APPENDIX I




























APPENDIX IV
















NONLINEAR ANALYSIS OF DYNAMIC STABILITY
OF ELASTIC SHELLS OF REVOLUTION









By

MARCUS GEORGE HENDRICKS


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA IN PARTIAL
FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA
1974


































TO MY WIFE,
GENEVIEVE,
AND MY DAUGHTER,
CARMEN














ACKNOWLEDGMENTS


The author wishes to express his sincere appreci-

ation to Dr. S. Y. Lu, chairman of his supervisory com-

mittee, for his continuous guidance and encouragement

throughout the entire period of this study. He also

wishes to express his gratitude to Dr. I. K. Ebcioglu,

Dr. U. H. Kurzweg, Dr. R. L. Sierakowski, and Dr. M. W.

Self for their helpful discussions with the author and

many valuable suggestions.














TABLE OF CONTENTS


Page


ACKNOWLEDGMENTS.................................. iii

LIST OF FIGURES................................. vi

NOMENCLATURE..................................... vii

ABSTRACT......................................... x

CHAPTER
1 INTRODUCTION............................ 1

2 THEORETICAL DEVELOPMENT OF
NONLINEAR SHELL EQUATIONS ................. 14
2.1 Shell Coordinates................... 14
2.2 Kinematic Relations................. 15
2.3 Constitutive Relations ............. 17
2.4 Kirchhoff's Hypothesis.............. 18
2.5 Variational Principle.............. 20
2.5.1 Hamilton's Principle........ 20
2.5.2 Equations of Motion ......... 21
2.5.3 Boundary conditions......... 26
2.6 Synthesis of Equations.............. 26
2.6.1 Shells of Revolution........ 27
2.6.2 Conical Shells.............. 27

3 DYNAMIC STABILITY EQUATIONS
OF CONICAL SHELLS....................... 35

4 METHOD OF SOLUTION...................... 39
4.1 Circumferential Modal Analysis..... 39
4.2 Elimination of
Meridional Derivatives ............. 42
4.2.1 Subdomain Method............ .45
4.2.2 Method of Satisfying
Boundary Conditions ......... 47











TABLE OF CONTENTS (Continued)


Page


5 NUMERICAL ANALYSIS FOR CONICAL SHELLS..... 52
5.1 Functional Description of the
Computer Program....................... 52
5.2 Checking the Computer Program........ 54
5.3 Dynamic Stability Results............ 55
5.3.1 Dynamic Modal Preference...... 56
5.3.2 Linearized Dynamic Stability.. 58
5.3.3 Nonlinear Dynamic Stability... 60
6 RESULTS AND CONCLUSIONS................... 69


APPENDIX I SHELL SPECIMEN AND BOUNDARY
CONDIT IONS ............................ 71

APPENDIX II CONVERGENCE.......................... 74

APPENDIX III FLOW DIAGRAM OF THE
COMPUTER PROGRAM..................... 77

APPENDIX IV COMPUTER PROGRAM SOURCE
LISTING................................ 81

BIBLIOGRAPHY........................................ 126

BIOGRAPHICAL SKETCH.................................. 131













LIST OF FIGURES


Figure Page


1 Development Paths in Constructing
Solution Methods for Problems of
Dynamic Stability of Shells............. 4

2 Shell Element............................. 25

3 Conical Shell Coordinates................ 29

4 Fourier Wave Number Dependence.......... 57

5 Flexural Mode Preference During
Dynamic Instability ..................... 59

6 Linearized Dynamic Stability for
High Axial Load ......................... 61

7 Linearized Dynamic Stability for
Intermediate Axial Load................. 62

8 Nonlinear Dynamic Stability for
Heaviside Pressure Loading............... 65

9 Nonlinear Dynamic Stability for
Intermediate Axial Load................. 66

10 Dynamic Stability Interaction Curves.... 68

11 Convergence Properties of Solution...... 76

12 Flow Diagram of the Computer Program.... 80















NOMENCLATURE


A,B

[B]

D

E

G

K

L

Ma MI, Ma8



N




P
P
'\
P

Q ,Q

R

R

R ,RB
T

U,V,W
',
W


Coefficients of the first fundamental form

Boundary condition matrix

Flexural stiffness

Young's modulus

Shear modulus

Membrane stiffness

Lagrangian Function

Stress couple components

Stress couple components at edges

Number of terms in power series

Stress resultants

Stress resultants at edges

Axial load

Potential energy

Shear stress resultants

Position Vector

Radius of latitude circle

Principal radii of curvature

Kinetic energy

Displacements of arbitrary point in the shell

Displacement function for the buckling mode


vii












V

an bnc n

aa' a ,an

ea e ,eB

f,g,h

h

n
j

m

n

p ,p ,p
PP
pn

q

r

s

s


t

,v,w

,v,w


x3

z

a,B

Y

E


Region

Temporal coefficients

Coefficients of thermal expansion

Strain components

Differential operators

Shell thickness

Normal unit vector

Number of essential boundary constraints

Number of time dependent unknowns

Circumferential wave number

Surface loads

Fourier component of normal pressure

Transformed coordinate

Radius vector

Meridional conical coordinate

Nondimensionalized meridional conical
coordinate

Time

Middle surface displacements

Nondimensionalized middle surface
displacements

Coordinate of axis of revolution

Distance along i
n
Curvalinear coordinates or conical angles

Strain tensor

Error


viii











Circumferential conical coordinate

Eigenvalue

Poisson's ratio

Density

Stress tensor

Nondimensional time

Spatial coefficient

Transformed circumferential conical
coordinate


Additional
Notations

[ ]


[ ]

( ),

( ') ,( )

L J

{ I


Number in brackets refers to reference
numbers in the Bibliography

Letter in brackets denotes matrix

Comma denotes partial differentiation

Denotes partial differentiations with
respect to time

Row vector

Column vector











Abstract of Dissertation Presented to the
Graduate Council of the University of Florida in Partial Fulfillment
of the Requirements for the Degree of Doctor of Philosophy


NONLINEAR ANALYSIS OF DYNAMIC STABILITY
OF ELASTIC SHELLS OF REVOLUTION

By

Marcus George Hendricks

December, 1974



Chairman: Dr. S. Y. Lu
Major Department: Engineering Sciences


It is the purpose of this dissertation to investi-

gate the dynamic stability of elastic shells of revolu-

tion. Two specific areas of this broad field are treated

in detail. First, this analytical study generates pre-

viously unavailable interaction relations for combined

dynamic loadings as they interact to cause passage from a

dynamically stable state to other states. Secondly, the

concept of linearized dynamic stability is extended to

include geometrically nonlinear effects. One role of these

effects is to allow the possibility of autoparametric ex-

citation of preferred flexural modes by the driven modes.

The question of whether such excitation can occur during

the primary dynamic instability motion sufficiently to af-

fect the magnitude of critical dynamic loads is studied.












A modified version of the subdomain method in combination

with circumferential modal analysis is developed for the

solution method. A computer program is constructed to

obtain numerical results and the dynamic stability charac-

teristics of a conical frustum are studied under a variety

of combined dynamic loadings. Interaction curves for

static stability, linearized dynamic stability, and non-

linear dynamic stability are generated.

For the particular loading conditions studied, the

results indicate that dynamic instability will occur at

loads below critical static loads. This reduction in crit-

ical dynamic loads was shown to be the result of both the

dynamic load factor effect and of autoparametric excitation.

The interaction curves which are generated illustrate these

effects quantitatively under conditions of combined dynamic

loading. A criterion for dynamic buckling is established

based on meridional mode shape changes. This ability to

detect sudden jumps in the meridional profile helps to verify

instability detected by divergence on displacement time

history curves and provides additional information about

the poststable state.


U














CHAPTER 1


INTRODUCTION



A dynamic stability problem can be defined as a

problem analyzed by Newton's equations of motion or any

equivalent method [1]. Dynamic stability problems for

continuous systems are nearly always governed by nonlinear

partial differential equations [2]. The solution to a

problem of dynamic stability in thin-walled structures may

be stated as the determination of what load-time combina-

tions cause the displacements of points in the material

body to increase sufficiently to either interfere with

operational specifications or cause a breakdown of the

structure. Methods for obtaining such solutions may be

categorized into three fundamental areas. The most obvi-

ous and usually most difficult method is the direct inte-

gration of the governing equations for sufficiently long

periods to allow direct observation of the ensuing motions.

A second technique is the investigation of the stability

of equilibrium points. The question here is whether a

slight perturbation of a dynamical system from an equilib-

rium state will produce motion confined to the neighborhood











of the equilibrium point or a motion tending to leave

that neighborhood [3]. This method is usually based on

either perturbation or characteristic equation techniques.

A third category of dynamic stability methods is on a

higher level of abstraction than the previous two and

relies on the existence and properties of variously de-

fined functionals. Energy functionals [4] and Liapunov

functionals [5] are two examples.

Definitions of dynamic stability may be categorized

by methods of approach to such problems. In situations

where the direct integration approach is employed, dynamic

instability may be defined to exist when some character-

istic parameter, such as deflection, stress, or strain,

becomes unbounded in one or more parts of the body. When

the method of small oscillations about equilibrium points

is used, instability may be defined as unbounded growth

of perturbations with time or by the existence of one or

more positive real parts of roots of a characteristic

equation. In problems where the solution method is of

the third above mentioned category a sufficient condition

for stability is defined according to the definite form

of the functional employed. In the case of energy func-

tionals, a boundary between stable and unstable regions

is defined by the vanishing of the second variation of

the energy functional. When Liapunov functionals are












employed, stability is defined when the functional is

positive definite and when its derivative is negative

definite.

The particular problem of interest in this study

is the dynamic buckling of shell structures, which rep-

resents a specific application of the more general theory

of dynamic stability principles. Just as the definitions

of dynamic stability are related to the solution methods,

the definitions of dynamic buckling of shells are related

to solution methods as well as the specifics of the phys-

ical problem such as geometry, material properties, or

loading rates.

Solution methods, falling within the first of the

three aforementioned categories, are direct temporal in-

tegration of equations of motion. They may be classified

into five procedural divisions. These are finite differ-

ence, finite element, Hamilton's principle, energy methods,

and methods of weighted residuals. Figure 1 illustrates

the procedural flow and interrelations between these meth-

ods for the study of dynamic stability of elastic shells

and is based on the most prominent articles dealing with

this subject.

First, let us consider recent significant contri-

butions falling within the finite difference method divi-

sion. Witmer et al. [6, 7, 8] have developed during the


_ __




















































Numerical Integration


* Shallow shell theory is often introduced here
** Bifurcation, if enforced, is usually introduced here

Figure 1. Development Paths in Constructing Solution
Methods for Problems of Dynamic Stability
of Shells











past decade a shell dynamics computer program known as

PETROS which utilizes spatial finite difference methods

in both the circumferential and meridional directions.

Utilizing PETROS the authors have studied a number of

explosive forming problems and have shown remarkable

agreement with experimental results. However, consider-

ing that after a ten-year development period, computer

run times are still measured in hours for moderately com-

plex problems, the practical usefulness of PETROS in study-

ing shell dynamic stability is somewhat limited. Using

shallow shell theory, Longhitano [9] developed numerical

solutions for the nonlinear dynamics of hemispherical shells

also utilizing finite difference solutions in two directions.

He showed that suddenly applied symmetic pressures may

couple the breathing mode into more preferred axisymmetric

or asymmetric vibratory modes causing instability at approx-

imately one half static buckling pressures. Again, however,

the two-dimensional discrete type of spatial treatment re-

sults in unusually long computer run times thus limiting

the number of cases treated.

Another class of finite difference solutions to

shell dynamic stability problems have appeared in the lit-

erature which require only approximately three fourths of

the computer run times as the previously cited two-dimen-

sional discrete methods and appear to give comparable











results. Such solutions based on elimination of the

circumferential coordinate via modal analysis prior to

finite differencing in the meridional direction have

been presented by Cromer [10], Ball [11, 12], and Kim

[13]. The results of Cromer and Ball also indicate that

suddenly applied axisymmetric pressures may couple breath-

ing mode response into preferred vibratory modes to cause

dynamic instabilities at pressure loadings lower than

static critical loads. These studies indicate that cou-

pling between parametrically excited flexural modes is

very weak and often may be neglected without compromising

the results.

Stricklin [14] and Wu [15] have made recent and

significant contributions to the nonlinear dynamic analy-

sis of shells by employing the finite element method. Re-

sults published by both of these investigators indicate

that accuracy comparable to methods emanating from the

equations of motion is obtainable if sufficiently complex

elements are used in large quantity. However, due to

longer computing times required for finite element tech-

niques, their major advantage clearly lies in the treat-

ment of complex structures where equations of motion are

not reasonably derivable. A seldom mentioned, yet almost

universal, appealing characteristic of finite element

techniques warrants a brief mention. This is that the











unknown dependent variables in a finite element solution

have a direct physical interpretation at all times as

the solution proceeds. Thus, to some extent, the analyst

never loses physical contact with his problem. This is

in contrast to most other analysis techniques applicable

to shell dynamics where the solution is carried forward

in some transformed mathematical space.

Klosner [16, 17] in a series of studies covering

a five-year period studied the effect of suddenly applied

axial loads on cylindrical shells. His solutions were ob-

tained by elimination of the circumferential coordinate

from nonlinear shallow shell equations and applying the

Ritz method or Hamilton's principle to the relations re-

sulting after introducing assumed solutions. His results

indicate that suddenly applied axial loads do not exhibit

the dynamic load factor effect found by others [9, 10, 11,

12] when suddenly applied normal pressures are considered.

These investigations reveal that a valuable indication of

the onset of dynamic instability can be found in the re-

versal in the trend of the time to first maximum deflec-

tion of the critical mode.

Another subcategory of solution methods in studying

dynamic shell stability by direct temporal integration is

known as the energy method. This technique circumvents

the derivation of the partial differential equations of











motion. The method involves deriving internal energy

expressions in conjunction with appropriate kinematic

relations, substituting assumed solutions into these

relations, and then applying Lagrange's equations to

the result. This procedure leads to either a coupled

system of nonlinear ordinary differential equations if

the spatial integrals are evaluated analytically or a

coupled system of nonlinear integro-differential equa-

tions if the spatial integrals are to be evaluated

numerically. In either case, the differentials are

temporal and must be solved numerically. Goodier [18],

Mclvor [19], and Lindberg [20] have presented solutions

to a particular type of dynamic instability of cylin-

drical shells utilizing this solution procedure. Hubka

[21] has presented alternate methods for numerical inte-

gration of the final equation sets for this problem.

This problem is concerned with the circumferential modal

coupling arising from the nonlinear membrane stiffness

terms in the equation sets obtained by this energy method

technique. The main results of these studies are that non-

linear coupling between circumferential modes may under

conditions of minor excitation or initial imperfections

in the preferred vibratory modes cause energy extraction

from the driven breathing mode into one or more preferred

modes such that the preferred mode becomes dynamically

unstable.


I











Utilizing this same energy method derivation,

Mente [22] has studied the dynamic nonlinear response

of cylindrical shells subjected to asymmetric pressure

loading. His computerized solution, known as DEPICS,

is able to handle a variety of coupled modes during the

temporal integration. The nonlinear stiffness is handled

by solving a linear eigenvalue-eigenvector problem at

each time step then adjusting the succeeding stiffness

value based on the nonlinear terms. Unfortunately this

rather extensive computer program has to date been unable

to satisfactorily define any regions of instability.

Significant Russian contributions to the study of

dynamic stability of shells have been made. Among these

are works by Shumik [23, 24] who also employed solutions

by direct temporal integration of expressions derived by

the energy method. These works deal with suddenly applied

loads using a linearized uncoupled dynamic stability theory.

His results indicate that cones with large taper buckle

dynamically at circumferential wave numbers higher than

the critical wave number for static loading.

Shiau [25] utilizing an energy solution studied

the effects of imperfections on the dynamic stability of

conical shells subject to axial impact. His results

agree with Klosner's [16, 17] in so much that a reversal

in the trend of the time to first maximum deflection of











the critical mode may be viewed as an indication of the

onset of dynamic buckling. The results, however, are

so critically dependent upon the choice of the single term

assumed buckling mode that the procedure appears to be

of limited general value.

A final division of solution methods in the cate-

gory of direct temporal integration is the method of

weighted residuals. The details of the procedure will

be covered in a later chapter. Nash [26] utilizing

shallow shell theory in conjunction with the Galerkin

method of weighted residuals investigated the response

of thin conical shells to dynamically applied axial

forces. He found that a rapid increase in end shorten-

ing may be used to denote the onset of dynamic instability.

Fisher [27] employed a similar solution method to

study the dynamic buckling of reinforced circular cylin-

ders subjected to suddenly applied axial compression load-

ings. He also found that a reversal of the trend of the

time to maximum deflection corresponds to onset of dynamic

buckling. Lakshmikantham [28] using shallow shell theory

in conjunction with smeared analysis and the Galerken tech-

nique investigated the same problem as Fisher [27]. His

solutions indicated that a jump in radial deflection was

the correct criteria for indicating dynamic instability.

An earlier work by Dietz [29] studying the same problem














by shallow shell theory and the Galerkin method indi-

cated that loss of stability is best defined as a sudden

drop in the axial load required to maintain a constant

end shortening rate.

By taking the results of the preceding literature

survey in context, several comments may be made concerning

the current state-of-the art in analytical studies of

dynamic shell stability. First, it appears that nonlinear

response problems which retain full circumferential modal

coupling yield the most reliable indicators of the onset

of dynamic buckling. Also the results appear to be highly

sensitive to theoretical approximations and to the quality

of the numerical solution procedures. Secondly, the study

of passage through bifurcation points is well developed

only for linearized dynamic stability techniques. The

few articles proposing to deal with nonlinear dynamic

stability unfortunately discarded the quadratic nonline-

arities, which are the most significant ones, retaining in-

stead only the cubic nonlinearities. This was done in order

to circumvent having to deal with inter-modal coupling. There-

fore, the results of these studies appear inconsistent.


1











It is the purpose of this dissertation to invest-

igate the dynamic stability of elastic shells of revolu-

tion. Two specific areas of this broad field are treated

in detail. First, this analytical study generates previ-

ously unavailable interaction relations for combined

dynamic loadings as they interact to cause passage from

a dynamically stable state to other states. Secondly, the

concept of linearized dynamic stability is extended to in-

clude geometrically nonlinear effects. One role of these

effects is to allow the possibility of autoparametric

excitation of preferred flexural modes by the driven modes.

The question of whether such excitation can occur during

the primary dynamic instability motion sufficiently to af-

fect the magnitude of critical dynamic loads is studied.

The theory derived herein is suitable for the study

of the dynamic stability of general shells of revolution.

In order to obtain numerical results, however, a specific

shape must be specified and the one used for numerical

analysis in this study is a truncated circular conical

shell structure. Conical frustum shells are in frequent

use as structural elements. Moreover, a conical shell

reduces to a cylindircal shell when the semivertex angle

of the cone becomes zero. Similarly, it reduces to a flat

circular plate when the semivertex angle of the cone ap-

proaches a right angle. Thus, the analysis and computer











program developed for cones can be generalized to handle

these two related problems.

A modified version of the subdomain method in

combination with circumferential modal analysis is de-

veloped for the solution method. A computer program is

constructed to obtain numerical results and the dynamic

stability characteristics of a conical frustum are stud-

ied under a variety of combined dynamic loadings. Inter-

action curves for static stability, linearized dynamic

stability, and nonlinear dynamic stability are generated.

A dynamic buckling criterion based on meridional mode

shape changes is established and the quantitative impor-

tance of nonlinear coupling effects is illustrated.















CHAPTER 2


THEORETICAL DEVELOPMENT OF NONLINEAR SHELL EQUATIONS



2.1 Shell Coordinates


A set of normal curvilinear coordinates (0, B, z)

is chosen in the shell space such that a and B lie on the

undeformed middle surface, z is perpendicular to this

middle surface, and z=O lies at the middle surface. The

equation of the undeformed middle surface is given in

terms of two independent parameters a and B by the radius

vector


r = r(a, ). (2.1)



To describe the location of an arbitrary point in the

space occupied by a shell, the position vector is defined

as


R(a,B,z) = r(a,B) + zin(a,8) (2.2)



where z measures the distance of the point from the corres-

ponding point on the middle surface along in. The unit

vector i is normal to the surface.











2.2 Kinematic Relations


Acceptable nonlinear strain-displacement relations

for shells may be obtained from one of two basically dif-

fering approaches. One approach is to utilize the finite

strain tensor of the three-dimensional theory of elasticity

in conjunction with the Gauss-Codazzi relations for a sur-

face [30, 31, 32, 33, 34]. Another method is the two-

dimensional approach to shell theory that evades the ques-

tions of the approximations involved in the descent from

three dimensions. This approach defines strain as one-

half of the difference between the material and spatial

metric surface tensors. Sanders [35] presented the first

consistent nonlinear kinematic relations derived by this

method using a continuum called a Cosserat surface.

Naghdi [36] discusses both of the above methods in detail.

Nonlinear kinematic relations are seldom utilized

in shell analysis in their complete form due to their

complexity. Therefore, simplifications are routinely made.

Throughout the following analysis it is assumed that the

strains in the middle surface and that the rotations about

coordinate axes are small. These assumptions imply that

an element of area on the deformed middle surface is iden-

tical in size to an element of area on the undeformed

middle surface and that the difference between the










Christoffel symbols of the deformed and undeformed coordi-

nate systems is zero. These assumptions result in a con-

siderable simplification in the resulting equations with-

out unduly restricting the applicability of the solutions.

The strain-displacement equations shown below were

derived using the finite strain description of the three-

dimensional theory of elasticity in a manner similar to

Ogibalov [32] and the nonlinear terms were regrouped into

surface rotation expressions. The resulting kinematic

relations are


ea = (1/(l+z/Ra))[U, /A+A, V/AB+W/Ra+(l/(2(1+z/R)))

x(-W,a/A+U/R )2


e = (1/(l+z/R ))[V,'/B+B, U/AB+W/R +(1/(2(1+z/R )))

x(-W, /B+V/R )2] (2.3)



e = (1/(2(1+z/R )) (V, /A-A, U/AB)

+(1/(2(1+z/RB )) (U,g/B-B,aV/AB)

+(I/(2(l+z/R )(I+z/R )))(-W,a/A+U/R )(-W,/B+V/R ).


In equations (2.3) the covariant components of the normal

and shear strains are denoted by e e e U respectively.











The components U, V, and W represent displacements of

an arbitrary point in the shell. R and R are the
a B
principal radii of curvature and A and B are the coeffi-

cients of the first fundamental form. Comma denotes

partial differentiation.


2.3 Constitutive Relations


The relations between the components of strain
and stress in an orthotropic, linearly elastic material

are given by Hooke's law [37] as


e= a/E -v a /E a n/E n+a T
a a

e = -V a /Ea+o -aBn/E non/E +aT


e = -Vna0a/Ea-n aB6/E B+n/En+anT
(2.4)
2eaB = oa6/GaB

2e = aB/G


an n/Gan.




Young's modulus in the a and B directions is de-

noted by E and E respectively. The Poisson Ratio vY

designates the contraction in the a direction caused by











a positive normal stress a in the B direction. G is
aB
the shear modulus in a plane which is tangent to the

(c,B) coordinate surface and (a a ) are the coeffi-

cients of thermal expansion in the a and B directions

respectively. These relationships are valid for materials

subjected to stresses below the proportional limit. By

introducing Kirchhoff's hypothesis and allowing for the

symmetry of the elastic parameter matrix [38] the above

equations may be written as



oa = [1/(1- VBa)][Ea e +1BaE e-(aa+ E a aE )T]

B = [1/(1- aV B )][E e +ve E e a (a E +aa E )T] (2.5)

aB = 2G aBe



2.4 Kirchhoff's Hypothesis


The reduction of the three-dimensional problem to

a two-dimensional one requires an assumption concerning

the variation of strain, displacement, or stress across

the shell thickness. To satisfy this requirement the

fourth assumption in Love's first approximation [39],

known as the Kirchhoff hypothesis, is introduced. This

hypothesis entails the vanishing of transverse shearing and

normal strains [33] and may be formulated as follows:











U(a,B,z) = u(- a, )+zea(a,B)

V(a,B,z) = v(a,3)+zeg(6 ,8)

W(a,B,z) = w(a,B) (2.6)

e = u/R -w,a/A = v/RB-w, /B


where u, v, and w are the components of displacement at

the middle surface in the a,B, and normal directions,

respectively, and 6 and 6 are the rotations of the

normal to the middle surface during deformation about

the B & a axes, respectively. The acceptance of this

assumption is due to its great clarity [40]. Although

Kirchhoff's hypothesis is a first approximation, its

applicability to nonlinear shell theory is well known

[33, 35]. The problem of determining the error intro-

duced by the hypothesis on the preservation of the normal

has thus far not been solved exactly [41]. Novozhilov

[42], Mushtari [43], and Koiter [44] have estimated that
the errors introduced by Kirchhoff's hypothesis are of

the order of h/R, which is small for thin shells.

Kirchhoff's hypothesis, modified to account for

transverse normal strain, has been employed in [31, 33,

34, 36, 45]. Effects of transverse normal strain were

not included in the present analysis.











2.5 Variational Principle



2.5.1 Hamilton's Principle


The Irish mathematician and physicist, Sir William

Hamilton (1805-1865), formulated his celebrated principle

in dynamics in which the governing equation depends ex-

plicitly on the energy of the system [46]. Hamilton's

principle is stated as an integral equation in which the

energy is integrated over an interval in time. In the

language of the calculus of variations, Hamilton's prin-

ciple states that the first variation of the time integral

of the difference between the kinetic energy T and the

potential energy P of a dynamical system is zero, that is,

t2
6 f (T P)dt = 0. (2.7)
t1



The equation is assumed to hold for all dynamical systems

whether they are conservative or nonconservative. The

quantity (T-P) is called the Lagrangian function and is

denoted by L. With this designation equation (2.7) be-

comes

t2
6 fLdt = 0 (2.8)
tI











and Hamilton's principle then asserts that the first vari-

ation of the time-integral of the Lagrangian function is

zero. Hamilton's principle is employed in the next sec-

tion to derive the equations of motion and the admissible

boundary conditions for shells.



2.5.2 Equations of Motion


Of the many procedures available for the derivation

of the equations of motion for a differential shell element,

Hamilton's principle is superior in nonlinear shell theory

due to its efficient treatment of problems involving curv-

ilinear coordinates and because it gives the admissible

boundary conditions, natural or forced, that are to be used

with the theory [47, 48]. With the total potential energy

P expressed as the difference between the internal strain

energy and the potential energy of the external forces, we

obtain from equation (2.8)



t2
6f { v [(P /2)Vivi-(1/2)siJYij]dv+/ si'Vids}dt = 0 (2.9)
t1 O 0



Here the density of the undeformed body and the Lagrangian

components of the displacement vector are p and V., re-

spectively. The symmetric Cauchy stress tensor sij is











measured per unit area of the undeformed body and must

be referred to the base vectors in the deformed body and

the strain tensor, Yij, is defined by equation (2.3).

The components of the surface force s are referred to

the base vectors in the undeformed body and the volume

and surface integrals in equation (2.9) must be extended

over the volume and surface of the body in its undeformed

state.

Introducing the kinematic, constitutive, and

Kirchhoff relations from previous sections, we eliminate

the time derivatives of the variations by integrating by

parts with respect to time and require the virtual dis-

placements to vanish at the end points of the arbitrary

interval tl t t2, thusobtaining the equations of motion,

stress resultants and couples, and boundary conditions.

The equations of motion derived in this section

reflect the assumptions of Kirchhoff's hypothesis. Small

strains and moderately small rotations were also assumed.

Thinness assumptions were delayed until after the equa-

tions were synthesized.









The equations of motion are:


(BN ), +(ANBa), +A, Nae-B, N +ABQA/R -AB(NQO +NB )/Ra

+ABpp = ABphu(l+h2/12RR )+ABph3(1/R +1/R )5 12


(BNB'a),+(AN) +B, NBa-A, Na+ABQS/R -AB(NB e+N50 )/R

+ABp = ABphV(I+h2/12R R )+ABph3(1/R +1/R ) /12


(BQ0) +(AQ), -AB(N'/R +NO/R )-(BO Na+BO ),
0 (2.10)
-(AeO +AO N ),+ABpn = ABphW(1+h2/12RaR )


(BM ) +(AMB), ,+A, Ma~-B, M-ABQ0

= ABph3 [(1/+l/R )U+ a]/12


(BMVa) a+(AMB) ,+B, aMa-A, M0-ABQB

= ABph3[(1/R +l/R )V+]/12



where (") denotes second partial differentiation with
respect to time and p p p represent the applied
surface loads.











The stress resultants and stress couples are given by


N0 h/2 a"
{N = / { B }(l+z/R )dz
N -h/2 a


NB h/2 ,B
{N } /2{ I l+z/Ra)dz
N -h/2 u


Mt h/2 a
M } = / { (l+z/R )zdz
MOB -h/2 oa

(2.11)

{MB h/2 Ba(1+z/R )zdz
Ma h/2 an6



Q } h/2 { l+z/RS}dz
0 -h/2 an 1+z/R

h/2
B = oadz
-h/2

h/2
M"B = aB zdz
-h/2




where h is the shell thickness.

Figure 2 illustrates the relations between the

stress resultants and couples to the generalized coordi-

nates and to the shell element.


1













































































Figure 2. Shell Element


I











2.5.3 Boundary Conditions


From equations (2.9), (2.10), and (2.11) we get

the following natural boundary conditions that arise

from the requirement of force balance. Along the edge

of constant a the boundary conditions are



N = N or u = u


N B+MB/R =


Qa+MB-N -N =
p a- pg


B or v = v


Q or w = w


M0 = M or = a
a .


The boundary conditions

from equation (2.12) by


along constant B may be obtained

interchanging a with B and u with


v.



2.6 Synthesis of Equations


The usual procedure followed is to reduce the

number of equations and unknowns to a more manageable

number by eliminating Qa and Q from the equations of

motion (2.10) which are thus reduced from five to three.

The force and moment resultant expressions (2.11) to-

gether with the constitutive relations (2.5) are then


(2.12)










substituted into the equations of motion. Finally, the

strain-displacement equations (2.3) and (2.6) are substi-

tuted, to yield three differential equations of motion

having u, v, and w as dependent variables and a, B, and

t as independent variables.


2.6.1 Shells of Revolution


In the preceding analysis R and R are arbitrary,

but in all subsequent applications these principal radii

will refer only to shells of revolution. It can be shown

[47] that for a shell of revolution the principal radii

may be expressed as


R = -[l+(aR o/ x 3)23/2/(D2R / x32
(2.13)
R = R /1+(Ro0/3x3)2


where R is the radius of the latitude circle and x is

the coordinate coincident with the axis of revolution.


2.6.2 Conical Shells


The above described synthesis may be carried out

using equation (2.13); however, the equations would be

extremely unwieldy. Thus, the procedure will be followed

only for a specific shell of revolution shape and the one











used in the remainder of this study is a circular conical

shell. A conical shell reduces to a cylindrical shell

when the semivertex angle becomes zero. Similarly, it

reduces to a flat circular plate when the semivertex

angle approaches a right angle. Therefore, the following

analysis can handle cones as well as the two related

problems. The conical coordinate system adopted is shown

in Figure 3. The coordinates, coefficients of the first

fundamental form, and principal radii become


a = s = 6

A = 1 B = s sin a (2.14)

R = R = s tan a.


In all subsequent relations, a will denote the semivertex

angle.

Assumptions relating to displacement magnitudes are now

introduced into the analysis. The squares of inplane

displacements are assumed to be small with respect to the

squares of normal displacements and the shell thickness

is assumed to be small compared to the radius.


















A

L.


Figure 3. Conical Shell Coordinates


I











Neglecting rotary inertia terms, the nonlinear

equations of motion are


s s 0 Os
N +sNs -N +N = sphu


-ctn a vN /s+ctna w, N /s+N +2NS+sN +ctna wsNso


+ctna M, /s+2ctna MS/s+ctna M,- = sphV


W, S+sw, sNs+sw,sNss-ctna N -ctna v, N6/s+w, N /s (2.15)


-ctna vN6 /s+w, N, /s-ctna v,s Ns+2w,s Nse-ctna vN0


+w, N, +w, sN,+2Ms,+sss-M, +M /s+2M1,/s+2M +sp



= sphw


where


i = 0 sina.


I










The nonlinear stress resultants and stress couples
are


Ns = [Eh/s(l-v2)]{su, +s(w, )2/2+v[v, +u+ctna w+(w, )2/2s

-ctna vw, /s]}


Nse = Nes = (Gh/s)(sv,s+u, -v+w,sw,-ctna vw,s)


N = [Eh/s2(1-v2)]{sv, +su+sctnc w+(w, )2/2-ctna vw,

+ctn2a v2/2+v[s2u,s+s2w,s)2/2]} (2.16)


Ms = [Eh3/12s(1-v2)]{-sw,ss+ctna us+ctna(w,s)2/2

+v[-w, /s-w,s+ctna(w, )2/2s2]


Ms- = Ms = (Gh3/12s)(-2w,s +2w, /s+ctna w, W,s/s)


Me = [Eh3/12s2(l-x2)](-w, -sws-ctna u-ctn2a w-vs2w, s)










Substituting equations (2.16) into equations
(2.15) yield three equations of motion. Noting that

Gh = K(1-v)/2

we have


K{[-u/s+u, s+su, ss+ ,/2s-3v, /2s+vs/2-ctna w/s

+(w,s/2s)(w, -ctna v, )+(w,s/2s)(w, -ctna v)+(w,) 2/2

-(w,)2/2s2 +ctna vw, /s2-ctn 2 v2/2s2+sw,sw,ss]
(2.17)
+v[-u, //2s+v, s/2s+ /2+ctn w,s-W, s/2(Ws+W,' /s

-ctna v, /s)+w, /s(w, s/2-w, /2s-ctna v,s+ctna v/s)

-ctna vw, s/2s]} = phsu



K{3u, /2s+u, s/2-v/2s+v,s/2+sv,ss/2+v, /s+ctn w,/s

+w, (w'ss/2+w, /s2)+w's('s +W/s)/2

+ctna[-sv(w,ss/2+w, /s2)+w,(-v+u, /2)+w, u/s]/s

(2.18)
+v[-u, /2s+u, s/2+v/2s-v,s/2-sv,ss/2+w,5sW s/2-w,s, /2s

-w,ssw, /2+ctna u,s/s(-ctna v+w, )+ctna w, (v-u, /2)/s

+ctna vw,ss/2]} = phsV










K{-ctna (u+v, +ctna w)/s+ctna[(w, )2/2+ww, ]/s2+su,sw,ss

+w,s(suss+V,/2+u, 4/2s)+w, v, /s2+w, (v,' /s2+v'ss/2

+u,'s2s)+w's ,(u, /s+v's)+u's5, s+uw, s2+u w ,/2s2



2 2 2 2
-vw,so/S-Vs W,/2s+vw, /2s2 -v, W,'s/2s+ctna(-v, 2/s2 -uv, /s2

-vv, /s2-u, v/2s2-V,s2/2-u,V,'s/2s+vV,s/s-vV,ss/2-u'sv/2s

-v2/2s2)+ctn2a (-v, w/s2)+v[-ctna u,s+W, (v, +ctna w)

+w, (u,s/2s-v,ss/2)+us ,/s+Ws(v, /2+ctna w,s/2-u, /2s)

+uw,ss+us, s-vsW,'s-u, W,s/s+vws /s+uW, /2s2 (2.19)

+V,s W/2s-vw, /2s2+v, W,s/2s+ctna (-u,s V,/s-U,s v/2s

+v,2/2+u, v,s/2s+vv, s/2-u, v/2s2-vv,/s+v2/2s2)]}

+D{-sw, ssss-2w,ss#/s-w, /s3+2w',sW/s2-2w sss-4w, /s3

+W,s/S-W,/S +Ctn(Usss+Wss ssss ss

+Ws 'sW, /s2W_'sW, /s3, 'ss'W,/s2 +(w, )2/s2

+v[(ctnm/s2)((-7/2s)(w, w,s )+3(w, )2/s2+W, s W,/2s

-Wss W, W,'sWsOO)]}+spn = phsW





34




where the membrane stiffness is


K = Eh/(l-v2) (2.20)



and the flexural stiffness is


D = Eh3/[12(1-v2)] (2.21)














CHAPTER 3


NONLINEAR DYNAMIC STABILITY
EQUATIONS OF CONICAL SHELLS



The nonlinear dynamic stability equations are

obtained from the nonlinear equations of motion by

employing a perturbation analysis. Substituting the

displacements


u = I + uB


v = v + vB (3.1)


w = I + wB



and the stress resultants and stress couple expressions



Ns= Ns + N 1Ms = Ms + M
I B I B


N = N + N M = M6 + M (3.2)
I B = M + M(B

I N+NB I B











and the load relation


P = PI + PB (3.3)


into equations (2.15) results in two coupled sets of equa-

tions. The quantities with subscript I are measured from

the undeformed state and the quantities with subscript B

are small but finite perturbations. This gives one equa-

tion set describing the motion in the stable space surround-

ing the undeformed state and another set, the stability

equations, describing the motion during passage thru the

nearest bifurcation point.

The resulting set of nonlinear dynamic stability

equations are as follows where the subscript B has been

dropped for convenience:



K[(-u/s+u,s+su, ss+u,/2s-3v,/2s+v,s /2-ctna w/s)


+v(-u,0 /2s+v, /2s+vs/2+ctna w,S)
(3.4)
+(w, /2s+w2s+w ,/2s+w /2-w. /2s 2+swsW,
Ss/2-w,/2s + sWss)


+v(-w,2/2-w, w ./2s+w w /2s-w 2/2s2)] = phsi
5 'ii 's) '= h










K[(3u, /2s+u, s/2-v/2s+vs/2+sv,ss/2+v, /s+ctn w,/s)

+v(-u, /2s+u, /2+v/2s-v,s/2-s,ss,/2)
(3.5)
+(w,W,'ss /2+w, w, /s 2+w, s s/2+Ws /2s+ctn 2 ww, /s2


+v(w,s W,s/2-W, sW /2s-w, ssW,/2)] = phsV




K{-ctna(u+v, +ctna w)/s+v(-ctna u, )

+ctna [w,2/2s2+ww, /s2+v(w, /2+ww,s)]}


3 22
+Df(-sw'ssss-2w'ss"/s-w, W /s 3 +2wsIK/S -2wsss

-4w, /s3 +Wss-/ss/ )+ctno[w, s2'ss W,/s2


+W,s2/S +W,sW,5s -W', W's/s -W,s,/s +w wss~/s2 (3.6)


+W,sW, s/s 2+v(-3w, W,'s/S +3w, /2s4 +WsW /s3


-Wss"W,/s2 -,s s/s2)]} + spn

ss +s'sss 's = phs
+wN I+sw, NI+sw I s+W VN Is w, Nis = phsP
- - - -- -- -- - - - -











The single underlined quantities represent non-

linear membrane stiffness terms. The double underlined

quantities represent nonlinear bending stiffness terms.

The nonlinear stability terms are denoted by a dashed

underline. These equations embody the assumptions that


wI < wB

and that


uB < wB

vB < wB


In the present study, a thin truncated conical

shell is loaded by a constant axial load and a nearly

axisymmetric Heaviside pressure load. The membrane-

state stress resultants are given by


Ns = -P/(ms sin2a)-pn(s tana)(1-cosx t)/2


N = -pn(stana)(l-cosX t)



where P is the applied axial load, pn is the magnitude

of the Heaviside pressure load, and A is the natural

frequency of the breathing mode of the shell.














CHAPTER 4


METHOD OF SOLUTION



4.1 Circumferential Modal Analysis


The nonlinear partial differential equations

derived in previous chapters are spatially two-dimen-

sional. The two directions are meridional and circum-

ferential. In this analysis, the two-dimensional equa-

tions are reduced to one-dimensional equations by

utilizing circumferential modal analysis. The inde-

pendent variable e is eliminated by expanding all

dependent variables into sine or cosine series in the

circumferential direction. As a result of the trig-

nometic series expansions, there is one set of govern-

ing equations for each circumferential wave number

considered. For a linearized analysis the sets are un-

coupled. For a nonlinear analysis the equation sets are

modally coupled through the quadratic terms. This cou-

pling arises when use is made of the trignometric prod-

uct identities


cos(ie)cos(je) = (1/2)[cos(i-j)6+cos(i+j)e]
(4.1)
sin(ie)sin(je) = (1/2)[cos(i-j)e-cos(i+j)e]











in order to facilitate the equating of like coefficients

of Fourier expansion terms which is required to identify


the equation sets.

Substituting


u(s,4,t) =


Q
v(s, ,t) = Z
n=o
Q
w(sp,t) = Z
n=o
Q
p(s,is ,t) = z
n=o


the expressions


u n(s,t) cos(np/sina)


v n(s,t) sin(np/sina)

w (s,t) cos(nJ/sina)


p (s,t) cos(n(/sina)


into the nonlinear dynamic stability equations (3.4 -

3.6) and equating like coefficients of Fourier expan-

sion terms yield the following:


(-l/s){l+[n2(l-v)/(2sin2 )]}u n+un's+sun'ss


+(n/2s sina)(-3+v)v +(n/2sina)(l+v)vns-(ctna wn/s) (4.3)


+vctna wn s = phs n/K


(4.2)










(n/2s sina)(-3+v)u -(n/2sina)(l+v)un's

+{-[(1-v)/2]-(n2/sin2a)}(vn/s)+[( -v)vn's/2] (4.4)

+[(1-v)svn,ss/2]-[n ctna wn/(s sina)] = phsVn/K



-ctna [(u /s)+vuns+(n/sina)(v n/s)+(ctn w n/s)]

+(D/K){[(4n2/sin2 )-(n4/sin4a)](wn/s3)+[(2n2/sin2 )+1]

x[(Wn'ss/S )-(wn,'s/s2)]-2wn sss- n ssss p n/K

-[P Wn'ss/(KFsin2a)]+(l-cosX ot)[-s tana Bcs
2 n / 2n 2n n/ s

-(s2tana ,ss/2)+(2n2 n/Sin2a)-(2nBn/sina)]/K
c- 2 2 2 2
+Tc{ctna{(-n2w2/4s2sin c)+v[(w ,2/2)+(w ,s/4) (4.5)

+ww ss+(w w /2)]}+(h2ctna/12){w 2+w 2/2
o o'ss n n'ss o'ss n'ss

-[n2/(2s2sin2a)](wn wnss-wn,2)}}+nc{ctna[(-n2W Wo /

(s2sin2a))

+v(w w +w w +w w ,s)]+(h2ctnx/12)[2w ,sW
(Wo'sWnso'ssn onss onss n'ss

2 2 sin2= /K
-(n /s sin C)w w]} = 0hsvi/K











where

n = circumferential wave number


-c i n=o
So n>o


c Z w /2[ (i+n)+ P i-n| c n o (4.6)



= E wi/2[np(.i+n)+cPinli_nlP1
i=o

n
B = Wi/2[nsp (i+n) sli-nIp li-n '],
1=0

s n=o
ni n>o



s 41 i n+l1
S=o i=n
1 isn-1




4.2 Elimination of Meridional Derivatives


The next phase of the analysis involves the re-

duction of the partial differential equations sets result-

ing from the circumferential modal analysis to ordinary

temporal differential equation sets which are then amen-

able to numerical solution.

Referring again to Figure 1 it is seen that the

available solution methods for this task fall generally

into four categories. These are finite element methods,

finite difference methods, variational principles, and












methods of weighted residuals. As noted by Finlayson

[49], for certain types of linear problems these methods

can be shown to be equivalent to each other. From the

literature abounding on shell dynamics one can safely

conclude that any of these methods when judiciously ap-

plied will give acceptable results. There are, however,

marked differences between the methods in the amount and

quality of manual mathematics and the quantity of machine

computing time required to give equivalent results.

The desired characteristics of the solution method

for the particular problem of interest in this disserta-

tion are now discussed. The solution method utilized

should yield an approximate direct solution where con-

vergence would be monotonic and assured. Spatial control

of error residuals was desired. Because the taper in the

conical shell results in nonsymmetric stiffness matrices,

a solution technique applicable to non-self-adjoint prob-

lems was required. Finally, it was hoped that the chosen

solution method would result in short computer compile

times because experience has shown that the majority of

computer time required for such a problem is in successive

compile times during development and debugging.

In matching these desired characteristics to the

known characteristics of the classical solution methods

an elimination process was begun which finally resulted

in the algorithm used herein.


I











Finite element techniques were eliminated from

consideration because this method is not competitive with

other methods in problems with regular geometries and

formulatable boundary conditions lying along coordinate

lines. This method requires significantly more manual

mathematics and computer execution time than equilibrium

equation solution methods in order to give comparable

results.

Finite difference methods have been shown to be

extremely powerful in solving nonlinear dynamic stability

problems involving shells. This method has been used with

particular success when coupled with circumferential mode

analysis. It was shown, however, by Radosta [50] that

convergence is slow when a meridional mesh is used with

a shell under high axial load. In the particular problem

at hand, i.e. a conical frustum subject to a high axial

load combined with a time dependent pressure load, it was

felt that a suitable selection of number and spacing of

meridional mesh points would degrade into a highly time

consuming trial and error effort.

The final phase of solution method selection con-

sisted of choosing one of the four methods of weighted

residuals: Galerkin, Least Squares, Collocation, and Sub-

domain. The least squares method was not seriously con-

sidered because it yields ordinary differential equations











of the second degree in the temporal operator. The

collocation technique was eliminated because it enhances

the same point location selection drawbacks as does the

finite difference method. Again, this is normally an

insignificant problem but was made significant here by

the expected buckled states resulting from the combined

type loadings being considered. The final choice between

the Galerkin and subdomain methods was based on the method

of error residual control. In the Galerkin technique the

error produced by each individual term in the assumed solu-

tion is forced to average to zero over the entire length

of the shell. In the subdomain method the error in the

total assumed solution is forced to average to zero over

each subdomain. This type of spatial control for error

residual was considered superior to the modal type of

error control offered by the Galerkin method and there-

fore, the subdomain technique was selected for use in this

investigation.



4.2.1 Subdomain Method


A description of the theoretical basis of the sub-

domain method is now in order. Consider the equation


L[u(x,t)] = 0 in V (4.7)

where L is a nonlinear, non-self-adjoint, ordinary











differential operator. If the approximation solution is

represented in the form
m
u = Z a (t) i(x) (4.8)
i= 1 1


and this is substituted into equation (4.7), we obtain

an approximation


L[-(x,t)] = e (4.9)


where E is the error. We now impose constraints on E by

requiring it to average to zero over a subdomain v of

the total domain V


iL[u(x,t)]dv = 0 (4.10)
v

The number of subdomains v are chosen to equal m the

number of time dependent unknowns in (4.8). The differen-

tial equation, integrated over the subdomain is then zero,

hence the name subdomain method. As m increases, the dif-

ferential equation is satisfied on the average in smaller

and smaller subdomains, and presumably approaches zero

everywhere. This process yields a dynamically coupled

second order set of nonlinear temporal differential equa-

tions which may in turn be solved by numerical methods.

Equation (4.8) is called the trial function and

may be selected in a variety of ways. It may be chosen











to satisfy the boundary conditions, but not the differen-

tial equations or to satisfy the differential equations

but not the boundary conditions. In order to simplify

the required hand calculations, a set of complete power

series were chosen as the trial functions for this prob-

lem. The trial functions were then forced to satisfy

the boundary conditions by a modified application of the

Lagrangian multiplier method which can be more properly

thought of as a coordinate transformation to require an

assumed power series solution to satisfy the essential

boundary conditions. An approximate satisfaction of the

differential equations was then obtained by the subdomain

method.


4.2.2 Method of Satisfying Boundary Conditions


A description of the technique utilized to satisfy

boundary conditions will now be presented. Consider the

trial function (4.8) in matrix form,


u = L J {a} (4.11)
(1xm)(mxl)


and the j essential boundary constraints,


[C.]{a} = 0. (4.12)
(jxm)(mx1)










Equation (4.12) may be partitioned as follows


E[ C1 C2
[jx(m-j)] ((jxj)


r 1
(m- 0.
(mx1)


Equation (4.13) may be rearranged as



{q2} = -[C2 ] [ C1 ] {q1
(jxl) (jxj) [jx(m-j)][(m-j)x1]


and noting that


{ql = [I]{ql}


we obtain


{a} = [ B ] {ql
(mxl) [mx(m-j)][(m-j)x ]



and (4.11) becomes


u = L4J[B]{ql}


where [B] may be considered as a projection operator.


(4.13)


(4.14)


(4.15)


(4.16)


(4.17)












After nondimensionalizing equations (4.3 4.5)

according to


s= s2s h = s2h

Un = 2Un I = PcrI
v = s2v
n n P = 2TEh2cos2 //3(1-v2)
c r
w = s cr
n 2n(
(4.18)


S= /E/p(l-v2) t/s2


P = Eh/(1-v2)s2 Pn



n = s2/p(1-v2)/E A


the tr

The tr


ial functions are

'ial functions are


u a
n n
vn N-1 b
n n

Wn =o Cn

Pn Pn


derivatives are likewise

nondimensionalized.


introduced into the equations.


i
i -i.
S. s .
i
i


(4.19)


After applying the transforms satisfying the

boundary conditions, the equations are then integrated

over (N-j) subdomains and the average residual error for


I












each subdomain equated to zero. During the original

formulation it was noted that this approach leads to a

singular mass matrix for equally spaced subdomains. To

circumvent this problem the equations were mutliplied

through by s. This is similar to a procedure known as

the method of moments first employed by Polhausen [51].

Having completed the above sequence of operations

results in Q sets of nonlinear second order temporal dif-

ferential equations. Each set contains 3 (N-j) equations

and the sets are coupled in the nonlinear terms. In ma-

trix form a typical set may be expressed as



[m][B]{q}(n)+[k][B]{q}(n)+[D][B]{q}(n)+{Nl.(n) = { n)

(4.20)


where

[m] mass matrix

[B] matrix requiring power series solution to

satisfy the prescribed boundary conditions

{q}(n) generalized coordinates of the nth cir-

cumferential mode

[k] stiffness matrix

[D] stability matrix

NLi LqmJ[B T[C ][Bn]{qn}- power series products

{P} load vector


I





51





As an example, for a problem retaining two circumferen-

tial modes with eight subdomains along the meridian,

equations (4.20) would represent a total set of forty-

eight coupled nonlinear second order ordinary differen-

tial equations to solve.














CHAPTER 5


NUMERICAL ANALYSIS FOR CONICAL SHELLS



The sets of equations (4.20) were coded into

computer language for numerical solution. A particular

shell specimen was selected for study. The salient

geometry and boundary conditions for the shell studied

are given in Appendix I. Fortran IV, level H machine

language was used and the program was constructed for

double precision arithmetic. A flow diagram of the oper-

ational characteristics is given in Appendix III. A list-

ing of the program is given in Appendix IV.



5.1 Functional Description of the Computer Program


If inplane inertia is considered ignorable with

respect to normal inertia, then equations (4.20) separate

into coupled sets of 2(N-j) nonlinear algebraic equations

and 2(N-j) nonlinear first order differential equations.

The flow of computations proceeds as follows. The

linear and nonlinear stiffness matrices, mass matrices,

boundary transform matrices, buckling matrices, and load-

ing vectors are generated by the program. At time equal

to zero the problem solution begins with known external











loading and initial conditions. The known normal deflec-

tions are substituted into the first two equations of

equilibrium and the resulting u, v vectors are generated

utilizing a Gauss elimination technique with complete

pivoting. If nonlinear effects in the first two equa-

tions of equilibrium are considered, this mode is itera-

tive in nature. Having generated the u's and v's for

time zero, they are substituted into the sets of 3rd

dynamic equilibrium equations for temporal integration.

The time integration is accomplished by a Hamming pre-

dictor-corrector integration technique. This generates

the w's for the next time step. Normal velocities are

also computed. This completes the sequence of advance

to the first time increment beyond time zero and provides

the required initial conditions in order to begin the

next advance in time. The sequence then recycles and the

temporal advance continues. At any time interval the an-

alyst chooses the solved for dependent variables are

transformed from the solution space to real space and

displayed as output. Each output consists of the com-

plete meridional profile for each mode in the solution.

The build up of errors in the numerical solution

is controlled by the following mechanisms. A loss of sig-

nificance control indicator is incorporated into the Gauss

elimination routine which warns the analyst when solutions












are being derived from a nearly singular coefficient

matrix. An error bound control is available in the

predictor-corrector integrator which limits this type

of error by automatically adjusting the size of the

time step. Round off errors are minimized by making

all calculations accurate to sixteen significant figures.

Such rigorous control of possible numerical errors is

essential in dynamic stability analysis because of the

possibility that numerical instability might be mistaken

for actual structural instability.



5.2 Checking the Computer Program


The control of clerical and conceptual errors in

the construction of the computer program was done by con-

tinually checking back to simpler known cases. In some

instances the known cases could be obtained by hand cal-

culations, in others, the results of other analysts were

used. For the shell specimen studied, the computer pro-

gram has been verified by checking against known solutions

for static stability, static deflection, free vibration

modes and frequencies [39], linear dynamic response, and

regions of parametric instability [52]. A convergence

study was also conducted to determine the convergence

properties of the solution for increasing numbers of











subdomains. The results of this study are presented in

Appendix II.



5.3 Dynamic Stability Results


After the above verifications were completed, the

program was used to study the linearized and the nonlinear

dynamic stability of a short conical frustum with simply

supported boundaries subject to various levels of constant

axial loads and critical levels of nearly axisymmetric

Heaviside pressure blasts. Eight subdomains along the

meridional direction were used. The driven breathing mode

and an arbitrary preferred flexural mode were retained in

the analysis. Complete coupling between the modes was re-

tained through the quadratic terms for the nonlinear case.

The critical static normal pressure for the speci-

men studied is 42.8 pounds per square inch and the critical

circumferential wave number is eight. This represents a

mid-span deflection in the breathing mode of 9 percent

of the thickness. The ordinate scale (W) of all figures

depicting time history responses is normalized by this

critical deflection. The critical static axial load is

40,400 pounds. The applied loads for time history graphs

will be noted as


(a, b, c)











where



a = P/P cr b = p/Pcr, c = P /Pcr.





Additionally, note that



P = P H(t)


Pn = PnH(t)




where



H(t)


denotes a Heaviside pressure pulse.


5.3.1 Dynamic Modal Preference


The program was used to determine the critical

flexural modes during dynamic instability. The results

are shown in Figure 4 and the critical static buckling

modes are indicated. Because only a few closely spaced

critical modes dominate the flexural response and because







57












5- 0
-- E

-0




~CO




















4 z
C
III ( rl

L1J o | C
0 0 0








Cci(



,aa)




o0CO C
1 I U r-


S|- fO Sa I I II 5- -


0 ^0) ro "0 |-

























CCJ 0
0








0
CL~u r a







a \\ I

011 I
r \\ I
a \\ O











coupling between these modes is very weak [10], the param-

eter study that follows is reduced by considering the in-

stability of only a sequence of single flexural modes to

find the mode indicating instability at the lowest load.



5.3.2 Linearized Dynamic Stability


The linearized dynamic analysis was obtained by

considering the vector {NL}(n) in equation (4.20) to be

zero. This means that the internal loads from the pre-

buckling state may interact with postbuckling deflections

to cause primary dynamic instability (i.e. snap through)

but internal parametric instabilities due to the pulsat-

ing nature of the membrane state are prohibited.

The critical circumferential wave number for

dynamic instability was found utilizing this analysis and

the results are shown in Figure 5.

While conducting a series of linearized dynamic

stability computer runs at various combinations of external

loadings, a feature of the numerical model was discovered

which represents an improvement over other computer programs

used for this purpose. This feature is that the meridional

profile undergoes a marked and rapid change of character

during runs at or above the critical dynamic load. For

loadings below critical, the meridional profile pulsates at






















0
1. a) a
,E0


O (u C


'I)



4-





C
O





E


E
O .


S-D -

co -0 -C











one half wave, slightly skewed due to the conical taper.

At or above the critical load the meridional profile

begins to pulsate as a half wave but later snaps to two

or more half waves. The time of snap, number and shape

of the waves in the buckled meridional profile depend on

the axial load to normal pressure ratio and their rela-

tion to static critical loads. Examples of this are shown

in Figures 6 and 7. It is felt that this improved ability

of detecting dynamic instabilities results from two areas

of the analysis. One being the spatial error residual con-

trol offered by the modified subdomain method employed and

the other being the strict control of error bound in the

numerical work. Additionally, it should be noted that

most published works on this type of problem utilize one

term assumed solutions for preferred modes and clearly

could not see this effect in their analysis.

A large number of linearized dynamic stability

runs were made at various combinations of static axial

and Heaviside pressure loads. The results were collated

into a dynamic stability interaction curve and are shown

in Figure 10.



5.3.3 Nonlinear Dynamic Stability


In the case of nonlinear dynamic stability, the

possibility of autoparametric excitation of preferred







































-0





0
*r
X
x









o



4-,
CC














cu



I-
5-














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o g





5-
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LU'
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u-





















cO
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x









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flexural modes by the pulsating breathing mode is allowed,

in addition to snap through or primary dynamic instability.

The question here is whether such excitation will occur

during the primary instability motion of the preferred mode

sufficiently to affect the magnitude of the external loads

required to induce dynamic instability. In contrast to the

external parametric excitation problems, the total energy

of the shell for the present problem is constant for all

t>O. Thus, the unstable modes when growing must derive

their energy from the breathing mode with a corresponding

decrease of energy in that mode. To evaluate this inter-

play a full nonlinear analysis is necessary.

Conceptually, the third equation of nonlinear dy-

namic stability may be expressed in an average sense as

follows:


+ % 2 2
+ 0 ow + k(uo) + f(w ) = oH(t)
(5.1 a,b)

n + [ g(o)w + (nVn) + (h(w)n = PnH(t)



where uo, w represent breathing mode deflections measured

from the undeformed state and u v n, w represent the

poststable growth of the buckling mode. X and n denote

the linear eigenvalues of the respective modes and f, g, h,











k, and 1 are differential operators. The superscript a

denotes an average over the shell length. Since w0 is

excited by a Heaviside pulse, the wo response is initially

periodic and equation (5.1 b) takes the character of an

inhomogeneous Mathieu equation. The primary buckling

terms denoted by a single underline cause an insignificant

shift to the left on a parametric stability diagram. In

the absence of imperfections in the preferred flexural mode,

denoted by the double underlined term, equation (5.1 b)

lies well within a stable zone of a Mathieu diagram and no

significant parametric growth of the buckling mode would

be expected.

For finite deterministic imperfections, however,

the location of stability boundaries becomes more difficult

and for significantly large pn H(t), solutions to equation

(5.1 b)may exhibit parametric growth at or below the dynamic

loads required to produce immediate snap through and may

lead to a delayed dynamic snap through. Whether or not

this beating type of growth occurs with sufficient quick-

ness to have an effect on the primary dynamic instability

can be determined only by direct temporal integration of

two coupled sets of equation (4.20).

A series of runs with the computer program was made

to determine the significance of this effect. Typical re-

sults are shown in Figures 8 and 9. As in the case of





















































C\j QJ
E

I-







66















0





SC o
_!









*r-
O 0 -0


no cr --











ar
0 0 -I-'

















E E
00











g5
---( *--

\ E






I-~

lo 5-
___1 _______ 1 i











linearized dynamic stability, a large number of computer

runs were made at various combinations of static axial and

Heaviside pressure loads and the results collated into

a nonlinear dynamic stability interaction curve shown

in Figure 10.

The linearized dynamic stability curve lies below

the static stability curve in Figure 10 due to the in-

creased severity of a suddenly applied load over that of

a statically applied load. The peak membrane forces de-

veloped by the breathing mode which is excited by a Heavi-

side pulse are twice as large as that of an equivalent

static case and contribute more to the developemnt of

instability in preferred flexural modes than a static

situation. This is known as the dynamic load factor effect.

The nonlinear dynamic stability curve lies below the

linearized dynamic stability curve because internal vibra-

tions interact between coupled modes so as to produce un-

stable beating resonances in the preferred buckling mode.

Such unstable vibrations lead to a delayed dynamic snap

through and this phenomenon is called autoparametric ex-

citation.











Static Stability
Linearized Dynamic
- Stability
SNonlinear Dynamic
Stability


n = 8


0 0.5


p H(t) / pcr


Figure 10. Dynamic Stability Interaction Curves


n

10















CHAPTER 6


RESULTS AND CONCLUSIONS



In the present analysis, a nonlinear shell theory

is derived and employed to study the nature of nonlinear

dynamic instability of a truncated thin circular conical

shell structure which is considered to be loaded by a

constant axial load and a nearly axisymmetric Heaviside

pressure load. A solution method is developed which sat-

isfies a boundary condition exactly and which converges

toward the exact solution of the governing equations with

increasing subdomains. The method avoids the necessity

of assuming the shapes of prebuckling or postbuckling

meridional profiles.

The results of this study confirm the feasibility

of the method of solution developed for this analysis. The

method converges rapidly yet can be applied to nonlinear

problems with minimum amounts of manual mathematics. The

trial functions are simple to manipulate yet satisfy any

formulatable boundary condition. The method is particu-

larly suitable for problems where poststable modes are not

known in advance as is usually the case for combined dynamic

loadings.











For the particular loading conditions studied,

the results indicate that dynamic instability will occur

at loads below critical static loads. This reduction in

critical dynamic loads was shown to be the result of both

the dynamic load factor effect and of autoparametric ex-

citation. The interaction curves which are generated

illustrate these effects quantitatively under conditions

of combined dynamic loading.

A criterion for dynamic buckling is established

based on meridional mode shape changes. This ability to

detect sudden jumps in the meridional profile aids to

verify instability detected by divergence on displacement

time history curves and provides additional information

about the poststable state.














APPENDIX I


SHELL SPECIMEN AND BOUNDARY CONDITIONS



The conical frustum utilized in this investigation

has the following properties (see Figure 3):


Material 1020 steel

a = 200

s1 = 5.85 inches

s2 = 14.22 inches

RI = 2.0 inches

R2 = 4.863 inches

h = 0.02 inches

v = 0.3 .



The boundary conditions assumed for the analysis

are:

u = v = w = Ms = 0 at sI and s2



This results in eight constraint equations which when ex-

pressed in terms of power series solutions for eight sub-

domains along the meridian become

10
I ais = 0 at s = sI and s = s2
i=o











10 -i
Z b.s = 0
1=o


12
-1
Z c.s = 0
1=o


at s = sl and s = s2




at s = s1 and s = s2


12
z c.[(i)(i-1) + v(-n2/sin2a
i=o


+ i)] i-2 = 0


at s = s1

and s = s2




























APPENDIX II















APPENDIX II


CONVERGENCE



In the use of the subdomain method, convergence

in the mean is desired. The main influence on convergence

is the choice of trial functions. For assured convergence,

the trial functions must be complete and linearly indepen-

dent [49]. The completeness property of a set of functions

insures that we can represent the exact solution provided

enough terms are used. The power series trial functions

used in this study are complete, for example, so that any

continuous function can be expanded in terms of them.

To demonstrate the rate of convergence for the

problem under study, a series of computer runs were con-

ducted to determine the dynamic perturbation response for

different numbers of subdomains. Dynamic stability response

for eight, ten, and sixteen subdomains was investigated.

The results are shown in Figure 11. The ordinate designa-

tion implies percent differences with respect to the best

approximation.


I _





76




















8






6

I I

o
O

- 4
L\


U


2






0 I
5 10

Number of Subdomains


Figure 11. Convergence Properties of Solution





























APPENDIX III















APPENDIX III


FLOW DIAGRAM OF THE COMPUTER PROGRAM



For a better understanding of the computer program,

a key which may be helpful in going from the theory to the

program is given below:



FORTRAN Name Theory Description


N m Number of subdomains

NW n Circumferential wave number

Q Q Total number of n

AP a Semivertex angle of cone

RO p Material density

E E Young's modulus

V v Poisson's ratio

F s2 Conical slant height

BTT B Boundary condition matrix

STT [k] Linear stiffness matrix

DND [m] Mass matrix

PV {p} Load vector

PLUG {NL} Nonlinear stiffness vectors

X T Nondimensional time











FORTRAN Name Theory Descriptions


U,V,Y u nv n, Nondimensional displacements

P P Axial load

PSD p Pressure level of Heaviside pulse

PRMT(2) T2 Upper time limit

PRMT(4) e Error bound

ORAN(I) -- Online core storage matrices

DHPCG -- Hamming's predictor-corrector
integration subroutine

DGELG -- Gauss-elimination subroutine



A flow diagram of the computer program is illustrated

in Figure 12.

















APPENDIX IV


FORTRAN IV SOURCE PROGRAM



The computer program developed to provide the

numerical solution to this dissertation problem is listed

below. The program requires 182,000 bytes computer core

storage space. A typical nonlinear dynamic stability run

executes in approximately in seven minutes. The compile

time is three seconds.







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BIBLIOGRAPHY


1. Hoff, N. J., "Dynamic Stability of Structures,"
Proceedings of an International Conference on
the Dynamic Stability of Structures, Northwest-
ern University, Evanston, Illinois, 1965, pp.
9-41.

2. Stoker, J. J., "Stability of Continuous Systems,"
ibid., pp. 46-52.

3. Meirovitch, L., Methods of Analytical Dynamics,
McGraw-Hill Book Co., New York, 1970.

4. Koiter, W. T., "General Equations of Elastic Stabil-
ity," Proceedings Symposium on the Theory of
Thin Shells, University of Houston, Houston,
Texas, 1967.

5. Meirovitch, L., op. cit.

6. Witmer, E. A., Leech, J. W., Pian, T. H. H., "Numer-
ical Calculation Technique for Large Elastic-
Plastic Transient Deformations of Thin Shells,"
AIAA Journal, Vol. 6, December 1968, pp. 2352-
2359.

7. Witmer, E. A., Leech, J. W., Morino, L., "PETROS 2,"
BRL Contract Report 12, 1969.

8. Witmer, E. A., Leech, J. W., Morino, L., "An Improved
Numerical Calculation Technique for Large Elastic-
Plastic Transient Deformations of Thin Shells,"
Journal of Applied Mechanics, Vol. 38, June, 1971,
pp. 423-436.

9. Longhitano, R., Klosner, J. M., "Nonlinear Dynamics
of Hemispherical Shells," AIAA Journal, Vol. 11,
August 1973, pp. 1117-1122.

10. Cromer, C. C., "An Investigation of the Nonlinear
Dynamic Response of Cylindrical Shells Under
Transient Pressure," Ph. D. Dissertation, Naval
Postgraduate School, Monterey, California, 1969.












11. Ball, R. E., "A Computer Program for the Geometric-
ally Nonlinear Static and Dynamic Analysis of
Arbitrarily Loaded Shells of Revolution," NASA
CR-1987, April 1972.

12. Ball, R. E., Burt, J. A., "Dynamic Buckling of Shallow
Spherical Shells," Journal of Applied Mechanics,Vol. 40,
June 1973, pp. 411-416.

13. Kim, C. W., "Static and Dynamic Buckling of Nonhomo-
geneous Cylindrical Shells," Ph. D. Dissertation,
University of Florida, Gainesville, Florida, 1972.

14. Stricklin, J. A., Martinez, J. E., Tillerson, J. R.,
Hong, J. H., Haisler, W. E., "Nonlinear Dynamic
Analysis of Shells of Revolution by Matrix Dis-
placement Method," N 70-30153, Texas A and M
University, College Station, Texas, February,
1970.

15. Wu, R. W. H., Witmer, E. A., "Nonlinear Transient
Responses of Structures by Spatial Finite-Element
Method," AIAA Journal, Vol. 11, August 1973,pp. 1110-1116.

16. Klosner, J. M., Roth, R. S., "Nonlinear Response of
Cylindrical Shells Subjected to Dynamic Axial
Loads," AIAA Journal, Vol. 2, October 1964, pp.
1788-1794.

17. Klosner, J. M., Zweben, C., "Dynamic Instability of
Circular Cylindrical Shells Having Viscoelastic
Cores," AIAA Journal, Vol. 5, June 1967, pp.
1128-1134.

18. Goodier, J. N., Mclvor, I. K., "The Elastic Cylindri-
cal Shells Under Nearly Uniform Radial Impulse,"
Journal of Applied Mechanics, Vol. 86, June 1964,
pp. 259-266.

19. Mclvor, I. K., Lovell, E. G., "Dynamic Response of
Finite-Length Cylindrical Shells to Nearly Uniform
Radial Impulse," AIAA Journal, Vol. 6, December
1968, pp. 2346-2351.

20. Lindberg, H. E., "Stress Amplification in a Ring
Caused by Dynamic Instability," Journal of Ap-
plied Mechanics, Vol. 41, June 1974, pp. 392-400.









21. Hubka, W. F., "Dynamic Buckling of the Elastic Cylindri-
cal Shell Subjected to Impulsive Loading," Journal of
Applied Mechanics, Vol. 41, June 1974, pp. 401-406.

22. Mente, L. J., "Dynamic Nonlinear Response of Cylindrical
Shells to Asymmetric Pressure Loading, AIAA Journal,
Vol. 11, June 1973, pp. 793-800.

23. Shumik, M. A., "Behavior of Conical Shells Under Dynamic
Loading," Prikladnaia Mekhanika, Vol. 5, 1969, pp. 15-
19 (in Russian).

24. Shumik, M. A., "On the Stability of Conical Shells Under
Dynamical Longitudinal Compression," Priklednaia Mek-
hanika, Vol. 6, 1971, pp. 122-126 (in Russian).

25. Shiau, A. C., "The Dynamic Buckling of a Truncated Coni-
cal Shell Structure," Ph. D. Dissertation, State Uni-
versity of New York at Buffalo, 1973.

26. Nash, W. A., Wilder, J. A., "Response of Thin Conical
Shells to Dynamically Applied Axial Force," Interna-
tional Journal of Non-Linear Mechanics, Vol. 7, 1972,
pp. 65-80.

27. Fisher, C. A., Bert, C. W., "Dynamic Buckling of an Axially
Compressed Cylindrical Shell With Discrete Rings and
Stringers," Journal of Applied Mechanics, Vol.40,September,
1973, pp. 736-740.

28. Lakshmikantham, C., Tsui, TY., "Dynamic Stability of
Axially-Stiffened Imperfect Cylindrical Shells Under
Axial Step Loading," AIAA Journal, Vol. 12, February
1974, pp. 163-169.

29. Dietz, W. K., "On the Dynamic Stability of Eccentrically
Reinforced Circular Cylindrical Shells," Syracuse
University Research Institute, Technical Report 1620,
January 1967, pp. 1245-1260.

30. Novozhilov, V. V., Foundations of the Nonlinear Theory
of Elasticity, Graylock Press, Rochester, New York,
1953.

31. Tsao, C. H., "Strain Displacement Relations in Large
Displacement Theory of Shells," AIAA Journal, Vol. 2,
1964, pp. 2060-2062.









32. Ogibalov, P. M., "Dynamics and Strength of Shells,"
NASA TT F-284, 1966.

33. Naghdi, P. M., Nordgren, R. P., "On the Nonlinear
Theory of Elastic Shells Under the Kirchhoff Hypothe-
sis," Quarterly of Applied Mathematics, Vol. 21, No. 1,
1963, pp. 49-60.

34. Biricikoglu, V., Kalnins, A., "Large Elastic Deformations
of Shells with the Inclusion of Transverse Normal
Strain," International Journal of Solids and Struc-
tures, Vol. 7, 1971, pp. 431-444.

35. Sanders, J. L., Jr., "Nonlinear Theories for Thin Shells,"
Quarterly of Applied Mathematics, Vol. 21, No. 1,
1963, pp. 21-36.

36. Naghdi, P. M., "The Theory of Shells and Plates," Report
No. AM-71-1, University of California, Berkeley, 1971.

37. Sokolnikoff, I. S., Mathematical Theory of Elasticity,
McGraw-Hill Book Co., New York, 2nd edition, 1956.

38. Kalnins, Arturs, "Static, Free Vibration, and Stability
Analysis of Thin, Elastic Shells of Revolution,"
Technical Report AFFDL-TR-68-144, Wright-Patterson
Air Force Base, 1969.

39. Leissa, Arthur W., "Vibration of Shells," NASA SP-288,
1973.

40. Novozhilov, V. V., The Theory of Thin Shells, P. Noord-
hoff Ltd., Groningen, 1959.

41. Gol Denveizer, A. L., Theory of Elastic Thin Shells,
Pergamon Press, New York, 1961.

42. Novozhilov, V. V., Finkel'shtein, R., "On the Incorrect-
ness of Kirchhoff's Hypothesis in the Theory of Shells,"
P. M. M., Vol. 7, 1943, pp. 331-340.

43. Mushtari, K. M., "Non-Linear Theory of Thin Elastic
Shells," N 62 72586, U. S. Department of Commerce,
Washington, D. C., 1962.

44. Koiter, W. T., "A Consistent First Approximation in the
General Theory of Thin Elastic Shells," I. U. T. A. M.
Proceedings of the Symposium on the Theory of Thin
Elastic Shells, Delft, North-Holland Publishing Co.,
Amsterdam, pp. 12-33.











45. Vlasov, V. S., "Basic Differential Equations in the
General Theory of Elastic Shells," NACA TM 1241,
1951.

46. Hurty, W. C., Rubinstein, M. F., Dynamics of Structures,
Prentice-Hall, Inc., New Jersey, 1964.

47. Meirovitch, L., op. cit.

48. Kraus, H., Thin Elastic Shells, Wiley and Sons, Inc.,
New York, 1967.

49. Finlayson, B. A., The Method of Weighted Residuals and
Variational Principles, Academic Press, New York and
London, Vol. 87, 1972.

50. Radosta, R. J., "Stability of Nonhomogeneous Shells of
Revolution," Ph. D. Dissertation, University of
Florida, Gainesville, Florida, 1971.

51. Polhausen, K., "The Approximate Integration of the Dif-
ferential Equation for the Boundary Layer," Z. Angew,
Mathematics and Mechanics, Vol. I, AD 64 5784 F. S.
T. I., Springfield, Va., pp. 252-268 (in English).

52. Tani, J., "Dynamic Instability of Truncated Conical
Shells Under Periodic Axial Load," International
Journal of Solids and Structures, Vol. 10, 1974,
pp. 169-176.














BIOGRAPHICAL SKETCH


Marcus George Hendricks was born May 9, 1940 at

Pickens, South Carolina. He graduated from Pickens High

School in May, 1958. He received a Bachelor of Science

degree in Civil Engineering from Clemson University in

June, 1962. Returning to Clemson University after serv-

ing two years in the U. S. Army as a commissioned officer,

the author was awarded a Master of Science degree in Civil

Engineering in August, 1966. In 1970 he enrolled in the

Graduate School at the University of Florida in order to

pursue work toward the degree of Doctor of Philosophy.

He is married to the former Genevieve J. Boggs and

has one daughter, Carmen.










I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.




Sung Y. Lu, Chairman
Associate Professor of
Engineering Sciences


I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.



brahim K. Ebcioglu
Professor of Engineering
Sciences


I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.




Ulrich H. Kurzweg
Associate Professor of
Engineering Sciences


I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.



Robert L. Sierakowski
Professor of Engineering
Sciences













I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.




Morris W. Self
P ofessor of Civil/
Engineering




This dissertation was submitted to the Graduate Faculty of
the College of Engineering and to the Graduate Council, and
was accepted as partial fulfillment of the requirements for
the degree of Doctor of Philosophy.

December, 1974


Dei, College of Engineering



e School

; Dean, Gradu te School


"















































UNIVERSITY OF FLORIDA


3 1262 08666 450 4




Nonlinear anlysis of dynamic stability of elastic shells of revolution
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 Material Information
Title: Nonlinear anlysis of dynamic stability of elastic shells of revolution
Physical Description: xi, 131 leaves. : illus. ; 28 cm.
Language: English
Creator: Hendricks, Marcus George, 1940-
Publication Date: 1974
Copyright Date: 1974
 Subjects
Subjects / Keywords: Nonlinear mechanics   ( lcsh )
Shells (Engineering)   ( lcsh )
Elastic plates and shells   ( lcsh )
Materials -- Dynamic testing   ( lcsh )
Buckling (Mechanics)   ( lcsh )
Engineering Sciences thesis Ph. D
Dissertations, Academic -- Engineering Sciences -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis -- University of Florida.
Bibliography: Bibliography: leaves 126-130.
General Note: Typescript.
General Note: Vita.
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Holding Location: University of Florida
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System ID: UF00098172:00001

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NONLINEAR ANALYSIS OF DYNAMIC STABILITY
OF ELASTIC SHELLS OF REVOLUTION









By

MARCUS GEORGE HENDRICKS


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA IN PARTIAL
FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA
1974

































TO MY WIFE,
GENEVIEVE,
AND MY DAUGHTER,
CARMEN














ACKNOWLEDGMENTS


The author wishes to express his sincere appreci-

ation to Dr. S. Y. Lu, chairman of his supervisory com-

mittee, for his continuous guidance and encouragement

throughout the entire period of this study. He also

wishes to express his gratitude to Dr. I. K. Ebcioglu,

Dr. U. H. Kurzweg, Dr. R. L. Sierakowski, and Dr. M. W.

Self for their helpful discussions with the author and

many valuable suggestions.














TABLE OF CONTENTS


Page


ACKNOWLEDGMENTS ................................. iii

LIST OF FIGURES................................. vi

NOMENCLATURE .................................... vii

ABSTRACT ........................................ x

CHAPTER
1 INTRODUCTION ............................ 1

2 THEORETICAL DEVELOPMENT OF
NONLINEAR SHELL EQUATIONS ............... 14
2.1 Shell Coordinates.................. 14
2.2 Kinematic Relations................ 15
2.3 Constitutive Relations............. 17
2.4 Kirchhoff's Hypothesis............. 18
2.5 Variational Principle.............. 20
2.5.1 Hamilton's Principle ........ 20
2.5.2 Equations of Motion......... 21
2.5.3 Boundary conditions ......... 26
2.6 Synthesis of Equations............. 26
2.6.1 Shells of Revolution ........ 27
2.6.2 Conical Shells .............. 27

3 DYNAMIC STABILITY EQUATIONS
OF CONICAL SHELLS ....................... 35

4 METHOD OF SOLUTION...................... 39
4.1 Circumferential Modal Analysis ..... 39
4.2 Elimination of
Meridional Derivatives............. 42
4.2.1 Subdomain Method............ 45
4.2.2 Method of Satisfying
Boundary Conditions ......... 47











TABLE OF CONTENTS (Continued)


Page


5 NUMERICAL ANALYSIS FOR CONICAL SHELLS..... 52
5.1 Functional Description of the
Computer Program..................... 52
5.2 Checking the Computer Program........ 54
5.3 Dynamic Stability Results............ 55
5.3.1 Dynamic Modal Preference...... 56
5.3.2 Linearized Dynamic Stability.. 58
5.3.3 Nonlinear Dynamic Stability... 60
6 RESULTS AND CONCLUSIONS................... 69


APPENDIX I SHELL SPECIMEN AND BOUNDARY
CONDITIONS ............................ 71

APPENDIX II CONVERGENCE .......................... 74

APPENDIX III FLOW DIAGRAM OF THE
COMPUTER PROGRAM .................... 77

APPENDIX IV COMPUTER PROGRAM SOURCE
LISTING .............................. 81

BIBLIOGRAPHY ....................................... 126

BIOGRAPHICAL SKETCH ................................ 131













LIST OF FIGURES


Figure Page


1 Development Paths in Constructing
Solution Methods for Problems of
Dynamic Stability of Shells............. 4

2 Shell Element........................... 25

3 Conical Shell Coordinates............... 29

4 Fourier Wave Number Dependence.......... 57

5 Flexural Mode Preference During
Dynamic Instability..................... 59

6 Linearized Dynamic Stability for
High Axial Load ......................... 61

7 Linearized Dynamic Stability for
Intermediate Axial Load ................. 62

8 Nonlinear Dynamic Stability for
Heaviside Pressure Loading.............. 65

9 Nonlinear Dynamic Stability for
Intermediate Axial Load................. 66

10 Dynamic Stability Interaction Curves.... 68

11 Convergence Properties of Solution ...... 76

12 Flow Diagram of the Computer Program.... 80














NOMENCLATURE


A,B

[B]

D

E

G

K

L
Mr ,MB,MaB



N

N ,N ,N


P

P

Q ,Q

R

Ro


T

U,V,W

W


Coefficients of the first fundamental form

Boundary condition matrix

Flexural stiffness

Young's modulus

Shear modulus

Membrane stiffness

Lagrangian Function

Stress couple components

Stress couple components at edges

Number of terms in power series

Stress resultants

Stress resultants at edges

Axial load

Potential energy

Shear stress resultants

Position Vector

Radius of latitude circle

Principal radii of curvature

Kinetic energy

Displacements of arbitrary point in the shell

Displacement function for the buckling mode


vii












V

an ,b ,c n

a a ,an

e ,e ,e

f,g,h

h

n
j

m

n
a 6n
p ,p ,p

Pn

q

r

s




t

u,v,w




x3

z

a,B

Y

C


Region

Temporal coefficients

Coefficients of thermal expansion

Strain components

Differential operators

Shell thickness

Normal unit vector

Number of essential boundary constraints

Number of time dependent unknowns

Circumferential wave number

Surface loads

Fourier component of normal pressure

Transformed coordinate

Radius vector

Meridional conical coordinate

Nondimensionalized meridional conical
coordinate

Time

Middle surface displacements

Nondimensionalized middle surface
displacements

Coordinate of axis of revolution

Distance along i

Curvalinear coordinates or conical angles

Strain tensor

Error


viii











Circumferential conical coordinate

Eigenvalue

Poisson's ratio

Density

Stress tensor

Nondimensional time

Spatial coefficient

Transformed circumferential conical
coordinate


Additional
Notations

[ ]

[ ]
( ),

(') ,( )

L J

{ }


Number in brackets refers to reference
numbers in the Bibliography

Letter in brackets denotes matrix
Comma denotes partial differentiation

Denotes partial differentiations with
respect to time

Row vector

Column vector











Abstract of Dissertation Presented to the
Graduate Council of the University of Florida in Partial Fulfillment
of the Requirements for the Degree of Doctor of Philosophy


NONLINEAR ANALYSIS OF DYNAMIC STABILITY
OF ELASTIC SHELLS OF REVOLUTION

By

Marcus George Hendricks

December, 1974


Chairman: Dr. S. Y. Lu
Major Department: Engineering Sciences


It is the purpose of this dissertation to investi-

gate the dynamic stability of elastic shells of revolu-

tion. Two specific areas of this broad field are treated

in detail. First, this analytical study generates pre-

viously unavailable interaction relations for combined

dynamic loadings as they interact to cause passage from a

dynamically stable state to other states. Secondly, the

concept of linearized dynamic stability is extended to

include geometrically nonlinear effects. One role of these

effects is to allow the possibility of autoparametric ex-

citation of preferred flexural modes by the driven modes.

The question of whether such excitation can occur during

the primary dynamic instability motion sufficiently to af-

fect the magnitude of critical dynamic loads is studied.











A modified version of the subdomain method in combination

with circumferential modal analysis is developed for the

solution method. A computer program is constructed to

obtain numerical results and the dynamic stability charac-

teristics of a conical frustum are studied under a variety

of combined dynamic loadings. Interaction curves for

static stability, linearized dynamic stability, and non-

linear dynamic stability are generated.

For the particular loading conditions studied, the

results indicate that dynamic instability will occur at

loads below critical static loads. This reduction in crit-

ical dynamic loads was shown to be the result of both the

dynamic load factor effect and of autoparametric excitation.

The interaction curves which are generated illustrate these

effects quantitatively under conditions of combined dynamic

loading. A criterion for dynamic buckling is established

based on meridional mode shape changes. This ability to

detect sudden jumps in the meridional profile helps to verify

instability detected by divergence on displacement time

history curves and provides additional information about

the poststable state.














CHAPTER 1


INTRODUCTION


A dynamic stability problem can be defined as a

problem analyzed by Newton's equations of motion or any

equivalent method [1]. Dynamic stability problems for

continuous systems are nearly always governed by nonlinear

partial differential equations [2]. The solution to a

problem of dynamic stability in thin-walled structures may

be stated as the determination of what load-time combina-

tions cause the displacements of points in the material

body to increase sufficiently to either interfere with

operational specifications or cause a breakdown of the

structure. Methods for obtaining such solutions may be

categorized into three fundamental areas. The most obvi-

ous and usually most difficult method is the direct inte-

gration of the governing equations for sufficiently long

periods to allow direct observation of the ensuing motions.

A second technique is the investigation of the stability

of equilibrium points. The question here is whether a

slight perturbation of a dynamical system from an equilib-

rium state will produce motion confined to the neighborhood











of the equilibrium point or a motion tending to leave

that neighborhood [3]. This method is usually based on

either perturbation or characteristic equation techniques.

A third category of dynamic stability methods is on a

higher level of abstraction than the previous two and

relies on the existence and properties of variously de-

fined functionals. Energy functionals [4] and Liapunov

functionals [5] are two examples.

Definitions of dynamic stability may be categorized

by methods of approach to such problems. In situations

where the direct integration approach is employed, dynamic

instability may be defined to exist when some character-

istic parameter, such as deflection, stress, or strain,

becomes unbounded in one or more parts of the body. When

the method of small oscillations about equilibrium points

is used, instability may be defined as unbounded growth

of perturbations with time or by the existence of one or

more positive real parts of roots of a characteristic

equation. In problems where the solution method is of

the third above mentioned category a sufficient condition

for stability is defined according to the definite form

of the functional employed. In the case of energy func-

tionals, a boundary between stable and unstable regions

is defined by the vanishing of the second variation of

the energy functional. When Liapunov functionals are












employed, stability is defined when the functional is

positive definite and when its derivative is negative

definite.

The particular problem of interest in this study

is the dynamic buckling of shell structures, which rep-

resents a specific application of the more general theory

of dynamic stability principles. Just as the definitions

of dynamic stability are related to the solution methods,

the definitions of dynamic buckling of shells are related

to solution methods as well as the specifics of the phys-

ical problem such as geometry, material properties, or

loading rates.

Solution methods, falling within the first of the

three aforementioned categories, are direct temporal in-

tegration of equations of motion. They may be classified

into five procedural divisions. These are finite differ-

ence, finite element, Hamilton's principle, energy methods,

and methods of weighted residuals. Figure 1 illustrates

the procedural flow and interrelations between these meth-

ods for the study of dynamic stability of elastic shells

and is based on the most prominent articles dealing with

this subject.

First, let us consider recent significant contri-

butions falling within the finite difference method divi-

sion. Witmer et al. [6, 7, 8] have developed during the


1_












Theory of Theory of Newton's Laws
Elasticity Surfaces or
Hamilton's Variati
Principle


Coupled Set of
Nonlinear Partial
Differential
Equations of
Equilibrium
---- tL

Modal Analysis
to Eliminate
Circumferential
Coordinate
I


- I
Coupled Set of
Nonlinear Partial
Differential
Expressions of
Internal Energy
--- *-----

Finite Difference
in Meridional and
Circumferential
Coordinates


Assumed Solution Finite
in Meridional in Meri
Coordinate Coordir
-it


Difference
dional
tate


Kinematic
Relations
onal of a
Finite
Element





Assemblage of
Elements in
Dynamic
Equilibrium


Coupled Nonlinear System(s)
of Ordinary Differential
Equations or of
Integro-Differential
Equations

Numerical Integration i

Shallow shell theory is often introduced here
Bifurcation, if enforced, is usually introduced here


Figure 1 .


Development Paths in Constructing Solution
Methods for Problems of Dynamic Stability
of Shells


L











past decade a shell dynamics computer program known as

PETROS which utilizes spatial finite difference methods

in both the circumferential and meridional directions.

Utilizing PETROS the authors have studied a number of

explosive forming problems and have shown remarkable

agreement with experimental results. However, consider-

ing that after a ten-year development period, computer

run times are still measured in hours for moderately com-

plex problems, the practical usefulness of PETROS in study-

ing shell dynamic stability is somewhat limited. Using

shallow shell theory, Longhitano [9] developed numerical

solutions for the nonlinear dynamics of hemispherical shells

also utilizing finite difference solutions in two directions.

He showed that suddenly applied symmetic pressures may

couple the breathing mode into more preferred axisymmetric

or asymmetric vibratory modes causing instability at approx-

imately one half static buckling pressures. Again, however,

the two-dimensional discrete type of spatial treatment re-

sults in unusually long computer run times thus limiting

the number of cases treated.

Another class of finite difference solutions to

shell dynamic stability problems have appeared in the lit-

erature which require only approximately three fourths of

the computer run times as the previously cited two-dimen-

sional discrete methods and appear to give comparable











results. Such solutions based on elimination of the

circumferential coordinate via modal analysis prior to

finite differencing in the meridional direction have

been presented by Cromer [10], Ball [11, 12], and Kim

[13]. The results of Cromer and Ball also indicate that

suddenly applied axisymmetric pressures may couple breath-

ing mode response into preferred vibratory modes to cause

dynamic instabilities at pressure loadings lower than

static critical loads. These studies indicate that cou-

pling between parametrically excited flexural modes is

very weak and often may be neglected without compromising

the results.

Stricklin [14] and Wu [15] have made recent and

significant contributions to the nonlinear dynamic analy-

sis of shells by employing the finite element method. Re-

sults published by both of these investigators indicate

that accuracy comparable to methods emanating from the

equations of motion is obtainable if sufficiently complex

elements are used in large quantity. However, due to

longer computing times required for finite element tech-

niques, their major advantage clearly lies in the treat-

ment of complex structures where equations of motion are

not reasonably derivable. A seldom mentioned, yet almost

universal, appealing characteristic of finite element

techniques warrants a brief mention. This is that the











unknown dependent variables in a finite element solution

have a direct physical interpretation at all times as

the solution proceeds. Thus, to some extent, the analyst

never loses physical contact with his problem. This is

in contrast to most other analysis techniques applicable

to shell dynamics where the solution is carried forward

in some transformed mathematical space.

Klosner [16, 17] in a series of studies covering

a five-year period studied the effect of suddenly applied

axial loads on cylindrical shells. His solutions were ob-

tained by elimination of the circumferential coordinate

from nonlinear shallow shell equations and applying the

Ritz method or Hamilton's principle to the relations re-

sulting after introducing assumed solutions. His results

indicate that suddenly applied axial loads do not exhibit

the dynamic load factor effect found by others [9, 10, 11,

12] when suddenly applied normal pressures are considered.

These investigations reveal that a valuable indication of

the onset of dynamic instability can be found in the re-

versal in the trend of the time to first maximum deflec-

tion of the critical mode.

Another subcategory of solution methods in studying

dynamic shell stability by direct temporal integration is

known as the energy method. This technique circumvents

the derivation of the partial differential equations of










motion. The method involves deriving internal energy

expressions in conjunction with appropriate kinematic

relations, substituting assumed solutions into these

relations, and then applying Lagrange's equations to

the result. This procedure leads to either a coupled

system of nonlinear ordinary differential equations if

the spatial integrals are evaluated analytically or a

coupled system of nonlinear integro-differential equa-

tions if the spatial integrals are to be evaluated

numerically. In either case, the differentials are

temporal and must be solved numerically. Goodier [18],

Mclvor [19], and Lindberg [20] have presented solutions

to a particular type of dynamic instability of cylin-

drical shells utilizing this solution procedure. Hubka

[21] has presented alternate methods for numerical inte-

gration of the final equation sets for this problem.

This problem is concerned with the circumferential modal

coupling arising from the nonlinear membrane stiffness

terms in the equation sets obtained by this energy method

technique. The main results of these studies are that non-

linear coupling between circumferential modes may under

conditions of minor excitation or initial imperfections

in the preferred vibratory modes cause energy extraction

from the driven breathing mode into one or more preferred

modes such that the preferred mode becomes dynamically

unstable.











Utilizing this same energy method derivation,

Mente [22] has studied the dynamic nonlinear response

of cylindrical shells subjected to asymmetric pressure

loading. His computerized solution, known as DEPICS,

is able to handle a variety of coupled modes during the

temporal integration. The nonlinear stiffness is handled

by solving a linear eigenvalue-eigenvector problem at

each time step then adjusting the succeeding stiffness

value based on the nonlinear terms. Unfortunately this

rather extensive computer program has to date been unable

to satisfactorily define any regions of instability.

Significant Russian contributions to the study of

dynamic stability of shells have been made. Among these

are works by Shumik [23, 24] who also employed solutions

by direct temporal integration of expressions derived by

the energy method. These works deal with suddenly applied

loads using a linearized uncoupled dynamic stability theory.

His results indicate that cones with large taper buckle

dynamically at circumferential wave numbers higher than

the critical wave number for static loading.

Shiau [25] utilizing an energy solution studied

the effects of imperfections on the dynamic stability of

conical shells subject to axial impact. His results

agree with Klosner's [16, 17] in so much that a reversal

in the trend of the time to first maximum deflection of











the critical mode may be viewed as an indication of the

onset of dynamic buckling. The results, however, are

so critically dependent upon the choice of the single term

assumed buckling mode that the procedure appears to be

of limited general value.

A final division of solution methods in the cate-

gory of direct temporal integration is the method of

weighted residuals. The details of the procedure will

be covered in a later chapter. Nash [26] utilizing

shallow shell theory in conjunction with the Galerkin

method of weighted residuals investigated the response

of thin conical shells to dynamically applied axial

forces. He found that a rapid increase in end shorten-

ing may be used to denote the onset of dynamic instability.

Fisher [27] employed a similar solution method to

study the dynamic buckling of reinforced circular cylin-

ders subjected to suddenly applied axial compression load-

ings. He also found that a reversal of the trend of the

time to maximum deflection corresponds to onset of dynamic

buckling. Lakshmikantham [28] using shallow shell theory

in conjunction with smeared analysis and the Galerken tech-

nique investigated the same problem as Fisher [27]. His

solutions indicated that a jump in radial deflection was

the correct criteria for indicating dynamic instability.

An earlier work by Dietz [29] studying the same problem














by shallow shell theory and the Galerkin method indi-

cated that loss of stability is best defined as a sudden

drop in the axial load required to maintain a constant

end shortening rate.

By taking the results of the preceding literature

survey in context, several comments may be made concerning

the current state-of-the art in analytical studies of

dynamic shell stability. First, it appears that nonlinear

response problems which retain full circumferential modal

coupling yield the most reliable indicators of the onset

of dynamic buckling. Also the results appear to be highly

sensitive to theoretical approximations and to the quality

of the numerical solution procedures. Secondly, the study

of passage through bifurcation points is well developed

only for linearized dynamic stability techniques. The

few articles proposing to deal with nonlinear dynamic

stability unfortunately discarded the quadratic nonline-

arities, which are the most significant ones, retaining in-

stead only the cubic nonlinearities. This was done in order

to circumvent having to deal with inter-modal coupling. There-

fore, the results of these studies appear inconsistent.











It is the purpose of this dissertation to invest-

igate the dynamic stability of elastic shells of revolu-

tion. Two specific areas of this broad field are treated

in detail. First, this analytical study generates previ-

ously unavailable interaction relations for combined

dynamic loadings as they interact to cause passage from

a dynamically stable state to other states. Secondly, the

concept of linearized dynamic stability is extended to in-

clude geometrically nonlinear effects. One role of these

effects is to allow the possibility of autoparametric

excitation of preferred flexural modes by the driven modes.

The question of whether such excitation can occur during

the primary dynamic instability motion sufficiently to af-

fect the magnitude of critical dynamic loads is studied.

The theory derived herein is suitable for the study

of the dynamic stability of general shells of revolution.

In order to obtain numerical results, however, a specific

shape must be specified and the one used for numerical

analysis in this study is a truncated circular conical

shell structure. Conical frustum shells are in frequent

use as structural elements. Moreover, a conical shell

reduces to a cylindircal shell when the semivertex angle

of the cone becomes zero. Similarly, it reduces to a flat

circular plate when the semivertex angle of the cone ap-

proaches a right angle. Thus, the analysis and computer











program developed for cones can be generalized to handle

these two related problems.

A modified version of the subdomain method in

combination with circumferential modal analysis is de-

veloped for the solution method. A computer program is

constructed to obtain numerical results and the dynamic

stability characteristics of a conical frustum are stud-

ied under a variety of combined dynamic loadings. Inter-

action curves for static stability, linearized dynamic

stability, and nonlinear dynamic stability are generated.

A dynamic buckling criterion based on meridional mode

shape changes is established and the quantitative impor-

tance of nonlinear coupling effects is illustrated.














CHAPTER 2


THEORETICAL DEVELOPMENT OF NONLINEAR SHELL EQUATIONS



2.1 Shell Coordinates


A set of normal curvilinear coordinates (x, 0, z)

is chosen in the shell space such that a and B lie on the

undeforme d middle surface, z is perpendicular to this

middle surface, and z=O lies at the middle surface. The

equation of the undeformed middle surface is given in

terms of two independent parameters a and 0 by the radius

vector


r = r(a,B). (2.1)


To describe the location of an arbitrary point in the

space occupied by a shell, the position vector is defined

as


R(a,B,z) = r(a,B) + zin(a,B) (2.2)


where z measures the distance of the point from the corres-

ponding point on the middle surface along i The unit

vector i is normal to the surface.











2.2 Kinematic Relations


Acceptable nonlinear strain-displacement relations

for shells may be obtained from one of two basically dif-

fering approaches. One approach is to utilize the finite

strain tensor of the three-dimensional theory of elasticity

in conjunction with the Gauss-Codazzi relations for a sur-

face [30, 31, 32, 33, 34]. Another method is the two-

dimensional approach to shell theory that evades the ques-

tions of the approximations involved in the descent from

three dimensions. This approach defines strain as one-

half of the difference between the material and spatial

metric surface tensors. Sanders [35] presented the first

consistent nonlinear kinematic relations derived by this

method using a continuum called a Cosserat surface.

Naghdi [36] discusses both of the above methods in detail.

Nonlinear kinematic relations are seldom utilized

in shell analysis in their complete form due to their

complexity. Therefore, simplifications are routinely made.

Throughout the following analysis it is assumed that the

strains in the middle surface and that the rotations about

coordinate axes are small. These assumptions imply that

an element of area on the deformed middle surface is iden-

tical in size to an element of area on the undeformed

middle surface and that the difference between the










Christoffel symbols of the deformed and undeformed coordi-

nate systems is zero. These assumptions result in a con-
siderable simplification in the resulting equations with-
out unduly restricting the applicability of the solutions.
The strain-displacement equations shown below were

derived using the finite strain description of the three-
dimensional theory of elasticity in a manner similar to
Ogibalov [32] and the nonlinear terms were regrouped into

surface rotation expressions. The resulting kinematic
relations are


ea = (1/(l+z/R ))[U, /A+A, V/AB+W/R +(l/(2(1+z/R )))

x(-W, /A+U/Ra 2


e = (1/(I+z/R ))[V, /B+B, U/AB+W/R +(l/(2(1+z/R )))

x(-W, /B+V/R )2] (2.3)



e = (1/(2(1+z/R )))(V, /A-A, U/AB)

+(l/(2(1+z/R )))(U, /B-B, V/AB)

+(1/(2(1+z/R a)(1+z/R )))(-W, a/A+U/Ra)(-WB/B+V/R ).


In equations (2.3) the covariant components of the normal
and shear strains are denoted by e e e a respectively.











The components U, V, and W represent displacements of

an arbitrary point in the shell. R and R are the

principal radii of curvature and A and B are the coeffi-

cients of the first fundamental form. Comma denotes

partial differentiation.


2.3 Constitutive Relations


The relations between the components of strain

and stress in an orthotropic, linearly elastic material

are given by Hooke's law [37] as


e = oa/E -v a /E V a n /E +a T
a a / -vcn n c

e = -v a /E +cA/E -v n/E +a T
e. / n n /(



en = -v n oa/E -v n /E +,n/E +aT
(2.4)
2eB oa/G



2e = an/G
an an.



Young's modulus in the a and 3 directions is de-

noted by Ea and E respectively. The Poisson Ratio v(B

designates the contraction in the a direction caused by











a positive normal stress a in the direction. G is

the shear modulus in a plane which is tangent to the

(a,6) coordinate surface and (a(, a ) are the coeffi-

cients of thermal expansion in the a and directions

respectively. These relationships are valid for materials

subjected to stresses below the proportional limit. By

introducing Kirchhoff's hypothesis and allowing for the

symmetry of the elastic parameter matrix [38] the above

equations may be written as



aa = [1/(1-v O )][E e +v e -(a E +a E )T]

aB = [1/(1- a V )][E +v Ee -(a E +a E )T] (2.5)


_a = 2G e



2.4 Kirchhoff's Hypothesis


The reduction of the three-dimensional problem to

a two-dimensional one requires an assumption concerning

the variation of strain, displacement, or stress across

the shell thickness. To satisfy this requirement the

fourth assumption in Love's first approximation [39],

known as the Kirchhoff hypothesis, is introduced. This

hypothesis entails the vanishing of transverse shearing and

normal strains [33] and may be formulated as follows:











U(a,,z) = u(a,8)+z6a(a,6)

V(a,6,z) = v(a,6)+ze (a,()

W(a,B,z) = w(a,) (2.6)

e = u/R -w, /A 6 = v/R -w, /B


where u, v, and w are the components of displacement at

the middle surface in the a,4, and normal directions,

respectively, and 0 and 6 are the rotations of the

normal to the middle surface during deformation about

the B & a axes, respectively. The acceptance of this

assumption is due to its great clarity [40]. Although

Kirchhoff's hypothesis is a first approximation, its

applicability to nonlinear shell theory is well known

[33, 35]. The problem of determining the error intro-

duced by the hypothesis on the preservation of the normal

has thus far not been solved exactly [41]. Novozhilov

[42], Mushtari [43], and Koiter [44] have estimated that

the errors introduced by Kirchhoff's hypothesis are of

the order of h/R, which is small for thin shells.

Kirchhoff's hypothesis, modified to account for

transverse normal strain, has been employed in [31, 33,

34, 36, 45]. Effects of transverse normal strain were

not included in the present analysis.











2.5 Variational Principle



2.5.1 Hamilton's Principle


The Irish mathematician and physicist, Sir William

Hamilton (1805-1865), formulated his celebrated principle

in dynamics in which the governing equation depends ex-

plicitly on the energy of the system [46]. Hamilton's

principle is stated as an integral equation in which the

energy is integrated over an interval in time. In the

language of the calculus of variations, Hamilton's prin-

ciple states that the first variation of the time integral

of the difference between the kinetic energy T and the

potential energy P of a dynamical system is zero, that is,

t2 %
6 f (T P)dt = 0 (2.7)
t1



The equation is assumed to hold for all dynamical systems

whether they are conservative or nonconservative. The

quantity (T-P) is called the Lagrangian function and is

denoted by L. With this designation equation (2.7) be-

comes

t2
6 f Ldt = 0 (2.8)
t1
1











and Hamilton's principle then asserts that the first vari-

ation of the time-integral of the Lagrangian function is

zero. Hamilton's principle is employed in the next sec-

tion to derive the equations of motion and the admissible

boundary conditions for shells.


2.5.2 Equations of Motion


Of the many procedures available for the derivation

of the equations of motion for a differential shell element,

Hamilton's principle is superior in nonlinear shell theory

due to its efficient treatment of problems involving curv-

ilinear coordinates and because it gives the admissible

boundary conditions, natural or forced, that are to be used

with the theory [47, 48]. With the total potential energy

P expressed as the difference between the internal strain

energy and the potential energy of the external forces, we

obtain from equation (2.8)


t
6f {f [(po/2)V i-(1/2)siJij]dv+f si Vdsldt = 0 (2.9)
t 0 0s



Here the density of the undeformed body and the Lagrangian

components of the displacement vector are p and V., re-

spectively. The symmetric Cauchy stress tensor sij is











measured per unit area of the undeformed body and must

be referred to the base vectors in the deformed body and

the strain tensor, Yij, is defined by equation (2.3).

The components of the surface force ss are referred to

the base vectors in the undeformed body and the volume

and surface integrals in equation (2.9) must be extended

over the volume and surface of the body in its undeformed

state.

Introducing the kinematic, constitutive, and

Kirchhoff relations from previous sections, we eliminate

the time derivatives of the variations by integrating by

parts with respect to time and require the virtual dis-

placements to vanish at the end points of the arbitrary

interval t1 t t2, thus obtaining the equations of motion,

stress resultants and couples, and boundary conditions.

The equations of motion derived in this section

reflect the assumptions of Kirchhoff's hypothesis. Small

strains and moderately small rotations were also assumed.

Thinness assumptions were delayed until after the equa-

tions were synthesized.










The equations of motion are:


(BNO), +(AN(), +A, N B-B, N +ABQc/R -AB(NO +Na O )/Ra

+ABpa = ABphu(l+h2/12R R )+ABph3(1/R +1/R ) /12


(BNaB), +(ANB), +B, Naa-A, Na+ABQ /R -CAB(N 3 +N Be )/R

+ABp1 = ABphV(I+h2/12R R )+ABph3(1/R +1/RB1 6/12


(BQa), +(AQ3), -AB(Na/R +NB/R )-(Be N+BN (2.10)
-(AG N a+AO^ N), +ABpn = ABphW(l+h2/12R R )


(BMO'),+(AMB1), +A, MaB-B, M -ABQa

= ABph3[(l/R +l/R 3 )U+]/12


(BMW3), +(AM3), +B, MBa-A,1Mc-ABQ3

= ABph3[(I/R +I/R )V+ ]/12



where (") denotes second partial differentiation with
respect to time and p p, pn represent the applied
surface loads.











The stress resultants and stress couples are given by


Nc h/2 a
{ = f { 1(1+z/R )dz
N a -h/2 ao0


{N }
N



{Mm }
M m


M
M }




QO




N

HS


h/2 B
= f { 1(1+z/R )dz
-h/2 az U


h/2 a
= f h{ }(1+z/R )zdz
-h/2 0c


h/2 a
-h/2 a}(1+z/R )zdz

h/2 n"

= / {0 1+z/RB}dz
-h/2 o5n 1+z/R


(2.11)


h/2
= f o~dz
-h/2

h/2
= f a zdz
-h/2


where h is the shell thickness.

Figure 2 illustrates the relations between the

stress resultants and couples to the generalized coordi-

nates and to the shell element.













Z,W


M B


C, u


Figure 2. Shell Element


7











2.5.3 Boundary Conditions


From equations (2.9), (2.10), and (2.11) we get

the following natural boundary conditions that arise

from the requirement of force balance. Along the edge

of constant a the boundary conditions are



N = N or u = u


N +M /R = N or v = v
(2.12)
Q U+Ma-N -N6 = Qa or w = w


Ma = Ma or 6 = 6




The boundary conditions along constant B may be obtained

from equation (2.12) by interchanging a with B and u with

V.



2.6 Synthesis of Equations


The usual procedure followed is to reduce the

number of equations and unknowns to a more manageable

number by eliminating Qa and Q from the equations of

motion (2.10) which are thus reduced from five to three.

The force and moment resultant expressions (2.11) to-

gether with the constitutive relations (2.5) are then










substituted into the equations of motion. Finally, the

strain-displacement equations (2.3) and (2.6) are substi-

tuted, to yield three differential equations of motion

having u, v, and w as dependent variables and a, B, and

t as independent variables.


2.6.1 Shells of Revolution

In the preceding analysis Ra and R are arbitrary,

but in all subsequent applications these principal radii

will refer only to shells of revolution. It can be shown

[47] that for a shell of revolution the principal radii

may be expressed as


Ra = -[l+(DR /x3) 2 3/2 /2R /x3 2)
(2.13)
R = R /l+(DR /Dx3)2


where R is the radius of the latitude circle and x is

the coordinate coincident with the axis of revolution.


2.6.2 Conical Shells

The above described synthesis may be carried out

using equation (2.13); however, the equations would be

extremely unwieldy. Thus, the procedure will be followed

only for a specific shell of revolution shape and the one











used in the remainder of this study is a circular conical

shell. A conical shell reduces to a cylindrical shell

when the semivertex angle becomes zero. Similarly, it

reduces to a flat circular plate when the semivertex

angle approaches a right angle. Therefore, the following

analysis can handle cones as well as the two related

problems. The conical coordinate system adopted is shown

in Figure 3. The coordinates, coefficients of the first

fundamental form, and principal radii become


a = s 8 = 0

A = 1 B = s sin a (2.14)

R = c R = s tan a.


In all subsequent relations, a will denote the semivertex

angle.

Assumptions relating to displacement magnitudes are now

introduced into the analysis. The squares of inplane

displacements are assumed to be small with respect to the

squares of normal displacements and the shell thickness

is assumed to be small compared to the radius.

















A

f--


A -A


Figure 3. Conical Shell Coordinates











Neglecting rotary inertia terms, the nonlinear

equations of motion are



Ns+sNs -N +N,0 = sphu



-ctn2 vN /s+ctna w, N /s+N, +2Ns+sNs+ctna w,sNs


0 so sO
+ctna M, /s+2ctnc M /s+ctna Ms = sphV


w N5s N0 0 w
w, sN+sw, ssN +sw,sNs-ctna N -ctna v, N /s+w,W N /s


0 0 sO se sO
-ctna vN, /s+w, N, /s-ctna v,NSN +2w, sN -ctna vN's


sO sO s s 0 0 s0 sO
+w, Ns +w, N, S+2Ms +SMss-Ms+M, /s+2M, /s+2Mse +sp
5'i 's s 5 s Iss- s p / s' P



= sphW


where


(2.15)


= 0 sina.










The nonlinear stress resultants and stress couples
are



Ns = [Eh/s(l-v 2)]{su,s+s(w,)2/2+v[v, +u+ctna w+(w, )2/2s

-ctna vw, /s]}


Nso = Nas = (Gh/s)(sv, +u, -v+w, W, -ctna vw, )


N6 = [Eh/s2(1-v2)]{sv, +su+sctna w+(w, )2/2-ctna vw,

+ctn2 v2/2+v[s2us+s2(w,s)2/2]} (2.16)


Ms = [Eh3/12s(l-v2 )]{-sw,ss+ctna u, s+ctna(w,) 2/2

+v[-w, 0/s-W,s+ctna(w, )2/2s2]


Mso = M -s = (Gh3/12s)(-2w, s+2w, /s+ctna w, sw, /s)


M0 = [Eh3/12s2(1-v2)](-w, -sw, -ctna u-ctn2 a w-vs2w,) .











Substituting equations (2.16) into equations

(2.15) yield three equations of motion. Noting that

Gh = K(I-v)/2

we have


K{[-u/s+u,s+su, ss+u, /2s-3v, /2s+v, s/2-ctna w/s

+(w,s/2s)(w, -ctna v, )+(w,s/2s)(w, -ctna v)+(w, )2/2

-(w, )2/2s2 +ctna vw, /s2-ctn2a v2/2s2+swsw,ss
(2.17)
+v[-u, /2s+v, /2s+v,s/2+ctn w,s-w, s/2(w, s+w,/s

-ctna v, /s)+w, /s(w, s/2-w, /2s-ctna v,s+ctna v/s)

-ctna vw, s/2s]} = phsu



K{3u, /2s+u, s/2-v/2s+v,s/2+sv, ss/2+v, /s+ctna w, /s

+w, (w, ss/2+w, /s2)+W, (w,+w, /s)/2

+ctna[-sv(w,ss/2+w, /s2)+W,s(-v+u, /2)+w, u/s]/s

(2.18)
+v[-u, /2s+u, s/2+v/2s-v,s/2-sv, ss/2+w,sW, s/2-w,sw, /2s

-w, ssw,/2+ctna u, s/s(-ctna v+w, )+ctna w,s(v-u, /2)/s

+ctn vw, ss/2]} = phsV











K{-ctna (u+v, +ctna w)/s+ctna[(w, )2/2+ww, ]/s2+su,s W,ss

+Ws(Uss+Vs/ '^ ^^ 't 9 9N ss
+w, (su, +v, /2+u,w /2s)+w, v, /s 2+w, (v, 4s 2+V,ss/2

+u, '/2s)+w,'s(u, /s+Vs)+u,s'W, +uw, /2+u 2 w,-/2s2

-vw, s/s-v, sW, /2s+vw, /2s 2-v, W,s/2s+ctnct(-v,2 /s2-uv, /S2

-vv, /s2-u, v/2 's -V,/2-uV, 's/2s+vv, s/s-v, ss/2-u, sv/2s

-v 2/2s2 )+ctn a2 (-v, w/s 2)+v[-ctna u, +w, s(v, +ctna w)

+w, (U,S /2s-v, ss/2)+us W, 4/s+w, s(Vs/2+ctn ws/2-u, w/2s)

+uw,+ss+U,sW, s-V, sW,s-u,W,'s/s+vw, /s+u, w, /2s2 (2.19)

+v,s W,/2s-vw, /2s2+v, Ws/2s+ctna (-u, sV,/s-us,sv/2s

+v,2/2+u, v, /2s+vv, /2-u ,v/2s2-vv, /s+v2/2s2)]}

+D{-swssss-2wss2 /s-w, /s3+2w'sq/s2-2w'sss-4w, /s3


'ss s s s^s ss s /sss ss, -4

+W,'s W, /s2-'sW, /s3+Wss W, /s2+(w, ) 2/s2

+v[(ctna/s2)((-7/2s)(w, w,'s)+3(w, )2/s2+ww,s W,/2s

-WssW,-W,'s W,'s)]}+spn = phs











where the membrane stiffness is


K = Eh/(1-v2)



and the flexural stiffness is


D = Eh3/[12(1-v2)1.


(2.20)


(2.21)














CHAPTER 3


NONLINEAR DYNAMIC STABILITY
EQUATIONS OF CONICAL SHELLS



The nonlinear dynamic stability equations are

obtained from the nonlinear equations of motion by

employing a perturbation analysis. Substituting the

displacements


u = uI + uB


v = vI + vB


w = I + wB


(3.1)


and the stress



Ns N s + Ns
= NI B


N = N + N


N N + N
I B


resultants and stress couple expressions



M1 = MS + Ms
I B


M = M1 + MB


Mso = Ms + Mf
I B


(3.2)











and the load relation


P = P + PB (3.3)


into equations (2.15) results in two coupled sets of equa-

tions. The quantities with subscript I are measured from

the undeformed state and the quantities with subscript B

are small but finite perturbations. This gives one equa-

tion set describing the motion in the stable space surround-

ing the undeformed state and another set, the stability

equations, describing the motion during passage thru the

nearest bifurcation point.

The resulting set of nonlinear dynamic stability

equations are as follows where the subscript B has been

dropped for convenience:


K[(-u/s+u,s+su,ss+u, /2s-3v, /2s+v, s/2-ctnc w/s)


+v(-u, /2s+v, /2s+vS/2+ctna w,s)
(3.4)
+(ww/2s+w2 2 2)
+(WsW, /2s+w's, w, /2s+w,s/2-w, /2s +sw,sW,ss


(-w /-w /sww /2s-w2/2s2)] = phsU
Ws/ sW, 2 w w










K[(3u, /2s+u, s/2-v/2s+v,s/2+sv, ss/2+v, /s+ctna w, /s)

+v(-u, /2s+u, /2+v/2s-v,s/2-sv, ss/2)
(3.5)
+(w, W,ss/2+w, w, /s2+wsW, /2+w, sW,/2s+ctn2a ww, /s2


+v(w,s W, /2-w,s W,/2s-w,ssW, /2)] = phsV




K{-ctna(u+v, +ctna w)/s+,v(-ctna u,)

+ctna [w/2s 2+ww, /s2+V(w, /2+ww, )]}


+D{(-swssss- 2wssq /S-W, O /s3+2wsw /s2-2sss

-4w, /s3+W, ss/s-W, s/s2)+ctn[w, 2+w, ssw /s2


+ws 2/S2 +W 5sss W,'s5/s3-ws '4W,/s3+w, Wss/s2 (3.6)


+W,'s W, /s2+v(-3w, w's /s +3w, /2s +Ws W, /s3


-w,ssWw/s2-, s Ws/s2)]} + spn


+w,sNs+sw, ssN+sw, sNIs wN/s+w,l N /s = phsw .











The single underlined quantities represent non-

linear membrane stiffness terms. The double underlined

quantities represent nonlinear bending stiffness terms.

The nonlinear stability terms are denoted by a dashed

underline. These equations embody the assumptions that


wI < wB

and that


uB < w


vB < WB



In the present study, a thin truncated conical

shell is loaded by a constant axial load and a nearly

axisymmetric Heaviside pressure load. The membrane-

state stress resultants are given by


NS = -P/(Trs sin2a)-pn(stanoa)(1-cosxot)/2



NI = -pn(stana)(l-cosx t)



where P is the applied axial load, pnI is the magnitude

of the Heaviside pressure load, and A is the natural

frequency of the breathing mode of the shell.














CHAPTER 4


METHOD OF SOLUTION



4.1 Circumferential Modal Analysis


The nonlinear partial differential equations

derived in previous chapters are spatially two-dimen-

sional. The two directions are meridional and circum-

ferential. In this analysis, the two-dimensional equa-

tions are reduced to one-dimensional equations by

utilizing circumferential modal analysis. The inde-

pendent variable 0 is eliminated by expanding all

dependent variables into sine or cosine series in the

circumferential direction. As a result of the trig-

nometic series expansions, there is one set of govern-

ing equations for each circumferential wave number

considered. For a linearized analysis the sets are un-

coupled. For a nonlinear analysis the equation sets are

modally coupled through the quadratic terms. This cou-

pling arises when use is made of the trignometric prod-

uct identities


cos(ie)cos(je) = (1/2)[cos(i-j)6+cos(i+j)6]
(4.1)
sin(ie)sin(je) = (l/2)[cos(i-j)e-cos(i+j)e]





40





in order to facilitate the equating of like coefficients

of Fourier expansion terms which is required to identify

the equation sets.

Substituting the expressions


u(s,i,t)


v(s,p,t)


w(s,4,t)


p(s ,t)


Q
n=o
Q
= z,
n=o
Q
= z,
n=o
Q
= o
n=o


u (s,t) cos(np/sina)


v (s,t) sin(ni /sina)


Wn (s,t) cos(ni/sina)


p (s,t) cos(n4/sina)


into the nonlinear dynamic stability equations (3.4 -

3.6) and equating like coefficients of Fourier expan-

sion terms yield the following:


(-l/s){l+[n2(l-v)/(2sin2 )] u +u n's +su n'ss


+(n/2s sina)(-3+v)v +(n/2sina)(l+v)v n -(ctna w /s) (4.3)


+vctna win's = phsu n/K


(4.2)










(n/2s sina)(-3+v)u (n/2sina)(l+v)u n's

+{-[(1-v)/2]-(n2/sin2 )}(v /s)+[(1-v)vn's/2] (4.4)

+[(1-v)svn',ss/2]-[n ctna w /(s since ] = phsV n/K



-ctna [(un/s)+vun's+(n/sina)(v /s)+(ctna wn/s)]

+(D/K){[(4n2/sin2a)-(n4/sin4 )](w /s3)+[(2n2/sin2a)+1]

x[(Wn'ss/s)-(Wn's/S2)]-2wn'sss-swn'ssss}+ p n/K

-[P w /(KTrsin2a)]+(1-cosX t)[-s tana B ,s
I n 'ss /2)+(2n c2 s

-(s2tana n, /2)+(2n2 n/sin2a)-(2n n/sina)]/K
c s cs/4) (4 5)

+nc{ctna{(-n2w2/4s2sin2a)+v\)[(w ,2/2)+(w n,2/4) (4.5)

+w w +(w w /2)]}+(h2ctna/12){w 2 +(w 2/2)
+o o'ss n n'ss O'ss n SS

-[n2/(2s2sin 2a)](nw n ss-w ,2)}}+n {ctna[(-n2 wo /

(s2sin2a))


-(no's n'sW oss n o n'ss 'ss nss

-(n2/s2sin2a)wo'ss Wn]} = phsOn/K











where

n = circumferential wave number

S= 1 n=o
S o n>o



n o
n zw2c c c o n=o (4.6)
c1= ( Pn) i-n P I n>o


1= w./2[ns p( i+n)+ sli-n p i nl],
s i=-0

s n=o
n>o



s = l in+l
p =o i=n
-1 isn-1




4.2 Elimination of Meridional Derivatives


The next phase of the analysis involves the re-

duction of the partial differential equations sets result-

ing from the circumferential modal analysis to ordinary

temporal differential equation sets which are then amen-

able to numerical solution.

Referring again to Figure 1 it is seen that the

available solution methods for this task fall generally

into four categories. These are finite element methods,

finite difference methods, variational principles, and












methods of weighted residuals. As noted by Finlayson

[49], for certain types of linear problems these methods

can be shown to be equivalent to each other. From the

literature abounding on shell dynamics one can safely

conclude that any of these methods when judiciously ap-

plied will give acceptable results. There are, however,

marked differences between the methods in the amount and

quality of manual mathematics and the quantity of machine

computing time required to give equivalent results.

The desired characteristics of the solution method

for the particular problem of interest in this disserta-

tion are now discussed. The solution method utilized

should yield an approximate direct solution where con-

vergence would be monotonic and assured. Spatial control

of error residuals was desired. Because the taper in the

conical shell results in nonsymmetric stiffness matrices,

a solution technique applicable to non-self-adjoint prob-

lems was required. Finally, it was hoped that the chosen

solution method would result in short computer compile

times because experience has shown that the majority of

computer time required for such a problem is in successive

compile times during development and debugging.

In matching these desired characteristics to the

known characteristics of the classical solution methods

an elimination process was begun which finally resulted

in the algorithm used herein.











Finite element techniques were eliminated from

consideration because this method is not competitive with

other methods in problems with regular geometries and

formulatable boundary conditions lying along coordinate

lines. This method requires significantly more manual

mathematics and computer execution time than equilibrium

equation solution methods in order to give comparable

results.

Finite difference methods have been shown to be

extremely powerful in solving nonlinear dynamic stability

problems involving shells. This method has been used with

particular success when coupled with circumferential mode

analysis. It was shown, however, by Radosta [50] that

convergence is slow when a meridional mesh is used with

a shell under high axial load. In the particular problem

at hand, i.e. a conical frustum subject to a high axial

load combined with a time dependent pressure load, it was

felt that a suitable selection of number and spacing of

meridional mesh points would degrade into a highly time

consuming trial and error effort.

The final phase of solution method selection con-

sisted of choosing one of the four methods of weighted

residuals: Galerkin, Least Squares, Collocation, and Sub-

domain. The least squares method was not seriously con-

sidered because it yields ordinary differential equations











of the second degree in the temporal operator. The

collocation technique was eliminated because it enhances

the same point location selection drawbacks as does the

finite difference method. Again, this is normally an

insignificant problem but was made significant here by

the expected buckled states resulting from the combined

type loadings being considered. The final choice between

the Galerkin and subdomain methods was based on the method

of error residual control. In the Galerkin technique the

error produced by each individual term in the assumed solu-

tion is forced to average to zero over the entire length

of the shell. In the subdomain method the error in the

total assumed solution is forced to average to zero over

each subdomain. This type of spatial control for error

residual was considered superior to the modal type of

error control offered by the Galerkin method and there-

fore, the subdomain technique was selected for use in this

investigation.



4.2.1 Subdomain Method


A description of the theoretical basis of the sub-

domain method is now in order. Consider the equation


L[u(x,t)] = 0 in V (4.7)

where L is a nonlinear, non-self-adjoint, ordinary











differential operator. If the approximation solution is

represented in the form
m
= Z ai (t)4i(x) (4.8)
i=l 1 1


and this is substituted into equation (4.7), we obtain

an approximation


L[u(x,t)] = e (4.9)


where E is the error. We now impose constraints on E by

requiring it to average to zero over a subdomain v of

the total domain V


fL[u(x,t)]dv = 0 (4.10)
v

The number of subdomains v are chosen to equal m the

number of time dependent unknowns in (4.8). The differen-

tial equation, integrated over the subdomain is then zero,

hence the name subdomain method. As m increases, the dif-

ferential equation is satisfied on the average in smaller

and smaller subdomains, and presumably approaches zero

everywhere. This process yields a dynamically coupled

second order set of nonlinear temporal differential equa-

tions which may in turn be solved by numerical methods.

Equation (4.8) is called the trial function and

may be selected in a variety of ways. It may be chosen











to satisfy the boundary conditions, but not the differen-

tial equations or to satisfy the differential equations

but not the boundary conditions. In order to simplify

the required hand calculations, a set of complete power

series were chosen as the trial functions for this prob-

lem. The trial functions were then forced to satisfy

the boundary conditions by a modified application of the

Lagrangian multiplier method which can be more properly

thought of as a coordinate transformation to require an

assumed power series solution to satisfy the essential

boundary conditions. An approximate satisfaction of the

differential equations was then obtained by the subdomain

method.


4.2.2 Method of Satisfying Boundary Conditions


A description of the technique utilized to satisfy

boundary conditions will now be presented. Consider the

trial function (4.8) in matrix form,


= L J {a} (4.11)
(1xm)(mx1)


and the j essential boundary constraints,

[C ]{a} = 0. (4.12)
(jxm) (mx1)










Equation (4.12) may be partitioned as follows


[ C1 C2 ] 1 q 1
[jx(m-j)] (jxj) q2
(mxl)



Equation (4.13) may be rearranged as



{q2} = -[C2 -I [ C1 ] {ql}
(jxl) (jxj) [jx(m-j)][(m-j)xl]


and noting that


{ql} = [I]{q }


we obtain


{a} = [ B ] {ql}
(mxl) [mx(m-j)][(m-j)xl ]



and (4.11) becomes


u = LpJ[B]{qI}


where [B] may be considered as a projection operator.


(4.13)


(4.14)


(4.15)


(4.16)


(4.17)





49





After nondimensionalizing equations (4.3 4.5)

according to


h = s2h

S = P P
I cr I


s = s2s

S= s2u


Wn = 2n
-Sw


P
cr


= 2rEh2cos2a//3(l-v2)


(4.18)


T = /E/p(1-v2) t/s2


Pn = Eh/(l-v )s2 Pn


\ = s2/p(1l-2)/E An


derivatives are likewise

nondimensionalized.


the trial functions are introduced into the equations.

The trial functions are

un an.
"n ni
v N- b .
n = ni s (4.19)
w io Ci

Pn Pni



After applying the transforms satisfying the

boundary conditions, the equations are then integrated

over (N-j) subdomains and the average residual error for











each subdomain equated to zero. During the original

formulation it was noted that this approach leads to a

singular mass matrix for equally spaced subdomains. To

circumvent this problem the equations were mutliplied

through by s. This is similar to a procedure known as

the method of moments first employed by Polhausen [51].

Having completed the above sequence of operations

results in Q sets of nonlinear second order temporal dif-

ferential equations. Each set contains 3 (N-j) equations

and the sets are coupled in the nonlinear terms. In ma-

trix form a typical set may be expressed as


[m][B]{q}(n)+[k][B]{q}(n)+[D][B]{q}(n)+{Nl.}(n) {p}(n)


(4.20)

where

[m] mass matrix

[B] matrix requiring power series solution to

satisfy the prescribed boundary conditions

{q}(n) generalized coordinates of the nth cir-

cumferential mode

[k] stiffness matrix

[D] stability matrix

NL. Lq J[B ]T[C ][B ]{q }- power series products

{P} load vector .





51





As an example, for a problem retaining two circumferen-

tial modes with eight subdomains along the meridian,

equations (4.20) would represent a total set of forty-

eight coupled nonlinear second order ordinary differen-

tial equations to solve.














CHAPTER 5

NUMERICAL ANALYSIS FOR CONICAL SHELLS


The sets of equations (4.20) were coded into

computer language for numerical solution. A particular

shell specimen was selected for study. The salient

geometry and boundary conditions for the shell studied

are given in Appendix I. Fortran IV, level H machine

language was used and the program was constructed for

double precision arithmetic. A flow diagram of the oper-

ational characteristics is given in Appendix III. A list-

ing of the program is given in Appendix IV.


5.1 Functional Description of the Computer Program


If inplane inertia is considered ignorable with

respect to normal inertia, then equations (4.20) separate

into coupled sets of 2(N-j) nonlinear algebraic equations

and 2(N-j) nonlinear first order differential equations.

The flow of computations proceeds as follows. The

linear and nonlinear stiffness matrices, mass matrices,

boundary transform matrices, buckling matrices, and load-

ing vectors are generated by the program. At time equal

to zero the problem solution begins with known external











loading and initial conditions. The known normal deflec-

tions are substituted into the first two equations of

equilibrium and the resulting u, v vectors are generated

utilizing a Gauss elimination technique with complete

pivoting. If nonlinear effects in the first two equa-

tions of equilibrium are considered, this mode is itera-

tive in nature. Having generated the u's and v's for

time zero, they are substituted into the sets of 3rd

dynamic equilibrium equations for temporal integration.

The time integration is accomplished by a Hamming pre-

dictor-corrector integration technique. This generates

the w's for the next time step. Normal velocities are

also computed. This completes the sequence of advance

to the first time increment beyond time zero and provides

the required initial conditions in order to begin the

next advance in time. The sequence then recycles and the

temporal advance continues. At any time interval the an-

alyst chooses the solved for dependent variables are

transformed from the solution space to real space and

displayed as output. Each output consists of the com-

plete meridional profile for each mode in the solution.

The build up of errors in the numerical solution

is controlled by the following mechanisms. A loss of sig-

nificance control indicator is incorporated into the Gauss

elimination routine which warns the analyst when solutions












are being derived from a nearly singular coefficient

matrix. An error bound control is available in the

predictor-corrector integrator which limits this type

of error by automatically adjusting the size of the

time step. Round off errors are minimized by making

all calculations accurate to sixteen significant figures.

Such rigorous control of possible numerical errors is

essential in dynamic stability analysis because of the

possibility that numerical instability might be mistaken

for actual structural instability.



5.2 Checking the Computer Program


The control of clerical and conceptual errors in

the construction of the computer program was done by con-

tinually checking back to simpler known cases. In some

instances the known cases could be obtained by hand cal-

culations, in others, the results of other analysts were

used. For the shell specimen studied, the computer pro-

gram has been verified by checking against known solutions

for static stability, static deflection, free vibration

modes and frequencies [39], linear dynamic response, and

regions of parametric instability [52]. A convergence

study was also conducted to determine the convergence

properties of the solution for increasing numbers of











subdomains. The results of this study are presented in

Appendix II.



5.3 Dynamic Stability Results


After the above verifications were completed, the

program was used to study the linearized and the nonlinear

dynamic stability of a short conical frustum with simply

supported boundaries subject to various levels of constant

axial loads and critical levels of nearly axisymmetric

Heaviside pressure blasts. Eight subdomains along the

meridional direction were used. The driven breathing mode

and an arbitrary preferred flexural mode were retained in

the analysis. Complete coupling between the modes was re-

tained through the quadratic terms for the nonlinear case.

The critical static normal pressure for the speci-

men studied is 42.8 pounds per square inch and the critical

circumferential wave number is eight. This represents a

mid-span deflection in the breathing mode of 9 percent

of the thickness. The ordinate scale (W) of all figures

depicting time history responses is normalized by this

critical deflection. The critical static axial load is

40,400 pounds. The applied loads for time history graphs

will be noted as


(a, b, c)





56





where



a = P/Pcr, b = p /p cr, c = p /pcr.





Additionally, note that



Po = POH(t)


Pn = Pn H(t)




where



H(t)



denotes a Heaviside pressure pulse.


5.3.1 Dynamic Modal Preference


The program was used to determine the critical

flexural modes during dynamic instability. The results

are shown in Figure 4 and the critical static buckling

modes are indicated. Because only a few closely spaced

critical modes dominate the flexural response and because


I














coupling between these modes is very weak [10], the param-

eter study that follows is reduced by considering the in-

stability of only a sequence of single flexural modes to

find the mode indicating instability at the lowest load.



5.3.2 Linearized Dynamic Stability


The linearized dynamic analysis was obtained by

considering the vector {NL}(n) in equation (4.20) to be

zero. This means that the internal loads from the pre-

buckling state may interact with postbuckling deflections

to cause primary dynamic instability (i.e. snap through)

but internal parametric instabilities due to the pulsat-

ing nature of the membrane state are prohibited.

The critical circumferential wave number for

dynamic instability was found utilizing this analysis and

the results are shown in Figure 5.

While conducting a series of linearized dynamic

stability computer runs at various combinations of external

loadings, a feature of the numerical model was discovered

which represents an improvement over other computer programs

used for this purpose. This feature is that the meridional

profile undergoes a marked and rapid change of character

during runs at or above the critical dynamic load. For

loadings below critical, the meridional profile pulsates at


I




















































































-0 -0
0 (1)
owo

E r-


4-) 3 (3
Co -=











one half wave, slightly skewed due to the conical taper.

At or above the critical load the meridional profile

begins to pulsate as a half wave but later snaps to two

or more half waves. The time of snap, number and shape

of the waves in the buckled meridional profile depend on

the axial load to normal pressure ratio and their rela-

tion to static critical loads. Examples of this are shown

in Figures 6 and 7. It is felt that this improved ability

of detecting dynamic instabilities results from two areas

of the analysis. One being the spatial error residual con-

trol offered by the modified subdomain method employed and

the other being the strict control of error bound in the

numerical work. Additionally, it should be noted that

most published works on this type of problem utilize one

term assumed solutions for preferred modes and clearly

could not see this effect in their analysis.

A large number of linearized dynamic stability

runs were made at various combinations of static axial

and Heaviside pressure loads. The results were collated

into a dynamic stability interaction curve and are shown

in Figure 10.


5.3.3 Nonlinear Dynamic Stability


In the case of nonlinear dynamic stability, the

possibility of autoparametric excitation of preferred







6]






















-0
m






0






4-







4-)
a, CD V
E
u- *
E




(Nj -0









S.-


.:I- CNJ


o 9
00 C


C-


0



00 0






V) E -C -C















o o
o o
Co C




C) C














cO o


SS-






4-



o


-4.




0 ---

\ E








CMt-~

5-







-,,


MENNEW











flexural modes by the pulsating breathing mode is allowed,

in addition to snap through or primary dynamic instability.

The question here is whether such excitation will occur

during the primary instability motion of the preferred mode

sufficiently to affect the magnitude of the external loads

required to induce dynamic instability. In contrast to the

external parametric excitation problems, the total energy

of the shell for the present problem is constant for all

t>0. Thus, the unstable modes when growing must derive

their energy from the breathing mode with a corresponding

decrease of energy in that mode. To evaluate this inter-

play a full nonlinear analysis is necessary.

Conceptually, the third equation of nonlinear dy-

namic stability may be expressed in an average sense as

follows:


2 2
w + w + k(u0 ) + f(w) p H(t)

(5.1 a,b)

[Xn g(w )]wn + 1(n ,vn) + h(w )wn = p H(t)



where u w0 represent breathing mode deflections measured

from the undeformed state and u v w represent the

poststable growth of the buckling mode. X and n denote

the linear eigenvalues of the respective modes and f, g, h,











k, and 1 are differential operators. The superscript a

denotes an average over the shell length. Since w0 is

excited by a Heaviside pulse, the w0 response is initially

periodic and equation (5.1 b) takes the character of an

inhomogeneous Mathieu equation. The primary buckling

terms denoted by a single underline cause an insignificant

shift to the left on a parametric stability diagram. In

the absence of imperfections in the preferred flexural mode,

denoted by the double underlined term, equation (5.1 b)

lies well within a stable zone of a Mathieu diagram and no

significant parametric growth of the buckling mode would

be expected.

For finite deterministic imperfections, however,

the location of stability boundaries becomes more difficult

and for significantly large p H(t), solutions to equation

(5.1 b)may exhibit parametric growth at or below the dynamic

loads required to produce immediate snap through and may

lead to a delayed dynamic snap through. Whether or not

this beating type of growth occurs with sufficient quick-

ness to have an effect on the primary dynamic instability

can be determined only by direct temporal integration of

two coupled sets of equation (4.20).

A series of runs with the computer program was made

to determine the significance of this effect. Typical re-

sults are shown in Figures 8 and 9. As in the case of
































t.0
cy')





















H~


cy*) C:

C

































0 0
o 0
0 0

o 0
C \J



o o
LO U)

0 0
OO)CY


COOc


moom











linearized dynamic stability, a large number of computer

runs were made at various combinations of static axial and

Heaviside pressure loads and the results collated into

a nonlinear dynamic stability interaction curve shown

in Figure 10.

The linearized dynamic stability curve lies below

the static stability curve in Figure 10 due to the in-

creased severity of a suddenly applied load over that of

a statically applied load. The peak membrane forces de-

veloped by the breathing mode which is excited by a Heavi-

side pulse are twice as large as that of an equivalent

static case and contribute more to the developemnt of

instability in preferred flexural modes than a static

situation. This is known as the dynamic load factor effect.

The nonlinear dynamic stability curve lies below the

linearized dynamic stability curve because internal vibra-

tions interact between coupled modes so as to produce un-

stable beating resonances in the preferred buckling mode.

Such unstable vibrations lead to a delayed dynamic snap

through and this phenomenon is called autoparametric ex-

citation.











Static Stability
Linearized Dynamic
.... Stability
-_.=_ Nonlinear Dynamic
Stability


n = 8


0 0.5


p H(t) / pc
cr


Dynamic Stability Interaction Curves


1.0










0.5










0


1.0


Figure 10.














CHAPTER 6


RESULTS AND CONCLUSIONS



In the present analysis, a nonlinear shell theory

is derived and employed to study the nature of nonlinear

dynamic instability of a truncated thin circular conical

shell structure which is considered to be loaded by a

constant axial load and a nearly axisymmetric Heaviside

pressure load. A solution method is developed which sat-

isfies a boundary condition exactly and which converges

toward the exact solution of the governing equations with

increasing subdomains. The method avoids the necessity

of assuming the shapes of prebuckling or postbuckling

meridional profiles.

The results of this study confirm the feasibility

of the method of solution developed for this analysis. The

method converges rapidly yet can be applied to nonlinear

problems with minimum amounts of manual mathematics. The

trial functions are simple to manipulate yet satisfy any

formulatable boundary condition. The method is particu-

larly suitable for problems where poststable modes are not

known in advance as is usually the case for combined dynamic

loadings.











For the particular loading conditions studied,

the results indicate that dynamic instability will occur

at loads below critical static loads. This reduction in

critical dynamic loads was shown to be the result of both

the dynamic load factor effect and of autoparametric ex-

citation. The interaction curves which are generated

illustrate these effects quantitatively under conditions

of combined dynamic loading.

A criterion for dynamic buckling is established

based on meridional mode shape changes. This ability to

detect sudden jumps in the meridional profile aids to

verify instability detected by divergence on displacement

time history curves and provides additional information

about the poststable state.






























APPENDIX I














APPENDIX I


SHELL SPECIMEN AND BOUNDARY CONDITIONS



The conical frustum utilized in this investigation

has the following properties (see Figure 3):


Material -

a = 200

sI = 5.85

s2 = 14.22

R1 = 2.0

R = 4.863

h = 0.02

v = 0.3 .


1020 steel



inches

inches

inches

inches

inches


The boundary conditions assumed for the analysis


are:


u = v = w = M = 0 at s and s
u = v = w = M = 0 at s I ands2


This results in eight constraint equations which when ex-

pressed in terms of power series solutions for eight sub-

domains along the meridian become

10 1
I a.s = 0 at s = sI and s = s2
i=o





73





10 -
Z b.s = 0 at s = s and s = s2
i=o


12
Sci.s = 0 at s = s1 and s = s 2
1=o


12 2 2 i2
E c.[(i)(i-1) + v(-n /sin 2 + i)]si- = 0
1=o
at s = s


and s = s2.




























APPENDIX II














APPENDIX II


CONVERGENCE


In the use of the subdomain method, convergence

in the mean is desired. The main influence on convergence

is the choice of trial functions. For assured convergence,

the trial functions must be complete and linearly indepen-

dent [49]. The completeness property of a set of functions

insures that we can represent the exact solution provided

enough terms are used. The power series trial functions

used in this study are complete, for example, so that any

continuous function can be expanded in terms of them.

To demonstrate the rate of convergence for the

problem under study, a series of computer runs were con-

ducted to determine the dynamic perturbation response for

different numbers of subdomains. Dynamic stability response

for eight, ten, and sixteen subdomains was investigated.

The results are shown in Figure 11. The ordinate designa-

tion implies percent differences with respect to the best

approximation.





76


















8

I

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.- 4
4P
uJ

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5 10

Number of Subdomains


Figure 11. Convergence Properties of Solution


15




























APPENDIX III















APPENDIX III


FLOW DIAGRAM OF THE COMPUTER PROGRAM



For a better understanding of the computer program,

a key which may be helpful in going from the theory to the

program is given below:


FORTRAN Name Theory Description


N m Number of subdomains

NW n Circumferential wave number

Q Q Total number of n

AP a Semivertex angle of cone

RO p Material density

E E Young's modulus

V V Poisson's ratio

F s2 Conical slant height

BTT B Boundary condition matrix

STT [k] Linear stiffness matrix

DND [m] Mass matrix

PV {p} Load vector

PLUG {NL} Nonlinear stiffness vectors

X T Nondimensional time











FORTRAN Name


P

PO
T2
r


Nondimensional displacements

Axial load

Pressure level of Heaviside pulse

Upper time limit

Error bound

Online core storage matrices

Hamming's predictor-corrector
integration subroutine

Gauss-elimination subroutine


A flow diagram of the computer program is illustrated

in Figure 12.


U,V,Y

P

PSD

PRMT(2)

PRMT(4)

ORAN(I)

DHPCG


DGELG


Theory


Descriptions







80




















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APPENDIX IV














APPENDIX IV


FORTRAN IV SOURCE PROGRAM



The computer program developed to provide the

numerical solution to this dissertation problem is listed

below. The program requires 182,000 bytes computer core

storage space. A typical nonlinear dynamic stability run

executes in approximately in seven minutes. The compile

time is three seconds.
















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