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Transitional instability of spherical shells under dynamic loadings

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Title:
Transitional instability of spherical shells under dynamic loadings
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Ho, Fang-Huai, 1934-
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English
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ix, 173 leaves : illus. ; 28 cm.

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Subjects / Keywords:
Buckling ( jstor )
Dynamic loads ( jstor )
Eggshells ( jstor )
Equation roots ( jstor )
Phase plane ( jstor )
Saddle points ( jstor )
Shallow shells ( jstor )
Spherical shells ( jstor )
Structural deflection ( jstor )
Trajectories ( jstor )
Buckling (Mechanics) ( lcsh )
Dissertations, Academic -- Engineering Mechanics -- UF
Engineering Mechanics thesis Ph. D
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bibliography ( marcgt )
non-fiction ( marcgt )

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Thesis:
Thesis--University of Florida, 1964.
Bibliography:
Bibliography: leaves 169-172.
General Note:
Manuscript copy.
General Note:
Vita.

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University of Florida
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University of Florida
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Copyright Fang-Huai Ho. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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13645080 ( OCLC )
ACZ5046 ( NOTIS )
AA00004948_00001 ( sobekcm )

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Full Text


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 REQUIREMENTS 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 Wo 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 Ho 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.
li


TABLE OF CONTENTS
Page
AC KNOWLE DGMENT S 00000000000. o.ooo.o. XX
LIST OF FIGURES OOOO.OOOOOO. .00000.0. V
ABSTRACT
viii
INTRODUCTION
I
lo A Historical Review and Recent Advancement . o 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 .o...oeooe. IB
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 Geometrical
Imperfections oo.o.ooooo..... 41
III.BUCKLING OF A CLAMPED SHALLOW SPHERICAL SHELL UNDER
A SUDDENLY APPLIED CONSTANT PRESSSURE OF
INFINITE TIME DURATION .............. 43
1. A Qualitative Study of the Instability
of the Structure .............. 43
2. Another Justification of the Buckling
Criterion ooo.oo.oo.o.doo.o 32
3. A Discussion of the Results ........ 54
iii


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 . 67
V,BUCKLING OF A COMPLETE SPHERICAL SHELL UNDER
SUDDENLY APPLIED UNIFORMLY DISTRIBUTED DYNAMIC
LOADINGS .0..0...0.0.0.C..0. 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
I II O FIGURE S ...OO.D. 00. 000000.00 136
IV. NOTATIONS .... ....... 167
REFERENCES . ............ 169
BIOGRAPHICAL SKETCH ... 173
iv


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 @ = 0.26) . 142
8. Response curves of the central deflection of
a shallow shell ( 0 = 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 6 = 0.26) under a suddenly
applied uniformly distributed dynamic loading 146
12. Relation between the critical pressure and the
geometrical parameter Q 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
v


LIST OF FIGURES (Continued)
Figure
14.
A justification of the buckling criterion
applied to the shallow shell with 0 = 0.26
Page
149
15.
Response curves for various values of \jr
when Q = 0.26 ..............
150
16.
Response curves for various values of 0
when = 1.07 .
o o
151
17.
Upper and lower values of "J, as a
function of 0
152
18.
T. 0 curves for various values of V .
cr T
0
152
19 o
sC
Critical D.O.L.F. -- vs. critical
c r
central deflection -- r) ........
1 cr
o
153
20.
Upper and lower critical deflections vs. Q
0
154
21.
k
Upper and lower values of Tcr vs. ^ .

155
22.
k
T vs curves for various values of Q
cr

156
23.
The projection of the trajectories on the
7) - rj plane, when 0 = 0.26, ij/ = 5
. .
157
24.
The projection of the trajectories on the
7) -T) plane, when 0 = 0.26, = 2 . .
0
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 ............
0
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
vi


LIST OF FIGURES (Continued)
Figure Page
31. Phase p!ane_trajectories when
0 = 0.44, q = 0.6823 ....... 164
32. Phase plane trajectories when 0 = 0.44,
q = 0.33 ................... 165
33. Buckling pressure vs. buckling region
parameter Q for a complete spherical shell . 166
vii


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.
ix


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). These two authors were interested
1
Underlined numbers in parentheses refer to the references.
1


2
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 ^+9) .
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


3
obtained the resonance curves shoving 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, (30) 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 1-degree-of-freedom nonlinear oscillatory system,^-
Results were obtained by numerical integrations 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
(j[) in 1959, using a similar technique as in the above mentioned
1
Refer to reference (36).


4
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 axisymmetric com
pressing of the shell. By removing the restriction on radial dis
placement presented in Q_), 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


5
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 3ystem 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 reasonings. 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


6
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:


7
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., \ < Q
where
A v'2o->'¡) T2/r<'h~ z/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 = hh0 <2)
i
Refer
to Figure
1.


8
By using this parameter, we shall have our dynamic equations in the
simplest form, i e., Q 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 Q related with X by the following equation
(3)
or, for V 0- 3 ,
(*)
hence, a larger 0 value implies a shallower shell. It will be
seen later that, for non-shallow shells, 0 becomes an awkward
measure for the geometrical shape. In the region of the validity
of the present theory, however, the 0 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
^Numbers in parentheses in the text which are not underlined
refer to the equation numbers.


9
function of time, numerical integrations will be used because the
system is then nonautononous.
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.
10


11
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.


12
1. Autonomous Conservative System
The typical dynamic equation of such a system is of the
following form
ij = -f ( v. I) (i.i)
where X is a parameter, e.g., the load parameter. An equivalent
form of (1.1) is the two dimensional system:
V = s
j = f ( V, *) (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
2
certain isolated points, i e the singular points, in the phase
plane of the system. These singular points are found by the condi
tion that 7) and £ vanish simultaneously, i.e., from equation (1,2),
^Refer to (9), (23) (29) and particularly (3) in which a
beautiful discussion of the "conservative system" has been given.
2
The names critical point and equilibrium points are also
used.


13
%
O
-fdU A ) = O
(1.3)
The first condition in (1.3) merely says that the singular points
are located on the 7] -axis (where £ = Q ) 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, 7), and 7] 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


14
"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 V -axis reflect the amplitudes
of the motions, because £ = 0 i.e., the T](z) 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 0-4 to og 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 ^ and then creeps to the magnitude equivalent
to OC Any motion outside the separatrix has its amplitude
larger than O C e.g., o) The severe change of the amplitude
during the loss of stability becomes apparent by comparing the
length of o B 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


15
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 r)i 71 and 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:


16
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
Z 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


17
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 xtend 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 TJ rj 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
jj = { ( 7) Z ) (1.4)
where t is the time variable. The similarity between (1.4) and
(1,1) is easily obtained by taking T equal to some definite value
of time, say \ ; 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


18
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 fi7], 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 t".
x
Notice the form of the trajectory in the y)-'7} plane. 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 at time r =
are determined by
(1.5)
0
this is shown in the figure as 7) Y)i V¡ By virtue of
(1.5), the following equation,
f ( V Z ) ~ 0
(1.6)
is simply the locus of all singular points (both stable and unstable
singular points) in 7)-rj plane. The intersection point of the curve
^We prefer not to use the term "phase plane" in this case.


19
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 = X = 0, the initial time. The first unstable
singular point in the plane when t = tQ certainly is the lower
measure of the critical deformation. In the case that f appears
only in the forcing function, the lower measure in the above defined
sense is simply bhe 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 5 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 and Au
according to the above discussions.


CHAPTER II
BUCKLING OF A CLAMPED SHALLOW SPHERICAL SHELL
UNDER A PURE IMPULSE
I. 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
(2.0)
This second order ordinary nonlinear differential equation,^ when
2
l) = 0.3, has the following form:
33 l_
\(o p h
(2.1)
^The derivation of this equation is attached in Appendix II.
O
Refer to Figure 1 for the definitions of each quantity
in this equation.
20


21
If the shell is sufficiently shallow so that the middle surface of
the shell can be approximated by a paraboloid, thus
Z ~ Z0 ( r / r0 ) 2
and the curvature of the middle surface has to satisfy the
condition:
dr2 Ro '
These last two equations give us the useful relation:
Zo ~
(2.2)
By virtue of equation (2.2), a dimensionless form of
(2.1) can be written as:
y¡ + (76 + 3.025 0 ) r) 5-11 r¡2
+ 2-1075 r/3
where the nondimensional quantities:
f *1
(2.3)
v = i=\frTp */r0 ,
2
Q h I Z0 %(?)= jT^z $ (t'> (2.4
have been used, and p is the mass density of the shell material;
the dot () now represents d/dt.
In that case the external pressure is an ideal rectangular
impulse as shown in Figure 6. For sufficiently small ^T, as


22
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- O ), with
certain initial conditions derived from the external disturbance.
It is conventional to take the displacement at the new initial
time, i.e., t = AT, 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.,
It is noted that equation (2.3) is equivalent to a nonlinear
spring-mass system with a unit mass (m = 1) and a constant force
in its nondimensional form,
Therefore, the above equa
tion reads:
e j ax = i (*t) o
where AX is the dimensionless form of the time duration AT, by
the definition in (2.4).


23
From the above discussion, we have our new initial condi
tions, thus
at Z = dt
V(*z) = 0
I (AX) = l) (4-c) = || 0 I*
(2.5)
where I* = q.(Atr) is the dimensionless impulse.
Because of the smallness of the time duration of the impulse
( AZ ta O ), it is conventional to solve this problem by taking
(2.5) to be the initial conditions at X : 0, thus
3?
= Tb
7] (0) 0
(2.6)
and a free oscillatory system with a dynamic equation obtained
from (2.3) by dropping the terra on the right-hand side, which is
involved with the external load q. Thus
7] + [i 2 76 t 3 025 0*)r) $-31 + 2 /o7- 0
The oscillatory motion of an autonomous system characterized by
equation (2.7) 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:
1 = l
(2.8)


24
Here it is apparent that £ is the dimensionless velocity of
the motion. Equation (28) can be integrated once and becomes
7 4 -3. £4 rf + ( 2-76 + 3-025 >2) V1
(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 7] and % equal to zero simul
taneously, i.e.
\ 0 and
7 [ 2-lo8 7)1 5-31 7 + (2-7 + 3 o2$ e2) ] = 0.
It is clear that the critical points are located on the "7 -axis.
One of them is the origin of the phase plane and the other two
points are defined by the equation:
77 =
'o
4-2/6
$-31 5-0$ Vo-'9103 0X
(2.11)
The stability nature of these singular points can be determined
1
from the following characteristic equation, (9), i.e.,
Refer to p. 317, reference (9).


25
Det (A)
-A f
-[(2-76 + 3-o25 02) lo (>2 \
+ \Z ] -A
(2.12)
It is given by the stability theorem^ that corresponding to a
pair of real eigen values of (2.12), an unstable singular point
will satisfy the following condition:
[ 6-324 *)02 10-62 \ +(276+3-025 62)] <0. (2
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 r]o (singular points)
provided,
01 4 o 19 303 or
@ 4 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.
^This is the characteristic equation of the linear approx
imation. The discussion of using this approximation is referred to in
Appendix Io


26
0 > 0-44, 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 Q 0-IU+. By virtue of (2.13), we know the
root:
1
4.2I&
$31
o 19303 9
(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 by I** from now on, T)'** in equation (2.16)
is read
This condition is equivalent to
defined in equation (4).
\< J37 where A is


27
From (2.9), for the separatrix, we have
- 1-0$38 I* 3-& Vj + (2 76 + 3 02$ e2) 712
/cr
~ C ] ~ (2.17)
It must be remembered that from (210), "?7cralso satisfies the
condition:^
( 276 + 3 02$ 02) = $-3lVtr- 2 1o8 Vet -
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 r,l 1.0538 V* (2.18)
where, from (2.16),
\ = 1-2$94$ 119181 V0-19303 Q2 (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.,
'*)(o) = 0 .
It should be noted that ^cr is subjected to the condition that
o i.e., v¡Lt is nontrivial.


28
It is apparent from this condition that the motion always starts at
the point where its trajectory intersects the £ 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:
or
(2.20)
where 1?tr is defined in (2.19).. These two equations will give the
critical impulse for any shallow spherical shell whose geometrical
parameter 0 is known and satisfies the condition in (2.14).
An example is given by taking 6 = 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 Q8) 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
1
They used a five-degree-of-freedom system.


29
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 0 r: 0-26 i (2.5), and (2.6) read:
#
V =
- 2-/075 V( V- o.S3$o3)(rj- 1-68453)
T)Co)
= O f yj (o) 0-536 2$ IA'.
(2.21)
Three singular points on the T) -axis are:
>? =
O a center
V =
0-£3503 a saddle point
T) =
1 6^4 53 a center.
Let 'r)Cr = 0.83503; from (2.20), we found
lcr = 1-34233 (2 o 22)
The equation of the separatrix is found as follows:
£2 = 1-0^2 Tj1* + 3 $4 rf 2-9645 i)1 + 0-SI83C.
This equation and other phase plane trajectories have been plotted
and are shown in Figure 7.
In Figure 7, when I < I cy given in (2.22), trajectories
of the motion are closed curves around the center point at the


30
origin, e.g., curve 1; when 1 '= Icr the motion is unstable and
* -r *
on the separatrix 2; when I > 1 cr 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 coordinate of these
points (e.g., 0A) is the measure of the maximum inward central
deflections of the shell. It is clear that the maximum inward
central deflection increases with I in a continuous fashion when
1 < lcr As soon as 1 is slightly larger than I tr the
maximum central deflection undergoes a severe change, from some
value less than 0.84 (e.g., 0A) 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 (26) 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 (27) However, when 0 is taken to be a definite
value, the solution of (2.7) can always be obtained. For an example,


31
in solving such type equations, we shall take the system defined
in equation (2.21); i.e., 0 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 0 = 0.26 in equation (2.9); we have the equation of
the trajectories for the system (2.21) in the following form:
} = J C 2-964$ Y i 3-54 T]3 1-0538 V* ,
(2.23)
where C is an integration constant. By using the second initial
condition in (2.21), C has the following expression:
C = O-S3625 I*)2 (2.24)
Our problem is to find the response zj (?) 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 = 1.2, and 1 = 1.5. We shall see, in contrast
to the small increment in the disturbance I the corresponding
responses will undergo characteristic changes.
When 1 = 1.2, the positive branch of (2.23) reads:
d V
dz
h
-I-03S ( T}~ 2-0049 )(>}-1 loStld-o + 0-3/47) .
(2.25)


32
This can be transformed into an elliptical integral of the Legendre's
standard form.^ Let us, first, formulate the quadratic equation:
Z-Sb \bb V2 4-7766 V + 1-fo2$3 = 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
2
will not be given here. The two roots of (2.26) are:
P = 0 21585 % = 7- 397 /S' (2-27>
Now, we use the following transformation:
= f* U
1 + z
0-27585 + 1 397 18- Z
1 + Z (2.28)
and
dl) = [ 1 ,2,3V(i +z)2 ] z ,
and then equation (2.25) can be transformed into the following form:
1
Refer to reference (41).
2
ecg., reference (43)


33
I.12133 oU
(1 + 2)2 dt
J
0-tz)2
J \ 0*55% (O y7721 Z2- / 4342)p 4272 Z2- 0-16983)
or
dz
dr
0-21196 fZ2- 8 o9$59)(22' 0-11899)
or
dz
0 46039 dr =
/"Z2-8 ?S£9)( 2J- 0-//899) .
1
(2.29)
Referring to reference (41)/ the function z can be written in the
form:
Z = 0-34495 Sn (u | m) ,
(2.30)
where
u = 1-3099J (r Co) *
m = o- o J469
(2.31)
Therefore, as we substitute (2.30) into equation (2.28), the
solution of the problem can be formally expressed as:
The expressions are on p. 26, reference (41).


34
V =
o- 2 7 fe % 5 + o 4S96 Snfu-I'fli)
l + o 3449fe Sn ("u ) m)
(2 o 32)
where U and m have their definitions in (2.31)o The value X
o
in (2.31) is determined by the initial condition: when X = 0,
xj = 0. In this fashion, we have XQ satisfying the following
equation
Sn (-1-30995 Zo | 0 01469 ) = 0-fe72 3* .
(2.33)
As an approximation,
Zo 0 469^ (2.34)
Solution (2.32) has the following general properties:
A. It is periodic because it involves the double periodic
function S'n(u|irn.) The real period of Sn(ulm) is
4K = 4(1.57658). The period, P of *] then, is equal to 4.81419
according to equation (2.31).
B. The 7} values are bounded in the interval -0.31464 4
V i 0.56344, because the value of Sti (u Im)varies between
-1 and +1. Therefore, the maximum amplitude of the dynamic response
V) corresponding to the disturbance I* = 1.2 is Y) = 0.56344.
' max
The next example; when I* = 1.5, the positive branch of
(2.23) reads:
~ = £ = J 0-64702 7964* + 3 *4 rf l-o$3? .


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
dr) f-
= y-i o538 crj-ot) C7)-fi)( r¡-r)( o) (2.35)
where
d 2. 0 = 0-38067 ,
= o-d- 0 28-537 t 6 = o-8M6 o-28*J7 i.
The counterpart of equations (2.26), (2.27), and (228) are
respectively
2
O 0170 3 If +
3 13426 V 2-6321$
O.
p- -I84-879 ft -3*87 (2.36)
and
^ -184 #79 + 0-83$&l Z
1 + 2
d7) = [ 17/u8] / (I-b Z)2 J dz
(2.37)
The transformation in (2.37) brings equation (2.35) into the simpli
fied form as in the following:


dz
A / f 22 -
i 0-999j
+
Mi
O O2IUH
)
y
where
A Ji U9993 ) o 081 uu
M, = 34,49) *6o$9 ,
M2 = 34, 429-9792 3 .
Therefore, we have
(2.38)
Z = I5l- 64249 Nc (aim) (2.39)
where
U. = 1 29092 (T- r0) ,
m = 0-94g,49.
(2.40)
Substituting (2.39) into (2.37), we have the solution
-I U2l9 Cn(u|m) + 126 -7534 I
C-n(uim) + 151-64249
(2.41)
where the definition:
Nc = < / Cn ,
has been used, and u, m are defined in (2.40).


37
In a similar way, To in (2.40) can be determined by
requiring 1](Z=o)= O thus
lQ -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:
A It is periodic, with period P = 9.03
Bo 7) values are in the interval -3.8079 £ 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
^ = 0563 to 2 .068) as I* value changed from 1,2 to 1.5. We can
max
conclude that the critical load Icr must be some value in between
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


38
3o 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 V -axis in the phase plane are the points
where the response curves reach their amplitudes, because at those
points, the velocity £ 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., 0 is taken to be the
value 0.26. Substituting (2.24) into equation (2.9) and using
0 = 0.26, we have the following equation for the trajectories
of this system:


39
%
2
|.o(54 r)l + %uS rf
02%1$G I*2 J
(2.43)
By virtue of the above discussion, the amplitudes of the response
curves, rj' s, are found by setting the velocities t,'s equal to
zero By doing so, from (2.-43) the following relation is obtained:
_J
o-2$7$6
- 3 54 7m + /%4$
(2.44)
For the solution to be physically meaningful, the positive branch of
the last expression should be used; thus,
- /3M32 7)* [ rj* 3.3*8-63 Vm + 2-WG\\ (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,, y term, equal to zero. Thus, the
three positions of rest (where ij = = E, = 0) are obtained:
iO il) 3> ,
r¡ 0 r) o-fa 7} l-GS .
(2.46)


40
In T)max I* plane, these curves will be straight lines parallel
to the I* axis. According to the criterion, the first unstable
equilibrium position ( ^ = 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 (^).
According to Figure 1 and equation (2.0), we have:
t
o
Therefore, certain definitions in reference (8) assume the following
expressions:
Z
and
By definition we have
A = C-/zc = V .
(2.47)
According to (2.47), the measure of buckling used in (8>) ,
/\ =1, corresponds to 7) = 1 in the notation of this paper,
max max
max


41
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 8 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 > 0 > 0.32 (or 3 = 87 < X c 453) that equation (2.20) agrees well
with the result given in (8) and appears almost the same as the
result of reference (13) when 0< 0.15 (or A > 6,6). It is believed
that this analysis is parallel to the presentation of (15), yet with
a cut-off point at a larger 0 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 "*7=0, i.e., on the axis. Any initial imperfection


42
(deflection) of the shell is equivalent to set the motion starting
at C 1 instead of at C as shown in Figure 7. It is seen from
S 3
this figure, that Cs' has a smaller ordinate than Cg 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
= 0-0$ z0
J t
or
7Ji 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
1. 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,
V = [(7 + J O25 02) 7] ~ 1-3 I 1JZ
+ 2 lo? I/3 1 l > 0 (3.1)
43


44
We have mentioned that the system is initially at rest, i.e.,
^(o) = 0
£ (0) = o
(3.2)
are the conditions at V = 0.
Note that the autonomous system in (3.1) can be integrated
once and results in the form:
*
2
A + (u-125 9 + 3 7)3 ,.0$4 7)4 Z > O ,
(3.3)
where A is the integration constant. Equation (3.3) is obviously
the equation of the phase plane trajectories.
The singular points (critical points) of the system in
equation (3.1) are found from:
V = \ = o
i.e., on the *) axis where
l 6f [ (2-76 + 3 02$ ?) 5.3! Tj1
(3.
+ 2 108- TJ? J Q _
Stability conditions in the neighborhood of these singular points
are determined by the following characteristic equation, thus
- /0 62 7) + (276 + 3 025 02)] = 0,
1 + f 6 7255 7)z
(3.5)


45
Corresponding to the real eigen values yielded by (3,5), the saddle
points satisfy the following condition:
3225 ^ ¡o 62 7jo + (21G 4 3 025 Q1) < 0 ,
(3.6)
where 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 (3d) 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
( = E, = 0). Therefore, the equation of this trajectory is
obtained from (3.3) by taking A = 0, thus
2
(3.7)
If, furthermore, this trajectory also passes through the saddle
point ( V) 0), it is obvious that the following condition has
to be satisfied:
or equivalently


46
for "70 1 0.
Attention is invited to the fact that this problem reduces
to finding common roots between equations (3.4) and (3.8).^ 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
_ t 11 I O5U
% =
0 2.062$ 9
Substituting (3.9) into (3.4), we have
3 162 rj* JOS' 7)2 + ( 2 76 + 3 02$ 92 ) ^
(3.9)
0 ,
(3.10)
which has three roots. The two nontrivial roots are solved from
the following equation:
rll
f. 11954
/ ( 13 2 1792 3*26020 Q2)
V 39 99291
(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
1Let us recall that V0 in (3.8) is one of the roots
of equation (3.4); it satisfies the condition (3.6).


47
time, that both ">7, and rj1 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 1) 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, V = r)l (Note that '*]2 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.,
0 0-630 7 1, (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 T) n 0
from (3.10), we have the following condition:
( ?-76 +3 02$ Q1) 7 .o% rj 3 162 (3 = 13)
^It corresponds to A > 3.2123, where A is defined
in equation (4).


48
which is satisfied by both Tjt and ^ in (3=11)= Substituting
(3.13) into (3.6), after combining terms in the same order of ?) ,
the following condition is obtained:
3 162 17 2 3 4 v) < 0 .
As we mentioned, both ?)]_ and ^ 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 77 is solved from
'o
equation (3.10), condition (3.6) is reduced to the following form:
V} 4 f. 11954
(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:
I I1954 -
( I 21792 38 26o2o Q )
39 99297
(3.13)
Therefore, without even going back to the phase plane, we can
write a general solution for the critical pressure by substituting
(3=13) into (3.9), i=e.,
2
' 77 Vtr
1-0*4. ^Icr
2.0625 e
(3=16)


49
where 7J 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 6 > 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 0 = 0.26, we have
7jir = 754 (3*17)
and from (3.16), we obtain immediately,
rcr 0 6*5 3?
(3.18)
Our question has been whether the value of q in (3.18) does cause
the instability of the system (3=1) when Q = 0.26, in other words,
to make the motion of the system on its separatrix, or equivalently,
does the value of Y) in (3.17) define the saddle point in the phase


50
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 9 = 026 and
q = qcr defined in (3.18); thus, equation (3.4) is in the form:*
2.lo? rjZ $-31 r]2 + 2 96449 V ~ 0 ?S1$Z 0 (3.19)
which has three real positive roots:
t] o nSoS rj2>- 0 "7 = 17863?. (3.20)
These are the singular points in the phase plane, while equation
2
(3.8) has the following form:
[ i o$u 7}3 3$4 72 +2-964$ r) o 73507 ] O, (3.21)
which has three roots:
Tj ?.2d3fe$ ,
and a double root at Tj o 567^4 (3.22)
Comparing (3.20) and (3.22), it is clear that 7 : 0.55754, i.e.,
the critical value defined in (3.17) is the common root between
We have multiplied the value (-1) through the original
equation (3.4).
2
Similarly, a quantity (-1) has been multiplied through
the original equation.


51
(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 7J in the first
equation of (3.8) by f) 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 "^-axis ( £> = 0)
Because of the preceding discussion, we conclude that the phase
plane trajectory for the motion of this particular shell ( 9 = 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 Q are found and given in Figure 12. Curves indicating
the variation of critical deflections with 0 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.


52
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 1, to zero, we obtain the following relation between the
nontrivial amplitude of the response curves and the corresponding
dynamic loadings.
( 2 76 + 3 o2^ 02 ) ^ 3 rf + I-te rf
4-2S 9
(3.23)
For the purpose of further discussions, let us take a
specific shell with B = CL26. From (3.23), we obtain:
1 ol2 5
(3.24)


53
The static load-deflection relation can be easily found from

equation (3.1) by taking the inertia term, i.e., the ^ term equal
to zero. For 6 = 0=26, we have
296U49 f) $-11 V2 + 2.108 V3
0-53625
(3 o 25)
Equations (3.24) and (3.25) are plotted in Figure 14. The inter
section point of these two curves is at Vm = 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 inteisection 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


54
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


55
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
unsyrametrical 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:


56
r¡ r 7^ Q% [(2-76 -h?2S92)V *T 3 I rf + 2jo7^^?]?
O < T < T ,
r¡(o) = ij(o) O
(3.26)
[(276 + 5 02$ e1)^ fr-3/ rf + 2 /07T 7? ] ,
t < r < 00 ,
(3.27)
where £ is the dimensionless forra of the quantity d t, 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)o 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


57
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
Z = Z o Immediately after Z = Z the motion of the shell is
on the trajectories characterized by equation (3.27). It is
obvious that the displacement ^ at t = I 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., ^(t) and £( z ) 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 Z Z > the equation of the trajectory^- is
Jb2 = (U 12$ 9f ) r) ~ ( 2-76 + 3 2$ 62 ) r)
+ 3 T4 V l-ob ,
(3.28)
~ 2
when 2 > Z > the equation of the trajectory of the motion reads:
(3.29)
1
Refer to equation (3.7).
2
Refer to equation (2.9).


58
The continuation property of the system at l = I requires:
% a) = yr(z)
^ (z) = y(z) .
Therefore, we have
C = 4-l2i> 0 l ) 7J(i) ,
(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
(3.31)
where C has been defined in (2*18) and (2.19). As an example,
when 0 = 0.26, according to the previous analysis in Chapter II,
Cc = 0.5183. Therefore,
O
cr = 8326
(3.32)
It should be noted that in equation (3.32), Zj (z ) 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:


59
A A time duration z 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 Q 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 = Z ,
i.e., 'T) (r ) can be found.
C. The critical pressure qcr is the one which satisfies
the condition (3.31), or when 0 = 0.26, the condition (3.32) is
satisfied. It is tedious, yet straightforward.


CHAPTER IV
BUCKLING OF A CLAMPED SHALLOW SPHERICAL SHELL UNDER A
UNIFORMLY 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:
(4.1)
2
If the following definitions are employed,
7 =
e = */z.
V
(4.2)
^Refer to Figure 1 for the geometry of the shell.
2
Refer to equations (2.2) and (2.4).
60


61
Then (41) can be written as
2
dl
+
[2H + 3-o2* 02) r¡ $ 31
+
2 log r¡
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 -j *,
(4.5)
where pQ is the nondimensional critical load for a complete shell
given by classic linear theory and has the following form:
- 2
f t/jo-v1;
for = 0.3, pQ = 1.21; Q is defined in the following:
q r;
e h
Q
2
9


62
which has a dimension: 1/sec. By utilizing (4.4) and (4.5),
equation (4.3) reads:
+ 2 I off r)3
(4.6)
where
Y
(4.7)
, a nondimensional quantity.
It is clear that the character of the solution of (4.6) depends
entirely upon the two parameters 'I/ and 0 We also like to
mention here that the transformed time variable 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 (46) is actually the load-
deflection relation of this problem. Let us assume the initial
conditions as follows:
7} ( Z* = 0 ) =
0
0
(4.8)
which imply an initially undistrubed shell.


63
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 ^ and 0 in (4.6), and
response curves were obtained by integrating the equations
numerically on the University's IBM-7090 computer.^ Dynamic
buckling loads were determined by use of the criterion proposed
in Chapter L The static load response curves were found by
n 2
dropping the inertia term, i.e., d^7? /d-£* 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., 0 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
(e g., \f/ = 100), the buckling of the shell approaches the static
case as it should be and the "creep pheonomenon" strongly indi
cates that loss of stability is of the "classic type." Followed
by several cycles of oscillation, the dynamic curves for '4/ = 100
converge to the static curve.
"Runge-Kutta method" was employed. The technique of
this method is found in (22).


64
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 Q values (or decreasing in
shallowness) As shown in Figure 18, no relative minimum corre
sponding to a certain 0 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 Q value
increases, corresponding to a decrease in critical pressure.
However, for a very shallow shell, i.e., a sufficiently large
value of Q e.g., 0 = 10 ( X as 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 Q i.e.,
the largest value of 9 for buckling to occur, has not been found.
It is the feeling of the author that the limiting value of 8
depends also upon the rate of the loading, i.e, the parameter i//
Therefore, the general answer may not be found without considerable
costly computations. A further discussion on this matter will be
given later.


65
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 ^cr 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 Jcr values will be. However, there exists a particular
shape of the shallow shell, which corresponds to 0 0.35. For
any other shape of the shell shallower than this value (i.e.,
0 > 0,35), the critical deflection decreases. This is to say,
there exists a maximum for the upper rJCY 9 curves as shown
in Figure 20. We do not know whether this value ( 0 s 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 0 < 0.35 in Figure 20. However, a contradiction arises for
Q > 0.35 (or, A < 4.31). Two possible hypotheses may be
provided: first, for all 0 > 0,35 there is no dynamic buckling


66
occurring because of the shallowness of the shell geometry; second,
for 0 ~ 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 thinkthat this value of Q 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 y on the critical dynamic
load is much more significant than that due to the geometrical
parameter Q We mentioned that the critical dynamic overload
factor !cr* is a function of two parameters (variables) Y and
0 i.e., it is a surface in the ( Y, 0 Z *) space. Based
upon the curves in Figures 17 to 21, we have the approximate
equations for the surface: (For, l) = 0.3)
/ -o8\
( 1 + V )
-7 3 9 7
Ur]
u
l.frl 4 2.0 Si e 132.
(27 +W
-99
I 31 1 o.79 6
(4.9)


67
We found that in the range of 0.5 £ V 3.5 and 0,15 £ 9 035,
results obtained from equation (A.9) agree well with that solved
from equation (A.6) and initial condition (A.8). The accuracy of
(A.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
-bw ^ K 9 n
Zcr = (a + V )( C t £ e )
(A.10)
with a, b, c, g, and k determined by experimental tests
The projection of the trajectories on the ^ V plane
are shown in Figures 23 and 2A. According to the criterion in
Chapter I, the critical state is bounded by the two points
indicated by L and U, ice, 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 "^-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 2A. We
found the functional relation between the critical load and the
geometry of the shall has a characteristic difference from the


68
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.^)
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 (_1) .
A functional relation between the critical dynamic overload pres
sure and the two parameters ^ and G was formulated in equa
tion (49); 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 1//" in (4.7), i.e,
which is a dimensionless quantity. The requirement of similitude
is very conveniently furnished by the quantity For an
example, if we choose the same geometric parameter Q for the
model and the prototype, then one can determine the nature of
Dynamic overload factor.


69
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 ^.
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 I, 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
70


71
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.^
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
lE.g., (16), (7), (27), (38), (39), and (40).


72
Kantian and Tsien, ie., the sphere is contracted to a sinaller
sphere and forms a single dimple after buckling. By virtue of
this buckling mechanism, Vol'mir allowed a uniform membrane
stress: - V-At. to be distributed in the buckling region before
the loss of stability and considered the buckling region as a
shallow spherical cap In other words, the differences between
Vol'mir1s method and Karman-Tsien's is that Vol'mir used a
Galcrkin'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


73
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.
1o On a New Mechanism of the Buckling of a Complete Spherical
Shell
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. When the external pressure q is much less than the
critical value, the shell contracts to a slightly smaller sphere;


74
as shown in Figure 25, Part A, the original shell contracts to
a sphere with radius equal to Rq 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-
1
Refer to Figure 25, Part C.


75
Co 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 O',
the small hole is contracted to an infinitesimal one. Under
such a condition, the membrane stress at O' 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 O', 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


76
region dilates (or the thickness of the boundary layer increases)
and forms the buckling region after the loss of stability of
the shello
By virtue of the a bove described buckling mechanism,
we arrive at the conclusion of the existence of a boundary
layer at the vicinity of a point 0' 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 O' is zero
and outside of this region the shell maintains a momentless
state with a membrane stress (Jr = 0/2h. By referring
to the stress distribution in a bent clamped circular plate,^
a reasonable assumption in the boundary layer region will be
a parabolic variation, i,e.,
(5.1)
^Refer to (34), pp 54-63.
^Refer to Figure 25, Part B.


77
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 (p.s.i.) in the following form:^
The equation of equilibrium,
D(V)
i iz
r <*r J 2
(5.2)
and the compatibility equation,
d_
dr
( V24>)
- £
\ J- ( )2 +
i d w j
[ 2r ( dr J
Ko dr J
)
where
VZ =
di
dr1
+ Ld.
r dr
(5.3)
1
Refer to equations (A.29) and (A.30) in Appendix II.


78
Since we have restricted ourselves in the problem of axi-
syrametrical deformations, then equation (2.0) can be used again
as a first approximation of the deflections in the buckling
region; thus, we have
w = £ f(r/re2)J (5.4)
Substituting (5.4) into (5.3) and integrating, we have
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 (fi and radial
stress (Tr and after including equation (5.1), the above equa
tion of the compatibility condition of deformations in the middle
surface of the shell reads
(5.5)


79
Substituting (5.5) into (5.2), the equation of equilibrium yields
the following form
a
(5-3W
i v
£ 4 rc
(7> -
' 0
6Ra i~v
2(2-V) ,r_,
( rj
Hkf
0
(5.6)
We shall use Galerkin's method. This method allows
equation (5.A) and (55) to be the approximate solutions of
equations (5.2) and (5.3), provided
ff G['-(r/r.)2J dA = 0
JJ a
(5.7)
is satisfied. In (5.7) "A" represents the area of the circular
1
region with radius r 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.
1
Refer to Figure 25.


80
r \ i E hZ
o-2) C
1
PF?o2 1 24
)
i-if )
-}]

E [ tn 75 if
R<>r02 4? (i-v)
(>219
2704
E r ff-jjv
b(l-v)
C
il
8
h 2 i ^ re2y
.3, equation (5=8) reads
f ' +
' Y 11
'0
? ) c
Rl ) t
to-62 2
Roro ^
£ 45
T
=
'n +
8
3 Ro /
2 r2 S
'o J
(5.8)
(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.,
20 T/zR0
0 = VZo .
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)o For the same reason, Q should be treated as a parameter
7 = V7 ,
f = W/Eh1 .


81
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 rQ
By utilizing (5.10), equation (59) takes the following
form
-[(li + 3 025 e2)7? 5 31
+ 2.1 off = 0 ; (5U)
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 1 as shown in the figure on the following page. Curve 1
constitutes three branches: the unbuckled stable branch 0A, 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


82


83
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 criterion*' is used. We would like also to
point out that it is incorrect when we have a P- S 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 (16)


84
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 ^-q plane; 6 is the
parameter. For each 0 value, equation (5.11) shows a possible
^Refer to reference (6), pp. 484-496.


85
load-deflection relation during the buckling process. Let us say
that the true P- £ relation during the buckling of the shell will
be the one with 6 = Cr anc* a ^orm similar to curve I, which has
been discussed above. We shall define the 0cr in the following
fashion: 0cr will make the system reach its "unstable equilibrium
position" with the smallest value of q0 By the fact that the state
of unstable equilibrium corresponds to a relative maximum position
on the qC7]) curve mathematically, the problem of finding Q 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 Q- 0.548. Therefore, we have
0cr = 0$48\ (5.12)
As Q = @cr = 0.548, equation (5.11) takes the following form:
| 3668,71 V ~ $31 rf + 2108'
0-US ( I-37& + 07 rj) (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 *1) = 1.22. Corresponding to 7= 0.418, we have the
critical load from equation (5.13):


86
= 0-82 1 (5.14)
I) cr i
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 ^ curve there is a
i = 0-32 (5 o 15)
oa
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 (RQ/h) ~ 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:
TT y
= 2itU-v) + j rc £ .
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:
Krenzke found experimentally, qcr = 0.84; refer to
reference (19).


87
AV', =
w r dr de
Let us define a dimensionless change of volume in the form:
AV = (
AV
fio lo
then, from the above equations, we have
A? = 2 TT ( I V) el | + T ^ ,
where 0cr = 0.548, defined in (6.12), and q, t) 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.,
1rCL ~ V, jK'-ll2)
Wei ~ 2n O-v) (Ro/h)(oW)2
Therefore, for V = 0.3,(R/h) = 20, we obtain the following expression:
r c l
A V
AV,
26-UI62 l + -o9k4 ^
"cl 31 96?
Together with equation (5.13), the relation between ( /a\/c\) anc^
(q/qcl) can fund> it as 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)


88
3o Buckling of the Shell under a Suddenly Applied Constant Pressure'*'
Let us assume that Reissner1s 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:
(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
( "0 = 0.3) complete spherical shell in the following form:
1
Refer to Figure 6, Part B


89
If the following dimensionless quantities, which have been employed
in the case of a shallow spherical shell are used^
(5.18)
then equation (5=17) takes the following nondimensional form:
jj + [( 2-76 + 3 02$ 92)y] £ 31 rf + 2 lo? rf ]
- +ze 7i].
(5.19)
If a shell is under such condition that it is initially
undisturbed, then at time X = X 0 = 0, we have the following
initial conditions
7] (o) = 0 and r¡ (O) = 0 .
(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
1
Refer
to equation
(2.4).


90
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:
(if = A + (-£ fe) 17 (?H + 302? e2-1 %e) T)2
+ 3-$u rf i $U 74 } r>o.
(5.21)
When q, 9 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 7 values solved from the following
equations:
= v = 0
(5.22)
or on the 7 axis where


Full Text

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