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European Journal of Mechanics A/Solids 49 (2015) 570e581

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European Journal of Mechanics A/Solids

journal homepage: www.elsevier .com/locate/ejmsol

Effects of non-uniform Winkler foundation and non-homogeneity onthe free vibration of an orthotropic elliptical cylindrical shell

Mousa KhalifaDepartment of Mathematics, Faculty of Science at Qena, South Valley University, Egypt

a r t i c l e i n f o

Article history:Received 2 May 2014Accepted 25 September 2014Available online 17 October 2014

Keywords:Vibration behaviorOrthotropic elliptical cylindrical shellsVariable Winkler foundation

E-mail address: mossa@dr.com.

http://dx.doi.org/10.1016/j.euromechsol.2014.09.0090997-7538/© 2014 Elsevier Masson SAS. All rights re

a b s t r a c t

Flügge's shell theory and solution for the vibration analysis of a non-homogeneous orthotropic ellipticalcylindrical shell resting on a non-uniform Winkler foundation are presented. The theoretical analysis ofthe governing equations of the shell is formulated to overcome the mathematical difficulties of modecoupling of variable curvature and homogeneity of shell. Using the transfer matrix of the shell, the vi-bration equations based on the variable Winkler foundation are written in a matrix differential equationof first order in the circumferential coordinate and solved numerically. The proposed model is applied toget the vibration frequencies and the corresponding mode shapes of the symmetrical and antisym-metrical vibration modes. The sensitivity of the vibration behavior and bending deformations to the non-uniform Winkler foundation moduli, homogeneity variation, elliptical and orthotropy of the shell isstudied for different type-modes of vibrations.

© 2014 Elsevier Masson SAS. All rights reserved.

1. Introduction

Materials and structural components are often non-homogeneous, either by design or because of the physicalcomposition and imperfections in the underlying materials. Ellip-tical shells represent one of the principal elements of aerospace andmarine structures, pressure vessels, process equipment and piping.The study of mechanical behavior of vibration for these shells canhelp designer achieve a reduction in the weight and an increase inthe stiffness, especially when those are in contact with elasticfoundations. Generally, the frequencies and mode shapes of vi-bration depend on some determining functions such as the radiusof the curvature of the neutral surface, the shell thickness, theshape of the shell edges, the elastic media, and the elastic proper-ties of shell and so on. In simple cases when these functions areconstant without foundations, the vibration deflection displace-ments occupy the entire shell surface. If the determining functionsvary from point to point of the neutral surface then localization ofthe vibration modes lies near the weakest lines on the shell surface.Mathematically, the consideration of non-homogeneity, ortho-tropy, variable elastic foundation and aspect ratio leads to a verycomplex problem involving several parameters. So, numerical orapproximate techniques are necessary for their analysis. Sincemoreattention is being paid to the analysis of shell's behavior embedded

served.

in elastic foundations through the WinklerePasternak model.There are different approaches to analyze the interaction between astructure and an ambient medium, see Pasternak (1954), Kerr(1964) and Bajenov (1975). However, very few models represent-ing the behavior of non-homogeneous materials have been re-ported in the literature. Vibrations of shells on elastic foundationshave been studied only recently and focused on the circular cylin-drical shell case; see Paliwal and Bhalla (1993), Paliwal et al. (1996),Paliwal and Pandey (1998, 2001), Paliwal and Singh (1999), Ng andLam (2000) and Gunawan et al. (2004, 2006). In most of thesestudies, the authors investigate the vibrations of homogeneousisotropic and orthotropic cylindrical shells on an elastic foundationusingmembrane theory under effect of Winkler and Pasternak typefoundations. On the other hand, the study of vibration of circularand non-circular cylindrical shells of non-homogenous materials isvery scarce, but there are a few important publications related tothis study such as Sofiyev and Keskin (2004) and Sofiyev et al.(2010, 2011). As it is found recently, a few researchers aredirected to devote their studies for the vibration behavior offunctionally graded elastic cylindrical and conical shells in additionto panel shells resting on elastic foundations such as Shah et al.(2010), Sofiyev and Kuruoglu (2012), Najafov et al. (2012),Tornabene (2011), Tornabene and Ceruti (2013) and they areemployed wave propagation method and Galerkin method to solvethe dynamical equations. In contrary, despite the great value ofengineering applications for the elliptical shells there are no pre-vious considerations for studying the vibration behavior for such

M. Khalifa / European Journal of Mechanics A/Solids 49 (2015) 570e581 571

shells under the effect of elastic foundations due to the difficultiesof mode coupling of terms in the solution model. But there aresome important studies of vibration for such shells without anymedia affected, see those Sewall and Pusey (1971), Yamada et al.(1985) and Soedel (2004) who are used the thin-shell theory intheir analysis. In spite of extensive works, which have been carriedout, it is felt that the combined effects of the orthotropy, homo-geneity variation, and the existence of elastic foundation on thevibration characteristics of elliptical cylindrical shells have not beenanalyzed. In the current study, an attempt is made to address thisproblem. The objective of this paper is to find the natural fre-quencies and corresponding mode shapes of vibration for non-homogenous orthotropic elliptical cylindrical shells resting on avariable Winkler foundation based on the Flügge's shell theory andusing the transfer matrix approach with Romberg integrationmethod. The results reveal that the variations of the elliptical cy-lindrical shell parameters have significant effects on the values ofthe natural frequencies and the mode shapes.

2. Theory and mathematical model of the problem

It is known for researchers that the studyof vibrationproblems inshells depends on the shell geometry and the model of shell theory.The present problem is modeled by Flügge's theory in existence of aWinkler-type foundation. This part is presented as follows:

2.1. Geometric formulation

The present shell is an orthotropic elliptical cylindrical shell andits material is non-homogeneously elastic. The curvature of cross-section profile of the elliptical shell is defined by the equationr ¼ r0 f(q), where r is the local varied radius along the cross-sectionmid-line, r0 is the reference radius of curvature, chosen to be theradius of a circle having the same circumference as the ellipticalprofile, and f(q) is a prescribed function of q and can be taken as:

f ðqÞ ¼ Affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 3cos2 q

p ; 0 � q � 2p; (1)

where 3is the elliptical parameter and measures the eccentricity ofthe cross section of the shell based on (A/B) value, and expressed by

Fig. 1. Coordinate system and geometry of an elliptical orthotropic

the semi-major, a, and semi-minor, b, axes of the elliptical cross-section profile as:

3¼ a2

b2� 1; B ¼ b=r0 and A ¼ a=r0: (2)

The position of a point on the middle surface of the shell is definedby the cylindrical coordinates (x, s, z), as shown in Fig. 1. The dis-placements of the middle surface of the shell are denoted by u, yand w in the axial, circumferential and transverse directions,respectively. The shell geometry is described by H, the thicknessand L, the axial length of the shell. If we suppose that the axial andcircumferential directions are principal axes of the orthotropicmaterial. Up to this point, the next relation is obtained:

nxEs ¼ nsEx; (3)

where Ex, Es are Young's moduli and nx, ns are Poisson's ratios in theaxial and circumferential directions, respectively.

2.2. Modal radius of curvature

Since the solution of the present problem drastically depends onthe variable curvature r(q), it follows that the radius of curvatureshould represent the actual geometry of the middle surface of theshell. In the present study, the radius of curvature can be got fromthis equation:

RðqÞ=r0 ¼�r2 þ r�2

�3=2=r0

�r2 þ 2r�2 � rr��

�: (4)

Using Eqs. (1) and (4), the expression C ¼ R(q)/r0 which will beappeared in Eqs. (25), can be found to take the form:

C ¼ Aða=bÞ�1� 31 cos2 q

1� 32 cos2 q

�3=2

; 31 ¼ 1� ðb=aÞ2;

32 ¼ 1� ðb=aÞ4(5)

which represents the dimensionless radius of curvature of theelliptical cylindrical shell.

cylindrical shell resting on a non-uniform Winkler foundation.

M. Khalifa / European Journal of Mechanics A/Solids 49 (2015) 570e581572

2.3. Model of non-homogeneity material

Non-homogeneous materials can frequently be found in natureand in manmade structures. In general, non-homogeneous mate-rials seem to be thosewith elastic constants varying continuously indifferent spatial directions. In most of available solutions to elasticnon-homogeneity, it is assumed that the material is isotropic ororthotropic, the Poisson's ratio is constant, and the Young's modulibeside the density of shell are either an exponential or a powerfunction of a spatial variable as in Khoroshun et al. (1988) and Zhangand Hasebe (1999). In the present study, the material properties ofthe non-homogeneous orthotropic elliptical shell is assumed toarise due to the variation of Young's moduli, shear modulus, anddensity along the hoop direction of the shell, q. However, for non-homogeneity of the elliptical shell material, we assumed that:

ðEx; Es;G; rÞ ¼ ðEx0; Es0;G0; r0Þ4ðqÞ; (6)

where Ex0 and Es0 are the Young's moduli in the x- and s-directions,respectively,G0 is the shear modulus, and r0 is the density of thehomogeneous orthotropic material of the shell. As 4 is a prescribedcontinuously function of q for non-homogeneity, and satisfies thecondition j4ðqÞj � 1.

For the present study, the non-homogeneous function is anexponential function taken the following form; see Lomakin (1976)and Roshan and Dhanpati (2007):

4ðqÞ ¼ ePðq�p=2Þ; (7)

whereP is a positive real number anddenotes to a non-homogeneityparameter of the material properties. Furthermore, we will be pro-posed a new ratio used for the first time to be expressed completelythe non-homogeneity of the shell. Since the function 4(q) has aminimum value for (q ¼ 0), and a maximum value for (q ¼ p/2).Consequently, the substituting of Eq. (7) into (6) yields to:

Ex Max

Ex Min¼ Es Max

Es Min¼ GMax

GMin¼ rMax

rMin¼ eP1 ; P1 ¼ ðp=2ÞP: (8)

Hereby, the dependence of the non-homogeneity ratio (z ¼ Max/Min) on P has the form:

z ¼ eP1 ; 0 � P1 � 1: (9)

2.4. Model of variable Winkler foundation

Shells supported by elastic foundations have been widelyadopted by many researchers to model various engineering prob-lems. To describe the interactions between the shell and foundationas possible, scientists have proposed various kinds of foundationmodels, as documented well in Kerr (1964). The present shell issurrounding by an elastic foundation over its circumference. Thefoundation medium is assumed to be nonlinear, homogenous andisotropic modeled by the Winkler model with one variableparameter, which describes the foundation by a series of closelyspaced, and distributed parabolically over the shell circumference.In this study, the normal reaction (force per unit area) of the elasticfoundation is assumed to be:

N ¼ XWðaÞw; (10)

where XW(a) is the variable Winkler foundation stiffness dependson the tangential angle a (see Fig. 1). The general equation form ofthis foundation is:

XWðaÞ ¼ XW0fðaÞ; (11)

where a is the angle between the tangent at the origin of s, locatedat the end of the semi-major axis, a, and the one at any point on thecenterline, and XW0 is the uniformWinkler foundation stiffness. Forthe present study, the function f(a) takes the parabolic form:

fðaÞ ¼ 1þ da2; (12)

where d is the amplitude of Winkler stiffness variation along thecircumference. The minimum value of f(a) occurs forXW0 ¼ XW(a ¼ 0), whereas XWMax ¼ XW(a ¼ p/2) is the maximumvalue of f(a), and the shell is resting on uniform Winkler founda-tion, XW0 when d ¼ 0. The dependence of the non-uniformity ratioc ¼ XWMam/XW0 on d has the form:

c ¼ 1þ 2:47d: (13)

2.5. Basic governing equations

When the above assumptions are taken into consideration, forgeneral circular cylindrical shells resting on elastic foundations andfor studying the free vibrations, the equilibrium equations of forces,based on Flügge (1973) take the forms:

N0xþN�

sxþrðsÞHu2u¼0; N0xsþN�

s þQs=RþrðsÞHu2y¼0;Q 0xþQ�

s �Ns=RþrðsÞHu2w�NðsÞ¼0; M0xþM�

sx�Qx¼0;M0

xsþM�s �Qs¼0; Ss�Qs�M0

sx¼0; Nxs�Nsx�Msx=R¼0;

(14)

where Nx, Ns and Qx, Qs are the normal and transverse shearingforces in the x- and s-directions, respectively, Nsx and Nxs are the in-plane shearing forces,Mx,Ms andMxs,Msx are the bending momentand the twisting moment, respectively, Ss is the equivalent shearingforce,u is the angular frequency of vibration, R is the variable radiusof curvature, ' ≡ v/vx and � ≡ v/vs. The relations between strains anddeflections for the cylindrical shells used here are taken fromNovozhilov (1964) and Uhrig (1973) as follows:

3x ¼ u0; 3s ¼ y� þw=R; gxs ¼ y0 þ u�; gxz ¼ w0 þ jx ¼ 0;gsz ¼ w� þ js � y=R ¼ 0; Kx ¼ j0

x; Ks ¼ j�s þ ðy� þw=RÞ=R;

Ksx ¼ j0s; Kxs ¼ j�

x þ u�=R;(15)

where 3x and 3s are the normal strains of the middle surface of theshell, gxs, gxz and gsz are the shear strains, and the quantities Kx, Ks,Ksx and Kxs representing the change of curvature and the twist ofthe middle surface, jx is the bending slope, and js is the angularrotation. The components of force and moment resultants in termsof Eq. (15) are given based on Flügge's theory as in on Flügge (1973)

Nx¼Dxð 3xþns 3sÞþðKx=RÞKx; Nxs¼DxsgxsþðKsx=RÞKsx;Ns¼Dsð 3sþnx 3xÞ�ðKs=RÞðKs� 3s=RÞ;Nsx¼DsxgsxþðKxs=RÞKxs; Msx¼2KxsðKxs�gxs=2RÞ;Mx¼KxðKxþnsKsþ 3x=RÞ; Ms¼KsðKsþnxKx� 3s=RÞ;

(16)

where the quantities Dx, Ds and Dxs are the extensional rigidities,and on considering the non-homogeneity of the shell, using Eq. (6),defined as:

Dx¼ExH=ð1�nxnsÞ¼ðEx0H=ð1�nxnsÞÞ4¼Dx04;Ds¼EsHð1�nsnxÞ¼ðEs0Hð1�nsnxÞÞ4¼Ds04; Dxs¼HG¼Dxs04;

(17)

M. Khalifa / European Journal of Mechanics A/Solids 49 (2015) 570e581 573

and Kx, Ks, and Kxs are the flexural rigidities defined as:

Kx ¼ ExH3.12ð1� nxnsÞ ¼

�Ex0H

3.12ð1� nxnsÞ

�4 ¼ Kx04;

Ks ¼ EsH3.ð1� nsnxÞ ¼ Ks04 and Kxs ¼ GH3

.12 ¼ Kxs04;

(18)

the components of subscript zero denote to the homogeneous ones.

3. Solution methodology

For complex geometries of non-uniform shells, exact solutionsare scarce. In this case, one has to make many approximations andsimplifications in order to reduce the governing differential equa-tions that reduce the original equations to a suitable form andincluding the effect of the non-homogeneity and orthotropy of thematerials and elastic foundations complicates the solution of thevibration problems considerably. This part consists of:

3.1. Matrix representation

From Eqs. (14)e(18) with eliminating the variables Qx, Qs,Nx,Nxs,Mx, Mxs and Msx which are not differentiated with respect to s, thevibration system of the partial differential equations for the statevariables u, y, w, js, Ms, Ss, Ns and Nsx of the shell are obtained asfollows:

u� ¼ð1=BDxsÞNsxþðG=RBÞj0s�y0; w� ¼ y=R�js;

y� ¼Ns=Ds�w=R�nxu0 �nxGj0x�ð1=RDsÞMs;

j�s ¼ðB=KsÞMs�nxBj0

x�ð1=RDsÞNsþðnx=RÞu0;M�

s ¼Ss�KxsðG=B�4Þj00s�ðRG=BÞN0

sx; N�s ¼�Ss=R�N0

sx�r0u2H4ðqÞy;

N�sx¼Dxð1�nxnsÞu00 �ðDxns=DsÞN0

sþðKxns=RKsÞM0s�Kxðð1�nxnsÞ=RÞj00

x�r0u2H4ðqÞu;

S�s ¼Ns=RþðnsBDx=DsÞM00s �Kxð1�nxnsBÞw000 �ðKx=RÞð1�nxnsÞu00 �ðnsDxG=DsÞN00

s�r0u2H4ðqÞwþXW04ðaÞw; B¼1þG; G¼H2

.12R2:

(19)

The solution of the system of Eqs. (19) which satisfies the boundarycondition for a simply supported shell takes the forms:

uðx; sÞ ¼ UðsÞcos bx; ðyðx; sÞ;wðx; sÞÞ ¼ �VðsÞ;WðsÞ�sin bx; jsðx; sÞ ¼ jsðsÞsin bx;

ðNxðx; sÞ;Nsðx; sÞ;Qsðx; sÞ; Ssðx; sÞÞ ¼�NxðsÞ;NsðsÞ;QsðsÞ; SsðsÞ

�sin bx;

ðNxsðx; sÞ;Nsxðx; sÞ;Qxðx; sÞÞ ¼�NxsðsÞ;NsxðsÞ;QxðsÞ

�cos bx;

ðMxðx; sÞ;Msðx; sÞÞ ¼�MxðsÞ;MsðsÞ

�sin bx;

ðMxsðx; sÞ;Msxðx; sÞÞ ¼�MxsðsÞ;MsxðsÞ

�cos bx; b ¼ mp=L; m ¼ 1;2;…

(20)

andm is the axial half wave number. The quantitiesUðsÞ;VðsÞ;… arethe state variables which are undetermined functions of s. Byintroducing Eqs. (20) into system (19), and after appropriate alge-braic operations, the systemof vibration equations can bewritten innon-linear ordinary differential equations referred to the variable sonly are obtained, in the following matrix form:

r0dds

8>>>>>>>>>><>>>>>>>>>>:

~U~V~W~js~Ms~Ss~Ns~Nsx

9>>>>>>>>>>=>>>>>>>>>>;

¼

266666666664

0 V12 0 V14 0 0 0 V18V21 0 V23 0 V25 0 V27 00 V32 0 V34 0 0 0 0V41 0 V43 0 V45 0 V47 00 0 0 V54 0 V56 0 V58V61 0 V63 0 V65 0 V67 00 V72 0 0 0 V76 0 V78V81 0 V83 0 V85 0 V87 0

377777777775

8>>>>>>>>>><>>>>>>>>>>:

~U~V~W~js~Ms~Ss~Ns~Nsx

9>>>>>>>>>>=>>>>>>>>>>;:

(21)

By using the state vector of fundamental unknowns Z(s), system(21) can be written as:

�r0

dds

�fZðsÞg ¼ ½V �fZðsÞg;

fZðsÞg ¼n~U; ~V ; ~W; ~js;

~Ms; ~Ss; ~Ns; ~Nsx

oT;�

~U; ~V ; ~W�¼ kx0

�U;V ;W

�; ~js ¼ ðkx0=bÞjs;

~Ms ¼�1.b2

�Ms;�

~Ss; ~Ns; ~Nsx

�¼

�1.b3

��Ss;Ns;Nsx

�:

(22)

For the non-circular cylindrical shell which cross-section profile isobtained by function r(q) ¼ r0 f(q), the hypotenuse (ds) of a right

triangle whose sides are infinitesimal distances along the surfacecoordinates of the shell takes the form: (ds)2 ¼ (dr)2 þ (r dq)2, thenwe have.

dsr0

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðf ðqÞÞ2 þ

�df ðqÞdq

�2s

dq: (23)

Using Eq. (23), the system of vibration Eqs. (22) takes the form:

M. Khalifa / European Journal of Mechanics A/Solids 49 (2015) 570e581574

�ddq

�fZðqÞg ¼ JðqÞ½V �fZðqÞg; (24)

whereJðqÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif 2ðqÞ þ f �2ðqÞ

p; and the variable coefficients matrix

[V] are given as:

V12 ¼ �b; V14 ¼ G1b242

.CB1; V18 ¼ G1b

3.4mB1; V21 ¼ nxb;

V23 ¼�1þ nxG1b

2�.

C; V25 ¼ G1b2.4h; V27 ¼ G1b

3.4h;

V32 ¼ 1=C; V34 ¼ �b; V41 ¼ �nx=C; V43 ¼ �nxb2B1;

V45 ¼ bB1=4h; V47 ¼ b2G1

.C4h; V54 ¼ mb4

�4� G1

.C2B1

�;

V56 ¼ �b; V58 ¼ b2G1

.C2B1; V61 ¼ 4ð1� nsnxÞ=C;

V63 ¼ b4ð1� nxnsÞ=2þ cW4ðaÞ.b3 � l24

.G1b

3; V65 ¼ B1b=h;

V67 ¼�1þ nsG1b

2.h�.

C; V72 ¼ �l24.G1b

3; V76 ¼ �1=C; V78 ¼ b;

V81 ¼ 4ð1� nxnsÞ=G1b� l24.G1b

3; V83 ¼ 4ð1� nxnsÞ=C; V85 ¼ ns=Ch; V87 ¼ �nsb=h

(25)

in terms of the following dimensionless parameters: natural fre-quency parameter l ¼ r0u

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir0ð1� nxnsÞ=Ex0

p, Winkler parameter

cW ¼ XW0ðr40=Kx0Þ, length ratio l ¼ L/r0, thickness ratio h ¼ H/r0,circumferential and axial orthotropic parameters h ¼ Ds0/Dx0 ¼ Ks0/Kx0 ¼ ns/nx and m ¼ Dxs0/Dx0 ¼ Kxs0/Kx0 ¼ G0(1 � nxns)/Ex0, whileB1 ¼ 1 þ G1/C2 and G1 ¼ h2/12. As well as the relation between thetangential angle, a and the central angle, q is given as: a¼ tan�1{(b/a)2tan q}, 0 � q � 2p.

By using the transfermatrix [T(q)] of the shell, see Tesar and Fillo(1988), the state vector Z(q) of the fundamental unknowns can beexpressed as:

ZðqÞ ¼ ½TðqÞ�Zð0Þ: (26)

The substitution of Z(q) into Eq. (22), yields to:

ðd=dqÞ½TðqÞ� ¼ JðqÞ½Vðq;aÞ�½TðqÞ�;½Tð0Þ� ¼ ½I�: (27)

The governing system of vibration (27) is too complicated to obtainany closed form solution, and this problem should be solved bynumerical methods. So, the matrix [T(q)] is obtained by using theRomberg integration method, with the starting value [T(0)] ¼ [I]which is given by taking q ¼ 0 in Eq. (26).

3.2. Mode types of vibration

For a plane passing through the central axis in a shell withstructural symmetry, symmetrical and antisymmetrical profiles canbe obtained, and then, only one-half of the shell circumference isconsidered with the boundary conditions at the ends taken to bethe symmetric or antisymmetric type of vibration modes. There-fore, the boundary conditions for symmetrical type-modes are:

~V ¼ ~js ¼ 0; ~Ss ¼ ~Nsx ¼ 0; (28)

whereas for the antisymmetrical type-modes they are:

~U ¼ ~W ¼ 0; ~Ns ¼ ~Ms ¼ 0: (29)

The natural modes of vibration of an elliptical cylindrical shellwhich has doubly symmetric profile may be classified as four

different types of vibration modes (SS, SA, AA, AS) based onwhether the modes are symmetric, S, or antisymmetric, A, withrespect to the minor and major axes of the shell at (q ¼ 0 or q ¼ p),and (q ¼ p/2 or q ¼ 3p/2), respectively. The substitution of Eqs. (28)and (29) into Eq. (26) yields to the vibration equations as follows:

26664T21 T23 T25 T27T41 T43 T45 T47T61 T63 T65 T67T81 T83 T85 T87

37775p

0BBB@

~U~W~Ms~Ns

1CCCA

ð0Þ

¼ 0

for symmetrical vibration; SS; SA;

(30)

26664T12 T14 T16 T18T32 T34 T36 T38T52 T45 T56 T58T72 T74 T76 T78

37775p

0BBB@

~U~W~Ms~Ns

1CCCA

ð0Þ

¼ 0

for antisymmetrical vibration; AA; AS:

(31)

Equations (30) and (31) give a set of linear homogenous equations

with unknown coefficients ð~U; ~W ; ~Ms; ~NsÞTð0Þ and ð~V ; ~js;~Ss; ~NsxÞTð0Þ at

q ¼ 0, and the determinant of the coefficient matrix should bevanished for the existence of a non-trivial solution. The non-trivialsolution is found by searching the values l in matrix [T(q ¼ p)]which make its determinant zero by using Lagrange interpolationprocedure. The mode shapes of vibration at any point of the cross-section of the shell are determined by calculating the eigenvectorscorresponding to l by using Gaussian elimination procedure foreach axial half mode m.

4. Presentation of results and discussion

The computer programs, based on the analysis described herein,are written by the author to find the frequency parameters and thecorresponding mode shapes of vibration for the present shellresting on a variable Winkler foundation. Our study is divided intofour parts in which the Poisson's ratio ns, the axial rigidity m andinitial Winkler modulus cW0 take the values 0.3, 0.35, and 50,respectively.

4.1. Verification of solution method

In references, Khalifa (2010, 2011a,b), number of comparisons ofnumerical results is presented to verify the validity of the used

Table 1Comparison of frequency parameters l for an elliptical cylindrical shell (l ¼ 3,h ¼ 0.01, n ¼ 0.3, a/b ¼ 2, m ¼ 1).

SS and SA modes AA and AS modes

lst Yamada et al. Present (h ¼ 1, m ¼ 0.35) Yamada et al. Present

1 0.0589 0.058252 0.0589 0.0582242 0.0623 0.062887 0.0642 0.0627883 0.0895 0.089277 0.0891 0.0892684 0.1023 0.097794 0.1025 0.0978615 0.1159 0.110502 0.1159 0.110763

M. Khalifa / European Journal of Mechanics A/Solids 49 (2015) 570e581 575

framework and approach method. The comparison studies of nat-ural frequencies calculated by different approaches have confirmedthe correctness of the proposed shell theory and solution methodfor studying the vibration behavior for such shells. In addition,Table 1 presents the frequency parameters l for an isotropic simplysupported elliptical cylindrical shell withoutWinkler foundation byYamada et al. (1985) who has obtained the solutions using the Ritzmethod, and the author who has obtained the transfer matrixapproach solutions based on the Flügge's shell theory using theRomberg integration method when (a/b ¼ 2, h ¼ 1 and m ¼ 0.35)that gives an isotropic cylindrical shell case. Another comparison ispresented in Table 2 under a uniform Winkler foundation byPaliwal et al. (1996) and Sofiyev et al. (2009) who are performedtheir solutions based on the thin-shell theory. Hereby, it is seenfrom these comparisons of natural frequencies calculated bydifferent approaches that the present solutions are valid and ingood agreement with these studies.

4.2. Vibration analysis

In general, the study of vibration is determined by finding thefrequency parameters l and the corresponding deflection dis-placements for each value of m, separately. To obtain the fre-quencies of vibration we will search the set of all eigenvalues, andto obtain the fundamental frequencies of vibration we will searchthe lowest values of this set which are very important for designingshell structures. The numerical results present the fundamentalfrequencies for the cases of (SS, SA, AA, AS) type-modes. Generally,elastic foundation makes significant changes on vibration charac-teristics and on mode shapes of shells; in addition, the frequencyparameters l increase continuously with an increase of the foun-dation modulus regardless the homogeneity/non-homogeneitymaterial of shells. The ones also decrease with the increase of theelliptical parameter value (a/b) and this result confirms the fact thatthe drastic change in the membrane and bending energies of shellwith an increasing of the non-circularity due to reducing the flatterregions of the ellipse. The effect of the Winker foundation on thefrequencies of vibration of homogeneous and non-homogeneousisotropic (h ¼ 1, m ¼ 0.35) or orthotropic (h s 1) elliptical cylin-drical shells, Table 3 gives the behavior of fundamental frequencyparameters l for different models of foundation and homogeneityratios (c, z). The numbers in the parentheses are the axial half wave

Table 2Comparison of frequency parameters l for a simply supported isotropic circularcylindrical shell (l ¼ 2, h ¼ 0.01, n ¼ 0.3, m ¼ 1, XW0 ¼ 10�4 N/m3).

n Paliwal et al. (1996) Sofiyev et al. (2009) Present (z ¼ h ¼ 1,m ¼ 0.35, a/b ¼ 1)

1 0.6792138 0.6788227 0.57549392 0.3646346 0.3639407 0.33283083 0.2080413 0.2052558 0.20528344 0.1382362 0.1274544 0.14654145 e e 0.1288563

numbers of themode in the axial direction. The results presented inthis table show that: (1) As the Winkler modulus ratio c and non-homogeneity ratio z increase, separately or together, an increasingin l values is occurred continuously for isotropic and orthotropiccases. (2) The effect of the Winkler foundation, regardless uniform/non-uniform, on the values of natural frequencies increases withnon-homogeneity material and for high values of the Winklermodulus the effect of the non-homogeneity is insignificant. (3) Theeffect of non-Winkler foundation and non-homogeneity model (c,z) on the natural frequencies of vibration becomes more significantwith an increase of circumferential rigidity h. (4) For the symmet-rical (SS, SA) and antisymmetrical (AA, AS) vibration modes, thevalues of l are more closed together with an decrease of ellipticalparameter (a/b) and become identical values for the case of circularcylindrical shells (a/b ¼ 1) under the effect of model (c, z ¼ 1, 1).Fig. 2 presents the effect of non-uniform Winkler foundation whenthe present shell is vibrating in the first mode of vibration. It isfound that the increase of the Winkler ratio c leads to increase in l

curves for isotropic/orthotropic and homogeneous/non-homogeneous cases. The frequency parameter curves for SS- andAA-modes are found to be identical modes for the effect of the non-uniformWinkler foundation and homogeneous model. In contrary,the ones have non-identical modes for the effect of the non-uniform Winkler foundation and non-homogeneous, isotropic/orthotropic cases. Fig. 3 presents the effect of non-homogeneitymaterial on the vibration behavior of an elliptical shell resting ona Winkler foundation. It is shown from these figures that the in-crease of non-homogeneity ratio leads to significant increase of thefrequency parameter curves regardless the shell is resting on theuniform/non-uniform Winkler foundation. It is also seen that themodel of non-homogeneity material and non-uniform Winklerfoundation (c, z), together, are causingmore effects on the vibrationbehavior of such shells. The effect of the orthotropy of ellipticalcylindrical shells resting on (c, z) model, together, Fig. 4 shows thatthe increase of orthotropy parameters leads to increase of the fre-quency parameter curves for the S- and A-modes of vibration.Furthermore, the influence of circumferential rigidity h on l curvesbecomes more significant with high values of h. In contrary, theones are more insignificant with higher m for the cases of theisotropic and orthotropic materials. The effect of the ellipticalparameter (a/b) of an isotropic and orthotropic shell resting on (c,z)model, together, to the frequency parameter is shown in Fig. 5. Itcan be opined from this figure that the increase in the values ofelliptical parameter leads to decrease the frequency curves and thisresult is compatible with (c, z ¼ 1,1) model. This is interpretedphysically as being due to localized bending in the flatter regions ofthe elliptical cylindrical shell.

4.3. Vibration mode shapes

The vibration mode of the shell structure can be defined as thedeflection of a structure at a particular frequency l. To obtain the(SS, AA) vibration modes corresponding to a particular frequencyparameter of the shell, we substitute by l into Eqs. (30) and (31) andusing Gaussian elimination technique to obtain the state vector atany point of the cross-section. It is found that the variable Winklerfoundation primarily affects on the vibration behavior of shells. Ingeneral, the vibrationmodes are distributed regularly over the shellsurface for the (c ¼ 1, z ¼ 1) model and are non-uniformly for the(c > 1, z > 1) model, moreover it becomes more pronounced overthe sections with less values of Winkler modulus ratio at (q ¼ 0, p).Figs. 6 and 7 show the influences of (c, z) models on the circum-ferential vibration modes of an isotropic and orthotropic ellipticalcylindrical shell for (SS, SA, AA, AS) type-modes. The thick curvesshow the composition of the circumferential and transverse

Table 3The first five fundamental frequency parameters l versus the circumferential rigidity hwith different a/b, under different modes of Winkler and non-homogeneity parameters(l ¼ 5, h ¼ 0.01, m ¼ 0.35, cW0 ¼ 50).

(a/b) ¼ 1Parameter ε

Symmetric vibration mode S Antisymmetric vibration mode A

(c,z)Y h ¼ 0.5 h ¼ 1 h ¼ 2 h ¼ 0.5 h ¼ 1 h ¼ 2

(1,1) 0.042585(1) 0.048331(1) 0.053493(1) 0.042585(1) 0.048330(1) 0.053486(1)0.044662(1) 0.052268(1) 0.067133(1) 0.044663(1) 0.052268(1) 0.067133(1)0.055128(1) 0.073189(1) 0.079780(1) 0.055098(1) 0.073145(1) 0.079780(1)0.075255(1) 0.078601(1) 0.099815(1) 0.075255(1) 0.078601(1) 0.099738(1)0.076616(1) 0.093205(2) 0.106262(2) 0.076616(1) 0.093158(2) 0.106260(2)

(2,1) 0.044127(1) 0.049510(1) 0.054571(1) 0.044018(1) 0.049676(1) 0.054719(1)0.045897(1) 0.053542(1) 0.068121(1) 0.046067(1) 0.053449(1) 0.068052(1)0.056344(1) 0.074091(1) 0.080594(1) 0.056380(1) 0.074085(1) 0.080378(1)0.076160(1) 0.079415(1) 0.100473(1) 0.076137(1) 0.079198(1) 0.100432(1)0.077462(1) 0.093904(2) 0.106951(2) 0.077228(1) 0.093889(2) 0.106855(2)

(3,1) 0.045507(1) 0.050599(1) 0.055587(1) 0.045316(1) 0.050907(1) 0.055875(1)0.046969(1) 0.054702(1) 0.069015(1) 0.047262(1) 0.054541(1) 0.068897(1)0.057571(1) 0.074978(1) 0.081422(1) 0.057698(1) 0.075024(1) 0.080985(1)0.077069(1) 0.080221(1) 0.101124(1) 0.076976(1) 0.079792(1) 0.101113(1)0.078348(1) 0.094568(2) 0.107516(2) 0.077879(1) 0.094585(2) 0.107421(2)

(1,1.5) 0.044463(1) 0.050042(1) 0.054817(1) 0.044526(1) 0.049884(1) 0.054710(1)0.046926(1) 0.053558(1) 0.067644(1) 0.046705(1) 0.053623(1) 0.067816(1)0.056949(1) 0.074445(1) 0.080957(1) 0.056878(1) 0.074376(1) 0.081460(1)0.076382(1) 0.079809(1) 0.100752(1) 0.076354(1) 0.080470(1) 0.100682(1)0.078756(1) 0.094145(2) 0.106834(2) 0.079505(1) 0.093991(2) 0.106971(2)

(1,2) 0.045939(1) 0.051208(1) 0.055849(1) 0.046148(1) 0.050843(1) 0.055572(1)0.048022(1) 0.054520(1) 0.067345(1) 0.047558(1) 0.054759(1) 0.068014(1)0.059395(1) 0.076161(1) 0.082073(1) 0.059314(1) 0.076106(1) 0.083573(1)0.077841(1) 0.080057(1) 0.102081(1) 0.077763(1) 0.082122(1) 0.102062(1)0.079144(1) 0.094791(2) 0.106880(2) 0.081231(2) 0.094501(2) 0.107301(2)

(1,2.5) 0.046662(1) 0.051763(1) 0.056380(1) 0.046992(1) 0.051246(1) 0.055970(1)0.048439(1) 0.054947(1) 0.067826(1) 0.047809(1) 0.055344(1) 0.068058(1)0.060978(1) 0.077306(1) 0.082823(1) 0.060894(1) 0.077269(1) 0.084922(1)0.078458(1) 0.080125(1) 0.102982(1) 0.078661(1) 0.083103(1) 0.103002(1)0.079516(1) 0.095117(2) 0.106990(2) 0.081617(2) 0.094727(2) 0.107425(2)

(2,1.5) 0.046453(1) 0.051615(1) 0.056231(1) 0.046377(1) 0.051640(1) 0.056287(1)0.048723(1) 0.055244(1) 0.069123(1) 0.048706(1) 0.055182(1) 0.069143(1)0.058160(1) 0.075451(1) 0.081724(1) 0.058118(1) 0.075413(1) 0.082049(1)0.077431(1) 0.080703(1) 0.101512(1) 0.077385(1) 0.081141(1) 0.101474(1)0.079681(1) 0.095056(2) 0.107661(2) 0.080194(1) 0.094939(2) 0.107743(2)

(3,2.5) 0.051726(1) 0.055984(1) 0.060138(1) 0.051739(1) 0.055799(1) 0.060021(1)0.053375(1) 0.059394(1) 0.071034(1) 0.053121(1) 0.059465(1) 0.071673(1)0.063567(1) 0.079641(1) 0.084460(1) 0.063470(1) 0.079599(1) 0.086278(1)0.081020(1) 0.082196(1) 0.104811(1) 0.081158(1) 0.084731(1) 0.104832(1)0.081584(1) 0.097551(2) 0.109030(2) 0.084070(1) 0.097246(2) 0.109521(2)

(a/b) ¼ 1.5(1,1) 0.040270(1) 0.044998(1) 0.050082(1) 0.040251(1) 0.044931(1) 0.050029(1)

0.041189(1) 0.049087(1) 0.061659(1) 0.041118(1) 0.049074(1) 0.061846(1)0.054552(1) 0.071549(1) 0.076033(1) 0.054524(1) 0.071519(1) 0.079033(1)0.070067(1) 0.073186(1) 0.096223(2) 0.071686(2) 0.076360(1) 0.096173(2)0.071717(2) 0.084813(2) 0.097031(1) 0.071835(1) 0.084768(2) 0.097022(1)

(2,1) 0.040871(1) 0.045483(1) 0.050560(1) 0.040812(1) 0.045466(1) 0.050561(1)0.041604(1) 0.049541(1) 0.061882(1) 0.041572(1) 0.049504(1) 0.062090(1)0.055368(1) 0.072114(1) 0.076648(1) 0.055416(1) 0.072135(1) 0.079616(1)0.070371(1) 0.073678(1) 0.096434(2) 0.071940(2) 0.076860(1) 0.096370(2)0.071961(1) 0.085016(2) 0.097425(1) 0.072157(1) 0.084979(2) 0.097432(1)

(3,1) 0.041423(1) 0.045938(1) 0.051016(1) 0.041331(1) 0.045967(1) 0.051067(1)0.041984(1) 0.049962(1) 0.062091(1) 0.041985(1) 0.049908(1) 0.062320(1)0.056157(1) 0.072668(1) 0.077245(1) 0.056280(1) 0.072735(1) 0.080184(1)0.070662(1) 0.074157(1) 0.096638(2) 0.072183(2) 0.077341(1) 0.096560(2)0.072194(2) 0.085212(2) 0.097808(1) 0.072478(1) 0.085182(2) 0.097852(1)

(1,1.5) 0.043376(1) 0.048198(1) 0.052943(1) 0.043051(1) 0.047616(1) 0.052405(1)0.045118(1) 0.052097(1) 0.065625(1) 0.044476(1) 0.051657(1) 0.064813(1)0.055583(1) 0.072493(1) 0.077787(1) 0.055617(1) 0.072544(1) 0.077742(1)0.072947(1) 0.076365(1) 0.097882(1) 0.073122(1) 0.076142(1) 0.097974(1)0.074854(2) 0.087745(2) 0.099519(2) 0.074508(2) 0.087314(2) 0.098657(2)

(1,2) 0.046063(1) 0.050716(1) 0.055264(1) 0.045580(1) 0.049891(1) 0.054502(1)0.047982(1) 0.054649(1) 0.068459(1) 0.047093(1) 0.054018(1) 0.067457(1)0.057156(1) 0.073888(1) 0.079576(1) 0.057102(1) 0.073901(1) 0.076859(1)

M. Khalifa / European Journal of Mechanics A/Solids 49 (2015) 570e581576

Table 3 (continued )

(a/b) ¼ 1Parameter ε

Symmetric vibration mode S Antisymmetric vibration mode A

(c,z)Y h ¼ 0.5 h ¼ 1 h ¼ 2 h ¼ 0.5 h ¼ 1 h ¼ 2

0.074622(1) 0.078669(1) 0.099125(1) 0.074073(1) 0.075956(1) 0.099223(1)0.077379(2) 0.090220(2) 0.102113(2) 0.076873(2) 0.090991(2) 0.100761(2)

(1,2.5) 0.047462(1) 0.051936(1) 0.056421(1) 0.046934(1) 0.051059(1) 0.055612(1)0.049223(1) 0.055952(1) 0.069640(1) 0.048321(1) 0.055277(1) 0.068774(1)0.058367(1) 0.074871(1) 0.080714(1) 0.058183(1) 0.074819(1) 0.076618(1)0.075489(1) 0.079779(1) 0.099970(1) 0.074478(1) 0.075958(1) 0.100042(1)0.078725(2) 0.091533(2) 0.103403(2) 0.077593(1) 0.090893(2) 0.101871(2)

(2,1.5) 0.044176(1) 0.048851(1) 0.053572(1) 0.043778(1) 0.048310(1) 0.053075(1)0.045715(1) 0.052734(1) 0.066024(1) 0.045103(1) 0.052231(1) 0.065161(1)0.056465(1) 0.073110(1) 0.078417(1) 0.056578(1) 0.073217(1) 0.078404(1)0.073326(1) 0.076886(1) 0.098323(1) 0.073532(1) 0.076705(1) 0.098453(1)0.075177(2) 0.088022(2) 0.099806(2) 0.074839(2) 0.087591(2) 0.098908(2)

(3,2.5) 0.049618(1) 0.053771(1) 0.058112(1) 0.048878(1) 0.052928(1) 0.057348(1)0.051204(1) 0.057762(1) 0.071137(1) 0.050271(1) 0.056880(1) 0.069932(1)0.060014(1) 0.076186(1) 0.081823(1) 0.060047(1) 0.076255(1) 0.077993(1)0.076544(1) 0.080895(1) 0.100961(1) 0.075542(1) 0.077211(1) 0.101112(1)0.079628(2) 0.092316(2) 0.104187(2) 0.079067(2) 0.091655(2) 0.102581(2)

(a/b) ¼ 2.5(1,1) 0.033920(1) 0.037037(1) 0.041429(1) 0.033593(1) 0.036734(1) 0.041256(1)

0.034770(1) 0.041393(1) 0.048836(1) 0.034695(1) 0.041458(1) 0.050515(1)0.049436(1) 0.056347(1) 0.062321(1) 0.049393(1) 0.064273(1) 0.070460(1)0.053169(1) 0.064129(1) 0.074257(2) 0.057019(2) 0.065217(2) 0.074220(2)0.056998(2) 0.065162(2) 0.079436(2) 0.057993(2) 0.065645(1) 0.079234(2)

(2,1) 0.034084(1) 0.037223(1) 0.041610(1) 0.033753(1) 0.036913(1) 0.041432(1)0.034938(1) 0.041495(1) 0.048870(1) 0.034863(1) 0.041575(1) 0.050580(1)0.049746(1) 0.056552(1) 0.062560(1) 0.049708(1) 0.064502(1) 0.070794(1)0.053313(1) 0.064346(1) 0.074326(2) 0.057091(2) 0.065278(2) 0.074287(2)0.057067(2) 0.065224(2) 0.079467(2) 0.058037(2) 0.065935(1) 0.079265(2)

(3,1) 0.034239(1) 0.037398(1) 0.041785(1) 0.033904(1) 0.037085(1) 0.041603(1)0.035095(1) 0.041590(1) 0.048903(1) 0.035024(1) 0.041687(1) 0.050643(1)0.050045(1) 0.056747(1) 0.062792(1) 0.050014(1) 0.064725(1) 0.071117(1)0.053452(1) 0.064556(1) 0.074392(2) 0.057161(2) 0.065337(2) 0.074352(2)0.057135(2) 0.065284(2) 0.079498(2) 0.058083(2) 0.066213(1) 0.079296(2)

(1,1.5) 0.038228(1) 0.041013(1) 0.045292(1) 0.036417(1) 0.039203(1) 0.043548(1)0.038401(1) 0.045487(1) 0.054715(1) 0.037458(1) 0.044082(1) 0.053054(1)0.050497(1) 0.059224(1) 0.063282(1) 0.050601(1) 0.064814(1) 0.069157(1)0.056602(1) 0.065035(1) 0.078311(2) 0.059430(2) 0.065393(1) 0.076398(2)0.060307(2) 0.068994(2) 0.084404(2) 0.060563(2) 0.067510(2) 0.081733(2)

(1,2) 0.041477(1) 0.044465(1) 0.048647(1) 0.038886(1) 0.041442(1) 0.045653(1)0.041918(1) 0.048788(1) 0.059864(1) 0.040059(1) 0.046597(1) 0.055397(1)0.051805(1) 0.062298(1) 0.064523(1) 0.051989(1) 0.064035(1) 0.068133(1)0.059515(1) 0.066210(1) 0.081691(2) 0.060758(1) 0.066668(1) 0.078372(2)0.063180(2) 0.072270(2) 0.087861(1) 0.061717(2) 0.069622(2) 0.083983(2)

(1,2.5) 0.043026(1) 0.046224(1) 0.50363(1) 0.040162(1) 0.042638(1) 0.046789(1)0.043726(1) 0.050362(1) 0.062169(1) 0.041493(1) 0.047996(1) 0.056656(1)0.052697(1) 0.064143(1) 0.065695(1) 0.052871(1) 0.063630(1) 0.067668(1)0.060997(1) 0.066985(1) 0.083403(2) 0.060770(1) 0.067460(1) 0.079442(2)0.064711(2) 0.075338(2) 0.088516(1) 0.063023(2) 0.070788(2) 0.085194(2)

(2,1.5) 0.038445(1) 0.041252(1) 0.045525(1) 0.036618(1) 0.039420(1) 0.043759(1)0.038637(1) 0.045652(1) 0.054767(1) 0.037659(1) 0.044226(1) 0.053134(1)0.050833(1) 0.059417(1) 0.063553(1) 0.050940(1) 0.065153(1) 0.069540(1)0.056732(1) 0.065271(1) 0.078400(2) 0.059519(2) 0.065648(1) 0.076480(2)0.060401(2) 0.069074(2) 0.084447(2) 0.060622(2) 0.067586(2) 0.081774(2)

(3,2.5) 0.043697(1) 0.046878(1) 0.050980(1) 0.040716(1) 0.043187(1) 0.047303(1)0.044389(1) 0.050906(1) 0.062492(1) 0.042020(1) 0.048363(1) 0.056867(1)0.053354(1) 0.064467(1) 0.066157(1) 0.053567(1) 0.064431(1) 0.068559(1)0.061261(1) 0.067477(1) 0.083643(2) 0.061327(1) 0.067989(1) 0.079652(2)0.064970(2) 0.074193(2) 0.088856(1) 0.063252(2) 0.070989(2) 0.085306(2)

M. Khalifa / European Journal of Mechanics A/Solids 49 (2015) 570e581 577

deflections on the shell surface and the dotted curves show theoriginal shell shape before the vibration case. The numbers in theparentheses are the axial half wave number corresponding to thefrequency parameters l listed in Table 3. It is observed from thesefigures that the adopted (c, z¼ 1) models are influenced enough on

the symmetric and antisymmetric vibration modes regardlessisotropic/orthotropic cases. Where the majority of symmetric andantisymmetric vibration modes are located near the weakest lineson the shell surface with less stiffness of Winker foundationmodulus, and become more pronounced with the increase of the

Fig. 2. Effects of non-uniformity Winkler ratio on frequency parameter of homogeneous isotropic and non-homogeneous orthotropic elliptical cylindrical shells for S & A-modes(m ¼ 1, m ¼ 0.35, a/b ¼ 2).

Fig. 3. Effects of non-homogeneity parameter on frequency parameter of orthotropic elliptical cylindrical shells resting on uniform and non-uniform Winkler modulus for S & A-modes (m ¼ 1, m ¼ 0.35, a/b ¼ 2).

Fig. 4. Effects of orthotropy parameters on frequency parameter of non-homogenous orthotropic elliptical cylindrical shells resting on non-uniform Winkler foundation for S & A-modes (m ¼ 1, a/b ¼ 2).

Fig. 5. First 1st frequency parameter versus elliptical parameter for non-homogeneous isotropic and orthotropic elliptical cylindrical shells resting on non-uniform Winklerfoundation for S & A-modes (m ¼ 1).

M. Khalifa / European Journal of Mechanics A/Solids 49 (2015) 570e581578

Fig. 6. Effects of non-homogeneity parameter and non-uniformity Winkler modulus on the circumferential symmetric and antisymmetric vibration modes of an isotropic andorthotropic elliptical cylindrical shell (m ¼ 0.35, a/b ¼ 1.5).

M. Khalifa / European Journal of Mechanics A/Solids 49 (2015) 570e581 579

Winkler foundation ratio c, and they reach the maximum at (q ¼ 0,p). The same condition is occurred for the adopted (c ¼ 1, z)models, but the effect of (c, z¼ 1) models on the vibrationmodes ismore significant than ones regardless isotropic/orthotropic cases.Also, from these figures, one may be concluded that the effects of(c, z) models are more significant and enough for considerationthan (c ¼ 1, z) and (c, z ¼ 1) models. Also, it is shown that the vi-bration modes become clearer at the area of ellipse centrode withthe increase of the elliptical parameter regardless the presence/absence of the (c, z) models. Themode shapes are similar in the sets

of the vibration modes having (m ¼ 1, 2) for each values of h and m

for different models of (c, z).

4.4. Particular case

We consider a special case for an orthotropic circular cylindricalshell (a/b ¼ 1) under the model of (c, z). Table 3, section 1, gives thefundamental frequency parameter l versus the circumferential ri-gidity h for different models of (c, z). From this section, we can seethat the frequency parameters increase with an increase in c and z

Fig. 7. The circumferential symmetric and antisymmetric vibration modes of homogeneous/non-homogeneous and isotropic/orthotropic elliptical cylindrical shells resting on auniform and non-uniform Winkler modulus (m ¼ 0.35, a/b ¼ 2.5).

Fig. 8. Effects of non-homogeneity parameter and non-uniformity Winkler modulus on the S & A-modes of isotropic and orthotropic circular cylindrical shells (a/b ¼ 1).

M. Khalifa / European Journal of Mechanics A/Solids 49 (2015) 570e581580

values separately or together. It is also seen that the values of thefrequency parameters of the symmetric and antisymmetric vibra-tions are found to be identical values for the case of (c ¼ 1, z ¼ 1),but for the (c > 1, z > 1) models those are closed to each other. Theinfluence of (c, z) model on the mode shapes of vibration, Fig. 8shows the circumferential symmetric and antisymmetric modesof a circular cylindrical shell under the (c, z) models. One can seefrom this figure that the vibration modes are uniformly distributedover the shell surface when the shell lies under (c¼ 1, z¼ 1) model.The deflections of the vibration modes are concentrated around(q ¼ 0, p) for the (c ¼ 1, z) and (c, z ¼ 1) models and become more

pronounced with larger c and z values. It is observed that the effectof (c¼ 1, z) model is more significant and dominated than (c, z¼ 1)model on the vibration mode, i.e. the vibration modes are moresensitive for variable Winkler foundation than the non-homogeneity material of shell.

5. Conclusions

A new approach to analyze the vibration behavior of an ortho-tropic elliptical cylindrical shell with non-homogeneous materialproperties and resting on a non-uniform Winkler-type foundation

M. Khalifa / European Journal of Mechanics A/Solids 49 (2015) 570e581 581

is presented. At first, the basic relations have been obtained fororthotropic elliptical shells, Young's moduli and density which varycontinuously in the circumferential direction. Based on the Flügge'sshell theory with using the transfer matrix of the shell, the gov-erning equations are formulated and reduced to eight first-orderdifferential equations in the circumferential coordinate, and canbe written in a matrix differential equation. The computed resultspresented herein pertain to find the fundamental vibration fre-quencies and the corresponding mode shapes of the deflectiondisplacements for four type-modes of vibration (SS, SA, AA, AS). Indetail, a parametric study is considered to demonstrate the effectsof various parameters, such as non-uniform Winkler foundation,the non-homogeneity and orthotropy of materials, the ellipticalparameter on the vibration behavior for such shells. From thisstudy, it could be concluded that:

� The computed results confirm that the variable Winkler foun-dation has significant effect on the vibration behavior of elasticelliptical cylindrical shells for the isotropic/orthotropic andhomogeneous/non-homogeneous cases.

� The non-homogeneity of material has significant effect on thevibration behavior of elastic elliptical cylindrical shells regard-less the isotropic/orthotropic of material and the presence/absence of Winkler foundation cases.

� The highest effect of (c, z) model on the vibration behavior ofshell is in together case, and the influence of c in existence of zconstant is more significant and dominated.

� As the Winkler modulus ratio increase c, the effect of the non-homogeneity on the vibration behavior decreases and for highvalues of c the effect of non-homogeneity is insignificant.

� The mode shapes of vibration concentrated around the areas(q¼ 0,p) where the effect ofWinkler foundation is the least, andbecome more pronounced with the increase of the c value.

� The mode shapes of vibration are more sensitive to the Winkermodulus ratio and elliptical parameter than the non-homogeneity ratio and orthotropy parameter of the shell.

� The vibration behavior of an isotropic and orthotropic ellipticalshell is found to be almost identical behavior under the effect of(c, z) models, and the mode shapes varied significantly fromthose of the circular cylinder cases.

� The effect of elastic foundation is more pronounced for (a/b ¼ 1)as compared to (a/b > 1) and this is confirmed the fact that thegeometry of shell in existence of elastic foundations plays animportant role in the mechanical behavior of vibration.

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