[American Institute of Aeronautics and Astronautics 48th AIAA/ASME/ASCE/AHS/ASC Structures,...

American Institute of Aeronautics and Astronautics

1

Numerical Integration Based Vibration Analysis of

Anisotropic Branched Shells of Revolution with Ring

Stiffeners

Altan Kayran* and Erdem Yavuzbalkan.†

Middle East Technical University, Ankara, 06531, TURKEY

In the present study, application of the multisegment numerical integration technique is extended to the solution of free vibration problem of macroscopically anisotropic branched shells of revolution with ring stiffeners. Solution procedure is based on a modified frequency trial method, which processes on the numerically integrated transformed fundamental shell equations which are obtained in terms of finite exponential Fourier transform of the fundamental shell variables. The governing shell equations are derived such that the full anisotropic form of the constitutive relations, including first order transverse shear deformation, and all components of translatory and rotary inertia are included in the analysis. To handle branched shells of revolution, modifications which are necessary to incorporate junctions are added to the numerical solution procedure. Comparison of the results for anisotropic branched shells of revolution has also been done with the results obtained by the finite element code Nastran. Inclusion of asymmetric circumferential stiffeners, with respect to the shell middle surface, into the semi-analytical solution method is also demonstrated by presenting two alternative method of analysis. It is deemed that present technique, besides providing an alternative solution methodology to study the modal characteristics of ring stiffened anisotropic branched shells of revolution, can be reliably used as an alternative computational tool to compare against the other solution methodologies.

Nomenclature

ijA = extensional stiffness coefficients )6,2,1,( =ji

ijAs = transverse shear stiffness coefficients )5,4,( =ji

ijB = bending-stretching coupling stiffness coefficients )6,2,1,( =ji

ijD = bending stiffness coefficients )6,2,1,( =ji

1E = Young’s modulus in the fiber direction

2E = Young’s modulus transverse to fiber

12G = In plane shear modulus

1323,GG = Transverse shear moduli

h = thickness of the shell wall

Rh = offset between the reference surfaces of the shell walls with and without stiffener

φθθθφφ NNN ,, = in-plane normal and shear stress resultants (per unit length)

φθθθφφ MMM ,, = moment resultants of in-plane normal and shear stresses (per unit length)

φQ = transverse shear stress resultant (per unit length)

R (= φθ sinR ) = distance of a point on the reference surface of the shell of revolution from the shell axis

φR = radius of curvature of the reference surface of the shell in the meridional (φ ) direction

* Assoc. Professor, Department of Aerospace Eng., [email protected], Associate Member of AIAA † Graduate Student, Department of Aerospace Eng., [email protected]

48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference<br> 15th23 - 26 April 2007, Honolulu, Hawaii

AIAA 2007-2114

Copyright © 2007 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.


2

θR = radius of curvature of the reference surface of the shell in the circumferential (θ ) direction 0φu = reference surface displacement of the shell wall along the shell meridian

0θu = reference surface displacement of the shell wall in the circumferential direction 0w = reference surface displacement of the shell wall in the transverse direction

x = meridional coordinate of the cylindrical shell

θφ ββ , = rotations of transverse normal about θ and φ curvilinear coordinates, respectively.

00 , θθφφ εε = reference surface normal strains along the φ and θ curvilinear coordinates, respectively

φ = meridional coordinate of the shell of revolution 0φθγ = in-plane shear strain of the reference surface

φζγ = transverse shear strain in ζφ − plane

θζγ = transverse shear strain in ζθ − plane

θθφφ κκ , = bending curvatures of the reference surface

φθκ = twisting curvature of the reference surface

ρ = Mass density

θ = circumferential coordinate of the shell of revolution

12ν = Poisson’s ratio

ζ = curvilinear coordinate in the thickness direction, measured from the reference surface

I. Introduction Several solution strategies have been used for the vibration analysis of shells of revolution. The most commonly

used approach is based on the representation of the shell variables by a Fourier series in the circumferential coordinate combined with the use of a numerical discretization technique such as finite elements or numerical integration in the meridional direction. Some examples of the application of finite element method used for the solution of anisotropic shells of revolution are given in Refs. [1-5]. Common point in these studies is the use of combination of Fourier series representation in the circumferential direction, and a finite element model for the discretization in the meridional direction by means of axisymmetric/conical finite elements.

Numerical integration method of analysis is basically a semi-analytical procedure with the inherent advantageous features of the applicability to any linear shell theory, and to general shells of revolution with no restriction on the type of boundary conditions at two ends of the shell6-8. In Ref. 6 the numerical integration method was applied to specially orthotropic laminated truncated conical shells to study the effect of transverse shear deformation on the natural frequencies of laminated shells of revolution. Reference 7 applies the numerical integration technique to the static solution of anisotropic shells of revolution by utilizing the classical shell equation in which transverse shear deformation is neglected. Reference 8 is the work of Kalnins who is one of the pioneers in demonstrating the application of the multisegment numerical integration technique to the solution of free vibration problem of isotropic shells of revolution by utilizing classical shell equations. Of utmost importance is the uniform convergence property of the numerical integration technique. Therefore, for shells of revolution complete circumferential properties, numerical integration method provides a powerful solution alternative.

Some examples of other methodologies used for the solution of vibration problem of anisotropic shells of revolution include the study of Heylinger and Jilani9, who incorporated Ritz method using a combination of power and Fourier series as the approximating functions for three displacement components. Tan10 presented an efficient substructuring analysis method for predicting the natural frequencies of shells of revolution utilizing the Sturm sequence method in conjunction with the massive substructuring technique. Tımarcı and Soldatos used the state space concept on the Love-type version of a unified shear deformable shell theory, to study the axisymmetric and flexural vibrations of angle-ply laminated circular cylindrical shells11.

It is noted that with the use of advanced composite materials, and various composite manufacturing techniques, production of circumferentially stiffened composite branched shells of revolution has become possible. Ring stiffened composite shells of revolution find applications in various areas such as rocket fuselages, pressure vessels,


3

submarines, aircraft, external stores, antenna etc. Ring stiffeners are not only used to reinforce the whole structure, but they are also used in the connection of shell segments to construct shells of revolution of one type with long meridian length, and to connect different shells of revolution at the junctions where the shell branches to a different type. Among some sample references, involving the dynamics of ring sitffened shells of revolution, is the work of Ruotolo12 who studied the dynamics of shells of revolution reinforced with ring stiffeners, and compared the Donnell’s, Love’s, Sanders’ and Flügge’s thin shell theories in the evaluation of natural frequencies of cylinders stiffened with rings. In his study, results are obtained for isotropic and specially orthotropic cylinders which are simply supported at the edges. Thus, by means of assumed displacement fields which satisfy the simply supported edge condition, natural frequencies and mode shapes could be determined from the stiffness matrix and mass matrix evaluated by including the smeared effect of stiffener over the entire surface of the cylinder. In another study, Xiang13 et al. presented an analytical method based on the state-space technique and domain decomposition approach to determine the natural frequencies of isotropic cylindrical shells having multiple intermediate ring supports and various combinations of end support conditions. In a recent study Wang and Lin14 presented an analytical methodology for the vibration analysis of ring stiffened cross-ply laminated cylindrical shells, and studied the effect of inner and outer ring on natural frequencies of ring-stiffened cross ply laminated cylindrical shells.

The main objective of the present paper is to combine the multisegment numerical integration technique with a modified frequency trial method for the solution of the natural frequencies, and the mode shapes of anisotropic branched shells of revolution with arbitrary discrete and continuous variation of the properties along the shell meridian. In this endeavor, extension of the method of solution to the vibration analysis of anisotropic branched shells of revolution, and analysis of anisotropic branched shells of revolution with asymmetric circumferential stiffeners with respect to the middle surface of the shell of revolution has been demonstrated. Although the method of solution is applicable to any linear shell theory, in the present study first order shear deformation theory based on Reissner-Naghdi shell equations is used. To handle branched shells of revolution, modifications which are necessary to incorporate junctions are added to the numerical solution procedure. Comparison of the solution of an anisotropic branched shell of revolution has also been done with the results obtained by the finite element code Nastran. In addition, vibration analysis of anisotropic shells of revolution reinforced by integral asymmetric ring stiffeners, with respect to the shell middle surface, has been explained by presenting two alternative method of analysis which are named as fundamental variable transformation method, and common reference surface method. It has been shown that, by either method, vibration analysis of anisotropic branched shells of revolution with integral ring stiffeners, which are placed asymmetrically with respect to middle surface of the shell of revolution, can be accomplished.

II. Governing Equations of Anisotropic Shells of Revolution In this section governing equations of free vibration of anisotropic shells of revolution will be reviewed briefly to

build the foundation for the extension of the method of solution to branched anisotropic shells of revolution with ring stiffeners. Figure 1 shows the geometry notation used for a shell of revolution.

Figure 1. Geometry and coordinate system of shell of revolution.


4

The starting point in the application of numerical integration technique for the solution of vibration problem of a shell of revolution is to express the equations governing the free vibration as a system of partial differential equations given in the form of Eq.(1).

( ) ( )

∂

∂∂∂=

∂∂

),,(2

2,),,(,),,(

),,(tttf

t θφψθ

θφψθ

θφψφθφψ

(1)

In Eq.(1), ψ is a vector representing the fundamental shell variables that enter into the appropriate boundary

conditions on a rotationally symmetric edge of the shell of revolution, and for the Reissner-Naghdi improved shell theory they are given by Eq.(2).15

=t,,( θφψ TMMNNQuuw φθφφφθφφφθφθφ ββ ,,,,,,,,, 000 (2)

The system of equations given by Eq.(1) is derived by the complex manipulation of three set of equations which are the strain displacement relations of the Reissner-Naghdi shell theory15,16, dynamic equilibrium equations15,16, and full anisotropic form of the constitutive relations relating the stress resultants to mid surface strains and curvatures of the laminated shell wall.17 The derivation of the fundamental system of equations, for anisotropic shells of revolution, in the form of Eq.(1) is explained in detail in Ref. 18. To aid the understanding in the next sections, the three set of equations are repeated below.

Eqs. (3-10) give the strain displacement relations for the Reissner-Naghdi shell theory15,16.

+

∂∂

= 00

0 1w

u

R φε φ

φφφ (3)

+

∂∂

+= φθ

φφ

ε θφ

θθθ sincos

sin

1 00

00 wu

uR

(4)

000

0 cot1

sin

1θ

θ

θ

φ

φ

θφθ

φφθφ

γ uR

u

R

u

R−

∂∂

+∂∂

= (5)

φβ

κ φ

φφφ ∂

∂=

R

1

(6)

∂∂

+=θ

βφβ

φκ θ

φθ

θθ cossin

1

R

(7)

θθ

θ

φ

φ

θφθ βφ

φβ

θ

β

φκ

RRR

cot1

sin

1 −∂

∂+

∂

∂=

(8)

φβγ

φφ

φφφζ ∂

∂+−=00

1 w

RR

u

(9)


5

θβγ

θ

θθθζ ∂

∂+−=00 1 w

RR

u

(10)

Dynamic equilibrium equations during free vibration of doubly curved shells of revolution are given by Eqs.(11-15)15,16.

φρφθ

φφφ φθφθφφ

φθφθθφφθ

φφsinsincos)(sin 0 RRuhRQR

NRNNR

N&&=+

∂∂

+−+∂

∂

(11)

φρφθ

φφφ φθθφθφ

θθφφθθ

φθsinsincos2sin 0 RRuhRQR

NRNR

N&&=+

∂∂

++∂

∂

(12)

φρφφφφφθ φθθφφφθθφφθφ

φθ sinsinsincossin 0 RRwhRNRNRQR

QR

Q&&=−−+

∂∂

+∂

∂

(13)

φβρφφθ

φφ φθφθφφφθθφφφ

φθθ

φφsin

12

1sincos)(sin 3 RRhRRQRMMR

MR

M&&=−−+

∂∂

+∂

∂

(14)

φβρφφθ

φφ φθθθφθφφθφ

θθθ

φθsin

12

1sincos2sin 3 RRhRRQRMR

MR

M&&=−+

∂∂

+∂

∂

(15)

For laminated composite structures, full anisotropic form of the constitutive equations relating the stress

resultants to mid surface strains ( 000 ,, φθθθφφ γεε ) and curvatures ( φθθθφφ κκκ ,, ) are given in matrix form by Eq.

(16)17.

=

φθ

θθ

φφ

φθ

θθ

φφ

φθ

θθ

φφ

φθ

θθ

φφ

κκ

κγεε

0

0

0

662616662616

262212262212

161211161211

662616662616

262212262212

161211161211

DDDBBB

DDDBBB

DDDBBB

BBBAAA

BBBAAA

BBBAAA

M

M

M

N

N

N

(16)

In addition, transverse shear stress resultants are related to transverse shear strains by Eq. (17).

=

0

0

5545

4544

φ

θφ

θ

γγ

AsAs

AsAs

Q

Q (17)


6

In Eqs.(16-17), the coefficients given in the square matrix expressions represent the stiffness coefficients of the shell wall which is built up from individual layers of arbitrary fiber orientation, and they are defined in the usual manner17.

In the derivation process of dynamic equilibrium equations, application of Hamilton’s principle also generates conditions on the boundary displacements, and boundary stress resultants applied on the constant meridional coordinate (φ =constant) for a shell of revolution. For the free vibration problem boundary conditions are given by19

Either φφN or 00 =φu (18)

Either φθN or 00 =θu (19)

Either φQ or 00 =w (20)

Either φφM or 0=φβ (21)

Either φθM or 0=θβ (22)

When the full anisotropic form of constitutive relations given by Eqs. (16-17) are utilized, the uncoupling of the circumferential coordinateθ , in Eq.(1), can not be achieved by the classical Fourier decomposition of the fundamental shell variables. Therefore, to accomplish the uncoupling of the circumferential coordinate from the fundamental system of equations given by Eq. (1), each fundamental variable is expanded in complex Fourier series.

θφψθφψ inn e∑=

+∞

∞−)(),( (23)

)()(),(2

1)(

2

0φψφψθφψ

πφψ θπ

nsncin

n ie −=∫= − (24)

where

θθθφψπ

φψπ

dnnc cos),(2

1)(

2

0∫= (25)

θθθφψπ

φψπ

dnns sin),(2

1)(

2

0∫= (26)

In Eqs.(23-24), it is assumed that for the free vibration analysis, the time dependence of each quantity in

synchronous motion appears in a factor tie ω , where ω is the natural frequency. Thus, the time variable can be eliminated from the fundamental system of equations given by Eq.(1). It is clearly seen that application of finite exponential Fourier transform results in doubling of the number of fundamental variables. Thus, finite exponential Fourier transform of the fundamental system of partial differential equations (Eq.(1)) yields a system of ordinary differential equations in terms of the transformed fundamental shell variables, and this set of equations are given in Eq.(27)

[ ]120

)2(

)1(

2020120

)2(

)1(

)(

)(),,(

)(

)(

xx

x

nKd

d

d

d

=

=φψφψφω

φψφψ

φψ

φ (27)


7

where n is the circumferential wave number and

=)()1( φψ Tnsncnsncnsncnsncnsnc uuuuww θθφφθθφφ ββββ ,,,,,,,,, 000000 (28)

=)()2( φψ Tnsncnsncnsncnsncnsnc MMMMNNNNQQ φθφθφφφφφθφθφφφφφφ ,,,,,,,,, (29)

In Eqs.(28) and (29), displacement and stress resultants with subscript ‘c’ refer to symmetric modes with respect to the tangential coordinate θ of the shell of revolution, and subscript ‘s’ refers to antisymmetric modes. The elements of the coefficient matrix K for anisotropic shells of revolution are given in Ref.18. Fundamental system of equations, Eq.(27), together with the finite exponential transform of the boundary conditions, Eqs. (18-22), specified at the two boundary edges of an anisotropic shell of revolution form an eigenvalue problem for the eigenvectors which are the transformed displacements, and stress resultants. It should be noted that finite exponential Fourier transform of the boundary conditions, on the physical variables, given by Eqs.(18-22), yields the same conditions on the sine and cosine parts of the fundamental shell variables calculated by Eqs.(25) and (26). Thus, for the first order shear deformation theory, because the fundamental system of equations are expressed in terms of transformed fundamental shell variables due to the use of full anisotropic constitutive relations (Eqs.(16-17)), total of ten boundary conditions need to be applied at an edge of the shell of revolution.

A. Boundary Conditions at the Apex of a Shell of Revolution

For shells of revolution which are closed at one end, the correct and independent conditions to apply at the apex are not the real boundary conditions since no physical boundary exists at the apex. Therefore, the conditions at the apex should be thought of as necessary conditions to ensure the existence of a finite solution at 0=R and 0=φ .

Careful study of the governing equations of shells of revolution reveals that depending on the value of the circumferential wave number, the boundary conditions to be imposed at the apex change. In the following, strain displacement relations (Eqs.(3-10)), and dynamic equilibrium equations (Eqs.(11-15)) will be utilized for the identification of the conditions to be applied at the apex of the shell of revolution. Because the final fundamental system of equations (Eq.(27)) is obtained in terms of transformed fundamental shell variables, finite exponential Fourier transform of the strain displacement relations, and dynamic equilibrium equations will be used in extracting the conditions on the transformed shell variables. It should be noted that application of the exponential transform

given by Eq.(24), results in a factor pn which multiplies the terms involving thp order derivatives of the shell

variables with respect to θ . 1. Circumferential wave number 0=n

When the circumferential wave number is zero, at the apex where R and φ are zero, the numerators of the strain

displacement relations (Eqs.(3-10)) must have corresponding order zeros to have finite strain values. Thus, from the

study of the strain displacement relations we can conclude that to have finite 0θθε at the apex 0φu must vanish, to

have finite 0φθε at the apex 0

θu must vanish, for finite θθκ at the apex φβ must vanish, and for finite φθκ at the apex

θβ must vanish. The fifth condition is extracted from dynamic equilibrium equation Eq.(13). In order for each term

to be finite, φQ must vanish. Thus, for 0=n , the ten boundary conditions that need to be applied, in terms of the

transformed fundamental shell variables, at the apex of a closed shell of revolution are given by

),,,,,,,,,( 0000nsncnsncnsncnsncnsnc QQuuuu φφθθφφθθφφ ββββ =0 (30)

2. Circumferential wave number 1=n

Finite exponential Fourier transform of the strain displacement relation for 0θθε yields two equations which are

given by Eq.(31).


8

[ ]φφφ

ε θφθ

θθ sincossin

1 0000ncnsncnc wnuu

R++=

[ ]φφφ

ε θφθ

θθ sincossin

1 0000nsncnsns wnuu

R+−=

(31)

Thus, for 1=n , at the apex of the shell of revolution, to have finite transformed strain components, the following conditions must hold.

00nsnc uu θφ + =0 and 00

ncns uu θφ − =0 (32)

Similarly, exponential transform of θθκ yields the following conditions at the apex of the shell of revolution.

nsnc θφ ββ + =0 and ncns θφ ββ − =0 (33)

It should be noted that the conditions given by Eqs.(32) and (33) are also the conditions for the exponential Fourier

transform of 0φθγ and φθκ to be finite at the apex of the shell of revolution. In addition, to have finite transverse

shear strain θζγ at the apex, cosine and sine parts of the transformed lateral displacement 0w must vanish at the

apex.

),( 00nsnc ww =0 (34)

So far six conditions are deduced from the study of the strain displacement relations. The remaining four conditions come from the requirement that the each term in the dynamic equilibrium equations also need to be finite. The finite exponential transform of Eq.(11) yields the conditions given by Eq.(35) on the transformed fundamental variables.

ncnc NN θθφφ = , nsns NN θθφφ = and nsnc NN φθφθ = =0 (35)

The finite exponential transform of Eq.(12) yields the conditions given by Eq.(36) on the transformed fundamental variables.

nsnc NN θθθθ = =0 (36)

Similarly, finite exponential Fourier transform of the remaining dynamic equilibrium equations Eqs.(13-15) yields the following conditions on the transformed fundamental shell variables.

0),,,( =nsncnsnc QQQQ θθφφ (from Eq.(13)) (37)

ncnc MM θθφφ = , nsns MM θθφφ = and nsnc MM φθφθ = =0 (from Eq.(14)) (38)

nsnc MM θθθθ = =0 (from Eq.(15)) (39)

It is noticed that the conditions Eq.(36) and Eq.(39) also dictate further conditions on the transformed stress resultants along the meridional direction, and on the transformed moment resultants of the in-plane stresses via Eqs.(35) and (38).


9

0),,,( =nsncnsnc MMNN φφφφφφφφ (40)

In addition, since through Eq.(34) the transformed lateral displacement vanishes at the apex, from Eq.(13) it is clear that the sine and cosine parts of the transformed transverse shear stress resultant φQ should also vanish at the apex of

the shell of revolution. However, the boundary conditions must be independent conditions, therefore the condition given by Eq.(34) on the lateral displacement will suffice. Finally, keeping in mind that the boundary conditions to be imposed at the apex must be selected among the fundamental shell variables (Eq.(2)), apart from the conditions given by Eqs.(32-34), the following conditions should be applied at the apex for .1=n

Either 0),( =nsnc NN φφφφ or 0),( =nsnc NN φθφθ

Either 0),( =nsnc MM φφφφ or 0),( =nsnc MM φθφθ

(41)

3. Circumferential wave number 2≥n For circumferential wave numbers greater than or equal to 2, the finite exponential Fourier transform of the

equations Eqs.(4,5), Eqs.(7,8) and Eq.(10) yields the following conditions on the transformed fundamental variables at the apex of the shell of revolution.

),,,,,,,,,( 000000nsncnsncnsncnsncnsnc uuuuww θθφφθθφφφφ ββββ =0 (42)

III. Analysis of Anisotropic Branched Shells of Revolution

A. Method of Solution for Anisotropic Branched Shells of Revolution

As proposed by Kalnins8, in the multisegment numerical integration technique, the shell is divided into M number of segments in the meridional direction, and the solution to Equation (27) can be written as

[ ] )(),,()( iinT φψφωφψ = , ),...2,1( Mi = (43)

where the transfer matrices iT are obtained from the initial value problems defined in each segment i by

[ ] [ ][ ]ii nTnKnTd

d),,(),,(),,( φωφωφω

φ= (44)

with the initial conditions [ ] [ ]InT ii =),,( φω (45)

In the solution process, the arbitrary variation of the geometric and material properties of the shell of revolution is handled during the numerical integration of Eq. (44) subject to initial conditions Eq. (45). The elements of the coefficient matrix K consist of the laminate stiffness coefficients ),,( ijijij DBA , radii of curvature of shell of

revolution ),( θφ RR , circumferential wave number (n ), and the coordinate information )(φ along the meridian of the

shell18. Numerical integration of Eq. (44) within each shell segment is performed by the IMSL subroutine DIVPAG, which uses a user supplied subroutine where all the elements of the coefficient matrix K are given. Therefore, as long as the discrete or continuous variation of the elements of the coefficient matrix along the meridian of the shell is coded accordingly, arbitrary variation of the shell properties can be handled, if the transformation of the fundamental variables are done properly at the junctions. This process does not require any additional modifications in the semi-analytical solution method.

Solution procedure for the anisotropic branched shell of revolution will be demonstrated for a branched shell consisting of two distinct regions. Figure 2 shows the reference surface of a branched shell of revolution comprising


10

two distinct regions S1 and S2. In Figure 2, ‘j’ denotes the location of the junction at which the shell geometry changes leading to a branched shell of revolution. T1 and T2 denote the tangents drawn to the mid planes of shells of revolution S1 and S2 at the junction ‘j’, respectively.

The governing fundamental system of equations represented by Eq. (27) can be written for the two regions S1

and S2 as

[ ]1

)2(

)1(

11

)2(

)1(

1)(

)(),,(

)(

)(

SS

SS nK

d

d

d

d

=

=φψφψφω

φψφψ

φψ

φ (46)

[ ]2

)2(

)1(

22

)2(

)1(

2)(

)(),,(

)(

)(

SS

SS nK

d

d

d

d

=

=φψφψφω

φψφψ

φψ

φ (47)

In Eqs. (46) and (47), the coefficient matrices have different elements depending on the geometry of the shell of revolution, characterized mainly by the two radii of curvature of the reference surface of the shell of revolution. In addition, any differences in the shell materials, and shell thicknesses in the distinct regions are assumed to be reflected in the coefficient matrices 1SK and 2SK . The fundamental variable vectors 1Sψ and 2Sψ are also

interpreted as the fundamental shell variables belonging to distinct shell regions S1 and S2. In order to analyze branched shells of revolution, the continuity relations of the fundamental variables at the end points of the shell segments must be rewritten. At the junction, the fundamental variables of the shell S1, and the fundamental variables of the shell S2 must obey the following transformation equation.

[ ] 1

)1(10102

)1( )()(SjxSj TR φψφψ = (48)

[ ] 1

)2(10102

)2( )()(SjxSj TR φψφψ = (49)

where the transformation matrix, which transforms the displacement and stress resultant vectors, is given in Appendix A. The elements of the transformation matrix are obtained by expressing the elements of the displacement

and stress resultant fundamental variable vectors )1(ψ and )2(ψ of shell S2, in terms of the fundamental variables of

shell S1, by utilizing the angle β between the tangents drawn to the reference surfaces of the shells S1 and S2 at the

Figure 2. Geometry of a general branched shell of revolution.


11

junction. By utilizing Eq.(43) and Eqs.(48-49), the continuity relations of the fundamental variables in partitioned form can be written at ends of the shell segments in two different shells S1 and S2, and at the junction as

1

)2(

)1(

11)4(

1)3(

1)2(

1)1(

11)2(

1)1(

)(

)(

)()(

)()(

)(

)(

Si

i

Siiii

iiii

Si

i

TT

TT

=

++

++

+

+

φψφψ

φφφφ

φψφψ

)1,...2,1( −= ji (50)

1

)2(

)1(

2

)2(

)1(

)(

)(

0

0

)(

)(

Sj

j

iSj

j

TR

TR

=

φψφψ

φψφψ

)( ji = (51)

2

)2(

)1(

21)4(

1)3(

1)2(

1)1(

21)2(

1)1(

)(

)(

)()(

)()(

)(

)(

Si

i

Siiii

iiii

Si

i

TT

TT

=

++

++

+

+

φψφψ

φφφφ

φψφψ

),....( Mji = (52)

For computational ease, the rows of the fundamental variable vector ψ at both ends of the shell, 1φ and 1+Mφ ,

are adjusted such that, for the anisotropic branched shell, the first 10 elements of 11)( Sφψ and the last 10 elements

of 21)( SM +φψ are the prescribed boundary conditions. In the following, to keep the uniformity of the notation used

for the partitioned fundamental variable vector, the boundary conditions are also represented by the same vector notation defined by Eqs. (28) and (29). Therefore, boundary condition at the ends of the branched shell of revolution is expressed by Eq. (53).

0()(2)1

)2(11

)1( == + SMS φψφψ (53)

By writing Eqs.(50-52) in each shell segment ‘i ’ separately, the whole equation set is brought into an upper triangular matrix equation by Gauss elimination. Eq. (54) shows the resulting matrix equation.

0

)(

)(

.

)(

)(

)(

)(

)(

)(

.

)(

)(

000000000.0

_0.........

............

0.0.......

0.00......

0..00.....

0...00....

0....00...

0.....00..

............

0.......00

0........0

1)1(2

)2(2

1)1(2

)2(2

)1(2

)2(1

)1(1

1)2(

1

2)1(1

1)2(

1

1

1

1

1

=

−−

−−

−−

−−

+

+

−

−

−

MS

MS

jS

jS

jS

jS

jS

jS

S

S

M

M

j

j

Z

Z

j

j

C

IE

IC

IE

IC

IE

IC

IE

IC

IE

φψφψ

φψφψφψφψφψφψ

φψφψ

(54)

In Eq.(54), iE and iC represent the 10X10 sub matrices which are expressed in terms of the partitioned transfer

matrices )4,3,2,1(iT . In the first shell S1

)2(11 TE = (55)

11

)4(11

−= ETC (56)


12

11

)1()2( −−+= iiii CTTE )1,...,3,2( −= ji (57)

( ) 111

)3()4( −−−+= iiiii ECTTC )1,...,3,2( −= ji (58)

The transition matrices ZE and ZC are calculated from

[ ] 11

−−= jZ CTRE (59)

[ ] 1−= ZZ ETRC (60)

In the second shell S2

1)1()2( −+= Zjjj CTTE (61)

( ) 11)3()4( −−+= ZZjjj ECTTC (62)

11

)1()2( −−+= iiii CTTE ),...,( Mji 1+= (63)

( ) 111

)3()4( −−−+= iiiii ECTTC ),...,( Mji 1+= (64)

For the anisotropic branched shell of revolution, the natural frequency is then determined by requiring the determinant of the coefficient matrix MC to vanish at the eigenvalue. Solution of the natural frequencies is

accomplished by evaluating the determinant of the characteristic matrix for incremented values of the frequency estimates within a frequency range of interest. When the finite exponential transform of the fundamental shell equations was used, it was observed that determinant of the characteristic matrix does not change sign, but rather it is always positive, and vanishes at the eigenvalue18. Therefore, a slope change detection algorithm in combination with inverse interpolations was devised to extract the natural frequency. The method essentially relies on checking the slope change of the determinant of the characteristic matrix, and detecting an interval where a natural frequency resides. Once an interval is determined, by successive inverse interpolations frequency is extracted.

For each natural frequency determined, the partitioned fundamental variable vector )( 1)1(2 +MS φψ at the end of the

branched shell of revolution is determined up to an arbitrary constant. The remaining unknown fundamental variables at each end of the shell segments, in shells S1 and S2, can then be calculated successively from

( ) ( )1)1(2

1)2(2 +

−= MSMMS E φψφψ (65)

( ) ( )1)2(

21

1)1(2 +−

−−+− = iMSiMiMS C φψφψ ),...,2,1( jMi −= (66)

( ) ( )1)1(2

1)2(2 +−

−−− = iMSiMiMS E φψφψ ),...,2,1( jMi −= (67)


13

( ) ( )jSZjS C φψφψ )2(2

1)1(2

−= (68)

( ) ( )jSZjS E φψφψ )1(2

1)2(1

−= (69)

( ) ( )1)2(

11

1)1(1 +−

−−+− = ijSijijS C φψφψ )1,...,2,1( −= ji (70)

( ) ( )1)1(1

1)2(1 +−

−−− = ijSijijS E φψφψ )1,...,2,1( −= ji (71)

For branched shells whose mid surfaces have the same tangent at the junction, 0=β . Thus, the transformation

matrix reduces to identity matrix, and in that case the following equality holds.

1

)2(

)1(

2

)2(

)1(

)(

)(

)(

)(

Sj

j

Sj

j

=

φψφψ

φψφψ

(72)

At the junction point ‘j’, the elements of the partitioned transformed fundamental variable vector ψ are calculated

twice for each distinct shells of revolution S1 and S2, successively from Eq.(67) for jMi −= , Eqs. (68-69) and

from Eq.(70) for 1=i . Thus, the variation of all the transformed fundamental variables along the meridian of the shell can be determined. The physical fundamental shell variables can then be determined from Eq.(23) for the particular circumferential vibration mode that is analyzed. B. Numerical Results

The solution procedure is demonstrated for a branched shell of revolution comprising a cylindrical and a conical part as shown in Figure 3.

In Figure 3, only the reference surface of the shell of revolution is shown, and the lengths of the meridian of the cylindrical and conical parts are shown by LS1, and LS2, respectively. In accordance with the theoretical derivation of

the general shell of revolution equations, the lateral displacements,01Sw and 0

2Sw , for distinct shell regions are

perpendicular to the reference surface of the shell. The particular anisotropic cylinder-cone combination analyzed has the following geometrical and material

properties. Young’s modulus in the fiber direction: 1E =213.74 GPa, Young’s modulus transverse to fiber: 2E =18.62 GPa

Shear moduli: 171.51312 == GG GPa, 137.423 =G GPa, Poisson’s ratio: 12ν =0.28

Mass density: ρ =2051.88 kg/m3

02Sw 0

1Sw

β

s

LS2

R

LS1

Axis of revolution

Reference surface

Figure 3. Geometry of a cylinder-cone combination.


14

Two layer shell with fiber orientation: 0o/45o, ply thickness: 1mm. The branched shell is assumed to be free at the cylindrical end, and clamped at the conical end. Vibration

analysis is performed for different cone anglesβ in the range of 0-30o. Figure 4 gives the variation of the scaled non-

dimensional fundamental frequency )10*( 3Ω , for different cone angles, and for three different non-symmetric

vibration modes. The non-dimensional frequency Ω is given by

11Eh ρω=Ω (73)

It is seen that beyond a certain circumferential wave number, the decrease of the natural with the cone angle is more evident, and this is clearly seen for the case of n =5. This behavior is attributed to the dominance of bending strain energy contribution to the total strain energy at high circumferential wave numbers20. As the cone angle gets higher, the circumferential length of the conical part increases, and at high circumferential wave numbers circumferential slices of the shell essentially behave like a longer span beam under bending. Thus, at high circumferential wave numbers, dominance of bending strain energy accounts for the decrease of the natural frequency with the cone angle. For n =1, which is essentially the global beam bending mode of the shell of revolution, initial increases in the cone angle results in an increase in the moment of inertia of the conical part, and this causes an increase in the natural frequency. When cone angle is increased further, the overall increase of the total mass of the branched shell, and the dominance of the extensional strain energy to the total strain energy at low circumferential wave numbers20, accounts for the decrease of the frequency.

The results obtained by the semi-analytical method are compared with the results of finite element solution, which is performed by Nastran21 utilizing the isoparametric curved thin shell element QUAD8, which contains 4 corner grid points and 4 edge grid points. The laminate is defined by the order and orientation of lamina with respect to a material coordinate system using the PCOMP entry. Layer materials are defined as two dimensional orthotropic materials through MAT8 entry, and transverse shear behavior is included in the material definition entry. The comparison of the variation of the fundamental natural frequency with the cone angle determined by the present method, and the finite element solution, which uses the Lanchzos algorithm, is presented in Figure 5 for n =1. The finite element mesh is made fine enough to achieve convergence of the results with respect to mesh size. Mesh consists of 18774 QUAD8 elements, and total node numbers used is 56850. Good agreement was observed between the semi-analytical solution and finite element solution by Nastran.

Figure 4. Variation of fundamental frequency with cone angle for the cylinder-cone combination.


15

For the anisotropic cylinder-cone combination, the normalized fundamental lateral displacement mode shape

( 0w ), which is obtained through the recursive relations given by Eqs.(65-71), is plotted in Figure 6. The normalized mode shape is obtained for the circumferential wave number of 3, and a cone angle of 20o. In Figure 6, the coordinate along the shell meridian ‘s ’ is normalized separately with respect to the meridian length of the

cylindrical and conical parts, 1SL and 2SL . Due to the anisotropic nature of the 0o/45o lay-up, the sine and cosine

components of the lateral displacement both are non-zero, and since the cone angle is 20o, a kink exists at the

cylinder–cone junction. Once the cosine (0ncw ) and sine ( 0nsw ) parts of the fundamental lateral displacement are

determined, the actual lateral displacement mode shape can then be constructed utilizing the complex Fourier series representation of the lateral displacement given by Eq. (23) in conjunction with Eq. (24), for the particular circumferential wave number.

Another sample result for an anisotropic branched shell of revolution, which is closed at the apex, is produced for a half sphere-cone combination given in Figure 7. In the numerical calculation, the branched shell is assumed to have a very small hole at the apex of the sphere to eliminate the singularity. The apex boundary conditions are then applied to the edge of the tiny hole at the apex. The conical end of the shell is let free. The particular anisotropic sphere-cone combination analyzed has the following geometrical and materials properties.

Figure 5. Variation of fundamental frequency with cone angle for the cylinder-cone combination (n=1).

Figure 6. Fundamental lateral displacement mode shape ( 0w ); n=3, =β 20o.


16

Lcone=0.315 m, R=0.21 m, cone angle β =30o, shell thickness h =2 mm

1E =213.74 GPa, 2E =18.62 GPa, 171.51312 == GG GPa, 137.423 =G GPa

12ν =0.28, ρ =2051.88 kg/m3

Two layer shell with fiber orientation: 0o/45o, ply thickness: 1mm.

Figure 8 shows the lateral displacement mode shape of the sphere-cone combination obtained by the numerical

integration method.

Figure 8. Fundamental lateral displacement mode shape; n=3, =β 30o.

The normalized mode shape is obtained for the circumferential wave number of 3. In Figure 8, the coordinate along the shell meridian is normalized separately with respect to the meridian length of the spherical and conical parts. Due to the anisotropic nature of the 0o/45o lay-up, the sine and cosine components of the lateral displacement are both non-zero. It should also be noted that all the fundamental variables are calculated twice from Eq. (54) at the junction jφ .Since the reference surfaces of the spherical and conical parts of the branched shell of revolution are not

tangent to each other at the junction, and there is a little jump in the lateral displacement at the shell junction. However, such a jump does not exist at the junction if the magnitude of the displacement was drawn. Once the cosine and sine parts of the fundamental lateral displacement are determined, the actual lateral displacement mode

Figure 7. Geometry of a sphere-cone combination.


17

shape can then be constructed utilizing the complex Fourier series representation of the lateral displacement given by Eq. (23) in conjunction with Eq. (24) for the particular circumferential wave number.

IV. Analysis of Anisotropic Branched Shells of Revolution with Ring Stiffeners Demonstration of the inclusion of circumferential stiffeners, into the vibration analysis of anisotropic branched

shells of revolution, is made for an anisotropic cylindrical shell with different integral stiffener configurations, and different number of stiffeners placed at different axial locations. Circumferential stiffeners are placed such that the shell is divided into axial segments of equal length. Figure 9 shows two different stiffener configurations; a central stiffener and an inner stiffener. Outer stiffener configuration is similar to the inner stiffener configuration with the stiffener placed on the outer surface of the shell, rather than on the inner surface.

In Figure 9, 1S and 3S denote the shell regions without any stiffener, and 2S denotes the shell region with an

integral stiffener. For the central stiffener configuration, the effect of the stiffener is integrated into the analysis by expressing the fundamental system of equations, given by Eq. (27), separately for the distinct shell regions with and without the stiffener. Fundamental shell variables are defined with respect to the middle surface of the shell, and for the stiffened part, different laminate stiffness coefficients are input in the coefficient matrix K of Eq. (27). The solution for the natural frequencies and the mode shapes then follows the same procedure, as in branched shell of revolution. However, due to the use of the same reference surface for the shell and the stiffener, there is no need for transforming the fundamental shell variables at the shell-stiffener junction unless the stiffener corresponds to a branch point.

It is expected that for moderately thicker shell and stiffener walls, the differences in frequencies of inner, central and outer stiffener configurations would be higher. Therefore, a methodology is needed to incorporate the effect of different configuration integral stiffeners into the analysis. For the inner and outer stiffener configurations, inclusion of the stiffener into the analysis can be accomplished by using two different approaches. In this study, these approaches are named as fundamental variable transformation, and common reference surface method.

A. Fundamental Variable Transformation Method

In this approach, the fundamental system of equations given by Eq. (27) are written such that, fundamental variables are defined on the reference surfaces of distinct shell regions with and without the stiffeners. The reference surfaces are taken as the mid planes of the shell walls with and without stiffeners. Figure 10 shows the schematic drawing of the fundamental variables defined for the shell regions with and without stiffener. In Figure 10 fundamental variables are shown in their positive senses, and Rh denotes the offset between the reference surfaces

of the shell walls with and without stiffener.

Figure 9. Stiffener configurations.


18

At the junctions, the fundamental variables are transformed by utilizing the following relations.

Preserved fundamental variables at the junctions: θθββ xxxx NNQw ,,,,,0

Inner stiffener case; 21 SS − junction:

110

20 )()()( SxRSxSx huu β−= 11

02

0 )()()( SRSS huu θθθ β−=

112 )()()( SxRSxSx NhMM += 112 )()()( SxRSxSx NhMM θθθ +=

(74)

Inner stiffener case; 32 SS − junction:

220

30 )()()( SxRSxSx huu β+= 22

03

0 )()()( SRSS huu θθθ β+=

223 )()()( SxRSxSx NhMM −= 223 )()()( SxRSxSx NhMM θθθ −=

(75)

Outer stiffener case; 21 SS − junction:

110

20 )()()( SxRSxSx huu β+= 11

02

0 )()()( SRSS huu θθθ β+=

112 )()()( SxRSxSx NhMM −= 112 )()()( SxRSxSx NhMM θθθ −=

(76)

Outer stiffener case; 32 SS − junction:

220

30 )()()( SxRSxSx huu β−= 22

03

0 )()()( SRSS huu θθθ β−=

223 )()()( SxRSxSx NhMM += 223 )()()( SxRSxSx NhMM θθθ +=

(76)

Thus, depending on the stiffener type, and type of junction ( 21 SS − , 32 SS − ) along the shell of revolution, by

utilizing the above relations among the fundamental variables, a transformation matrix can be written at each junction. Thus, the continuity relations in partitioned form can be written at the junctions in the same manner as in

Axis of revolution

S1 xQw ,0

xx Nu ,0

xx Nu ,0

xx M,β

xQw ,0

S2

S3

hR

x

Mid planes

xx M,β

Figure 10. Fundamental variables of shell regions with and without stiffener.

θθβ xM,

θθ xNu ,0

S1,S3 Ref.Surface

θθβ xM,

θθ xNu ,0

hR S2 Ref. Surface

θ


19

Eq. (51). As long as the transition matrices ZE and ZC in Eq. (54) are defined appropriately at each junction, the

remaining solution procedure is same as the solution for a branched shell of revolution.

B. Common Reference Surface Method

In this method, the mid plane of the shell wall is also used as the reference surface for the stiffener. Thus, all the fundamental variables of the shell of revolution are defined with respect to this common reference surface. In this approach, since the stiffness coefficients of the shell wall are defined with respect to the common reference surface, for symmetric ply lay-outs, with respect to the mid plane of the stiffener, bending-stretching coupling coefficients will be non zero. In this method, since all the fundamental variables of the shell regions with and without stiffeners are defined with respect to the same reference surface, as long as same shell geometry is preserved, no transformation of the fundamental variables is necessary. Therefore, whether a central stiffener, inner or outer stiffener is utilized, the only action that needs to be taken is the modification of the coefficient matrix K in Eq. (44), to reflect the changes in stiffness coefficients of the shell regions with and without stiffener.

C. Numerical Results

Numerical results are performed for an anisotropic cylindrical shell with the following geometric, material properties, and stiffener configurations. Geometric properties: Cylinder length (L ): 1.2 m, radius of shell reference surface with no stiffener ),( 31 SS RR :

0.21 m, shell wall thickness without stiffener: 1.92 mm, shell wall thickness with stiffener: 5.76 mm, stiffener width: 60 mm Material properties: Ply materials: High modulus graphite epoxy with

1E = 207.348 GPa, 2E = 5.183 GPa, 132312 GGG == =3.11 GPa, 12ν =0.25, =ρ 1524.474 kg m-3

Ply thickness: 0.24 mm, shell wall without stiffener consist of 8 plies with 50o fiber orientation, shell wall with stiffener consists of 24 plies with 50o fiber orientation Stiffener configurations: Central, inner, and outer stiffener configurations with one, two and three stiffeners placed at axial locations, such that total shell length is divided into equal length segments in each case. Boundary condition: Shell is clamped at left end; x =0, and simply supported at right end; x = L

Comparison of the natural frequencies (Eq. (73)), for a single stiffener configuration, determined by the fundamental variable transformation and common reference surface methods is made in Table 1. Table 1 lists the

scaled, fundamental non-dimensional frequencies ( 310*Ω ) corresponding to different circumferential wave numbers, for the inner and outer stiffener configurations. As it is expected, the results of both approaches are very close to each other. It should be noted that in the common reference surface method, bending stretching coupling coefficients of the region with the stiffener are non-zero because of the offset of the reference surface from the mid plane. On the other hand, in the fundamental variable transformation method, the bending stretching coupling coefficients of the region with the stiffener are zero. It is deemed that both methods can be used in the analysis of anisotropic shells of revolution. However, fundamental variable transformation method requires additional definition of transformation matrices for the fundamental variables at each junction.

Table 1. Comparison of the scaled non-dimensional frequencies ( 310*Ω ). Inner Stiffener Outer Stiffener

n Ref. Surface

Variable Trans.

Ref. Surface

Variable Trans.

0 0.4746 0.4744 0.4757 0.4753 1 0.3583 0.3581 0.3587 0.3585 2 0.1626 0.1628 0.1575 0.1576 3 0.1687 0.1697 0.1541 0.1543 4 0.2406 0.2413 0.2252 0.2254 5 0.3297 0.3303 0.3163 0.3166 6 0.4310 0.4317 0.4209 0.4214 7 0.5394 0.5399 0.5331 0.5336


20

Scaled, fundamental non-dimensional frequencies )10*( 3Ω , which are determined for different number of

stiffeners along the shell axis, are compared in Figure 11 for various circumferential modes. Stiffener configuration is taken as central in each case.

Figure 11 shows that the effect of increasing the number of stiffeners along the shell axis is felt more and more as the circumferential wave number is increased. This behavior is attributed to the dominance of bending strain energy at high circumferential wave numbers. At the higher circumferential wave numbers, circumferential slices of shell of revolution behave as beams under bending, and therefore, increase in stiffener number results in an increase in frequency. However, at low circumferential wave numbers, extensional strain energy starts to dominate, and it is deemed that due to the additional mass effect of the stiffeners, natural frequency decreases when circumferential stiffeners are added. At the axisymmetric vibration mode (n =0), the difference in the frequencies is felt most strongly.

The effect of inner and outer stiffener configurations on the natural frequencies is compared in Figure 12.

Figure 11. Effect of stiffeners on the natural frequencies (central stiffeners).

Figure 12. Effect of stiffener configurations on the natural frequencies.


21

It is seen that at higher circumferential modes, inner stiffener configuration has slightly higher frequencies compared to the outer stiffener configuration, and the increase in the frequency is higher when the number of stiffeners used is increased. At high circumferential wave numbers, circumferential slices taken from the inner stiffener region resembles to a slightly shorter span beam under bending, compared to the slices taken from the outer stiffener region. Thus, the slight increase of the natural frequency of the inner stiffener configuration is justified. At low circumferential wave numbers, because of the dominance of extensional strain energy, the natural frequencies of inner and outer stiffener configurations get very close to each other. However, it is also noticed that as the circumferential wave number is increased further, the natural frequencies of inner and outer stiffener configurations gradually merge together. Physically, high circumferential wave number means many nodal points around the circumference. Therefore, beyond a certain number of nodal points around the circumference, the effect of the stiffener configuration on the frequencies is felt less and less, and frequencies start to merge together.

Normalized vibration mode shape associated with the fundamental frequency corresponding to n =2 is plotted in Figure 13, for the single/central stiffener configuration along the axis of the cylinder. For the particular shell, the

dominant vibration mode was found to be the lateral displacement 0w .

Figure 13. Fundamental lateral displacement mode shape ( 0w ); single/central stiffener (n=2).

Figure 13 shows the cosine and sine parts of the lateral displacement, and due to the anisotropic nature of the 50o lay-up, both cosine and sine part of the lateral displacement is non-zero. Plot given in Figure 13 clearly shows the flattening effect of the stiffener which is placed at the middle of the cylinder. The actual lateral displacement mode shape can be constructed by utilizing the cosine and sine parts of the lateral displacement in the complex Fourier series representation, which is given by Eqs. (23,24), for n =2.

V. Conclusion A methodology based on numerical integration technique is presented for the vibration analysis of shear

deformable anisotropic branched shells of revolution with ring stiffeners. The main contribution of the present work is the extension of the multisegment numerical integration technique to the vibration analysis of branched anisotropic shells of revolution. The applicability of the method of solution is further extended to asymmetrically ring stiffened anisotropic branched shells of revolution, and two alternative method of analysis is presented. It has been shown that, either by the fundamental variable transformation method or by the common reference surface method, circumferential stiffeners can be placed asymmetrically with respect to the middle surface of the shell, and free vibration analysis of ring stiffened anisotropic branched shells of revolution can be performed by the semi-analytical solution method. It has also been shown that the effect of number of circumferential stiffeners added, and the stiffener configurations, such as inner or outer, on the fundamental natural frequencies is closely linked to the particular circumferential vibration mode.


22

It is deemed that present technique, besides providing an alternative solution methodology to study the modal characteristics of ring stiffened anisotropic branched shells of revolution, can be reliably used as an alternative computational tool to compare against the other solution methodologies.

Appendix The non-zero components of the transformation matrix TR are:

βcos)10,10()9,9()4,4()3,3()2,2()1,1( ====== TRTRTRTRTRTR 1)8,8()7,7()6,6()5,5( ==== TRTRTRTR

βsin)4,2()3,1( −== TRTR , βsin)2,4()1,3( == TRTR

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Laminated 3D Axisymmetric Shells: Bending, Free Vibration and Buckling,” Composite and Structures, Vol.71, 2005, pp. 273-281.

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7 Lestingi,J., and Padovan, J., “Numerical Analysis of Anisotropic Rotational Shells Subjected to Nonsymmetric Loads,” Computers and Structures, Vol. 3, 1974, pp.133-147.

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9 Heyliger, P.R., and Jilani, A., “Free Vibrations of Laminated Anisotropic Cylindrical Shells,” Journal of Engineering Mechanics, Vol. 119, No. 5: 1993, pp. 1062-1077.

10 Tan, D.Y., “Free Vibration Analysis of Shells of Revolution,” Journal of Sound and Vibration, Vol. 213, No.1, 1998, pp. 15-33.

11 Tımarcı, T., and Soldatos, K.P., “Vibrations of Angle-Ply Laminated Circular Cylindrical Shells Subjected to Different Sets of Edge Boundary Conditions,” Journal of Engineering Mathematics, Vol. 37, 2000, pp. 211-230.

12 Ruotolo, R., “A Comparison of Some Thin Shell Theories Used for the Dynamic Analysis of Stiffened Cylinders,” Journal of Sound and Vibration, Vol. 243, No.5, 2001, pp. 847-860.

13 Xiang, Y., Ma, Y.F., Kitipornchai, S., Lim, C.W., and Lau, C.W.H., “Exact Solutions for Vibration of Cylindrical Shells with Intermediate Ring Supports,” International Journal of Mechanical Sciences, Vol. 44, 2002, pp. 1907-1924.

14 Wang, R.T., and Lin, Z.X., “Vibration Analysis of Ring-Stiffened Cross-Ply Laminated Cylindrical Shells,” Journal of Sound and Vibration, Vol. 295, 2006, pp. 964-987.

15 Kayran, A., and Vinson, J.R., “The Effect of Transverse Shear Deformation on the Natural Frequencies of Layered Composite Paraboloidal Shells,” Journal of Vibration and Acoustics, Vol. 112, 1990, pp. 429-439.

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19 Soedel, W., Vibration of Shells and Plates, 3rd ed., Marcel Dekker, Dordrecht, New York, 2004. 20 Arnold, R.N., and Warburton, G.B.,“ Flexural Vibrations of the Walls of Thin Cylindrical Shells Having Freely Supported

Ends,”Proceedings of the Royal Society of London, Series A, Mathematical and Physical Sciences, Vol. 197, 1949, pp.238-256. 21 Schaeffer, H.G., MSC.Nastran Primer for Linear Static Analysis, MSC Software Corporation, Santa Ana, California 2001.

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