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Bifurcation and snap-through phenomena in asymmetric dynamic analysis of shallow spherical shells
Computers& Stnrcturrs,Vol. 6, pp. 241-251. PergamonPress 1976. Prided in Chat Britain
BIFURCATION AND SNAP-THROUGH PHENOMENA IN ASYMMETRIC DYNAMIC ANALYSIS OF SHALLOW SPHERICAL SHELLS? NURI AKKASS Department of Civil Engineering, Middle East Technical University, Ankara, Turkey (Receiued
6 hoember
1975)
Abstract-The asymmetric dynamic behavior of clamped shallow spherical shells under a uniform step pressure of infinite duration is investigated. The solution of a linear eigenvalue problem yields the bifurcation paths and also the lower bound for the asymmetric dynamic snap-through buckling pressure. The asymmetric dynamic response of shells with a shape imperfection is studied. The asymmetric dynamic snap-through buckling load is defined to be the threshold value of the step pressure at which the asymmetric response shows significant growth rate. The snap-through buckling loads are obtained for a few shell parameters. The numerical results are compared with the available experimental results and they are in good agreement. Finally, a preliminary study of the phase planes is presented.
1. INTRODUCTION Buckling analysis of received considerable well-knawn fact that applied loads usually
When the loading is static, one can distinguish between two types of buckling: (a) Axisymmetric snap-through buckling results when the static response is constrained to be symmetric. It corresponds to the limit point of the axisymmetric fundamental equilibrium path which is represented by point A in Fig. 1. (b) Asymmetric buckling corresponds to the bifurcation point of the axisymmetric fundamental path (point B in Fig. 1). At this point the shell bifurcates into an asymmetrical deformation mode. Using the terminology of Ahnroth, Meller and Brogan[21, depending upon whether the equilibrium on the bifurcated path is stable or unstable, the asymmetric buckling may be called gentle or violent, respectively. When one talks about the behavior of a statically loaded shell, he talks about an axisymmetric fundamental equilibrium path, a bifurcation point and a bifurcated path. The reason for underlining the words “path” and “point” will be clarified in the following section.
shells under dynamic loads has attention in the literature. It is a shells subjected to dynamically
buckle at load levels which are smaller than the corresponding quasi-static buckling load. The reduction in dynamic buckling load from the corresponding static buckling load depends upon the duration of the dynamic loading. The studies show that the most severe type of dynamic loading is the step load of infinite duration. The purpose of the present investigation is to study the bifurcation and snap-through phenomena in the asymmetric dynamic analysis of clamped shallow spherical shells subjected to a uniform step pressure of infinite duration. It is realized that actual structures are more complicated than spherical caps, but to study the behavior of such simple structures is not completely useless. Investigations on the behavior of simple structures under dynamic loading form the basis of understanding the behavior of complicated structures under the same type of loading. The problem of asymmetric dynamic buckling of shells is so complicated that the criteria for buckling are not even reasonably well understood. It is hoped that the present investigation on this simple model will shed new light on the problem. At this point, it is necessary to summarize some general concepts of buckling of shells under static loading; because, later in the paper, some analogies between the static and dynamic buckling phenomena will be drawn. Buckling under static loads Buckling of shells under static loads has attracted much attention in the past few decades, and an extensive literature has been developed. ‘One is referred to the excellent survey paper of Hutchins& and Koiter[ll for a discussion of these studies. Here, it will suffice to summarize the general concepts of static buckling of shells. tpresented at the Second National Symposium on Computerized Structural Analysis and Design at the School of Engineering and Applied Science, George Washington University, Washington, DC., 29-31 March 1576. *Assistant Professor.
Buckling under dynamic loads As in the case of static loading, buckling of shells under dynamic loads has also received much attention. Here, it will not be attempted to present a reasonable survey of the investigations made on this subject; however, some of them will be referred to in the paper, as the need arises. To study the behavior of dynamically loaded shells, one must include time in the equations as an additional parameter and this complicates matters considerably. However, when the loading is dynamic (step loading, for instance) one would still expect to distinguish between two types of buckling: (a) Axisymmetric snap-through buckling results when the dynamic response is constrained to be symmetric. In contrast to the statically loaded shell, the behavior of which can be depicted on a two-dimensipnal sketch as shown in Fig. 1, the deflection versus load versus time behavior of the dynamically loaded shell can be depicted only on a three-dimensional sketch. Therefore, the axisymmetric behavior of a dynamically loaded shell can be described by an axisymmetric dynamic fundamental equilibrium surface. The projection of this surface onto
241
NURIAKKAS r
Bifurcation
point
/
,--! Limit point
A/-
B
Axisymmetric
Equlibrium
fundamental
path
static
path
: E : ii ! LOCUS of dynamic peak deflection
DEFLECTION
Fig. 1. Axisymmetricstaticfundamentalequilibriumpath. the deflection versus time plane is shown in Fig. 2 for different values of the load. Considering a step load of infinite duration, it is seen in Fig. 2 that, for load levels less than a critical load, the shell oscillates about its static equilibrium position with relatively small amplitudes. However, when the load exceeds the critical level, the oscillations of the shell become very large and the axisymmetric dynamic snap-through buckling is said to have occurred. To determine the axisymmetric dynamic snapthrough buckling load, the peak in time history of a characteristic deflection (the volume of the shell deformation or the apex deflection) is plotted against the amplitude of the applied load. The load which corresponds to a sudden large increase in the characteristic deflection is said to be the axisymmetric dynamic snap-through buckling load. This is illustrated in Fig. 3 where it is also seen that the axisymmetric dynamic snap-through buckling occurs when the locus of the peak characteristic deflection intersects the unstable branch of the axisymmetric static load deflection curve. This definition of the axisymmetric dynamic snap-through buckling load has first been suggested by Budiansky and Roth[3] and used by other investigators[4-131 for the dynamic buckling analysis of spherical shells. Most of the numerical results available on axisymmetric dynamic snap-through buckling loads of sphericalcaps under a uniform step pressure are now in reasonable agreement. It may be said that the criterion
Projection
Fig. 3. Definition of axisymmetric dynamic snap-through buckling load.
suggested by Budiansky and Roth[3] has generally been accepted. (b) Asymmetric dynamic buckling results when the dynamic response is not constrained to be symmetric. Although there are very few investigations made on the asymmetric dynamic analysis of spherical caps under a uniform step pressure of infinite duration, the investigations on the asymmetric dynamic instability of cylindricals shells are more extensive. The equations of motion for an impulsively loaded cylindrical shell assuming small asymmetric (flexural) amplitudes are given by Goodier and McIvor [14] as follows: iiotao=o,
ii. + F(ao)a, = 0,
(1) (2)
in which a,, is the Fourier coefficient of the symmetric (radial) mode and a,, is the Fourier coefficient of the n th asymmetric mode. Dots indicate differentiation with re-
of
Fig. 2. Projections of axisymmetric dynamic fundamental equilibriumsurface and bifurcation paths.
243
Asymmetricdynamicanalysisof shallowsphericalshells spect to time. Lindberg[lS], Anderson and Lindberg[l6] and Cromer and Ball[17] use the same type of equations in their analysis of the same shell. Goodier and McIvor [ 141reduce eqn (2) to the classical Mathieu equation and study the stability of the so-called Mathieu modes throu8h the use of the Mathieu stability chart. The same type of problem was solved by McIvor and Sonsteg~d[l8] for a closed spherical shell. On the other hand, Lindberg [ 151and Anderson and Lindberg [ 161study the so-called hyperbolic modes by determining the conditions under which the coefficient F(ao) of a, in eqn (2) becomes negative at least part of the time. The method of determining the conditions under which F(ao) ceases to be positive definite has been used by other investigators. Ahnroth, Meller and Brogan!21 call this a “quasi-static” approach. Ho and Nash 1191, Abr~amson[2OJ and Carlson[21] use the same approach for the solution of their problems. Cromer and Ball1171 study both the hyperbolic and Mathieu modes and they show that, near the dynamic buckling threshold, the hyperbolic modes completely dominate the Mathieu modes. As mentioned above the investigations on the asymmetric dynamic buckling analysis of spherical caps under a uniform step pressure are few. Stricklin and Martinez[7] were probably the first ones to present a numerical study on this subject. They encountered no evidence of asymmetric dynamic buckling for the shell parameter A = 7.5 when they used the Fourier harmonics n = 0, 1, 2. Later Stricklin et aL[8] presented some additional numerical results for A = 6, 7 and 9. Ball and Burt[ll] determined the asymmetric dynamic buckling loads of spherical caps for a large range of shell rises. In both of these works[l, ll] an imperfection was assumed in the step pressure to excite the asymmetric modes, and the criterion used for defining the asymmetric dynamic bucking is identical to the criterion for the ~isymmet~c dynamic snap-through buckling. In other words, when a small change in the nearly axisymmetric step pressure causes a sudden large increase in the characteristic deflection of the shell, asymmetric dynamic buckling is said to have occurred. Klosner and Longhitano[lO] studied the asymmetric response of a dynamically loaded hemispherical shell introducing an asymmetric initial velocity to the shell. However, numerical results on asymmetric dynamic buckling loads are not presented due to the amount of computer time required. Fulton and Bartont221, in their study of the asymmetric dynamic buckling of shallow arches, let very small numerical roundoff errors play the role of imperfections in the calculations and excite asymmetric dynamic response. They claim that “the presence and growth of the antisymmetric component of response is sufficient, in this case, to trigger symmetric snap buckling”, and “a criterion which considers only sudden growth in the symmetric response component is not sufficiently sensitive to identify asymmetric .buckling”. Accordingly, they define the asymmetric dynamic buckling load as the threshold value of load beyond which the asymmetric component of displacement response shows significant growth rate. It seems that, by using such a definition, Fulton and Barton[22] actually determined the bifurcation points of the axisymmetric dynamic path. According to McIvor [23], however, the criterion used in Ref. [22] is conservative. The short discussion of the literature on the subject presented above shows that, at present, there is no well-understood and generally accepted criterion avaiiable for the asymmetric dynamic buckling analysis of
shells. The available numerical results, although very few, are not in good agreement. The criteria which seem to be reasonable result in asymmetric dynamic buckling loads much above the experimental results[7,9, 111.Generally speaking, to obtain the asymmetric dynamic behavior of a shell one must carry out a complete transient response analysis of the shell with very small imperfections. On the other hand, this results in the si~ficant increase in computer time required. According to Ahnroth, Meller and Brogani “it is important, therefore, to pursue the possibilities of finding methods that can give results of acceptable accuracy with less effort”. Here, it is suggested that the axisymmetric dynamic fundamental equilibrium surface mentioned above intersects an asymmetric surface which will be called the bifurcated sltrface. The intersection of these two surfaces will generate a b~f~cation path in the Ioad versus deflection versus time space. The projection of the b~~~tion path onto the deflection versus time plane is shown in Fig. 2 for different wave numbers M and n. Therefore, the axisymmetric fundamental equilibrium path, the bifurcated path and the bifurcation point of the static analysis correspond to the axisymmetric fundamental equilibrium surface, the bifurcated surface and the bifurcation path of the dynamic analysis, respectively. It is the purpose of this investigation to see if the b~~cation paths do, indeed, exist. Then, using the concept of the b~urcation path, the bifurcation loads for spherical caps under a uniform step pressure will be determined. These bifurcation loads will be defined to be lower bounds for the asymmetric dynamic snap-through buckling. Later in the paper, whether the lower bounds obtained so actually correspond to the dynamic snap-through buckling of the shell will be investigated. To determine the shape of the bifurcated surface is the subject of dynamic ~stbuc~ing (or, at least, dynamic initial ~stbucklin~ theory. Only then it is possible to study if dynamically loaded shells are imperfection sensitive. 2. MATHEMATICAL MODEL
Basic equations The behavior of a thin clamped shallow spherical shell undergoing moderately large deflections due to a uniform step pressure of infinite duration can be described by Marguerre’s equations [24] in non~mensional form
-2($+($
-+)+4,~
-$$
(3)
with boundary conditions at the clamped edge (x = ,J) w
=o,
w’=O,
(5) (6)
244
NURI AK&Q
and the initial conditions for step iioading at r = 0 w =o,
(xaq
w
The nondimensional quantities appearing in eqns (3~(10) are related to the corresponding physical quantities through the relations h
0 =&W,
f=4E.2hF,
A = [48(1- v2)]‘“(Hlh)‘“,
(16)
(9)
am-0 z*
x =;r,
-+a+xw;= - gwa,
06(0,~) = wS(O,7) = wc(h,7) = o&I, 7) = 0, cp(O,7)= W(h, 7) -$h,
?) = 0
(17) (18)
and the initial conditions for the step loading are (11)
p = &4.
7 = (2N /u~)(E~~)“~~.
in which Cp= fb and V”() are for the axisymmetric case. The boundary conditions are
I
In these expressions a, H and h are, respectively, the base plane radius, apex rise and shell thickness (Fig. 4). Young’s modulus is denoted by E and Poisson’s ratio by V.The quantities W and F are the physical transverse deflection and Airy stress functions, respectively. The uniform pressure is 4, physical time is t and rn is the mass per unit voiume of the shell. The radial and circumferential coordinates are denoted by r and 8, respectively. Because the shell is assumed to be shallow, only the effect of the vertical inertia force is included in the analysis. Finally, the notations employed to indicate differentiation are as follows:
WO(X, 0) = F
(x, 0) = 0.
(19)
The apex conditions (x = 0) follow from the symmetry and boundedness of the membrane forces at the apex. The average deflection of the shell will be represented by the following dimensionless quantity. I P(T)=+
I0
wo(n, T)X
dx.
(20)
Equations (IS)-(19) define the axisymmetric nonlinear dynamic behavior of the shell completely. Their numerical solution will be summarized later in the paper. Asymmetry behavior
The equations governing the asymmetric dynamic perturbations of the shell are
The boundary and initial value problem described by eqns (S)-(lO) is completed by assigning appropriate reg~a~ty conditions at the apex. A solution of eqns (3)-(10) is sought in the following form: w(x, a, 7) = wo(x,7) t ml(x, 7) cos ?I&
(13)
f(x, e, 7) = f& 7) +f&x, 7) cos rse,
(14)
in which on and fn are due to the asymmetric behavior and they are assumed to be small. Substitutive eqns (13) and (14) into eqns (3~(~0~, one will obtain the equations governing the axisymmetric and asymmetric behavior of the shell. Axisymmetr~c behavior The equations governing the axisymmet~c non~in~ dynamic behavior of the shell are{41
and boundary conditions at the clamped edge (x = h) 0” =CO:=or
(23)
hf”‘-l[1-v+(2+v)n2]~‘+~:f =o- (24) ” n “A” To the system of eqns (2~~(24), there are added the conditions of regularity at the apex Fr$ (ofI,ill, XC&xf:f)= 0.
(2%
The problem is completed by assigning appropriate initial conditions which will be presented in the following sections. In the equations above 9 = -of, and the differential operators are defined by
t------o& Fig. 4. Clamped shallow spherical shell.
245
Asymmetricdynamicanalysisof shallowsphericalshells For simplicity in the discussion, eqns (21) and (22) will be written in the following form: $+
F(W,f”,B, (P)= 0,
(27)
G(o,, f., Y, @)= 0.
(28)
Again for simplicity in the discussion, it is assumed that eqn (28) has been used to eliminate fn from eqn (27). Accordingly, the asymme~c d~arni~ behavior of the clamped shallow spherical shell under a step pressure of infinite duration is governed by the following symbolic form: ~+H(o,,Y,@)=O,
(29)
with the appropriate boundary and initial conditions. Equation (29) is similar to Hill’s equation and it is not apparent that the equation can be reduced to the classical lvfathieu equation. Accordingly, for its solution one will have to resort to numerical techniques. Here, it is suggested that, by solving eqn (29) with appropriate initial condition, one can obtain the bifurcation points and the asymmetric dynamic snap-through buckling loads of the shell. Therefore, there are two different types of problems solved in this investigation. Problem I: Bi~~rcut~~points Both the shell geometry and the loading are assumed to be perfect. The initial conditions are WI(x, 0) =$x,
0) = 0.
(30)
By determining the conditions under which the coefficient of on in the functions H of eqn (29) ceases to be positive definite, one can obtain the b~m~ation path described previously and shown in Fig. 2. Problem I is governed by eqns (21)-(25) and (30) without the inertia term, and it is a linear eigenvalue problem. For a given load, one may obtain more than one bifurcation point corresponding to different wave numbers at different times. This is done for a specific shell geometry and will be presented later in the paper. However, what is important is the lowest load level for which there is at least one bif~cation point. A~cordin~y, in this paper, the critical b~ur~tion pressure pE is defined as the lowest pressure for which the eigenvalue problem governed by eqns (21)-(25) and (30) possesses a nontrivial solution for any integer n at any time. The critical bifurcation pressure pC will not necessarily lead to snapthrough buckling. In any case, at pC some asymmetric infinitesimal disturbances will lead to finite asymmetric oscillations which, in turn, may lead to snap-through bu~kIing. Therefore, the critical bif~cation pressure is the lower bound for the asymme~c dynamic snapthrough buckling pressure. Problem I was first solved by Akkag [25] for the spherical cap. The approach is similar to that which is used for studying the hyperbolic modes of cylindrical shells[lS-11. In Refs. [19-211 similar approaches are used for different problems. Problem II: Asymmetric dynamic snap -through buckling IOU&
To determine the asymme~c dynamic snap-grout buckling loads for spherical caps under a uniform step
pressure of infinite duration, the asy~e~ic motion must be excited somehow and its behavior must be investigated. The governing equations and the boundary conditions are still those given by eqns (21)-(25), the inertia term included. However, the initial conditions (30) are replaced by the following initial conditions: (31) ITl:
~(x,O)=Cx(~
-x)‘,
(32)
in which C is a constant which is irrelevant because the problem solved is a linear problem. Equation (32) shows that an initial imperfection in the shape of the buckling mode has been assumed. The only requirement for selecting an imperfection in the form given by eqn (32) was that it satisfy the necessary geometrical boundary conditions. From now on, the imperfection given by eqn (32), shall be called “~perfection type 1 (ITl)“. To see if the shape of the initial imperfection assumed affects the asymmetric behavior of the cap, a second type of imperfection (IT2) will also be used in the numerical investigation. This is given by IT2:
w. (x, 0) = &‘(A - x)~,
(33)
in which C is again an irrelevant constant. In the present investigation only an initial shape imperfection has been assumed. However, it is expected that other imperfections will have a similar effect in initiating asymmetric growth. To measure the response in each asymmetric harmonic the following nondimensional parameter will be used. * I*
wn(x, 7)x dx.
(34)
This is similar to eqn (20). Although it does not represent the average asymmetric deflection of the shell, it can be used to measure the relative response of each asymmetric harmonic. 3.NUMERICAL PROCEDURE
In this section, the numerical procedures employed in the solution of the governing equations will be described briefly. Axisymmetric behavior The numerical procedure employed in this study in the solution of the eqns (1+(19) which govern the axisymmetric nonlinear dynamic behavior of the shell has been described in detail by Huang[4]. For the sake of completeness, a brief account of the present application of this procedure is presented here. The governs nonlinear eqns (15j( 19) are solved numerically by a finite difference method with an iterative procedure. First of all, it is assumed that odx, T) is known. The function cP(x,7) is obtained from eqn (16) numerically using central-difference formulas for the derivatives. Now, with 0(x, T) known, 00(x, T) is obtained from eqn (15). In this last step, the second time derivative of w. is approximated by the Houbolt expression[26]:
246
NURIAKKQ
in which (67) is the equal time increment in rj = I, and i and j indicate spatial and time stations, respectively, If the values of ooii in the three preceeding time steps are known, then eqn (1.5)can be reduced to the form [A lb1 = IbX
(36)
where [A f is a band matrix with the width of band equal to 5 and {b) is a known vector. From the matrix eqn (36), the values of wo” for the present time are evaluated. The average dispiacem~nt P(T) is obtained using eqn (20). Iterations terminate if the relative error in the values of P(T) in two successive iterations is less than 1 percent. In the numerical computation, Poisson’s ratio is taken as one-third, the equal spatial interval A = 0.5 and the time interval (Ar)=O.2. Finally, the response time used is 7 = 10. The axis~mmetric dynamic buckling loads using the criterion described in the Introduction and the procedure outlined above were obtained previously[6]. However, Stephens and FultonlS] employed a longer response time, so their results are expected to be more accurate. Accordingly, the axisymmetric dynamic buckling loads of the spherical cap under a step pressure used in this investigation for comparison purposes are the ones given in Ref. 1% Problem I: ~jfurcation
points
The linear e~genvalue problem, eqns (21~(25) and (30) which describe the bifurcation points of the cap, was replaced by a finite difference analogue using central difference formulas. At each time step, o. (hence Y) and @ are obtained from the ~isymmetric problem. They are substituted into the eigenvalue problem. Potters’ algo~t~[2?], with the modi~catiou suggested by Blum and Fulton[ZS] which ehminates sign changes associated with the singularities of the bifurcation determinant, was used in determining the b~~~ation points. The value of the modified bifurcation determinant was pfotted vs time for different values of the loading parameter p and the wave number n. A change in sign of the modified bifurcation determinant corresponds to a bifurcation point for given p and FL
fined as the threshold value of the load in these m ip j vs p curves. The irrelevant constant G was taken to be 0.00001. All of the preceding calculations were carried out on the IBM 370/145computer using double precision throughout. In the fo~o~ng, the numerical results will be presented. 4. ~ER~C~ RESULTS The axisymmetric average deflection p vs time r curves for the spherical cap with h =‘7 under a uniform step pressure of infinite duration are given in Fig. 5 for different values of the pressure p. It is seen that, for r < 5, it is possible to go through an almost thorough search for the bifurcation points to pIot the projections of the bifurcation paths on the p vs T plane. On the other hand, for T >5, the p vs T curves given are either so close to each other or so far apart that much smaller load intervals are necessary to be able to plot the projections of the bifurcation paths. This would, obviously, require a tremendous amount of computer time. A~cor~n~y, in the present work, only the region for r < 5 has been inspected for the projections, and they are presented in Fig. 6, Figure 6 shows the projections of the bifurcation paths on the p vs r plane for the same shell (h = 7). The projections are for the wave numbers n = 2,3,4,.5,6. The figure proves that the b~ur~ation paths, described in ~n~~uction, do, indeed, exist. Moreover, it is seen in Fig. 6 that the bifurcated surface and the axisymmetric dynamic fund~en~l equ~Iibrium surface may intersect more than once. For instance, the bifurcated surface corresponding to n = 2 intersects the axisymmetric surface along three separate bifurcation paths in the region considered. At present, it is not known if there is actually more than one bifurcated surface corresponding to, for instance, n = 2. It is clear from Fig. 6 that for some values of the load, for instance for p ~0.36, the axisymmetric dynamic equilibrium path does not intersect the b~urcat~ surface at all. On the other hand, at some load levels, for instance at p = 1.00, the axisymmetric dynamic equilibrium path intersects all of the bifurcated surfaces considered with different wave numbers. Therefore, there is a load level at which the axisymmetric dynamic equilibrium path intersects only one of the b~urcat~ surfaces. This Ioadievel has 5
Probiem II is governed by eqns (21~(2.9 and (31)_(32) for ITI. For XT2,it is governed by eqns (21)-(29, (31) and (33). Problem II is not an eigenvalue problem but an initial value problem; however, it is still linear. As in the case for Problem I, at each time step, o. (hence Y) and Q are obtained from the axisymmetric problem. They are substituted into eqns (21) and (22). The second time derivative of o, in eqn (21) is approximated again by the Houbolt expression[26], given by eqn (35) replacing w. with on. Following the same procedure as for the ax~symmet~c behavior, if the values of ctm”’in the three preceding time steps are known, then the values of w? for the present time can be evaluated. There is no iteration process in this probiem. Once w,ii are obtained, the so-called average displacement WI of the asymmetric harmonic under consideration is computed using eqn (34). Then, p vs T curves are plotted. The response time used is, again, I = 10. Finally, to determine the asymmetric dynamic snap-through buckling load, the max~um absolute value of pa, m Inn1,vs load p~ameter p curves are plotted. Tbe asymmetric dynamic snap-through buckling load is de-
9 4
Fig. 5. Axisymmetric average deflection vs time curves.
247
Asymmetricdynamicanalysisof shallowsphericalshells
-12-lb-
Fig.6. Projectionsof bifurcationpaths. previously been defined to be the lower bound for the asymmetric dynamic snap-through buckling. The lower bounds have been obtained for a larger range of shell rises and they are presented in Fig. 13 and will be discussed later. So far, the results on Problem I have been presented. Figures 7-12 present the results on Problem II. Fignres 7 and 8 show the so-called average displacement p vs time T curves for A = 6 and R = 2 with IT1 and IT2, respectively, for different values of the pressure p. A comparison of these two figures reveals that the type of the imperfection considered does not affect the asymmetric behavior of the shell significantly. Of course, the magnitudes of p for IT1 and IT2 are different, but this is not relevant for our purposes. It can be said, however, that the asymmet~c dynamic behavior is sensitive to the magnitude of the initial imperfection but not to its type. Similar conclusions have been obtained by Anderson and Lindberg[ 161and Cromer and Ball[l7] for the h~rboli~ modes of cylindrical shells. Moreover, in both of these works]16,17l, it was found that the maximum amplitude attained by a given hyperbolic mode is essentially linearly proportional to the magnitude of the imperfection in that mode. A comparison of Figs. 7 and 8 reveals that the same conclusion is valid also for the present problem. The p vs T curves were plotted for different values of A,n and p. Then these curves were used to plot the maximum absolute vaIue of p, m [p 1,vs load parameter p curves for different values of h and n, The results are shown in Figs. 9-12. The corresponding values of A and n are given in the figures. Figures 9-l 1 are for IT1 and Fig. 12 is for IT2 which is presented only for comparison purposes. All of the figures are self-explanatory. The conclusions obtained by studyi~ these figures are very similar to the conclusions of Fulton and Barton[22] for shallow arches. The figures show that for low values of the step pressure, the asymmetric ~brations are quite small. For these low values of p, RI/P] vs p curves are almost horizontal straight lines. Beyond some threshold step pressure, however, the asymmetric response shows
-16 * -18 -2o..22-
111
_2&-
%=6
-26f
n-2
012365678
z
9
10
Fig.7. Asymmetricdynamicresponsefor imperfectiontype I. significant growth rate. The threshold value of the pressure is defined to be the asymmetric dynamic snapthrough buckling load. This criterion is the same as that used Fulton and Barton[23]. The littIe solid circles on the curves in Figs. P-11 give the lower bounds for the asy~e~c dynamic buckting obtained from Problem I. The comparison of these lower bounds with the threshold values will be made later. It is seen in Fig. 9 that the threshold value of the pressure is quite well defined for A = 6. On the other hand, for A =7, 7.5 and 8, as shown in Pigs. 10 and 11, the ~esholds are not as distinct. For these cases, the threshold values of the step pressure were obtained as foIlows: As seen in the figures, the mlpn/ vs p curves contain two portions which are almost straight lines. The intersection of the two straight lines was defined to be the threshold and the asymmetric dynamic snap-through buckling loads were obtained accordingly. However, taking refuge behind the results obtained by Fulton and Barton[23], it may be claimed that the thresholds would be more distinct if the time response were larger than 10 used in this study. A comparison of Figs. 12 and 9 reveals that the type of imperfection considered does not change the asymmetric dynamic snap-bough buckling Ioad. The results obtained from Problems I and II are summarized in Fig. 13. The present results and the results of
248
NUIU AKIGQ x
-3
to
2e 24 22
IT1
20 f+“iI
18 16 14 12 la
t \ t
-lOO-
f
I t
\ \
\\
-140-160-
xf2
_160-
X=6
-200
n= 2
3
f
k
-120 -
6
P Fig. 10. Asymmetric dynamic snap-throrrghbuckling for h - 7.
Fig. 8. Asymmetric dynamic response for imperfectiontype II.
ml&
ol-. 0
’
*
O*l 0.2 a3
*
8
1
*
I
0.0 0.5 0.6 0.7 0.6 P Fig. 9. Asymmetric dynamic snap-through buckling for h = 6.
the previous investigators are given also in the Appendix. In Fig. 13, the cout~uuous light curve which is denoted by n = 0 is the ~symme~e dynamic snap-through buckling curve of Stephens and FultonlS1. The curves with n # 0 give the lower bounds for the asymmetric dynamic buckling which are obtained from Problem I, They were presented by Akkag{25]+The full circles and triangles denote the asymmetric dynamic snap-through buckling
Fig. f 1. Asymmetric dynamic snap-through buckling for i = 4.9 alldn =8.
loads for R = 2 and pt = 3, respectively, for different values of A. They are obtained from Problem II. The result of the experimental work on the dynamic buckling behavior of the shell with A = 7.5 is also shown on the figurel91. A study of Fig, 13 reveals the following conclusions: The bifurcation loads obtained from Problem I are always lower than the asymmetric dynamic snap-through buckling loads, as expected. The difference between the lower bound and the snap-through buckling Ioad is not significant for some values of h and n (for instance, for
249
Asymmetric dynamic analysis of shallow spherical shells *1o-2 22 20-
X=6
16-
IT2
Il.2
16. m&I
1412 10 06L20 0
0.1
0.2
0.3
0.4
0.6
0.6
0.7
0.6
Fig. 12. Asymmetricdynamic snap-though buckling for A =6 with imperfection type II.
06.
05.
04. 03 t
I
023
I 4
5
6
7
8
9
IO
II
12
13
x
For a given A, the critical wave number is the same for both the bifurcation buckling and the snap-through buckling. Lock, Okubo and Whittier[9] determined the critical step pressure loads experimentally required to produce snapping for the shell with A = 7.5. In the discussion of their experimental results, Lock, Okubo and Whitter[91 suggest that the shell geometry A = 7.5 is close to the transition between the axisymmetric and asymmetric snapping regimes. The present results show that A = 7.5 is, indeed, very close to this transition. Moreover, the experimental results of Ref. [9] and the results of the present work, whether obtained from Problem I or Problem II, are in very good agreement. The numerical results on the asymmetric dynamic buckling of spherical caps obtained by the previous investigators are not shown in Fig. 13. They are given in the Appendix with some short discussions. It will suffice to say, however, that they are all above the present results and that the present result for A = 7.5 is the only one which is in agreement with the experimental result, so far. Finally, in Figs. 14 and 15 the phase planes (2” vs p curves) are given for A = 6 for two different values of the pressure p. Figure 14 is for p = 0.3 which is below the critical pressure and Fig. 15 is for p = 0.5 which is above the critical pressure. When the pressure is subcritical, it is seen from Fig. 14 that the shell vibrates about an equilibrium position in its asymmetric mode. On the other hand, when the pressure is supercritical (Fig. 15) it appears that there is more than one equilibrium position about which the shell vibrates. The study of the phase planes will probably reveal some interesting facts about the asymmetric dynamic behavior of shells; however, this should be the subject of further research. As a last observation, it should be pointed out that the phase plane of Fig. 15 contains a cusp. As known from the axisymmetric dynamic analysis of shells, a cusp implies a snap-through buckling.
Fig. 13. Criticalsteppressurevs shellparameter. A = 6, n = 2 and for A = 8, n = 3). For some values of A and n, however, the difference is quite large; the largest difference being for A = 6, n = 3. It is suspected that the gap would be closed by using a larger response time.
5. SUMMARY AND CONCLUSIONS asymmetric dynamic behavior of clamped shallow spherical shells under a uniform step pressure of infinite duration has been investigated. The conclusions obtained The
X=6 n=2 (2At)i”
IT1
X10-a
p = 0.3
x10
Fig. 14. Phase plane for subcritical pressure.
-3
250
NURIAKKAS
(2Ar)il, x IU3
A~kaowfedgemen~-The author is indebted to Bo Almroth for some private discussion on the subject. REFERENCES 1. 1. W. Hutchinson and W. T. Koiter, Postbuckling theory. Appf. Me&. Rev. 23, 1353-1366(1970). 2. B. 0. Almroth, E. Melfer and F. A. Brogan, Computer solutions for static and dynamic buckling of shells. IUTAM Symposium on Buckfirzgof Structures, Harvard University (1974). 3. B. Budiansky and R. S. Roth, Axisymmetric dynamic buckling of clamped. shallow spherical shells. Colfecded Papers on lnsf~bility of She@ Structures-J%2 NASA Langley Research Center TN D-1510(1962). 4. N. C. Huang, Axisymmet~c dynamic snap-through of elastic clammed shallow snherical shells. A.I.A.A. J. 7. 215-220
(I%!$.
-6 -28
-24
-20
-16
-12
-8
-4
0
4
9"
9 x lo-
Fig. 15. Phase plane for supercritical pressure.
may be summarized as follows: 1. The axisymmetric dynamic. fundamental equilibrium surface intersects the bifurcated surface along a bifurcation path. 2. The bifurcation path can be obtained from the solution of a linear eigenvalue problem. 3. The lowest value of the step pressure at which the axisymmetric dynamic equilibrium path intersects the bifurcated surface is defined to be the critical bifurcation pressure which is the lower bound for the asymmetric dynamic snap-through buckling pressure. 4. If the shell geometry has an imperfection, the asymmetric response is excited. The threshold value of the pressure at which the asymmetric response shows significant growth rate is defined to be the asymmetric dynamic snap-through buckling load. 5. The maximum amplitude attained by a given asymmetric mode is essentially linearly proportional to the magnitude of the imperfection in that mode. 6. The shape of the imperfection assumed does not affect the asymmetric dynamic snap-through buckling load. 7. The asymmetric dynamic sna~through buckling pressure and its lower bound are close to each other for some she11parameters. 8. The numerical and experimental results on the asymmetric dynamic buckling pressure for the shell with A = 7.5 are in good agreement. 9. When the step pressure is supercritical, the phase plane of the asymmetric behavior contains a cusp implying a snap-through buckling. 10. To prove or disprove the validity of the suggestions on the asymmetric dynamic buckling theory of shells made in this work, more experimental investigations are needed. il. To study the imperfection sensitivity of dynamically loaded shells, a dynamic postbuckling theory, similar to Koiter’s static postbuckling theory, should be developed.
-
5. W. B. Stephens and R. E. Futton, Axisymmet~c static and dynamic buckling of spherical caps due to centrally distributed pressures. A.I.A.A. 3. 7, 2120-2126(1959). 6. N, AkkaSand N. R. Bauld, Jr,, Axisymmetric dynamic buckiing of clamped shallow spherical and conical shells under step loads. A.1.A.A. J. 8, 2276-2277(1970). 7. J. A. Stricklin and J. E. Martinez, Dynamic buckling of clamped spherical caps under step pressure loadings. A.I.A.A, .I. 7, 1212-1213(1969). 8. J. A. Stricklin, J. E. Martinez, J. R. Tillerson, J. H. Hong and W, E. Haisier, Nonlinear dynamic analysis of shells of revolution by matrix displacement method. A.I.A.A. $. 9, 629-636 (1971). 9. M. &I.Lock, S. Okubo and J. S. Whittier, Experiments on the snapping of a shallow dome under a step pressure load. A.I.A.A. J. 6, 1320-1326(1%8), 10. J. M. Kiosner and R. Longhitano, Nonlinear dynamics of hemispherical shells. A.I.A.A. J. 11, 1117-1122(1973). 11. R. E. Ball and J. A. Burt, Dynamic buckling of shallow spherical shells. J. Appl. Me& 40, 411-416 (1973). 12. R. E. Ball, A program for the nonlinear static and dynamic analysis of arbitrarily loaded shells of revolution. Computers and Structures, 2, 141-162(1972). 13. J. Mescall and T. Tsui, Intluence of damping on the dynamic stability of spherical caps under step pressure loading. A.1.A.A. J; 9, 1244-1247(1971). 14, 3. N. Goodier and I. K. M&or, The elastic cylindrical shell under nearly uniform radial impulse. J. Appl. Me& 31, 259-266 (1964). 1.5.Ii. E. Lindberg, Bucklingof a very thin cylindrical shell due to an impulsive uressure. J. AvpL Me&. 31, 267-272(1964). 16. D. L. ‘Andersdnand H. E. L&dberg, Dynamic pulse buckling of cylindrical shells under transient lateral pressures. A.J.A.A, J. 6, 589-598(1968). 17. C. C. Cromer and R. E. Bail, Dynamic buckling of cylindrical shell, 1. Engng Mech. Div. EM3, 657-671(1971). 18. I. K. Mclvor and D. A. Sonstegard, Axisymmetric response of a closed spherical shell to a nearly uniform radial impulse. 1. .&oust. S&. Am. 48, 1540-1547(1966). 19. F. H. Ho and W. A. Nash, Dynamic buckling of spherical shells. Proc. Symp. No~-C~ass~co~ Problems. Intemation~ Association of Shell Structures, Warsaw (1963). 20. G. R. Abrahamson, Critical velocity for collapse of a shell of circular cross section without bu&ing. J. Appl. Mech. 41, 407-411 (1974). 21. R. L. Carlson, Structural instability induced by creep. Recent Progress in Applied Mechanics, The Folke Odqvist Vofume (Edited by B. Broberg, J. Hult and F. Niordson), pp. 161-179. klmqvist‘$ Wiksell,.Stockholm (1967). 22. R. E. Fulton and F. W. Barton, Dvnamic buckling of shallow arches. J. Engng Me&. Diu. EMj, 865-877(19’7i). 23. I. K. Mclvor, Discussion of dynamic buckling of shaiiow arches, 1. Engtw Me&. Div. EM2, 478-481 (1972). 24. K. Marguerre, &r Theorie der gekriimmten Platte grosser Formanderun& Pruc. Sfh Inter. Con& Appl. Me&. p. 93 (1938). 25. N. Akkag, Asymmetric buckling behavior of spherical caps
Asymmetric dynamic analysis of shallow spherical shells under uniform step pressures. .I. Appl. Mech. 39, 293-294 (1972). 26. J. C. Houbolt, A recurrence matrix solution for the dynamic response of aircraft in gusts. Rept. 1010,NACA (1951). 27. M. L. Potters, A matrix method for the solution of a secondorder difference equation in two variables. Reporf MR19, Mathematich Centrum, Amsterdam, Holland (1955). 28. R. E. Blum and R. E. Fulton, A modification of Potters’
251
method for solving eigenvalue problems involving tridiagonal matrices. A.I.A.A. J. 4,2231-2232 (1966). APPENDIX
Asymmetric dynamic buckling loads for clamped shallow spherical shells under a uniform step pressure of infinite duration are given in this Appendix. The notation NS stands for “not specified”.
Table 1. Problem
1 I
12
10.37
1
!
I
I a /Problem
3
/ I
II bi
Rcif.(7,81
1
j I
I
Ref.{111
0.511 I
I
5,e
I
1 Ref.
1 I
(91 =
I
I
I
I
J
“Bifurcation loads, lower bound. bAsymmetric dynamic snap-through buckling loads. ‘Of the two shells tested experimentally, one buckled axisymmetrically and the other asymmetricaUy. dIf n = 1 checked is not known. ‘It seems that n = 3 is not checked. ‘The threshold is at p = 0.47. ‘Taken from Ref. [ll]. “It seems that n = 3 is not checked. In Ref. [ll] it is given that Stricklin et al. [8] encountered asymmetric dynamic buckling also for A = 9, whereas in the present investigation axisymmetric dynamic buckling governed for this case.
BIFURCATION AND SNAP-THROUGH PHENOMENA IN ASYMMETRIC DYNAMIC ANALYSIS OF SHALLOW SPHERICAL SHELLS? NURI AKKASS Department of Civil Engineering, Middle East Technical University, Ankara, Turkey (Receiued
6 hoember
1975)
Abstract-The asymmetric dynamic behavior of clamped shallow spherical shells under a uniform step pressure of infinite duration is investigated. The solution of a linear eigenvalue problem yields the bifurcation paths and also the lower bound for the asymmetric dynamic snap-through buckling pressure. The asymmetric dynamic response of shells with a shape imperfection is studied. The asymmetric dynamic snap-through buckling load is defined to be the threshold value of the step pressure at which the asymmetric response shows significant growth rate. The snap-through buckling loads are obtained for a few shell parameters. The numerical results are compared with the available experimental results and they are in good agreement. Finally, a preliminary study of the phase planes is presented.
1. INTRODUCTION Buckling analysis of received considerable well-knawn fact that applied loads usually
When the loading is static, one can distinguish between two types of buckling: (a) Axisymmetric snap-through buckling results when the static response is constrained to be symmetric. It corresponds to the limit point of the axisymmetric fundamental equilibrium path which is represented by point A in Fig. 1. (b) Asymmetric buckling corresponds to the bifurcation point of the axisymmetric fundamental path (point B in Fig. 1). At this point the shell bifurcates into an asymmetrical deformation mode. Using the terminology of Ahnroth, Meller and Brogan[21, depending upon whether the equilibrium on the bifurcated path is stable or unstable, the asymmetric buckling may be called gentle or violent, respectively. When one talks about the behavior of a statically loaded shell, he talks about an axisymmetric fundamental equilibrium path, a bifurcation point and a bifurcated path. The reason for underlining the words “path” and “point” will be clarified in the following section.
shells under dynamic loads has attention in the literature. It is a shells subjected to dynamically
buckle at load levels which are smaller than the corresponding quasi-static buckling load. The reduction in dynamic buckling load from the corresponding static buckling load depends upon the duration of the dynamic loading. The studies show that the most severe type of dynamic loading is the step load of infinite duration. The purpose of the present investigation is to study the bifurcation and snap-through phenomena in the asymmetric dynamic analysis of clamped shallow spherical shells subjected to a uniform step pressure of infinite duration. It is realized that actual structures are more complicated than spherical caps, but to study the behavior of such simple structures is not completely useless. Investigations on the behavior of simple structures under dynamic loading form the basis of understanding the behavior of complicated structures under the same type of loading. The problem of asymmetric dynamic buckling of shells is so complicated that the criteria for buckling are not even reasonably well understood. It is hoped that the present investigation on this simple model will shed new light on the problem. At this point, it is necessary to summarize some general concepts of buckling of shells under static loading; because, later in the paper, some analogies between the static and dynamic buckling phenomena will be drawn. Buckling under static loads Buckling of shells under static loads has attracted much attention in the past few decades, and an extensive literature has been developed. ‘One is referred to the excellent survey paper of Hutchins& and Koiter[ll for a discussion of these studies. Here, it will suffice to summarize the general concepts of static buckling of shells. tpresented at the Second National Symposium on Computerized Structural Analysis and Design at the School of Engineering and Applied Science, George Washington University, Washington, DC., 29-31 March 1576. *Assistant Professor.
Buckling under dynamic loads As in the case of static loading, buckling of shells under dynamic loads has also received much attention. Here, it will not be attempted to present a reasonable survey of the investigations made on this subject; however, some of them will be referred to in the paper, as the need arises. To study the behavior of dynamically loaded shells, one must include time in the equations as an additional parameter and this complicates matters considerably. However, when the loading is dynamic (step loading, for instance) one would still expect to distinguish between two types of buckling: (a) Axisymmetric snap-through buckling results when the dynamic response is constrained to be symmetric. In contrast to the statically loaded shell, the behavior of which can be depicted on a two-dimensipnal sketch as shown in Fig. 1, the deflection versus load versus time behavior of the dynamically loaded shell can be depicted only on a three-dimensional sketch. Therefore, the axisymmetric behavior of a dynamically loaded shell can be described by an axisymmetric dynamic fundamental equilibrium surface. The projection of this surface onto
241
NURIAKKAS r
Bifurcation
point
/
,--! Limit point
A/-
B
Axisymmetric
Equlibrium
fundamental
path
static
path
: E : ii ! LOCUS of dynamic peak deflection
DEFLECTION
Fig. 1. Axisymmetricstaticfundamentalequilibriumpath. the deflection versus time plane is shown in Fig. 2 for different values of the load. Considering a step load of infinite duration, it is seen in Fig. 2 that, for load levels less than a critical load, the shell oscillates about its static equilibrium position with relatively small amplitudes. However, when the load exceeds the critical level, the oscillations of the shell become very large and the axisymmetric dynamic snap-through buckling is said to have occurred. To determine the axisymmetric dynamic snapthrough buckling load, the peak in time history of a characteristic deflection (the volume of the shell deformation or the apex deflection) is plotted against the amplitude of the applied load. The load which corresponds to a sudden large increase in the characteristic deflection is said to be the axisymmetric dynamic snap-through buckling load. This is illustrated in Fig. 3 where it is also seen that the axisymmetric dynamic snap-through buckling occurs when the locus of the peak characteristic deflection intersects the unstable branch of the axisymmetric static load deflection curve. This definition of the axisymmetric dynamic snap-through buckling load has first been suggested by Budiansky and Roth[3] and used by other investigators[4-131 for the dynamic buckling analysis of spherical shells. Most of the numerical results available on axisymmetric dynamic snap-through buckling loads of sphericalcaps under a uniform step pressure are now in reasonable agreement. It may be said that the criterion
Projection
Fig. 3. Definition of axisymmetric dynamic snap-through buckling load.
suggested by Budiansky and Roth[3] has generally been accepted. (b) Asymmetric dynamic buckling results when the dynamic response is not constrained to be symmetric. Although there are very few investigations made on the asymmetric dynamic analysis of spherical caps under a uniform step pressure of infinite duration, the investigations on the asymmetric dynamic instability of cylindricals shells are more extensive. The equations of motion for an impulsively loaded cylindrical shell assuming small asymmetric (flexural) amplitudes are given by Goodier and McIvor [14] as follows: iiotao=o,
ii. + F(ao)a, = 0,
(1) (2)
in which a,, is the Fourier coefficient of the symmetric (radial) mode and a,, is the Fourier coefficient of the n th asymmetric mode. Dots indicate differentiation with re-
of
Fig. 2. Projections of axisymmetric dynamic fundamental equilibriumsurface and bifurcation paths.
243
Asymmetricdynamicanalysisof shallowsphericalshells spect to time. Lindberg[lS], Anderson and Lindberg[l6] and Cromer and Ball[17] use the same type of equations in their analysis of the same shell. Goodier and McIvor [ 141reduce eqn (2) to the classical Mathieu equation and study the stability of the so-called Mathieu modes throu8h the use of the Mathieu stability chart. The same type of problem was solved by McIvor and Sonsteg~d[l8] for a closed spherical shell. On the other hand, Lindberg [ 151and Anderson and Lindberg [ 161study the so-called hyperbolic modes by determining the conditions under which the coefficient F(ao) of a, in eqn (2) becomes negative at least part of the time. The method of determining the conditions under which F(ao) ceases to be positive definite has been used by other investigators. Ahnroth, Meller and Brogan!21 call this a “quasi-static” approach. Ho and Nash 1191, Abr~amson[2OJ and Carlson[21] use the same approach for the solution of their problems. Cromer and Ball1171 study both the hyperbolic and Mathieu modes and they show that, near the dynamic buckling threshold, the hyperbolic modes completely dominate the Mathieu modes. As mentioned above the investigations on the asymmetric dynamic buckling analysis of spherical caps under a uniform step pressure are few. Stricklin and Martinez[7] were probably the first ones to present a numerical study on this subject. They encountered no evidence of asymmetric dynamic buckling for the shell parameter A = 7.5 when they used the Fourier harmonics n = 0, 1, 2. Later Stricklin et aL[8] presented some additional numerical results for A = 6, 7 and 9. Ball and Burt[ll] determined the asymmetric dynamic buckling loads of spherical caps for a large range of shell rises. In both of these works[l, ll] an imperfection was assumed in the step pressure to excite the asymmetric modes, and the criterion used for defining the asymmetric dynamic bucking is identical to the criterion for the ~isymmet~c dynamic snap-through buckling. In other words, when a small change in the nearly axisymmetric step pressure causes a sudden large increase in the characteristic deflection of the shell, asymmetric dynamic buckling is said to have occurred. Klosner and Longhitano[lO] studied the asymmetric response of a dynamically loaded hemispherical shell introducing an asymmetric initial velocity to the shell. However, numerical results on asymmetric dynamic buckling loads are not presented due to the amount of computer time required. Fulton and Bartont221, in their study of the asymmetric dynamic buckling of shallow arches, let very small numerical roundoff errors play the role of imperfections in the calculations and excite asymmetric dynamic response. They claim that “the presence and growth of the antisymmetric component of response is sufficient, in this case, to trigger symmetric snap buckling”, and “a criterion which considers only sudden growth in the symmetric response component is not sufficiently sensitive to identify asymmetric .buckling”. Accordingly, they define the asymmetric dynamic buckling load as the threshold value of load beyond which the asymmetric component of displacement response shows significant growth rate. It seems that, by using such a definition, Fulton and Barton[22] actually determined the bifurcation points of the axisymmetric dynamic path. According to McIvor [23], however, the criterion used in Ref. [22] is conservative. The short discussion of the literature on the subject presented above shows that, at present, there is no well-understood and generally accepted criterion avaiiable for the asymmetric dynamic buckling analysis of
shells. The available numerical results, although very few, are not in good agreement. The criteria which seem to be reasonable result in asymmetric dynamic buckling loads much above the experimental results[7,9, 111.Generally speaking, to obtain the asymmetric dynamic behavior of a shell one must carry out a complete transient response analysis of the shell with very small imperfections. On the other hand, this results in the si~ficant increase in computer time required. According to Ahnroth, Meller and Brogani “it is important, therefore, to pursue the possibilities of finding methods that can give results of acceptable accuracy with less effort”. Here, it is suggested that the axisymmetric dynamic fundamental equilibrium surface mentioned above intersects an asymmetric surface which will be called the bifurcated sltrface. The intersection of these two surfaces will generate a b~f~cation path in the Ioad versus deflection versus time space. The projection of the b~~~tion path onto the deflection versus time plane is shown in Fig. 2 for different wave numbers M and n. Therefore, the axisymmetric fundamental equilibrium path, the bifurcated path and the bifurcation point of the static analysis correspond to the axisymmetric fundamental equilibrium surface, the bifurcated surface and the bifurcation path of the dynamic analysis, respectively. It is the purpose of this investigation to see if the b~~cation paths do, indeed, exist. Then, using the concept of the b~urcation path, the bifurcation loads for spherical caps under a uniform step pressure will be determined. These bifurcation loads will be defined to be lower bounds for the asymmetric dynamic snap-through buckling. Later in the paper, whether the lower bounds obtained so actually correspond to the dynamic snap-through buckling of the shell will be investigated. To determine the shape of the bifurcated surface is the subject of dynamic ~stbuc~ing (or, at least, dynamic initial ~stbucklin~ theory. Only then it is possible to study if dynamically loaded shells are imperfection sensitive. 2. MATHEMATICAL MODEL
Basic equations The behavior of a thin clamped shallow spherical shell undergoing moderately large deflections due to a uniform step pressure of infinite duration can be described by Marguerre’s equations [24] in non~mensional form
-2($+($
-+)+4,~
-$$
(3)
with boundary conditions at the clamped edge (x = ,J) w
=o,
w’=O,
(5) (6)
244
NURI AK&Q
and the initial conditions for step iioading at r = 0 w =o,
(xaq
w
The nondimensional quantities appearing in eqns (3~(10) are related to the corresponding physical quantities through the relations h
0 =&W,
f=4E.2hF,
A = [48(1- v2)]‘“(Hlh)‘“,
(16)
(9)
am-0 z*
x =;r,
-+a+xw;= - gwa,
06(0,~) = wS(O,7) = wc(h,7) = o&I, 7) = 0, cp(O,7)= W(h, 7) -$h,
?) = 0
(17) (18)
and the initial conditions for the step loading are (11)
p = &4.
7 = (2N /u~)(E~~)“~~.
in which Cp= fb and V”() are for the axisymmetric case. The boundary conditions are
I
In these expressions a, H and h are, respectively, the base plane radius, apex rise and shell thickness (Fig. 4). Young’s modulus is denoted by E and Poisson’s ratio by V.The quantities W and F are the physical transverse deflection and Airy stress functions, respectively. The uniform pressure is 4, physical time is t and rn is the mass per unit voiume of the shell. The radial and circumferential coordinates are denoted by r and 8, respectively. Because the shell is assumed to be shallow, only the effect of the vertical inertia force is included in the analysis. Finally, the notations employed to indicate differentiation are as follows:
WO(X, 0) = F
(x, 0) = 0.
(19)
The apex conditions (x = 0) follow from the symmetry and boundedness of the membrane forces at the apex. The average deflection of the shell will be represented by the following dimensionless quantity. I P(T)=+
I0
wo(n, T)X
dx.
(20)
Equations (IS)-(19) define the axisymmetric nonlinear dynamic behavior of the shell completely. Their numerical solution will be summarized later in the paper. Asymmetry behavior
The equations governing the asymmetric dynamic perturbations of the shell are
The boundary and initial value problem described by eqns (S)-(lO) is completed by assigning appropriate reg~a~ty conditions at the apex. A solution of eqns (3)-(10) is sought in the following form: w(x, a, 7) = wo(x,7) t ml(x, 7) cos ?I&
(13)
f(x, e, 7) = f& 7) +f&x, 7) cos rse,
(14)
in which on and fn are due to the asymmetric behavior and they are assumed to be small. Substitutive eqns (13) and (14) into eqns (3~(~0~, one will obtain the equations governing the axisymmetric and asymmetric behavior of the shell. Axisymmetr~c behavior The equations governing the axisymmet~c non~in~ dynamic behavior of the shell are{41
and boundary conditions at the clamped edge (x = h) 0” =CO:=or
(23)
hf”‘-l[1-v+(2+v)n2]~‘+~:f =o- (24) ” n “A” To the system of eqns (2~~(24), there are added the conditions of regularity at the apex Fr$ (ofI,ill, XC&xf:f)= 0.
(2%
The problem is completed by assigning appropriate initial conditions which will be presented in the following sections. In the equations above 9 = -of, and the differential operators are defined by
t------o& Fig. 4. Clamped shallow spherical shell.
245
Asymmetricdynamicanalysisof shallowsphericalshells For simplicity in the discussion, eqns (21) and (22) will be written in the following form: $+
F(W,f”,B, (P)= 0,
(27)
G(o,, f., Y, @)= 0.
(28)
Again for simplicity in the discussion, it is assumed that eqn (28) has been used to eliminate fn from eqn (27). Accordingly, the asymme~c d~arni~ behavior of the clamped shallow spherical shell under a step pressure of infinite duration is governed by the following symbolic form: ~+H(o,,Y,@)=O,
(29)
with the appropriate boundary and initial conditions. Equation (29) is similar to Hill’s equation and it is not apparent that the equation can be reduced to the classical lvfathieu equation. Accordingly, for its solution one will have to resort to numerical techniques. Here, it is suggested that, by solving eqn (29) with appropriate initial condition, one can obtain the bifurcation points and the asymmetric dynamic snap-through buckling loads of the shell. Therefore, there are two different types of problems solved in this investigation. Problem I: Bi~~rcut~~points Both the shell geometry and the loading are assumed to be perfect. The initial conditions are WI(x, 0) =$x,
0) = 0.
(30)
By determining the conditions under which the coefficient of on in the functions H of eqn (29) ceases to be positive definite, one can obtain the b~m~ation path described previously and shown in Fig. 2. Problem I is governed by eqns (21)-(25) and (30) without the inertia term, and it is a linear eigenvalue problem. For a given load, one may obtain more than one bifurcation point corresponding to different wave numbers at different times. This is done for a specific shell geometry and will be presented later in the paper. However, what is important is the lowest load level for which there is at least one bif~cation point. A~cordin~y, in this paper, the critical b~ur~tion pressure pE is defined as the lowest pressure for which the eigenvalue problem governed by eqns (21)-(25) and (30) possesses a nontrivial solution for any integer n at any time. The critical bifurcation pressure pC will not necessarily lead to snapthrough buckling. In any case, at pC some asymmetric infinitesimal disturbances will lead to finite asymmetric oscillations which, in turn, may lead to snap-through bu~kIing. Therefore, the critical bif~cation pressure is the lower bound for the asymme~c dynamic snapthrough buckling pressure. Problem I was first solved by Akkag [25] for the spherical cap. The approach is similar to that which is used for studying the hyperbolic modes of cylindrical shells[lS-11. In Refs. [19-211 similar approaches are used for different problems. Problem II: Asymmetric dynamic snap -through buckling IOU&
To determine the asymme~c dynamic snap-grout buckling loads for spherical caps under a uniform step
pressure of infinite duration, the asy~e~ic motion must be excited somehow and its behavior must be investigated. The governing equations and the boundary conditions are still those given by eqns (21)-(25), the inertia term included. However, the initial conditions (30) are replaced by the following initial conditions: (31) ITl:
~(x,O)=Cx(~
-x)‘,
(32)
in which C is a constant which is irrelevant because the problem solved is a linear problem. Equation (32) shows that an initial imperfection in the shape of the buckling mode has been assumed. The only requirement for selecting an imperfection in the form given by eqn (32) was that it satisfy the necessary geometrical boundary conditions. From now on, the imperfection given by eqn (32), shall be called “~perfection type 1 (ITl)“. To see if the shape of the initial imperfection assumed affects the asymmetric behavior of the cap, a second type of imperfection (IT2) will also be used in the numerical investigation. This is given by IT2:
w. (x, 0) = &‘(A - x)~,
(33)
in which C is again an irrelevant constant. In the present investigation only an initial shape imperfection has been assumed. However, it is expected that other imperfections will have a similar effect in initiating asymmetric growth. To measure the response in each asymmetric harmonic the following nondimensional parameter will be used. * I*
wn(x, 7)x dx.
(34)
This is similar to eqn (20). Although it does not represent the average asymmetric deflection of the shell, it can be used to measure the relative response of each asymmetric harmonic. 3.NUMERICAL PROCEDURE
In this section, the numerical procedures employed in the solution of the governing equations will be described briefly. Axisymmetric behavior The numerical procedure employed in this study in the solution of the eqns (1+(19) which govern the axisymmetric nonlinear dynamic behavior of the shell has been described in detail by Huang[4]. For the sake of completeness, a brief account of the present application of this procedure is presented here. The governs nonlinear eqns (15j( 19) are solved numerically by a finite difference method with an iterative procedure. First of all, it is assumed that odx, T) is known. The function cP(x,7) is obtained from eqn (16) numerically using central-difference formulas for the derivatives. Now, with 0(x, T) known, 00(x, T) is obtained from eqn (15). In this last step, the second time derivative of w. is approximated by the Houbolt expression[26]:
246
NURIAKKQ
in which (67) is the equal time increment in rj = I, and i and j indicate spatial and time stations, respectively, If the values of ooii in the three preceeding time steps are known, then eqn (1.5)can be reduced to the form [A lb1 = IbX
(36)
where [A f is a band matrix with the width of band equal to 5 and {b) is a known vector. From the matrix eqn (36), the values of wo” for the present time are evaluated. The average dispiacem~nt P(T) is obtained using eqn (20). Iterations terminate if the relative error in the values of P(T) in two successive iterations is less than 1 percent. In the numerical computation, Poisson’s ratio is taken as one-third, the equal spatial interval A = 0.5 and the time interval (Ar)=O.2. Finally, the response time used is 7 = 10. The axis~mmetric dynamic buckling loads using the criterion described in the Introduction and the procedure outlined above were obtained previously[6]. However, Stephens and FultonlS] employed a longer response time, so their results are expected to be more accurate. Accordingly, the axisymmetric dynamic buckling loads of the spherical cap under a step pressure used in this investigation for comparison purposes are the ones given in Ref. 1% Problem I: ~jfurcation
points
The linear e~genvalue problem, eqns (21~(25) and (30) which describe the bifurcation points of the cap, was replaced by a finite difference analogue using central difference formulas. At each time step, o. (hence Y) and @ are obtained from the ~isymmetric problem. They are substituted into the eigenvalue problem. Potters’ algo~t~[2?], with the modi~catiou suggested by Blum and Fulton[ZS] which ehminates sign changes associated with the singularities of the bifurcation determinant, was used in determining the b~~~ation points. The value of the modified bifurcation determinant was pfotted vs time for different values of the loading parameter p and the wave number n. A change in sign of the modified bifurcation determinant corresponds to a bifurcation point for given p and FL
fined as the threshold value of the load in these m ip j vs p curves. The irrelevant constant G was taken to be 0.00001. All of the preceding calculations were carried out on the IBM 370/145computer using double precision throughout. In the fo~o~ng, the numerical results will be presented. 4. ~ER~C~ RESULTS The axisymmetric average deflection p vs time r curves for the spherical cap with h =‘7 under a uniform step pressure of infinite duration are given in Fig. 5 for different values of the pressure p. It is seen that, for r < 5, it is possible to go through an almost thorough search for the bifurcation points to pIot the projections of the bifurcation paths on the p vs T plane. On the other hand, for T >5, the p vs T curves given are either so close to each other or so far apart that much smaller load intervals are necessary to be able to plot the projections of the bifurcation paths. This would, obviously, require a tremendous amount of computer time. A~cor~n~y, in the present work, only the region for r < 5 has been inspected for the projections, and they are presented in Fig. 6, Figure 6 shows the projections of the bifurcation paths on the p vs r plane for the same shell (h = 7). The projections are for the wave numbers n = 2,3,4,.5,6. The figure proves that the b~ur~ation paths, described in ~n~~uction, do, indeed, exist. Moreover, it is seen in Fig. 6 that the bifurcated surface and the axisymmetric dynamic fund~en~l equ~Iibrium surface may intersect more than once. For instance, the bifurcated surface corresponding to n = 2 intersects the axisymmetric surface along three separate bifurcation paths in the region considered. At present, it is not known if there is actually more than one bifurcated surface corresponding to, for instance, n = 2. It is clear from Fig. 6 that for some values of the load, for instance for p ~0.36, the axisymmetric dynamic equilibrium path does not intersect the b~urcat~ surface at all. On the other hand, at some load levels, for instance at p = 1.00, the axisymmetric dynamic equilibrium path intersects all of the bifurcated surfaces considered with different wave numbers. Therefore, there is a load level at which the axisymmetric dynamic equilibrium path intersects only one of the b~urcat~ surfaces. This Ioadievel has 5
Probiem II is governed by eqns (21~(2.9 and (31)_(32) for ITI. For XT2,it is governed by eqns (21)-(29, (31) and (33). Problem II is not an eigenvalue problem but an initial value problem; however, it is still linear. As in the case for Problem I, at each time step, o. (hence Y) and Q are obtained from the axisymmetric problem. They are substituted into eqns (21) and (22). The second time derivative of o, in eqn (21) is approximated again by the Houbolt expression[26], given by eqn (35) replacing w. with on. Following the same procedure as for the ax~symmet~c behavior, if the values of ctm”’in the three preceding time steps are known, then the values of w? for the present time can be evaluated. There is no iteration process in this probiem. Once w,ii are obtained, the so-called average displacement WI of the asymmetric harmonic under consideration is computed using eqn (34). Then, p vs T curves are plotted. The response time used is, again, I = 10. Finally, to determine the asymmetric dynamic snap-through buckling load, the max~um absolute value of pa, m Inn1,vs load p~ameter p curves are plotted. Tbe asymmetric dynamic snap-through buckling load is de-
9 4
Fig. 5. Axisymmetric average deflection vs time curves.
247
Asymmetricdynamicanalysisof shallowsphericalshells
-12-lb-
Fig.6. Projectionsof bifurcationpaths. previously been defined to be the lower bound for the asymmetric dynamic snap-through buckling. The lower bounds have been obtained for a larger range of shell rises and they are presented in Fig. 13 and will be discussed later. So far, the results on Problem I have been presented. Figures 7-12 present the results on Problem II. Fignres 7 and 8 show the so-called average displacement p vs time T curves for A = 6 and R = 2 with IT1 and IT2, respectively, for different values of the pressure p. A comparison of these two figures reveals that the type of the imperfection considered does not affect the asymmetric behavior of the shell significantly. Of course, the magnitudes of p for IT1 and IT2 are different, but this is not relevant for our purposes. It can be said, however, that the asymmet~c dynamic behavior is sensitive to the magnitude of the initial imperfection but not to its type. Similar conclusions have been obtained by Anderson and Lindberg[ 161and Cromer and Ball[l7] for the h~rboli~ modes of cylindrical shells. Moreover, in both of these works]16,17l, it was found that the maximum amplitude attained by a given hyperbolic mode is essentially linearly proportional to the magnitude of the imperfection in that mode. A comparison of Figs. 7 and 8 reveals that the same conclusion is valid also for the present problem. The p vs T curves were plotted for different values of A,n and p. Then these curves were used to plot the maximum absolute vaIue of p, m [p 1,vs load parameter p curves for different values of h and n, The results are shown in Figs. 9-12. The corresponding values of A and n are given in the figures. Figures 9-l 1 are for IT1 and Fig. 12 is for IT2 which is presented only for comparison purposes. All of the figures are self-explanatory. The conclusions obtained by studyi~ these figures are very similar to the conclusions of Fulton and Barton[22] for shallow arches. The figures show that for low values of the step pressure, the asymmetric ~brations are quite small. For these low values of p, RI/P] vs p curves are almost horizontal straight lines. Beyond some threshold step pressure, however, the asymmetric response shows
-16 * -18 -2o..22-
111
_2&-
%=6
-26f
n-2
012365678
z
9
10
Fig.7. Asymmetricdynamicresponsefor imperfectiontype I. significant growth rate. The threshold value of the pressure is defined to be the asymmetric dynamic snapthrough buckling load. This criterion is the same as that used Fulton and Barton[23]. The littIe solid circles on the curves in Figs. P-11 give the lower bounds for the asy~e~c dynamic buckting obtained from Problem I. The comparison of these lower bounds with the threshold values will be made later. It is seen in Fig. 9 that the threshold value of the pressure is quite well defined for A = 6. On the other hand, for A =7, 7.5 and 8, as shown in Pigs. 10 and 11, the ~esholds are not as distinct. For these cases, the threshold values of the step pressure were obtained as foIlows: As seen in the figures, the mlpn/ vs p curves contain two portions which are almost straight lines. The intersection of the two straight lines was defined to be the threshold and the asymmetric dynamic snap-through buckling loads were obtained accordingly. However, taking refuge behind the results obtained by Fulton and Barton[23], it may be claimed that the thresholds would be more distinct if the time response were larger than 10 used in this study. A comparison of Figs. 12 and 9 reveals that the type of imperfection considered does not change the asymmetric dynamic snap-bough buckling Ioad. The results obtained from Problems I and II are summarized in Fig. 13. The present results and the results of
248
NUIU AKIGQ x
-3
to
2e 24 22
IT1
20 f+“iI
18 16 14 12 la
t \ t
-lOO-
f
I t
\ \
\\
-140-160-
xf2
_160-
X=6
-200
n= 2
3
f
k
-120 -
6
P Fig. 10. Asymmetric dynamic snap-throrrghbuckling for h - 7.
Fig. 8. Asymmetric dynamic response for imperfectiontype II.
ml&
ol-. 0
’
*
O*l 0.2 a3
*
8
1
*
I
0.0 0.5 0.6 0.7 0.6 P Fig. 9. Asymmetric dynamic snap-through buckling for h = 6.
the previous investigators are given also in the Appendix. In Fig. 13, the cout~uuous light curve which is denoted by n = 0 is the ~symme~e dynamic snap-through buckling curve of Stephens and FultonlS1. The curves with n # 0 give the lower bounds for the asymmetric dynamic buckling which are obtained from Problem I, They were presented by Akkag{25]+The full circles and triangles denote the asymmetric dynamic snap-through buckling
Fig. f 1. Asymmetric dynamic snap-through buckling for i = 4.9 alldn =8.
loads for R = 2 and pt = 3, respectively, for different values of A. They are obtained from Problem II. The result of the experimental work on the dynamic buckling behavior of the shell with A = 7.5 is also shown on the figurel91. A study of Fig, 13 reveals the following conclusions: The bifurcation loads obtained from Problem I are always lower than the asymmetric dynamic snap-through buckling loads, as expected. The difference between the lower bound and the snap-through buckling Ioad is not significant for some values of h and n (for instance, for
249
Asymmetric dynamic analysis of shallow spherical shells *1o-2 22 20-
X=6
16-
IT2
Il.2
16. m&I
1412 10 06L20 0
0.1
0.2
0.3
0.4
0.6
0.6
0.7
0.6
Fig. 12. Asymmetricdynamic snap-though buckling for A =6 with imperfection type II.
06.
05.
04. 03 t
I
023
I 4
5
6
7
8
9
IO
II
12
13
x
For a given A, the critical wave number is the same for both the bifurcation buckling and the snap-through buckling. Lock, Okubo and Whittier[9] determined the critical step pressure loads experimentally required to produce snapping for the shell with A = 7.5. In the discussion of their experimental results, Lock, Okubo and Whitter[91 suggest that the shell geometry A = 7.5 is close to the transition between the axisymmetric and asymmetric snapping regimes. The present results show that A = 7.5 is, indeed, very close to this transition. Moreover, the experimental results of Ref. [9] and the results of the present work, whether obtained from Problem I or Problem II, are in very good agreement. The numerical results on the asymmetric dynamic buckling of spherical caps obtained by the previous investigators are not shown in Fig. 13. They are given in the Appendix with some short discussions. It will suffice to say, however, that they are all above the present results and that the present result for A = 7.5 is the only one which is in agreement with the experimental result, so far. Finally, in Figs. 14 and 15 the phase planes (2” vs p curves) are given for A = 6 for two different values of the pressure p. Figure 14 is for p = 0.3 which is below the critical pressure and Fig. 15 is for p = 0.5 which is above the critical pressure. When the pressure is subcritical, it is seen from Fig. 14 that the shell vibrates about an equilibrium position in its asymmetric mode. On the other hand, when the pressure is supercritical (Fig. 15) it appears that there is more than one equilibrium position about which the shell vibrates. The study of the phase planes will probably reveal some interesting facts about the asymmetric dynamic behavior of shells; however, this should be the subject of further research. As a last observation, it should be pointed out that the phase plane of Fig. 15 contains a cusp. As known from the axisymmetric dynamic analysis of shells, a cusp implies a snap-through buckling.
Fig. 13. Criticalsteppressurevs shellparameter. A = 6, n = 2 and for A = 8, n = 3). For some values of A and n, however, the difference is quite large; the largest difference being for A = 6, n = 3. It is suspected that the gap would be closed by using a larger response time.
5. SUMMARY AND CONCLUSIONS asymmetric dynamic behavior of clamped shallow spherical shells under a uniform step pressure of infinite duration has been investigated. The conclusions obtained The
X=6 n=2 (2At)i”
IT1
X10-a
p = 0.3
x10
Fig. 14. Phase plane for subcritical pressure.
-3
250
NURIAKKAS
(2Ar)il, x IU3
A~kaowfedgemen~-The author is indebted to Bo Almroth for some private discussion on the subject. REFERENCES 1. 1. W. Hutchinson and W. T. Koiter, Postbuckling theory. Appf. Me&. Rev. 23, 1353-1366(1970). 2. B. 0. Almroth, E. Melfer and F. A. Brogan, Computer solutions for static and dynamic buckling of shells. IUTAM Symposium on Buckfirzgof Structures, Harvard University (1974). 3. B. Budiansky and R. S. Roth, Axisymmetric dynamic buckling of clamped. shallow spherical shells. Colfecded Papers on lnsf~bility of She@ Structures-J%2 NASA Langley Research Center TN D-1510(1962). 4. N. C. Huang, Axisymmet~c dynamic snap-through of elastic clammed shallow snherical shells. A.I.A.A. J. 7. 215-220
(I%!$.
-6 -28
-24
-20
-16
-12
-8
-4
0
4
9"
9 x lo-
Fig. 15. Phase plane for supercritical pressure.
may be summarized as follows: 1. The axisymmetric dynamic. fundamental equilibrium surface intersects the bifurcated surface along a bifurcation path. 2. The bifurcation path can be obtained from the solution of a linear eigenvalue problem. 3. The lowest value of the step pressure at which the axisymmetric dynamic equilibrium path intersects the bifurcated surface is defined to be the critical bifurcation pressure which is the lower bound for the asymmetric dynamic snap-through buckling pressure. 4. If the shell geometry has an imperfection, the asymmetric response is excited. The threshold value of the pressure at which the asymmetric response shows significant growth rate is defined to be the asymmetric dynamic snap-through buckling load. 5. The maximum amplitude attained by a given asymmetric mode is essentially linearly proportional to the magnitude of the imperfection in that mode. 6. The shape of the imperfection assumed does not affect the asymmetric dynamic snap-through buckling load. 7. The asymmetric dynamic sna~through buckling pressure and its lower bound are close to each other for some she11parameters. 8. The numerical and experimental results on the asymmetric dynamic buckling pressure for the shell with A = 7.5 are in good agreement. 9. When the step pressure is supercritical, the phase plane of the asymmetric behavior contains a cusp implying a snap-through buckling. 10. To prove or disprove the validity of the suggestions on the asymmetric dynamic buckling theory of shells made in this work, more experimental investigations are needed. il. To study the imperfection sensitivity of dynamically loaded shells, a dynamic postbuckling theory, similar to Koiter’s static postbuckling theory, should be developed.
-
5. W. B. Stephens and R. E. Futton, Axisymmet~c static and dynamic buckling of spherical caps due to centrally distributed pressures. A.I.A.A. 3. 7, 2120-2126(1959). 6. N, AkkaSand N. R. Bauld, Jr,, Axisymmetric dynamic buckiing of clamped shallow spherical and conical shells under step loads. A.1.A.A. J. 8, 2276-2277(1970). 7. J. A. Stricklin and J. E. Martinez, Dynamic buckling of clamped spherical caps under step pressure loadings. A.I.A.A, .I. 7, 1212-1213(1969). 8. J. A. Stricklin, J. E. Martinez, J. R. Tillerson, J. H. Hong and W, E. Haisier, Nonlinear dynamic analysis of shells of revolution by matrix displacement method. A.I.A.A. $. 9, 629-636 (1971). 9. M. &I.Lock, S. Okubo and J. S. Whittier, Experiments on the snapping of a shallow dome under a step pressure load. A.I.A.A. J. 6, 1320-1326(1%8), 10. J. M. Kiosner and R. Longhitano, Nonlinear dynamics of hemispherical shells. A.I.A.A. J. 11, 1117-1122(1973). 11. R. E. Ball and J. A. Burt, Dynamic buckling of shallow spherical shells. J. Appl. Me& 40, 411-416 (1973). 12. R. E. Ball, A program for the nonlinear static and dynamic analysis of arbitrarily loaded shells of revolution. Computers and Structures, 2, 141-162(1972). 13. J. Mescall and T. Tsui, Intluence of damping on the dynamic stability of spherical caps under step pressure loading. A.1.A.A. J; 9, 1244-1247(1971). 14, 3. N. Goodier and I. K. M&or, The elastic cylindrical shell under nearly uniform radial impulse. J. Appl. Me& 31, 259-266 (1964). 1.5.Ii. E. Lindberg, Bucklingof a very thin cylindrical shell due to an impulsive uressure. J. AvpL Me&. 31, 267-272(1964). 16. D. L. ‘Andersdnand H. E. L&dberg, Dynamic pulse buckling of cylindrical shells under transient lateral pressures. A.J.A.A, J. 6, 589-598(1968). 17. C. C. Cromer and R. E. Bail, Dynamic buckling of cylindrical shell, 1. Engng Mech. Div. EM3, 657-671(1971). 18. I. K. Mclvor and D. A. Sonstegard, Axisymmetric response of a closed spherical shell to a nearly uniform radial impulse. 1. .&oust. S&. Am. 48, 1540-1547(1966). 19. F. H. Ho and W. A. Nash, Dynamic buckling of spherical shells. Proc. Symp. No~-C~ass~co~ Problems. Intemation~ Association of Shell Structures, Warsaw (1963). 20. G. R. Abrahamson, Critical velocity for collapse of a shell of circular cross section without bu&ing. J. Appl. Mech. 41, 407-411 (1974). 21. R. L. Carlson, Structural instability induced by creep. Recent Progress in Applied Mechanics, The Folke Odqvist Vofume (Edited by B. Broberg, J. Hult and F. Niordson), pp. 161-179. klmqvist‘$ Wiksell,.Stockholm (1967). 22. R. E. Fulton and F. W. Barton, Dvnamic buckling of shallow arches. J. Engng Me&. Diu. EMj, 865-877(19’7i). 23. I. K. Mclvor, Discussion of dynamic buckling of shaiiow arches, 1. Engtw Me&. Div. EM2, 478-481 (1972). 24. K. Marguerre, &r Theorie der gekriimmten Platte grosser Formanderun& Pruc. Sfh Inter. Con& Appl. Me&. p. 93 (1938). 25. N. Akkag, Asymmetric buckling behavior of spherical caps
Asymmetric dynamic analysis of shallow spherical shells under uniform step pressures. .I. Appl. Mech. 39, 293-294 (1972). 26. J. C. Houbolt, A recurrence matrix solution for the dynamic response of aircraft in gusts. Rept. 1010,NACA (1951). 27. M. L. Potters, A matrix method for the solution of a secondorder difference equation in two variables. Reporf MR19, Mathematich Centrum, Amsterdam, Holland (1955). 28. R. E. Blum and R. E. Fulton, A modification of Potters’
251
method for solving eigenvalue problems involving tridiagonal matrices. A.I.A.A. J. 4,2231-2232 (1966). APPENDIX
Asymmetric dynamic buckling loads for clamped shallow spherical shells under a uniform step pressure of infinite duration are given in this Appendix. The notation NS stands for “not specified”.
Table 1. Problem
1 I
12
10.37
1
!
I
I a /Problem
3
/ I
II bi
Rcif.(7,81
1
j I
I
Ref.{111
0.511 I
I
5,e
I
1 Ref.
1 I
(91 =
I
I
I
I
J
“Bifurcation loads, lower bound. bAsymmetric dynamic snap-through buckling loads. ‘Of the two shells tested experimentally, one buckled axisymmetrically and the other asymmetricaUy. dIf n = 1 checked is not known. ‘It seems that n = 3 is not checked. ‘The threshold is at p = 0.47. ‘Taken from Ref. [ll]. “It seems that n = 3 is not checked. In Ref. [ll] it is given that Stricklin et al. [8] encountered asymmetric dynamic buckling also for A = 9, whereas in the present investigation axisymmetric dynamic buckling governed for this case.