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On the instabilities of an externally loaded shallow spherical shell∗
Inr. 1. Non-Linear Mechanics. Vol. 17. No. 2. pp. 85403. Printed in Great Britain.
1981.
ON THE INSTABILITIES OF AN EXTERNALLY LOADED SHALLOW SPHERICAL SHELL*
Lawrence
Livermore
HARRY P. ALESSO National Laboratory, University of Catifornia, Livermore, U.S.A.
CA 94550.
Abstract-The simple von Kbrman model of a clamped shallow elastic cap subjected to external pressure is reformulated as an elementary catastrophe. Conceptual understanding of load deflection behavior is substantially improved as a result. Three distinct modes of deflection behavior are identified. One snap-through type behavior is substabtiated by comparison to experimental data. I. INTRODUCTION
This paper discusses the application of a new mathematical tool called Elementary Catastrophe Theory (ECT) to the analysis of instabilities in shells. ECT includes a visual model that illustrates underlying phenomenological processes that occur near singular and critical points. In addition, ECT is capable of providing significant qualitative and quantitative information about regions near points of instability. After an overview of ECT, we will apply the method to instabilities of a shallow spherical shell subjected to an external load. The next section has a brief introduction to catastrophe theory, a general technique in non-linear mathematics that explores those solution regions where sudden jumpsin solutions occur while the parameters of the equation are changing smoothly. In the mid-1960s RenC Thorn suggested using Poincare’s qualitative dynamics as a basis for a topological theory of dynamical systems. Thorn’s famous list of seven elementary catastrophes now classifies discontinuous behavior in natural phenomena. Next, early ECT models will illustrate the behavior of Euler arches and struts to provide background for the discussion of instabilities in spherical shells. The first example is Zeeman’s [I] ECT model of a simple Euler arch and is followed by a model for an arch with pinned supports. Next, Zeeman’s [l] simple Euler strut demonstrates how elasticity of a beam contributes to the problem. Finally, Zeeman’s [l] pinned Euler strut demonstrates how higher modes of buckling may be explored. This last case provides insights into the behavior of higher buckling modes of spherical shells. Following these basic examples, we will reformulate the total potential energy of a shallow elastic spherical shell under external load, modeled by von K&man [2], as an elementary catastrophe. This alternative representation will make the behavior of deflections in a loaded spherical shell more readily understandable. One distinct advantage is that the ECT model visually presents distinct types of buckling behavior for the spherical shell-behavior that depends on geometric characteristics of the shell and on loading conditions. The concluding section presents experimental data that substantiate the theory developed. 2. CATASTROPHE
THEORY
2. I Background Catastrophe theory, proposed by RenC Thorn in the technique in non-linear mathematics. Thorn’s [3] work, bifurcations, is based on two assumptions. 1. The solution system arises from a potential function 2. The entire morphology of solutions is determined by to define the potential function. *This work was performed Livermore National Laboratory
NLM
Vol.
17. No. 2-B
under the auspices of the U.S. under Contract W-7405-Eng-48.
196Os, is a powerful new a method of classifying V and its derivatives.
values of parameters
Department
of Energy
used
by Lawrence
86
H. P. ALESSO
In catastrophe theory, one first defines a potential function V for the physical phenomenon being studied; i.e., V = V(ai, Xi) where Oi are parameters and xj are behavior variables, any of which may be functions of time. From the potential function, catastrophe theory defines a topological form consisting of a behavior surface B (described by the behavior variables) and a control plane C (described by the parameters). Figure 1, to be discussed more fully later, shows the relationship of these two surfaces. The objective of catastrophe theory, somewhat different from those of singularity and bifurcation theory, is to determine characteristics of morphological events that take place in transitions at bifurcation points of the potential function. In the neighborhood of a singularity, the locus of solutions are called catastrophes and form hypersurfaces, which allow discontinuous jumps between surfaces. Visualizations using the topological form are often quite valuable in understanding processes that take place at singularity points of the potential function, where stability characteristics are likely to change suddenly. Solutions exist as long as the potential function V does not vary explicitly with time. When a hypersurface is crossed, a switch in the dynamics of the system occurs. Under certain circumstances, established symmetry and dynamical behavior of a solution system spontaneously change into a new symmetry with a new behavior. When such symmetry change is eminent (i.e. the limit of structural stability is reached), multiple solutions or behavior may become available. At that time, the original solution is about to branch at a critical point. To establish the nature of the singularity’s stability, examine the trajectories in its neighborhood. If all trajectories close to the singularity approach it asymptotically (as f + w), the singularity is said to be asymptotically stable and is defined as an atlractor. If trajectories move away as t + Q), the singularity is asymptotically unstable and is defined as a repeller. In applying catastrophe theory, one constructs the behavior surface that includes all the potential function’s attractors, repellers, saddle points and rest points, and then defines how the solution changes as it moves from attractor to attractor along the surface [4-g]. 2.2. Elementary catastrophe theory (ECT) General catastrophe theory at present is a concept with limited applications because too little is known about the behavior of arbitrary vector fields to rigorously define attractors in most problems. However, by restricting problems to those whose
Fig. 1. Model of the cusp, one of the simpler elementary catastrophes.
On the instabilities of an externally loaded shallow spherical shell
87
behavior surfaces are defined by gradient vector fields, the mathematically rigorous theory of ECT has been developed. Gradient vector fields are of great interest because nearly all trajectories on the behavior surface tend to approach a point attractor, and an attractor point is, in general, a minimum of the potential function V of the system. The most useful applications of ECT are to systems whose states are determined by relative minima of an explicit potential function V. The parameters used in defining V determine the locations of those relative minima. In ECT, the potential function must be infinitely differentiable with respect to the behavior variables when the parameters are held constant, (i.e., any cross-section of the behavior surface generated by holding parameters constant will be continuous). Under these conditions, a smooth change in the parameters can give rise to a discontinuous jump on the behavior surface. ECT is valuable in that every structurally stable way in which a smooth change in potential V can give rise to a discontinuous change of state is described by one of the elementary catastrophes. Thorn found that under static conditions there are exactly seven elementary catastrophes possible if the potential function contains no more than two behavior variables (which ultimately determine the behavior surface) and no more than four parameters (which determine the control plane geometry). If the potential with one behavior variable, x, shows a local minimum --I d2V =c>o 2 dx2 I X=0 c may change its sign when certain parameters of the system under consideration are changed. This turns the stable point x = 0 into an unstable point for c < 0 or into a point of neutral stability for c = 0. In the neighborhood of such a point the behavior of V(x) is determined by the next non-vanishing power of x. We will call a point where c = 0 an instability point. Assume c#Osothat
V(x)=cx3+....
We shall show later that in practical cases, V(x) may be disturbed either by external causes, which in mechanical engineering may be loads, or internally by imperfections. Let us assume that these perturbations are small. Which of them will change the character of V(x) the most? Very close to c = 0 higher powers of x, e.g. x4, are much smaller so that such a term presents an unimportant change of V(x). On the other hand, imperfections or other perturbations may lead to lower powers of x than cubic so that these can become dangerous in perturbing V(x). Here we mean by ‘dangerous’ that the state of the system is changed appreciably. The most general case would be to include all lower powers leading to V(x) = A + Bx + Dx2 + cx3. Adding all perturbations which change the original singularity V(x) in a non-trivial way are called according to Thorn ‘unfoldings’. In order to classify all possible unfoldings of V(x) we must do away with superfluous constants. First, by an appropriate choice of the scale of the x-axis we can choose the coefficient c equal to I. Furthermore,‘we may shift the origin of the x axis by the transformation x=x’+6
to do away the quadratic term. Finally, we may shift the zero point of the potential so that the constant term vanishes. We are thus left with the ‘normal’ form of V(x) V(x) =
x3+ UlX.
88
H. P. ALESSO
Readers interested in identification and discussion of elementary catastrophes other than the cusp, which will be discussed in the next section, are referred to a previous paper by Alesso on the subject [9]. 2.3. The elementary cusp catastrophe In this section we will discuss the cusp, one of the simpler elementary catastrophes. Later, we will reformulate in terms of a cusp catastrophe the problem of instabilities of a shallow spherical shell subjected to an external load. Figure 1, previously used in presenting basic terminology of catastrophe theory, shows a cusp. The cusp model is formulated from a potential function of the form: V(x, a,, a& = x4/4 + a,x2/2 + a2x.
(1)
The cusp has two control parameters and one behavior variable. The behavior surface for the cusp consists of all points that satisfy the equation: d V/dx = x3 + a,x + a2 = 0.
(2)
In constructing the visual model of a cusp, as in Fig. 1, the two control parameters (a,, a2) are plotted on the control plane. The parameter that appears as the coefficient of x in the equation of the behaviour surface has an axis nearly parallel to the folds in the cusp. This parameter will be called the splitting parameter or factor. The parameter that appears as a constant has an axis nearly normal to the folds and will be called the normal parameter or factor. The behavior variable x is plotted on the vertical axis (normal to the control plane). As may be seen in Fig. 1, the behavior surface contains a fold, which means that for certain combinations of the control parameters, there are multiple points on the behavior surface. By taking the discriminant of the cubic in equation (2) and setting it equal to zero, we obtain: (a,/3)‘+ (a2/2)’ = 0.
(3)
Points on the control plane that satisfy equation (3) form a set of bifurcation points K in the form of a cusp-from which the catastrophe gets its name, cusp. Inside the cusp all points lie below multiple sheets of the behavior surface. That is, the folded region of the behavior surface generated by equation (2) projects onto the control plane as the region within the envelope defined by equation (3). In terms of stability, inside the cusp there are two stable states in competition; outside the cusp there is only one stable state. Points on the cusp itself are projections of those points on the behavior surface at which a catastrophe is imminent-that is, at points on the fold lines of the behavior surface a sudden change from one stable state to another is possible. The cusp model has one singularity, which is located at the origin of the x, a,, a2 coordinate system. Locating the origin at the singularity is possible through a linear translation of axes applied to the potential function V. To locate points on the behavior surface of the cusp, select numerical values for a, and oz. Then, determine what value or values of x, with these values of a, and u2, will satisfy d V/dx = 0 and plot this value (or values) on the normal to the control plane through the point (or, a2) of the plane. The behavior surface, when completed, will be aunique, smooth surface with one singularity. Each point (a,, a2) of the control plane has at least one point of the behavior surface directly above it. If there is only one point above, the solution system is stable for that combination of a, and u2. However, in the cusp model, a section of the behavior surface folds to form a pleated surface with three sheets that narrow from front to back and then vanish at the singular point. All points on the pleated surface represent possible solutions. Points on the top and bottom sheets represent minimum points of the potential function V, while points on the middle sheet represent maximum values of V. Points
On the instabilities of an externally loaded shallow spherical
shell
89
on both the upper and lower sheets represent possible system behavior. However, since the middle sheet represents maximum potential, it is inaccessible as a solution. The fold lines between sheets contain the inflection points of the potential function V and project onto the control plane as the bifurcation set K, a cusp in the present example. As long as the projection of a solution path on the behavior surface lies outside the cusp, behavior of the solution system is smooth. When the projection of the path crosses the cusp, a catastrophe occurs and the behavior path jumps from the top to the bottom sheet of the behavior surface, as in path A in Fig. 1. Similarly, a path along the lower sheet whose projection crosses the cusp will include a catastrophe in which the jump is from the lower to the upper sheet. Catastrophes occur because energy minima in a physical system are attractors. The catastrophes just described, being in a system influenced by attractors, are part of typical paths followed as the system assumes a new state of static equilibrium. In summary, the cusp model has five characteristics of note. 1. Within the cusp, the potential function V is bimodal. The two stable solution states are represented by the upper and lower sheets. 2. The middle sheet is inaccessible as a possible solution to V if V represents a physical problem. 3. Sudden transitions take place between stable states on the upper and lower sheets when a fold line is reached. 4. A hysteresis type path of stable solutions can be generated along the trace of the intersection of the behavior surface with any plane normal to the control plane. 5. If the system approaches the singularity in the direction of the fold region, the singular point presents a dilemma; namely, the divergent choice between the upper and lower sheets of the behavior surface. The dual cusp, a related model, is defined as a cusp elementary catastrophe with the middle sheet as the stable equilibrium surface and the upper and lower sheets as unstable equilibrium surfaces. The potential function V for the dual cusp is: v = - (y4/4 + cryv2 + Ey) and the behavior
(4)
surface is defined by: d V/dy = - (y-‘+ cry + E) = 0.
In Zeeman [I], the dual cusp was demonstrated as the behavior of the second harmonic mode introduced into the solution by the presence of the parameter E, the physical setting to represent an imperfection. He also gave the relation between the cusp and its dual as: xz + 4y= = 9 where x = Behavior variable of the cusp = first harmonic mode under a perturbation; y = Behavior variable of the dual cusp = second harmonic mode under a perturbation r = Relative behavior variable = first harmonic mode under no perturbation. 3. ECT
MODELS
QF THE
INSTABILITIES
OF ARCHES
AND
STRUTS
3. I. Background The static instability of a structure under load is, in general, analyzed by the structure’s non-linear load-deflection behavior. Euler in 1744 first analyzed the buckling of a beam under increasing load and found the deflection to have diverging behavior. Many similar problems have been solved in catastrophe language [I, 10, I I]. Although most of these examples were already well understood in terms of prior investigations using bifurcation theory, many of the ECT formulations provided conceptual improvements. This section will provide a mathematical entry point for studying buckling in terms
90
H. P.
ALESO
of ECT. First, the simple Euler arch [l, 11-131 will be modeled by two cases that demonstrate the cusp catastrophe. Second, the simple Euler strut 11, 111 will be used to show how elasticity of a beam affects the cusp catastrophe when compression and load act as control parameters. Abrupt changes in behavior of the strut will be seen as the strut changes from straight to upbuckled and then to downbuckled as the control parameters vary. Third, the pinned Euler strut [l, 11-131 will be used to examine harmonics that may occur in buckling. The pinned case, however, is not only an ordinary cusp, but includes a dual cusp. The dual cusp has the stability of its topological form reversed; normally stable areas are unstable. An interesting result of this case is that specific harmonic states are associated with discrete values of loading. 3.2. The simple Euler arch Case 1. The Euler arch [ 1, 111 is used as a simple entry point into beam instability. It displays the five characteristics of the cusp catastrophe previously listed in Section 2.3. The configuration of the arch can change radically, depending on the path followed along the behavior surface of the associated cusp. The arch (see Fig. 2) consists of two rigid arms of equal length joined at P by a spring with the spring constant ~1. The arms, which are initially collinear (i.e., 8 = MO”), are supported at the ends and a compressive force 6 is applied as shown. The two rigid arms will remain at 180” until the force @ overcomes the resistance of the spring and a new value of 8 is assumed (shown in Fig. 2 as buckled position 1). If /3 is then held constant and a gradually increasing load a is applied at P, a critical load a will be reached, at which time the arms will catastrophically snap into the opposite buckling position, shown as buckled position 2. To model this behavior by ECT, first sum the moments of the system by taking one of the arms as a freebody and summing about P to obtain: MP = 2@ + (a/2)f. cos x - /3L sin x where x is the angle the arms make with the original position and L is the length of the arm; To obtain the potential function for one arm, integrate over x, assume L to be of unit length, and apply the boundary condition V(0) = 0. When both arms are considered, the resulting potential function V for the arch is: V(x) = 2j~x* + a sin x - 2p( 1 - cos x).
(5)
This equation represents the total potential energy of the system: the energy of the springis 2&, the energy of the load is a sin x, and the energy stored in compression is -2P(lcos x).
Fig. 2. Model of the simple Euler arch, consisting of two rigid arms connected by a spring, used with the ECT method.
On the instabilities of an externally loaded shallow spherical shell
91
has one singularity point found by taking V’(x) = V”(x) = ,d/dx. Points where V’(x) = 0 form the behavior surface. Points where V”(x) = 0 form the fold lines and the point where V”‘(x) = 0 is the singularity point. By. combining derivatives of equation (5) appropriately and letting /3 = 2~ + b (where b is a translational constant), the singularity point can be located at a = 0, /3 = 2~, and x = 0 [l]. By a Taylor expansion, equation (5) becomes The behavior
surface
V”‘(x) = 0, where the primes represent
V(x) = 2~’
where 05 and 06 represent
+ a(x - x3/6 + ti) - 2p[ 1 - (1 - x2/2 + x4/24 + 06)] higher order terms. Regrouping and eliminating /3 gives
V(x) = ax - bx2 + /~x’/6 - ax3/6 + bx’/6 + @ + 0’.
Note that @‘/6 is independent of the control parameters a and b and p > 0. Further, since x is small, -ax3/6 and +bx’/6 are relatively insignificant in comparison to the fist two terms. Therefore V(X) = /.Ax’/~- bx* +
ax.
(6)
Equation (6) has the form of equation (1) and is therefore the potential function of a cusp catastrophe with load a and compression -b as control parameters. The angle x is the behavioral variable. The singular point is located at the origin. Figure 3 shows the cusp model. In Fig. 3 the dashed trajectory illustrates one possible behavior of the rigid arms. At point 1 (the singular point), the arms are at 180”, there is no a load, and /3 is at the critical value of 2~. With any increase in p (and consequently 6). the arms buckle (in our example using Fig. 2 they buckle up) and begin to follow a path indicated by 2. Along path 2, increasing b smoothly increases x on the behavior surface. At point 3, b is held constant and a load a is applied and increased until the path along the behavior surface reaches the fold line. At the fold line, a critical value has been reached. The system again becomes unstable and buckles down to point 4 on the behavior surface. By holding /3 constant and varying the load a, the buckling position can be flipped back and forth in a cyclic manner shown in Fig. 4. This illustrates the hysteresis property of the cusp catastrophe.
Fig. 3. The cusp catastrophe model for the simple Euler arch of Fig. 2.
92
H. P. ALESSO
X
a
-I
Fig. 4. Hysteresis
type behavior possible with the simple Euler arch shown in Fig. 2.
Case 2. Another simple arch, shown in Fig. 5 [ll-131, consists of two arms composed of linear extensional springs pinned together at one end and fixed to rigid supports at the other. The stiffness coefficient of the springs is p, the distance between supports is 2R and a load a is placed where the arms are joined. The angle x is variable and has an initial value of x,. The strain energy of the springs is U(x). The original length of the spring is (R/cos ~0) and the length becomes (R/cos x) as the load a is applied; thus U(x) = ~(R/cos x0- R/cos x)* [12]. If x0 and x are small then
Thus the strain energy becomes U(x) = ; ~R’(x: - 2xhx2+ x4)+ 06. The potential energy of the load is -aD(x) in which the deflection due to the load a is D(x) = R tan x0- R tan x. For small angles the deflection can be taken as D(x) = R(x, - x).
The potential function V of the system is then V(x, a, x0) = U(x) - all(x),
Fig. 5. Model of the simple Euler arch, consisting of two linear extensional the ECT method.
springs, used with
On the instabilities of an externally loaded shallow spherical shell
93
Of
where pR* > 0. The equation of the behavior surface, upon extracting
a constant multiple, is
dV x=xJ-x;x++o. This equation is of the form of equation (1) and therefore a cusp catastrophe, with - xi and +a/*R as the independent control variables, and with x as the behavior variable (see Fig. 6). The dashed path on Fig. 6 shows how first increasing x0 (path 2) and then load a (path 3) until a critical point is reached causes a catastrophe in which the arch changes from an upbuckled to a downbuckled position. 3.3. The simple Euler strut Replacing the springs and rigid arms of the last section with an elastic beam complicates the problem. To now account for the action of the entire beam it is necessary to analyze and sum a collection of beam cross-sections. Figure 7 shows the Euler strut [l, 111 for which L denotes the length, p the spring constant, s the distance along the arched strut, and f(s) the height at distance s from the left end. In this section, x is the vertical displacement at the center of the strut. (See Zeeman [l] for solution of this problem.) The compressive force a acts as the cause, and the shape f(s) is the ‘effect’, found to be x sin (?rs/L). Also p = p(.lrlL)* is the critical compressive force that results in initial buckling from a straight beam. In Fig. 8 the strut is stable from fl = 0 to p = ~(?rlL)*. At that point the strut can no longer absorb the compressive force internally and is at the limit of stability. Any increase in p causes the strut to become unstable and forces it to seek a stable equilibrium position (either up- or downbuckling). Further increases in p cause increases in the amplitude of the sine wave shape of the strut. In Zeeman[l], the total potential energy of the system is used to
\ Upbuckle
Fig. 6. The cusp catastrophe
model for the Euler arch of Fig. 5.
H. P. ALESSO
Fig. 7. Model of the simple Euler strut, as formulated by the ECT method.
a. Stable 6-h b. Unstable c. Stable
(Critical point Fii. 8. At critical load for an Euler strut, two stable states are equally possible if the load is
increased.
form the cusp potential function
x4 b v=gj-zx2+ax [where b = vr(/3 - l)]. Then dV x3 x=16-bx+a=O. Letting a = 0 and solving for x yields
which forms the parabola in Fig. 8. Until now a has been taken as zero. If a is now applied to an upbuckled strut, it will remain in an upbuckled position until a critical value for a is reached. Beyond this value the strut will become unstable again and snap to the downbuckled position. The Euler strut exhibits the same characteristics of the cusp catastrophe as the Euler arch. 3.4. The pinned Euler strut The pinned Euler strut [l, 11, 121 illustrates how imperfections of a strut affect its capacity to take loads. The model shown in Fig. 9 is that of an Euler strut of length L pinned between supports d apart. In Fig. 9: x = the centerline height of the strut for the first harmonic. y = the center height of the second harmonic. r = the difference between the length of the strut, L, and the distance d between supports; i.e., r = L - d. f(s) = s sin s + y sin 2s. The imperfection parameter B locates the distance from the centerline to where the load a is applied and represents a manufacturing imperfection. Though Golubitsky
On the instabilities of an externally loaded shallow spherical shell
95
Fig. 9. Model of the pinned Euler strut showing typical first and second harmonic modes in a strut having imperfections.
[14] suggests that the imperfection parameter E may be more complicated than this, we will use this simplified approach adopted by Zeeman. Figure 10 shows that the behavior surface of the strut modeled in Fig. 9 is a dual cusp catastrophe with e and +a as control parameters, and y as the behavior variable. Since this is a dual cusp, the middle sheet represents stable behavior, while the upper and lower sheets are unstable. At point A in Fig. 10(a), the load is initially a = 0 and the shape of the strut is a maximum first harmonic with a zero second harmonic (i.e., x is maximum and y is zero). As a increases (along path AB), the second harmonic grows
V~-(v4-ey+ayz) Control plane E
A. C
b)
V
\b8
C
A
CY
rr critical
Fig. IO. (a) Model of the dual cusp catastrophe that represents the pinned Euler strut with imperfections. (b) By carefully loading and unloading, the unstable second harmonic position C can be reached.
96
H. P. ALESSO
until the system reaches a,-hticsrat point B, where the system is about to snap catastrophically. Assume now that the load is not increased further, but rather is slowly decreased. The result is that the strut now follows the path BC on the upper sheet, the configuration of the second harmonic instead of the first harmonic. Point C represents a maximum second harmonic and zero first harmonic. Although the strut can remain at point C, any very slight perturbation will cause the strut to snap back to the first harmonic maximum position A on the middle sheet from the unstable upper surface. Referring now to Fig. 10(b), point A (on the stable middle sheet) is the unloaded shape with maximum first harmonic x and zero second harmonic y. The second harmonic height y increases with increasing load a until point B is reached. Point B is on the fold line and directly above the point p of the bifurcation set on the control plane. If only the first harmonic is considered, the behavior would snap through when B is reached. However, as the second harmonic is now assumed to dominate and the load is reduced, the behavior will follow the curve BC instead of snapping through. At C the behavior is in unstable equilibrium, ready to snap back to the original first harmonic shape. An alternative qualitative behavior would be to follow path AB to B as the load is increased, and then return to A along the path BA as the load is removed. The interrelationship between the first mode cusp and the second mode dual cusp is shown in Fig. 13 (see Zeeman [l]).
4. THE
INSTABILITIES
OF AN EXTERNALLY
LOADED
SPHERICAL
SHELL
4.1.
Overview The difficulties encountered in the study of cylindrical and spherical shells arise more from the perversity of the problem itself [2, 12-251 than from the particular mathematical formulation. However, considerable insight can be gained by formulating such problems in terms of ECT. Alesso [9] formulated an ended ring in a rigid cavity as a catastrophe and cited sample calculations that agreed with experimental data. Zoelly and Leibenson used linear theories in early investigations of thin elastic spherical shells under outside pressure. In the 193Os, Sechler and Bollay carried out experiments on buckling pressures that gave data within 25% of pressures predicted by linear theories. In 1939, von Karman and Tsien used the Ritz method to obtain a non-linear and non-monotone relation between load and deflection. They found that until buckling occurs, a perfectly spherical shell under an increasing pressure will contract by a constant amount 6,. A general review of buckling of a shallow spherical shell can be found in von K&man [2]. In that reference, the von Karman non-linear equations for a shallow spherical shell segment are given as:
DV2V2w=[w,fj+~V2f+, and the compatibility
(7)
equation,
& V2V2f= w;,-
w3,w’,
-+v2w
where
[w,fl = K.&, + wyyfxx - 2w*,fx,
On the instabilities of an externally loaded shallow spherical shell
91
and E = modulus of elasticity fY = bending stiffness of plate 4 = periodic impressed force w = radial deflection of middle surface f = stress function t = shell thickness R = radius of middle surface and subscripts indicate partial derivatives. Note that as R + 00 these reduce to plate theory [l 11. 4.2. ECT formulation of von Khndn model The non-linear differential equation (7) has no known exact, general solution. Although Hyman [20], Ball [26] (for the dynamic case), Kalnins [27], and Varadan [24] have made efforts to calculate approximate solutions for variations of these equations, a comprehensive qualitative and quantitative picture has not emerged. In fact, Hyman [20] introduced higher mode buckling behavior of a new qualitative type, which added to the complexity of understanding the overall behavior. Modeling complex problems in the form of non-linear algebraic equations rather than non-linear differential equations is often useful. von Karmin and Kerr [2] in 1965 suggested the following model for the externally loaded shallow cap. As the model, a bent beam clamped at the supports (see Fig. 11) is assumed to contract or extend axially except in the shaded regions at the supports and in the middle, in which regions all bending is assumed to occur. Straight parts of the beam (unshaded regions) are assumed to remain rigid. For this model, von KBrmin and Kerr formulated the following non-linear algebraic relationship for the stationary solutions of the potential energy: aV* _=
as*
0 =
ktp*(l
- h*2)1’2-2(h*
involving the non-dimensional
- 6*)k(l
-k)-$(l
- h*2)“2 arccos (’ -
f*‘*)
quantities
where p = external uniform load, h = undeformed depth of shallow shell, S = deflection from undeformed height, L = length, t = von Karman’s expansion term to define regions of pure bending, V= potential energy, E = modulus of elasticity, a = area of beam cross-section and k = (1 + 8*2-2h*6*)“2.
Undeformed positi
Fig. Il. The von KBrm6n model of an externally loaded shallow cap. As loading is applied, bending occurs in the shaded areas only and axial changes in length in the unshaded areas only.
98
H.P.
ALESSO
Equation (8) will now be formulated as an ECT cusp representation given later by equation (11). First, expand the arccos as a Taylor Series. Then, let F = l/k and treat F as a Legendre polynomial generating function in which we neglect terms of order 0(6*‘) and higher. This gives, av* as*
-=O=A++h*
F*-B(l-
-2S*-2F(h*-a*)+?
h*cS*)F$!(l
-h*6*)‘F5 + 0(6*‘),
where A = p*d/( 1 - h*‘); B = - (2:/3L)d(l-
h**);
F = (1 _ 2h*S* + &49-1/*= g (6*)‘P,(h*); P,(h*) = Legendre polynomials.
Taking only the I= 0 term in Fs and neglecting 0(6*‘) gives:
av*
-=0=A+2h*-2S*-2F(h*-S*)+~~-B(1-h*6*)F3-~(1-h*6*)3
as*
where F = 1 + 6*P, + S**P* + S*3P3+ 0(6*‘) F = 1 + 2S*P, + S**(2P2+ P:) + 6*32(P3+ PIP*) + O(S*‘) F3 = I + 3S*PI + 8**(3P2 + 3P:) +
S*3(3P3
+
6PlP2 + Pi) + O(S*‘).
By multiplying out polynomials and gathering terms, we find,
aV* as*
-
= 0 = b, + 2b28* + 3b3S** + 4b4S*3 + 0’
(9)
where
2b2=-2h*P,+B
3b3= B(3P,h*y+
> rP2+;
4b4= - B 3P3+6P,P2+P:-
7rP,P2- “p,+7
and P,= h*
P2 = ; (3h** - 1)
- 2P,h* + 2P2 + 3P,Bh* + 3BP:h*
On the instabilities
Physical
restraints:
of an externally loaded shallow spherical shell
t’:Ot?
Note: b4 # 0 for t = h* = 0. Equation (9) is 4-determinate. Now apply a change of coordinates to equation (9) such that x = CT*-+-c.
Substituting equation (10) into equation coefficient of 8** goes to zero) gives:
(9) and picking C = (bJ46,)
(10) (so that the
aV* = x3 + a,x + uz = 0 i)X
where a,
=$-$4
:_ 4
j
Jr, L, h*)
4.3. Predicted qualitative behavior of the cusp model Equation (11) has the form of an elementary catastrophe, the cusp described in Section 2.3. From the topological shape of the cusp (see Fig. 1) and the functional relationships of the parameters a, and a z, we will now make qualitative predictions about the behavior of the deflection 6 as we vary arch height h* and load p*. In equation (11), changes in parametric conditions (t, L, h*, p*) cause the behavior variable x (which equals 6* + C, where C is a constant) to move along the topological surface of the cusp elementary catastrophe represented by the equation. If we let L be a constant and assume that t is zero in the area of interest, we find that the potential can be represented by:
Note that in equation (12) the load p* appears only in a2. The mutual dependence of a, and a2 on h* is a problem we will not discuss herein. The underlying similarity between equation (12) and the models presented in Sections 3.2-3.4 is apparent. In each model the external load appears exclusively in the normal factor a2. Likewise, in each case the splitting factor a, contains a term that represents a measure of arch of the structure. The cusp of equation (12) predicts first mode behavior, for which there are three distinct qualitative possibilities. These possibilities are represented by paths A, B and C of Fig. 12. Path A is typical of the type behavior in which there is a slight compression under external load (1 to 2), followed by a snap-through phenomena (jump from 2 to 3). Path B illustrates direct snap-through behavior. The distinction between paths A and B depends on the geometry of the shell (t, L, h*). Path C, which represents the case where the internal pressure is greater than the external pressure, produces continuous deformation in an outward direction, 4.4. Behavior of an externally loaded shallow spherical shell with an imperfection Buckling of the pressure loaded shallow shell is usually analyzed for symmetric
loo
H. P.
ALESSO
Fig. 12. The cusp catastrophe model obtained by reformulatkg the VORGrm9n r,lodel ~-41: .. shallow cap by the ECT method. Paths A, B and C represent the three b, ;ic types of deformations that can occur.
behavior as in Sections 4.1-4.3. Analyses such 1 that of Huang [I81 show that asymmetric buckling modes can exist at pressure: rlow critical pressures found for static symmetric snap-buckling, and especially if ‘;: perfections in the shape’ of the shell exist. An equation analogous to equation (11) for the asymmetric dual cusp (see Section 3.4) is given by: aV* -=y3+ay+r=0 JY
(13)
where y is the second mode behavior variable, a = f3(t, L, h*, p*), and E is an imperfection parameter of the sphere. The second harmonic y is significant in the von KBrmPn model when an offset load (due to imperfection 6) is included. Note that the external load p* now appears in the splitting factor rather than the normal factor. Figure 13 illustrates the interrelationship between the cusp and the dual cusp. The change in the way that p* appears in the controlling equations is reflected by the rotation in behavior space between the cusp and dual cusp. For a more complete discussion of the relationship shown in Fig. 13, see Zeeman [l]. In Hyman 1201, the behavior of the second mode of shallow shells was explored and hypothesized to be a continuous ‘closed loop’. I speculatively suggest that the behavior of the dual cusp model of equation (13) might account for Hyman’s closed loop behavior pattern. 4.5. Supportive evidence Preliminary computer calculations of the stationary solutions of the potential energy given in equation (11) gave good agreement with von KBrman and Kerr [2] as reported in their Figs. 9 and 11. Incidently, Fig. 11 in [2] defines the control plane characteristic cusp in terms of p* and h*. The cusp model of equation (ll), shown as Fig. 12, predicts the behavior of S*, the non-dimensionalized deflection of an elastic, externally loaded shallow cap. Figure 14 shows how S* behaves as the non-dimensionalized pressure p* and cap height h* vary along the three types of path shown in Fig. 12. Curves relating deflections x to p* and h* are similar but are translated along the deflection axis according to the
On the instabilities of an externally loaded shallow spherical shell Path A
IO1
Path C
Path B
Fig. 13. Diagrams representing the variation of deflection 6 with pressure p* and cap height h* for the three types of deformation shown by the paths in Fig. 12.
Behavior variable Dual
Dual cusp Fig. 14. Relationships of behavior surfaces and control planes for the
cusp and
dual cusp.
relationship x = S* + C, where C is the constant that locates the singular point of the cusp at the origin. Experimental data presented by Stephens and Fulton [28] and by Huang [18], and reviewed by Svalbonas and Kalnins [22], are plotted as load deflection curves in Fig. 15. The experimental curves compare favorably with the deflection behavior of a type A path on the cusp model shown in Figs. 12 and 14. 4.4. Conclusions The ECT method we have discussed provides improved conceptual insight into the behavior of shallow shells. After an overview of elementary catastrophe modeling of beam buckling, we have reformulated the simple von Karmin-Kerr model of a clamped, elastic shallow cap under external load as a cusp catastrophe. The resulting model presents three qualitatively different behavior types. NLM
Vol. 17. No. 2-C
H. P. ALESSO
102
01 0.20
I
I
0.30
0.40 Loading
Fig. 15. Experimental
I 0.50
I 0.60
(p’)
results of two studies support the qualitative behavior predicted by the ECT formulation of the von Karman model.
1. Initial continuous deformation followed by a discontinuous jump in deformation. 2. Initial discontinuous jump deformation. 3. Continuous deformation (under a net internal load). Experimental evidence in support of the first behavior type was presented. In addition, we have suggested a modeling approach for analyzing the second-mode behavior of a shallow shell.
Acknowledgement-It is a pleasure to acknowledge the valuable criticisms gestions of Prof. Hugh Keedy in the preparation of this manuscript.
and numerous
helpful sug-
REFERENCES 1. E. C. Zeeman, The buckling beam. Lecture Notes in Mathematics, Vol. 525, pp. 374-394. Springer, New York (1976). Collected Works of Theodore uon Kdrmdn, 1952-1963. von Kbrmln Institute, Belgium (1975). :: R. Thorn. Structural Sfobility and Morphogensis. W. A. Benjamin, Reading, MA (1975). 4. T. H. Brocker, Differential germs and catastrophes. London Mathematical Society Lecfure Notes Series. Cambridge University Press, Cambridge (1975). Y. Lu, Singularity Theory and an Introduction to Catastrophe Theory. Springer, New York (1976). i: D. O’Shea, An exposition of catastrophe theory and its application to phase transition. Queen’s Pap. pure oppl. Mafh. 47, IO-12 (1976). 7. D. H. Satlinger, Topics in stability and bifurcation theory. Lecrure Notes in Mathematics, Vol. 309. Springer, New York (1974). 8. G. Wasserman, Stability of unfoldings. Lecture Notes in Mathematics, Vol. 393. Springer, New York (1974). 9. H. P. Alesso. Elementary catastrophe theory modeling of an end loaded ring in a rigid cavity. Nucl. Eng &sign 53. 55-70 (1979). 10. C. Chillingworth, The catastrophe of the buckling beam. Dynamical Systems, Lecture Notes in Mafhematks, Vol. 468. Springer; New York (1974). II. T. Poston and 1. Stewart. Catastroohe Theorv and Ifs Aoolications. Pitman. San Francisco (1978). . , 12. J. M. T. Thompson, Phil.. Trans. RI Sot. L&d. 292. I-2j il979). 13. J. M. T. Thompson and G. W. Hunt, A General Theory of Elastic Stability. John Wiley, New York (1973).
103
On the instabilities of an externally loaded shallow spherical shell
14. M. Golubitsky and D. Schaeffer. Imperfect bifurcation in the presence of symmetry, Communs Pure appl. Math. 32, 21-98 (1979). IS. B. Bidiansky, Ed., Buckling of Structures. Springer, New York (1976). 16. B. Bidiansky. Theory of buckling and post buckling of lelastic structures. Adoonces in Appled Mechanics, Vol. 14. Academic Press, New York (1974). 17. J. S. Hansen, AIAA J. 15, 1638-1644 (1977). 18. N. Huang, Air Force Report AFOSR, 68-0469, TR7. February (1968). 19. P. T. Hus, J. Elkon and T. H. H. Pian. Note on the instability of circular rings confined to a rigid cavity. 1. oppl. Mech. X.559-562 (1964). 20. B. I. Hyman, Non-linear mechanics. Trans. Am. Sot. mech. Engrs 34.49-55 (1967). 21. T. H. H. Pian and L. L. Bucciarelli, Jr., Buckling of radially constrained circular ring under distributed loading. Inr. J. Solids Strucfures 3, 7 IS-730 (1967). 22. V. Svalbonas and A. Kalnins, Dynamic buckling of shells: evaluation of various methods, Nucl. Engng Design 44, 331-336 (1977). 23. J. J. Tuma and R. K. Munshi, Advanced Structural Analysis. McGraw-Hill, New York (1971). 24. T. K. Varadan, Snap-buckling of orthotropic shallow spherical shells, 1. appl. Mech. 45,44547 (1968). 25. E. A. Zagustin and Cl. Hermann, Stability of an elastic ring in a rigid cavity. 1. oppl. hfech. 34, 263-270 (1967). 26. R. E. Ball and J. A. Burt, 1. appl. Mech. 40.411-416 (1973). 27. A. Kalnins and V. Biricikoglus. J appl. Mech. 37, 629-634 (1970). 28. W. Stephens and R. Fulton, AIAA J. 7. 2120 (1969).
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1981.
ON THE INSTABILITIES OF AN EXTERNALLY LOADED SHALLOW SPHERICAL SHELL*
Lawrence
Livermore
HARRY P. ALESSO National Laboratory, University of Catifornia, Livermore, U.S.A.
CA 94550.
Abstract-The simple von Kbrman model of a clamped shallow elastic cap subjected to external pressure is reformulated as an elementary catastrophe. Conceptual understanding of load deflection behavior is substantially improved as a result. Three distinct modes of deflection behavior are identified. One snap-through type behavior is substabtiated by comparison to experimental data. I. INTRODUCTION
This paper discusses the application of a new mathematical tool called Elementary Catastrophe Theory (ECT) to the analysis of instabilities in shells. ECT includes a visual model that illustrates underlying phenomenological processes that occur near singular and critical points. In addition, ECT is capable of providing significant qualitative and quantitative information about regions near points of instability. After an overview of ECT, we will apply the method to instabilities of a shallow spherical shell subjected to an external load. The next section has a brief introduction to catastrophe theory, a general technique in non-linear mathematics that explores those solution regions where sudden jumpsin solutions occur while the parameters of the equation are changing smoothly. In the mid-1960s RenC Thorn suggested using Poincare’s qualitative dynamics as a basis for a topological theory of dynamical systems. Thorn’s famous list of seven elementary catastrophes now classifies discontinuous behavior in natural phenomena. Next, early ECT models will illustrate the behavior of Euler arches and struts to provide background for the discussion of instabilities in spherical shells. The first example is Zeeman’s [I] ECT model of a simple Euler arch and is followed by a model for an arch with pinned supports. Next, Zeeman’s [l] simple Euler strut demonstrates how elasticity of a beam contributes to the problem. Finally, Zeeman’s [l] pinned Euler strut demonstrates how higher modes of buckling may be explored. This last case provides insights into the behavior of higher buckling modes of spherical shells. Following these basic examples, we will reformulate the total potential energy of a shallow elastic spherical shell under external load, modeled by von K&man [2], as an elementary catastrophe. This alternative representation will make the behavior of deflections in a loaded spherical shell more readily understandable. One distinct advantage is that the ECT model visually presents distinct types of buckling behavior for the spherical shell-behavior that depends on geometric characteristics of the shell and on loading conditions. The concluding section presents experimental data that substantiate the theory developed. 2. CATASTROPHE
THEORY
2. I Background Catastrophe theory, proposed by RenC Thorn in the technique in non-linear mathematics. Thorn’s [3] work, bifurcations, is based on two assumptions. 1. The solution system arises from a potential function 2. The entire morphology of solutions is determined by to define the potential function. *This work was performed Livermore National Laboratory
NLM
Vol.
17. No. 2-B
under the auspices of the U.S. under Contract W-7405-Eng-48.
196Os, is a powerful new a method of classifying V and its derivatives.
values of parameters
Department
of Energy
used
by Lawrence
86
H. P. ALESSO
In catastrophe theory, one first defines a potential function V for the physical phenomenon being studied; i.e., V = V(ai, Xi) where Oi are parameters and xj are behavior variables, any of which may be functions of time. From the potential function, catastrophe theory defines a topological form consisting of a behavior surface B (described by the behavior variables) and a control plane C (described by the parameters). Figure 1, to be discussed more fully later, shows the relationship of these two surfaces. The objective of catastrophe theory, somewhat different from those of singularity and bifurcation theory, is to determine characteristics of morphological events that take place in transitions at bifurcation points of the potential function. In the neighborhood of a singularity, the locus of solutions are called catastrophes and form hypersurfaces, which allow discontinuous jumps between surfaces. Visualizations using the topological form are often quite valuable in understanding processes that take place at singularity points of the potential function, where stability characteristics are likely to change suddenly. Solutions exist as long as the potential function V does not vary explicitly with time. When a hypersurface is crossed, a switch in the dynamics of the system occurs. Under certain circumstances, established symmetry and dynamical behavior of a solution system spontaneously change into a new symmetry with a new behavior. When such symmetry change is eminent (i.e. the limit of structural stability is reached), multiple solutions or behavior may become available. At that time, the original solution is about to branch at a critical point. To establish the nature of the singularity’s stability, examine the trajectories in its neighborhood. If all trajectories close to the singularity approach it asymptotically (as f + w), the singularity is said to be asymptotically stable and is defined as an atlractor. If trajectories move away as t + Q), the singularity is asymptotically unstable and is defined as a repeller. In applying catastrophe theory, one constructs the behavior surface that includes all the potential function’s attractors, repellers, saddle points and rest points, and then defines how the solution changes as it moves from attractor to attractor along the surface [4-g]. 2.2. Elementary catastrophe theory (ECT) General catastrophe theory at present is a concept with limited applications because too little is known about the behavior of arbitrary vector fields to rigorously define attractors in most problems. However, by restricting problems to those whose
Fig. 1. Model of the cusp, one of the simpler elementary catastrophes.
On the instabilities of an externally loaded shallow spherical shell
87
behavior surfaces are defined by gradient vector fields, the mathematically rigorous theory of ECT has been developed. Gradient vector fields are of great interest because nearly all trajectories on the behavior surface tend to approach a point attractor, and an attractor point is, in general, a minimum of the potential function V of the system. The most useful applications of ECT are to systems whose states are determined by relative minima of an explicit potential function V. The parameters used in defining V determine the locations of those relative minima. In ECT, the potential function must be infinitely differentiable with respect to the behavior variables when the parameters are held constant, (i.e., any cross-section of the behavior surface generated by holding parameters constant will be continuous). Under these conditions, a smooth change in the parameters can give rise to a discontinuous jump on the behavior surface. ECT is valuable in that every structurally stable way in which a smooth change in potential V can give rise to a discontinuous change of state is described by one of the elementary catastrophes. Thorn found that under static conditions there are exactly seven elementary catastrophes possible if the potential function contains no more than two behavior variables (which ultimately determine the behavior surface) and no more than four parameters (which determine the control plane geometry). If the potential with one behavior variable, x, shows a local minimum --I d2V =c>o 2 dx2 I X=0 c may change its sign when certain parameters of the system under consideration are changed. This turns the stable point x = 0 into an unstable point for c < 0 or into a point of neutral stability for c = 0. In the neighborhood of such a point the behavior of V(x) is determined by the next non-vanishing power of x. We will call a point where c = 0 an instability point. Assume c#Osothat
V(x)=cx3+....
We shall show later that in practical cases, V(x) may be disturbed either by external causes, which in mechanical engineering may be loads, or internally by imperfections. Let us assume that these perturbations are small. Which of them will change the character of V(x) the most? Very close to c = 0 higher powers of x, e.g. x4, are much smaller so that such a term presents an unimportant change of V(x). On the other hand, imperfections or other perturbations may lead to lower powers of x than cubic so that these can become dangerous in perturbing V(x). Here we mean by ‘dangerous’ that the state of the system is changed appreciably. The most general case would be to include all lower powers leading to V(x) = A + Bx + Dx2 + cx3. Adding all perturbations which change the original singularity V(x) in a non-trivial way are called according to Thorn ‘unfoldings’. In order to classify all possible unfoldings of V(x) we must do away with superfluous constants. First, by an appropriate choice of the scale of the x-axis we can choose the coefficient c equal to I. Furthermore,‘we may shift the origin of the x axis by the transformation x=x’+6
to do away the quadratic term. Finally, we may shift the zero point of the potential so that the constant term vanishes. We are thus left with the ‘normal’ form of V(x) V(x) =
x3+ UlX.
88
H. P. ALESSO
Readers interested in identification and discussion of elementary catastrophes other than the cusp, which will be discussed in the next section, are referred to a previous paper by Alesso on the subject [9]. 2.3. The elementary cusp catastrophe In this section we will discuss the cusp, one of the simpler elementary catastrophes. Later, we will reformulate in terms of a cusp catastrophe the problem of instabilities of a shallow spherical shell subjected to an external load. Figure 1, previously used in presenting basic terminology of catastrophe theory, shows a cusp. The cusp model is formulated from a potential function of the form: V(x, a,, a& = x4/4 + a,x2/2 + a2x.
(1)
The cusp has two control parameters and one behavior variable. The behavior surface for the cusp consists of all points that satisfy the equation: d V/dx = x3 + a,x + a2 = 0.
(2)
In constructing the visual model of a cusp, as in Fig. 1, the two control parameters (a,, a2) are plotted on the control plane. The parameter that appears as the coefficient of x in the equation of the behaviour surface has an axis nearly parallel to the folds in the cusp. This parameter will be called the splitting parameter or factor. The parameter that appears as a constant has an axis nearly normal to the folds and will be called the normal parameter or factor. The behavior variable x is plotted on the vertical axis (normal to the control plane). As may be seen in Fig. 1, the behavior surface contains a fold, which means that for certain combinations of the control parameters, there are multiple points on the behavior surface. By taking the discriminant of the cubic in equation (2) and setting it equal to zero, we obtain: (a,/3)‘+ (a2/2)’ = 0.
(3)
Points on the control plane that satisfy equation (3) form a set of bifurcation points K in the form of a cusp-from which the catastrophe gets its name, cusp. Inside the cusp all points lie below multiple sheets of the behavior surface. That is, the folded region of the behavior surface generated by equation (2) projects onto the control plane as the region within the envelope defined by equation (3). In terms of stability, inside the cusp there are two stable states in competition; outside the cusp there is only one stable state. Points on the cusp itself are projections of those points on the behavior surface at which a catastrophe is imminent-that is, at points on the fold lines of the behavior surface a sudden change from one stable state to another is possible. The cusp model has one singularity, which is located at the origin of the x, a,, a2 coordinate system. Locating the origin at the singularity is possible through a linear translation of axes applied to the potential function V. To locate points on the behavior surface of the cusp, select numerical values for a, and oz. Then, determine what value or values of x, with these values of a, and u2, will satisfy d V/dx = 0 and plot this value (or values) on the normal to the control plane through the point (or, a2) of the plane. The behavior surface, when completed, will be aunique, smooth surface with one singularity. Each point (a,, a2) of the control plane has at least one point of the behavior surface directly above it. If there is only one point above, the solution system is stable for that combination of a, and u2. However, in the cusp model, a section of the behavior surface folds to form a pleated surface with three sheets that narrow from front to back and then vanish at the singular point. All points on the pleated surface represent possible solutions. Points on the top and bottom sheets represent minimum points of the potential function V, while points on the middle sheet represent maximum values of V. Points
On the instabilities of an externally loaded shallow spherical
shell
89
on both the upper and lower sheets represent possible system behavior. However, since the middle sheet represents maximum potential, it is inaccessible as a solution. The fold lines between sheets contain the inflection points of the potential function V and project onto the control plane as the bifurcation set K, a cusp in the present example. As long as the projection of a solution path on the behavior surface lies outside the cusp, behavior of the solution system is smooth. When the projection of the path crosses the cusp, a catastrophe occurs and the behavior path jumps from the top to the bottom sheet of the behavior surface, as in path A in Fig. 1. Similarly, a path along the lower sheet whose projection crosses the cusp will include a catastrophe in which the jump is from the lower to the upper sheet. Catastrophes occur because energy minima in a physical system are attractors. The catastrophes just described, being in a system influenced by attractors, are part of typical paths followed as the system assumes a new state of static equilibrium. In summary, the cusp model has five characteristics of note. 1. Within the cusp, the potential function V is bimodal. The two stable solution states are represented by the upper and lower sheets. 2. The middle sheet is inaccessible as a possible solution to V if V represents a physical problem. 3. Sudden transitions take place between stable states on the upper and lower sheets when a fold line is reached. 4. A hysteresis type path of stable solutions can be generated along the trace of the intersection of the behavior surface with any plane normal to the control plane. 5. If the system approaches the singularity in the direction of the fold region, the singular point presents a dilemma; namely, the divergent choice between the upper and lower sheets of the behavior surface. The dual cusp, a related model, is defined as a cusp elementary catastrophe with the middle sheet as the stable equilibrium surface and the upper and lower sheets as unstable equilibrium surfaces. The potential function V for the dual cusp is: v = - (y4/4 + cryv2 + Ey) and the behavior
(4)
surface is defined by: d V/dy = - (y-‘+ cry + E) = 0.
In Zeeman [I], the dual cusp was demonstrated as the behavior of the second harmonic mode introduced into the solution by the presence of the parameter E, the physical setting to represent an imperfection. He also gave the relation between the cusp and its dual as: xz + 4y= = 9 where x = Behavior variable of the cusp = first harmonic mode under a perturbation; y = Behavior variable of the dual cusp = second harmonic mode under a perturbation r = Relative behavior variable = first harmonic mode under no perturbation. 3. ECT
MODELS
QF THE
INSTABILITIES
OF ARCHES
AND
STRUTS
3. I. Background The static instability of a structure under load is, in general, analyzed by the structure’s non-linear load-deflection behavior. Euler in 1744 first analyzed the buckling of a beam under increasing load and found the deflection to have diverging behavior. Many similar problems have been solved in catastrophe language [I, 10, I I]. Although most of these examples were already well understood in terms of prior investigations using bifurcation theory, many of the ECT formulations provided conceptual improvements. This section will provide a mathematical entry point for studying buckling in terms
90
H. P.
ALESO
of ECT. First, the simple Euler arch [l, 11-131 will be modeled by two cases that demonstrate the cusp catastrophe. Second, the simple Euler strut 11, 111 will be used to show how elasticity of a beam affects the cusp catastrophe when compression and load act as control parameters. Abrupt changes in behavior of the strut will be seen as the strut changes from straight to upbuckled and then to downbuckled as the control parameters vary. Third, the pinned Euler strut [l, 11-131 will be used to examine harmonics that may occur in buckling. The pinned case, however, is not only an ordinary cusp, but includes a dual cusp. The dual cusp has the stability of its topological form reversed; normally stable areas are unstable. An interesting result of this case is that specific harmonic states are associated with discrete values of loading. 3.2. The simple Euler arch Case 1. The Euler arch [ 1, 111 is used as a simple entry point into beam instability. It displays the five characteristics of the cusp catastrophe previously listed in Section 2.3. The configuration of the arch can change radically, depending on the path followed along the behavior surface of the associated cusp. The arch (see Fig. 2) consists of two rigid arms of equal length joined at P by a spring with the spring constant ~1. The arms, which are initially collinear (i.e., 8 = MO”), are supported at the ends and a compressive force 6 is applied as shown. The two rigid arms will remain at 180” until the force @ overcomes the resistance of the spring and a new value of 8 is assumed (shown in Fig. 2 as buckled position 1). If /3 is then held constant and a gradually increasing load a is applied at P, a critical load a will be reached, at which time the arms will catastrophically snap into the opposite buckling position, shown as buckled position 2. To model this behavior by ECT, first sum the moments of the system by taking one of the arms as a freebody and summing about P to obtain: MP = 2@ + (a/2)f. cos x - /3L sin x where x is the angle the arms make with the original position and L is the length of the arm; To obtain the potential function for one arm, integrate over x, assume L to be of unit length, and apply the boundary condition V(0) = 0. When both arms are considered, the resulting potential function V for the arch is: V(x) = 2j~x* + a sin x - 2p( 1 - cos x).
(5)
This equation represents the total potential energy of the system: the energy of the springis 2&, the energy of the load is a sin x, and the energy stored in compression is -2P(lcos x).
Fig. 2. Model of the simple Euler arch, consisting of two rigid arms connected by a spring, used with the ECT method.
On the instabilities of an externally loaded shallow spherical shell
91
has one singularity point found by taking V’(x) = V”(x) = ,d/dx. Points where V’(x) = 0 form the behavior surface. Points where V”(x) = 0 form the fold lines and the point where V”‘(x) = 0 is the singularity point. By. combining derivatives of equation (5) appropriately and letting /3 = 2~ + b (where b is a translational constant), the singularity point can be located at a = 0, /3 = 2~, and x = 0 [l]. By a Taylor expansion, equation (5) becomes The behavior
surface
V”‘(x) = 0, where the primes represent
V(x) = 2~’
where 05 and 06 represent
+ a(x - x3/6 + ti) - 2p[ 1 - (1 - x2/2 + x4/24 + 06)] higher order terms. Regrouping and eliminating /3 gives
V(x) = ax - bx2 + /~x’/6 - ax3/6 + bx’/6 + @ + 0’.
Note that @‘/6 is independent of the control parameters a and b and p > 0. Further, since x is small, -ax3/6 and +bx’/6 are relatively insignificant in comparison to the fist two terms. Therefore V(X) = /.Ax’/~- bx* +
ax.
(6)
Equation (6) has the form of equation (1) and is therefore the potential function of a cusp catastrophe with load a and compression -b as control parameters. The angle x is the behavioral variable. The singular point is located at the origin. Figure 3 shows the cusp model. In Fig. 3 the dashed trajectory illustrates one possible behavior of the rigid arms. At point 1 (the singular point), the arms are at 180”, there is no a load, and /3 is at the critical value of 2~. With any increase in p (and consequently 6). the arms buckle (in our example using Fig. 2 they buckle up) and begin to follow a path indicated by 2. Along path 2, increasing b smoothly increases x on the behavior surface. At point 3, b is held constant and a load a is applied and increased until the path along the behavior surface reaches the fold line. At the fold line, a critical value has been reached. The system again becomes unstable and buckles down to point 4 on the behavior surface. By holding /3 constant and varying the load a, the buckling position can be flipped back and forth in a cyclic manner shown in Fig. 4. This illustrates the hysteresis property of the cusp catastrophe.
Fig. 3. The cusp catastrophe model for the simple Euler arch of Fig. 2.
92
H. P. ALESSO
X
a
-I
Fig. 4. Hysteresis
type behavior possible with the simple Euler arch shown in Fig. 2.
Case 2. Another simple arch, shown in Fig. 5 [ll-131, consists of two arms composed of linear extensional springs pinned together at one end and fixed to rigid supports at the other. The stiffness coefficient of the springs is p, the distance between supports is 2R and a load a is placed where the arms are joined. The angle x is variable and has an initial value of x,. The strain energy of the springs is U(x). The original length of the spring is (R/cos ~0) and the length becomes (R/cos x) as the load a is applied; thus U(x) = ~(R/cos x0- R/cos x)* [12]. If x0 and x are small then
Thus the strain energy becomes U(x) = ; ~R’(x: - 2xhx2+ x4)+ 06. The potential energy of the load is -aD(x) in which the deflection due to the load a is D(x) = R tan x0- R tan x. For small angles the deflection can be taken as D(x) = R(x, - x).
The potential function V of the system is then V(x, a, x0) = U(x) - all(x),
Fig. 5. Model of the simple Euler arch, consisting of two linear extensional the ECT method.
springs, used with
On the instabilities of an externally loaded shallow spherical shell
93
Of
where pR* > 0. The equation of the behavior surface, upon extracting
a constant multiple, is
dV x=xJ-x;x++o. This equation is of the form of equation (1) and therefore a cusp catastrophe, with - xi and +a/*R as the independent control variables, and with x as the behavior variable (see Fig. 6). The dashed path on Fig. 6 shows how first increasing x0 (path 2) and then load a (path 3) until a critical point is reached causes a catastrophe in which the arch changes from an upbuckled to a downbuckled position. 3.3. The simple Euler strut Replacing the springs and rigid arms of the last section with an elastic beam complicates the problem. To now account for the action of the entire beam it is necessary to analyze and sum a collection of beam cross-sections. Figure 7 shows the Euler strut [l, 111 for which L denotes the length, p the spring constant, s the distance along the arched strut, and f(s) the height at distance s from the left end. In this section, x is the vertical displacement at the center of the strut. (See Zeeman [l] for solution of this problem.) The compressive force a acts as the cause, and the shape f(s) is the ‘effect’, found to be x sin (?rs/L). Also p = p(.lrlL)* is the critical compressive force that results in initial buckling from a straight beam. In Fig. 8 the strut is stable from fl = 0 to p = ~(?rlL)*. At that point the strut can no longer absorb the compressive force internally and is at the limit of stability. Any increase in p causes the strut to become unstable and forces it to seek a stable equilibrium position (either up- or downbuckling). Further increases in p cause increases in the amplitude of the sine wave shape of the strut. In Zeeman[l], the total potential energy of the system is used to
\ Upbuckle
Fig. 6. The cusp catastrophe
model for the Euler arch of Fig. 5.
H. P. ALESSO
Fig. 7. Model of the simple Euler strut, as formulated by the ECT method.
a. Stable 6-h b. Unstable c. Stable
(Critical point Fii. 8. At critical load for an Euler strut, two stable states are equally possible if the load is
increased.
form the cusp potential function
x4 b v=gj-zx2+ax [where b = vr(/3 - l)]. Then dV x3 x=16-bx+a=O. Letting a = 0 and solving for x yields
which forms the parabola in Fig. 8. Until now a has been taken as zero. If a is now applied to an upbuckled strut, it will remain in an upbuckled position until a critical value for a is reached. Beyond this value the strut will become unstable again and snap to the downbuckled position. The Euler strut exhibits the same characteristics of the cusp catastrophe as the Euler arch. 3.4. The pinned Euler strut The pinned Euler strut [l, 11, 121 illustrates how imperfections of a strut affect its capacity to take loads. The model shown in Fig. 9 is that of an Euler strut of length L pinned between supports d apart. In Fig. 9: x = the centerline height of the strut for the first harmonic. y = the center height of the second harmonic. r = the difference between the length of the strut, L, and the distance d between supports; i.e., r = L - d. f(s) = s sin s + y sin 2s. The imperfection parameter B locates the distance from the centerline to where the load a is applied and represents a manufacturing imperfection. Though Golubitsky
On the instabilities of an externally loaded shallow spherical shell
95
Fig. 9. Model of the pinned Euler strut showing typical first and second harmonic modes in a strut having imperfections.
[14] suggests that the imperfection parameter E may be more complicated than this, we will use this simplified approach adopted by Zeeman. Figure 10 shows that the behavior surface of the strut modeled in Fig. 9 is a dual cusp catastrophe with e and +a as control parameters, and y as the behavior variable. Since this is a dual cusp, the middle sheet represents stable behavior, while the upper and lower sheets are unstable. At point A in Fig. 10(a), the load is initially a = 0 and the shape of the strut is a maximum first harmonic with a zero second harmonic (i.e., x is maximum and y is zero). As a increases (along path AB), the second harmonic grows
V~-(v4-ey+ayz) Control plane E
A. C
b)
V
\b8
C
A
CY
rr critical
Fig. IO. (a) Model of the dual cusp catastrophe that represents the pinned Euler strut with imperfections. (b) By carefully loading and unloading, the unstable second harmonic position C can be reached.
96
H. P. ALESSO
until the system reaches a,-hticsrat point B, where the system is about to snap catastrophically. Assume now that the load is not increased further, but rather is slowly decreased. The result is that the strut now follows the path BC on the upper sheet, the configuration of the second harmonic instead of the first harmonic. Point C represents a maximum second harmonic and zero first harmonic. Although the strut can remain at point C, any very slight perturbation will cause the strut to snap back to the first harmonic maximum position A on the middle sheet from the unstable upper surface. Referring now to Fig. 10(b), point A (on the stable middle sheet) is the unloaded shape with maximum first harmonic x and zero second harmonic y. The second harmonic height y increases with increasing load a until point B is reached. Point B is on the fold line and directly above the point p of the bifurcation set on the control plane. If only the first harmonic is considered, the behavior would snap through when B is reached. However, as the second harmonic is now assumed to dominate and the load is reduced, the behavior will follow the curve BC instead of snapping through. At C the behavior is in unstable equilibrium, ready to snap back to the original first harmonic shape. An alternative qualitative behavior would be to follow path AB to B as the load is increased, and then return to A along the path BA as the load is removed. The interrelationship between the first mode cusp and the second mode dual cusp is shown in Fig. 13 (see Zeeman [l]).
4. THE
INSTABILITIES
OF AN EXTERNALLY
LOADED
SPHERICAL
SHELL
4.1.
Overview The difficulties encountered in the study of cylindrical and spherical shells arise more from the perversity of the problem itself [2, 12-251 than from the particular mathematical formulation. However, considerable insight can be gained by formulating such problems in terms of ECT. Alesso [9] formulated an ended ring in a rigid cavity as a catastrophe and cited sample calculations that agreed with experimental data. Zoelly and Leibenson used linear theories in early investigations of thin elastic spherical shells under outside pressure. In the 193Os, Sechler and Bollay carried out experiments on buckling pressures that gave data within 25% of pressures predicted by linear theories. In 1939, von Karman and Tsien used the Ritz method to obtain a non-linear and non-monotone relation between load and deflection. They found that until buckling occurs, a perfectly spherical shell under an increasing pressure will contract by a constant amount 6,. A general review of buckling of a shallow spherical shell can be found in von K&man [2]. In that reference, the von Karman non-linear equations for a shallow spherical shell segment are given as:
DV2V2w=[w,fj+~V2f+, and the compatibility
(7)
equation,
& V2V2f= w;,-
w3,w’,
-+v2w
where
[w,fl = K.&, + wyyfxx - 2w*,fx,
On the instabilities of an externally loaded shallow spherical shell
91
and E = modulus of elasticity fY = bending stiffness of plate 4 = periodic impressed force w = radial deflection of middle surface f = stress function t = shell thickness R = radius of middle surface and subscripts indicate partial derivatives. Note that as R + 00 these reduce to plate theory [l 11. 4.2. ECT formulation of von Khndn model The non-linear differential equation (7) has no known exact, general solution. Although Hyman [20], Ball [26] (for the dynamic case), Kalnins [27], and Varadan [24] have made efforts to calculate approximate solutions for variations of these equations, a comprehensive qualitative and quantitative picture has not emerged. In fact, Hyman [20] introduced higher mode buckling behavior of a new qualitative type, which added to the complexity of understanding the overall behavior. Modeling complex problems in the form of non-linear algebraic equations rather than non-linear differential equations is often useful. von Karmin and Kerr [2] in 1965 suggested the following model for the externally loaded shallow cap. As the model, a bent beam clamped at the supports (see Fig. 11) is assumed to contract or extend axially except in the shaded regions at the supports and in the middle, in which regions all bending is assumed to occur. Straight parts of the beam (unshaded regions) are assumed to remain rigid. For this model, von KBrmin and Kerr formulated the following non-linear algebraic relationship for the stationary solutions of the potential energy: aV* _=
as*
0 =
ktp*(l
- h*2)1’2-2(h*
involving the non-dimensional
- 6*)k(l
-k)-$(l
- h*2)“2 arccos (’ -
f*‘*)
quantities
where p = external uniform load, h = undeformed depth of shallow shell, S = deflection from undeformed height, L = length, t = von Karman’s expansion term to define regions of pure bending, V= potential energy, E = modulus of elasticity, a = area of beam cross-section and k = (1 + 8*2-2h*6*)“2.
Undeformed positi
Fig. Il. The von KBrm6n model of an externally loaded shallow cap. As loading is applied, bending occurs in the shaded areas only and axial changes in length in the unshaded areas only.
98
H.P.
ALESSO
Equation (8) will now be formulated as an ECT cusp representation given later by equation (11). First, expand the arccos as a Taylor Series. Then, let F = l/k and treat F as a Legendre polynomial generating function in which we neglect terms of order 0(6*‘) and higher. This gives, av* as*
-=O=A++h*
F*-B(l-
-2S*-2F(h*-a*)+?
h*cS*)F$!(l
-h*6*)‘F5 + 0(6*‘),
where A = p*d/( 1 - h*‘); B = - (2:/3L)d(l-
h**);
F = (1 _ 2h*S* + &49-1/*= g (6*)‘P,(h*); P,(h*) = Legendre polynomials.
Taking only the I= 0 term in Fs and neglecting 0(6*‘) gives:
av*
-=0=A+2h*-2S*-2F(h*-S*)+~~-B(1-h*6*)F3-~(1-h*6*)3
as*
where F = 1 + 6*P, + S**P* + S*3P3+ 0(6*‘) F = 1 + 2S*P, + S**(2P2+ P:) + 6*32(P3+ PIP*) + O(S*‘) F3 = I + 3S*PI + 8**(3P2 + 3P:) +
S*3(3P3
+
6PlP2 + Pi) + O(S*‘).
By multiplying out polynomials and gathering terms, we find,
aV* as*
-
= 0 = b, + 2b28* + 3b3S** + 4b4S*3 + 0’
(9)
where
2b2=-2h*P,+B
3b3= B(3P,h*y+
> rP2+;
4b4= - B 3P3+6P,P2+P:-
7rP,P2- “p,+7
and P,= h*
P2 = ; (3h** - 1)
- 2P,h* + 2P2 + 3P,Bh* + 3BP:h*
On the instabilities
Physical
restraints:
of an externally loaded shallow spherical shell
t’:Ot?
Note: b4 # 0 for t = h* = 0. Equation (9) is 4-determinate. Now apply a change of coordinates to equation (9) such that x = CT*-+-c.
Substituting equation (10) into equation coefficient of 8** goes to zero) gives:
(9) and picking C = (bJ46,)
(10) (so that the
aV* = x3 + a,x + uz = 0 i)X
where a,
=$-$4
:_ 4
j
Jr, L, h*)
4.3. Predicted qualitative behavior of the cusp model Equation (11) has the form of an elementary catastrophe, the cusp described in Section 2.3. From the topological shape of the cusp (see Fig. 1) and the functional relationships of the parameters a, and a z, we will now make qualitative predictions about the behavior of the deflection 6 as we vary arch height h* and load p*. In equation (11), changes in parametric conditions (t, L, h*, p*) cause the behavior variable x (which equals 6* + C, where C is a constant) to move along the topological surface of the cusp elementary catastrophe represented by the equation. If we let L be a constant and assume that t is zero in the area of interest, we find that the potential can be represented by:
Note that in equation (12) the load p* appears only in a2. The mutual dependence of a, and a2 on h* is a problem we will not discuss herein. The underlying similarity between equation (12) and the models presented in Sections 3.2-3.4 is apparent. In each model the external load appears exclusively in the normal factor a2. Likewise, in each case the splitting factor a, contains a term that represents a measure of arch of the structure. The cusp of equation (12) predicts first mode behavior, for which there are three distinct qualitative possibilities. These possibilities are represented by paths A, B and C of Fig. 12. Path A is typical of the type behavior in which there is a slight compression under external load (1 to 2), followed by a snap-through phenomena (jump from 2 to 3). Path B illustrates direct snap-through behavior. The distinction between paths A and B depends on the geometry of the shell (t, L, h*). Path C, which represents the case where the internal pressure is greater than the external pressure, produces continuous deformation in an outward direction, 4.4. Behavior of an externally loaded shallow spherical shell with an imperfection Buckling of the pressure loaded shallow shell is usually analyzed for symmetric
loo
H. P.
ALESSO
Fig. 12. The cusp catastrophe model obtained by reformulatkg the VORGrm9n r,lodel ~-41: .. shallow cap by the ECT method. Paths A, B and C represent the three b, ;ic types of deformations that can occur.
behavior as in Sections 4.1-4.3. Analyses such 1 that of Huang [I81 show that asymmetric buckling modes can exist at pressure: rlow critical pressures found for static symmetric snap-buckling, and especially if ‘;: perfections in the shape’ of the shell exist. An equation analogous to equation (11) for the asymmetric dual cusp (see Section 3.4) is given by: aV* -=y3+ay+r=0 JY
(13)
where y is the second mode behavior variable, a = f3(t, L, h*, p*), and E is an imperfection parameter of the sphere. The second harmonic y is significant in the von KBrmPn model when an offset load (due to imperfection 6) is included. Note that the external load p* now appears in the splitting factor rather than the normal factor. Figure 13 illustrates the interrelationship between the cusp and the dual cusp. The change in the way that p* appears in the controlling equations is reflected by the rotation in behavior space between the cusp and dual cusp. For a more complete discussion of the relationship shown in Fig. 13, see Zeeman [l]. In Hyman 1201, the behavior of the second mode of shallow shells was explored and hypothesized to be a continuous ‘closed loop’. I speculatively suggest that the behavior of the dual cusp model of equation (13) might account for Hyman’s closed loop behavior pattern. 4.5. Supportive evidence Preliminary computer calculations of the stationary solutions of the potential energy given in equation (11) gave good agreement with von KBrman and Kerr [2] as reported in their Figs. 9 and 11. Incidently, Fig. 11 in [2] defines the control plane characteristic cusp in terms of p* and h*. The cusp model of equation (ll), shown as Fig. 12, predicts the behavior of S*, the non-dimensionalized deflection of an elastic, externally loaded shallow cap. Figure 14 shows how S* behaves as the non-dimensionalized pressure p* and cap height h* vary along the three types of path shown in Fig. 12. Curves relating deflections x to p* and h* are similar but are translated along the deflection axis according to the
On the instabilities of an externally loaded shallow spherical shell Path A
IO1
Path C
Path B
Fig. 13. Diagrams representing the variation of deflection 6 with pressure p* and cap height h* for the three types of deformation shown by the paths in Fig. 12.
Behavior variable Dual
Dual cusp Fig. 14. Relationships of behavior surfaces and control planes for the
cusp and
dual cusp.
relationship x = S* + C, where C is the constant that locates the singular point of the cusp at the origin. Experimental data presented by Stephens and Fulton [28] and by Huang [18], and reviewed by Svalbonas and Kalnins [22], are plotted as load deflection curves in Fig. 15. The experimental curves compare favorably with the deflection behavior of a type A path on the cusp model shown in Figs. 12 and 14. 4.4. Conclusions The ECT method we have discussed provides improved conceptual insight into the behavior of shallow shells. After an overview of elementary catastrophe modeling of beam buckling, we have reformulated the simple von Karmin-Kerr model of a clamped, elastic shallow cap under external load as a cusp catastrophe. The resulting model presents three qualitatively different behavior types. NLM
Vol. 17. No. 2-C
H. P. ALESSO
102
01 0.20
I
I
0.30
0.40 Loading
Fig. 15. Experimental
I 0.50
I 0.60
(p’)
results of two studies support the qualitative behavior predicted by the ECT formulation of the von Karman model.
1. Initial continuous deformation followed by a discontinuous jump in deformation. 2. Initial discontinuous jump deformation. 3. Continuous deformation (under a net internal load). Experimental evidence in support of the first behavior type was presented. In addition, we have suggested a modeling approach for analyzing the second-mode behavior of a shallow shell.
Acknowledgement-It is a pleasure to acknowledge the valuable criticisms gestions of Prof. Hugh Keedy in the preparation of this manuscript.
and numerous
helpful sug-
REFERENCES 1. E. C. Zeeman, The buckling beam. Lecture Notes in Mathematics, Vol. 525, pp. 374-394. Springer, New York (1976). Collected Works of Theodore uon Kdrmdn, 1952-1963. von Kbrmln Institute, Belgium (1975). :: R. Thorn. Structural Sfobility and Morphogensis. W. A. Benjamin, Reading, MA (1975). 4. T. H. Brocker, Differential germs and catastrophes. London Mathematical Society Lecfure Notes Series. Cambridge University Press, Cambridge (1975). Y. Lu, Singularity Theory and an Introduction to Catastrophe Theory. Springer, New York (1976). i: D. O’Shea, An exposition of catastrophe theory and its application to phase transition. Queen’s Pap. pure oppl. Mafh. 47, IO-12 (1976). 7. D. H. Satlinger, Topics in stability and bifurcation theory. Lecrure Notes in Mathematics, Vol. 309. Springer, New York (1974). 8. G. Wasserman, Stability of unfoldings. Lecture Notes in Mathematics, Vol. 393. Springer, New York (1974). 9. H. P. Alesso. Elementary catastrophe theory modeling of an end loaded ring in a rigid cavity. Nucl. Eng &sign 53. 55-70 (1979). 10. C. Chillingworth, The catastrophe of the buckling beam. Dynamical Systems, Lecture Notes in Mafhematks, Vol. 468. Springer; New York (1974). II. T. Poston and 1. Stewart. Catastroohe Theorv and Ifs Aoolications. Pitman. San Francisco (1978). . , 12. J. M. T. Thompson, Phil.. Trans. RI Sot. L&d. 292. I-2j il979). 13. J. M. T. Thompson and G. W. Hunt, A General Theory of Elastic Stability. John Wiley, New York (1973).
103
On the instabilities of an externally loaded shallow spherical shell
14. M. Golubitsky and D. Schaeffer. Imperfect bifurcation in the presence of symmetry, Communs Pure appl. Math. 32, 21-98 (1979). IS. B. Bidiansky, Ed., Buckling of Structures. Springer, New York (1976). 16. B. Bidiansky. Theory of buckling and post buckling of lelastic structures. Adoonces in Appled Mechanics, Vol. 14. Academic Press, New York (1974). 17. J. S. Hansen, AIAA J. 15, 1638-1644 (1977). 18. N. Huang, Air Force Report AFOSR, 68-0469, TR7. February (1968). 19. P. T. Hus, J. Elkon and T. H. H. Pian. Note on the instability of circular rings confined to a rigid cavity. 1. oppl. Mech. X.559-562 (1964). 20. B. I. Hyman, Non-linear mechanics. Trans. Am. Sot. mech. Engrs 34.49-55 (1967). 21. T. H. H. Pian and L. L. Bucciarelli, Jr., Buckling of radially constrained circular ring under distributed loading. Inr. J. Solids Strucfures 3, 7 IS-730 (1967). 22. V. Svalbonas and A. Kalnins, Dynamic buckling of shells: evaluation of various methods, Nucl. Engng Design 44, 331-336 (1977). 23. J. J. Tuma and R. K. Munshi, Advanced Structural Analysis. McGraw-Hill, New York (1971). 24. T. K. Varadan, Snap-buckling of orthotropic shallow spherical shells, 1. appl. Mech. 45,44547 (1968). 25. E. A. Zagustin and Cl. Hermann, Stability of an elastic ring in a rigid cavity. 1. oppl. hfech. 34, 263-270 (1967). 26. R. E. Ball and J. A. Burt, 1. appl. Mech. 40.411-416 (1973). 27. A. Kalnins and V. Biricikoglus. J appl. Mech. 37, 629-634 (1970). 28. W. Stephens and R. Fulton, AIAA J. 7. 2120 (1969).
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Zusammenfassunq: In dieser Arbeit werden Probleme aufgeftihrt, die im Zusammenhang mit der nichtlinearen Behandlung diinner StB'be auftreten, wobei die Stabe sich in reiner Biegung mit konstanter Anfangskriimnung und Verdrehung und mit willkiirlichen kinetischen Querschnittsbedingungen befinden. NutDrehmomente, die an den Stabenden angreifen, werden beriicksichtigt. Die hier entwickelten Losungen, welche die Kriimmungskomponenten und die Verdrehung des Stabes nach der Verformung bestinunen, sind exakt und in der Form elliptischer Integrale.