0.01).


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Fig. 3. Effect of thickness to radius ratio h/R on the first

unstable region for a simply supported FGM cylindricalshell under combined static axial extensional loading

and periodic axial load (m=1,2 , n=1,2 , crNN 5.00= ,

L/R=1.0 , T=300K, =1.0 ).

whereL

mm

= , n represents the number of

circumferential waves and m represents he number of axial

half-waves.

Substituting Eq. (20) into Eqs.(14)-(18) and letting 0=dN

in Eq.(1), yields

=

+

+

0

0

00

0

000

000

0000000

000

32

32

1

21

21

2

5554535251

4544434241

35340

2

333131

2524230

2

1221

1514131211

mn

mn

mn

mn

mn

m

m

K

H

CB

A

II

II

III

II

TTTTT

TTTTT

TTNTTT

TTTNTT

TTTTT

(21)

where ijT is given in appendix A.

To solve the equations of motion containing the dynamic

load dN , a solution is sought in the form shown below

=

=

=

=

=

=

=

5

1 1 1

5

1 1 1

coscos)(

),()(

j m n

mmnjmnj

j m n

mnjmnj

nxAtq

xUtqu

(22)

=

=

=

=

=

=

=

=

5

1 1 1

5

1 1 1

sinsin)(

),()(

j m n

mmnjmnj

j m n

mnjmnj

nxBtq

xVtqv

(23)

=

=

=

=

=

=

=

=

5

1 1 1

5

1 1 1

cossin)(

),()(

j m n

mmnjmnj

j m n

mnjmnj

nxCtq

xWtqw

(24)

=

=

=

=5

1 1 1

1 ),()(j m n

xmnjmnj xtq

=

=

=

=5

1 1 1

coscos)(j m n

mmnjmnj nxHtq

(25)

=

=

=

=

=

=

=

=

5

1 1 1

5

1 1 1

2

sinsin)(

),()(

j m n

mmnjmnj

j m n

mnjmnj

nxKtq

xtq

(26)

where )(tqmnj is a generalized co-ordinate, and mnjU , mnjV

, mnjW , xmnj and mnj are the modal function of FGM

cylindrical shells with simply supported ends, under the

axially static load 0N . Substituting Eqs.(22)-(26) into

Eqs.(14)-(18), yields

mnjmnjmnjmnj

mnjxmnjmnjmnjmnj

HIUI

WVUL

2

2

2

1

1 ),,,,(

=(27)

mnjmnjmnjmnjmnjm

mnjxmnjmnjmnjmnj

IVIVN

WVUL

22

21

20

2 ),,,,(

=

(28)

mnjmnjmnjm

mnjxmnjmnjmnjmnj

WIWN

WVUL

2

1

2

0

3 ),,,,(

=

(29)

xmnjmnjmnj

mnjxmnjmnjmnjmnj

IUI

WVUL

3

2

2

4 ),,,,(

=(30)

mnjmnjmnj

mnjxmnjmnjmnjmnj

IVI

WVUL

3

2

2

5 ),,,,(

=(31)

Eqs. (14)- (18) may be rewritten as


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Fig. 4. Unstable region versus axial compressive loading crNN0 for a simply supported FGM cylindrical shell under

combined static axial compressive loading and periodic axial loading ( m=1,2 , n=1,2 , L/R=1.0 , T=300K,

=1.0 , h/R=0.01).

Table 1. Comparison of the points of origin P1 for a simply supported silicon nitride-nickel FGM cylindrical shell underaxial extensional loading

P1 (Type A) P1 (Type B)

Present Ng et al. (2001) Present Ng et al. (2001)

0 10.955 10.956 10.778 10.774

0.5 10.896 10.894 10.849 10.849

1.0 1.0867 10.865 10.883 10.883

5.0 10.809 10.805 10.936 10.937

10 10.795 10.791 10.945 10.946

10.778 10.773 10.955 10.946

( )( )=

=

=

=++5

1 1 1

21

20coscos

j m n

mmnjmnjmnjmnjmnj nxHIAIqq (32)

( )( )=

=

=

++5

1 1 1

21

2sinsin

j m n

mmnjmnjmnjmnjmnj nxKIBIqq

=

=

==+

5

1 1 1

2

0sinsincosj m n

mmnjmnjmd nxqBPtN

(33)

( )=

=

=

+5

1 1 1

12 cossin

j m n

mmnjmnjmnjmnj nxCIqq

=

=

=

=+5

1 1 1

2 0cossincosj m n

mmnjmnjmd nxqCPtN

(34)

( )( )=

=

=

=++5

1 1 1

32

20coscos

j m n

mmnjmnjmnjmnjmnj nxHIAIqq (35)

( )( )=

=

=

=++5

1 1 1 32

20sinsin

j m n mmnjmnjmnjmnjmnj

nxKIBIqq (36)


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Making use of the orthogonality condition, Eqs.(32)-(36) are

simplified to

0

000

000

000

000

cos

000

000

000

000

000

000

000

000

22

2

22

5

2

1

5

2

1

5

2

1

5

2

1

5

1

1

=

+

NN

d

N

NN

q

M

q

q

Q

Q

Q

PtN

k

k

k

q

M

q

q

m

m

m

(37)where

In the above formula, the coefficients of mode shapes IA, IB , IC , IH and IK can be obtained from Eq. (21).

Based the CST and the orthogonality condition, Eq.(37) can

be simplified to

=

+

0

0

0

000

000

000

000

cos

000

000

000

000

000

000

000

000

22

222

3

2

1

3

2

1

3

2

1

3

2

1

3

1

1

M

q

M

q

q

Q

Q

Q

PtN

k

k

k

q

M

q

q

m

m

m

NN

d

NNN

(40)

where

The coefficients IA , IB , IC in Eqs.(40) can be obtained

from Eq.(24). Equations (37) and (40) are in the form of a

second order differential equations with periodic coefficients

of the Mathieu-Hill type. Using the method presented byBolotin (1964), the regions of unstable solutions are

separated by periodic solutions. As a first approximation, the

periodic solutions with period 2T can be sought in the form

where Ia and Ib are arbitrary constants. Substituting

equation (42) into equations (37) and (40), and equating the

coefficients of the 2sinPt and 2cos Pt terms, a set of

linear homogeneous algebraic equations in terms of Ia and

Ib can be obtained. The conditions for non-trivial solutions

for the linear homogeneous algebraic equations are

Each unstable region is bounded by two lines which originate

from a common point from the P-axis. The branches emanate

( ) ( )[

( ) ( ) ( )]

( )[ ]2

1212

2

2

32

2

321

2

21

2

21

2

,

2

IIIImI

III

IIIII

IIIII

CIKIBIBLQ

mk

KIBIHIAICI

HIAIKIBIl

m

++=

=

++++

++++=

(38)

);5(5),2(35

),1(45:),(),(

);5(55),2(25

),1(15:)1,2(),(

);5(5),2()3(5

),1()4(5:),1(),(

);5(10),4(9),3(8

),2(7),1(6:)2,1(),(

);5(5),4(4),3(3

),2(2),1(1:)1,1(),(

22

2

==

===

=+=+

=+==

==

===

===

====

===

====

jNjN

jNINNnm

jNjN

jNInm

jNjN

jNINnm

jjj

jjInm

jjj

jjInm

(39)

( ) ( ) ( )[ ]

( )[ ]2

2

11

2

2

21

21

21

3,...,3,2,1

,2

,

,2

NI

CIBIBL

Q

mk

CIAIBIlm

IIImI

III

IIII

=

+=

=

++=

(41)

2cos

2sin

Ptb

Ptaq III += (42)

02

1

4

1 2=+ IdIII QNkmP

(43)

02

1

4

1 2=++ IdIII QNkmP (44)


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at 0=dN from the I2 . The left and right branch

correspond with

)0(24

24

>

=

+=

d

I

IdII

I

IdII

Nm

QNkP

andm

QNkP

)0(24

24

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