Transitional instability of spherical shells under dynamic loadings

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Transitional instability of spherical shells under dynamic loadings
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Ho, Fang-Huai, 1934-
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Engineering Mechanics thesis Ph. D
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Thesis--University of Florida, 1964.
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Bibliography: leaves 169-172.
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Manuscript copy.
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Vita.

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TRANSITIONAL INSTABILITY OF SPHERICAL

SHELLS UNDER DYNAMIC LOADINGS












By
FANG-HUAI HO


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











UNIVERSITY OF FLORIDA


April, 1964














ACKNOWLEDGMENTS


The author wishes to express his gratitude to the members

of his supervisory committee: Professor W. A. Nash, chairman

of his committee, for suggesting the dissertation topic and for

the encouragement and advice received from him through the period

of this research; Professor W. L. Sawyer for reading the complete

manuscript and making many corrections; Professor I. K. Ebcioglu

and Professor C. B. Smith for their encouragement and advice; and

the late Professor H. A. Meyer for his many suggestions in the

numerical solution of the problem in Chapter IV.

He is also indebted to Dr. S. Y. Lu of the Department of

Engineering Science and Mechanics for reading Chapter V of this

research and for providing the opportunity to discuss several

questions in that chapter with him.

The author is indebted to the Office of Ordnance Research,

U.S. Army, for their sponsorship of this study.
















TABLE OF CONTENTS


Page

ACKNOWLEDGMENTS . . ii

LIST OF FIGURES . .... v

ABSTRACT .a a a a a a a a a a a a a a a a a a a a a a a viii

INTRODUCTION . ...... .. 1

1. A Historical Review and Recent Advancement .. .. 1
2. The Scope of the Present Research ....... 5

CHAPTER

I. A CRITERION FOR DYNAMIC BUCKLING 10

1. Autonomous Conservative System ....... 12
2. Nonautonomous System a a ... o 16

II. BUCKLING OF A CLAMPED SHALLOW SPHERICAL
SHELL UNDER A PURE IMPULSE . 20

1. A Qualitative Discussion of the Loss
of Stability of the Structure ....... 20
2. A Study of the Dynamic Response ..... 30
3. A Justification of the Buckling Criterion 38
4. A Note on the Effect of Initial Gdometrical
Imperfections .a .......... 41

III. BUCKLING OF A CLAMPED SHALLOW SPHERICAL SHELL UNDER
A SUDDENLY APPLIED CONSTANT PRESSURE OF
INFINITE TIME DURATION . 43

1. A Qualitative Study of the Instability
of the Structure ............. 43
2. Another Justification of the Buckling
Criterion a a a a a a a a .... 52
3. A Discussion of the Results .... 54













TABLE OF CONTENTS (Continued)


CHAPTER Page

IV. BUCKLING OF A CLAMPED SHALLOW SPHERICAL SHELL
UNDER A UNIFORMLY DISTRIBUTED PRESSURE
LINEARLY INCREASED WITH TIME . 60

1. The Solution of the Problem 60
2. A Conclusive Discussion of the Problem o 67

V, BUCKLING OF A COMPLETE SPHERICAL SHELL UNDER
SUDDENLY APPLIED UNIFORMLY DISTRIBUTED DYNAMIC
LOADINGS ..... .......... 70

1. Introduction ............. 70
1. On a New Mechanism of the Buckling
of a Complete Spherical Shell 73
2. Buckling of the Shell under a Static Load 77
3. Buckling of the Shell under a Suddenly
Applied Constant Pressure ........ 88
4. Buckling under a Pure Impulse. 105
5. Conclusions and Discussions 109

VI. A CONCLUSIVE REMARK . 114

APPENDICES

I. THE STABILITY THEOREM OF NONLINEAR MECHANICS 119

II. THE EQUATION OF MOTION FOR TRANSVERSE
VIBRATION OF SHALLOW SPHERICAL SHELLS 124

III. FIGURES . . 136

IV. NOTATIONS ................... 167

REFERENCES ........... ... ..... 169

BIOGRAPHICAL SKETCH .................. 173
















LIST OF FIGURES


Figure Page

1. Geometry and deformations of the shell 136

2. Phase plane trajectories and the variation
of potential energy .......... 137

3. Phase space trajectory and its projection 138

4. The dynamic buckling criterions .... 139

5. Comparison of axisymmetrical theories on the
static buckling of shallow spherical shells 140

6. Dynamic loadings ............. 141

7. Phase plane trajectories (when E = 0.26) 142

8. Response curves of the central deflection of
a shallow shell ( G = 0.26) under the action
of impulses. ............... 143

9. Critical impulse determined by buckling
criterions ........... 144

10. Comparison of analytical theories on dynamic
buckling of shallow shells under pure impulse 145

11. The threshold of instability of a shallow
shell (with e = 0.26) under a suddenly
applied uniformly distributed dynamic loading 146

12. Relation between the critical pressure and the
geometrical parameter e for shallow shells
under the action of uniformly distributed
static and dynamic loadings ... ..... 147

13. Relation between the critical deflection and
the geometrical parameter 0 for shallow shells
under various dynamic and static loadings 148












LIST OF FIGURES (Continued)


Figure Page

14. A justification of the buckling criterion
applied to the shallow shell with 6 = 0.26 149

15. Response curves for various values of W
when e = 0.26 ... ... .. 150

16. Response curves for various values of E
when V = 1.07 ................. 151

17. Upper and lower values of -7r as a
function of 9 ............... 152

18. cr curves for various values of .4/ 152

19. Critical DoO.L.F. -- 2* vs. critical
central deflection -- c- . 153

20. Upper and lower critical deflections vs. 154

21. Upper and lower values of Zcr vs. V 155

22. 7 vs. Yi/ curves for various values of e 156
cr
23. The projection of the trajectories on the
plane, when E = 0.26, \1= 5 157

24. The projection of the trajectories on the
'9 4) plane, when 8 = 0.26, 4' = 2 158

25. Buckling region of a complete spherical shell 159

26 Load deflection curve for a complete
spherical shell ... ........ .... 160

27. Comparison of theoretical result and
experimental tests ............. 161

28. Sketched phase plane trajectories for the
motion of a shell ........... 162

29. Graphical solution of equation (5.29) ... 163

30. Frequency curves of the normal and
log-normal distributions ....... 163













LIST OF FIGURES (Continued)
1


Figure Page

31. Phase planetrajectories when
S= 0.44, q = 0.6823 .. .. 164

32. Phase plane trajectories when Q = 0.44,
q 0.35 ........ .. .. 165

33. Buckling pressure vs. buckling region
parameter a for a complete spherical shell 166












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


TRANSITIONAL INSTABILITY OF SPHERICAL SHELLS UNDER
DYNAMIC LOADINGS


By


Fang-huai Ho


April 18, 1964


Chairman: Dr. W. A, Nash

Major Department: Engineering Science and Mechanics


The dynamic instability of thin shallow spherical shells

under uniformly distributed impulsive loadings has been discussed

by use of the stability theory of nonlinear mechanics together

with a topological method.

The axisymmetrical buckling of the shallow shells under

three types of dynamic pressure, namely, an impulse function, a

step function, and a linear function of time, has been solved

based upon a first approximation and Galerkin's method. The

results of the first two cases, within the range of the validity

of the theorem, are compared respectively with those obtained by

Humphreys and Bodner, Budiansky and Roth, and Jiro Suhara.

A buckling criterion based upon the relation between the


viii














dynamic load-deformation curve and its counterpart in the static

case has been established through the comparison of the results

obtained by this criterion with those obtained by the application

of the stability theorem to autonomous systems.

An extension of the same techniques to the solution of

problems in the case of a complete spherical shell has also been

made. Based upon an assumed new buckling mechanism, the static

buckling pressure for such a shell obtained by a first approx-

imation and axisymmetrical deformation theory, agrees well with

recent experiments conducted individually by Krenzke and

Thompson. Transitional instabilities of a complete spherical

shell under uniformly distributed dynamic pressures in the form

of a step function and an impulse function were also discussed.

Results are presented in the form of figures.













INTRODUCTION


1. A Historical Review and Recent Advancement


The problem of dynamic instability of a thin spherical

shell under time dependent external forces is inseparable from the

problem of the transverse vibrations of such a structure. Studies

on the latter problem, as in the works of Mathieu, Lamb, Lord

Rayleigh and Love, date back to 1882. As may be found either in

Love's Theory of Elasticity or Rayleigh's The Theory of Sound,

these classic works have been mainly devoted to finding the

smallest natural frequency or the "gravest tone" of the system.

The method used by Lord Rayleigh was essentially an energy method

plus an assumed displacement pattern; the bending and membrane

energies were considered separately, depending on whether the

middle surface of the shell is extensible or inextensible.

The first rational study of the transverse vibration of

a shallow spherical shell, using three simultaneous equations of

motion for the three displacement components, was due to Feder-

hofer (44). Federhofer's problem was discussed later by Reissner

using a different approach (45)o These two authors were interested





1Underlined numbers in parentheses refer to the references.











in the transverse free oscillations of a shallow shell. In Reissner's

paper, certain conventional simplifications usually employed in the

static loading case had been introduced because of the thinness and

shallowness of the structure; the frequency equation was expressed in

the form of a determinant involved with Bessel functions; numerical

results were obtained by a Galerkin approximation method. A great

contribution to the problem of transverse vibration of thin shallow

elastic shells was also due to Reissner in 1955. In his paper (46),

by an order of magnitude analysis, Reissner justified an important

simplification for the problem; i.e., the tangential (longitudinal)

inertia terms may be omitted with negligible errors. This simplifi-

cation has made possible the solutions of other shell oscillation

problems (47, 48 and 49).

The work of Eric Reissner and others, as mentioned above,

dealt mainly with linear, free vibrations of a thin shallow elastic

spherical shell. The first investigation of the problem of forced

vibration and the problem of stability of such a structure were

probably due to Grigoliuk (11). In the work by Grigoliuk, a non-

linear oscillation system was considered for the first time.

A great amount of work in the nonlinear vibration of shell

structures and their stability under periodic forces has been done

by V. V. Bolotin. In his paper of 1958 (4), he first discussed

the problem of forced oscillation and stability of a complete

spherical shell under a periodic external loading. He considered

the oscillations of both nonlinear and linear cases, that is, the

system vibrating with both finite and infinitesimal amplitudes and










obtained the resonance curves showing that "hard excitation" occurs

until the critical frequency is reached. This is in contrast with

the oscillation of a flat plate where "soft excitation" starts at

the lower boundary of the instability region. A rather complete

collection of the problems of parametric stability of elastic

systems was also due to Bolotin, His two books, (50) and (51),

are no doubt valuable contributions to this field.

Along with the development of aerospace science, another type

of shell buckling problem has become ever so important, i.e., buck-

ling under impulsive and blast loadings. As an extension of Hoff's

work (14) on the stability problem of a column, in 1958, A. S. Vol'mir

solved a problem of dynamic buckling of a hinged cylindrical panel

under axial pressure. In his 1958 paper, Vol'mir considered the end

shortening of the structure as a linear function of time. Using a

first approximation of deflection form for both total and initial

deflections, plus a Galerkin method, he was able to solve the problem

by considering a i-degree-of-freedom nonlinear oscillatory system.1

Results were obtained by numerical integration and the critical condi-

tion of the system was determined from the response curves. As an

example, he showed that the dynamic critical load was about 1.7 times

higher than the upper static critical load. The problem of buckling

of a cylinder under external uniformly distributed load increased

linearly with time was considered by V. L. Agamirov and A. S. Vol'mir

(1) in 1959, using a similar technique as in the above mentioned


IRefer to reference (36).











paper by Vol'mir. This same problem was considered again by

Kadashevich and Pertseve (18) in 1960. In contrast to Agamirov and

Vol'mir's work, where transverse inertia was considered alone, these

two authors have also considered the inertia of axisymanetric com-

pressing of the shell. By removing the restriction on radial dis-

placement presented in (1), they considered a nonlinear dynamic system

of three degrees of freedom. Three types of dynamic load were

considered in this paper: a suddenly applied load with constant

pressure, an impulsive load with finite time durations, and a uni-

form load increasing linearly with time. For a very rapidly applied

dynamic loading, they found the contribution of the inertia of axi-

symmetric compression is essential.

In 1962, two papers treating the problem of dynamic buckling

of shallow spherical shells under uniformly applied impulsive load-

ings appeared in the open literature in the western world. The first'

paper was given by J. S. Humphreys and S. R. Bodner (15), where the

critical condition of the system was determined by an energy method,

and nonlinear strain-displacement relations were employed. To the

author's knowledge, this paper is unique in that it presents a solu-

tion of dynamic buckling problems in this fashion. Although the

behavior of the dynamic response could not be obtained by this method,

the general relation between the geometric shape and critical impulses

as well as critical deflections were obtained through a much easier

and clearer discussion, The second paper was published by

B. Budiansky and R. S. Roth (8) in December 1962. Snapping of a










shallow spherical shell under an impulsive loading with various

time durations was used as an example to establish a buckling cri-

terion proposed by these two authors. Using a higher order of sym-

metrical modes, Budiansky and Roth integrated a five-degree of

freedom dynamic system numerically, and the critical condition of

the system was determined by a proposed buckling criterion which

has a physically significant basis. It should be noted that before

Budiansky and Roth, the condition of the threshold of the shell

buckling was determined in a rather arbitrary fashion; i.e., having

neither physical nor mathematical reasoning. The future trend of

the investigation of the buckling behavior of a dynamically loaded

shell seems to be such that the governing differential equations

are integrated by various numerical means. It is for this reason

that the contribution of Budiansky and Roth is of particular interest.


2, The Scope of the Present Research


It is well known that the investigation of dynamic buckling

of a shell under impulsive or blast pressures is mainly on determining

the magnitude of buckling deflection in the process of loading.

Usually, on one hand, we have a dynamic system which is essentially

"unstable"; i.e., the deformation is unbounded as time increases, and

we are at the position to determine the threshold of the buckling

from the response curves; on the other hand, we have a dynamically

stable system, and we have to determine whether the buckling of the












structure occurs or not, because of the significantly larger displace-

ment in the transient region. The conventional method where the

critical condition was determined by the first amplitude of the

response curve, or by the state where the dynamic response has severe

changes, may be used to solve the first type of problem although it

will not be precise. It may introduce serious errors if the problem

is of the nature of the second type. As we have mentioned, a buckling

criterion has been proposed by Budiansky and Roth for a certain type

of dynamic load. These two authors have based their work upon a

certain physical picture of the deformation of the shell during the

loading process and established a certain "measure" which defines the

critical state of the structure in a characteristic load-response

diagram. The philosophy of this criterion is fresh and remarkable;

however, the difficulties in extending to the other cases is obvious.

In a rigorous manner, the correct "measure" of buckling can only be

obtained from the sample of experiments. Errors introduced from

plausible assumptions may become significant when the load-response

curve does not present a change in the form of a "Jump." Therefore,

the following questions arose: Is it possible to propose a buckling

criterion with a more general sense? Is there any relation at all

between the dynamic load-deflection curve and its counterpart in the

static case? Let us seek positive answers to both of these two

questions It is on this basis that the present research will be

devoted to the following two purposes:









A. A dynamic buckling criterion in a general sense; i.e.,

independent of the type of loading and geometry of the shell, will

be proposed from a comparative basis. Naturally, the critical con-

ditions of the statically loaded shell and a free oscillating shell

will be good measure of the critical condition of the shell under

dynamic loadings.

B. An attempt will be made to unify the two methods, i.e.,

the energy method and the dynamic response method, in the study of

the dynamic instability of shell structures. Therefore, a qualita-

tive discussion of the motion of the dynamic system as referred to the

change of total energy level is desired.

Because of the second purpose mentioned above, we shall

restrict ourselves to the problem of considering a single deformation

mode, i.e., a first-degree-of-freedom system. It is well known that

in the static case, the above restriction will make the result of the

theory applicable only for sufficiently shallow shells, e.g., <6

where


A = 2-v) T R/ 4V3( (/h (1)

which is a standard geometrical parameter used in the shallow shell

theory (16). It should be mentioned that, in this research, we

shall use a different geometrical parameter, which has the following

definition:

e = zo (2)


Refer to Figur. 1













By using this parameter, we shall have our dynamic equations in the

simplest form, i.e., e appears only in the linear term in the

differential equation. However, as shown in (2), this parameter

is rather ambiguous. Therefore, for a proper interpretation, we

always consider e related with X by the following equation


e 4.V3)- V) X X2 (3)

or, for ).= 0.3 ,


0 6.609 / A: (4)

hence, a larger E value implies a shallower shell. It will be

seen later that, for non-shallow shells, E becomes an awkward

measure for the geometrical shape. In the region of the validity

of the present theory, however, the e defined in equation (2) may

be satisfactorily used as a geometrical parameter. The transitional

instability of a shallow spherical cap under three types of impul-

sive loadings will be investigated. The dynamic equations of the

system will be obtained by using a Galerkin's approximation method,

and a phase plane method will be employed to discuss the stability

of the system. However, for the case that the load is a linear





1Numbers in parentheses in the text which are not underlined
refer to the equation numbers.





9



function of time, numerical integration will be used because the

system is then nonautonomous.

An extension of the same technique to the solution of

problems in the case of a complete spherical shell will also be

made.













CHAPTER I


A CRITERION FOR DYNAMIC BUCKLING


As well as in the static case, the loss of stability of a

shell determined by the deformation of the structure, i.e., a

buckling state, can be determined from the load-deformation rela-

tions. In the static case, to determine the state of buckling

usually does not add any trouble, because there always is a point

of relative maximum on the load-deformation curve, which is shown

by the increasing of deflection with a decreasing load, hence the

state of instability is very clear. In the dynamic case, however,

as we know, the shell may have buckled before the system reaches its

first amplitude in the nonlinear oscillation due to a severe change

in the deformation. Therefore, a stable oscillation can cause the

critical condition for the structure as well as an unstable oscil-

lation. This situation happens particularly when the load is

rapidly applied,and with short time durations, or an impulsive

type. It is for this reason that a physically significant buckling

criterion should not be based upon the stability nature of the oscil-

latory system alone; it should be safeguarded by a certain fixed

value which satisfactorily measures the danger of the structure.











However, it is well known that an unstable vibration system will have

its amplitude increasing indefinitely with time. Therefore, the

transitional point for an original stable system to an unstable system

will always represent a critical condition for the structure. This

is to say that the study of dynamic stability of the oscillatory

system is still the most important consideration in the investigation

of the dynamic buckling of shell structures, although it becomes

impossible for some cases; for example, when the system is eventually

unstable, then other techniques have to be used.

It is the purpose of this section to establish a new buckling

criterion based upon the very nature of the dynamic stability

theorems. Certain measures of the buckling of the shell of this

nature will be provided after the following discussion. The danger

of overestimation of the critical loadings will also be safeguarded

through the comparison of the characteristic load-deformation curves

for some structures under other situations, whose stability nature

are well known.

The proof of such a criterion is impossible at this stage,

yet its physical significance is not difficult to observe and will

be established through the examples given in the following chapters.














1. Autonomous Conservative SystemI


The typical dynamic equation of such a system is of the

following form

17= $f f^ ) (1.1)


where X is a parameter, e.g., the load parameter. An equivalent

form of (1.1) is the two dimensional system:




|= f (r, X) (1.2)

It is well-known that the discussion of the stability of all the

possible motions described by (1.1) is essentially the same as

discussion the stability of the motion in the neighborhood of

certain isolated points, i.e., the singular points,2 in the phase

plane of the system. These singular points are found by the condi-

tion that 1 and vanish simultaneously, i.e., from equation (1.2),




1Refer to (9), (23), (29), and particularly (3), in which a
beautiful discussion of the "conservative system" has been given.

2The names critical point and equilibrium points are also


used.










= 0



(1.3)

The first condition in (1.3) merely says that the singular points

are located on the 17 -axis (where = 0 ) It is the second condi-

tion in (1.3) that determines the singular points in the phase

plane. For a system as (1.1), we can have only two types of singu-

larity, namely, the center and the saddle point. The trajectories

around a center and around a saddle point have a characteristic

difference, and this is shown in Figure 2.

In Figure 2, fl, and 7)3 are centers; motion around these

two points is described by simple closed trajectories, which is

stable in character. The trajectory passing through the saddle

point is called a separatrix, which, less rigorously speaking, is

the partition between two motions with different characteristics.

It is also seen from the same figure that a trajectory lies outside

of the separatrix and has a higher energy level than the one located

inside of it.

By virtue of.the above discussion, we may say that the study

of the stability of the dynamic system is essentially equivalent to

finding the character of the trajectory of the system, and the loss

of stability of the system is equivalent to the condition that the

system moves on the separatrix in the phase plane. A further exam-

ination of the phase plane sketch will make it clear that the sense of











"loss of stability" mentioned above has the same nature as the usual

dynamic buckling criterions, i.e., the characteristic deformation

undergoes a severe change (increased). In the phase plane, all points

where the trajectories intersect the 77 -axis reflect the amplitudes

of the motions, because g = 0 i.e., the 77(r) curves of the

motions have a horizontal tangent at that point. The magnitude of

the amplitudes are measured relatively by the length from the

origin 0 of the phase plane. For any motion moving on a trajectory

inside of the separatrix, the amplitude increases gradually as the

total energy level increases, i.e., due to the increasing of the

external pressures. This is shown as from OA to 0 8 Once the

external pressure reaches the critical value which causes the motion

on the separatrix, the amplitude undergoes a characteristic change.

It first reaches 0 *12 and then creeps to the magnitude equivalent

to OC Any motion outside the separatrix has its amplitude

larger than 0 C e.g., OnD The severe change of the amplitude

during the loss of stability becomes apparent by comparing the

length of o08 with OC .

Let us summarize the above discussion and make a useful

conclusion. We have reached the point that the determination of

the dynamic instability of the system (1.1) is equivalent to

finding the motion on the separatrix in the phase plane of the

system. It will become clear in the later examples that the










equation of the separatrix is determined solely by the unstable

singular point of the phase plane, hence, by one of the roots of

equation (1.3). Comparing (1.3) with (1.1), we immediately found,

by its very nature, that equation (1.3) is simply the state of static

equilibrium, i.e., the counterpart of equation (1.1) in the static

state. Moreover, we have 7r7 77, and 173 the possible states

of static equilibrium, in the phase plane sketch. By possible states

of static equilibrium, we mean the deformation (or deflection)

determined by the position of these points would be a state of

static equilibrium if the external distrubance is a static one.

Thus far, we are able to state that the loss of dynamic stability

is characterized by the load-deformation relation reaching a

possible state of static equilibrium. In most problems of dynamic

buckling of shells, the singular points are interior to a closed

path. There is a theorem due to Poincare;

In a conservative system, the singular points interior to

a closed path are saddle points and centers. Their total number is

odd and the number of centers exceeds the number of saddle points

by one.

By virtue of the above theorem, since in most of the cases

of the shell buckling, the first equilibrium position always

corresponds to the trivial solution of the undeformed state,

we may state a criterion for the instability of the system has a

nature as equation (1.1), which is as follows:










Criterion. The threshold of the dynamic instability (or

buckling) is defined by a point on the characteristic load-

deformation curve, where the deformation of the dynamic system

reached the first unstable state of static equilibrium.

It is noted that, for a single degree of freedom system

(1.1), this criterion of instability should give the same

result as would be obtained directly from the dynamic stability

theorems, i.e., the phase-plane method. However, there is no

restriction in the application of the above criterion to the

systems of higher degrees of freedom, while the topological

method, in general, does not apply in such cases.


2. Nonautonomous System


In general, the topologic method cannot be used to solve

the problem of a nonautonomous system, i.e., when the time variable

r expressly appears in the dynamic equation, because the trajec-

tory of a motion is in a space rather than in a plane. For a

certain class of equations, Minorsky (24) developed a method which

he called the "stroboscopic method." By finding an identical trans-

formation, the original nonautonomous system can be transformed into

a stroboscopic system which is autonomous. Therefore, the stability

problem of a periodic motion of the original system is equivalent

to the problem of investigating the stability of singular points in

its stroboscopic system. Unfortunately, this clever method cannot

be applied to the type of problem which has nonperiodic motions and

with large nonlinearity, mainly due to the difficulties of finding









the stroboscopic transformation. Furthermore, for certain problems,

in which we are interested, the motion is known to be unstable as

time increases indefinitely. As far as buckling is concerned, we

are merely interested in knowing where the deformation begins to

increase violently or attains dangerous magnitude. We have seen

in the last case, i.e., the autonomous system, that the beginning

of the violent increment of deformation is defined by the initia-

tion of the instability of the dynamic system, and as a matter of

fact, they are identical. However, it is impossible to extend the

same logic to the nonautonomous system, for some of them eventually

reach a state of "unstable motion," e.g., a system under a forcing

function with a magnitude increasing linearly with time. No matter

what conditions we have, however, the projection of the trajectory

of motion onto a r7 7 plane still offers us some information

regarding the "violent increment of deformations," as we shall see

in the following.

Let us take the following system


j= ( 71,Z ) (1.4)


where T is the time variable. The similarity between (1.4) and

(1i1) is easily obtained by taking r equal to some definite value

of time, say A ; i.e., at the certain time X the motion of (1,4)

is on a trajectory in the phase plane characterized by (1.1).

Therefore, the motion of (1.4) can be treated instantaneously as









a motion of an autonomous system of the form of (1.1). However,

it passes only one point on the trajectory of each phase plane.

Let us specify, furthermore, that the form of f( 7, t) in (1.4) is

increasing in magnitude together with t, i.e., the energy level

becomes higher and higher as t increases. The space trajectory of

the motion of (1.4), in this case, can be visualized as in

Figure 3. We can project the trajectory onto a plane similar to

the phase plane and it will be in a form as shown in the above

mentioned figure. It should be remembered that we have specified

the forcing function to be a monotonous increasing function of '.

Notice the form of the trajectory in the 1)-i plane.I It is very

similar to the form of an autonomous system with a negative damping

term; with the only difference that the "unstable focus" changes

with time. All the unstable singular points Vi at time t Ti

are determined by


S( I ; ( 015)

this is shown in the figure as 7 7 7 '"' By virtue of

(1.5), the following equation,

Sf7, ) = 0 (1.6)


is simply the locus of all singular points (both stable and unstable

singular points) in 77- plane. The intersection point of the curve


We prefer not to use the term "phase plane" in this case.










defined by equation (1.6) and the response curve, i.e., the solution

of equation (1.4) is simply the inflection point of the response

curve. At the first inflection point of the curve, the change of

the slope is zero and the slope of the curve is a maximum; there-

fore, it is the upper measure of the "violent increment of the

deformation." There is another significant singular point from

(1.5), i.e., when -i = o = 0, the initial time. The first unstable

singular point in the plane when t = to certainly is the lower

measure of the critical deformation. In the case that t appears

only in the forcing function, the lower measure in the above defined

sense is simply the critical amplitude of the free vibration of the

system (1.4). Therefore, the middle point of these two bounds is a

reasonable measure of the dynamic buckling.

As a conclusion, we summarize the criterion proposed in the

last section in the graphs shown in Figure 4. In Figure 4.P repre-

sents characteristic load and S is designated to be the characteristic

deformation. D represents the typical dynamic curves, while S repre-

sents typical static curves; AL is the critical amplitude of the

free oscillation of the dynamic system. The critical condition of

each system is determined by Acr or bounded by AL and Au

according to the above discussions.












CHAPTER II


BUCKLING OF A CLAMPED SHALLOW SPHERICAL SHELL
UNDER A PURE IMPULSE


1. A Qualitative Discussion of the Loss of
Stability of the Structure


It has been shown in Appendix II that the dynamic problem

of a clamped shallow spherical shell under the action of a

uniformly distributed load q(t) can be reduced to a single non-

linear second order ordinary differential equation if a first

order approximation of deflections in the following form is

employed; thus
)2

() (t) = [ (~) .J (2.0)



This second order ordinary nonlinear differential equation,1 when

V 0.3, has the following form:2

d24 + E i ( 12."1 2 + 2.76 1 o.62 C2



+ 8-43 3 3
Tor4 1 6 ph (2.1)


1The derivation of this equation is attached in Appendix II.

2Refer to Figure 1 for the definitions of each quantity
in this equation.









If the shell is sufficiently shallow so that the middle surface of
the shell can be approximated by a paraboloid, thus
z Zo (r r)2
and the curvature of the middle surface has to satisfy the
condition:


dC Z
d. T2


These last two equations give us the useful relation:


Zo /22Ro


By virtue of equation
(2.1) can be written as:

S+ [ (.76 + 3.0o2

+ 2.1o;


(2.2), a dimensionless form of


S02) 7 6.31 2

71 3]- 3] ,
16


(2.3)


where the nondimensional quantities:


= E/p t/R


0e= /Zo


S(t) (2.4)


have been used, and p is the mass density of the shell material;
the dot (') now represents d/dr.
In that case the external pressure is an ideal rectangular
impulse as shown in Figure 6. For sufficiently small 6T, as


R,


(2.2)


^p 2









discussed in reference (28), this problem can be formulated in the
following fashion: the oscillation of the system is essentially

a free oscillation started at the time t = AT (i.e., AT-# 0 ), with

certain initial conditions derived from the external disturbance.

It is conventional to take the displacement at the new initial

time, i.e., t = A T, still being the same as the one at t = 0.

For an initially undisturbed system, the new initial deflection

will remain zero because of the smallness of AT. The velocity at

t = AT, however, will not be the same as that at t = 0, because

of the sudden and instantaneous external disturbance. If the

system is originally undisturbed, i.e., velocity is zero at

t 0, then the velocity at the new initial time can be easily

found by use of "the principle of linear impulse and linear momen-

tum," i.e.,


I T dt M[= 4(T)- (0o)]



It is noted that equation (2.3) is equivalent to a nonlinear

spring-mass system with a unit mass (m m 1) and a constant force

in its nondimensional form, e 9 Therefore, the above equa-

tion reads:


i ^-6 = [ (At)- OJ

where A7; is the dimensionless form of the time duration 4T, by

the definition in (2.4).










From the above discussion, we have our new initial condi-

tions, thus,

at t =at

'7(,t) = 0 ;

33 *
(41 (41 (2.5)

where I* = q.(At) is the dimensionless impulse.

Because of the smallness of the time duration of the impulse

( 4t C 0 ), it is conventional to solve this problem by taking

(2.5) to be the initial conditions at tr 0, thus



77(0) = 0 o) 1 e
(2.6)

and a free oscillatory system with a dynamic equation obtained

from (2.3) by dropping the term on the right-hand side, which is

involved with the external load q. Thus


S+ 2.7+ 2 5.0o2 +)- 3.31 r' + 2.1o7 0. (2.7)

The oscillatory motion of an autonomous system characterized by

equation (27) and initial conditions (2.5) can be discussed quali-

tatively by a phase plane method.

We shall follow reference (23) and write (2.7) in the

following form:




S- [(2.76 + 3.o02 8o) .31 r+ 2.108 i 3 (2.8)









Here it is apparent that is the dimensionless velocity of
the motion. Equation (2.8) can be integrated once and becomes


= [l.o3s I -k. 4 (+ 2.76 + 3.o2 52) 17

C (2.9)
where C is an integration constant. Equation (2.9) is the equa-

tion of the trajectories in the phase plane. It describes all

possible motions of a system characterized by (2.8). The stability

of the motion will be analyzed in the following,

The singular points (or critical points) of (2.8) are found

from the equation (2.8) by putting 77 and 4 equal to zero simul-

taneously, i.e.

= = 0 and


?r2.o8 i2f S.31 7 + (2.76 +3.o2 )] = (2.10)

It is clear that the critical points are located on the 17-axis.

One of them is the origin of the phase plane and the other two

points are defined by the equation:


= 4.216 [ 3 (2.11)

The stability nature of these singular points can be determined

from the following characteristic equation, (9), i.e.,


Refer to p. 317, reference (9).










-x
Bet (X)
-[(2.7+ 3.o2 92)- lo62')
+ 6.324 7] -

0 (2.12)

It is given by the stability theorem1 that corresponding to a

pair of real eigen values of (2.12), an unstable singular point

will satisfy the following condition:


[ .324 2 o062z + (2.76 +302 2)] < 0. .13)

It is also apparent from equation (2.12) that there are only two

types of singular points for this system, namely, the stable

critical points of "center" type and the unstable points of the

type of "saddle points."

Let us return to equation (2.11). It is clear that there

exist real positive nontrival values of ro (singular points)

provided,

e1 o0,19303 or


S 0.-44 (2.14)

Physical significance of this condition is that if a shell is

sufficiently shallow such that its geometrical parameter 6 is

larger than a certain limit, i.e.



1This is the characteristic equation of the linear approx-
imation. The discussion of using this approximation is referred to in
Appendix I.











e ; o.44, 1 (2.15)

There will be no "snap buckling" under the action of an impulse.

For this reason, we shall be interested only in those shells with

geometrical parameter 8 -: 044.. By virtue of (2.13), we know the

root:


".216[bl- oi 3o -

(2.16)

of (2.10) is the unstable saddle point. The trajectory passing

through this point is called a separatrix. The motion on the

separatrix is essentially unstable, and the motion described by a

trajectory inside of the separatrix, in general, has different

character than the one described by a trajectory outside of the

separatrix. Therefore, the problem of determining the critical

condition of the system reduces to one of find the motion whose

trajectory is the separatrix.

Because the separatrix passes through the singular point

defined in (2.16), then by using equation (2.9) we can find its

equation. For the purpose of emphasis, let us replace the

symbol (7 by ,7r i.e., from now on, 772) in equation (2.16)
0 cr 17o
is read

/o (1 7cr



This condition is equivalent to A4 3.87, where A is
defined in equation (4).









From (2.9), for the separatrix, we have


-[ I 038 1 +3 73, + (2.76 + 3.02 0) 2

-C ] 0 (2.17)

It must be remembered that from (2.10), 77,,also satisfies the
condition:1

(2276 + 3.025 92) = 1 7,, 2.108 7 .

Using this equality and (2.17), we can determine the constant
C = Cs, which will yield equation (2.9) as the equation of the
separatrix. Thus,


Cs = 1.77 1f.0o38S (2.18)

where, from (2.16),


7C = 1.2941 1.197t 1 -.19303 (2.19)
Let us return to (2.5), the initial conditions which
define the motion of the system under the action of an impulse.
From the first condition in (2.5), i.e.,


7(0o) = 0 .




It should be noted that rc, is subjected to the condition that
Mr: o i.e., Xir is nontrivial.










It is apparent from this condition that the motion always starts at

the point where its trajectory intersects the j, axis. In case

the motion is on the separatrix, i.e., at the critical condition,

from equation (2.9), (2.18),and with condition (2.5), we have the

following result:

(3/y it o = -) Cs ,
or

I; = FCS/33e
= (16/33) / 1.77 1. j.o-3 ,3
(2.20)

where 1), is defined in (2.19). These two equations will give the

critical impulse for any shallow spherical shell whose geometrical

parameter E is known and satisfies the condition in (2.14).

An example is given by taking e = 0.26, (A 5). We shall

see, particularly in this numerical example, that the result obtained

by using a phase plane method will be the same as obtained by using

the buckling criterion proposed in the last chapter. Furthermore,

the same result may be obtained if Budiansky-Roth's criterion and

techniques in (8) are employed. It should also be mentioned that

the result reported in reference (8) is numerically more accurate

than that given by equation (2.20) because of higher order approxi-

mations used by those authors. However, the problem solved by


1They used a five-degree-of-freedom system.










this simple but precise method will permit certain qualitative

conclusions which could not be obtained, or would cause much labor

in calculations if other methods are employed.

When e = 0-26 (2.5), and (2.6) read:



77 = 2.1075 i)(t- o.93o03)( 77 1-643)


7r(o) = 0


; 7j(0) = o. 362 _I1'


(2.21)


Three singular points on the 77 -axis are:


71 = 0 ,


77 = 0- 3503


77 = 1.694S3


a saddle point


a center.


Let 77cr = 0.83503; from (2.20), we found


1.34233


(2.22)


The equation of the separatrix is found as follows:


2= I- 38 74 + 3S4 r7 2.9645 )1 + 0o1830.


This equation and other phase plane trajectories have been plotted

and are shown in Figure 7.

In Figure 7, when I < 1I given in (2.22), trajectories

of the motion are closed curves around the center point at the


a center










origin, e.g., curve 1; when = the motion is unstable and

on the separatrix 2; when I > Itcr the motion is on a trajectory

such as 3. It is also easy to explain the occurrence of the

"buckling" from this figure. The points where trajectories intersect

the ) -axis correspond to the situation that the response curve

reaches its amplitude. Therefore, the '1 coordinate of these

points (e.g., OA) is the measure of the maximum inward central

deflections of the shell. It is clear that the maximum inward

central deflection increases with 1* in a continuous fashion when

I < Icr As soon as 1 is slightly larger than I~r the

maximum central deflection undergoes a severe change, from some

value less than 0.84 (e.g., OA) to some value greater than 2

(e.g., OC). Because of this severe change of deflection, snap-

buckling of the shell occurs.


2. A Study of the Dynamic Response


It is noted that the differential equation has the form

of (2.7), and with initial conditions (2.6) can be integrated. The

solution of such an equation, in general, is involved with Jacobian

Elliptical Functions. It is still impossible to give a nontrivial

expression for the solution of the equation, which is of the same

form as equation (2.7). However, when e is taken to be a definite

value, the solution of (2.7) can always be obtained. For an example,











in solving such type equations, we shall take the system defined

in equation (2.21); i.e., 9 is taken to be 0.26 in equation (2.7)

and (2.6). All numerical work involved will be presented in detail.

We feel that the result of this section will clarify certain impor-

tant points in both the last section and the following section on

the justification of the buckling criterion.

Let 6 = 0.26 in equation (2.9); we have the equation of

the trajectories for the system (2.21) in the following form:


A = /C 2.9645 7" + 3,S. 773 038 4 (2.23)
(2.23)

where C is an integration constant. By using the second initial

condition in (2.21), C has the following expression:

C = (0i362 ) I)2 (2.24)

Our problem is to find the response 77(t) corresponding to each

disturbance I*. It is still impossible to obtain the general

expression and only particular cases will be given.

We shall study the responses corresponding to two individual

disturbances: 1 i 1.2, and I = 1.5. We shall see, in contrast

to the small increment in the disturbance 1 the corresponding

responses will undergo characteristic changes.

When I= 1.2, the positive branch of (2.23) reads:





/-I-jo~3 (. 2.0049)(7-1. Io?)(17- o-634)(17 + 0.314 7)
(2.25)











This can be transformed into an elliptical integral of the Legendre's

standard form.1 Let us, first, formulate the quadratic equation:

2.86166 v2 47-766 if + 1.10293 = 0 (2.26)

Its coefficients are related to the zeros of the algebraic equa-

tion under the radical sign in (2.25) in definite ways which can be

found in almost any textbook treating on elliptical functions and

will not be given here. The two roots of (2.26) are:

P = 0-27585 = 1-3971a (2.27)
Now, we use the following transformation:



1+ Z
= o-27tSiF + 1.39718 Z
4 + 2 (2.28)

and
d = I '2f33/(I+z)2 dZ ,

and then equation (2.25) can be transformed into the following form:





1
Refer to reference (41).

e.g., reference (43).







33




1. 12133 o(Z

(1+2)2 dt


( i+Z)2 /l.038 ( Z7721 z2- 1 4.342)(.2) 27 Z o016983),


or

-- 2119, (Z- S.OM9)(2'- 0,11899)


or
o o9z == .(f2 -_899dzo
0.46039 dz


(2.29)

Referring to reference (41),I the function z can be written in the

form:
Z = 0-34495 Sn (4 Im) (2.30)

where
U = 1.30993 (Z-Z) ,


w = 0.) o169.
(2.31)

Therefore, as we substitute (2.30) into equation (2.28), the

solution of the problem can be formally expressed as:




1
The expressions are on p. 26, reference (41).












= o-27955 + 0-49196 SnLIn)
I + o.34495 5,(ulm) (
(2.32)

where U1 and WT have their definitions in (2.31). The value T

in (2.31) is determined by the initial condition: when r = 0,

7q 0. In this fashion, we have Zto satisfying the following

equation.

Sn (-1-30o995 To I o.o0469) 0.5723
(2.33)

As an approximation,
To = 0o469 (2.34)

Solution (2.32) has the following general properties:

Ao It is periodic because it involves the double periodic

function 5n(ulI'm7) The real period of Sn(ujIm) is

4K = 4(1.57658). The period, P of 11 then, is equal to 4.81419

according to equation (2.31).

B. The 77 values are bounded in the interval -0.31464 A

-7 0.56344, because the value of Sn (ulm) varies between

-1 and +1. Therefore, the maximum amplitude of the dynamic response

c17 corresponding to the disturbance I* 1.2 is max 0.56344.

The next example; when I* = 1.5, the positive branch of

(2.23) reads:


17 = = /0.64702 2.964k 2 + 3.4 77 -I-o3 ?7.






35



Different from the last case, the rational function in the radical

sign in the above equation has complex roots. As a counterpart of

equation (2.25), we can express the above equation in the following

form; thus


(2.35)


where


(3 = o.3go67,


I = 0-83M + o 28937 c


The counterpart

respectively


of equations (2.26), (2.27), and (2.28) are


0o01703 if + 3.13426 V -2.6321 = O,


p= -184.8179




-IM4. 79 +

1 "


% = o-3587 ,


o.83&'7 Z


ct = j ^ (44S 7/ .+z)2 dz

(2.37)

The transformation in (2.37) brings equation (2.35) into the simpli-

fied form as in the following:


6 = o085g6 028W37 Z.


(2.36)


J lo53 (7- )(7-f)(r-r)( ) ,












A/( Z2- 2 )(Z + M2
)49993 0-o0o14A




/1. o38( 149993) o.ogl0t91 /18. 7/4872


34,49 1. f6o9 ,


34,499-97923 .


(2.38)


Therefore,


have


"= I -.64249 Nc (ualm)


where


= 1.29092 (T-to) ,

= 0.94a49.


(2.40)


Substituting


77


(2.39) into (2.37), we have


-14.9.8g9 Cn(ulm)

C'n(ul m)


solution


126 7534 1

151- 64249


where the definition:

Nc = /Cn

has been used, and u, m are defined in (2.40).


dz


where


(2.39)


(2.41)











In a similar way, To in (2.40) can be determined by

requiring J(Z=o)= 0 ; thus


-- -0.70755 (2.42)


It is obvious that the solution in equations (2.41), (2.40), and

(2.42) is characteristically different from the solution represented

by equations (2.32), (3.31), and (2.34). Solution in (2.41) has the

following characteristics:

Ao It is periodic, with period P = 9.03,

B, '7 values are in the interval -3.8079 7 -S 2.06868.

The maximum amplitude of the dynamic response,

?max = 2.06868.

Notice the characteristic change in the form of the dynamic

response and the severe increment in the amplitude (from

max 0.563 to 2.068) as I* value changed from 1.2 to 1.5. We can
max
conclude that the critical load I* must be some value in between
cr

the two values. One gets a satisfactory justification by referring

back to equation (2.22), where the critical impulse was found to

be 1.34233.

Response curves corresponding to I* = 1.2 and I* = 1.5 are

presented in Figure 8.












3. A Justification of the Buckling Criterion


In the last section, we have seen that the dynamic response

for a system defined by (2.21) can be found by integrating the

differential equation directly, and the solution in terms of Jacobian

Elliptic Functions. The dynamic responses corresponding to other

external impulses than those given in the last section may also be

obtained in a similar manner, yet the procedure is laborious. If

merely the amplitudes of the response curves are desired, then for

a one-degree-of-freedom system, as equation (2.21), the difficulty

of integrating the differential equations can be removed by use of

the information obtained from the previously discussed topological

method.

We have mentioned that the intersection points of the

trajectories and the 77 -axis in the phase plane are the points

where the response curiies reach their amplitudes, because at those

points, the velocity E is equal to zero. This fact suggests that

we obtain the amplitude-impulse relation for the system (2.6) and

(2.7),or their special case (2.21), from the equation of trajectories,

i.e., equation (2.9). Let us restrict ourselves to dealing with the

special case defined in equation (2.21); i.e., 8 is taken to be the

value 0.26. Substituting (2.24) into equation (2.9) and using

E8 0.26, we have the following equation for the trajectories

of this system:











= i.o4 4 -.4 753 + 2.964S '7' 0-28756 1*2j

(2.43)
By virtue of the above discussion, the amplitudes of the response

curves, 'qm's, are found by setting the velocities ,'S equal to

zero. By doing so, from (2.43) the following relation is obtained:


2 2 2
o= -.276 [io4 -3.54,, 2-2964]5 % .
(2.44)

For the solution to be physically meaningful, the positive branch of

the last expression should be used; thus,


I-= /3.-632 f[ 3.35 lj + 1261]6. (2.45)

This relation is shown in Figure 9. Good agreement between the

result presented in Figure 9 and the result for the two cases worked

out in the last section indicates the correctness of this technique.

In Chapter II we have proposed that the shell will buckle

when the characteristic load-deformation curve reaches the first

unstable state of static equilibrium. There are three particular

static equilibrium states, i.e., the positions of rest of the system

(2.21) which can be found from the first equation of (2.21) by

setting the inertia force, i.e., '7 term, equal to zero. Thus, the

three positions of rest (where 7 7 fi = 0) are obtained:

(I) (2) o3)
7'= 7 o= 4 7 I.6g (2.46)










In 7max I* plane, these curves will be straight lines parallel

to the I* axis. According to the criterion, the first unstable

equilibrium position ( -7 0.84) defines the critical condition.

As shown in Figure 9, the I*cr thus found is identical with that

found previously by the phase plane method.

It is also of interest to see the comparison between the

present criterion and that proposed by Budiansky and Roth in (8).

According to Figure 1 and equation (2.0), we have:





o = 4 l- (r/r)2]


Therefore, certain definitions in reference (8) assume the following

expressions:



(Z.O)AVE ; fZordodr = 4
0 o
and


IWn) = Tl f vaordeodr = /2


By definition we have


:= /Z, = 7 (2.47)


According to (2.47), the measure of buckling used in (8),

A max= 1, corresponds to 77ax 1 in the notation of this paper.
malX m~aX










It is apparent from Figure 9 that the same critical impulse will

be obtained if the criterion proposed in reference (8) is employed.

Furthermore, it is also indicated in the same figure that the measure

used by Budiansky and Roth falls into the unstable branch of the

load-deformation curve and is close to the point of instability.

Therefore, the criterion proposed by these two authors is proved to

give satisfactory accuracy for this specific problem.

Critical impulses for other values of e based upon equa-

tions (2.19) and (2.20) have also been calculated. Results are

presented in Figure 10. It is shown in the figure, for

0.44> 8 > 0.32 (or 3.87 /. i4 4.53) that equation (2.20) agrees well

with the result given in () and appears almost the same as the

result of reference (15), when 8< 0.15 (or ; > 6.6). It is believed

that this analysis is parallel to the presentation of (15), yet with

a cut-off point at a larger E value, i.e., the present analysis

admitted a shallower shell to buckle under the applied impulse. This

tendency seems to be correct as compared with the result of using a

higher degree of approximation given in reference (8).


4. A Note on the Effect of Initial Geometrical Imperfections


It is rather interesting that we may conclude, on the basis

of Figure 7, that any axially symmetric geometrical imperfection will

give a deduction of the critical impulse for the shell. It is the

nature of the equation of the separatrix of having a relative max-

imum when 1 = 0, i.e., on the axis. Any initial imperfection









(deflection) of the shell is equivalent to set the motion starting

at C instead of at C. as shown in Figure 7. It is seen from

this figure, that Cg' has a smaller ordinate than C5. Therefore,

the critical impulse based upon the former will have a lower value.

For an example, let us assume that the initial deflection of the

shell is axially symmetrical and has the same form as the deflection

of the shell, i.e., can be described by equation (2.0); further-

more, it has a central deflection

S= O.OS Zo

or
O; 0.0 (2.48)

Based upon this value, the critical impulse will be 6 per cent less

than that directly given by equation (2.20).














CHAPTER III


BUCKLING OF A CLAMPED SHALLOW SPHERICAL SHELL UNDER A
SUDDENLY APPLIED CONSTANT PRESSURE OF INFINITE
TIME DURATION


i. A Qualitative Study of the Instability of the Structure


We have seen, from the last section, the solution of the

shell buckling under a pure impulse with infinitesimal time dura-

tion did not answer the buckling problem of the shell under the

impulsive pressures with sufficiently long time durations. In

this section we shall solve the problem, which is another

limiting case of the above-mentioned problem; the time duration

of the applied impulsive loading is infinite.

Let us assume, without loss of generality, that before

the time t = 0 the system is at rest, and at time t = 0 a

uniformly distributed constant pressure q is suddenly applied on

the surface of the shell. The history of such a load is shown in

Part B of Figure 6. The dynamic equation of this problem can be

obtained directly from (2.3) by taking q as a constant. Thus,


S= j [(.27 + 3.o2e) -.3,1 2

+ 2 log 77'1 > 0. (3.1)









We have mentioned that the system is initially at rest, i.e.,

7(0) = 0
(0o) = (3.2)

are the conditions at T 0.

Note that the autonomous system in (3.1ol) can be integrated

once and results in the form:

2 = A + (u. 125 e0 ) ( (2.76+ 3.o20 o2) 7

+ 3. 4? 73- J.o4 4 > t>o ,
(3.3)

where A is the integration constant. Equation (3.3) is obviously

the equation of the phase plane trajectories.

The singular points (critical points) of the system in
equation (3.1) are found from:


S= = 0

i.e., on the 77 axis where


T6 e L(2.76 + 3o.02 ) e) 5.3,1 7
+2.108'73 3 Q. (3.4)

Stability conditions in the neighborhood of these singular points
are determined by the following characteristic equation, thus


a2 + [6-82 71- 10-62 77 +(2.76+ 3.02 82)] = o,


(3.5)









Corresponding to the real eigen values yielded by (3.5), the saddle

points satisfy the following condition:


6.322 1)' 10.62 77, + (276 + 3o2 ) 0 (3.6)


where 7o, is the root of equation (3.4).

The problem of finding the critical load q which will make

the motion of the system described by (3.1) and (3.2) lose its

stability is equivalent to the problem of finding the q value which

will put the motion of the system on a trajectory passing through

the saddle point in the phase plane, i.e., on the separatrix. By

virtue of its initial conditions, the trajectory of the motion of

the system (3.1) and (3.2) passes through the phase plane origin

( r7 = > = 0). Therefore, the equation of this trajectory is

obtained from (3.3)" by taking A = 0, thus

12 = fi./2 e (2.76 + J.02 0e2)

+ 3 4 7-' I o ~173 ] 7. (3.7)

If, furthermore, this trajectory also passes through the saddle

point ( To 0), it is obvious that the following condition has

to be satisfied:

0o [f.'2, 12 (276 + 3-o2t e') o + 3.-, io-,. 7]3
0 ,
or equivalently


33 76 + 302 ) 3 + .oi,3
(3.8)









for 7o 0 O.

Attention is invited to the fact that this problem reduces

to finding common roots between equations (3.4) and (3.8).1 A

solution to a similar problem for a complete spherical shell is

given in Chapter V, where the technique is discussed in detail.

Equations (3.4) and (3.8) will have common roots provided

17 qn f 3oSu r3
-- 06 77 3 (3.9)
2.o62 6

Substituting (3.9) into (3.4), we have

3.162 7 .3- o8' 7 + (2.76 + 3.02t Q) = 0,

(3.10)


which has three roots. The two nontrivial roots are solved from

the following equation:


,, 1. +/ (J '.21792 38.26020 82)
7] I IM194 -t .99--------
39.99297


(3.11)


It should be noted that it is not true that both solutions in

(3.11) are the singular points of the system, because they are

not solved directly from (3.4). It merely says, at the present



Let us recall that 17 in (3.8) is one of the roots
of equation (3.4); it satisfies the condition (3.6).











time, that both 77, and 77, in (3.11) are the possible common roots

between (3.4) and (3.8) when the corresponding q value in (3.9)

is taken. In other words, if 77, is substituted into (3.9) to

yield a particular value of q and this value is used to replace

the parameter q in both (3.4) and (3.8), then these two equations

will possess a common root, 77 = r) (Note that '72 may not be

a root of either of these two equations.) Since equation (3.10)

gives the common root of (3.4) and (3.8) which specifies the

buckling of the shell, we may deduce one of the important con-

clusions, i.e.,


e 0.6307 (3.12)


has to be satisfied; otherwise, no buckling will occur, because

equation (3.10) has no real positive nontrivial roots, as indicated

in (3.11).

We shall see that only one of the two nontrivial roots of

(3.11) will satisfy condition (3.6), i.e., be a singularity in the

form of a saddle point. Discarding the trivial solution q. 0

from (3.10), we have the following condition:


( 276 + 3.o2 02) = 7.09' r7 3.162 72 (3.13)




pIt corresponds to A > 3.2123, where A is defined
in equation (4).











which is satisfied by both 7 and 17 in (3.11). Substituting

(3.13) into (3.6), after combining terms in the same order of 77,

the following condition is obtained:

3.162 772 4 77 < 0

As we mentioned, both ?), and y2 are real positive values,

because we have put the restriction (3.12) into our problem. If

this is the case, and it is noted that 7r is solved from

equation (3.10), condition (3.6) is reduced to the following form:


'r / 1.I19.4 (3.14)

It is apparent that only the smaller root in (3.11) will satisfy

the condition (3.14) and be a possibly unstable saddle point for

the dynamic system (3.1). More definitely, let us put the critical

deflection as follows:


S= i.i9 ( I.21792 3.-262 2)(3.15)
39 99297 (3.)

Therefore, without even going back to the phase plane, we can

write a general solution for the critical pressure by substituting

(3.15) into (3.9), i.e.,
2 3
= '.71 1r 10 cr
cr 2.0625 0 (3.16)














where '7cr has its definition in (3.15).

It may be apparent that certain techniques and their logical

foundations have not been made clear in the above discussions, and

they are very difficult, if not impossible, to be discussed on a

general basis. Therefore, we shall give an example using a

specific shell with 68 0.26, which has been used in the previous

problem.

Let us first find the solution, i.e., the critical deflec-

tion and load, and then go back to verify that these critical

quantities do put the motion of the system on its separatrix and

cause the shell to reach the threshold of instability.

From (3.15), when 8 = 0.26, we have


S= 754 (3.17)

and from (3.16), we obtain immediately,


S= 068538 (3.18)

Our question has been whether the value of q in (3.18) does cause

the instability of the system (3.1) when e = 0.26, in other words,

to make the motion of the system on its separatrix, or equivalently,

does the value of 7) in (3.17) define the saddle point in the phase











plane when q takes the value in (3.18)? To answer this question, let

us write down equation (3.4) and (3.8) by taking e = 0.26 and

q qr defined in (3.18); thus, equation (3.4) is in the form:


2.10o 773 .f( )12 + 2.96449 ,7 o.367!i3 = 0,


(3.19)


which has three real positive roots:


(I)
r7 = o.17o05


(2)
77 = O W54


(3)
17 = 178639. (3.20)


These are

(3.8) has


the singular points in the phase plane, while equation
2
the following form:


[ lo ~ -3 3.4 72 + 2.9W n 0-735o73 = 0, (3.21)


which has three roots:

= 2243 ,


and a double root at


S= 0-74 .


(3.22)


Comparing (3.20) and (3.22), it is clear that I7 : 0.55754, i.e.,

the critical value defined in (3.17) is the common root between





1We have multiplied the value (-1) through the original
equation (3.4).

2Similarly, a quantity (-I) has been multiplied through
the original equation.











(3.19) and (3.21). Furthermore, tested by condition (3.6), this

value ( 7? = 0.55754) defines a saddle point in the phase plane.

A further remark on the equations (3.21) or (3.8) will completely

answer the above question. Let us -replace the 71o in the first

equation of (3.8) by 7, and comparing the final equation with

equation (3.7), we find (3.8) is merely the condition of the inter-

section of a phase plane trajectory with the '7-axis ( p = 0).

Because of the preceding discussion, we conclude that the phase

plane trajectory for the motion of this particular shell ( e = 0.26)

will pass through the saddle point if the parameter q is defined in

(3.18); therefore, the value of q in (3.18), or in general in (3.16),

is the critical value for the external pressure. The phase plane

trajectory of the motion of the shell at the threshold of the

instability is shown in Figure 11. Based upon equation (3.16) and

(3.15), relations between the critical load and the geometrical

parameter e are found and given in Figure 12. Curves indicating

the variation of critical deflections with e values are given

in Figure 13. Comparison between this problem and its counterpart

in the static case has been made and is also shown in the

figures mentioned.











2. Another Justification of the Buckling Criterion


In Section 3, Chapter II, we have discussed a method of

obtaining the load-deformation curve. For a different problem

in this chapter, the same method may still be used to obtain the

relation between the characteristic load and the characteristic

deformation. The critical condition of this system, according

to the discussion in Chapter I, can be determined, and the result

thus found will be compared with that obtained in the last

section as a justification of the proposed criterion.

If the characteristic deformation is chosen as the ampli-

tude of the central deflection following Section 3, Chapter II,

the amplitude-pressure relation for system (3.1) can be obtained

from the equation of trajectories, i.e., equation (3.7). By

equating to zero, we obtain the following relation between the

nontrivial amplitude of the response curves and the corresponding

dynamic loadings.


S (2.76 + 3.o2 e ) 77 +. 4 2 + t-.o4 73
d ~ 4..f2t 0 (3.23)


For the purpose of further discussions, let us take a

specific shell with 0 = 0.26. From (3.23), we obtain:


= 296449 7) 3.t4 + 1.o 7
d .o 072& (3.24)











The static load-deflection relation can be easily found from

equation (3.1) by taking the inertia term, i.e., the 17 term equal

to zero. For e = 0.26, we have


2.964149 77 .31 7 2 + 2.108' r"3
S o.362S (3.25)


Equations (3.24) and (3.25) are plotted in Figure 14. The inter-

section point of these two curves is at Tml = 0.55754, where the

dynamic curve has a relative maximum. It is obvious that the inter-

section point falls into the unstable branch of the static curve.

According to Case 1 of the buckling criterion in Figure 4, the

corresponding pressure at the point of intersection is the critical

dynamic load, thus qcr = 0.68539. These results are identical

with those given by (3.15) and (3.16) based upon the stability

theorem.

The identical results obtained via two different approaches

have established the following facts:

A. In certain cases the dynamic criterion proposed in

Chapter I is identical with the stability theorem.

B. Without given explanations, we have taken the specific

expression in equation (3.9), which led to finding the solution

of the problem. This turns out to be correct, since the solu-

tion in (3.23) is entirely independent of (3.9), yet the same

result was yielded.

A comparison between the present theorem and reference (30)

is presented in both Figure 12 and Figure 14. It is interesting










that the critical dynamic load given by the present theorem falls

in between the two values for the critical dynamic load obtained

by use of different methods given in reference (30). The critical

deflections given by (30) are seen to be larger than those given

by equation (3.15). This is partially due to the fact that a

different deformation form was chosen in reference (30). However,

a similar relative relation between static and dynamic curves,

as indicated in Figure 14 was also seen in reference (30). From

the resulting curves presented in (30), by utilizing the buckling

criterion in Chapter I, the critical dynamic pressure can be

obtained with negligible errors as compared with the solution.

From this point of view, the result of the analysis in (30) may

be used as another justification of the proposed criterion.


3. A Discussion of the Results


In both Chapters I and II, a qualitative method has been

used to discuss the motion of the system and to determine the

critical condition of the dynamically loaded structure. It is

also evident in these two chapters that good agreement exists

between the result obtained in this way and the result by the use

of the buckling criterion outlined in the first chapter. This gives,

at least qualitatively, a justification of the proposed criterion.

From the point of view of the applications, the proposed criterion

is subjected to no restrictions of any sort, while the topological

discussion would meet certain difficulties when the system is

nonautonomous or of higher degrees of freedom. However, it is












rather convincing that the phase plane method is suitable for use

in discussing the dynamic buckling problems. The accuracy of this

method may be restricted by the fact of using a single degree of

freedom system. The general behavior of the motion, however, is

much clearer as being plotted in trajectories. Furthermore, through

the examples given in the last two chapters, one can see the direct

connection between the energy method and the method of response

curves, which were usually employed in solving the dynamic buckling

problems. This fact was clearly shown in Figures 7, 8, and 9. The

motion on the separatrix, which passes a saddle point corresponding

to a certain extreme of the energy level is the threshold of the

substantial change in the amplitude of the dynamic response.

In the application of the proposed criterion to dynamic

buckling problems considering higher axisymmetrical modes or

unsymmetrical forms of deformations, a suitable characteristic

deformation has to be chosen. One of the examples is to take the

mean deflection of the shall as the authors of reference (8) did.

A more general problem is that of rectangular loadings

characterized by the application of pressure q at time t = 0,

which is held constant for a time duration A t and then suddenly

removed. This would require the simultaneous solution of two

nonlinear differential equations of the following form:











Z. [(2i- 76l +6-50.o2et9 -3 173 4 + 2.b07f ]3,

0 < Tr < 9

) o = 0 (3.26)



S- [(2-76 + 3.o2 ez)-)7 031 1' + 2.o07t 17 7,

tr< < 00 (3.27)


where Z is the dimensionless form of the quantity 4t, according

to the definition in (2.4). The stability problem of this system

may not be solved without having a general solution of (3.26).

It should be noted that it is not possible to give a general

expression for the response of the load q in (3.26) in a nontrivial

form. This point has also been mentioned in Section 2, Chapter 2.

It is for this reason that only discussion of obtaining the

results will be given in the following. It is also understood that

the shell buckling occurs after the load is removed, i.e., the time

duration of the applied loading is sufficiently small. Therefore,

the loss of stability of the system is largely due to equation

(3.27). The present problem, by virtue of the above discussion,

has the same characteristics as the problem that has been con-

sidered in Chapter II; in fact, the latter is merely a limiting

case of the former.

Much as we have done in the previous sections, this

problem also can be phrased in the language of the topological










method. After doing this, the condition of instability can be

formulated in a straightforward manner. The motion of the shell

under the action of the said dynamic loading is described by both

equation (3.26) and (3.27). Specified clearly in these two equa-

tions, the motion will be on the trajectories of (3.26) until

=L t Immediately after 7 = t the motion of the shell is

on the trajectories characterized by equation (3.27). It is

obvious that the displacement 17 at t = Z is the common solution

of both (3.26) and (3.27). The critical condition of the struc-

ture will correspond to the following situation in the phase plane:

the response of q and its time derivative in equation (3.26) at

time Z = Z i.e., (r) and (iJ), which gives the initial

conditions for (3.27), will put the motion of the structure on the

separatrix of system (3.27). We shall discuss this matter as

follows: when i L the equation of the trajectoryI is

S= (42 eg } (2.76 +3.02& e2) 72

+ J3.1 1l 1o-I ?74
(3.28)


when Z the equation of the trajectory of the motion2 reads:


S= -[(2.76 + 3025 01) 7*- 7.4 3

+ io4 C 1
(3.29)




Refer to equation (3.7).

2Refer to equation (2.9).











The continuation property of the system at L = t requires:

() = Z) 7 (i ) = 7 .
Therefore, we have

C = (4..2.e )Y7() (3.30)


by comparing equation (3.28) and (3.29). The critical condition

of the system under load q now depends entirely upon the value of C.

Reference is made to equation (2.18), which defines the equation

of the separatrix; the condition of instability of the system

characterized by equations (3.26) and (3.27), or alternatively,

equations (3.28) and (3.29) then turns out to be


S(r = C/4 o e (3.31)

where Cs has been defined in (2,18) and (2.19). As an example,

when 6 = 0.26, according to the previous analysis in Chapter II,

Cs = 0.5183. Therefore,


--f) = 0.L48326 (3.32)

It should be noted that in equation (3.32), 7((i7) is also a func-

tion of q. This is obvious as shown in equation (3.26). Since

no analytical form of the solution of (3.26) can be given, further

discussions would require a great number of calculations. A

procedure for determining the critical pressure, qcr, is suggested

as follows:











A. A time duration i was preassigned, based on the

external impulsive loading.

B. Assign also a series of numerical values for the

loading q in (3.26); these values are arranged in an ascending

order of magnitude and with sufficiently small increment. By

taking e as a certain value, e.g., 0.26, corresponding to each q,

every equation in the form of (3.26) can be integrated either

analytically (in terms of Jacobian Elliptic Functions), or

numerically. Therefore, the response of each q at time Z = t ,

i.e., '? (r) can be found.

Co The critical pressure qcr is the one which satisfies

the condition (3.31), or when e = 0.26, the condition (3.32) is

satisfied. It is tedious, yet straightforward.












CHAPTER IV


BUCKLING OF
UNI FORMLY


A CLAMPED SHALLOW SPHERICAL SHELL UNDER A
DISTRIBUTED PRESSURE LINEARLY INCREASED
WITH TIME1


1. The Solution of the Problem


Equation (2.1) in Chapter II can be written in the

following form:


+ [ 2.76

o. 62 Ro
r2


+ o
YO


42 + 8.43 2
-f- ^1
y;h


3]


33
16


(4.1)


If the following definitions are employed,2


z7 = /Z2Ro ,
-ro 2/
Zo 2Ro


e = /Z.


v =


(4.2)


IRefer to Figure 1 for the geometry of the shell.

2Refer to equations (2.2) and (2.4).


(PR
E










Then (4.1) can be written as


RO)2 ) 17
* t


+ [(276+ 302 02e) 1 -31 '2


+3
+ 2.109 17]_


(4.3)

If the load q is a function which is linearly increasing with
time t, i.e., the form as shown in Part C of Figure 6, then it
can be represented by the expression:

--- Q ;t (4.4)


where Q is the pressure increasing rate and has the dimension

psi. Following reference (1), we shall use a "transformed" time
variable *,


where po is the nondimensional
given by classic linear theory


'(/o* (4.5)


critical load for a complete shell
and has the following form:


2
10 V2)

for V = 0.3, po = 1.21; Q is defined in the following:


Q = QR/Eh2


16 @* (E '










which has a dimension: i/sec. By utilizing (4.4) and (4.5),

equation (4.3) reads:

d2 } 21 r2 2
(dt)2 + (276 + .02 31


-2..o 3] (4.6)


where
V
Vo & (4,7)
a nondimensional quantity.


It is clear that the character of the solution of (4.6) depends

entirely upon the two parameters T and 9 We also like to

mention here that the transformed time variable Z* is the

"dynamic overload factor"; i.e., the ratio of the critical dynamic

load to the corresponding critical static load of a complete

sphere, which has been defined in equation (4.5). Therefore, the

response curve obtained by integration (4.6) is actually the load-

deflection relation of this problem. Let us assume the initial

conditions as follows:


( = o) = 0




(4.8)


which imply an initially undistrubed shell.










The nonautonomous system (4.6) with initial conditions

(4.8) is best solved by a numerical method. Different values of

the rates of dynamic loading and geometrical shape of shells have

been selected to substitute the parameters 4 and 9 in (4.6), and

response curves were obtained by integrating the equations

numerically on the University's IBM-7090 computer.1 Dynamic

buckling loads were determined by use of the criterion proposed

in Chapter Io The static load response curves were found by

dropping the inertia term, i.e., d2 /dZ*2 from (4.6), and the

typical form of those curves was shown in Figure 15.

In Figure 15, the form and the nature of the response

curves are very similar to their counterparts in reference (1).

For a rapidly applied load with larger Q(e.g., V = 0.3), deflec-

tion increases slowly at the beginning and has a vigorous change

at the time of buckling. For a certain shell, i.e., 9 E,

etc., are fixed, the faster the rate of increasing the dynamic

load, the higher the dynamic overload pressure will be. It is

also seen in the same figure, for a very slow rate of load

(eog., V = 100), the buckling of the shell approaches the static

case as it should be and the "creep phenomenon" strongly indi-

cates that loss of stability is of the "classic type." Followed

by several cycles of oscillation, the dynamic curves for V/= 100

converge to the static curve.




1A "Runge-Kutta method" was employed. The technique of
this method is found in (22).











Another set of curves was presented in Figure 16, based

upon various geometrical parameters. As a different feature from

the static case, it is found that the critical load decreases

monotonically with the increasing 6 values (or decreasing in

shallowness). As shown in Figure 18, no relative minimum corre-

sponding to a certain 8 value seems to exist as it usually does

in the static case. (Refer to Figure 5.) Also as indicated in

Figure 16, the curves move toward the left as the 8 value

increases, corresponding to a decrease in critical pressure.

However, for a very shallow shell, i.e., a sufficiently large

value of 8 e.g., 9 = 10 ( AI 0.806), as indicated in

Figure 16, the curve does not follow the above argument and falls

to the right of the curves with Q values smaller than 10.

Because this curve remains at very small deflections at a very

large pressure, it is clear that the failure of this structure

will not be by buckling. As is well known in hydrostatic loading

problems, buckling will not occur for a very shallow shell which

has a geometry close to a circular plate. This is also observed

in this dynamic loading problem. The limiting value of $ i.e.,

the largest value of 0 for buckling to occur, has not been found.

It is the feeling of the author that the limiting value of e

depends also upon the rate of the loading, i.e., the parameter 4f.

Therefore, the general answer may not be found without considerable

costly computations. A further discussion on this matter will be

given later.










Figure 19 shows the relation between the critical dynamic

overload factor and the critical central deflection of the shell,

i.e., the deflection at the time of the buckling. In this figure

we observe the rate of change of the upper 7cr is decreasing as

z *cr increases. This phenomenon can be explained as due to the

development of membrane stresses which usually play an important

role in the large deflection theory of plates and shells. Further-

more, these curves seem to approach asymptotically to different

limits. These limits depend solely upon the geometry of the

shell; the shallower the shell is the higher the limit of the

upper Tcr values will be. However, there exists a particular

shape of the shallow shell, which corresponds to e v 0.35. For

any other shape of the shell shallower than this value (i.e.,

e > 0.35), the critical deflection decreases. This is to say,

there exists a maximum for the upper 7cr 0e curves as shown

in Figure 20. We do not know whether this value ( e 0.35) is

the common maximum point for all values of V ; it would require

much computation to answer this question. We are rather interested

in the significance of the existence of such a point. Intuitively,

it is reasonable to believe that a shallow shell would permit more

severe deflection, as compared to the height of its raise, than

a nonshallow one would, This is noted as being true for all cases

where 8 < 0.35 in Figure 20. However, a contradiction arises for

e > 0.35 (or, X ( 4.31). Two possible hypotheses may be

provided: first, for all 9 > 0.35 there is no dynamic buckling










occurring because of the shallowness of the shell geometry; second,

for 9 : 0.35 the deflection of the shell has the highest sensi-

tivity, or the weakest shape with respect to the dynamic load.

Therefore, one may think'that this value of 8 may have something

to do with the size of the buckling region of a complete sphere.

Both Figures 19 and 20 indicate that if we want to select

a certain value of characteristic deflection as the measure of the

buckling of the shell, then it may be necessary to take a

different value of this quantity for different geometrical shapes

of the shell as well as for various dynamic loads.

Comparing Figures 17 and 21, we are able to conclude that

the effect of the load-rate parameter r on the critical dynamic

load is much more significant than that due to the geometrical

parameter 4 We mentioned that the critical dynamic overload

factor Zcr* is a function of two parameters (variables) 'V and

Q i.e., it is a surface in the ( /, e Z *) space. Based

upon the curves in Figures 17 to 21, we have the approximate

equations for the surface: (For, ) = 0.3)






= i. 'I o7o e- -].

jT r IL ,,


(4.9)











We found that in the range of 0.5 K 4 3.5 and 0.15 4 9 0.35,

results obtained from equation (4.9) agree well with that solved

from equation (4.6) and initial condition (4.8). The accuracy of

(4.9) can be proved only by experiment. We believe, however, that

a design formula for different values of Poisson's ratio can also

be established in the following form



Zr = (a +* r )(c ) (4.10)


with a, b, c, g, and k determined by experimental tests.

The projection of the trajectories on the 7 1Y plane

are shown in Figures 23 and 24. According to the criterion in

Chapter I, the critical state is bounded by the two points

indicated by L and U, i.e., the lower and the upper bounds. It

is seen from these trajectories that the deformation 7 increases

very slightly after passing the point U and oscillates about

different equilibrium points on the '7-axis with an increase in

amplitude.


2. A Conclusive Discussion of the Problem


The nature of the response of shallow spherical shells

to a high-speed dynamic load with linearly increasing pressure

intensity were found and represented in Figures 15 to 24. We

found the functional relation between the critical load and the

geometry of the shall has a characteristic difference from the











static case and this is shown by comparing the curves in Figure 17

with their counterpart in Figure 5.

Differing from the result obtained by Agamirov and

Vol'mir, we found the critical dynamic load (or, D.O.L.F.1)

depends upon the ratio of the speed of elastic waves in the shell

material and the product of the shell radius with the increasing

rate of the intensity of the external pressure, instead of

solely upon the increasing rate of the load as presented in ().

A functional relation between the critical dynamic overload pres-

sure and the two parameters V"f and e was formulated in equa-

tion (4.9); it gives the critical dynamic load from the given

geometric shape, the material properties, and the increasing

rate of the dynamic loading.

Another suggestion was also offered by formula (4.9).

Let us return to the definition of f in (4.7), i.e.,





which is a dimensionless quantity. The requirement of similitude

is very conveniently furnished by the quantity 41 For an

example, if we choose the same geometric parameter e for the

model and the prototype, then one can determine the nature of


IDynamic overload factor.












dynamic buckling of the prototype under a very high rate of

dynamic loading, i.e., a very large Q, from the test of a model

with lower Young's Modulus and a relatively lower rate of the

dynamic load, provided they have the same value of 4V.

Discussion of the buckling criterion was also made. It

is seen from the results that both the increasing rate of the

dynamic load and the geometrical shape of the shell have influ-

ences on the critical deflection of the structure. Therefore,

the criterion proposed in Chapter 1, which permitted "the measure

of the critical state," to change along with different dynamic

loadings and geometric shapes of the shall, has definite advantages.















CHAPTER V


BUCKLING OF A COMPLETE SPHERICAL SHELL UNDER SUDDENLY
APPLIED UNIFORMLY DISTRIBUTED DYNAMIC LOADINGS


Introduction


The nonlinear problem of a complete spherical shell has

been discussed by numerous people since the article by von

Karman and Tsien first appeared in 1941 (17). The progress toward

a substantiated explanation of the discrepancy between the classic

linear theory of Zolley and experiments has been rather slow.

This is partially due to the lack of reliable experimental data

on the buckling load for a complete spherical shell. For many

years Tsien's (31) energy criterion and the "lower buckling

load" (32) have been used to determine the load-carrying capacity

of a thin elastic spherical shell. In the year 1962, two inde-

pendent experimental tests contributing to the problem were

reported by Thompson (33) and Krenzke (19). Both of these two

experiments show that the critical load of a complete spherical

shell can be much higher than the lower buckling load offered by

Tsien; it ranges from 45 per cent to 70 per cent of the critical

pressure predicted by classic linear theory. A fairly precise

theoretical analysis based upon a large-deflection strain energy

theory was also given by Thompson in reference (33). Thompson











ended his research with the conclusions:

A. The initial buckling (pre-buckling) was seen to be

classical in nature, i.e., the load deflection process is con-

tinuous.

B. The buckling load was about 75 per cent of Zolley's

"upper buckling load."

C. The stable post-buckling states are observed to be

rotationally symmetrical.

Both Thompson and Krenzke's tests suggested further

theoretical investigations. It seems that more precise work on

the numerical investigation of the governing differential equa-

tions as has been done for shallow shells is very much to be

desired.1

Among all the works on the stability problem of complete

spherical shells, Vol'mir's first approximation (35) remains

unknown to most of the western scholars, mainly because it was not

a successful one. He attempted to investigate the post-buckling

behavior of a complete spherical shell by using a very simple form

of deformation, i.e., the form in equation (2.0) and integrated

the equation approximately by a Galerkin method. His assumption

on the buckling process was the same as the one assumed by von


E.g., (16), (), (2), (38), (39), and (4O).
















Karman and Tsien, i.e., the sphere is contracted to a smaller

sphere and forms a single dimple after buckling. By virtue of

this buckling mechanism, Vol'mir allowed a uniform membrane

stress: 6 = "R/2h, to be distributed in the buckling region before

thL loss of stability and considered the buckling region as a

shallow spherical cap. In other words, the differences between

Vol'mir's method and Karman-Tsien's is that Vol'mir used a

Galerkin's method based upon a variation of the governing

differential equations and also considered the variation only of

a single parameter, which is the central deflection, while Karman-

Tsien based their method upon the variation of the total energy

with two parameters. In a limiting case of small deflection

theory, Vol'mir found a buckling load twice as large in magnitude

as predicted by the classic linear theory and failed to obtain

a lower buckling load.

In Vol'mir's approximation method, we find certain ques-

tions which may have led to his failure in obtaining an approx-

imate solution. First, he considered that buckling occurs at the

transition of the membrane state and the bending state of the

shell; therefore, his effort was concentrated on finding a post-

buckling load. Second, the assumption of a uniform membrane
















stress in the buckling region before the loss of stability

requires an abrupt change of stress state, which seems

impossible under a continuous loading process. If the

structure is perfect in geometrical shape, this change of

stress distribution would require different equations to

describe the equilibrium of the shell and result in

mathematical difficulties of integrating the differential

equations. It seems to us that Vol'mir's method may

be restudied using a different contemplation of the buckling

mechanism of a complete spherical shell.


1. On a New Mechanism of the Buckling of a Complete Spherical
Ah.ll


We shall assume that the buckling of the shell follows

a possible mechanism which permits the transition from a membrane

state of stress to a bending stress state occurring in a con-

tinuous fashion and the transition occurs before the loss of

stability of the shell. This is described in the following

paragraphs.

A. Whcn the external pressure q is much less than the

critical value, the shell contracts to a slightly smaller sphere;















as shown in Figure 25, Part A, the original shell contracts to

a sphere with radius equal to Ro A R. As q increases, AR

increases and a significant change in curvature occurs because

of the change in radius.

B. Let us take another assumption that the structure has

a resistant nature against the higher pressure and has a tendency

to resume its original curvature. Based on experimental evidence

that spheres form a single dent after buckling, we think that the

resumption of the curvature starts at a small region, or we might

say at a point. The effect of the resumption of the curvature

from a larger one to a smaller one has introduced a pure moment,


which will be in the same direction as that caused by the external

pressure. In other words, we consider that the initiation of

bending stress in the shell is due to the imperfect nature of the

structure; however, the external pressure will certainly help to

increase the magnitude of the bending stress and build up the

inward deflections.


1Refer to Figure 25, Part C.
















C. The existence of such a single point for the first

resumption of the original curvature may be explained as due to

the "imperfections." Let us assume a spherical shell with perfect

geometrical shape all around except a very small hole at point 0.


(The advantage of the assumption of a small hole is that we

do not have to make any other assumptions on the form of the

imperfections.) When the original shell contracts to a smaller

one (refer to Figure 25) so that A moves to A', and 0 to 0',

the small hole is contracted to an infinitesimal one. Under

such a condition, the membrane stress at 0' is certainly

zero,

Do If we allow the existence of such an infinitesimal

hole at the point 0 in Figure 25, then the bending state is

inherent in the problem itself. As shown in Figure 25, Part C,

in the immediate vicinity of 0', the situation is very similar

to a clamped circular plate with a central hole. The idea of

"boundary layer" may be best fitted into this particular

circular region; outside of this region, a pure membrane state

remains. When the external pressure q increases, this circular









region dilates (or the thickness of the boundary layer increases)

and forms the buckling region after the loss of stability of

the shell.

By virtue of the above described buckling mechanism,

we arrive at the conclusion of the existence of a boundary

layer at the vicinity of a point O'o In this region, both

membrane and bending stresses exist at the time of the stabil-

ity of the shell. We are interested in the distribution of the

membrane stress in the boundary layer region during the load-

ing process. As we have mentioned, the stress at 0' is zero

and outside of this region the shell maintains a momentless

state with a membrane stress 0. = oR~o/2h. By referring

to the stress distribution in a bent clamped circular plate,1

a reasonable assumption in the boundary layer region will be

a parabolic variation, i.e.,




A ( /R ) (5.1)






iRefer to (34), pp. 54-63,

2Refer to Figure 25, Part B.











We shall analyze a nonlinear problem of the loss of

stability of a complete sphere by taking the buckling region as

a clamped shallow spherical segment with a nonuniform membrane

stress in the form of (5.1) distributed in the middle surface

before its loss of stability.


2. Buckling of the Shell under a Static Load


We shall take the buckling region of a complete spherical

shell as a shallow spherical segment clamped along a circular

boundary, as shown in Figure 1, Part B. From the discussion in

Appendix II for shallow spherical shells, we have the governing

differential equations for such a shell under a uniformly

distributed static load q (posi.) in the following form:i

The equation of equilibrium,


Dd'72 ) = h I+ 1 +
dr dr o r or 2 (
(5.2)


and the compatibility equation,


L (v) JW )2 7 01w
d-r dr Ro (r
(5.3)

where

V2 d2 +
drz r dr


IRefer to equations (Ao29) and (A.30) in Appendix II.










Since we have restricted ourselves in the problem of axi-

symmetrical deformations, then equation (2.0) can be used again

as a first approximation of the deflections in the buckling

region; thus, we have

/= o 2 r )] (5.4)


Substituting (5.4) into (5.3) and integrating, we have


3p Y r3 07

E 0 fj 2(2-V(' rr -3( 3
6R, (/-;P) '0r, _r ij
which is the condition that the strains or stresses in the middle

surface due to large deflections have to satisfy. However, it

should be remembered that there is a membrane stress already in

the middle surface due to the contraction effect of the rest of

the spherical shell outside of the boundary layer region. By

using the relation between the stress function 4p and radial

stress U. and after including equation (5.1), the above equa-

tion of the compatibility condition of deformations in the middle

surface of the shell reads

E;f3+ 4r)7)- ]
-6 ro I(fL -

E~.Y F 2(2-Y) R.Y ,3 V 3
6R, t ) (Y r95 2 5)
(5.5)











Substituting (5.5) into (5.2), the equation of equilibrium yields

the following form


32 Sr E Y- r )3
4 0 -)





^^ J 2h roJL R r2^J


2D (5,6)


We shall use Galerkin's method. This method allows

equation (5.4) and (5.5) to be the approximate solutions of

equations (52) and (5.3), provided


JJ G -(y)]2 hd = o0 (7
0A

is satisfied In (5.7) "A" represents the area of the circular
1
region with radius ro. It should be noted that r is not a
o o
constant; it is the thickness of the boundary layer, which

varies with the external pressure.

After performing the above integration, we find that the

central deflection 4 has to satisfy the following equation,

which describes the equilibrium conditions.


IRefer to Figure 25









S"EH + __ t+ 33 2-
rPRo2: l24 >-

zE rJ-275 ) 6279
Ro oL 49'(0-V) 2704


4 -E -r 33 1 33
7 [4 (,(I-) 13 ^


'U '


- 5r, 8)
(5.8)


For V = 0.3, equation (5.8) reads


I 12.1 42


+ 2.76


lo.62
Ro ,r


.43
T4


43)


3 Ro
2 r.2


(5.9)


As in the case of shallow shells, we shall employ the approx-
imations and dimensionless quantities described in equations
(2.2) and (2.4), i.e.,

o 7 r2Ro 7 = /4 ,
e = ~/zo i, =
e 2= /


It should be noted that among these quantities, z is neither a
o
fixed constant nor a given value as in the case of shallow shells,
It depends upon the size of the buckling region as indicated in
(5.10). For the same reason, e should be treated as a parameter


(5.10)


EJ\Pr r rr8













in the sequel, and it will be used as a measure of the size of the

buckling region, or the thickness of the boundary layer, to replace

the variable ro.

By utilizing (5.10), equation (5.9) takes the following

form



S [ [ o + 2,. e]- ?.76 + 3.0o2 2e) 531 '


+ 2.1o J = 0 ; (511)

this is the load-deflection relation when the shell is under

static equilibrium.

Let us take this opportunity to discuss the nature of

the load-deflection relation and its associated stability proper-

ties. A typical curve of equation (5.11) is in the form of

curve i as shown in the figure on the following page. Curve 1

constitutes three branches: the unbuckled stable branch OA, the

unstable branch AB, and the buckled stable branch BA'. Instead

of calling point A and B the bifurcation points or branch points,

we shall directly call them the critical points. The feature of

the loss of stability is such that, during the loading process,

the equilibrium position of the structure moves from 0 to B' and

then to A in a continuous and monotonously increasing fashion;

any slight increment of the pressure at the equilibrium position

at A would cause a sudden and large increase in deflection, which




































































- -4- -


- -- -- 9- -- -











brings the equilibrium position from A to some point above A'

on the buckled stable branch. Therefore, the equilibrium condi-

tion at A is certainly a "critical" situation, and the load

corresponding to the equilibrium condition at A on curve 1 deserves

the name of "critical load." In the static loading analysis, we

shall permit only one type of buckling of the shell that is due

to the loss of stability after passing the point A; therefore,

the critical load at point A is also the buckling load of the

structure. The significance of the equilibrium situation at the

critical point B defines the equilibrium condition where the

"outward snap" of the shell occurs. For the purpose of emphasis,

we shall repeat the argument that equilibrium condition at

point B has no significance to the instability of the shells, if

a classic buckling criterionI is used. We would like also to

point out that it is incorrect when we have a P- 8 curve in the

form of curve 1; this then implies the necessity of using an

"energy criterion," The main difference between these two

criterions is the method of determining the buckling load. The

classic criterion defines the buckling load by having a hori-

zontal tangent at the critical point, i.e., the buckling of the





We use the definition given by Kaplan and Fung in
reference (L6).











shell is solely due to the loss of stability and the load-deflection

curve usually is in the form of curve II in the figure referred to.

On the other hand, energy criterion permits a "jump" from the state

at B' to B during the inward deformation process; thus, a lower

buckling load corresponding to the pressure at the equilibrium

state at B is defined. Curves I and II indeed represent two

different types of instability. According to Biezeno and Grammel (6),

the instability represented by curve I is called "transitional

instability" and the other is called "complete instability," which

does not have the monotonously increasing branch beyond the range of

instability. An example based upon a classic criterion was given in

reference (6), where the load deflection relation was in the form

of curve I. We shall investigate the instability of the shell, basing

the investigation upon a classic criterion. The condition of equili-

brium corresponding to point A in the above-mentioned figure will be

referred to as the "unstable equilibrium position" or the "critical

position" because it defines the loss of stability of the system as

well as the buckling of the shell.

Let us return to equation (5.11). Geometrically, it repre-

sents a one parameter family of curves in the 7)-q plane; 0 is the

parameter. For each 0 value, equation (5.11) shows a possible





Refer to reference (6), pp. 484-496.











load-deflection relation during the buckling process. Let us say

that the true P- S relation during the buckling of the shell will

be the one with e = cr' and in a form similar to curve I, which has

been discussed above. We shall define the ecr in the following

fashion: 6cr will make the system reach its "unstable equilibrium

position" with the smallest value of qo By the fact that the state

of unstable equilibrium corresponds to a relative maximum position

on the q(?7) curve mathematically, the problem of finding 0cr is

equivalent to seeking a least maximum for the family of curves in

equation (5.11).

After calculations, the locus of the "unstable equilibrium

position" was found to be as shown in Figure 26. It has a relative

minimum when e 0.548. Therefore, we have


= cor 0. (5.12)


As c= r = 0.548, equation (5.11) takes the following form:


n 3-668971 r .31 -?I + 2.lo 7
0.48 .37 + 0.7. 71) (5.13)

This equation describes the curve shown in Figure 26, which repre-

sents the "best possible" relation between the load and deflection

in the sense of yielding a smallest critical load. It posses a

maximum at 7 = 0.418 corresponding to the critical position and a

minimum at-7 = 1.22. Corresponding to 7= 0.418, we have the

critical load from equation (5.13):










Icr = .21 (5.14)

which is about 68 per cent of Zolley's result based on a classic

linear theory and matches very well with both Thompson and Krenzke's

experimental results. It also should be noted that corresponding

to the minimum position on the -q curve there is a


,= (5.15)

which is about 26 per cent of the result of the classic linear theory.

As we have mentioned, this load corresponds to the outward snapping

load in the unloading process. The experimental test by Thompson

gave an outward snapping load for shells with (Ro/h) a 20 of the

magnitude about 22 per cent of the linear classic result. The change

of volume during the loading process can also be obtained from the

analysis and has the following form:

ho 3 f5
Av = 27r(-2)) E yr0

The first term in the above equation was due to the membrane contrac-

tion, while the second term was the volume developed by inward

deformation in the buckling region and was found by the following

equation:




I
Krenzke found experimentally, qcr = 0.84; refer to
reference (19).











V2 J w rdr de
Jo 0o

Let us define a dimensionless change of volume in the form:

S= L--


then, from the above equations, we have

'V = 2,r" (I-i) E2 a)c +

where r = 0.548, defined in (6.12), and q, 77 were also given

previously. We shall use the subscript "cl" to indicate quantities

corresponding to the result obtained from the classic linear

theory, i.e.,


IcL = 2/13 2)

VCL = 2Tr (-Y)(Ro/h)(o.-4)2 1-cL

Therefore, for V = O.A(R/h) = 20, we obtain the following expression:


3.6 L V [26,4162 t + 2.o09La ]7

Together with equation (5.13), the relation between ( V/ /Vcl) and

(q/qcl) can be found; it is given in Figure 27. As far as the
critical loads are concerned, the theoretical result is qualitatively

good as compared with experiment. The rate of increasing of pressure

in the post-buckling region was seen to be faster than the experi-

mental results given by (33).










3. Buckling of the Shell under a Suddenly Applied Constant PressureI


Let us assume that Reissner's simplified theory on transverse

vibration of a thin shallow elastic shell, i.e., the inertial forces

in the middle surface are neglected as being small compared to the

transverse inertial force does also hold true in the case of a

complete spherical shell. We, therefore, obtain the equation of

motion for such a shell by adding one term:


(ph j) (5.16)

which is due to the transverse inertial effects, to the right-hand

side of the equation of equilibrium (5.2). Equations of compati-

bility are kept in the same form as equation (5.3). By taking the

same form of equation (5.4) for the axisymmetrical dynamic deforma-

tions of the shell and considering the central deflection varying

along with time, after performing a similar integration of the

Galerkin's functional in the form of (5.7), we obtain the counter-

part of equation (5.9), i.e., the dynamic equation of a metallic

( 1 = 0.3) complete spherical shell in the following form:


4Ld24 + E 12-1 + 2.76 i 62 2
dtL+ PR2 2

+ ^ 8 3 f '1 4 R]. (5.17)
y4 8> 1 ~ ph ^ 2 -, .


Refer to Figure 6, Part B.












If the following

in the case of a


dimensionless quantities, which have been employed

shallow spherical shell are used'


- t
RO
.. d2.


S- ,
Zo


(5.18)


then equation (5.17) takes the following nondimensional form:



+ [(2.76 + 3.02o 2)j .3J1 + 2zIo" 'Z73

r BB 3 l


(5.19)


If a shell is under such condition that it is initially

undisturbed, then at time I = To = 0, we have the following

initial conditions


7 (o) =


and () =


(5.20)


2
Because of the '7 term appearing in the nonlinear part,

equation (5.19) does not belong to any well-known class of equations


iRefer to equation (2.4).


E

V2 _E
P











whose behavior has been systematically discussed. In the case of a

given load q, the dynamic response can be obtained by integrating

(5.19) and the solution, in general, in terms of elliptical functions.

Now, in this problem, q is taken as an unknown parameter.

We are looking for the critical value of q which will result in the

motion of the system being unstable. Since the system in (5.19)

appears to be autonomous, then a qualitative discussion of the

motion is possible by use of a topological method.

Equation (5.19) can be integrated once when q is taken as a

constant, and the following result is obtained:


()-2 = A + ( ~e? (2.76 + 3.02 62- ~ o


+ 3.' 7)3- 1-/o4 74 2 >0.

(5.21)


When q, 0 and the arbitrary constant A are assigned certain values,

equation (5.21) is the equation of the trajectories in the phase

plane of the system (5.19). This equation, in general, posses

three (3) singular points at the I values solved from the following

equations:




j = (j = o
(5.22)


or on the I axis where














S- (2.6 +3.02 e92 -3 ) + 231 ff


2108 173 = 0.

(5.23)


Let '?1, ,2' and 173, arranged in an order of increasing magni-

tude, be the roots solved from equation (5.23) when the values of

q and 8 are given. Tested by the stability theorem (9), we

shall see there are, in general, two centers and one saddle point.

Because of the initial conditions in (5.20), the trajectory which

passes through the origin (0,0) of the phase plane does represent
2
the oscillatory motion of the shell. Therefore, the problem of

finding the critical qcr for equation (5.19) is equivalent to

finding the q value which will make the separatrix pass through
3
the origin (0,0) of the phase plane. Equation (5.21) suggests

that for the trajectory passing through the origin, the value of A

in (5.21) should be taken equal to zero. Thus,





iRefer to reference (9), p. 317.

2Refer to the discussions in Chapter III.


3Refer to Figure 28.