. 1

L and 2

L are the axial and

circumferential lengths of the middle su rface of the shell,

and the thickness

H

varying continuously in the

circumferential direction. The cylindrical coordinates

,,

x

sz are taken to define the position of a point on the

middle surface of the shell, as shown in Figure 1(a), and

Figure 1(b) shows the three-lobed cross-section profile of

the middle surface, with the apothem denoted by1

A

, and

the radius of curvature at the lobed corners by 1

R. While

,u

and w are the deflection displacements of the mid-

dle surface of the shell in the longitudinal, circumferen-

tial and transverse directions, respectively. We suppose

Figure 1. Coordinate system and geometry of a three-lobed cross-section cylindrical shell with circumferential variable

thickness.

(a)

(b) (c)

M. K. AHMED

Copyright © 2011 SciRes. AM

331

that the shell thickness

H

at any point along the cir-

cumference is small and depends on the coordinate

and takes the following form:

0

H h

(1)

where 0

h is a small parameter, chosen to be the average

thickness of the shell over the length 2

L. For the cylin-

drical shell which cross-section is obtained by the cuta-

way the circle of the radius 0

r from the circle of the

radius 0

R(see Figure 1(c)) functio n

have the form:

11cos

where

is the amplitude of thi-

ckness variation, 0

dh

, and d is the distance be-

tween the circles centers. In general case

00hH

is the minimum value of

while

mHh

is

the maximu m value of

, and in case of 0d

the

shell has constant thickness 0

h. The dependence of the

shell thickness ratio 0m

hh

on

has the form

12

.

For studying the free harmonic vibrations and buck-

ling of the shell under consideration, the equilibrium

equations of forces for the cylindrical shell, subjected to

an axial compressive load P, based on the Goldenveiz-

er-Novozhilov theory are taken from [27,28] as follows:

22

2

0, 0,

0, 0,

0, 0,0,

xsxxs ss

xs sxsxx

xssss ssxxssxsx

NN PuHuNNQRPH

QQNRPwHwMM Q

MMQ SQMNNMR

(2)

where Nx, Ns and Qx, Qs are the normal and transverse

shearing forces in the x and s directions, respectively, Nsx

and Nxs are the in-plane shearing forces, Mx, Ms and Mxs,

Msx are the bending moment and the twisting moment,

respectively, Ss is the equivalent (Kelvin-Kirchoff) shea-

ring force, P is the axial force per unit length, constant

along the circumference,

is the mass density, R is the

radius of curvature of the middle surface,

is the an-

gular frequency of vibration, 'x , and

s

.

The relations between strains and deflections for the

cylindrical shells used here are taken from [29] as fol-

lows:

,,, 0,

0, ,,

xsxsxz x

szsx xs ssx sxsx

uwRuw

wRkkwRR,kk R

(3)

where

x

and

s

are the normal strains of the middle

surface of the shell, ,

x

sxz

and

s

z

are the shear

strains, and the quantities ,,

x

ssx

kkk and

x

s

k represen-

ting the change of curvature and the twist of the middle

surface,

x

is the bending slope, and

s

is the angu-

lar rotation.

The components of force and moment resultants in

terms of Equation (3) are given as:

,,12

,,1

xxs ssxxsxs

xxsss xsxsx

NDNDN D,

MKkkMKkkM kk

(4)

From Equations (2)-(4 ), with eliminating the variables

,,,, ,

x

sxxs xxs

QQN NMM and

s

x

M

wh ich are not dif-

fer-rentiated with respect to

s

, the system of the partial

differential equations fo r the state variables ,,, ,

s

uw

,,

s

ss

M

SN and

s

x

N of the shell are obtained as fol-

lows:

2

22 2

22

2( (1))6,,,

,21,

1, ,

1.

sx sss

ss xssss

sssss sx

sx s

uD NHRNDwRuwr

MKNRDRuMSK

SNRM KwPwHwNPSRNH

NDuPu NHu

(5)

The quantities D and K, respectively, are the ex-

tensional and flexural rigidities expressed in terms of the

Young’s modulus E, Poisson’s ratio

and the wall

thickness

H

as the form:

2

1DEH

and

32

12 1KEH

, and under the variable thickness

case, using Equation (1), those are written as follows:

M. K. AHMED

Copyright © 2011 SciRes. AM

332

2

00

1DEh D

(6)

323 3

00

1KEh K

(7)

where 0

D and 0

K

are the reference extensional and

flexural rigidities of the shell, chosen to be the averages

on the middle surface of the shell over the length 2

L.

For the free vibration of a simply supported shell, the

solution of the system of Equation (5) is sought as fol-

lows:

111

1

1

1

,cos,,,,,sin,, sin,

,,,,,,,,,,sin,

,,,, ,,,cos,

,,,,sin,

,,, c

ss

xsss xsss

xssxxxssxx

xs xs

xssxxs sx

mmm

u xsUsxxswxsVsWsxxssx

LLL

m

NxsNxsQxsSxsNsNsQsSsx

L

m

N xsN xsQxsNsNsQsx

L

m

MxsMxsMsMsx

L

M xsM x,sM sM s

1

os ,1,2,

mxm

L

(8)

where m is the axial half wave number and the quanti-

ties

,UsV s are the state variables and undeter-

mined functions of

s

.

3. Matrix Form of the Basic Equations

The differential equations as shown previously are mod-

ified to a suitable form and solved numerically. Hence,

by substituting Equation (8) into Equation (5) and take

relations (6) and (7) into account, the system of vibration

equations of the shell can be written in non-linear ordi-

nary differential equations referred to the variable s only

are obtained, in the following matrix form:

12 1418

21 2327

32 34

41 43 45 47

54 56

63 65 67

7276 78

81 87

00000

0 0000

000000

0000

000 0 00

00 00 0

0000 0

000000

s s

s s

s

s

sx

VV V

UU

VVV

VV

VV

WW

VVVV

d

aVV

ds MM

VVV

S

VVV

N

VV

N

s

s

sx

.

S

N

N

(9)

By using the state vector of fundamental unknowns

Z

s, system (9) can be written as follows:

d

aZsVsZs

ds

(10)

,, ,,,,,,

T

sssssx

ZsUVWM S N N

0

2

0

,,,, ,

,1,

s

sss

UVWk UVW

kM M

3

1

,,1,, ,

sssxsssxm

SNN SNN.

L

For the noncircular cylindrical shell which cross-sec-

tion profile is obtained by function (

raf

), the

hypotenuse

ds of a right triangle whose sides are

infinitesimal distances along the surface coordinates of

the shell takes the form

22 2

ds drrd

, then we

have

2

2df

ds

f

d

ad

(11)

By using Equation (11), the system of vibration Equa-

tions (10) takes the form:

dZVZ

d

(12)

M. K. AHMED

Copyright © 2011 SciRes. AM

333

Where

2

2,

df

fd

and the coefficients matrix

V

are give n as:

12

Vml

,

22

14 6Vmlh

,

32

18 61Vmlh

,

21

Vml

,

23 1Vc ,

32

27 12Vmlh

, 32 1Vc

,

34

Vml

, 41

Vc

2

43

Vml

, 3

45 1Vh

, 46 12Vhc

,

22

54 21Vhml

56 1V

,

43

232

63 1212Vml hmlPml

,

65

Vml

,

67

Vm/lc

,

3

2

72 12VPml hml

,

76 1Vc , 78

Vml

,

3

22 2

81 11212VhmlPmlhml

,

87

Vml

in terms of the following dimensionless shell parameters:

frequency parameter222

00

haD

, load factor

20

PaK P, 1

lLa and 0

hha.

As the function

f

formulates the profile of a shell

with a three lobed cross section, and expressed as in [15]

2

22

111 111

0 0

11 11

2

022 000

111 1111

00

11 1

2cos4 sin,for0

cosec30, ,for120

() ,

2cos1204sin120,for 120120

sec, ,for120180

acc ac

ac

f

acc ac

ac

1

111111 1111

,, ,tan343aAacRacacac

The state vector

Z

of fundamental unknowns can

be expressed as in [30]

0ZYZ

(13)

by using the transfer matrix

Y

of the shell, and the

substitution of the expression into Equation (10) yields

,

0

dd YVY

YI.

(14)

The governing system (14) is too complicated to ob-

tain any closed form solution, and this problem is highly

favorable for solving by numerical methods. Hence, the

matrix

Y

is obtained by using numerical integra-

tion, by use of the Runge-kutta integration method of

forth order, with the starting value

0YI

(unit

matrix) which is given by taking 0

in Equation (13),

and its solution depends only on the geometric and mar-

tial properties of the shell, and the same solution can be

used for appropriate boundary conditions imposed at the

shell circumference.

For a plane passing through the central axis in a shell

with structural symmetry, symmetrical and antisymme-

trical profiles can be obtained, and consequently, only

one-half of the shell circumference is considered with the

boundary conditions at the ends taken to be the symme-

tric or antisymmetric conditions. Therefore, the boundary

conditions for symmetrical and antisymmetrical vibra-

tions are:

0, 0,

0,0,respectively

sssx

ss

VSN

UWN M

(15)

4. Natural Frequencies and Modes

A shell with (

1f

) represents a circular

cylindrical shell of constant thickness. The substitution

of Equation (14) into Equation (12) results the frequency

equations

21 23 25 27

41 43 45 47

61 63 65 67

81 83 85 870

0

s

s

YYYYU

YYYYW

YYYYM

YYYYN

for symmetrical vibration, (16)

M. K. AHMED

Copyright © 2011 SciRes. AM

334

12 14 16 18

32 34 36 38

52 45 56 58

72 74 76 780

0

s

s

sx

YYYY V

YYYY

YYYY S

YYYY N

for antisymmetrical vibration. (17)

The matrix [V] depends on the frequency parameter

and the load factor Pin addition to

,

Y

is also

a function of these parameters. Equations (16) and (17)

give a set of linear homogenous equations with unknown

coefficients

0

T

ss

U,W,M ,N

and

0

T

ss sx

V,,S,N

,

respectively, at 0

. For the existence of a nontrivial

solution of these coefficients, the determinant of the co-

efficient matrix should be vanished. The standard proce-

dures cannot be employed for obtaining the eigenvalues

of the frequency equations. The nontrivial solution is

found by searching the values

which make the deter-

minant zero by using Lagrange interpolation procedure.

The eigenvalues of

which make the determinant zero

give the natural frequencies of the shell. The mode sha-

pes at any point of the cross-section of the shell, over the

length 2

L, are determined by calculating the eigenvec-

tors corresponding to the eigenvalues by using Gaussian

elimination procedure. For free vibration problems with-

out prestress the natural frequencies result when 0P

,

and for vibration problems with prestress the natural

frequencies result for non-zero values of P. For buck-

ling problems, the buckling load results when 0

in

Equations (16) and (17), and the lowest values of the

buckling loads give the critical loads of the shell.

5. Numerical Results and Discussion

A computer program based on the analysis described

herein has been developed to study the symmetrical and

antisymmetrical vibrations of the considered shell, and

the vibration frequencies and the critical buckling loads

of the shell with circumferential variable thickness, are

calculated numerically. Results for uniform thickness of

circular cylindrical shells (1

and 1

) are obtained.

The correctness of applied method is cited in [31] with

other literature. Our study is divided into two parts:

5.1. Vibrations

The study of vibration s is determined by finding the nat-

ural frequencies

and corresponding mode shapes at

0P

. The numerical results presented herein pertain to

the minimum frequencies and the associated mode shapes

of the shell.

The effect of variation in thickness on the minimum

frequencies of vibration, Tables 1 and 2 give the funda-

Table 1. The fundamental frequencies λ for symmetric modes of a three-lobed cross-section cylindrical shell with variable

thickness, for which (=1,= 0.3,= 4,=0.02mνlh ).

1 2 3 4 5

0.0 0.019568 0.028312 0.035171 0.041386 0.047378

0.2 0.019736 0.027475 0.034116 0.040315 0.046313

0.4 0.029985 0.035825 0.040712 0.045741 0.051032

0.6 0.039787 0.050925 0.056946 0.061702 0.066208

0.8 0.053618 0.068977 0.079662 0.086458 0.091184

1.0 0.074135 0.087694 0.100293 0.110095 0.115710

Table 2. The fundamental frequenc ies λ for antisymmetric modes of a three-lobed cross-section cylindrical shell with vari-

able thickness, for which (=1,=0.3,= 4,=0.02mνlh ).

1 2 3 4 5

0.0 0.019568 0.026504 0.033007 0.039530 0.046143

0.2 0.019736 0.026235 0.032464 0.038710 0.045019

0.4 0.029985 0.036643 0.042989 0.049309 0.055683

0.6 0.045266 0.054520 0.062245 0.069595 0.076761

0.8 0.070167 0.082516 0.091719 0.100270 0.108014

1.0 0.074135 0.087694 0.100293 0.110095 0.115710

M. K. AHMED

Copyright © 2011 SciRes. AM

335

mental frequencies for symmetric and antisymmetric vi-

brations of a three-lobed cross-section cylindrical shell,

with the same circumferential length, versus the radius

ratio

for different values of

. The results presen-

ted in these tables show that the increase of the thickness

ratio resulted in an increase in the fundamental frequen-

cies for each value of the radius ratio. These results con-

firm the fact that the effect of increasing the shell flexur-

al rigidity becomes larger than that of increasing the shell

mass when the thickness ratio in creases. Also, the incr-

ease of the radius ratio for the chosen values of the thick-

ness ratio results in an increase in the frequencies

and they become larger at 1

(circular cylinder). In

case of a constant thickness (1

), the symmetric and

antisymmetric type vibrations have the same values ver-

sus the radius ratio, except for the symmetric and antisy-

mmetric ones with respect to three planes passing through

a corner and an axial mid-line of the opposite wall give

unidentical values. For a circular cylindrical shell (1

),

the symmetric and antisymmetric type vibrations give the

same values versus the thickness ratio. In Tables 3 and 4,

the first five frequencies

of symmetric and antisym-

metric type vibrations are presented for different values

of radius ratios (05,07,10...

) and thickness ratios

(1,2,5

). The numbers in the parentheses are the axi-

al half wave numbers of the mode in the

x

-direction.

An ordering change of the mode in the

x

-direction can

be seen for certain values of

and

.

In Table 5, the comparison of the fundamental fre-

quencies

for symmetric and antisymmetric vibrations

of a shell with radius ratio (05.

) is cited for different

axial half wave numbers and thickness ratio. With an in-

crease of axial half wave number aspect the thickness

ratio, the fundamental frequencies increase. It is shown by

this table that the values of fundamental frequencies

for symmetric and antisymmetric modes are very close to

each other for the large mode number, m.

Figures (2-4) show the first five circumferential mode

shapes of a three-lobed cross-section cylindrical shell of

variable thickness for symmetric and antisymmetric vi-

bration modes corresponding to the frequencies

listed

in Tables 3 and 4. The thick lines show the composi tion

of the circumferential and transverse deflections, and the

numbers in the parentheses are the axial half wave num-

Table 3. The first five frequencies λ for symmetric modes of the shell, with (=0.3,=4,=0.02νlh ).

Modes

First Second Third Forth Fifth

1 0.036513(1) 0.037069(1) 0.046870(2) 0.052419(2) 0.065142(3)

[15]0.03653 0.03716 0.04691 0.05248 -

0.5 2 0.043201(1) 0.056216(1) 0.061026(2) 0.082699(3) 0.086711(2)

5 0.057253(1) 0.092093(2) 0.100173(1) 0.125091(3) 0.162623(2)

0.7 2 0.059096(1) 0.075956(1) 0.086524(2) 0.116652(3) 0.121253(2)

5 0.077638(1) 0.116732(1) 0.125442(2) 0.170122(3) 0.200812(1)

1.0 2 0.087641(1) 0.114102(1) 0.130261(1) 0.177653(2) 0.201403(1)

5 0.114922(1) 0.133672(1) 0.210313(2) 0.218261(1) 0.246320(1)

Table 4. The first five frequencies λ for antisymmetric modes of the shell, with (=0.3,=4,=0.02νlh ).

Modes

First Second Third Forth Fifth

1 0.037069(1) 0.052419(2) 0.064586(1) 0.068488(3)0.089686(1)

[15]0.03716 0.05240 0.06470 - 0.8982

0.5 2 0.044870(1) 0.066191(2) 0.080338(1) 0.086261(3)0.124450(1)

5 0.065098(1) 0.099283(2) 0.121120(1) 0.131662(3)0.194172(1)

0.7 2 0.066803(1) 0.092671(2) 0.093671(1) 0.117432(3)0.139351(1)

5 0.091152(1) 0.131372(1) 0.136280(2) 0.172388(3)0.201140(1)

1.0 2 0.087659(1) 0.114122(1) 0.130263(1) 0.177656(2)0.201405(1)

5 0.115203(1) 0.133701(1) 0.210395(2) 0.218264(1)0.246371(1)

M. K. AHMED

Copyright © 2011 SciRes. AM

336

Table 5. Comparison of f undamental fr equencie s for symmetric and anti symmetric modes of the shell, with (0.5,=0.3,ζ=ν

= 4,= 0.02lh )

Symmetric modes Antisymmetric modes

m 1 2 3 1 2 3

1 0.036513 0.043201 0.057253 0.037069 0.044870 0.065098

2 0.046870 0.061026 0.092093 0.052419 0.066191 0.099283

3 0.065142 0.082699 0.125091 0.068488 0.086261 0.131662

4 0.089935 0.112703 0.168702 0.091517 0.114570 0.173476

5 0.121390 0.151092 0.222726 0.122015 0.151953 0.225840

6 0.159833 0.197871 0.285843 0.160012 0.198190 0.287713

7 0.205271 0.252890 0.357090 0.205380 0.252920 0.358031

8 0.258043 0.315931 0.435938 0.258211 0.315819 0.436230

9 0.318039 0.386746 0.522161 0.318140 0.386557 0.522053

10 0.385308 0.464975 0.616405 0.385370 0.464749 0.615964

Figure 2. The first five symmetrical mode shapes of a three-lobed cross-section cylindrical shell with variable thickness. {(a)

1η=, (b) 5η=, 0.5ζ=}; {(c) 2η=, 0.7ζ=}.

(3)

(2)

M. K. AHMED

Copyright © 2011 SciRes. AM

337

Figure 3. The first five antisymmetrical mode shapes of a three-lobed cross-section cylindrical shell with variable thickness.

{(a) 1η=, (b) 5η=, 0.5ζ=}; {(c) 2η=,0.7ζ=}.

ber corresponding to the fundamental frequencies

.

There are considerable differences between the modes of

1

and 1

for symmetric and antisymmetric mo-

des. For 1

, the majority of symmetric and antisym-

metric modes, the displacements at the thinner edge are

larger than those at the thicker edge i.e. the vibration mo-

des are localized near the top generatrix 0

. It is also

shown by these figures that ordering changes of the mo-

des, which have the same axial half wave number, and

the mode shapes are similar in the sets of the vibration

modes having m = 1, 2 and 3 for 1

.

5.2. Buckling

Consider the buckling of a three-lobed cross-section cy-

lindrical shell with circumferential variable thickness un-

der axial compressive loads, constant over the length2

L.

To obtain the buckling loads we will search the zero va-

M. K. AHMED

Copyright © 2011 SciRes. AM

338

Figure 4. The first five symmetrical mode shapes of a circular cylindrical shell with variable thickness. {(a)

5η=, 0.7ζ=}; {(b) 2η=, (c) 5η=, 1.0ζ=}.

lue of the eigenvalues

, and to obtain the critical loads

we will search the lowest values of the buckling loads.

In Figures (5-7 ), the eigenvalues of vibration

ver-

sus the axial compression loads Pare shown of a three-

lobed cross-section cylindrical shell, with variable thick-

ness, for symmetric and antisy mmetr ic mo de s. Th e nu m-

ber in the parentheses is the axial half wave number cor-

responding to each one of the first five eigenvalues listed

in Tables 3 and 4. With an increase in axial load, each

eigenvalue monotonically decreases and finally becomes

zero. The values on the ordinate represent the eigenva-

lues of the shell without the action of axial load. The

loads that make the eigenvalues zero are called buckling

loads, beyond which the shell beco mes unstable and then

can not keep its shape. It is shown from theses figures

that the buckling loads increase with an increase of the

thickness ratio

for each value of

, and also for

each value of

the buckling loads increase with an in-

M. K. AHMED

Copyright © 2011 SciRes. AM

339

crease of

, and for 1

they become larger. The

critical loads for symmetric and antisymmetric vibrations

are occurred with 3m.

Figure 7 shows the eigenvalues of vibration

ver-

sus the axial compression loads for a circular cylindrical

shell with circumferential variable thickness. In Figure

7(a), the plotted eigencurve gives the critical buckling

load of a circular cylindrical shell of a constant thickness,

(266 579P.), and good agreement result with [14], and

it is occurred with 1m. While the critical buckling

loads of a circular cylindrical shell having variable thick-

ness are occurred with 2m.

Figure 8 shows the buckling loads versus the axial

half wave number of a shell with radius ratio (05.

)

for symmetric and antisymmetric modes. It is shown from

this figure, symmetric case gives lower buckling loads

than antisymmetric one, and the buckling loads are very

close to each other for the large mode number m. Also,

with an increase of axial half wave number, the buckling

loads increase after once decreasing. For each value of

, the critical buckling loads of the shell are occurred

with 3m

for the symmetric and antisymmetric mod-

es.

6. Conclusions

An approximate analysis for studying the free vibration

and buckling of a three-lobed cross-section cylindrical

shell having circumferential varying thickness under

axial membrane loads is presented. The numerical results

presented herein pertain to the fundamental frequencies

and buckling loads as well as the associated mode shapes

Figure 5. Eigenvalues of vibration versus axial load of a three-lobed cross-section cylindrical shell with variable

thickness, symmetric case. (= 0.3,= 0.02νh).

M. K. AHMED

Copyright © 2011 SciRes. AM

340

Figure 6. Eigenvalues of vibration versus axial load of a three-lobed cross-section cylindrical shell with variable

thickness, antisymmetric case. (=0.3, =0.02νh).

Figure 7. Eigenvalues of vibration versus axial compression load of a circular cylindrical shell with circumferen-

tial vari abl e thickness. (=0.3, =0.02νh).

M. K. AHMED

Copyright © 2011 SciRes. AM

341

Figure 8. Buckling loads versus axial half wave number of a

three-lobed cross-section cylindrical shell with variable thi-

ckness. (=0.3,= 0.02νh).

by using the transfer matrix method. The method is

based on thin shell theory and applied to a shell of sym-

metric and antisymmetric vibration modes, and the anal-

ysis is formulated to overcome the mathematical difficul-

ties associated with mode coupling caused by variable

shell wall curvature and thickness. The first five funda-

mental frequencies and mode shapes as well as critical

buckling loads have been presented, and the effects of

the thickness ratio of the cross-section on the natural fre-

quencies, mode shapes, and buckling loads were exa-

mined. For the thickness ratio 1

, the vibration mod-

es are distributed regularly over the shell surface, but for

1

the modes are localized near the weakest genera-

trix 0

, (thinner edge), in most cases of the vibration

modes. However, the critical buckling loads increase

with either increasing radius ratio or increasing thickness

ratio and become larger for a circular cylindrical shell.

For the cylindrical shell of (1

and 1

), the criti-

cal loads are occurred with 2m

, but for the shell un-

der consideration of (1

and 1

) they occurred

with 3m for the symmetric and antisymmetric vibra-

tion modes.

7. Acknowledgments

The author is grateful to anonymous reviewers for their

good efforts and valuable comments which helped impr-

ove the quality of this paper.

8. References

[1] W. Flügge, “Die Stabilität der Kreiszylinderschale,” In-

genieur Archiv, Vol. 3, No. 5, 1932, pp. 463-506.

[2] L. H. Donnell, “Stability of Thin-Walled Tubes under

Torsion,” NACA Report, No. NACA TR-479, 1933.

[3] H. Becker and G. Gerard, “Elastic Stability of Orthotrop-

ic Shells,” Journal of Aerospace Science, Vol. 29, No. 5,

1962, pp. 505-512.

[4] G. Gerard, “Compressive Stability of Orthotropic Cy-

linders,” Journal of Aerospace Science, Vol. 29, 1962,

pp. 1171-1179.

[5] S. Cheng and B. Ho, “Stability of Heterogeneous Aeolo-

tropic Cylindrical Shells under Combined Loading,”

AIAA Journal, Vol. 1, No. 40, 1963, pp. 892-898.

[6] R. M. Jones, “Buckling of Circular Cylindrical Shells

with Multiple Orthotropic Layers and Eccentric Stiffen-

ers,” AIAA Journal, Vol. 6, No. 12, 1968, pp. 2301-2305.

doi:10.2514/3.4986

[7] Y. Stavsky and S. Friedl, “Stability of Heterogeneous

Orthotropic Cylindrical Shells in Axial Compression,”

Israel Journal of Technology, Vol. 7, 1969, pp. 111-119.

[8] M. M. Lei and S. Cheng, “Buckling of Composite and

Homogeneous Isotropic Cylindrical Shells under Axial

and Radial Loading,” Journal of Applied Mechanics, Vol.

36, No. 4, 1969, pp. 791-798.

[9] J. B. Greenberg and Y. Stavsky, “Buckling and Vibration

of Orthotropic Composite Cylindrical Shells,” Acta Me-

chanica, Vol. 36, No. 12, 1980, pp. 15-29.

doi:10.1007/BF01178233

[10] J. B. Greenberg and Y. Stavsky, “Vibrations of Axially

Compressed Laminated Orthotropic Cylindrical Shells,

Including Transverse Shear Deformation,” Acta Mecha-

nica, Vol. 37, No. 1-2, 1980, pp. 13-28.

doi:10.1007/BF01441240

[11] J. B. Greenberg and Y. Stavsky, “Stability and Vibrations

of Compressed Aeolotropic Composite Cylindrical

Shells,” Journal of Applied Mechanics, Vol. 49, No. 4,

1982, pp. 843-848. doi:10.1115/1.3162625

[12] J. B. Greenberg and Y. Stavsky, “Vibrations and Buck-

ling of Composite Orthotropic Cylindrical Shells with

Nonuniform Axial Loads,” Composites Part B: Engi-

neering, Vol. 29, No. 6, 1998, pp. 695-702.

doi:10.1016/S1359-8368(98)00029-8

[13] A. Rosen and J. Singer, “Vibrations of Axially Loaded

Stiffened Cylindrical Shells,” Journal of Sound and Vi-

bration, Vol. 34, No. 3, 1974, pp. 357-378.

doi:10.1016/S0022-460X(74)80317-2

[14] G. Yamada, T. Irie and M. Tsushima, “Vibration and

Stability of Orthotropic Circular Cylindrical Shells Sub-

jected to Axial Load,” Journal of Acoustical Society of

America, Vol. 75, No. 3, 1984, pp. 842-848.

doi:10.1121/1.390594

[15] G. Yamada, T. Irie and Y. Tagawa, “Free Vibration of

Non-Circular Cylindrical Shells with Variable Circumfe-

rential Profile,” Journal of Sound and Vibration, Vol. 95,

No. 1, 1984, pp. 117-126.

doi:10.1016/0022-460X(84)90264-5

[16] K. Suzuki and A. W. Leissa, “Free Vibrations of Noncir-

cular Cylindrical Shells Having Circumferentially Vary-

ing Thickness,” Journal of Applied Mechanics, Vol. 52,

M. K. AHMED

Copyright © 2011 SciRes. AM

342

No. 1, 1985, pp. 149-154. doi:10.1115/1.3168986

[17] V. Kumar and A. V. Singh, “Approximate Vibrational

Analysis of Noncircular Cylinders Having Varying Thi-

ckness,” AIAA Journal, Vol. 30, No. 7, 1991, pp. 1929-

1931. doi:10.2514/3.11161

[18] O. Mitao, T. Hideki and S. Tsunemi, “Vibration Analysis

of Curved Panels with Variable Thickness,” Engineering

Computations, Vol. 13, No. 2, 1996, pp. 226-239.

doi:10.1108/02644409610114549

[19] R. F. Tonin and D. A. Bies, “Free Vibration of Circular

Cylinders of Variable Thickness,” Journal of Sound and

Vibration, Vol. 62, No. 2, 1979, pp. 165-180.

doi:10.1016/0022-460X(79)90019-1

[20] R. M. Bergman, S. A. Sidorin and P. E. Tovstik, “Con-

struction of Solutions of the Equations for Free Vibration

of a Cylindrical Shell of Variable Thickness along the

Directrix,” Mechanics of Solids, Vol. 14, No. 4, 1979, pp.

127-134.

[21] T. Irie, G. Yamada and Y. Kaneko, “Free Vibration of a

Conical Shell with Variable Thickness,” Journal of Sound

and Vibration, Vol. 82, No. 1, 1982, pp. 83-94.

doi:10.1016/0022-460X(82)90544-2

[22] S. Takahashi, K. Suzuki and T. Kosawada, “Vibrations of

Conical Shells with Varying Thickness,” Japan Society of

Mechanical Engineering, Vol. 28, No. 235, 1985, pp.

117-123.

[23] W. I. Koiter, I. Elishakoff, Y. W. Li and J. H. Starness,

“Buckling of an Axially Compressed Cylindrical Shell of

Variable Thickness,” International Journal of Solids and

Structures, Vol. 31, No. 6, 1994, pp. 797-805.

doi:10.1016/0020-7683(94)90078-7

[24] H. Abdullah and H. Erdem, “The Stability of Non-Ho-

mogenous Elastic Cylindrical Thin Shells with Variable

Thickness under a Dynamic External Pressure,” Turkish

Journal of Engineering and Environmental Sciences, In

Turkish, Vol. 26, No. 2, 2002, pp. 155-164.

[25] S. L. Eliseeva and S. B. Filippov, “Buckling and Vibra-

tions of Cylindrical Shell of Variable Thickness with

Slanted Edge,” Vestnik Sankt-Peterskogo Universiteta, In

Russian, No. 3, 2003, pp. 84-91.

[26] S. B. Filippov, D. N. Ivanov and N. V. Naumova, “Free

Vibrations and Buckling of a Thin Cylindrical Shell of

Variable Thickness with Curelinear Edge,” Technische

Mechanik, Vol. 25, No. 1, 2005, pp. 1-8.

[27] A. L. Goldenveizer, “Theory of Thin Shells,” Pergamon

Press, New York, 1961.

[28] V. V. Novozhilov, “The Theory of Thin Elastic Shells,” P.

Noordhoff Ltd., Groningen, 1964.

[29] R. Uhrig, “Elastostatik und Elastokinetik in Matrizen-

schreibweise,” Springer-Verlag, Berlin, 1973.

[30] A. Tesar and L. Fillo, “Transfer Matrix Method,” Kluwer

Academic, Dordrecht, 1988.

[31] M. Khalifa, “A Study of Free Vibration of a Circumfe-

rentially Non-Uniform Cylindrical Shell with a Four

Lobed Cross Section,” In Press, Journal of Vibration and

Control, 2010.