University of Nigeria Research Publications ETTE, Anthony Monday PG/Ph.D-82-1731 Author Title Dynamic Buckling Of Imperfection- Sensitive Elastic Structures Under Slowly- Varying Time Dependent Loading Faculty Physical Science Department Mathematics Date 1990
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University of Nigeria Research Publications
ETTE, Anthony Monday PG/Ph.D-82-1731
Aut
hor
Title
Dynamic Buckling Of Imperfection-Sensitive Elastic Structures Under Slowly-
Varying Time Dependent Loading
Facu
lty
Physical Science
Dep
artm
ent
Mathematics
Dat
e
1990
DYNAMIC BUCKLING OF nPERFECTION-SENSITIVE ELASTIC STRUCTURES
UNDER SLOWLY-VARYING TIME DEPENDENT LOADING
by
AhrnONY MONDAY E m
PG/Ph.D/82/173 1
SUBMITTED IN PARTIAL FULFlLMENT OF THE REQUIREMENT FOR THE
AWARD OF THE DECREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS OF
THE UNIVERSITY OF NIGERLA
SUPERVISOR: PROFESSOR J.C. AMAUGO, FAS
ANTHONY MONDAY ETTE, a Postgraduate student in the Department of Mathematics
and with Reg. No. PG/Ph.D/82/1731 has satisfactorily completed the requirements for
course and research work for the degree of DOCTOR OF PHILOSOPHY in
MATHEMATICS. The work embodied in this thesis is original and has not been
submitted in part or full for any other diploma or degree of this or any other University.
, I/ DR: G.C. CHUKWUMAH Head of Department
R J.C. AMAZIGO Supervisor
1 ACKVOWLEDGEMENT
I
I I wish to express my gratitude to my supervisor, Professor J.C. Amazigo for
methodically directing and guiding my sense of direction in the cause of the research
reported in this thesis. 1 wish in particular to express my heart-felt appreciation to him for
stimulating and motivating my interest in Applied Mathematics and for his deep-seated
understanding of the predicaments and constraint facing a typical Third World research
environment. He is to me, in every unqualified sense a complete gentleman, a friend, a
teacher and above all, a mathematician per excellence.
My unreserved indebtedness also goes to my friend Dr. Moses Oludotun Oyesanya, his
wife, Jumoke, and their children for their unqualified hospitality to me. Words cannot
adequately describe the numerous helps, motivational advice and scholarly encouragement
which this noble compatible pair and other members of their household rendered to me
especially in the darkest moment of greatest despair. They proved to me, in all honesty,
friends in need and friends indeed.
TABLE OF CONTENTS
TITLE PAGE
CERTIFICATION AND APPROVAL PAGE DEDICATION
ACKNOWLEDGEMENT
ABSTRACT
INTRODUCI'ION
CHAPTER
1 DYNAMIC BUCKLING OF A CUBIC MODEL
1 . 1 Formulation of the equation
1.2 Static Problem
1.3 Dynamic Problem - Step Loading
1.4 Slowly-varying Loading
2 DYNAMIC BUCLKING OF A CUBIC MODEL - SPECIAL CASES
2.1 Determination of the dynamic buckling load of the imperfect model structure for the case 6 = c.
2.2 Determination of the dynamic buckling load of the imperfect model structure for the case 6 = f 2 .
3 DETERMINATION OF THE DYNAMIC BUCKLING LOAD OF A
SPHERICAL CAP SUBJECTED TO A SLOWLY VARYING TIME
DEPENDENT LOADING
3.1 Derivation of the equation
3.2 Solution of the problem
4 DISCUSSION OF RESULTS
REFERENCES
FIGURES
TABLES
ABSTRACT
The dynamic buckling loads of some imperfection-sensitive elastic structures subjected
to slowly varying time dependent loading are determined using perturbation procedures.
First, we consider an elastically imperfect column resting on a softening nonlinear elastic
foundation. The governing differential equation has two small parameters. We determine
the dynamic buckling load of this column subjected to the stipulated loading for three
different cases. The cases are when the small parameters are not related and when they are
related first linearly and next quadratically in some way.
This idea is next applied to an elastically imperfect spherical cap and the dynamic
buckling load of the cap subjected to a slowly varying time dependent loading is
determined. The result shows, among other things, that for the case of the cap, the
coupling term has no significant contribution to the initial post-buckling phenomenon.
By assuming, in the results, that the slowly varying loading function is numerically
unity, we obtain the associated step loading results for both the column and the spherical
cap. These latter results confirm existing results for columns under step loading and
establish new ones for the spherical cap.
INTRODUCTION
The determination of the dynamic buckling load of an imperfection-sensitive elastic
structure under various time dependent loading histories has been an area of intense study
ever since Budiansky and Hutchinson, [2, 4, 61 extended the original work of Koiter on
static theory of post buckhg behaviour of elastic structures to the case of dynamic loading.
So far many of the studies in this area have concentrated primarily on the cases where the
time dependent loading is either step loading [2, 3,4, 7, 8, 11-17], impulsive loading [4,
61, or periodic loading [7]. Besides the periodic case, there has been a dearth of analytic
studies for the cases where the loading is essentially time dependent. The exceptions are
contained in [4,6] where rectangular and triangular loadings are considered.
Existing literature on the subject shows that dynamic buckling of imperfection-sensitive
elastic materials is usually modelled by nonlinear differential equations and that geometrical
imperfections in these materials are responsible for large scale reductions in their structural
strength. Our aim is to calculate the dynamic buckling loads of these structures from the
associated dynamic differential equations under certain prescribed initial and boundary
conditions and from these results, predict to what extent these geometrical imperfections
influence the buckling strength of the structures. Many of the earlier works on dynamic
buckling of elastic structures have sought to correlate reductions in buckling strength with
assumed initial imperfections of various sizes and shapes. The primary aim in such studies
has been that the results so obtained should provide qualitative information needed for a
statistical theory of buckling.
In many analytical studies [2, 4, 6 , 7, 81, it has become the practice to relate the
dynamic buckling stre.ngth of a given imperfect structure to its static strength so as to avoid
the repetitions of solving the problems on dynamic buckling for different imperfecrion-
sensitive structures under various imperfections for each different kind of loading.
Although the static buckling theory of an imperfection-sensitive structure is well
understood the accurate theory and proper understanding as well as the universal definition
of the mechanism of dynamic buckling are yet to be fully appreciated and formulated. As a
result, there is no consensus as to what constitutes "dynamic buckling".
As in other branches of Mechanics, analytical and numerical methods are the two basic
methods of solution of problems on dynamic buckling. Notable among solutions that have
used essentially analytical method are the works of Budiansky and Hutchinson [2, 4, 61,
Amazigo and Lockhart [l 1, 13, 141, Amazigo [12], Amazigo and Frank [3] and Danielson
[8]. It must however be pointed out that because of the non-linearity that has so far
characterized the resultant dynamic differential equations involved, the analytical
determination of the dynamic buckling load may present formidable difficulty. In some
cases [2,4], the analytical determination of the dynamic buckling load is preceded by first
simplifying the governing differential equations and discretizing the already continuous
system. The danger here as pointed out by Tamura and Babcock [16], is that the discrete
models tend to display static equilibrium positions which the continuous structure did not
possess. In some cases, analytical determination of the dynamic buckling load is first
accomplished by simplifying the modelling as done by Budiansky and Hutchinson [2 ,4]
and Danielson [8] while in some other cases this is done by enacting some simplifying
assumptions on the differential equations as done by Danielson [8]:
There have been several numerical solutions for some dynamic buckling problems.
Notable is the work of Svalbonas and Kalnins [7] who developed new computer
programmes for dynamic buckling loads of shells under general time dependent loading
and them compared and evaluated existing :~nalytical and numerical solurions for a specific
problem - the dynamic stability of spherical caps subjected to uniform step loading. They
showed that for certain classes of structures, the analytic solutions based on simple
dynamic buckling model approach of Budiansky and Hutchinson [2,4] and Danielson [8]
may be appropriately modified to give accurate buckling predictions. Other notable
numerical works in this regard include Tamura and Babcock [16], Fisher and Bert [17],
and Roth and Klosner [13].
In this work, we first consider a model problem consisting of a two-bar simply
supported column subjected to an axially applied step loading and next, the same model
under a slowly varying time dependent loading Af (6t) which varies only slightly over a
natural period of oscillation of the structure. Using phase plane analysis, we determine the
dynamic buckling load of the structure under the applied step loading in the first case. The
resulting differential equations in the second case contain two small parameters f and 6
where 5 << 1, 6 << 1 and A is the load parameters. Here
(0.01) f(0) = 1
(0.02) If(Gt)l s 1, r 2 O
By using a novel generalization of Lindstedt-Poincare method and initially assuming
that Gand 5 are not related, a two timing regular perturbation expansion in G and 5 gives a
uniformly valid solution of the dynamic response (displacement) of the system. We next
find the maximum displacement, reverse the series of maximum displacement in a manner
suggested in [ I , 12,201 and finally invoke the condition for dynamic buckling to determine
the dynamic buckling load of the model structure under slowly varying time dependent
loading.
We next assume that Gand 5 are related such that
(0.03) s = p = where a is real and constant. We determine the dynamic buckling loads for a = 1 and a =
2 and show in each case that the dynamic buckling load of the model structure under step
loading can be derived from these two cases by setting f (61) = 1 in the results for a = 1
and a = 2.
Lastly, we consider an imperfect spherical cap subjected to a time dependent slowly
varying loading ilf(6r). The imperfection is assumed to be both axisymmetric and
unsymmetric. By neglecting both the prebuckling inertia and the axisymmetric
imperfection and assuming homogeneous initial conditions, we determine analytically the
dynamic buckling load of the spherical cap under the stipulated loading. By setting
f (St) = 1 in the results obtained we specialize these results to the step loading case. We
compare these step loading results with the only existing analytical (approximate) results
obtained earlier by Budiansky and Hutchinson [2,4] and show that the results obtained in
the work reported here are conservative.
CHAPTER ONE
DYNAMIC BUCKLING OF A CUBIC MODEL
A simple model which is typical of this type of structure is a two-bar simply supported
column subjected to an axial load P(T) applied at T = 0. The bars are of length L, rigid and
weightless and carrying a mass M at the centre hinge whose motion is restrained by a
nonlinear softening spring that provides a restoring force KL(x - P x ~ ) , P > 0 where x is
the central hinges displacement. The initial displacement X plays the role of an
imperfection (See Fig. 1).
1.1 Formulation of the Equation
Let Q be the tension on each of the two arms of the column and 8 be a small angular
displacement of either bar from the neutral vertical position. We note that for small angular
displacement
For equilibrium of the point of application of the dynamic load in the axial direction, we get
For motion of the mass M, we have
(1.103) MX = 2 ~ s i n 8 - KL(X -px3), d'
('1 = -0. dT
Neglecting nonlinear geometric effects (cos 8 = 1, sin 8 -- 8) and eliminating Q between
(1.102) and (1.103) using (1.101) gives
We introduce the following non-dimensional quantities
where 5 << 1, 6 << 1.
The differential equation (1.104) and associated initial conditions become
(1.1OSa) ii + (1 - Af (6t))5 - b t 3 = Acf(6r),
(1.1OSb) ~ ( 0 ) = ((0) = 0
where b = f l ~ ~ > 0.
A is the load parameter. f (6t) is a slowly varying continuous function of t with right hand
derivatives of all orders at t = 0 and satisfying
(1.106) f(O)=l, If(6t)l<-l, r 20 .
The problem (1.105) is said to be that of a cubic model because of the cubic
nonlinearity. It models many important physical structures and the method of its solution
will be used to solve the problem on a spherical cap which essentially is a quadratic
structure.
We shall first find a solution for the static case corresponding to (1.105).
1.2 Static Problem
The static problem corresponding to (1.1 05) is
(1.201) (1 -1) t - b t 3 = A t where, in this case f (St) E 1 and 4' = 0.
Equation (1.201) is a nonlinear algebraic equation and for = 0 it becomes an eigenvalue
problem with solutions
(1.202a) 5 = 0
for all A and
(1.202b) ~ = l - b ( ~
3
a = 1 is called the classical buckling load for the perfect structure. For this case 5 = 0, the
curve of A versus 5 is symmetrical in 5 about the axis with zero slope at 5 = 0, while for
5 + 0 , the static buckling of the imperfect structure is independent of the sign of 5. By
setting a = 0 in (1.201) we get 4
where A, denotes the static buckling load of the imperfect structure.
1.3 Dynamic Problem - S t e ~ load in^
We now study the problem (1.105) where the loading is step loading. In this case f(&)
= 1 and the resulting differential equation is
(1.301a) j ' + ( l - ~ ) ( - b ( ~ = ~ 5 , r > O
(1.301 b) {(o) = ((0) = 0
The solution of (1.301) is important because step loading is a special case of a slowly
varying loading. Hence, an analytical solution of (1.301) will automatically shed some
light on the solution of (1.105).
A first integral of (1.301a) gives
The maximum value, 5, of 5 is obtained when 6 = 0. Thus, for 5, t 0, we obtain
There exists a maximum value of A for which a bounded value for the displacement 5, exists and it is this maximum value that is defined as the dynamic buckling load AD. This
dl, critical' value itD satisfies = 0 and f o r il greater than itD, the response is
d~ a
monotonic and unbounded. Thus from (1.304) or (1.303) we obtain
We shall now develop a perturbation scheme for the analytic solution of (1.301). We
proceed by first obtaining a uniformly valid perturbation solution of (1.301) in terms of the
load parameter A. The amplitude of the bounded solution is then obtained and A is next
maximized with respect to this amplitude. Following Lindstedt-Poincare method [I, 51, we
introduce a time scale i defined by
(1.306) i = (1 -1)"~(1+ w2c2 + U ~ F ~ + - - ) ~
Thus, we get
Substituting (1.306) and (1.307) into (1.301) we obtain
We shall now expand 6 in a Taylor series in the parameter namely
Substituting (1.309) into (1.308) and equating like powers of 5, we get
d 5 Requiring that -;i;-(tc) = 0, implies from (2.203a) that
The substitution of (2.216) into (2.21 8) gives -3 (2.219) ~ ( - y ~ s i n r 2 c + q l ~ ~ ~ t 2 c ) + ~ [ - y 3 ~ i n 1 2 c + t l j ~ ~ ~ t 2 c
We note that every function of 72 in (2.219) is to be evaluated at 72 = z2,.
Next, we expand each function of t2c and 72c of (2.219) in a Taylor series expansion
about (t2c, rZc) = ((j2),0) and equate to zero like cofficients of 5 and obtain
(2.220a) sin ti2) = 0
This gives
(2.220b) tL2) = nz, n= l , 2 , .... We shall however take n = 1 for definiteness. Therefore we get
(2.220~) p) 0
Similarly, we obtain
where to is evaluated in a manner similar to the determination of To in (2.124g) and in fact
has the same value as To which is --"-- By expanding each function of t2 and z2 of (I -A) ' '~
(2.216) in a Taylor series about (r2,r2) = (rA2),0) and thereafter collecting terms in powers
of 5 , we obtain the maximum displacement 6, of 6 in the following way r
Further simplification of (2.221) gives
By performing an analysis similar to the one used from (2.127) to (2.134) we'see that at
buckling
where cl and c3 are now evaluated at il = AD.
From (2.223) we get
where
f ' ( 0 ) = F'(0).
The following salient points are to be noted.
(a) The dynamic buckling load for the case 6 = f can be calculated from (2.224) 2
(b) The load degradation is of order 7of the imperfection parameter 5.
(c) The dynamic buckling load in this case depends on f '(0).
(d) For f '(0) a 0 the result (2.224) gives the step loading result which is
(2.225) ( 1 ~ ~ ) ~ ~ - 36 D I 51 2
(e) We conclude that, in general, the dynamic buckling load of the imperfect cubic
structure depends on 6 and c. If 6 and 5 are related linearly, then the dy narnic
buckling load is evaluated from (2.134) whereas if there is a quadratic relationship
33
between them in the form 6 = F 2 , the dynamic buckling load is evaluated from
(2.224). These two results, are not equal unless in a situation of step loading.
(f) The results (2.224) is asymptotically equal to that obtained from (1.451) if in the
latter we replace 6 by F 2 .
CHAPTER THREE
DETERMINATION OF THE DYNAMIC BUCKLING LOAD OF A SPHERICAL CAP
SUBJECTED TO A SLOWLY VARYING TIME DEPENDENT LOADING
fl Derivation of Equations
We shall consider a shallow section So of a spherical cap and take Cartesian coordinates
x and y in the base of the plane and z coordinate normal to this plane. We shall give the
membrane strains E,, E>. and E~ in terms of the tangential displacement U, V and the
normal displacement as W which acts in the radial direction. We shall similarly let K,, K,
and K~ be the components of curvature in the indicated directions while N,, Ny and Nxl
represent the stress components. The couple is represented in its component form as
M,, My and Mq. We emphasize that the suffices appearing in E, K, N and M above are
not to be interpreted as indicating differentiation. In static situation, all the above space
variables depend only on the spatial variables x and y while in the dynamic case, time is an
additional independent variable. The strains and curvature are given in terms of the
variables U, V and W as
&Y
(3.101) 1::
where expression such as W, indicates partial differentiation of W with respect to a.
Similarly, the stress-strain relationship is given by
(3.102)
where
E is Young's modulus and v and h are the Poisson's ratio and the shell thickness
respectively. As before, surfices following N, M, E and K in (3.102) do not indicate
differentiation.
For the classical static buckling analysis, we consider the case in which the strains,
stress and displacement are time independent and in which the spherical cap is considered
perfect. We shall let P be the time independent lateral pressure acting on the spherical cap .
Using calculus of variation and the principle of virtual work, we can derive the three
differential equations of equilibrium in U , V and W. If however, we introduce Airy's srress
function F ( x , y ) for stress resultants which gives
(3.103) Nx = Fyy, Ny = Fxx, Nxy = -Fxy
and finally take the variational equation given by
(3.104) j j [ ~ ~ 6 ~ ~ + My6rcy + 2 M , 6 ~ ~ + NX6zx + N y 6 ~ y + 2 ~ ~ 6 ~ ~ ~ ] y l d ~
= external virtual work,
we obtain the following equilibrium equation
and the compatibility equation 1 1
(3.106) - V ~ F --V'W + W,Wyy -(w,,)' = 0 Eh R
where V4 and v2 are the two-dimensional bihlumonic and Laplacian operators
respectively. Before static buckling sets in the perfect shell is in a uniform membrane
stress state where
where R is the radius of curvature of the shell. For subsequent state, we take
where f and w are zero prior to buckling. The linear buckling equations are obtained by
substituting (3.108) into (3.105) and (3.106) and linearising with respect to f and w to get
(3.109) 1 1 D V ~ W + - V ' ~ + - P R V ~ W = O R 2
We obtain periodic solutions of these homogeneous eigenvalue equations by taking
Substituting (3.1 11) into (3.109) and (3.1 10) gives
(3.112a) B = - ~ h ( ~ , ~ + gy2)-1
(3.1 12b) 2Eh
P =-[(gx2 R +gy2)-I + ~ ~ ( g ~ ~ + gy2)]
where '
and suffices in g, and gy do not indicate differentiation. To find the critical buckling load
PC, we minimise P with respect to g, and gy and obtain
(3.1 13) g? + gy2 = e2 as the condition for the attainment of the critical load PC. For the condition (3.11 3), the
critical pressure PC becomes
The dynamic equi!ibrium equations for the case of an imperfect spherical cap subjected
to step loading was similarly obtained by Danielson [8]. His work was in turn based on an
earlier work on static buckling of an imperfect spherical cap by Hutchinson 191. In this
case, the stress-strain relation and displacement now depend on both the spatial variables x
and y as well as on time t . The displacement of any point on the spherical cap was taken by
where Wo is the prebuckling radially symmetric mode, W, is the axisymmetric buckling
mode and W2 is an arbitrary non-axisymmetric buckling mode. The imperfection was
taken (by Danielson) in the form
(3.1 16) W = @$ + F2w2. In a manner similar to the derivation in the static case [7,8,9], the following multiple mode
dynamic equilibrium equations were derived for the step loading case, where t > 0
where A is the "amplitude" of the applied step load and K~ and K2 are scalars. It is to be
noted that 5; << 1, z2 << 1 and o O , w, and w, are the circular frequencies of the
prebuckling and axisymmetric modes and the circular frequency of the non-axisymmetric
mode respectively all of whose numerical values are embodied in the derivation of (3.1 17)
[BI .
3.2 Solution of the Problem -
For our problem we shall determine the dynamic buckling load of an imperfect
spherical cap subjected to a slowly varying time-dependent loading, where the time
dependent loading is taken in the form Af (St) and f (0) = 1, 1 f (&)I < 1, t > 0, S c< 1.
The modification of (3.1 17) to this new problem gives
An analytical solution of (3.201) will now be sought for the case in which the prebuckling
wme number wo is infinitely large. This would also be the case if we set the prebuckling
inertia term to be iero i.e.
This then means that
(3.203) to( t> = Af (61)
Hence,'from (3.201), we get
In (3.204) we have already asserted that f = f (6r). For our solution we shall set 5, = 0.
This choice is informed by earlier works [7, 81, that lower values of dynamic buckling
loads (for step loading case) are oblained if is numerically set equal to zero. We expect
this to be true for our case since step loading is a particular case of slowly varying loading.
The analysis to be presented here will follow the following procedures.
Calculating uniformly valid expressions for the displacements of the two buckling
modes t1 and t2 since that of to( t ) has already been known as in (3.203),
finding maximum displacement for each of these modes and hence evaluating the
effective maximum displacement for thz whole spherical cap,
reversing the series for effective maximum displacement,
determining the dynamic buckling load and
making pertinent deductions from the rzsult obtained in (d).
We shall let
In the analysis to follow, we shall l e ~ I I'.
We now generalize the Lindstedt-Poixare method by defining i by
Then we have
where n = 1,2 and henceforth, partial differentiation with respect to t and z such as
6,- and tnr, shall not be indicated with the use of a comma.
Thus we get
We substitute (3.207) into (3.204a) and get
where we have used the fact that 5 - 0.
By substituting (3.207) into (3.204b) for the case n = 2, we can derive an equation similar
to (3.208a) for t2. We shall let 6, and t2 be expanded in the following double series.
where aik = a"(;, z) and bir = bir(;, T) and the superscripts appearing in aik and bjr are
understood not to be powers. By substituting (3.209) into (3.208) and equating both - i k coefficients of 5 6 and F i g r (at the respective appropriate instances), we have the
following sequence of equations:
(3.210b) ~ a ~ ' = -2 ~ ~ ( 1 - Af ) - 112 20 - 1 A ) a;,
etc. The initial conditions for are
(3.2 1 1 a) a2k(0,0) = 0 for all k .
(3.21 1b) aY(0,o) = 0
etc.
The initial conditions for bir are
(3.213a) bjr (0,0) = 0 for all j, r,
(3.213b) bi O (0, 0) = 0,
(3.213~) bi '(o,o) + (1 - A)-112 b:' (0, 0) = 0,
0 2
We have not included ~a~~ and Lb2' because calculation shows that a3k and b2' ( k , r
= 0, 1, 2, ...) are numerically zero.
We first solve (3.212a) and obtain
t 3.214a) b1'(?,r) = alO(r)cosi +p,,(r)sini + B ( r )
'The use of initial condition (3.213) on (3.214a) gives
t 3.214~) alo(0) = -B(O), PI&> = 0.
We now substitute (3.214a) into (3.212b) and obtain
~ b " = - 2(1- (3.2 15) (- aio sin i + &'o cos i)
0 2
+ y'(1 - af )"I2 (-alo sin i +fro cosi)
202
To ensure uniformly valid solution for b1 in respect of i , we equate to zero the
coefficients of cost and sin i in (3.125). This gives respectively
The solutions of (3.216), bearing in mind (3.214~) are
(3.217b) Plo(z) e 0.
Thus, we obtain from (3.215)
(3.2 18a) bl'(i,z) = all(z)cosi +a1(z)sini .
The use of initial condition (3.213) on (3.218a) gives
Next, we substitute (3.2 l4a), (3.21 8a) into (3,.212c) bearing in mind (3.217a) and
(3.217b) and get
(3.2 19) 2(1- A , ) - ] / ~ (-ail sin i + pil sin i)
0 2
To ensure uniformly valid solution for bI2 in terms of i , we equate to zero the
coefficients of cosi and sin; in (3.219) and obtain respectively
and
The solutions of (3.220), bearing in mind (3.218b) are
and
(3.221 b) a,,(O) EO.
We now solve for bI2 in (3.219) and obtain
(3.222) (1 - ~f )-I B"
bI2(i,r) = a12(r)cosi +PI2(r)sini - 4 On using (3.213), we obtain from (3.222)
So far. we obtain
(3.2243) blO(i, 5) = a l 0 ( ~ ) c o s i + B(5)
(3.224b) b 1 ' ( ~ , ~ ) = / l l I ( ~ ) s i n f
Further analysis gives
(3.224~) b12(i, 5) = a12(r)cosi - (1 - 2f )-I B"
4 We note that, using (3.224a),
Substituting (3.225) into (3.210a), we obtain
Thus, we solve for a20 in (3.226) and obtain
. (3.227a) a20 (i. 7) = ( r ) c o s ( ~ ) i + ty(r)sin(%)i 6'2
r
The use of initial conditions (3.21 1) on (3.227a) gives ,-
We note the following, using (3.224a,b) 1 0 1 1 1 (3.228) b b = - aloP1 sin 2i + BPl sin i.
2
Thus, substituting (3.228) into (3.210b), we obtain
(3.229) ~ a ~ ' = - - ~f [(: 1 - - ei0 s in(z ) i + ty;o co s (5 ) i ) a 2 w2
#
-[(l-~f)-la?o] sin2i 2Ba1,(l-)if)-1sini 2
- 2 (2) - 4 (2) - I
To ensure a uniformly valid solution for a2' in terms of i , we equate to zero, the
coefficients of cos($)i and sin(%);. This gives
and
(3.230b) a f y i - af)-312
202
Since (2) + 0 , we solve (3.230) bearing (3.227) in mind and obtain
We now solve for a2 ' ( i , T ) in (3.229) and get
(3.232) n2'(i,r)=821(r)~~~(~)i+y21(r)sin(~)i r
2
- ~2 (2) ( 1 - af )"I2 2Balo sin
202
Using the initial condition (3.21 I), we obtain
(3.233a) (0) = 0
where (3.233b) is evaluated at z= 0. A detailed simplification of (3.233b) gives r
We note that, using (3.224), the following simplification holds:
We now substitute (3.224b), (3.234), (3.232) and (3.227a) into (3.210~) and get (using
a& cos 2i + 2Ba10 cosi
4 {(%i'-I} I To ensure uniformly valid solution for a", we equate to zero, the coefficients of
cos($)i and sin(?); in (3.235) and obtain
The solutions of (3.236), using (3.233) are 114
(3.237a) ylzl(7) = y1,l
(3.237b) 92, (z) 0
22 - We do not need the explicit determination of a ( t ,z) in this work. So far, we h~ivt:
obtained the following
+- I a', cos2i + 2B:10 cosi 2 2 ) - 4 3
r 1
2 B q o sin i +
I 2Wa1o(l- ~f
aloj311 sin 2; BPll s in i 2{($r - + {(g -
+
K ( 1 - A - 1 alo 2 sin 2i
As indicated before, a3j ( i , r ) r 0 V j. We now note the following multiplication:
' - 1 1 2 202 i{(?r -4y
Substituting (3.240) and (3.212d) in (3.212d), w e get 2
(3.141) ~ b ~ ~ = - ( I - ~ f ) - " ~ ~ j a , ~ c o s ~ - ( l - ~ f ) - ~ 6'2
2 +cos(i - W. . + B&, cos (2 ) i - {(a) aIo q ( 1 - ?J)-'
cost
- 3 a&8(1- Af )-' cos 2 i 3{2jZ~2{($)2 }
To ensure a uniformly valid solution for b30 in terms of i , we equate to zero the
coefficients of cosi in (3.241) and get
Thus, we get r
We now solve for b30 in (3.241) and get
The use of initial condition (3.213) on (3.244) gives -
We note that the expression on the right hand side of (3.245a) is to be evaluated at z = 0.
A further simplification of (3.245a) yields
We now note the following multiplications
[("+ 2 B ~ ) ~ ~ .in; + a&pl (sin 3i - sin i ) B& lalo sin 2i 4 { Q 2 - 41
+ w, 2 ( ) -1 I
2 + sin i) B P I , sin 2i - 2 ) a lo t l - .~f +
aloj31 sin 2; BPI, sin i . 2 J. 2 - 4 + 1 - I}
We now substitute (3.224a,b), (3.244), (3.246a) and (3.246b) into (3.212e), usins
(3.246) and obtain 2 2
(3.247) ~ b ~ ' = -(l - ~f )-1'2p;~l sini +=(I - ~f ~ f ) ' ' ~ j a ; ~ sin i 0 2 6'2
W - W 2 W, 2 ((1 - ~f I-' ~ e ~ ~ ) ' + n ( k ) t ((-) ~ ~ a : ~ ( l - . z . ) - * ) ~ ( ( ~ ) - 3) +
W, 2 + O2 sin 2t
1 - (& W , 2 0, 2 { - { - 4)
W 2 3($) K2 {a:o (I - Af)-2)'sin 3i + - a 3 ~ sin i 32{(2)' - 4)
+p30 cos i + (2 - 3
a:o(sin 3; + sin i) Bat, sin 2; 2 +
6) 2 {(a) - 4) {Q2 - 1)
{[*+B2hl1sini+ a:op, , (sin 3 i - sin i) + BP] 0, ,alo 2 sin 2; ( 1 -1
To ensure a uniformly valid solution for b3' in terms of i , we equate to zero the
coefficients of cosi and sin ; in (3.247) and obtain
The solution of (3.248a), using (3.245b) is
(3.249a) P30(~) = O .
We now solve for b31 from (3.247) and get
01 2 0 2 - ( ) 2 1 - ) 2 ) ' { ( ) - 3) sin 2i
W, 2 6'1 2 3{(q) - } { - 4)
aloPl, sin 3; B& sin 2; + 32{( 2)2 - 2) 6{(?)2 - 1)
Using the initial conditior. ('3.21 3), we get from (3.250)
(3.25 1 ) a3,(@. = 0
60
We shall now expand in a Taylor series about (id1),0;5;,6). Carrying this out. we
where each aV in (3.255a) is to be evaluated at (iil), 7:)) = (iA1),O).
-(a (2) In a similar way, we expand e2. in a Taylor series about (to , z, ) = (d2),0) and gel
( 1 ) where n2j , j = 1,2 and their partid derivatives are evaluated at ( t , , ra ) = (;A1),0).
Solving (3.257a) using (3.238), we get - -
I afo sin 2ii1) 2 a l o sin iil) (3.258a) (2) rc2 (1 - Af 1-l @, +
0 2 - o~~ (0) sin(2)ih1) = 0. - 4 - 1
Similarly, from (3.257b) and using (3.238) and @.239), we get
(3.258b) + + (1 - 1)-"2 a:0]
The expression (3.258) so far being evaluated at ( i i l ) ,r i l ) ) = ($),o). We note the
following simplifications
( 3 . 2 5 8 ~ ) a ~ ~ ( i h l ) , ~ ) = - ( % ) ~ ~ o ( ~ ) s i n ( ~ ) i ! l ) wz 0,
-
Further simplification gives the following:
where
and
The use of (3.258) will enable fit) to be determined for specific values of ;A1) found from
(3.258a). We similarly determine a2' (fil),O) and obtain r r
We shall however represent u ~ ~ ( $ ' ) , o ) simply as
where A1(&,fd1)) and A2(3,id1)) are defined in an obvious manner from (3.258j). The 0 2 0 2
evaluation of ty2' (0) gives r 1
We next evaluate U ~ ~ ( ; ~ ( ' ) , O ) and get
3 f cos 24') 2 cos id1) -{-+ 0 1 2 - 2 ( ) - 4 ( 3 ) 2 - 1
6'2 0 2
Thus, following (3.255a) and (3.258), we obtain cl, as:
In deriving (3.259), we have used the fact, following (3.257a) that ay(;f),0)= 0 .
We are yet to determine t i 1 ) in (3.259).
We next expand (3.256b) in a Taylor series about (i,(2),e))= (fh2),0) bearing
(3.255b) in mind and equate to zero coefficients of 5d6' (j = I , 2 ,...; r = 0, I , 2 ,... ).
From the coefficient of (c2,1), we get
(3.260a) 10 (2) 0) = 0 b; (fo 9
This means
(31260b) -al0(0)sin id2) = 0, ;A2) = nn, (n = O,1,2, . . .). We shall however take n = 1. Hence we get ;A2' = n.
By equating the coefficient of 58, we get
(3 .'260c) '0 -(2)b10 ii + 0 p)b!0 17 + b! I 1 - (1 - A )- 112 br 10 = 0
where (3.260~) is to be evaluated at (iL2), rL2)) = (n,O). This gives
If we equate respectively the coefficients of ( g . 1) and ( C S ) , we get
(3.260e) -(2) = -(a = 0 '10 '11
By equaring the coefficient of (::,I), we get
With (3.260) in mind, we now rewrite (3.255b) with only the non-vanishing terms thus 3 30 (3.26 1 ) = C2[b1O + bi0tA2)6 + . - a ] + t 2 [ b + 6(b:'t$;) + b ~ ~ f ~ ~ ) ? ~ ~ )
where bjr and partial derivatives are evaluated at (iA2),0) = (z,O) in (3.261). We shall
now evaluate each of the terms appearing in (3.261). Thus from (3.249~) we get
If we substitute into (3.262a) we get the following lengthy result.
We also evaluate the following at zo = 0:
Thus, we have
If we substitute for a;o(0) from (3.262j) we get
where
We now evaluate from (3.250) and get after simplification
where
From (3.244) we get
Thus, using (3.260d), we now evaluate b:0(ii2),0)?$2) and get:
where ~ ~ ( 3 ) represents the terms inside the square bracket of (3.266e). Using (3.218b) '"2
and (3.2600, we evaluate b:' (iA2).0)$i) thus
(3.266e) by (?h2.0)?$) = K 3 { - 2
4u2(1 - -1 * - 4 ( ) 0, 2 -1 +I}. 2 '"a
Similarly we evaluate
where
From (3.244), we evaluate b30(fo(2),0) and obtain
where -
( 2 ) We shall now evaluate t f ) , t i 2 ) and t20 .
From (3.206a), we get (4
1.
(3 .267) ca) = w J (I - 2f (6ts))ll2 dts
Using (3.252), we get
112 ( 1 ) 112 ( 2 ) 112 ( a ) (3.268a) f 1 = 1 -A) to , ih2) = a2(1 - 2 ) to , iit) = u 2 ( l - 2 ) t20
Using (3.206b) on (3.268a) we get
- 1 = - 1 = 1 112 ( 1 ) '0 '0 2 ( - 2 ) to .
' Therefore
Similarly, we obtain
Using (3.268a), (3.206b), (3.243) and (3 .260f) , we get
112 ( 2 ) g = a ( I - A ) t20 + pi (0 )
from which we evaluate
where
dynamic stability of the structure at this stage. This is essentially the main
conmbution from our first level of approximation (3.281a). Thus, strictly speaking,
as far as the initial post buckling is concerned, the coupling term plays no part.
This means that at this stage the coupling term can be neglected in the problem
compared to ~ ~ 5 : This is an important result that has never been obtained before.
(vii) (3.288) is certainly a "refinement" of (3.284) and clearly shows the contribution of
the coupling term 5,<2 on later post buckling phenomenon.
(viii) As the results (3.284) and (3.288) show, it is possible in this problem to relate the
dynamic buckling load AD under slowly varying loading to the static buckling load
As and thus by-pass the labour of repeating the calculation for various imperfection
parameters. This is done by noting that the static buckling load As of the spherical
shell (with Kl<? neglected) is given by
(4.01) 3
(1 - 2,)' = 2&JF& If we eliminate I&/ from (3.284) (a similar procedure can be done for (3.288)) we get
I + fif'(0) m. -(I) (I) A3(-,fO ,to .
4(l- A D ) @2
-(1) (1) ( t o t o ) is evaluated at a = I D .
If we set f'(0) = 0 in (3.284), (3.288) and (4.02), we get the following results
corresponding to the two levels of approximation in the step loading case:
sin sn
Evaluating b:Ot$g) at (fi2),0) = (z,O), we get
Let
(3.2680 G9 (s) = AGl (s) + G2 (s)
Then
We shall further let
(3.268h) sin sn.
Then, we have by(fd2),0)f$) as
We now assemble all the individual terms in (3.261) (which have been evaluated) anti
determine the maximum displacement c2a for r2. The result is
+(I - +)G* + 2 ~ , - 6, + c3} + 0(2j2)]
where we have, for simplicity, deleted the dependence of G: on 5. We note that, from 0 1
(3.259), the corresponding maximum displacement for 5, can be rewritten as
(3.270)
where
(3.27 1)
K 2 ( 3 2 ~ 2 n [{ - 2 i ( l )
5'; = - ~ ) 3 w 2 "I 2 +- cos-
0, } O (A) - 4 (y) -1 2
On account of (3.203), we see that the maximum displacement for the prebuckling
mode gc! is
(3.272) jOa = A
Now, following (3.1 15), the effective maximum displacement for the spherical shell is 5,
and is given t.1
Using (3.2691. (3.270) and (3.272), we get
+(I- 4)G8 + 2G7 - 7rG9 + G3}].
It is to be emphasized that for any allowable choice of 2, iil) must necessarily satisfy
We shall now determine the dynamic buckling load of the imperfect spherical cap
subjected to a slowly varying time dependent loading. We shall however give the results in
two separate levels of approximation. It is anticipated that each of these results will
automatically establish some far reaching consequencies.
We now write the following
Following (3.274), we recast (3.273b) in the form
We shall for simplicity write (3.275) as -
(3.276) 6 , = 2 + d 1 t 2 +d2c22+d3c2+...
On account of (3.203), we see that the maximum displacement eOa for the prebuckling
mode is
(3.272) 50, = 2
Now, followi~g (3.115), the effective maximum displacement for the spherical shell is 6 ,
Using (3.2691, (3.270) and (3.272), we get
(3.273b) C
I-A 8(1-
74
where dl , d2 and d3 are the coefficients of c2, 5: and 5: in (3.275). The series (3.276) is
similar to (2.127) of the model problem and so, its reversal automatically follows the same
procedure adopted from (2.128) to (2.135) of the preceding model problem analysis.
Reversing the series (3.276), we write
(3.277a) c2 =sl( + s 2 t 2 +s3t3 +- - -
where we have set
(3.277b) 6=5,-a and s1, s2 and .Q are coefficients to be determined. By finding f2, c: and using
(3.277a) and substituting into (3.276) and thereafter equating coefficients of 5, we get
As in the analysis leading to (2.130), we expect that at the initiation of dynamic buckling,
there is a a = AD such that $(AD) = 0. This implies
(3.279) s1 +25& +3s3t2 = O
where the'coefficients of si (i = 1,2,3) are evaluated at A = AD. Thus, we have
where the appropriate sign will be taken. For our first level of approximation, we notice
that a dynamic buckling procedure is established if we limit (3.276) to
(3.28 1 a) ( = d l c + d2f22 + In this case, the reversed series is
. (3.281b) & = s ~ { + s ~ ~ ~ + - .
where, in fact, dl and d2 still have the same value as in (3.278). The invocation of the
dynamic buckling criterion and subsequent simplification gives
where and are evaluated at 1 = AD. On evaluating (3.282), we get
(3.283) (1 - = ~ A D I K ~ ~ ( ~ ) 0, 2 11 + + 6f'(0))~1)I.
Further simplification gives
valid for Wl W1 1<-c2pc->2. 0 2 a 2
So far, (3.284) gives the result of the fust level of approximation. We shall now determine
the result of the second level of approximation. In this case we use (3.277a) - (3.280).
Using an analysis similar to the derivation of (2.133a) - (2.1 33c), we get
where the right hand side of (3.285) is evaluated at 2 = AD which is the dynamic buckling
load sought. We shall now evaluate each of the terms in (3.285) and (3.280). Following
(3.275) and (3.276), we get
(3.286a) 2 1
dl = -(I + 6f '(0)Rll) I-a
Thus, we have
Thus, we have
The evaluation of 6 , from (3.280) gives
By substituting for 6 , in (3.285) and simplifying thereafter, we obtain the following
lengthy expression
where
0 1 0 1 Equation 3.288 is valid for 1 < - < 2 or - > 2. All functions of A in (3.288) are 0 2 0 2
evaluated at AD and we have taken the negative square root sign where root occurs, this
choice being informed by the same reason leading to the determination of (2.134).
CHAPTER FOUR
DISCUSSION OF RESULTS
The results (3.284) and (3.288) give the expressions for determining the dynamic
buckling load AD of the spherical cap. In either case, we see that the load
degradation is of order 5 of the imperfection parameter 5;. The dynamic buckling load AD depends, among other things, on the ratios of the
circular frequencies of the buckling nodes and t2 except hat 2 # 1.2.
The two results (3.284) and (3.288) hold with no additional restriction on the slowly
varying time dependent loading function f (6t).
We observe that to order F36 in the effective maximum displacement (such as in
(3.273b), and generally to any nonlinear order in f2 (but linear in 4 in which
the effective maximum displacement of the spherical cap is determined, the
corresponding dynamic buckling load depends, as far as f (at) is concerned, on
f'(0). If however the determination of the maximum displacement is g i ~ e n to an
accuracy that is nonlinear in 6, the dynamic buckling load, in this case, will definitely
depend on f '(0) and on higher derivatives of f (7) evaluated at 2 = 0.
Up to the order c36 in the maximum displacement, there is no tacit dependence of
the two results (3.284) and (3.288) on the quadratic term ~ ~ 5 : and this confirms
Koiter's assertion [7, 81, that the term K~<: can be neglected compared to the effect
of coupling between the two nodes as far as initial post buckling behaviour is 1
concerned. I
I
The result (3.284) shows that to order c26 in the determination of the effective i maximum displacement, there is no contribution of the coupling term 5,:- . - on the i
The results (4.03) - (4.05) are novel derivations with possible applicability in
Engineering. They have long been sought for but never obtained analytically. The
only known analytical attempt is that in a similar work by Budiansky and Hutchinson
[2,4] in which they sought the dynamic buckling load of an imperfect cylindrical
shell subjected to step loading. The governing differential equation in this ciise is
equivalent neglecting the quadratic term ~ ~ e : , setting F, = 0 in (3.204) and setting
f(&) E 1. Budiansky and Hutchinson obtained approximate solutions by neglecting d'5 the inertia term --& in (3.204e). The approximate solutions are
from which it follows that
In what follows, we shall undertake a case study of (4.02) - (4.07) for 2 = ,I
112 (integer), v = 0.3 and k2 = &[3(1- v2)] . Table I summarizes the salient points.
From the Table 1, we make the following deductions:
In general, the dynamic buckling loads calculated from (4.06) give higher values
compared to the ones calculated from (4.03) and the disparity becomes greater with 0 increasing 1 . 0 1
The dynamic buckling load decreases with increased 2 and c2. The approximate values of dynamic buckling load AD = A I D , say resulting from
(4.06) can be several times higher than the values AD = A20, say, from (4.03) W 0 particularly when is very high. A I D is only comparable with j / 2 D when 2 is W2 %
exceedingly s~nall. -
For all values of t2 and 2 tested, the expression 6G5 . - 5 ~ ; b in (4.04) was "2
negative. We thus conclude that the coupling term 5,c2 does not necessarily lead to
buckling. Thus the structure still maintains its load carrying capability when the
influence of the coupling term t 1 j 2 is called to play. Thus, it is only the quadridtic
term that dominates initial post buckling phenomenon; does not. This result is
also true of (3.288).
REFERENCES
J.C. Amazigo, "Buckling of stochastically imperfect columns on nonlinear elastic
foundations", Quart. Appl. Math (1971), pp. 403-409.
B. Budiansky and J.W. Hutchinson, "Dynamic buckling of imperfection sensitive
structures", Proceedings of XIth Inter. Conm. of Appl. Mech., Springer Verlag, Berlin
(1966).
J.C. Amazigo and D. Frank, "Dynamic buckling of imperfect column on nonlinear
foundations", Ouart. Appl. Math, Vo1.31, No. 1 (1973), pp. 1-9.
B. Budiansky, Dynamic buckling of elastic structures: Criteria and estimates. In
Dvnamic Stabilitv of Structures, Pergamon, New York (1966).
J.D. Cole, Perturbation Methods in Applied Mathemati~, Blaisdell Waltham 1966.
J.W. Hutchinson and B. Budiansky, "Dynamic buckling estimates", A.I.A.A. J. Vol.
4, No. 3 (1966) pp. 525-530.
V. Svalbonas and A. Kalnins, "Dynamic buckling of shells: evaluation of various
methods", Nuclear Enpineering and Design, Vol. 44 (1977), pp. 331-356.
D. Danielson, "Dynamic buckling loads of imperfection-sensitive structures from
perturbation procedures", A.I.A.A. J. Vol. 7, No. 8 (1969), pp. 1506-1510.
J.W. Hutchinson, "Imperfection sensitivity of externally pressurized spherical shells",
J. Appl. Mech. (March 1967), pp. 49-55.
10 B. Budiansky and J.C. Amazigo, "Initial post-buckling behaviour of cylindrical shells
under external pressure", J. of Math. and Phvscs, Vol. 47 No. 43 (1968),. pp. 223-
235.
8 3
1 1 D. Lockhart and J.C. Amazigo, "Dynamic buckling of externally pressurized imperfect
cylindrical shells", J. Appl. Mech,, Vol. 42 (1973), pp. 316-320.
12 J.C. Amazigo, "Dynamic buckling of structures with random imperfections". In
Stochastic ~roblems in Mechanics, Ed. H. Leiphelz, University of Waterloo Press
(1974), pp. 243-254.
13 R.S. Roth and J.M. Klosner, "Nonlinear response of cylindrical shells subjected to