1

DYNAMIC ANALYSIS

OF SHELLS OF REVOLUTION

by

Timothy Akanbi Ibidapo Akeju

A thesis submitted for

the degree of Doctor of Philosophy

in the Faculty of Engineering of

the University of London

Concrete Structures and Technology,

Civil Engineering Department,

imperial College of Science and Technology,

London, May, 1972.

ABSTRACT

The free vibration problem of a shell of revolution

with arbitrary meridional configuration is formulated from

the general equations of motion of an undamped vibrating .

shell. The problem is reduced to that of any particular

shell shape by using results presented from differential

geometry of the surfaces by method of vector analysis.

The thesis deals with the application of four linear

shell theories to the dynamic problem. A brief discussion

of the outstanding differences in the underlying assumptions

of the shell theories proposed by Novozhilov(351 , Reissner(391 ,

Sanders(491 and Vlasov1491 is given. Love's First Approximation

Theory ( 2 6 ) is discussed as the basic foundation of all the four

theories.

The partial differential equations of motion are reduced

to linear differential equations by seeking the solution in

terms of displacements with sinusoidal circumferential components.

The representation is adequate because of the closed form of

the shells considered. The definition of the displacement

vector lead to the formulation of the problem in matrix notation.

Finite difference expressions are used to integrate the equations

along the meridian.

The use of the displacement vector leads to one equation

being left over at each boundary. This equation is brought into

consideration by use of difference interpolation formulas and

the whole set of equation is subsequently reduced to a system

of eigenvalue equation.

The eigenvalue problem is solved by the QR algorithm

of Francis(141 which is presented as an improvement on the

LR algorithm of Rantishauser(371 An economic method of

calculating the eigenvectors of the band matrix resulting

from the problem by utilizing an algorithm due to Martin and

Wilkinson(291 is also given. The essential features of the

Fortran program based on this work is discussed.

The work has revealed that certain shell dynamic

problems can be solved adequately by Vlasov Shallow Shell

Theory. It is shown that the theory becomes inadequate in

solving non-shallow shell problems in cases where bending

.action contributes substantially compared to membrane action

to the strain energy of the shell.

3

To

4

Bisi and Lola

ACKNOWLEDGEMENT

The author would like to thank Professor A.L.L. Baker

for the opportunity to undertake this study in the Concrete

Structures and Technology Section of the Department of Civil

Engineering at Imperial College.

The author would also like to express his profound

gratitude to his supervisor, Dr. J.Munro for his unflinching

support and encouragement throughout the duration of this work.

Many thanks are due to the author's colleagues, Mr. J.

Oliveira and Mr. D.L. Smith for their useful suggestions and

discussions on the work.

The computer program was developed simultaneously on

the University of London Computer Centre CDC 6600 computer

and the Imperial College Computer Centre CDC 6400 computer.

The author is grateful to the members of staff of both centres

for their help and co-operation.

This work was done whilst the author was the tenure of

a Commonwealth Scholarship. The author would like to convey

his gratitude to the Commonwealth Scholarship Commission in

the U.K. for the offer of the scholarship and to the Federal

Nigeria Government Scholarship Board who nominated him for

the award.

Finally the author would like to thank Miss M. Ming

for making such a success of typing the manuscript with its

difficult equations and notations.

6

C O N T E N T S

Abstract 2

Acknowledgement 5

Contents 6

Notations 8

CHAPTER 1

Introduction 12

CHAPTER 2

Geometry of Surfaces of Revolution 18

CHAPTER 3

Linear Shell Theories 30

CHAPTER 4

Equations of Motion 44

CHAPTER 5

Finite Difference Representation of Free

Vibration Equations 54

CHAPTER 6

Numerical Solution of Eigenproblem 68

CHAPTER 7

Discussion of Results 74

CHAPTER 8

Summary and General Conclusions 106

APPENDIX 1

Vlasov Theory Equations 110

APPENDIX 2

Novozhilov Theory Eouations 121

7

APPENDIX 3

Reissner Theory Equations

129

APPENDIX 4

Sanders Theory Equations 133

APPENDIX 5

The Computer Program 141

References 151

7 7 7 11 12 1 13 Unit vector bases

8

NOTATIONS

a Cylinder radius

Throat radius of a hyperboloid of revolution

A Metric coefficient of middle surface

A1 ,A2 Coefficient of First Fundamental Quadratic

Form for a surface with orthogonal parametric axes.

A11rA22 Al2 Coefficient of First Fundamental Quadratic

Form for a general surface.

b Curvature parameter of a hyperboloid of revolution.

Bll►B22 B12 COefficient of the Second Fundamental Quadratic

Form for a general surface

D Flexural stiffness

E Young's modulus of elasticity

h Shell thickness

i Station number i = o, 1,

k11 k22 k12 k21 Middle surface bending strains

K Extensional stiffness

K1 K2 Principal normal curvatures

KG Gaussian curvature

Characteristic length

Lij Linear differential operator-

m11 m22 m12 m21 Couple stress resultants

n Number of circumferential waves

nil n22 n12 n21 stress resultants

9

n Normal vector to a surface at a point

N Total number of intervals along the meridian

P Position vector of a generic point on a surface

PI P2 surface loading

qi q2 Transverse shear force resultant

r Radius of parallel circle of a surface of revolution

R1 R2 Principal radii of curvature

s Total meridian arc length

t Time

T Kinetic energy

ul u2 Displacements in the meridian and circumferentiaL

directions of the undeformed middle surface of the •

shell.

Vb Strain energy due to bending action

Vm Strain energy due to membrane action

Vt Total strain energy

X1 X2 X3 Rectangular catesian co-ordinate axes

al a2 Co-ordinate parameters in the meridian and

circumferential directions

611 622 612 621 Middle surface stresses

A Forward difference operator

V Backward difference operator

Frequency parameter = Xt 2

Kinetic energy integral

Poisson ratio

p Mass density

all 0.22 012 0.21 Middle surface stresses

Half angle subtended by generators of cone at apex

10

Bending strain energy integral

Membrane strain energy integral

Phase angle corresponding to n-th circumferential

harmonic

Natural frequency corresponding to the n-th

circumferential harmonic

NOTES

(1) An underlined upper case letter denotes a matrix.

In general all the matrices except the matrix T

in chapter 5 are (3 x 3) for the general case and

(2 x 2) for the axisymmetric vibrations.

(2) An underlined lower case letter denote a vector

e.g.

the displacement vector

the null vector

112

w

0

0

(3) Comma before a subscript denotes a partial

differentiation with respect to the variable a,

or a2

e.g. u2 ,1

@u2 @al

U112

3a2

11

(4) Dot over a symbol denotes partial differentiation

of the quantity with respect to time

e.g. . u = au

at ••

u =

(5) Dash over a symbol denotes ordinary differentiation

of the quantity with respect to al

e.g. r. = dr dal

A" = d2A dal

12

CHAPTER ONE

INTRODUCTION

1.1 General

Shells of revolution are used in civil engineering

practice in the form of cooling towers, industrial chimneys,

silos and similar structures. In the aerospace industry

they are being used amongst others as the structural elements

of launch vehicles, spacecraft and aircraft fuselage.

The failure of three cooling towers at Ferrybridge,

England, in 1965 has since stimulated the interest of civil

engineering analysts on the design of shells of revolution.

It has now been generally a.ccegted,that by far the major

form of loading on these structures are dynamic in nature.

An assessment of their frequencies and mode shapes is of

paramount importance in determining the required dynamic

characteristics for their design.

However their complex configuration coupled with

complex governing differential equations often result in

problems which are not soluble in closed forms. Therefore

analysts have taken solace in the use of numerical techniques.

A brief review of• some important contributions on the subject

will be given below and this will be followed by a definition

of the objectives of the present work.

1.2 Review of Numerical Analysis of Shells of Revolution

An abundant wealth of literature has resulted from the

application of numerical techniques to the solution of the

equations of shells of revolution. The three most often used

13

methods of solution are numerical integration, finite

difference and finite element methods. As well as in

the case of static and stability analysis, the three

numerical techniques have been used with several variants

by several people in the analysis of shells of revolution

for dynamic loads.

Martin and Scriven(281 numerically solved Timoshenko's

shell membrane equations. Their solution was inadequate to

describe the state of stress near the base because of the

existence of bending moments caused by ring beams, which the

membrane theory neglects. Another solution developed baC

Martin, Maddock and Scriven(271 to the membrane displacement

in hyperbolic shells based on Novozhilov equations of equi-

librium was also inadequate to describe the actual boundary

conditions at the lower edge.

Budiansky and Radkowski(61 reduced the equations of

Sanders' theory to four second order linear differential

equations which they eventually solved by finite difference

method. UzSoy(451 applied the method to several practical

examples and also formulated a displacement solution to the

equations of axisymmetric vibrations of shells of revolution.

Martin and Albasinyt2' proposed a method for deter-

mining the bending solution for hyperbolic shells of

revolution by solving Novozhilov's equations using finite

difference technique. Abu-Sitta(11 applied a modified

finite difference technique to solve the general Novozhilov

equations of equilibrium in terms of displacements. This was

extended to the dynamic problem by Hashishi and Abu-Sitta(191.

14

Numerous authors (5,18,22,42,51) have applied the

finite element method to solve problems of thin shells of

revolution with various features such as curved elements,

branched elements, variable thickness, asymmetric loadings

and non isothermal loadings.

Goldberg, Bogdanoff and Marcus(16) used a general-

isation of Holzer method for the free torsional vibrations

of shafts to determine the axisymmetric vibrations of short

conical shells. The method was extended by Goldberg and

Bogdanoffil " to include the nonsymmetric vibrations of a

cone. There was an inherent problems of the growth of

extraneous solutions when the method is applied to longer

shells. Kalnins(23) and Cohen") resolved this problem

by dividing the shell into short segments, integrating over

each segment and combining the solution to satisfy continuity'

requirements.

Some early treatment of vibration characteristics of

shells of revolution were based on the inextensional theory

of Rayleigh(38), a simplified theory in which the extension

of the middle surface is neglected. Prominent among such

work were the analysis of fixed-free conical shell by Strutt

(44) and the determination of natural frequency of hyperboloids

by Neal1341.

The inextensional theory predicts results which agree

very well with experimental results for some restricted

boundary conditions like the sphere-cone combination of Saunders

and othersr"). However Arnold and Warburton(31.4) Weingarten

(40) and Platusl36) have shown that the inextensional theory

arbitrary shells

Schnobrichr71.

1.3 Scope of Present Work

of revolution by Carter, Robinson and

15

alone is not very adequate for the case of a completely

rigid edge.

Zarghamee and Robinson152I used Goldenveizerisr171

asymptotic integration method to study a freely vibrating

steep spherical shell. The method was generalised to include

In the studies carried out by Hashishir191 and Carter

(71, numerical results were published for the frequencies and

mode shapes of shells of revolution obtained using different

numerical techniques to solve equations from different theories.

In this work solutions are presented for shell dynamic problems

based on different linear shell theories using the same

numerical technique. The object of which is to determine if

the assumptions on which the theories are based give rise to

any significant differences in the numerical results. Of

special interest is the fact that numerical results is being

presented for vibrating doubly curved shells solved with

Vlasov shallow shell theory equations.

Often numerical results like those mentioned above are

obtained at the expense of a large number of computations

resulting from complicated algorithms and iterations with the

logical consequence of a lot of computer time usage. Moreover

some of them can only identify a few modes of vibration and

analysts have often dismissed this inadequacy by arguing that

higher modes are insignificant. In the present work such

dismissal is not necessary as the simple numerical approach

16

used always results in the finite number of modes which

corresponds to the numbers of degree of freedom used to

represent the continous structure.

These objectives are achieved by first setting out

the geometry of the shells of revolution in Chapter Two

and particularizing the parameters for cylinders, cones

and hyperboloids of revolution. Chapter Three is a brief

discussion of the pertinent differences in the linear

theories of thin shells as credited to Novozhilov(351 ,

Reissner(39), Sanders 11+ 0) and Vlasov(46). In Chapter

Four the equations of motion of a shell of revolution with

arbitrary meridional configuration are formulated using as

basis Novozhilov's equations. The partial differential

equations are then reduced to linear differential equations

which are written concisely in operator form. Chapter Five

introduces the representation of these equations in matrix

form as a.prelude to integration by finite difference method

and a formulation of the classical eigenvalue problem. Chapter

Six contains the discussion of the algorithms used to solve

the problem defined in the previous chapter. A discussion of

the results obtained for sample problems is presented in

Chapter Seven.

In order to reduce the unwieldinesscaused by equations

in Chapter Four and Chapter Five, the elements of the matrices

have been defined in a separate appendix for each of the

theories considered in the thesis. These appendices are

numbered one to four. Appendix Five is a concise description

of the computer program based on the work.

17

The problem has been formulated for shells of

revolution clamped at the base and free at the top as

this set of boundary conditions is considered to be the

closest theoretical representation of edge conditions in

chimney stacks and cooling towers. It is however pointed

out in the thesis that the method can be used for any set

of homogeneous boundary conditions.

18

CHAPTER TWO

GEOMETRY OF SURFACES OF REVOLUTION

2.1 General

In discussing the geometry of a given surface, two

methods of approach are possible. One could use the very

precise and elegant notations of tensor analysis or

alternatively employ vectorial method. The latter method

will be used in this thesis as it proves much more simple

and direct to apply to the type of shells considered.

In what follows relevant results from differential

geometry will be summarised for a general surface. These

are presented merely to define the notations used as

systematic development and proof of the relations can be

found in the literature(12,13r43,49). These results will

be applied to a surface of revolution by choosing a part-

icular parametric set as a basis. By defining a specific

generator the results can further be applied to each type

of surface of revolution considered in the thesis.

2.2 Relevant Results from Differential Geometry

With a rectangular cartesian co-ordinate axes (X1 ,

X2 , X3 ) and unit vector basis 12 and 13 any arbitrary

point on a surface can be specified by the position vector

P (Figure 2.1) which is given by

7 „ 7 7 = Xill -I- A212 A313 (2.1)

where the components X1 , X2 and X3 are defined in terms of

the parameters (al a2) by

19

XI X1 (a1,a2)

X2 X2 (alfa2) (2.2)

x3 X3 (al,a2)

alb al t

al (2.3)

o a2 2ir (2.4)

The notation 'A'represent a vector A.

The differential change dP in P from a point A to an

infinitesmally close point B (Figure 2.1) on the surface

can be written as

dP

P,Idal + P,2da2

(2.5)

The notation ( )1i signifies partial differentiation

with respect to the parameter ai(i= 1,2). The square of the

distance between any two neighbouring points is

ds2 = A211da1 + 2Al2daida2 + A22da22 (2.6)

Equation (2.6) is the first fundamental quadratic form of

the surface with coefficients All , A22 and Al2 given by

- - Ail = 13,1 • P,1

A222 = P,2 • P,2

Al2 = A11A22 COST = P,fP,2

T is the angle between the parametric lines, it follows

that for an orthogonal parametric set.

Al2 = 0 (2.10)

The unit normal to the surface at A is given by

1,- ,1 x P,2 (2.11)

1 13,1 X P,2 I

where the notations A.B. and A x B represent the scalar

product and vector product of A and B respectively and cl

represents the absolute value of C.

Prl • n,1 ▪ P,11 . n

. n,2 + P,2 • nrli=

. n, 2 = P -,22.n

1311

2B1 2

B22 2

20

The curvatures of the normal section through an

arbitrary point A are useful tools for examining the nature

of the neighbourhood of the point. These curvatures K are

given by

Kn = II (2.12)

where II is the second fundamental quadratic form, like the

first fundamental quadratic form/ the second fundamental

quadratic form can be expressed in terms of its coefficient

B11, B22 and B12 as

II = B11 (dal)2 + 2B12dalda2 +B22(da2)2(2.13)

where

For the parametric lines to be lines of curvatures

the conidtion

B12 = o (2.17)

must be satisfied in addition to (2.10). In this case the

curvatures of the parametric lines are referred to as normal

principal curvature and are given by

1 1

B11 KI

R A2 (2.18) 11

K2 = 1 B 22 A2 (2.19) 22

The development of the equations of the theory of shells is

considerably simplified by choosing the parametric lines to

be lines of curvature.

An arbitrary choice of the four parameters (for the

case of parametric lines being lines of curvature) A11, A22

21

Kl and K2 can only define a valid surface if they satisfy

three fundamental relations in the theory of surfaces known

as Gauss-Codazzi relations. They are obtained from the

equality of the mixed second derivative of the unit vectors.

The two Codazzi relations are

K2 A11 / 2 = (K1A11)/2 (2.20a)

K1 A22, 1 = (K2A22)/1 (2.20b)

whilst the Gauss relation is

(ATI A22/1)/1 -1

+ (A22A11,2),2= GAl1A22 (2.21)

where KG KIK2 (2.22)

is Gaussian curvature of the surface.

2.3 Application of these Results to surface of Revolution

A surface of revolution is formed by rotating a plane

curve called a meridian about an axis, the axis of revolution.

The plane containing the meridian and axis of revolution is

called the meridional plane. A cross section of the surface

perpendicular to the axis is a parallel circle or latitude.

If the parameters al and a2 are chosen such that al is linear

function of the distance along a meridian and a2 is angular

distance along the parallel circle of radius "r(a1 ), the

equation of the surface can be written as

X1 = r (c41 ) cosa 2 (2.22)

X2 = r(al ) sina2 (2.23)

X3 = x3(a1) (2.24)

r(al) and x3(a1) are functions of a1 only. The position vector

of any arbitrary point on the surface (equation 2.1) is

7 '7 = r(al) cosazii r (al ) (..1)13 (2.25)

22

whilst the coefficient of the First Fundamental Quadratic

Form (2.7), (2.8) and (2.9) are

Ail = {r /(a1)}2+ {x3-(a1)}2

A2 (2.26)

M2 = {r (a1 )}2

(2.27)

A1 2 (2.28)

The notation ( )- signifies total differentiation,

with respect to the parameter al

From (2.28) it is evident that (al , a2) as defined

by (2.22), (2.23) and (2.24) is an orthogonal parametric

set.

The coefficients of the Second Fundamental Quadratic

Form (2.14), (2.15), (2.16) are as follows for a shell of

-revolution.

Bil (Ot 1 ) X3' (a1) (a1 )x3' (a1) (2 . 29 )

B22 = r (al) x3" (al) (2.30) A

B12 o (2.31)

Equations (2.28) and (2.31) show that the parametriC

set (al,a2) defined by (2.2a) --bd. (2.2+) for a shell of

revolution are lines of curvature.

The principal normal curvatures to the surface (2.18)

and (2.19) are

Ki 1 = r.(a 1 ) x3" (ct ) (cti ) x3" ) (2.32) A

K.2 1 x3' (al)

R2 A r (al )

(2.33)

The signs of the principal radii of curvature depend

on the choice of the parmeter al. If al is chosen such that

23

X3 (al >0) the principal radius of curvature R2 is positive

and as its centre of curvature is on the axis, a principal

radius of curvature may then be defined to be positive when

its centre of curvature lies on the same side of the surface

as the axis of revolution. With this definition RI will be

positive when the meridian is concave inward.

The first of the two Codazzi relations(2.20a),(2.2Ob)

is identically satisfied whilst the second reduces to

Kir' = K2''r K2r'

(2.34)

whilst the Gauss relation reduces to

-KGAr r,-

A r'A' A

(2.35)

The choice of the parameter al and consequently the

.form of the parameters r, A, Ki and K2 are summarised below

for the three surfaces - cylinder,cone and hyperboloid.

2.4 Cylinder

A circular cylinder is generated by rotating a straight

line parallel to the axis about the axis. The middle surface

of the cylinder is completely specified in space by the radius

a and the height The parameter (al, a2) can be chosen

with reference to the rectangular co-ordinate set (X1,X2,X3)

(Figure 2.3) as

Xi = a COSa2

X2 = a sina2

X3 = lal

Hence

A = 1

r = a

K1 = o

K2 - 1 a

2.5 Cone

A conical surface is generated by revolving a straight

line which is not parallel to the axis about it. The generator

and the axis should not coincide within the region defined by

(2.3). This implies that the analysis is restricted to only

truncated cone.

r > o

The middle surface of the cone .can be completely

specified by the height, the upper and lower radii or diameters.

From these values the angle of inclination of the generator

(I) with the axis (Figure 2.4) can be calculated. The parametrib

set (al,a2) is defined with respect to the axes (X1,X2,X3) as

XI = allS11-1(DCOSa2

X2 Sina2

X3 = a1lcosa2

and the geometric parameters are :

A = 1

K1

K2 = cots all

2.6 Hyperholoid of One Sheet

A hyperboloid is the surface generated by revolving a

hyperbola about the axis of revolution. The equation of the

hyperboloid can be written in terms of the rectangular cart-

esian set (X1,X2,X3) as

X12 + X22 - X32 = 1 (2.37) a` 132

24

25

where a is the throat radius and b is a curvature parameter.

A hyperboloid can be specified completely by either the

values HI and H2 (Figure 2.5) with the parameter a and b

or instead of HI and H2 the value of the parameter al at

the upper and lower end of the hyperboloid. The set

(al ra2 ) is defined by

X1 = acoShalcosa2

X2 = acoshaisina2 •

X3 = b sinhal

Hence

A = A2sinh2a1+ b2cosh2al

a coshal

K1 = - ab A3

K2 aA

gG

FIG. 2.1

ZI

A

FIG. 2.3

Z8

X. 3

FIG. 2.4

'.3

7C5

ZI

FIG. 2.5

30

CHAPTER THREE

LINEAR SHELL THEORIES

3.1 General

The several variants of shell theories available to

the analysts today are based on improved version of a set

of hypotheses first put forward by Love, commonly referred

to as Love's first-approximation and second-approximation

theories. The inclusion of the effects of transverse

shear and normal strain in the second approximation theory

constitute its basic difference from the first-approximation

theory. The theories discussed in this thesis are those

derived from Love's first-approximation theory. The theories

of Novozhilov, Reissner, Vlasov and Sanders are considered to

be representative of these theories. The main differences be-

tween the resulting equations of the first-approximation theory

are consequences of the stage of the derivation at which certain

assumptions of Love's first-approximation are incorporated.

Sanders' theory differs from the others as it removes aR in-

consistency present in the others. This being the fact that

the strains do not all vanish for small rigid body motion of

the shell except in the special case of axisymmetric loading

of shells of revolution.

In this chapter the differences in the equations of

these four theories will be discussed for a general shell

element - whose middle surface is defined by the first fund-

amental quadratic form

ds2 -- A l 2da l 2 + A22da22 (3.1)

31

and the principal radii of curvature are R1 and R2.

Prior to doing this, Love's first-approximation theory and

the implications of the basic underlying assumptions will

be discussed. Other class of shell theories resulting

from the relaxation of one or two of Love's assumptions are

mentioned.

3.2 Love's Postulates

A set of hypotheses which forms the foundation of the

Theory of Thin Elastic Shells was proposed by Love in a paper

(25) in 1888 in which he improved upon previous attempts by

Aron and Mathieu. This work which was later reproduced in

Love's Treatise on the Mathematical Theory of Elasticity(26)

was an extension to the case of thin shells of Kirchhoff's

plate hypothesis. The assumptions which form the basis of

Love's theory have been summarised in different ways by

different authors. In this thesis these assumptions will

be discussed under two broad groups, the assumptions of

linearity and shell assumptions.

Under the first group are assumptions which are not

only peculiar to shells but are also usually made in all

cases of linear structural analysis. These assumptions are

as follows :

1. small displacements

2. small deformations

3. linear constitutive relations

The assumption of small displacements makes it possible

to refer all derivations and calculations to the original

32

configuration of the structure whilst that of small deform-

ation leads to strains which are linear functions of the

displacements. Linear constitutive relations implies that

stresses are linear functions of the strains.

Group under the shell assumptions are the following:

1. The shell is thin

2. The transverse normal stress is negligible

3. Normals to the reference surface of the shell remain

normal to it and undergo no change in length during

deformation.

As no specific definition of thinness exists, it has

been suggested that the theory is only applicable to shells

whose thickness is everywhere less than one tenth of the

radius of curvature of the reference surface.

The second and third assumptions are further simpli-

fications of the constitutive relations and effectively lead

to a two dimensional elastic theory in place of the general

three dimensional theory of elasticity. The restriction to

the consideration of thin shells also justifies neglecting

the transverse normal stress. The last assumption which

implies that all the strain components in the direction of

the normal to the reference surface vanish is an extension

of the Bernoulli Euler hypothesis of beam theory to shells.

3.3 Linear Shell Theory

Shell theory based on the proceeding assumptions is

called first approximation theory for thin shells or first

order linear theory of thin shells. This differs from the

second approximation theory of thin shells or higher order

33

linear theory in which the assumption of small deflection

is preserved but as mentioned above the assumption neglect-

ing transverse normal stress is relaxed. The relaxation of

the assumption of small deflection leads to the non-linear

theory of shell. The non-linearity can be present in the

strain displacement relations, in the constitutive relations

or both. Very comprehensive discussions of shell‘ theories

under these heading abound in the literaturer32,33,2".

The first approximation theory as presented by many

people including Novozhilov, Reissner, Vlasov and Sanders

produces different equations. This thesis is concerned-with

application of these four theories to the dynamic analysis of

shells of revolution. Since all of the assumptions of Love's

first approximation theory are preserved, the differences in

the theory are in the expression for the strain displacement

relations, the equations of equilibrium or motion and the

boundary conditions. In the remainder of this chapter, these

differences will be discussed under their respective headings.

3.4 Novozhilov's Theory

By a systematic application of the assumptions of Love's

first approximation theory to the state of strain in a small

element of the shell, Novozhilov(35I concluded that the deform-

ation of the element can be completely characterized by specify-

ing six parameters, which have to do with the variations in the

dimensions and the distortions of the element. These parameters

otherwise known as the strain displacement relations of the

middle surface are as follows beginning with the linear strain

in the direction of al

k22 = 1 W,2 112 N A2,1 J (Ell+ RI) A l R1 A2 (K2 R2, 2 A1A2 (3.6)

34

C11 A1 ,2112 Al AlA2 R I

The linear strain in the direction of a2

6 22 22.'2 + A2,1111 A2 A1A2 R2

The membrane shear strain

(3.2)

(3.3)

2512 A 2 (t),1

Al (3a1) A l A2 K1,2

(3.4)

The change in curvature of the middle surface in the

direction of al

k11 = 1 W,1+ Ul. A1,2 01,2 4' 112 A l (2k1 RI ,1 A1A2 A2 R2 i (3.5)

The change in curvature of the middle surface in the

direction of a2

The twist of the middle surface

k12 = k21 = 1 A1,2W,1 A2,1W,2) A1A2 Al A2

1 (u1,2 A1,2u11 + (u2,1 - A2,1112 )(3.7) R A2 A1A2 'I R2 ‘T1 A1A2

The introduction of the definition of the stress

resultants in place of the stresses is made possible by

the application of Love's third shell assumption. A

consideration of the equilibrium of the shell element in

terms of these generalised stresses gives rise to the

following equations of equilibrium for this theory :

35

(A2n1 I f 1 + (A1n21),2 + A1,2n12- A2,1n22

- AlA2q1 + A1A2p1 = 0 (3.8) RI

(A2n12),1 + (2k1n22) r 2

A2',11121- Alf2n11

- A1A2q2 R2

(A2q1),1

A1A2p2 = o (3.9)

+ (259q2),2 + A1A2 nll + n22)_ RI R2

+ A1A2p = 0 (3.10)

(A2m11)11 + (A1M21),2 Alr2M12 A2r1M22

- A1A2(11 = o (3.11)

(A2m1.2),1 + (A1m22),2 + A2,1M21 A1,2m11

- A1A2q2 = 0 (3.12)

nI2 n21 M12 + M21 = o (3.13) Rl R2

The basic physical assumptions, which lead to the

definition of the stress resultants and stress couples in

place of the stresses give five independent physical boundary

conditions at each edge. However the conventional

assumptions of Love lead to the above equations of equilibrium

which is an eight order system of equations requiring the

specification of only four conditions at each edge for a unique

solution of the system. The justification of the reduction of

the five boundary conditions to four at each edge by mechanical

considerations is discussed by Novozhilov(351. In this thesis

only the boundary conditions for a free edge and for a fixed

edge are of interest. These are given by Novozhilov's theory

as follows

a free edge along al = constant.

36

11n = 0

n12 7-712

o (3.14)

q1 ,12,2— 0

m 0 11

b. a fixed edge along al = constant

0

u2 0 (3.15)

w = o

_!,1 u l A l

RI 0

3.5 Reissner's Theory

Eric Reissner in oner391 of his many papers on the

subject of shell theory maintained that simpler and more

direct set of formulas for the strain displacement relations

can be derived by beginning with the strain components for

curvillinear orthogonal co-ordinates and introducing into

them-the assumption that the normal to the undeformed middle

surface is deformed into the normal to the deformed middle

surface. This is followed by the introduction of the

assumption that the shell is thin. Since Reissner's six

strain displacement relations differs from those of Novozhilov

only in the expression for the twist of the middle surface,

only that expression will be given here as

1A4IR1)f2.

Al (A

fl k12 k21 = A2 (W,2 U2 )

Al A22 (3.16) A2R2 0 A2 Al2

In like manner Reissner derived a system of equations

of equilibrium which written in the notation of this thesis

are the same as those of Novozhilov. Having used a vectorial

37

approach in his derivation he did not arrive at the boundary

conditions. However Krausr241 using Reissner's version of

Love's first approximation theory and a variational approach

derived the boundary conditions which being the same as those

of Novozhilov will not be repeated in this section.

3.6 Sanders' Theory

J.L. Sanders showed that shell theory equations

like those of Novozhilov and Reissner derived from Love's

first approximation theory give strains which, except for

the special case of axisymmetric loading of shells of

revolution, do not all vanish for small rigid body motions

of the shell. Sanders. reduced the number of strain-displace-

ment relations from ten to eight by setting the transverse

shear strain to zero. From the form of the rotation about

the normal in terms of the displacements he showed that the

two membrane shear strains of his theory are equal. A further

reduction of the strain quantities to six is achieved by

defining an average membrane shear stress resultant, average

twist couple resultant and an average twist. The following

are Sanders' six strain displacement relations beginning with

the linear strain in the direction of al

• u1, 1 ▪ A1,2 U2 - W E11 A 1 A IA2 R1 (3.17)

The linear strain in the direction of a2

• U2r 2 ▪ A2,1 Ul W £22 A 2 A lA 2 R 2 (3.18)

The membrane shear strain

• U2f1 • 111,2 Al Ul A2,1 U2 2e12 (3.19) Al A2 A1A2 A1A2

38

The change of curvature of the middle surface in the direction

of al

Al 'A l RI ,1 + A1,2 ,2 U2) kll = 1 tH,14- al 0.1

A2 R2 I )

A1A2

The change of curvature of the middle surface in the direction

of a2

k22 = 1 Wr2+ U2 A2,1 (Al (— A2 A2 ,2 AlA2 Al RI

(3.21)

The twist of the middle surface

k12 = 1 [A2 (/12 U2) Al 12.7_,1 12j) A1,2[Wrl AIA2 A2 T2,1 ‘Al R1 ,2 ‘k1 ,- R1

— A2,1 ( d12 U2) -I- 1/2 ) 1 — 1 (A2U2) (A1U2) ,2 }

A2 R2 (3.22)

Neglecting the quantity - 1/R2) (m12 - m21)

in comparison to z (n12 + n21) in the expression for the

virtual change of energy he arrived at a set of conditions

of equilibrium which does not involve the conventional sixth

equation. The equation is nevertheless usually suppressed

on the basis of the equality of shear stress resultants and

the definition of the twist couples. The following are

Sanders' five conditions of equilibrium.

(A2n3.1) ,1

AIA2q1

1

+

+ (A0-112

2

A2 2

(A11122

R2

(="-

),2

),2

+

1 Ri

+

1 R2

A1lin12

)m12.1

A —2,0112

7

)M12]sl

,2

A2,11122

+ A1A2p1

A1,2nil

+ A1A2P2

= o

= 0

(3.23)

(3.24)

121.

(A2n1 2 )

- A1A2q2 R2

(3.20)

39

(A2(10,1 (A s_1(42) t2 (nil + n22 AJA2 11

+ AlA2Pz (3.25)

(A2m10,1

(A1M12)/2 A1/2M12 A2/1M22

A1A2q1 = o (3.26)

(A2M12)/1

(A1M22)/2 A2/1M12 — A1/211111

AI A2q2 (3.27)

The same virtual change of energy yield the

following boundary conditions. For the edge al = constant

free

nli mll = o R1

nI2 + 1 - 3 N m12 = 0 '2R1 7R2)

ql M12/2 = 0

A2

mil = 0

For the edge al = constant fixed

ul = 0

U2 = 0

w = 0

Wt1 0

(3.28)

(3.29)

3.7 Vlasov's Theory

In his very early attempt at the construction of a

theory of shells on the basis of the hypothesis that normals

to the undeformed middle surface remain normal to the middle

40

surface, Aron disregarded terms depending on the tangential

displacements ul and u2 in the formula for the change of

curvature and twist. On purely intuitive grounds it is

obvious that the displacement wplays a big role comparatively

on this terms and led to Aron's error being accepted as a

simplification. Vlasov (46r47) obtained the same set of

results on the basis of the, argument that the terms h 2/12R1

h 2/12R2 and their derivative are of second order values for

the displacement ul and U2 for thin shells with the relative

thickness h/Rmin less than one thirtieth. The theory has

been used in the analysis of shallow shells and in this-respect

notably among others by Donnell (10,11) in the stability of

cylinderical shells, Jenkins (21) in the static analysis of

cylinderical shells and Munro (30) in the analysis of thin

shells. The work of Vlasov referred to earlier is considered

to be adequately representative of this group and the follow-

ing are the strain displacement relations of the theory

The linear strain in the direction of al

Ell • UI,I ▪ A 1 ,2 U2 - W

(3.30)

Al A1A2

The linear strain in the direction of a 2

E22 • U2,2 + A2,1 ul W

A2 A1A2 R2 (3.31)

The membrane shear strain

2E12 • A2 ( u2 ) Al A2 ,1

+ Al A2 (

11 ) A ' 1 2

(3.32)

A1A2 ( nil + n22) Ri R2

• AlA2Pz = 0 (3.38)

(A2q1),1 + (A1(12),2

41

The change in curvature of the middle surface in the

direction of al

kll = 1 W 1) • A112 (W/2)

Al Al ,1 A1A2 A2 (3.33)

The change in curvature of the middle surface in the

direction of a 2

k22 1 (E/2) ▪ A2,1 Wr 1) A2 A2 ,2 A1A2 Al

The twist of the middle surface

(3.34)

k12 = k21-2 (W1 12 A1A2

- A1r2wr1 Al

A_2, 1W/2) A2

(3.35)

With the simplifications discussed above the equations of

equilibrium reduce to the following

(A2n1 1 ),1 + (A1n12),2 + A1/2n12

A2/1n22 A1A2p1 = o (3.36)

(A2ni2),1 + (A1r122),2+ A2,1n21

- A1,2n11 + A1A2p2 = 0 (3.37)

(A2M11)/1 • (A1M21)/2 ▪ A1r2M12

A2r1M22 - AIA.2q1 = o (3.39)

(A2M12)/1 ▪ (A111122),1 ▪ A2r1M12

A1r2M11

AIA2q2 = o (3.40)

n12 n21 = o (3.41)

42

The boundary conditions for a free edge a1 = constant are

rill

ni 2

0

0

qi + M12,2 r

ml

(3.42)

whilst those for a fixed edge al = constants are

Ul

U2

w

- w,1 - ul Al RI

0

0

0

0

(3.43)

3.8 Closure

Emphasis has been placed in this chapter on the

outstanding differences in the equations of four linear

shell theories by comparing their strain displacement

relations, equations of equilibrium and boundary conditions.

It is to be noted that the strain displacement relation

terms are only meaningful by specifying an appropriate

definition of a set of constitutive relations. The consti-

tutive relations employed in this thesis are the same as

those of Love's first approximation for a homogeneous, isotropic

elastic material. This relations are given in chapter 4.

The specification of the constitutive relation along

with the above equations completely defines the problems on

the theorectical side and poses it for solution by analytical

and numerical procedures. This is the subject of the next

few chapters.

43

The existence of other linear theories of shell

different from those considered above need being mentioned.

However a chronicle of these is not intended here. Good

reviews of such theories have been produced by Naghdi(32'331

and Hildebrand et a1.(201

44

CHAPTER FOUR

EQUATIONS OF MOTION

4.1 General

The equations of motion of a vibrating shell of

revolution will be derived in this chapter. The equations

of equilibrium in terms of stress resultants for a general

shell element have been presented in the previous chapter.

For purposes of completeness this stress resultants are

defined in the present chapter. The equations of motion

for a shell of revolution are obtained from the equations

of equilibrium by adding the appropriate dynamic terms and

using the geometric parameters given for the surface of

revolution in chapter two. The equations of motion are

further expressed in terms of the displacement components

by using the constitutive relations for the shell material.

The purpose of the present chapter is to illustrate

the method of derivation of the equations and hence the

strain - displacement relations are those appropriate to one

of the theories considered. The procedure can of course be

repeated for any of the other three theories.

4.2 Definition of Stress Resultants

Figure 4.1 shows the stresses acting on a differential

element of the shell that is formed by the surface al =

constant, a2 = constant and a3 = ± h/2. In thin shell theory

it is convenient to replace the usual consideration of stresses

by equivalent stress resultants and stress couples, which are

ml

45

obtained by integrating the stress distributions through

the thickness of the shell. In effect these stress

resultants are forces per unit length of the middle surface

and their consideration reduces the problem to one of two

dimensions along the co-ordinate lines. Figure 4.2 and

4.3 show the positive sense of the stress resultant and

stress couples respectively. The stress resultant and

stress couples are defined from the stresses as

nu

n22

n1 2

n21

q h/2

Gil

-h/2

h/2

622

1 -h/2

r111/2

al2

-h/2

h/2

0.21

-11/2.

(1

(1

(1

(1

- z — R2

- z RI

z R2

-

)

)

dz

dz

dz

dz

(4.1-)

(4.2)

(4.3)

(4.4)

(1 R2 ) zdz (4.5)

-h/2

46

h/2

M22 a 22 (1 z

' R1 zdz (4.6)

-h/2

rh/2

(1 z )

R.2

m12

612 zdz (4.7)

\J-h/2

h/2

m21 1 - z a21 zdz

v -h/2

(4.8)

4.3 Constitutive Relations

The appropriate set of stress-strain relations depends

on the mechanical properties of the material of the shell.

This thesis is restricted to the special case of an isotropic

homogeneous elastic material. The stress strain relation

is the same as those of Love's first approximation given for

a two dimensional body as

E (Eli + ve22) (177777) (4.9)

0- 22 (1 - v• 2-

) (e22 vEll)

(4.10)

GI 2

(1 -v) £12

(1 - v 2 )

C521

E (1 -v) 621

(1 - v2 )

47

The constitutive relations in terms of stress result-

ants are obtained by substituting equations .(4.9), to (4.12)

inclusive in equations (4.1) to (4.8) which after consider-

ation of thinness and evaluation of the appropriate definite

integerals reduce to the following equations:

nil = K (E11 + vE22) (4.13)

n22 = K (E22 + veil) (4.14)

nit = K (1 - V) E12 (4.15)

n2I . K (1 - v) E21 (4.16)

mil = D (k11 + vk22) (4.17)

M22 = D (k22 + vkl1) (4.18)

mI2 = D (1 - v) k12 (4.19)

M21 = D (1 - v) k21 (4.20)

where the extensional and flexural-stiffnesses are given

respectively by

K Eh (1 - v) (4.21)

D - Eh3 12(1 - v2)

(4.22)

4.4 Equationsof Motion in terms of stress resultant

The equations of equilibrium of an element of a given

shell surface have been given for the four linear theories

of interest in Chapter 3. The equations of motion can be

obtained from these equations by replacing the body forces

and the static loads by the dynamic loads.

48

Using Novozhilov's theory relations and

including only in-plane inertia terms, the equation of motion

in terms of stress-resultants are given for a shell of

revolution with the geometric parameters defined in Chapter

2 by

(rnii),1 + An21,2 r _ n22 - Arql - Arphia l Ri

+ ArP1 = o (4.23)

(rni2),1 + An22,2 + r'n21 - Arq2 - Arph112 R2

+ ArP 2 = o (4.24)

(Aqi), (rq2),1 Ar nil a22 ) R1 R2

- Arph* + ArPz :.--- o (4.25)

(rmil),1 + (Am12 ),2 - r. m22 - Arql = o (4.26)

(rm12),1 + (Am22 ),2 + r'm12 - Arq 2 = o (4.27)

where the notation 2 = 3 2Z 5T2 (4.28)

The last equation is identically satisfied by

substitution of the expressions for the stress resultants -

and the remaining five equations can be reduced to the

following three equations by eliminating qi and q2 :

*(rnii),1 + An21,2 - r.r - -22 - 1 - (rmli),1 + Am21,2 RI

+ r, m22 - Arph'il l + ArP1 = o (4.29)

(rni 2) rl + An22,2 + r-.1121 - 1 - (rm12),1 + AM22,2 R2

4- . r M2 1 - ArphZ12 + ArP2 = o (4.30)

- r1m 22(11+ l(rm12),1

Ar ,r1 1 n22 RI R2

71 + Am22,2

(rm11),1 Am21,1

+ r m2I

49

Arp, = o (4.31) z

4.5 Equations of motion in terms of Displacement

The three equations above for any of the linear shell

theories considered in this thesis can be reduced to three

equations in terms of the displacement ul , u2 and w by

substitution of the appropriate stress-resultant-displacement

relations in the equations. Symbolically the three equations

can be written as

Lii{ul} ▪ 1,12{u2} • 1,13{w} Ar(1 - v2)(Pi- Eh •

= o (4.32)

L21{111} L22{U2} • 1.23{w} + Ar(l - v2)(P2- phil2) Eh

= o (4.33)

L 31{111} + L32{112} • L33{w} + Ar(1 - v2)(P2- phci ) Eh

o (4.34)

where Lij (i, j = 1,2,3) are linear functions of partial

differential operators in the variable al and a2. These

operators are defined for each theory in Appendix One to

Appendix Four.

4.6 Closure

The procedure for deriving the equations of motion

in terms of displacements for a general shell of revolution

has been illustrated with the equations of Novozhilov's

theory. By making use of the expressions for the definition

of the stress-resultants and the constitutive relations given

above, the equivalent equations for the other theories

discussed in chapter three can he derived.

The differential operators of section 4.5 are defined

for each of the theories in the Appendices without repeating

the steps leading to their derivation. The operators for

Novozhilov's theory are derived as those of Vlasov's theory

plus the required additional terms to illustrate the extent

of the difference between both theories. The operators for

Reissner's and Sanders' theories are defined from those of

Novozhilov's theory to provide comparison in like manner

between them.

50

51.

SIGN CONVENTION FOR STRESSES

FIG. 4.1

52

SIGN CONVENTION FOR STRESS RESULTANTS

FIG. 4.2

55

SIGN CONVENTION FOR STRESS COUPLES

FIG. 4.3

54

CHAPTER FIVE

FINITE DIFFERENCE REPRESENTATION OF FREE VIBRATION EQUATIONS

5.1 Introduction

In the present chapter the problems of free vibration

of a shell of revolution will be formulated from the general

dynamic problem defined in the last chapter by equating the

external forces on the shell to zero. The system of partial

differential equations will be reduced to a system of ordinary

differential equations by assuming displacements with sinusoidal

circumferential components. These equations are then integrated

by use of finite difference expressions.

Whilst the definition of the displacement vector is

instrumental in the development of the matrix equations which

lead to the eigenvalue problem, its use in expressing the

boundary conditions create an extra equation at each edge which

has to be brought into the formulation by providing supplement

equations by use of interpolation formulas. The application

of central difference expressions involves defining fictitous

points beyond the edges. Displacements at these points are

eliminated before the final system of equations is obtained.

5.2 Matrix Governing Equations

The differential equations for a freely vibrating shell

of revolution are determined from the equations of motion

(4.32, 4.33, 4.34) by setting the external load to zero given

equations of the form

U2

- n w

55

••

LiI{Ul} + L1 2fu21 + L13{w} - rphui = (5.1)

•• L21{111} + 1,22{u2} + L23{w} - rphu2 = (5.2)

•• L31{u1} + 1,32{u2} + L33{w} - rphw = (5.3)

As the shells are closed in the circumferential

direction, the solution of the equations (5.1), (5.2) and

(5.3)may besought in the following series form

ul cosna2cos( wnt - fin)

u2 = .T111 sinna2cos( wnt - fin)

(5.4)

cosna2cos( wnt n)

Term by term substitution of the set of equationd

(5.4) in the free vibration equations (5.1), (5.2) and

(5.3) lead to equations which can be written compactly in

matrix form for any harmonic n as

+ B V,111 + C V,ii + D v,1 + E v

Av

(5.5)

where v is a displacement vector defined by

n Ui

(5.6) V =

A, B, C, D and E are (3 x 3) matrices defined in Appendix One

through to Appendix Four. Similarly for the boundary conditions

the first three equations of the set (2.14) can be written as

F V,111 + G v,Il + H v I K v o (5.7)

F, G, H and K are defined in the Appendices and o is a null

vector.

The first three equations for the other set of boundary

conditions (2.15) can be written as

v = o (5.8)

5.3 Finite Difference and Interpolation Formulas

Since the highest order of governing equations (5.5),

(5.7) and (5.8) is four, the accuracy of finite difference

expressions used in this thesis is correspondingly four.

For a point (i)thederivativesofavariablez.in difference

form up to and including the fourth order are given by the

following difference formulasr"

zi

Z. 1

=

1 (zi-z 12(Sal -

1 (-z1. -, A + 12(da l ) 2

-z. 2)

1 + 7(6a1)3 1-2

1 (z. - 1-2 (-Ea 1 )

1-1(-z.

8zi- +

16zi-1 -

2z. - 221.

4z.

8z1 i+1

30zi

+1

- z i+2

+ 16zi+1

(5.9)

z. ) 1-r2

1+1 ) 11-2

where 6a is the interval between any two consecutive stations 1

and z. is the value of the variable z at station i.

Newton's forward difference and backward difference

formulas are as follows(311

z (x + py) = z(x) + pAz(x) + 1/2 p(p-1)A2z(x) +.•••

(5.10)

56

57

z(x + py) = z(x) + pVz(x) + 1/2p(p+1)V 2z(x) +.,..,.

(5.11)

where A and V are forward and backward difference operators

respectively.

The value of the variable z at the fictitious point

(-2) , z--2 can be expressed in terms of the value of the same

variable at the next fictitious point (-1) by making

x = -y

in (5.10) and also the variable zn+2 in terms of Zn+1 by

making

x =

1

in (5.11) resulting respectively in the following

z-2 = (1 - A + A2 - A3 + A4 - A5 + ...) z-1

(5.12)

zn+2= (1 + V + V2 + V3 + V4 + V5 ...) zn+1

(5.13)

Substituting the expressions for differences up to

and including the fourth order in (5.12) and (5.13) give

the following interpolation formulas:

z 5z - 10z + 10z - 5z + z (5.14) -2 0 -11 2 3

zn+2 =

5z n+1 n n-1

- 10z + 10z - 5z + z (5.15) n-2 n-3

58

5.4 Representation of Equation in Difference Forms

By substituting the appropriate difference expressions

(5.9) for derivatives of the vector v in equation of motion

(5.5) for the station (i), the following equation is obtained.

AX v. + Bx.v. + Cx.v. + DX v.

- 1-1-2 - 1 -1-1 - 1-1 - 1-11

where

E .v., = Av. - -1 (5.16)

Xi A i = 1 Ai - 1 Bi - 1 Ci + 1 Di (75-air4 2(6ai)3 ff(65-4-0 2 12(óC,1) (5.17)

Xi E = 1 Ai + 1 Di - 1 Ci - 1 Di — TO7i) 4 2 (Ta 1 ) 3 12 Oa i )2 12 (6a i) (5.21)

Similarly equation (5.7) can be written for a free

boundary in difference form as

F,xv. + G.v. + H.v. + .v.,

-2 -1-2 —1-1-1 K-1 -1-t-1

+ J.Xv., = 0 (5.22) -2 -1-T-2

where

F.X - 1 Fi - 1 Gi + 1 Hi

-1 (6c71 ) 3 2 (6Cii ) 2 1260l (5.23)

Bx = -4 Ai + Bi + 4 Ci - 2 Di (Tai) 4 (Tai) 3 3(Tai) 2 3(Sal)

Cxi = 6 Ai - 5 Ci + Ei (Sa l)4 7(6070 2

Dix = -4 Ai - Bi + 4 Ci + 2 Di (Tair (Ecti) 3 3(Sal) 2 -3(Sal)

59

G. = Fi + 4 Gi - 2 Hi -a (6a1)3 3(Sal)2 36a1

H. = - 5 Gi + Ki -a 2-(6.-ct-i) 2 -

K. = - Fi + 4 Gi + 2 Hi 1 (Ta1)3 3(601,1)2 Hai

J.x = - 1 Fi - 1 Gi 1 Hi 2(6a1)3 12(dal )2 126a1

(5.24)

(5.25)

(5.26)

(5.27)

Since the matrix equations (5.7) and (5.8) incorporates

three each of the four conditions at the boundary, it is

necessary to supplement these equations by equations which 1

will involve the fourth of either of these conditions. Such

equations can be obtained by using the interpolations formulas

(5.14) and (5.15) in the following manner. For the free

boundary station (o) the fourth equation of the system of

equations (3.14) is written in finite difference form as the

third component of a matrix equation- whose other two components

are obtained using the interpolation formula (5.14) for ul

and u2 respectively. The same procedure is used to obtain

a matrix equation for the -Clamped condition at boundary (n)

involving the use of the interpolation formula (5.15) and the

fourth equation of the system of equations (3.15). The result-

ing matrix equations from these operations are as follows

Lxv + MXv + Nxv + OXv + Pxv + QXv = o

-0 --2 -0 --1 -0-0 -0-1 -0-2 -0-3 - (5.28)

Rxv Sxv + Txv + 1.1/ 1 + Vxv wxv

-n-n-3 + -n-n-1 -n-n -n-n+ 1 -n-n+ 2 = o

(5.29)

60

The elements of the (3x3) matrices in the equation

(5.28) and (5.29) are defined in Appendix One through to

Appendix Four.

5.5 Formulation of the Matrix Eigenproblem

The problems of free vibrations of a shell of

revolution is defined for n + 1 finite difference stations

numbered o, 1, 2...n by application of the following equations

(i) (5.22) at boundary station (o) ;

(ii) (5.28) at boundary station (o) ;

(iii) (5.16) at stations o to n inclusive ;

(iv) (5.8 ) at boundary station (n) ;

(v) (5.29) at boundary station (n)'4

The problem thus consists of the following n + 5

system of matrix equations

F xv -o --2

LXv

A.v. -1-1-2

Rxv -n-n-3

+

+

+

+

G xv + -0

Mxv + -o--1

B.v. +

( i = o,

Xn

Sxv + -n-n-2

H xv -0 -o

N xv -0 -o

C.v. -1-1

1, 2, ...

= o

T xv -n -n-1

+ K xv -o -1

+ 0 xv -o -1

+ D.v. -1-1+1

n)

+ Uxv -n-n

+ J xv -0 -2

+ P xv -o -2

+ Q xv -0 -3

+ E.v. -1-1+2

+ V xv -n -n+1

+ w v —n—n+2

=

=

=

=

o

o

Xv. -1

o

(5.22)

(5.28)

(5.16)

(5.8)

(5.29)

61

The system contains four displacement vectors v_2,

v-1 , 7- -

3 n+1

and vn+2 introduced at fictitious points beyond

the boundaries of the shell to facilitate the use of

interpolation formulas. The elimination of these vectors

constitute the subject matter of the next subsections.

Equation (5.8) is used in eliminating the null vector v

from the system.

5.5.1 Elimination of v-2 and v-1

The vectors v-2 andv-1 have non-zero coefficients

in the matrix equations (5.22), (5.28) and equation (5.16)

for i = o and 1. The equations are repeated below for clarity

F xv+ G xv + Hxv + KXv + X • o (5.22) -o -o --I -0-0 -0-1 -1.) -2

Lxv + M Xv...1 + N Xv + OXv + PXv -0 --2 - -o -o --0-1 .

+ Q 0 xv- • o (5.23) - -3

Axv + Bxv, + Cxv + Dxv + E xv = Xv (5.16a) -o-- 2 --0--1 -0-0 -0-1 -0 -2 -0

Axv + Bxv + Cxv + Dxv + Exv = AIv (5.16b) -1-o -1-1 -1-3 - .

The first three equations can be reduced into two in

which the vector v-2 has been eliminated by adding to equation

(5.23) the equation produced by premultiplying equation (5.22)

by-LDF0x)-1 and by adding to equation (5.16a) the equation

produced by premultiplying equation (5.22) by -A(F ox)-1 o

After this elimination the four boundary equations reduce to

the following three equations

+'

+

1\1"."'v —0-0

C"v

+

+

—0-1

IYAr

+

+

—0-2

E'Ar —0-0 —0-1 —0-2

B'v

+ Q""ci 0 —0-3

= ,xy

62

Axv- + B

xv + C

xv + D

xv

—1--1 —1-0 —1-1 —1-2

where

Mx.— Lx (F —1X Gx —o —0 —o —o —0

• Nx -1 Lx(Fx) Hx

—o —0 —0 —o —o

• 2o Lx(Fx) —1 Kx

—o —o —o —o

— P" Lx(Fx) 1 jx —o • —0 —o —o —o

-o Qx -o

Bx B— . — Ax(Fx) —1 Gx

—0 —o —o —0 —o

x = C -1 C" — Ax Hx —o —o —o —0 —o

x = D -1 D— — Ax(Fx) Kx

—o

—o —o —o —o

Ex = E -1 E— — Ax(Fx) jx —o —0 —o —o —o

+ Exv = (5.16b)

(5.32)

(5.33).

(5.34)

(5.35)

(5.36)

(5.37)

(5.38)

(5.39)

(5.40)

Similarly the vector v_ 1 can be eliminated from equations

(5.30), (5.31) and (5.16b) by adding to (5.31) and (5.16b) the

resulting equations from premultication of equation (5.30) by

-B(W)-1 :and -Ax(W) 1 -o -o -1 -o follows :-

The final two equations are as

C/v + D"v -0-0 -0-1

B"v + C/V -1-0

-1-1

+ E/v + F"v = Xv -0-2 -0-3 -0

+ D/v + E"v -1-2 -1-3 = Av

-1

63

where

C" = C" - B"(M")-1N" -o -0 -o -o -o

D" = D" - B"(M")-10"." -0 -o -o o -o

E" = E" - B-(M1-1P" -0 -0 -o o -o

F = 0 o -o

x = B 13" - AX (M1-1N.. -1 -1 -1 -o -0

x = C C" - Ax (M-1 -10" 7-1 -1 -1 -o -0

DX = D D" - Ax (M") -1P" -1 -1 -1 -o -o

E" - = Ex Ax (M") -IQ'''. -1 -1 -1 -o -o

(5.43)

(5.44)

(5.45)

(5.1(6)

(5.47)

(5.48)

(5.49)

(5.50)

5.5.2 Elimination of v and v n+2 -fl+1

In the system of equations defining the problem, the

coefficients of the vectors v r1+1 and vn+2 can have non zero

values in equation (5.16) for i = n-1 and n and in equation

(5.29). These equations with vn eliminated are as follows

Ax v + Bx v + Cx EX + E v v -fl-1-n-3 -n-1-n-2 -n-1--n-1 -n-i-n+1

= Xv -n-1 (5.16c)

64

Axv +

xv + Dxv + E

xv = o (5.16d) -n-n-2 -n-n-1 -n-n+1 -n-n+2

Rxv S

xv Txv + V

xv -n-n-9 -n -n-2 -n-n-1 -n-n+1

+ Wxv = o (5.29) n-n+2

The elimination of 171n+2 is carried out by adding to

(5.14 the equation obtained from the premultiplication of

equation (5.29) by -E(W);1.) 1 giving the following equation

R"-k7 + A .'V B'cr D^ir = o (5.51) -n-n-3 -n-n-2 -n-n-1 -n-n+1

where

R" -n

-n

-n

D" -n

= - Ex(Wx)-1Rx -n -n -n

= Ax - Extwxlisx

-n -n \-n1 -n

Bx - Ex(Wx)-n(

= -n -n -n -n

Dx = - Ex(Wx)-1 Vx -n -n -n -n

(5.52)

(5.53)

(5.54)

(5.55)

n+1 can then be eliminated from equations (5.51) and -

(5.16c) by adding to the latter the result of premultiplying

the former by -En---1 (D-1 1 to obtain n

A' v + 13' v + v = Av (5.56) -n -1 -n -3 -n-r-n -2 -n-1-n -2 -n-1

where

Ex A" (D -n-1 -n-1 -n -n (5.58)

65

x Ex B' = B - (D "1-1A" (5.59) -n-1 -n-1 -n-1 -n -n

C" = C x - Ex " (D) 1Bit (5.60) -n-1 -n-1 -n-1 -n -n

5.5.3 Eigenvalue Equation

The elimination of vectors at fictitious points beyond

the boundaries reduce the original system of n+5 equation to

the following n equations

C"v • D"v + E"v + F'v = Xv -o-o -0-1 -0-2 -0-3 -o (5.41)

B"v -1-0 + C"v -1

+ D'v -1-2 + E"v -1-3

= Xv

A.v. • B.v. + C.v. + D.v. -1-1-2 -1-1-1 -1-1 -1-1+1

+ E.v., = Xv. -a

(i = 1,2,3 - n-3) Ax v + Bx v + Cx -n-2-n-4 -n-2-n-3 _n-2-n-2

(5.16)

A" v -n -1 -n -3 + B"v

-n -n -2

D Vin- + Y-n-2 -n-1 -n-1 (5.16e)

+ Cv = A--n-1 (5.56)

This system constitute an eigenvalue equation

T z Az (5.61)

where T is (3n x 3n) band matrix of the coefficients in the

form

T

66

C' E' -o -o -o —0

Bi Cl Di El

A2 nx C2 D2 E2

o2 x — — — —

X X X D3 .,,X

A3 B3 C3 o3 _ — — — —

AX BX DX -n-2 -n-2 -n-2 -n-2 0

A". -n-1 -n-1 -n-1

whose eigenvalue is (5.62)

A pw2(1 - v2) E (5.63)

(5.64)

V2

V3

• V n-2

n-i

5.6 Closure

Expression for the displacement components with

sinusoidal circumferential modes have been used to reduce the

partial differential equations of motion to a linear differential

set. The equations were then integrated in the meridional

direction by fourth order central finite difference expressions.

and eigenvector is

z V

v i

67

The boundary conditions at the edges of the shell

were introduced by using Newton's forward and backward

interpolation formulas which necessitated defining fictious

vectors beyond the boundaries. It must be noted that the

procedure described in this chapter for eliminating the vectors

at these points are peculiar to free-clamped boundary conditions.

However the method can be used for any set of homogeneous

boundary conditions. U2soy(451 has discussed the method for

a variety of homogeneous boundary conditions for the special

case of axisymmetric vibrations.

68

CHAPTER SIX

NUMERICAL SOLUTION OF EIGENPROBLEM

6.1 General

The fundamental eigenproblem consists of the deter-

mination of those values of A which make the set of n

homogeneous linear equations

A v

in n unknown to have a non-trivial solution. That this

was an important classical problem has been shown by the

tremendous attention given to it by mathematicians

now and in the past. A detailed treatment of some numerical

aspect of the subject can be found in Wilkinsonr5". What

is intended here is a brief summary of some numerical procedures

on which the computer programs used in the solution of the

problem are based.

The main program used is an application of the QR

algorithm of Francis(141 which itself is an improvement on the

LR algorithm of Rautishauserr371. Both algorithms are described

below.

Conventional QR and LR algorithm programs do not often

take cognisance of the band form in narrow band matrices_ and

can therefore involve the usage of a large computer storage when

a matrix of large order is to be solved. Martin and Wilkinson

(29) developed an algorithm which calculates the eigenvectors

of band matrices from a compact storage of the non-zero elements

of the matrix. However the program suffers from the limitation

69

of not being able to determine an eigenvalue for the band

matrix. The user is required to guess an eigenvalue for

the band matrix and after several iterations, the number

of which depends on how near to an exact value the guess

is, the eigenvector is determined and an improved eigenvalue

is calculated from the e envectors by using the Rayleigh

quotient.

6.2 The LR Algorithm

The LR algorithm is based on the triangular decomposition

of a matrix A into

A = L R (6.2)

where L is a unit lower triangular matrix and R is an upper

triangular matrix.

A similarity transformation of the matrix A results

in the following relation.

L 1 A L

L 1 ( L R) L = R L

(6.3)

This in effect consist of decomposing A and multiplying

the factors in the reverse order. If the original matrix is

denoted by A and the resultant matrix A derived from the —s

matrix As-1 in the (s-1)th LR operation, given by

A —s-1 —s-1 —s-1

AS Rs-1 Ls-1

(6.4)

(6.5)

As is similar to As-1 and by induction to A . —

Rautishauserr371 showed that under certain restrictions after

an indefinite repetetion of the process Rs tends to an upper

triangular matrix whose diagonal elements are the eigenvalues

in the order of the modulus.

A -s R ss

A -s+1 = Q A Q = QRQ s s s -sT -Qs s-s

(6.7)

(6.8)

70

The LR algorithm is often not very successful when

applied to a general unsymmetric matrix with large dimensions

of the type obtained from the formulation described in chapter

5. This is because the triangular decomposition is not

always numerically stable and the amount of computation required

is likely to be very great.

6.3 The QR Algorithm

In place of the triangular decomposition described above

for the LR algorithm Francis(141 used a factorisation into the

product of a unitary matrix Q and an upper triangular matrix

U. A matrix Q is unitary if it satisfies the relation.

QT

(6.6)

where I is a unit matrix and Q is the complex conjugate

transpose (Hermitian Transpose matrix) of the complex valued

matrix Q. Orthorgonal matrix, a term more frequently used

in the literature, refers to the special case of a real

unitary matrix.

The QR algorthms can be defined by the relations.

In this way the QR algorithm can be used to obtain

better numerical stability for large nonsymmetric matrix.

The amount of computations that the method involves in one

iteration on a general matrix is often considerably lowered

if the matrix is first reduced to a bessenberg form

71

6.4 Determination of Eigenvectors of Band Matrices

Martin and Wilkinson(291 published an ALGOL program

which determines the eigenvectors of a general band matrix.

A FORTRAN translation of this program was used in the present

work.

The algorithm which takes advantage of the band form to

economise on arithmetric operations determines the value of x

for an assumed X in the equation.

Ax = ax (6.8)

by factorising the matrix (A - XI) into the product of 'a lower

triangular matrix and an upper triangular matrix.

The factors are used to perform an inverse iteration

by solving the system of equation

(A - XI) z r+1 x —r (6.9)

2sr+1 Z /a r+1 r+1

where ar+1 is the element of the vector 2r+1 having the largest —

modulus. Therefore each vector xr contains maximum element

of unit value.

After each iteration X + r+1 is the improved estimate

of the eigenvalue where 6r+1 is the element of the vector Z r+1

in the same position as the unit element of r

The iteration process is terminated if the residual:.

vector corresponding to the current approximation to the

eigenvector and eigenvalue (xr-1 - 6 Z r)/ar is smaller than

some pre-assigned tolerance, or if the number of iterations

72

performed is equal to a limit set to the maximum number of

iteration permitted for any eigenvector.

An improved eigenvalue is determined by using the

Rayleigh quotient for a symmetric matrix or the generalised

Rayleigh quotient for the non-symmetric matrix. The

difference between these is a consequence of the equality of

the right and left eigenvectors in the case of the symmetric

matrix. The general Rayleigh quotient is defined by

x7Ax/17

x (6.10)

where y is the left eigenvector of the matrix A and x is the

right eigenvector of the same matrix.

6.5 Closure

A brief description of the algorithms on which the

computer program used in the solution of the problem defined

in chapter 5 has been given in the present chapter. All the

algorithms set out to determine the eigenvalues of a given

matrix by first decomposing the matrix into products involving

triangular matrices. The QR algorithm produces better numeric-

ally stable result in fewer steps of computations than the LR

algorithm by using the unitary matrix as a factor.

The third algorithm described uses smaller computer

storage as it takes the band form into consideration in the

determination of the eigenvectors of a band matrix. Partial

pivoting ensures the numerical stability of the decomposition

whilst the generalised Rayleigh quotient is used to obtain a

better approximation to the eigenvalue. The method was used

in conjunction with the eigenvalues obtained from the QR

73

algorithm to obtain eigenvectors in very few number of

iterations and improved eigenvalues. In the case where

the lowest natural frequency is sought, the guessed value

of A is made zero.

74

CHAPTER SEVEN

DISCUSSION OF RESULTS

7.1 General

The fact that free vibration characteristics are

essential ingredients for a study of forced vibrations of

structural systems has been stressed in chapter one. Further-

more a study of free vibration characteristics in themselves

is necessary for a general understanding of the behaviour of

a vibrating shell. In industrial applications a knowledge

of the natural frequencies is useful in preventing destructive

effects of resonance with nearby rotating or oscillating

piece of machinery. Hence the importance of the frequencies

and modes of free vibrations cannot be overemphasised.

The object of this chapter is to study the free vibration

characteristics of shells of revolution with the analysis out-

lined in the previous chapters of this thesis. As a prelude

to the study, the results are first compared with results

available in the literature to demonstrate the accuracy of the

method and strengthen the discussion following.

The limit of application of shallow shell theory in

solving a non-shallow shell dynamic problem is investigated

by comparing frequency parameters for given problems using

different shell theories. The influence of shell thickness

and poisson ratio on the frequency is also investigated.

The three sample shells used for the investigation are

shown in Figures 7.01, 7.02 and 7.03. They are a model

75

cylinder , a model cone and a full scale hyperboloid of

revolution. These samples have been chosen because solutions

for them are readily available in the literature (7) I (48)

7.2 Comparison of Results with those available in the

Literature

Table 7.01 and 7.02 are comparison of results for the

cylinder and the hyperboloid with the work of Carter, Robinson

and Schnobrich171. Very good agreement was noticed for the

fundamental frequency parameters of a cylinder obtained from

the present work and those of the above reference as illustrated

in Table 7.01. In the case of the hyperboloid of revolution

the results of the present work are consistently higher than

those of the referenced papers for the first mode in each

harmonic. However very good agreement exists for higher modes.

7.3 Mode Shapes

The modes of the hyperboloid of revolution are

investigated for a 5 in thick concrete (v=0.15) shell using

Novozhilov theory equations. The first four axisymmetric

(n = o) modes are shown in Figure 7.04. The mode shapes are

in agreement with those of Carter et al (7) . The motion in

the first mode is predominantly axial whilst those of higher

modes are transverse in nature. In the second mode the

transverse displacement attains a large amplitude near the base

but elsewhere the displacement is small. In higher modes more

large amplitudes are formed over larger portion of the base

with a tendency to spread upwards. For modes other than the

first the tangential displacements are of smaller order than

76

the transverse displacement.

In the skew-symmetric modes (n = 1) both the transverse

displacement and the tangential displacements are of the same

order for the first three modes. Short waves with large

amplitude near the base like the type discussed above for the

axisymmetric modes occur in the transverse displacement for

the third and fourth modes. The formation of these large

amplitude short waves near the base is found to be only

characteristic of the hyperboloid as will be seen from a

comparison of the mode shapes of the other shell shapes given.

A knowledge of the modes in which these large amplitude'waves

with short length are present is useful when considering the

forced vibration of the structure. Their inclusion or non

inclusion in modal analysis can affect the speed of convergence

of a solution. This very useful decision is dependent of

course on the characteristics of the dynamic load being

considered.

Figure 7.06 shows the axisymmetric mode of the cylinder.

The motion of the first mode is axial while those of higher

modes are transverse or normal. For higher modes the motion

is represented by waves of almost equal length whose wavelength

increase as n increases.

The skew-symmetric modes of the cylinder shown in

Figure 7.07 show that the motion of the tangential displacement

along the meridian is axial in nature whilst the motion of the

other two displacement components as in the axisymmetric case

are made up of almost equal length waves. The regular shape

77

of the modes can be linked to the constant nature of the

geometric parameters of the cylinder along the meridian.

The attainments of mode shapes that look alike is pointer

to quick convergence in forced vibration modal analysis.

This is a useful point in assessing the adequacy of the

shallow shell theory as would be discussed later.

The axisymmetric modes and the skew-symmetric modes

of the truncated cone are shown in Figures 7.08 and 7.09

respectively. For all modes the tangential displacements

are of a lower order to the transverse displacement. Just

like the previous two shell configurations, the motion in

the first modes are predominantly axial and this is true of

the tangential components for higher modes. The fact that

the waves of the transverse displacements for modes other

than the first are like the cylinder's but not just as regular

as those of the latter confirms the assertion that the regular-

ity is connected with the nature of the geometric parameters.

7.4 Frequency, Shell thickness and Poisson Ratio

On purely engineering considerations it could be reasoned

that because of the dependence of the stiffnesses of a shell on

its thickness, the latter should be an important factor influenc-

ing the frequency of vibration of the shell. That this is not

so is clearly demonstrated by Table 7.05 in which the sample

cylinder is analysed for the thicknesses of 0.04 in and 0.06 in

keeping all other dimensions and properties as in Figure 7.01.

It is also further confirmed by Table 7.06 for a 5 in and 7 in

thick hyperboloid whose other dimensions and properties are as

in Figure 7.03. These tables demonstrate that the thickness

of the shell has very little influence on the frequency parameters

78

at low values of n but increases with higher n.

In Table 7.07 the poisson ratio used to analyse the

cylinder of Figure 7.01 is decreased from 0.30 to 0.25 and

in table 7.08 that of hyperboloid of revolution of Figure

7.03 is increased from 0.15 to 0.20. The purpose of this

exercise is to investigate what effect slight errors in the

assumed poisson ratio has on the frequency determined. The

results tabulated reveal that the frequency is not very much

affected by such errors.

7.5 Adequacy of Shallow Shell Theory

The frequency parameters for the first four modes for

n = o through to n = 6 are tabulated for a cylinder and a

hyperboloid of revolution in Tables 7.03 and 7.04 respectively.

They show that all the theories agree for low values of n and

that the shallow shell theory gives results which are clearly

different from those produced by the three non-shallow shell

theories for higher values of n. Where the discrepancies

exist it is noticed that they are more pronounced for frequency

parameters for lower modes than for higher modes. A plausible

explanation of this behaviour can be given through the dependence

of the frequency parameter on thickness by an energy consideration

as follows.

The total strain energy of a thin elastic shell of

revolution Vt can be split into two components ; Vm the strain

energy due to membrane action and Vb the strain energy due to

bending action

Vt Vm + Vb (7.1)

79

and for a shell of revolution with notations of previous

chapters

t a1 27r

b al o

Vt Eh 2(1 - v2) (Ell E22) 2

2(1 - v) (6 11E22 E12 2 )

Arda 2 da l

Eh 3 kk11

b + k22) 2

al

i

2(1 - v) (k11k22 - k122)

Arda 2da l

(7.2)

In terms of mode shapes and time dependent component

in the form of the system of equations (5.4) the equation

(7.2) can be reduced to

Vt TrEh + h2,4) ) Cost (wt - (P) (7.3) 2(1 - v2) m 12 b

where the membrane integral (Pm is given by

t a1

b a 1

(1) .m {

(3.1 + E22) 2 — - 2(1 — v) (e11E22-612)

(7.4)

rda 1

and the bending integral cbb is given by

80

(1) 1,= 1 (7-11 + 1.22)2 - 2(1 - v) (T11T22 - T122)

Arda1

(7.5)

The term c and (i,j = 1,2) in (7.4) and '(7.5) ij

are the meridional component of the membrane strain s. and lj

bending strain k,. respectively.

The kinetic energy of a thin elastic shell of revolution

is given by

1 2 ph

t t a 1 27r

b al o

Cif W2 )Ardaida

(7.6)

In terms of mode shapes and time dependent component

as above

T u phw2nsin2 (wt -

(7.7) 2

where the kinetic energy integral n is given by

11

t a l

i al b

2 1712 .)Arda l (7.8)

From the law of conservation of energy

V constant

(7.9)

from which

(T )max(Vt)max - (7.10)

and by substituting (7.3) and (7.7) in (7.10)

81

3-1 pw2(1 - v2)2,2 =tcpm b + h2cpj 9,2 E 12 n (7.11)

Equation (7.11) shows that the first term representing

the membrane contribution to the frequency parameter is

independent of h whilst the second term representing the bend-

ing contribution is directly proportional to h2. The same

conclusion can indirectly be arrived at from a close exam-

ination of the equation in appendix one to appendix four.

The membrane strain displacement relations given in

chapter three are the same for all the four theories whilst

the shallow shell theory of Vlasov differs markedly from the

other three theories in the bending strain displacement relations.

However equation (7.11) has shown that the frequency parameters

dependence on h2 is directly related to the bending contribution

to the strain energy. Hence the frequency parameters obtained

from the shallow shell theory and the non shallow shell theories

should vary in the same pattern as the variation of frequency

parameter with thickness.

In tables 7.05 and 7.06 the frequency parameters was

seen to vary only slightly with h for very low n but to have

varied more when h increases. That the same fact is observed

to be true of tables (7.03) and 7.04) is in accordance with the

conclusions deduced from equation (7.11).

82

The important conclusion which can be arrived at from

the foregoing for a dynamic solution is that the shallow shell

theory is unlikely to give very accurate results for a modal

analysis in which a large number of harmonics needs to be

considered. However if faster convergence is achieved and

only three or four harmonics are being used, the extra work

implied in using much more elaborate theory is undesirable.

7.6 Closure

Frequency parameters and mode shapes for shells of

revolution based on the analysis presented in earlier chapters

have been compared with those available in the literature.

There was good agreements between the results.

The effect of shell thickness on the frequency parameters

has been investigated. This has been used as an indirect means

of explaining the inadequacy of the shallow shell theory to

predict results of the same order as other linear shell theory

for n beyond certain range. The frequency of thin elastic

shell of revolution has been found to show little sensitivity

to slight error in the value of the poisson ratio.

A consideration of energy of the shell has been used to

derive an expression which shows that the main differences in

the results of the shallow and non shallow shell theory is a

direct consequence of the bending contribution to the strain

energy. Frequency parameters tabulated for a model cylinder

and a full scale hyperboloid are in agreement with this

deduction.,

83

TABLE 7.01

COMPARISON WITH PUBLISHED RESULTS

FUNDAMENTAL FREQUENCY-PARAMETERS FOR A CYLINDER

n Present work

Carter et al

0 0.4234 0.4233

1 0.05822 0.05821

2 0.01358 0.01356

3 0.004500 0.004467

4 0.003386 0.003326

5 0.005519 0.005425

84

TABLE 7.02

COMPARISON WITH PUBLISHED RESULTS

NATURAL FREQUENCY PARAMETER FOR A HYPERBOLOID

n Mode Present work Carter et al

0 1 0.1682 0.1766

2 0.3737 0.3833

3 0.4006 0.4166

1 1 0.7628 0.3180

2 0.1244 0.1356

3 0.3208 0.3255

2 1 0.02956 0.009165

2 0.04121 0.04011

3 0.1424 0.1423

3 1 0.01212 0.005559

2 0.02176 0.01165

3 0.05395 0.05502

4 1 0.007250 0.004100

2 0.01440 0.006162

3 0.02397 0.02269

5 1 0.006781 0.003149

2 0.007894 0.006008

3 0.01741 0.01243

85

TABLE 7.03

FREQUENCY PARAMETERS FOR A CYLINDER

OBTAINED FROM DIFFERENT THEORIES

n Mode Vlasov Novozhilov Reissner Sanders

o 1 0.4239 0.4239 0.4239 0.4239

2 0.8920 0.8920 0.8920 0.8920

3 0.9065 0.9065 0.9065 0.9065

4 0.9108 0.9108 0.9108 0.9108

1 1 0.05829 0.05827 0.05827 0.05827

2 0.4131 0.4130 0.4130 0.4130

3 0.6802 0.6802 0.6802 0.6802

4 0.8371 0.7992 0.7993 0.7992

2 1 0.01368 0.01362 0.01362 0.01362

2 0.1629 0.1628 0.1628 0.1628

3 0.4616 0.4614 0.4614 0.4614

4 0.6504 0.6502 0.6502 0.6502

3 1 0.004687 0.004541 0.004547 0.004541

2 0.07039 0.07019 0.07020 0.07019

3 0.2626 0.2623 0.2624 0.2623

4 0.4597 0.4593 0.4594 0.4593

86

Table 7.03 (Cont.)

n Mode Vlasov Novozhilov Reissner Sanders

4 1 0.003704 0.003453 0.003449 0.003448

2 0.03604 0.03569 0.03572 0.03569

3 0.1524 0.1520 0.1520 0.1520

4 0.3142 0.3136 0.3138 0.3136

5 1 0.006008 0.005581 0.005585 0.005581

2 0.02403 0.02354 0.02357 0.02354

3 0.09668 0.09605 0.09612 0.09605

4 0.2190 0.2182 0.2184 0.2182

1 0.01130 0.01071 0.01071 0.01071

2 0.02293 0.02224 0.02227 0.02224

3 0.07120 0.07037 0.07044 0.07035

4 0.1630 0.1620 0.1621 0.1620

87

TABLE 7.04

FREQUENCY PARAMETERS FOR A HYPERBOLOID

OBTAINED FROM DIFFERENT THEORIES

n Mode Vlasov Novozhilov 'Reissner Sanders

o 1 0.1682 0.1682 0.1682 0.1682

2 0.3737 0.3737 0.3737 0.3737

3 0.4007 0.4006 0.4006 0.4006

4 0.4318 0.4318 0.4318 0.4318

1 1 0.07629 0.07628 0.07629 0.07628

2 0.1244 0.1244 0.1244 0.1244

3 0.3208 0.3207 0.3208 0.3208

4 0.3718 0.3718 0.3718 0.3718

2 1 0.02961 0.02960 0.02960 0.02960

2 0.04120 0.04119 0.04119 0.04119

3 0.1424 0.1424 0.1424 0.1424

4 0.2658 0.2658 0.2658 0.2658

3 1 0.01220 0.01217 0.01217 0.01217

2 0.02176 0.02172 0.02172 0.02172

3 0.05400 0.05397 0.05398 0.05398

4 0.1337 0.1337 0.01337 0.01337

Table 7.04 (cont,)

n Mode Vlasov Novozhilov Reissner Sanders

88

4 1 0.007303 0.007253 0.007256 0.007253

2 0.01448 0.01441 0.01441 0.01441

3 0.02403 0.02398 0.02398 0.02398

4 0.06395 0.06398 0.06390 0.06398

5 1 0.007032 0.006807 0.006903 0.006927

2 0.007851 0.007900 0.007824 0.007767

3 0.01751 0.01743 0.01742 0.01744

4 0.03195 0.03187 0.03188 0.03186

89

TABLE 7.05

EFFECT OF THICKNESS ON

FREQUENCY PARAMETER FOR A CYLINDER

n Mode 0.04in 0.06in

0 1 0.4239 0.4248

2 0.8920 0.8934

3 0.9065 0.9078

4 0.9108 0.9153

1 1 0.5827 0.5837

2 0.4130 0.4132

3 0.6802 0.6826

4 0.7992 0.8015

2 1 0.01362 0.01377

2 0.1628 0.1636

3 0.4614 0.4649

4 0.6502 0.6590

3 1 0.04541 0.05247

2 0.07019 0.07194

3 0.2623 0.2675

4 0.4593 0.4714

Table 7.05 (cont.)

n Mode 0.04in 0.06in

90

4 1 0.003454 0.005827

2 0.03569 0.03959

3 0.1520 0.1603

4 0.3136 0.3307

5 1 0.005581 0.01162

2 0.02354 0.03175

3 0.09606 0.1098

4 0.2182 0.2431

6 1 0.01071 0.02339

2 0.02224 0.03807

3 0.07037 0.09334

4 0.1619 0.1986

91

TABLE 7.06

EFFECT OF THICKNESS ON

FREQUENCY PARAMETER FOR A HYPERBOLOID

n Mode 5 in 7 in

o 1 0.1682 0.1679

2 0.3737 0.3812

3 0.4006 0.4104

4 0.4318 0.4422

1 1 0.07628 0.07626

2 0.1244 0.01244

3 0.3207 0.3214

4 0.3718 0.3793

2 .1 0.02960 0.02959

2 0.04119 0.04122

3 0.1424 0.1425

4 0.2658 0.2664

3 1 0.01217 0.01224

2 0.02172 0.02193

3 0.05397 0.05414

4 0.1337 0.1341

Table 7:06 (Cont.)

n Mode 5 in 7 in

92

4 1 0.007253 0.007630

2 0.01441 0.01492

3 0.02398 0.02443

4 0.06389 0.06460

5 1 0.006807 0.007590

2 0.007900 0.009238

3 0.01743 0.01853

4 0.03187 0.03317

93

TAELE 7.07

EFFECT OF POISSON RATIO ON

THE FREQUENCY OF A CYLINDER

n Mode No. v = 0.25 v = 0.30

0 1 5529 5483

2 7978 7953

3 8024 8018

4 8040 8037

1 1 2061 2033

2 5403 5412

3 7016 6945

4 7553 7529

2 1 989.3 983.0

2 3411 3398

3 5730 5721

4 6791 6790

3 1 567.4 567.7

2 2241 2231

3 4322 4313

4 5710 5708

Table 7;07 (Cont.)

n Mode No. v = 0.30 v = 0.25

94

4 1 489.4 494.4

2 1595 1591

3 3289 3283

4 4719 4716

5 1 621.6 629.2

2 1290 1292

3 2612 2610

4 3935 3934

6 1 858.7 871.1

1247 1256

3 2230 2234

4 3386 3389

Values of the frequencies are in cycles/second

95

TABLE 7.08

EFFECT OF POISSON RATIO ON THE

FREQUENCY OF A HYPERBOLOID

n Mode No. V = 0.15 V = 0.20

1 7.563 7.302

2 11.27 11.30

3 11.67 11.70

4 12.12 12.15

1 1 5.093 4.998

2 6.504 6.357

3 10.44 10.35

4 11.24 11.26

2 1 3.172 3.101

2 3.742 3.669

3 6.959 6.918

4 9.507 9.494

3 1 2.034 1.983

2 2.718 2.675

3 4.284 4.250

4 6.742 6.721

Table 7.08 (Cont.)

n Mode No. v = 0.15 v = 0.20

96

4 1 1.571 1.538

2 2.214 2.177

3 2.856 2.837

4 4.661 4.642

5 1 1.521 1.539

2 1.639 1.580

3 2.434 2.419

4 3.292 3.280

Values of the frequencies are in cycles/second.

97

0.04'

4.001

FIG. 7.01 MODEL CYLINDER

= 0.3 E = 3.0 x /0 7 /6 in - 2

7.36:: /0 4 /b %/2 -` sec 2 4.o

5.00"

FREE

RS

CLAMPED

2.00# • 1

FIG. 7.02

MODEL CONE

0.25 E = /05 lb i/7-2

3 ,e /0-6 /b /n -4 secs

L

5.00

FREE

CLAMPED

J i

FIG. 7.03

HYPERBOLOID OF REVOLUTION

9 _., - 0.15 E -,-- 3x106 3 i/7 -2

---; 2.25 x /0-4v /6 h7 -4 sect I. .--- 1008 b ...-„,_ 25/6

100

n.o mode / n.o mode 2

14 10

oc,1 ocb a/A

(a) (6 )

n.o mode 3 n .o mode 4

—/

(C)

/0

or,6

(d)

FIG. 7.04

AXISYMMETRIC MODES FOR A HYPERBOLOID

Liz 0

0 41

mode I mode 2

-/ oct

(a)

(b)

mode 3 mode 4

0

-1

(c)

FIG. 705

SKEW SYMMETRIC MODES FOR A HYPERBOLO1D

I n =0 mode 3

0

*10

(C)

ab

(d)

102

0 mode / /7 a= 0

mode 2

0 0

at (a)

(b)

FIG. 7.06

AXISYMMETRIC MODES FOR A CYLINDER

no I mode /

0

ut

••

(a)

mode 0

(b)

103

n. mode 3 mode 4

a t o(;' (c)

(d)

FIG. 7.07

SKEW—SYMMETRIC MODES FOR A CYLINDER

(d) or/

104

n .o mode /7 a. 0 mode 2

a/4

oe/b

(a)

(b)

/7 0 mode 3 n.o mode 4

0

Nib

(c)

FIG. 7.08

AXISYMMETR IC MODES FOR A TRUNCATED CONE

0 110 10

105

n il mode / mock 2

£10

ot b alit (a)

(b)

n=1 mode 3 n. mode if

b cx,t oc,t

(c)

(d)

FIG. 7.09

SKEW—SYMMETRIC MODES FOR A TRUNCATED CONE

106

CHAPTER EIGHT

SUMMARY AND GENERAL CONCLUSIONS

8.1 The Problem

In this thesis the problem of free vibration of a shell

of revolution with arbitrary meridional configuration was

formulated from the general equations of motions of an undamped

elastic thin shell by equating the forcing function to zero.

The closed form of the shell in the circumferential direction

made it possible to represent the displacement in series form

with sinusoidal circumferential component. This was useful

in the reduction of the partial differential equations to three

fourth order linear differential equations. These equations

written in matrix form, employing a displacment formulation,

are integrated by substitution of fourth order central finite

difference equations. Newton's interpolation formula was used

to derive equations to supplement conditions at the edges of the

shell. The whole system of equations is identified as an

algebraic eigenvalue equation and solved as such.

8.2 Shell Theories

An investigation of the adequacy of shallow shell theory

to predict dynamic characteristics of shells of revolution was

carried out by deriving the equations using Vlasov shallow shell

theory and non shallow shell theory due to Novozhilov, Reissner

and Sanders. The four theories are traced to a single source

Love's First Approximation Theory. Having developed along

different lines, they have led to slightly different strain-

107

displacement relations which are summarised in the thesis.

Vlasov shallow shell theory which is an"engineering theory"

differs from the others in proposing less cumbersome equations

which have been found to be useful in solving shallow shell

problems. The soundness of the structural, arguments which.

lead to the derivation has made the present author view that

these simplified equations could also be applicable with

limitations to non-shallow shells. This limitation is defined

for dynamics of shells of revolution. The three non shallow

shell theory have been considered for purposes of comparison.

Novozhilov and Reissner theories are chosen because of their

general acceptability amongst analysts as averagely adequate

linear shell theories. Sanders theory which is the latest of the

theories lays claim to having got rid of all the inconsistencies

in the other theories. It is interesting that the results of

the present work confirm that there is very little difference

between the three non-shallow shell theory in engineering sense

at .least.

8.3 Solution Algorithms

The soltition of the eigenvalue problem was obtained by

application of either the QR algorithm of Francis or Martin

and Wilkinson's algorithm. Two programs based on the former

and one based on the latter are used. The three programs

provide adequate independent checks on the solution. Exactly

the same numerical results are obtained by separate applications

of these programs on any single problem. Computer core storage

and fast turn around requirements at development stage dictated

108

the choice of program to be used for a specific problem.

The results tabulated in chapter seven are obtained by using

the Harwell QR algorithm program.

8.4 General Conclusions

The following general conclusions are deduced from

this work.

1. The shallow shell theory is adequate for solving

dynamic problems of shells of revolution if convergence

is expected in a few number of harmonics but would

certainly not be recommended if a large number of

harmonic is required to obtain convergence.

2. The frequency parameter is not dependent on the thickness

of the shell for low harmonics but increasingly vary with

the thickness as higher number of harmonics are considered.

3. The effect of thickness on the frequency parameters and

the investigation of the adequacy of shallow shell

theory has been linked with the bending contribution to

the strain energy of the shell.

4. The frequency of the shell is found to vary little with

slight variations in poisson ratio of the shell.

5. A study of the mode shapes has revealed that constant

geometric parameters of the cylinder is reponsible for

the almost sinusoidal wave form of the transverse motion

of the cylinder. The transverse displacement of the

hyperboloid is found to possess a characteristic large

109

amplitude short wave near *the base which spreads to

the top as n increases.

8.5 Suggestions for Further Research

In line with the present work the following suggestions

are put forward for further research :

1. An extension of the method of analysis proposed in this

thesis to forced vibration analysis of shells of

revolution with modal analysis.

2. An extension of the method to include other boundary

conditions with particular reference to the case of

partial fixity at base instead of clamped edge considered.

It is considered that this may be useful in applications

in regions prone to seismic actions.

3. A formulation of the same problem for translational

shell to define the limit of applicability of shallow

shell theory to dynamics of translational shell.

4. A development of the computer program to include variable

thickness along the meridian of the shell.

5. A development of the computer program to handle sandwich

shells of revolution and other shell shapes like the

torus and paraboloid of revolution.

+ V a

v r

R1 3a2

R2 3a 2 K2 1R1} L3 +

V • - A L23 R2 {.1

110

A1.1

APPENDIX ONE

VLASOV THEORY EQUATIONS

Definition of Operators in Equations 4.32; 4.33; 4.34

The operators Lij (i,j = 1,2,3) are represented by

Lij for Vlasov's theory and are defined as follows :

L11 (1 - v)A32 vAr - r 3a1 l A Bali 2r aa22- R1R2 Ar

LV 2 = (1 + v) a2 (3 - v) r"3 L I3 Pg.120 2 acti3a2 2 r 3a 2

V J--1T

2 1 (a. — v) D 2 (3 - v) r"3 2 Da l 3a 2 2 r 3a2

V T 1-Q2 • A a2 ▪ (1 - fr / Ar r"

r Taa2 2 3al A @ak R1R2 Ar

T V -L, 3 2 112 + v 3

RI Da2

L33 = Ar 1 2v 1 - h2 r 32 171 2 R 2 2 1.2 A Ta R1R2

+[k• + r - 3r2k.',: • kA 3) Ki3

as

Tal3

+ f - 2 (E'" ( Ar2

+

+

111

3rA'" ) ~ -p:rt

(1 - v) A rR I R2

Al.2 Definition of Elements of Matrices in Equation 5.5

A =

B =

c =

D =

b 3 I

C I I

C21 C22

C31

d 3 1

. ::: 1 a33 J

C23

C 33

" I

......

112

ell e12 e13

e21 e22 e23

e31 e32 e33

The elements of the matrices are distinguished for

each theory by superscripts e.g. aij for Vlasov theory is

aYij. The following are the non-zero elements of the matrices

for Vlasov's theory.

V eli = -n2A (1 - v) - r-2 - vAr

2r Ar R1R2

e12 = - n (3 - v) r 2

e13 = r + r rRi R2 RI R1

e21 = - n (3 - v) r' 2

= -An2 (1 - v)Ar - (I - v)r-2 r 2 R1R2 Ar

E

V

V e22

e2 3 = An 1R2

r e31 vr" R2 R1

e32 = An

f

1 R2

113

e33 Ar + 1 —2

R11722

An ` n2 I2r" - 4r'2 - 2r'A' ( r1RTRv2 )il 3 Ar 2 ifiT3

- h2 12

V r rA' A

d12 = n (1 + v) 2

r + vr RI R2

,V u21 = - n (1 + v)

2

d22 = (1 - - r2V- 2

1/4,1.23

0

d31 vr‘

1 R2

V , U32 = 0

,V u33 = - h2 2n2 I r' + A' 12 Kr2 A 2 r

2r - rA" _Tr A

+ 10rA"A — - r" A" + 7r 'A.' 2 - 15rA" 3 — 2r -r" A b AAs A b A'r

U13

114

+ r.3 T7r2

+ 3r'2A- A r

-

+ vrA' A7RTR2

+ yrR; AITTYR2

AR1R2

] + vrRi.

AR11222

V C11 r

A

C22 = (1 - V) r

2 A

C33 = -h2 - 2n2 + r".". - 7r-A- + 15rA...2

17 Ar A

- 4rA" - r' 2 - vr

A ATr

- h2 6rAl 12 A

- h2 17 -3 A

A1.3 Definition of Elements of Matrices in Equation 5.7

f11

f12 f13

F

f21 £22 £23

f 3 1 f 3 2 f 3 3 _

gil g12 g13

G

g21 g22 g23

g31 g32 g33

,V D33

V a3 3

h12 h13

h22 h23

h32 h33

115

h 1 1

H

h21

h31

k13

K

k11 kli

k21 k22 k23

k3 k32 k33

The non-zero elements of the matrices for Vlasov's

theory are defined as follows :

k21 . vr' Ar

V k12 vn r

kiV - 1 - v

R1 R2

hV 1 A

P-21 r

kV2 - r 2 Ar

h22 = 1 A

V k33

1133

116.

(3 - v) n2r -2

f- n2 (2 - r"A' + rA" - 3rA-2

A PT.7

r ---

+ r vr .2712r R1R2

g3 3V

3rA' A

J-33 r

A2

A1.4 Definition of Elements of Matrices in Equations

5.28 and 5.29

111 112 113

LX -0

121 122 123

131 132 133

MX -0 mil m12 m13

m21 m22 m23

M31 m32 M33

Nx

--o /1/1

n2I

n31

1112

n22

n32

nI3

n23

1133

Qx

—0

Rx

—n

117

012 013

021 022 023

031 032

i pli

P21piz

P13

P22 P23

p31 P32

qi2 q13

q21 q22 q23

q31 q32 q33

rii r12 r13

r2 1 r22 r23

r31 r32 r33

x —S

n

T X

—n

S11

S21

S31 4-

t11

t21

t31

S12

S22

S32

t12

t22

t32

S13

S23

S33

t13

t23

t33

Ox

r— • 0 1 1

033

-X - o

P33

qi 1

= x U —u

= x V —n

= wx —n

118

Ull U12 U13

U21 U22 U23

U31 U32 U33

V11 VI2 V13

V21 V22 V23

V31 V32 V33

W11 WI2 W13

W21 W22 W23

W31 W32 W33

The non-zero elements of these matrices are defined

for Vlasov's theory as follows :

li t =

V J-i22 =

iV .L.33

V mil

V M22

V M33

V nil

V n22

V n33

=

=

=

=

=

=

=

1 vr' - A' A3

- A' A3

vn2 r

-

+

1 17(Sal) [1.72.r

- 5

- 5

- 8 vr'

12(6a 1 )2A2

16 12(Sai Air

10

10

- 30 -

12(Sa 1 )2A2

12(6a1)2A2

1

1

119

011 = - 10

V

012 = - 10

V

033 = 8 vr" - A' + 16 12 (Sal) {Air A 3 12 ((Sal ) 2A 2

131.V

1 = 5

p22 5

p33 = - 1 yr" - A" 12 ( Sal) {Ar A 3

- 1 12(Sal)2A2

q

V V C122 1 0133

ril l 1

r32 = 1

V r 3 3 - 1 12(Oai )

V Si 1

V S22

5

5

V S33 8 1f (Sal )

V t 1

„LV t22

L.33

10

10

0

O

L13 vr R2

V

V tii 1

V U2 2

V U33

120

viV 5

v22 = 5

V3 V

3 = 1 12 ((Sal )

V

wi 1 = - 1

Wit = - 1

V

W33 = 0

121

APPENDIX TWO

NOVOZHILOV THEORY EQUATIONS

A2.1 Definition of Operators in Equations 4.32; 4.33; 4.34.

For Novozhilov's theory the operators Lij (i,j=1,2,3)

are represented by Lij and are defined with respect to the

Vlasov's V. theory operators Lij of Appendix One as follows :

Lil LVil + h2 r .

a2 + (r - rRI a

12R1 AR1 Da12 TR1 AR12 aa l

f

+ (1 7 v) A _ rill

a2 7a22

rRl' r (AR12 ArR1

(

r"+ vr-Ri- TR' AR12

,N 1-1TV 12 + h2 1 a2

12R1 K2 Da2Dal + [v(1 yr,

rR2

D rR2 9a2

N -1-J

, = 13 L13 h2 [r a3 r -3rA- a2

12R1 A2 Tal3 A-2 77A7 22/

az + rA- + r + vr a Ted KT T2r R1R2 acti

2r-2 a2

Ta22

h 2 r a 3 -I- [(E 12 :P RI Tal3 A2

N V L31 L31 1 (E A AR]. 1

122

,N 1.121 = L2 V

1 h2 12R2

1 a 2 R1 7a 2 3a 1

Ri Ri 2

T N 1-122 = L22

(2 - v ) r' rR1

h2 12R 2

(1

kc2

- v)r32 + (1 - v)(1. )" 2 k AR 2 3al AR2

rR2 a 2 TaL

(1 - V) r' AR2

+ (1 - v) r'2 ArR2

L 23 =_- V 1JT23 h 2 1 D 3 r A' 32

17112 Fs TaDa2 T2 3alacc2

- A a 3 + (1 — v )A3 172 Ta2 3 R1R2 Dal]

rR1" 32 Ta l 2 A7R1 2

+ /r \ ' - ( TR )

rRi TriV

k01 "2

-F

r V rA2 R1 A

1

r' '+ TR'

- .2

rA2R1

vr'Ri( +

r' A RI

(2 A'RL I

v A

(rR1T A AR1 2

- v)32 rR1 3a22 3a 1

Rz + (2 - v ) r' a 2 rRI 2 R r4 as

- vr'A' —2— A RI r'R' AR'

+ rRi -A" + 2rRi'2 rRI AR1 2 A`RE AR1

123

,N T L32 J-J3

V 2 h2 (2 - v) 93 +I( 1 \ + 1

2 T2 AR2 3a1 23a2 UNK2) A - (11

vr" 2r" + (1 - v) r \" ArR2 ArR2 r AR21 9a19a2

A D3 - 2 r R2 Wa23 2)

+ vr - 2r. ArR2 ArR2

(1 - V) r

E. AR2

( ( ). .2)1. ArR2 Ta2

L33 , • LI 3 3

A2.2 Definition of Elements of Matrices in Equation. 5.5

The elements which are defined as in section A1.2 of

Appendix One are given with respect to the Vlasov's theory

corresponding elements of the same section.

• elV l + h2 f- (1 - v) Ant - r vr'' ell 12R1 rRi ArR1 AR1

ei 2 eve h2n .vR'2 vr + 2r' i 12R1 f R22 rR2 rR2

= el V s + h2 2n2r'

12R1 -Tr-

• diV

l + h2 r' - 2rR1 " - 12R1 AR1

e ta

N u 1.

AR1 2 Az-R1

124

n R2

}

V d21 d21

d12 = k ,,11V 2 h2

'17R1

di3 = u.13 h2 n2 ▪ r'A'

+ rA" - 3rA"2 1-2-R1 r A"

A3- ---A-4-

r'2 ▪ vr

K2r R1R2

V cli • cli + h2

12R1 I AR1

V C13 = C13 h2 r 3rA"

17R1 27, 2 -3*

,N V b13 = KJ h13 h2

17R 1 A2

e21 = e21 ▪ h2n Rc - (2 - v)r" 12R2 RI2

e22 • e2V 2 h2

12R2 n2A + (1 - v) r '" r-A" rR2 AR2 T7172

- r"R".2 + r"2]. AR22 ArR2

e23 = e23 • h2 f n3A - (1 - v)Ani 12R2 R1R2

- h2 12R2 n

125

a22 "22 + h2 (1 - v) f r' - rA' rRi 12R2 AR 2 KTR2 AR2 2

d2 3 "23 h 2

A' 12R 2 Ar

C 23 = C23 h 2 n 1fR 2 A

N V C22 = C22 h 2 (1 - V) r 12R 2 AR 2

N e3/ V e31 h2 (2 - v) n2R1," 12 rR 1 2

2 °v . - n r r 2R 1

(1 + v) r"R( yrKT2 - 2r"12'

KTRT2 + (3 + v) r'A'R1

A3121 2

rR I "A" A3 R 1 2

+ 3rR I "A' A3R / 2

- 3rA'2R; A4R 1 2

- 6rA"RI 2 A2R 1 3

4r'R( 2 r' 2 Ri" 6rR I 'R I A2R 1 3 A 2 rR 1 2 A2R 1 3

6rR 1 "3 + vr" T171 4 A21212 YTIR.1

- 3vr"A' A3 R 1

- vr'A" Y2 111

3vr'A'2 - 2r'r" + A-7Zr- rA 2 R1

2r'2A' r rA3 R 1 r2A2R1

126

e32 = e32 - h2 . n3 A - vnR2 + vR2 ?A". 12

z r R2 AR2 2 A R2

2vnR2 '2 AR2

- (3 - 2v ) n + nr ArR2 Ar 2 R2

(3 - 2v) nr"A" + (3 - 2v) r —z--

A rR2 ArR2 2

AN u31 d3V

rA"." —t A Ri

6rRi —2— 3 Ri

A d32 u3V

2

c31 = C31

C3 2 C32

h2 (1 + - (3 + v)r'A". - 4r -"Ri" 12 A2R1 RT2

3rk. 2 - + 6rA'111" -27171 7C-271 1:377.

(2

h2 n 12

h2 12

h

- v) n2 - r'2

T.2rRi

+

3rRi.

2R2 + ArR2

3rA" :-.A.-3Tt-

2r"

rRi

(2 - v) A" i A 2 R2 AR2 2

n

TrRi

(2 - v)

A2R2

12 AR2

bai b31 - h2 r 12 • A2R1

V .1..)3 3 = b33

a3 3 V a33

127

A2.3 Definition of Elements of Matrices in Equation 5.7

The elements are as given in Section A1.3 and are

defined for Novozhilov's theory in terms of the corresponding

Vlasov's theory elements of Section A1.3 of Appendix One as

follows :

k21 = k21 n , h2 6R1 R2

k22 = k22 r" h2 Ar 611:

k23 = A23 • h2 nr" 6R2 Are

h23 = h23 h2 n 6R2 Ar

h22

hie h2 1

6R72.( A

N 1,17 A-31 A.31 - 2n2(1 - v)A - r"Ri.

rR1 AR12 -

AR12

+ 2r11172 rRI" A" AR1 3 AzR1`

.2

rTiR i - vAr — R2R1 2

A32 A- 1,1732 n (2v - 3) r"

R22 rR2

, h3, 113V 1 ▪ r" - rA" 1E12‘

AR1 TrR1 A-A14

128

,N h 32 • h32 ▪ n(2 - V)

R2

g31 = g31 + r AR1

A2.4 Definition of Elements of matrices in Equation 5.28

The elements are defined as in Section A1.4 of Appendix

One and are given with respect to Vlasov's theory corresponding

elements as follows :

N „ „ 1-31 • J.3

V 1 1

T2(Sal)

M31 = m31 - 8 1 T2(Sa 1 ) AR1

n31 • n31 21.1' AR1 2 ArR 1

N n32 • n3

V 2 ▪ vn

rR2

031 = 031 + 8 1 12 (Sal) AR1

P31 = P31 1 12 (Sa l) AR1

A2.5 Note

Any element not defined in this Appendix for Novozhilov's

theory should be assumed to take the same value as the corres-

ponding elements defined for Vlasov's theory in Appendix One.

L11

+4,12

Lil

4-1N ,

2 1

h2 12R1

h 2 12R1

(1 - v) A a2

Ta22

f 1 + R2 Ta1

rR1 2

(1 v) 2

R N L13 L131

T R 4-122 L22 h (1 - v) 32

r rAR2

12R2 2 AR2 3a1 2

+ 2r"R; AR22 AR2L

r'1- a R2 2 rR2 Ta2

1 R2

+ Ar R1R2

129

APPENDIX THREE

REISSNER'S THEORY EQUATIONS

A3.1 Definition of operators in Equations 4.32;4.33;4.34.

The operators Lij (i,j = 1,2,3) are represented by

Lij for Reissner's theory and are defined with respect to

Novozhilov's theory operators Lij of Appendix Two as follows:

,N L21 J-J21 h2 - v) 1 3 + r" - Ri

I2R2 2 Ri 3a1 rR1 R1 2

,R ,N L23 1,23

R , 1,31 L3 1 h2 (1 - v) IR( 1

12 rR12 a 2

130

L32 = L32 + h2 (1 - v) 12 f

1 3 - A" 3 AR2 3a12 T2R2 Wal

A ArR2 AR22

R11222 5q2

,R , L33 1.13

N 3

A3.2 Definition of Elements in Matrices of Equation, 5.5

The elements defined as in Section A1.2 of Appendix

One are given in terms of the corresponding Novozhilov's

theory elements of section A2.2 of Appendix Two as folldws:

R el i ell + h2

12R1 n2A (1 - v) rR1 2

e112 = el2 + h2 12R1

n (1 - v) r rR2

- R" R2

2 2

R (112 di 2 h2 (1 - v)n

12R1 2 R2

e21 = e21 + h2 n (1 - v) r E.; 2 12R2 2

e22 = e22 h2 (1 - v) 3r"11; + rR2 " - ,2

12R2 2 AR2 AR22 ArR2

- 2rR2 A2 AR2 3

- rR2 'A' AzR2'

+ Ar 2

R1R2

2R22 - R2"-A". — 3 AR2 A —2-2 R2

d331 = d31 h2 (1 12

,R U32 d32 h2 (1

T2

C32 = C32 h2 (1

A 2 7▪ 1"4:1112

- v)n2 {1 -\ TR/

- v)n A- A —2 R2

- v) n 12 AR2

131

,R U21 = d21 h2 (1 - v)n

12R2 2 R1

I d22 = d22 + h2 (1 v) rA". r'

12R2 2 T7R2 TR2

C22 = C22 h2 (1 - v) r 12R2 2 AR2

e31 = e31 + h2 (1 - v) n2 Rc 12 rR12

e32 = e32 ▪ h2 (1 - v)n 2r*".1S + RI" 12 ArR2 AR22

A3.3 Definition of Elements of Matrices in Equation, 5.7

The elements defined as in Section A1.3 of Appendix

One are given below for Reissner's theory with respect to

the corresponding Novozhilov's theory elements defined in

Section A2.3 of Appendix Two.

k21

„R A-22

=

=

k21

kN

-22

h2

12R2

h2

12R2

132

(1 - v) n rRl

c

r"' R2A

ArR2 AR22

(1

2

- V) 2

k1723 = K23 X23

,N k31 • X31 - v) Ant rR1

x32 N ,

• K32 (1 - v) n f r" rR2 1722

h22 = h22 h2 (1 - v) 1 12R2 2 AR2

1132 • 1132 (1 v) n

R2

A3.4 Note

Any element not defined in this Appendix for Reissner's

theory should be assumed to take the same value as the

corresponding elements defined for Novozhilov's theory in

.Appendix Two.

- h2 (1 - v) 11[13 12 8 R2 K1 f172

1 D2 RI Da23al

S L12

N T 1,12

133

APPENDIX FOUR

SANDERS THEORY EQUATIONS

A4.1 Definition of Operators in Equations 4.32;4.33;4.34

The operators Lij (i j = 1,2,3) are represented by

Lij for Sanders theory equations and are defined with respect

to Novozhilov's theory equation operators Lij of Appendix Two

as follows :

L L11 h2 — v) A 1 — 1112 — 1 a 2 12 8 r R2 RI RI "R2 3a22

ir' rR2

+ r" + R2. , a rR1 1722 3a2

T S TN J-113 -1-113 h2 (1 - V) 1 - 11 a' — 1

12 2 R2 RI r Ta1Da22 T2

32

Ta22

32 pa0a2

L21 = L21 h2 (1 - v) 1 12 8 R1 K2 11-1 R2

1(11-.2

- El )(3r' + 311c - r.

1722 171 2 rRlR12 rR2 R2

- 1 ) (3r' - 1 )1 3 R2 R1 K2 5a2

L22 = L2 2 - h 2 (1 - v) r 12 8

(R2" 12 4".2) ( 3 R2 2 R I AR2

11-R1

fR2 AR2

1 (I- - 1 ) AR R1 R2

(3A' ArR2

-

a 2 AR 1 5ec 1 2

1-1T S31 = L31

134

+ 5R'2 + r' AR2 2

ArR2 ARI AR1 2 5a1

- 1 1 + r.2 + R2 ) 1T1R2 2 Ar2R2

r AR; A- ArR22

1 '+ 121 2R 2

r'2 Ar 2R1

r ArRi 2

R AR2 2

+ R,"A' + 2R2 ' - A 2R2 2 AR2 3

(R2 2 ". R2

- R1")( r- 2 R1 ArR2

r ArR1 I2 :AIR 2

L23 -1-1 -1-12 3 - lit (1 - ) (1 12 2 Ki

) 1 a 3 R 2 A Ta22 aot

- (11

1 ) 2r" + A" A 2 ) R22172 R12 A Da2@ct 172 Ar

(

R2 ) (1-

R1 R2 + 2r' ) (R2 -

R22 RI") r" R I 2 Ar 3(121 Ar e

- /11 (1 - 1 + 1 a 3 2r [

R R2 5a 2 2 3a l

▪ R2 A K2 2

a 1 aaa2

135

1.02 , 1-13N

+

2 -

A" -2 A R2

h2

12

+

(1 - V) [-

2r"

(1 RI

-

+ 1 Rz

2r" -

1 A

R2

AR22

' a 762361 2

RI -"CRT'I

a2

-- 29al

2

A2 -2 A R1

+ ArR2 ArRI

k.

+ L. + r'121 " + 2R2" + 2R2"A" + 4R2"2 R12R2 ArR1 2 AR22 A2R2 AR23

- 5r"R2'

3A a ArR22

RIR/ Ta2

L33 N „

• 1-1 3 3

A4.2 Definition of Elements of Matrices in Equation 5.5

The elements are as defined in Section A1.2 of Appendix

One and are given below for Sanders' Theory in terms of

corresponding Novozhilov's theory elements defined in Section

A2.2 of Appendix One.

ellS

• ell - h2n2 (1 - v) A 1 + h2 n2 (1 - v) A 12R2 8r R2 12RI 8r R2

e12 • e12 + h2 (1 - v) 1 r' + 2rR 2 12R2 Tr T2 R2

7

h2 n (1 - v) 1 r' + 2rR2' T2RI 8r RI 1,7—r

= ei3 - h2 n2 (1 - v) 12R2

= diz - h 2 n (1 - v) 12R2

di3 h 2 n2 (1 - v)

e13

diz

d13

136

r" + h2 n2 (1 - v) r" 2r 2 12R1 2r2

3 + h2 n(1 - v)1 4 - 1 8r T2R1 8 R2 Ri

1 - h2 n2 (1 - v) 1 12R2 2r 12R1 2r

e21 = e21 - h2 n (1 - v) 1 r" + 2R2 12Rz 8r Rz R2`

- h 2 n (1 - v) 3r" + 6R1" - 4r' - 4R1 -- 12R1 8 rR1 R12 rR2 R22 R1R2

- 3R2 R1R2

e22 = e22 h2 (1 - v) r"2 + 2r"I12 " + 2rR 2 "A" 12R2 8 ArR2 K-172.7 A.1124

+ 2r1212 - 2rR2"" h2 (1 - v) r"2 - Ar AR2 3 AR2 2 12111 8 ArR1

+ Ar + 2r"R1 " - 2rR2" + 2rR2 "A" R2R1 2 AR1

—_-7 AR2 2 A zRi

4rRz"2 2rR2 ' RI " AR2 3 AR2' R1

e23 = e23 h2 n (1 - v) 2r"2 r'R"2 12R2 Ar e ArR2

d2 S L.A. 21

S e31 e31

S e32

N e 3 2

h2 n T2R 2

(1 - 2r' + A" + R2 AR2 2 -2 Ar TA

h2 n - v) 2r" + + R1 " 12R1 2 Ar A2 AR1

+ h2 - v) 3r - h2 (1 - v)r 4 - 1 T2R2 8AR2 12R1 8 AR2 AR1

h2 n (1 - + h2 n (1 - v) T2R2 2A 12R1 2A

+ h2 n (1 - v) R2 h2 n2(1 - R1 12R2 2rR2 12R1 2rR1

- h 2 n (1 - v) 4R 2 -2 + 2R 2 "A' - 12R2 2 AR2 3 A2R2 AR22

AS 3 = d23

c23 = C2 T 3

N C2 2 C22

S

137

h2 n (1 - v) - A + 2r -2 + r"R _L 12R1 2 R R2 1-12 2 Ar2 ArR1

- h2 n - v) + h2 n(_1 - v) 3 _ A

T2R2 8R2 12R1 8 R1 R2

S A kA.2 2 = d22 h2 (1 - r 3A- + 8R2 + 12R2 8 ArR2 AR2 2 ArR2

+ h2 (1 - v) r 4A- + 6R2 - - 2R1' T2R1 8 172-R 2 AR2 2 1 AR12

+ 4R " AR2R1 ArR1

12 kS

138

- 5r- 112" - h2 n (1 - v ) A ArR22 12R1 2 RI R2

3A + r' R " R22 ArR1

d31 = d31 + h2 n2(1 - v) + h2 n2(1 - v) 12R2 2r 12R1 2r

d32 = d32 - h2 n (1 - v) 12R2 2

2r' - R2 ". Ar AR2

+ K2 }

h2 n (1 - v) R 2r" 1.2R 1 2 AR1 Ar

+ A' K2 fr.

C32 c32 h2 n (1 - v) + h2 n (1 - T2R2 2A T2R1 2A

A4.3 Definition of Elements of Matrices in Equation 5.7

The elements are as given in Section A1.3 and are

defined for Sanders' theory in terms of the corresponding

Novozhilov's theory elements of Section A2.3 of Appendix

Two as follows :

1,N k11 - + h2 - R1.

12R1 vr" 1

12R1 ArRi AR]. 2

kS

2 + h2 1 12R1 rR2

S k13 k13 hz 2.r12

12R1 r2

140

,S K32 = k32 3 (1 v) n (1 - v) r'- n (1 - v)

2 rR2 R2 L

h32 „i I132 n • - v) (1 - v)

2R2 2R1

A4.4 Note

Any element not defined in this Appendix for Sanders'

theory should be assumed to take the same value as the

corresponding elements defined for Novozhilov's theory in

Appendix Two.

141

APPENDIX FIVE

THE COMPUTER PROGRAM

A5.1 General

The main features of a FORTRAN program DANSOR

(Dynamic Analysis of shellsof Revolution) written to solve

numerically the problem set out in this thesis is described

in this appendix The program was developed simultaneously

on the FUN compiler of the University of London CDC 6600

computer and the Imperial College CDC 6400 computer. It

shotld work on other compilers on these machines.

A list of the source deck of the program will not be

included in this thesis. The program can readily be obtained

in the Civil Engineering Departmental program library. How-

ever this appendix contains a flow chart which gives an in-

sight into the logic of the program and should facilitate

updating it if necessary.

Brief description of the functions of each subroutine

is given, this include description of subroutines obtained

from the library, all of which have been slightly modified

by the present author to suit the general pattern of DANSOR.

In doing this the original objectives of their respective

anthOri have not been destroyed:

A5.2 Flow Chart

Figure A5.1 (a) and A5.1 (b) give the flow chart of the

program DANSOR.

142

A5.3 Summary of Functions of Subprograms

The subprogram DANSOR is written as a general calling

program for all the required subroutines in a specific

analysis. As the problem was formulated in terms of the

displacement vector two separate versions of the program

are required the axisymmetric version in which all the

working matrices are (2 x 2) and the non-axisymmetric or

general version in which the working matrices are (3 x 3).

Other than calling the subroutines, this subprogram reads

the data, initialises some working locations and prints the

final results.at the end of the calculations for each harmonic.

The other subroutines will be summarised under the

following broad headings - subroutines common to the axi-

symmetri.cand general version, subroutines which have axi-

symmetric and general version and the library subroutines.

A5.3.1 Subroutines common to both versions

PRINT called from DANSOR writes the data read which

consist of type of shell of revolution, dimensions and

material properties of the shell.

GEOMX calculates the geometric parameters for each

station on the shell using the results of chapter two. EQUATV,

EQUATN, EQUATR and EQUATS compute the elements of the matrices

in equation of motion based on Vlasov, Novozhilov, Reissner

and Sanders theory respectively. These subroutines are not

independent in the sense that EQUATR and EQUATS compute the

differences between the terms of the Novozhilov and Reissner

or Sanders theory respectively whilst EQUATN computes the

differences between the expressions for the elements of Vlasov

143

and Novozhilov equations. The expressions coded are given

in Appendix one to four inclusive. BOUNDV, BOUNDN,BOUNDR and

BOUNDS equivalent in the same order to the four last sub-

routines used the method described above to compute expressions

for the elements of the matrices for the imposition of free-

clamped boundary conditions at the edges of the shell based on

the appropriate theory.

A5.3.2 Subroutines with Axisymmetric and General version

Each of the subroutines described in this section has

a general version in which the working matrices are (3 x 3)

and an axisymmetric version in which these matrices are (2 x 2)

CALLFN is used to call FIN which performs the subsitution

of the fourth order finite difference expressions. For purposes

of economy in storage and computation the sequence of finite

differences substitution is made possible after computing terms

of the boundary matrices by calling EQMAT, whose duty is to

transfer elements into temporary storage positions or to positions

in common block for the use of FIN in the process of substitution.

MINV is a simple Cramers Rule matrix inversion subroutine,

It identifies a singular matrix and prints a warning error

message but does not cause the program to terminate execution.

Any such error message which can only occur if there has been

mistakes in the data or in execution of the program itself must

give inaccurate results for the problem.

CALELM carries out the sequence of elimination of

fictitious vectors described in chapter five through ELIM which

calls each time MAM- a simple matrix multiplication subroutine

144

and MAS - a simple matrix subtraction subroutine. The use

of the sequence involved in CALELM to cover all the occasions

that eliminations are done in the analysis is made possible by

creation and application of a very useful null matrix in the

program.

ASSEM1 assembles the elements of the matrices at the end

of the computation for each station in the appropriate elements

of a big matrix for the eigenvalue calculations. The Martin-

Wilkinson algorithmr2" requires the final matrix to be arranged

in a special form which involves using the band form of the

matrix to advantage. ASWST is designed by the present-author

to do this job.

A5.3.3 Library Subroutines

Any of three library packages each employing one of the

algorithms described in chapter six is used to complete the

solution. The packages are from the SHARE Library, the HARWELL

Library and a Fortram translation of Martin and Wilkinson pro-

gram(291.

The SHARE 360 package consists of QREIG and QRT which

determine the n eigenvalues of a Hessenberg matrix resulting

from HESSEN, by the QR transformation.

In the Harwell Package, the subroutines EBO1A, EBO4A,

EBO5A, MEO4B and MCO3AS calculate the eigenvalues and eigenvectors

of the Hessenberg matrix produced by MCO8A. The present author

edited the package to calculate a specific number of eigenvectors

and to use the same locations for both the right and left eigen-

vectors. Whenever this is necessary it requires calling the

145

package twice as against one entry in the original version.

Since the formulation of the problem gives rise to an

unsymmetric final matrix, the subroutines used in Martin and

Wilkinson Package are those subroutines which handles unsymmetric

matrix. The documentation of UNSRAY, BASL1 and BANDT1 are

given elsewhere (291 , it will therefore not be repeated here.

However minor editing which involves lowering the core storage

required and slight adjustment to suit this program and the

CDC compilers was carried out by the present author.

The eigenvalues and eigenvectors obtained from the Share

and Harwell Packages are not always in the order of magnitude

of modulus. The subroutine ARRANK and ARRANG arrange the

eigenvalues in order of magnitude of modulus with the least

first for the two packages respectively.

A5.4 Data and Results

Data for a given problem are read from three cards.

The first card specifies the type of shell of revolution and

the properties of the material of the shell. The second

specifies the solution options whilst the third specifies the

dimensions of the shell.

The following characters read on the first card are.

defined by format No. 1001 of DANSOR.

NNN specifies the type of shell of revolution to

be analysed.

1 cylinder

2 cone

3 hyperboloid of revolution of one sheet.

146

YOUNG

Youngs modulus of the shell material

DENS

Mass density of shell material

POISS - Poisson ratio of shell material

The following characters are read in the second card

and are defined by format No. 1002 of DANSOR.

NEQUAT -

=

specifies the theory to be used

1 Vlasov's theory

2 Novozhilov's theory

3 Sanders' theory

4 Reissner's theory

NMS Value of the first harmonic to be analysed

NMSS Value of the last harmonic to be analysed

(The last two characters can take any values within the given

format for the axisymmetric case as they are never used).

MVECT - The number of mode shape3corresponding to least

frequencies to be determined.

(This should be zero when only frequencies are required from

the Harwell Package).

NN

last problem indicator

o for last problem

otherwise any integer value within the format.

The third card which specifies the geometry of the shell

is read in subroutine GEOMX and obviously requires different

formats for each shape. For the cylinder Format 101 of

GEOMX defines the character A, B and HH where

A a of Figure 2.3

of Figure 2.3

147

HH thickness of shell

DL Referenced length

For the cone Format 201 of GEOMX defines the character RS1,

RS2, H and HH where

RS1 - rl of Figure 2.4

RS2 - r2 of Figure 2.4

H - h of Figure 2.4.

HH thickness of shell

DL Referenced length

For the hyperboloid Format 301 of GEOMX defines the character

A, B, H1, H2, and HH where

A - throat radius ; a of Figure 2.5

B - curvature parameter ; b of Equation (2.37)

H1 - height of top from throat as in Figure 2.5

H2 - height of throat from base, h2 of Figure 2.5

DL - Referenced length

The output are self explicit as the subroutine PRINT

has been designed to write out brief description of each group

of figures printed.

A5.5 Closure.;

The program DANSOR described above is written to determine

the natural frequencies and mode shapes of shells of revolution

(cylinder, cone and hyperboloid) by'use of any of four linear

shell theory. The main object of the author for the use of

such a large number of subroutines has been to facilitate editing

and further development of the program. This should present n0

difficulties since all that is required is by-passing a subroutine

148

or a set of subroutines in DANSOR and calling in their place

new subroutines developed to carry out the same or modified

analysis.

The type of shells of revolution to be analysed can be

increased to include other shapes like the torus or paraboloid

of revolution by coding results for them from the analysis in

chapter two in the form of subroutine GEOMX. Other forms of

boundary conditions can be handled provided as stated in chapter

four they are homogeneous boundary conditions.

(I Substitute finite Difference V.2. =displacement vector

at station i

(Eliminate v-1)

Expressions

FIGURE A5.1A

149

START

NPROB=o NNN=Type of shell

NSP=Total no of stations

1 NN =Last problem control

NMS=Last harmonic no NPROB=NPROB+1-

STAGE READ

Read NNN.NSP, NN

( HH,NMS etc. )

(Calculate values of

geometric constants

\s, at each station

I=I+1

Calculate elements of,

matrix in equations of

motion

M =no of harmonic

I =no of station

STAGE FIN

Write Frequency

parameter and mode

shapes J

150

Store Results o

stage FIN

Calcutate Element of Matrix

in Equation for Imposition

of Boundary conditions.

NO -> (Eliminate 0

( Eliminate vn+2and

Assemble matrix

NO

Determine Eigenvalues2

and Eigenvectors

vn+1

FIGURE A5.1B

151

REFERENCES

1. ABU-SITTA, S.H.

'A Finite Difference Solution of the General

Novozhilov Equations' IASS International Colloquim

Madrid. September - October 1969.

2. ALBASINY, E.L.;MARTIN,, D.W.

'Bending and Membrane Equilibrium in Cooling

Towers' J. of Eng. Mech. Division, Proceedings

ASCE Vol. 93 No. EM3 June 1967,1-17.

3. ARNOLD, R.N.; WARBURTON, G.R.

'Flexural Vibrations of Walls of Thin Cylindrical

Shells Having Freely Supported Ends' Proc. Roy. Soc.

of London Series A vol. 197,1949 1 pp 238-256.

4. ARNOLD, R.N.; WARBURTON, G.R.

'The Flexural Vibration of Thin Cylinders'

J. Proc. of Inst. of Mech. Engrs. vol. 167 1953 62-74.

5. BOGNER, F.K.; FOX, R.L.; SCHMIT, L.A.

'A Cylindrical Shell Discrete Element' Journal

AIAA vol. 5 No. 4 April 1967 745-750.

6. BUDIANSKY, B. ; RADKOWSKI, P.P.

'Numerical Analysis of Unsymmetrical Bending of

Shells of Revolution.'-AIAA J, vol 1 No. 8 August 1963.

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