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## Material Information- Title:
- Transitional instability of spherical shells under dynamic loadings
- Creator:
- Ho, Fang-Huai, 1934-
- Publication Date:
- 1964
- Language:
- English
- Physical Description:
- ix, 173 leaves : illus. ; 28 cm.
## Subjects- Subjects / Keywords:
- Buckling ( jstor )
Dynamic loads ( jstor ) Eggshells ( jstor ) Equation roots ( jstor ) Phase plane ( jstor ) Saddle points ( jstor ) Shallow shells ( jstor ) Spherical shells ( jstor ) Structural deflection ( jstor ) Trajectories ( jstor ) Buckling (Mechanics) ( lcsh ) Dissertations, Academic -- Engineering Mechanics -- UF Engineering Mechanics thesis Ph. D - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Thesis:
- Thesis--University of Florida, 1964.
- Bibliography:
- Bibliography: leaves 169-172.
- General Note:
- Manuscript copy.
- General Note:
- Vita.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright Fang-Huai Ho. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 000568321 ( ALEPH )
13645080 ( OCLC ) ACZ5046 ( NOTIS ) AA00004948_00001 ( sobekcm )
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TRANSITIONAL INSTABILITY OF SPHERICAL SHELLS UNDER DYNAMIC LOADINGS By FANG-HUAI HO A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA April, 1964 ACKNOWLEDGMENTS The author wishes to express his gratitude to the members of his supervisory committee: Professor Wo A. Nash, chairman of his committee, for suggesting the dissertation topic and for the encouragement and advice received from him through the period of this research; Professor W. L. Sawyer for reading the complete manuscript and making many corrections; Professor I. K. Ebcioglu and Professor C. B. Smith for their encouragement and advice; and the late Professor Ho A. Meyer for his many suggestions in the numerical solution of the problem in Chapter IV. He is also indebted to Dr. S. Y. Lu of the Department of Engineering Science and Mechanics for reading Chapter V of this research and for providing the opportunity to discuss several questions in that chapter with him. The author is indebted to the Office of Ordnance Research, U.S. Army, for their sponsorship of this study. li TABLE OF CONTENTS Page AC KNOWLE DGMENT S 00000000000. o.ooo.o. XX LIST OF FIGURES OOOO.OOOOOO. .00000.0. V ABSTRACT viii INTRODUCTION I lo A Historical Review and Recent Advancement . o 1 2.The Scope of the Present Research <> . . . 5 CHAPTER I.A CRITERION FOR DYNAMIC BUCKLING ......... 10 1. Autonomous Conservative System ....... 12 2. Nonautonomous System .o...oeooe. IB II.BUCKLING OF A CLAMPED SHALLOW SPHERICAL SHELL UNDER A PURE IMPULSE ............ 20 1. A Qualitative Discussion of the Loss of Stability of the Structure ....... 20 2. A Study of the Dynamic Response ...... 30 3. A Justification of the Buckling Criterion 38 4. A Note on the Effect of Initial Geometrical Imperfections oo.o.ooooo..... 41 III.BUCKLING OF A CLAMPED SHALLOW SPHERICAL SHELL UNDER A SUDDENLY APPLIED CONSTANT PRESSSURE OF INFINITE TIME DURATION .............. 43 1. A Qualitative Study of the Instability of the Structure .............. 43 2. Another Justification of the Buckling Criterion ooo.oo.oo.o.doo.o 32 3. A Discussion of the Results ........ 54 iii TABLE OF CONTENTS (Continued) CHAPTER Page IV.BUCKLING OF A CLAMPED SHALLOW SPHERICAL SHELL UNDER A UNIFORMLY DISTRIBUTED PRESSURE LINEARLY INCREASED WITH TIME .......... 60 1. The Solution of the Problem ....... 60 2. A Conclusive Discussion of the Problem . 67 V,BUCKLING OF A COMPLETE SPHERICAL SHELL UNDER SUDDENLY APPLIED UNIFORMLY DISTRIBUTED DYNAMIC LOADINGS .0..0...0.0.0.C..0. 70 1. Introduction ............... 70 1. On a New Mechanism of the Buckling of a Complete Spherical Shell ...... 73 2. Buckling of the Shell under a Static Load 77 3. Buckling of the Shell under a Suddenly Applied Constant Pressure ........ 88 4. Buckling under a Pure Impulse. ...... 105 5. Conclusions and Discussions ....... 109 VI.A CONCLUSIVE REMARK .............. 114 APPENDICES I. THE STABILITY THEOREM OF NONLINEAR MECHANICS 119 II. THE EQUATION OF MOTION FOR TRANSVERSE VIBRATION OF SHALLOW SPHERICAL SHELLS ..... 124 I II O FIGURE S ...OO.D. 00. 000000.00 136 IV. NOTATIONS .... ....... 167 REFERENCES . ............ 169 BIOGRAPHICAL SKETCH ... 173 iv LIST OF FIGURES Figure Page 1. Geometry and deformations of the shell .... 136 2. Phase plane trajectories and the variation of potential energy 137 3. Phase space trajectory and its projection 138 4. The dynamic buckling criterions .... . 139 5. Comparison of axisymmetrical theories on the static buckling of shallow spherical shells 140 6. Dynamic loadings ..... 141 7. Phase plane trajectories (when @ = 0.26) . 142 8. Response curves of the central deflection of a shallow shell ( 0 = 0.26) under the action of impulses 143 9. Critical impulse determined by buckling criterions ........... 144 10. Comparison of analytical theories on dynamic buckling of shallow shells under pure impulse 145 11. The threshold of instability of a shallow shell (with 6 = 0.26) under a suddenly applied uniformly distributed dynamic loading 146 12. Relation between the critical pressure and the geometrical parameter Q for shallow shells under the action of uniformly distributed static and dynamic loadings .... 147 13. Relation between the critical deflection and the geometrical parameter 0 for shallow shells under various dynamic and static loadings . 148 v LIST OF FIGURES (Continued) Figure 14. A justification of the buckling criterion applied to the shallow shell with 0 = 0.26 Page 149 15. Response curves for various values of \jr when Q = 0.26 .............. 150 16. Response curves for various values of 0 when = 1.07 . o o 151 17. Upper and lower values of "J, as a function of 0 152 18. T. 0 curves for various values of V . cr T 0 152 19 o sC Critical D.O.L.F. -- vs. critical c r central deflection -- r) ........ 1 cr o 153 20. Upper and lower critical deflections vs. Q 0 154 21. k Upper and lower values of Tcr vs. ^ . 155 22. k T vs curves for various values of Q cr 156 23. The projection of the trajectories on the 7) - rj plane, when 0 = 0.26, ij/ = 5 . . 157 24. The projection of the trajectories on the 7) -T) plane, when 0 = 0.26, = 2 . . 0 158 25. Buckling region of a complete spherical shell . 159 26. Load deflection curve for a complete spherical shell .............. 160 27. Comparison of theoretical result and experimental tests ............ 0 161 28. Sketched phase plane trajectories for the motion of a shell , 162 29. Graphical solution of equation (5.29) . . . . 163 30. Frequency curves of the normal and log-normal distributions .... 163 vi LIST OF FIGURES (Continued) Figure Page 31. Phase p!ane_trajectories when 0 = 0.44, q = 0.6823 ....... 164 32. Phase plane trajectories when 0 = 0.44, q = 0.33 ................... 165 33. Buckling pressure vs. buckling region parameter Q for a complete spherical shell . 166 vii Abstract of Dissertation Presented to the Graduate Council in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy TRANSITIONAL INSTABILITY OF SPHERICAL SHELLS UNDER DYNAMIC LOADINGS By Fang-huai Ho April 18, 1964 Chairman: Dr. W. A. Nash Major Department: Engineering Science and Mechanics The dynamic instability of thin shallow spherical shells under uniformly distributed impulsive loadings has been discussed by use of the stability theory of nonlinear mechanics together with a topological method. The axisymmetrical buckling of the shallow shells under three types of dynamic pressure, namely, an impulse function, a step function, and a linear function of time, has been solved based upon a first approximation and Galerkin's method. The results of the first two cases, within the range of the validity of the theorem, are compared respectively with those obtained by Humphreys and Bodner, Budiansky and Roth, and Jiro Suhara. A buckling criterion based upon the relation between the viii dynamic load-deformation curve and its counterpart in the static case has been established through the comparison of the results obtained by this criterion with those obtained by the application of the stability theorem to autonomous systems. An extension of the same techniques to the solution of problems in the case of a complete spherical shell has also been made. Based upon an assumed new buckling mechanism, the static buckling pressure for such a shell obtained by a first approx imation and axisymmetrical deformation theory, agrees well with recent experiments conducted individually by Krenzke and Thompson Transitional instabilities of a complete spherical shell under uniformly distributed dynamic pressures in the form of a step function and an impulse function were also discussed. Results are presented in the form of figures. ix INTRODUCTION 1. A Historical Review and Recent Advancement The problem of dynamic instability of a thin spherical shell under time dependent external forces is inseparable from the problem of the transverse vibrations of such a structure. Studies on the latter problem, as in the works of Mathieu, Lamb, Lord Rayleigh and Love, date back to 1882, As may be found either in Love's Theory of Elasticity or Rayleigh's The Theory of Sound, these classic works have been mainly devoted to finding the smallest natural frequency or the "gravest tone" of the system. The method used by Lord Rayleigh was essentially an energy method plus an assumed displacement pattern; the bending and membrane energies were considered separately, depending on whether the middle surface of the shell is extensible or inextensible. The first rational study of the transverse vibration of a shallow spherical shell, using three simultaneous equations of motion for the three displacement components, was due to Feder- hofer (44)Federhofer's problem was discussed later by Reissner using a different approach (45). These two authors were interested 1 Underlined numbers in parentheses refer to the references. 1 2 in the transverse free oscillations of a shallow shell. In Reissner's paper, certain conventional simplifications usually employed in the static loading case had been introduced because of the thinness and shallowness of the structure; the frequency equation was expressed in the form of a determinant involved with Bessel functions; numerical results were obtained by a Galerkin approximation method. A great contribution to the problem of transverse vibration of thin shallow elastic shells was also due to Reissner in 1955. In his paper (46), by an order of magnitude analysis, Reissner justified an important simplification for the problem; i e the tangential (longitudinal) inertia terms may be omitted with negligible errors. This simplifi cation has made possible the solutions of other shell oscillation problems (47, 48 and ^+9) . The work of Eric Reissner and others, as mentioned above, dealt mainly with linear, free vibrations of a thin shallow elastic spherical shell. The first investigation of the problem of forced vibration and the problem of stability of such a structure were probably due to Grigoliuk (11) In the work by Grigoliuk, a non linear oscillation system was considered for the first time. A great amount of work in the nonlinear vibration of shell structures and their stability under periodic forces has been done by V. V. Bolotin. In his paper of 1958 (4), he first discussed the problem of forced oscillation and stability of a complete spherical shell under a periodic external loading. He considered the oscillations of both nonlinear and linear cases, that is, the system vibrating with both finite and infinitesimal amplitudes and 3 obtained the resonance curves shoving that "hard excitation" occurs until the critical frequency is reached. This is in contrast with the oscillation of a flat plate where "soft excitation" starts at the lower boundary of the instability region. A rather complete collection of the problems of parametric stability of elastic systems was also due to Bolotin. His two books, (30) and (51), are no doubt valuable contributions to this field. Along with the development of aerospace science, another type of shell buckling problem has become ever so important, i.e=, buck ling under impulsive and blast loadings. As an extension of Hoff's work (14) on the stability problem of a column, in 1958, A. S. Vol'mir solved a problem of dynamic buckling of a hinged cylindrical panel under axial pressure. In his 1958 paper, Vol'mir considered the end shortening of the structure as a linear function of time. Using a first approximation of deflection form for both total and initial deflections, plus a Galerkin method, he was able to solve the problem by considering a 1-degree-of-freedom nonlinear oscillatory system,^- Results were obtained by numerical integrations and the critical condi tion of the system was determined from the response curves. As an example, he showed that the dynamic critical load was about 1.7 times higher than the upper static critical load. The problem of buckling of a cylinder under external uniformly distributed load increased linearly with time was considered by V. L. Agamirov and A. S. Vol'mir (j[) in 1959, using a similar technique as in the above mentioned 1 Refer to reference (36). 4 paper by Vol'mir, This same problem was considered again by Kadashevich and Pertseve (18) in 1960. In contrast to Agamirov and Vol'mir's work, where transverse inertia was considered alone, these two authors have also considered the inertia of axisymmetric com pressing of the shell. By removing the restriction on radial dis placement presented in Q_), they considered a nonlinear dynamic system of three degrees of freedom Three types of dynamic load were considered in this paper: a suddenly applied load with constant pressure, an impulsive load with finite time durations, and a uni form load increasing linearly with time. For a very rapidly applied dynamic loading, they found the contribution of the inertia of axi symmetric compression is essential. In 1962, two papers treating the problem of dynamic buckling of shallow spherical shells under uniformly applied impulsive load ings appeared in the open literature in the western world. The first paper was given by J. S Humphreys and S. R, Bodner (15), where the critical condition of the system was determined by an energy method, and nonlinear strain-displacement relations were employed* To the author's knowledge, this paper is unique in that it presents a solu tion of dynamic buckling problems in this fashion. Although the behavior of the dynamic response could not be obtained by this method, the general relation between the geometric shape and critical impulses as well as critical deflections were obtained through a much easier and clearer discussion. The second paper was published by B Budiansky and R. S. Roth (8) in December 1962. Snapping of a 5 shallow spherical shell under an impulsive loading with various time durations was used as an example to establish a buckling cri terion proposed by these two authors. Using a higher order of sym metrical modes, Budiansky and Roth integrated a five-degree of freedom dynamic 3ystem numerically, and the critical condition of the system was determined by a proposed buckling criterion which has a physically significant basis. It should be noted that before Budiansky and Roth, the condition of the threshold of the shell buckling was determined in a rather arbitrary fashion; i.e., having neither physical nor mathematical reasonings. The future trend of the investigation of the buckling behavior of a dynamically loaded shell seems to be such that the governing differential equations are integrated by various numerical means. It is for this reason that the contribution of Budiansky and Roth is of particular interest. 2, The Scope of the Present Research It is well known that the investigation of dynamic buckling of a shell under impulsive or blast pressures is mainly on determining the magnitude of buckling deflection in the process of loading. Usually, on one hand, we have a dynamic system which is essentially "unstable"; i.e., the deformation is unbounded as time increases, and we are at the position to determine the threshold of the buckling from the response curves; on the other hand, we have a dynamically stable system, and we have to determine whether the buckling of the 6 structure occurs or not, because of the significantly larger displace ment in the transient region. The conventional method where the critical condition was determined by the first amplitude of the response curve, or by the state where the dynamic response has severe changes, may be used to solve the first type of problem although it will not be precise It may introduce serious errors if the problem is of the nature of the second type As we have mentioned, a buckling criterion has been proposed by Budiansky and Roth for a certain type of dynamic load These two authors have based their work upon a certain physical picture of the deformation of the shell during the loading process and established a certain "measure" which defines the critical state of the structure in a characteristic load-response diagram The philosophy of this criterion is fresh and remarkable; however, the difficulties in extending to the other cases is obvious In a rigorous manner, the correct "measure" of buckling can only be obtained from the sample of experiments. Errors introduced from plausible assumptions may become significant when the load-response curve does not present a change in the form of a "jump." Therefore, the following questions arose: Is it possible to propose a buckling criterion with a more general sense? Is there any relation at all between the dynamic load-deflection curve and its counterpart in the static case? Let us seek positive answers to both of these two questions It is on this basis that the present research will be devoted to the following two purposes: 7 A, A dynamic buckling criterion in a general sense; i.e., independent of the type of loading and geometry of the shell, will be proposed from a comparative basis. Naturally, the critical con ditions of the statically loaded shell and a free oscillating shell will be good measure of the critical condition of the shell under dynamic loadings. B. An attempt will be made to unify the two methods, i.e., the energy method and the dynamic response method, in the study of the dynamic instability of shell structures. Therefore, a qualita tive discussion of the motion of the dynamic system as referred to the change of total energy level is desired. Because of the second purpose mentioned above, we shall restrict ourselves to the problem of considering a single deformation mode, i.e., a first-degree-of-freedom system. It is well known that in the static case, the above restriction will make the result of the theory applicable only for sufficiently shallow shells, e.g., \ < Q where A v'2o->'Â¡) T2/r<'h~ z/h (1) which is a standard geometrical parameter used in the shallow shell theory (16). It should be mentioned that, in this research, we shall use a different geometrical parameter, which has the following definition: e = hh0 <2) i Refer to Figure 1. 8 By using this parameter, we shall have our dynamic equations in the simplest form, i e., Q appears only in the linear term in the differential equation. However, as shown in (2),^ this parameter is rather ambiguous. Therefore, for a proper interpretation, we always consider Q related with X by the following equation (3) or, for V 0- 3 , (*) hence, a larger 0 value implies a shallower shell. It will be seen later that, for non-shallow shells, 0 becomes an awkward measure for the geometrical shape. In the region of the validity of the present theory, however, the 0 defined in equation (2) may be satisfactorily used as a geometrical parameter. The transitional instability of a shallow spherical cap under three types of impul sive loadings will be investigated. The dynamic equations of the system will be obtained by using a Galerkin's approximation method, and a phase plane method will be employed to discuss the stability of the system. However, for the case that the load is a linear ^Numbers in parentheses in the text which are not underlined refer to the equation numbers. 9 function of time, numerical integrations will be used because the system is then nonautononous. An extension of the same technique to the solution of problems in the case of a complete spherical shell will also be made. CHAPTER I A CRITERION FOR DYNAMIC BUCKLING As well as in the static case, the loss of stability of a shell determined by the deformation of the structure, i.e., a buckling state, can be determined from the load-deformation rela tions In the static case, to determine the state of buckling usually does not add any trouble, because there always is a point of relative maximum on the load-deformation curve, which is shown by the increasing of deflection with a decreasing load, hence the state of instability is very clear, In the dynamic case, however, as we know, the shell may have buckled before the system reaches its first amplitude in the nonlinear oscillation due to a severe change in the deformation. Therefore, a stable oscillation can cause the critical condition for the structure as well as an unstable oscil lation. This situation happens particularly when the load is rapidly applied, and with short time durations, or an impulsive type. It is for this reason that a physically significant buckling criterion should not be based upon the stability nature of the oscil latory system alone; it should be safeguarded by a certain fixed value which satisfactorily measures the danger of the structure. 10 11 However, it is well known that an unstable vibration system will have its amplitude increasing indefinitely with time Therefore, the transitional point for an original stable system to an unstable system will always represent a critical condition for the structure This is to say that the study of dynamic stability of the oscillatory system is still the most important consideration in the investigation of the dynamic buckling of shell structures, although it becomes impossible for some cases; for example, when the system is eventually unstable, then other techniques have to be used. It is the purpose of this section to establish a new buckling criterion based upon the very nature of the dynamic stability theorems Certain measures of the buckling of the shell of this nature will be provided after the following discussion. The danger of overestimation of the critical loadings will also be safeguarded through the comparison of the characteristic load-deformation curves for some structures under other situations, whose stability nature are well known. The proof of such a criterion is impossible at this stage, yet its physical significance is not difficult to observe and will be established through the examples given in the following chapters. 12 1. Autonomous Conservative System The typical dynamic equation of such a system is of the following form ij = -f ( v. I) (i.i) where X is a parameter, e.g., the load parameter. An equivalent form of (1.1) is the two dimensional system: V = s j = f ( V, *) (1.2) It is well-known that the discussion of the stability of all the possible motions described by (1.1) is essentially the same as discussion the stability of the motion in the neighborhood of 2 certain isolated points, i e the singular points, in the phase plane of the system. These singular points are found by the condi tion that 7) and Â£ vanish simultaneously, i.e., from equation (1,2), ^Refer to (9), (23) (29) and particularly (3) in which a beautiful discussion of the "conservative system" has been given. 2 The names critical point and equilibrium points are also used. 13 % O -fdU A ) = O (1.3) The first condition in (1.3) merely says that the singular points are located on the 7] -axis (where Â£ = Q ) It is the second condi tion in (1.3) that determines the singular points in the phase plane. For a system as (1.1), we can have only two types of singu larity, namely, the center and the saddle point. The trajectories around a center and around a saddle point have a characteristic difference, and this is shown in Figure 2. In Figure 2, 7), and 7] are centers; motion around these two points is described by simple closed trajectories, which is stable in character. The trajectory passing through the saddle point is called a separatrix, which, less rigorously speaking, is the partition between two motions with different characteristics. It is also seen from the same figure that a trajectory lies outside of the separatrix and has a higher energy level than the one located inside of it. By virtue of.the above discussion, we may say that the study of the stability of the dynamic system is essentially equivalent to finding the character of the trajectory of the system, and the loss of stability of the system is equivalent to the condition that the system moves on the separatrix in the phase plane. A further exam ination of the phase plane sketch will make it clear that the sense of 14 "loss of stability" mentioned above has the same nature as the usual dynamic buckling criterions, i.e., the characteristic deformation undergoes a severe change (increased). In the phase plane, all points where the trajectories intersect the V -axis reflect the amplitudes of the motions, because Â£ = 0 i.e., the T](z) curves of the motions have a horizontal tangent at that point. The magnitude of the amplitudes are measured relatively by the length from the origin 0 of the phase plane. For any motion moving on a trajectory inside of the separatrix, the amplitude increases gradually as the total energy level increases, i e., due to the increasing of the external pressures. This is shown as from 0-4 to og Once the external pressure reaches the critical value which causes the motion on the separatrix, the amplitude undergoes a characteristic change. It first reaches 0 ^ and then creeps to the magnitude equivalent to OC Any motion outside the separatrix has its amplitude larger than O C e.g., o) The severe change of the amplitude during the loss of stability becomes apparent by comparing the length of o B with OC . Let us summarize the above discussion and make a useful conclusion. We have reached the point that the determination of the dynamic instability of the system (1.1) is equivalent to finding the motion on the separatrix in the phase plane of the system. It will become clear in the later examples that the 15 equation of the separatrix is determined solely by the unstable singular point of the phase plane, hence, by one of the roots of equation (1.3) Comparing (1.3) with (1.1), we immediately found, by its very nature, that equation (1.3) is simply the state of static equilibrium, i.e., the counterpart of equation (1.1) in the static state. Moreover, we have r)i 71 and the possible states of static equilibrium, in the phase plane sketch. By possible states of static equilibrium, we mean the deformation (or deflection) determined by the position of these points would be a state of static equilibrium if the external distrubance is a static one. Thus far, we are able to state that the loss of dynamic stability is characterized by the load-deformation relation reaching a possible state of static equilibrium. In most problems of dynamic buckling of shells, the singular points are interior to a closed path. There is a theorem due to Poincare; In a conservative system, the singular points interior to a closed path are saddle points and centers. Their total number is odd and the number of centers exceeds the number of saddle points by one. By virtue of the above theorem, since in most of the cases of the shell buckling, the first equilibrium position always corresponds to the trivial solution of the undeformed state, we may state a criterion for the instability of the system has a nature as equation (1.1), which is as follows: 16 Criterion. The threshold of the dynamic instability (or buckling) is defined by a point on the characteristic load- deformation curve, where the deformation of the dynamic system reached the first unstable state of static equilibrium. It is noted that, for a single degree of freedom system (1.1), this criterion of instability should give the same result as would be obtained directly from the dynamic stability theorems, i.e., the phase-plane method. However, there is no restriction in the application of the above criterion to the systems of higher degrees of freedom, while the topological method, in general, does not apply in such cases. 2. Nonautonomous System In general, the topologic method cannot be used to solve the problem of a nonautonomous system, i,e., when the time variable Z expressly appears in the dynamic equation, because the trajec tory of a motion is in a space rather than in a plane. For a certain class of equations, Minorsky (24) developed a method which he called the "stroboscopic method." By finding an identical trans formation, the original nonautonomous system can be transformed into a stroboscopic system which is autonomous. Therefore, the stability problem of a periodic motion of the original system is equivalent to the problem of investigating the stability of singular points in its stroboscopic system, Unfortunately, this clever method cannot be applied to the type of problem which has nonperiodic motions and with large nonlinearity, mainly due to the difficulties of finding 17 the stroboscopic transformation. Furthermore, for certain problems, in which we are interested, the motion is known to be unstable as time increases indefinitely. As far as buckling is concerned, we are merely interested in knowing where the deformation begins to increase violently or attains dangerous magnitude. We have seen in the last case, i.e., the autonomous system, that the beginning of the violent increment of deformation is defined by the initia tion of the instability of the dynamic system, and as a matter of fact, they are identical. However, it is impossible to xtend the same logic to the nonautonomous system, for some of them eventually reach a state of "unstable motion," e.g., a system under a forcing function with a magnitude increasing linearly with time. No matter what conditions we have, however, the projection of the trajectory of motion onto a TJ rj plane still offers us some information regarding the "violent increment of deformations," as we shall see in the following. Let us take the following system jj = { ( 7) Z ) (1.4) where t is the time variable. The similarity between (1.4) and (1,1) is easily obtained by taking T equal to some definite value of time, say \ ; i.e., at the certain time X the motion of (1.4) is on a trajectory in the phase plane characterized by (1.1). Therefore, the motion of (1.4) can be treated instantaneously as 18 a motion of an autonomous system of the form of (1.1). However, it passes only one point on the trajectory of each phase plane. Let us specify, furthermore, that the form of fi7], T) in (1.4) is increasing in magnitude together with t, i.e., the energy level becomes higher and higher as t increases. The space trajectory of the motion of (1.4), in this case, can be visualized as in Figure 3. We can project the trajectory onto a plane similar to the phase plane and it will be in a form as shown in the above mentioned figure. It should be remembered that we have specified the forcing function to be a monotonous increasing function of t". x Notice the form of the trajectory in the y)-'7} plane. It is very similar to the form of an autonomous system with a negative damping term; with the only difference that the "unstable focus" changes with time. All the unstable singular points at time r = are determined by (1.5) 0 this is shown in the figure as 7) Y)i VÂ¡ By virtue of (1.5), the following equation, f ( V Z ) ~ 0 (1.6) is simply the locus of all singular points (both stable and unstable singular points) in 7)-rj plane. The intersection point of the curve ^We prefer not to use the term "phase plane" in this case. 19 defined by equation (1.6) and the response curve, i.e., the solution of equation (1.4) is simply the inflection point of the response curve. At the first inflection point of the curve, the change of the slope is zero and the slope of the curve is a maximum; there fore, it is the upper measure of the "violent increment of the deformation." There is another significant singular point from (1.5), i.e., when = X = 0, the initial time. The first unstable singular point in the plane when t = tQ certainly is the lower measure of the critical deformation. In the case that f appears only in the forcing function, the lower measure in the above defined sense is simply bhe critical amplitude of the free vibration of the system (1.4). Therefore, the middle point of these two bounds is a reasonable measure of the dynamic buckling. As a conclusion, we summarize the criterion proposed in the last section in the graphs shown in Figure 4. In Figure 4 P repre sents characteristic load and 5 is designated to be the characteristic deformation. D represents the typical dynamic curves, while S repre sents typical static curves; AL is the critical amplitude of the free oscillation of the dynamic system. The critical condition of each system is determined by Acr or bounded by and Au according to the above discussions. CHAPTER II BUCKLING OF A CLAMPED SHALLOW SPHERICAL SHELL UNDER A PURE IMPULSE I. A Qualitative Discussion of the Loss of Stability of the Structure It has been shown in Appendix II that the dynamic problem of a clamped shallow spherical shell under the action of a uniformly distributed load q(t) can be reduced to a single non linear second order ordinary differential equation if a first order approximation of deflections in the following form is employed; thus 2 (2.0) This second order ordinary nonlinear differential equation,^ when 2 l) = 0.3, has the following form: 33 l_ \(o p h (2.1) ^The derivation of this equation is attached in Appendix II. O Refer to Figure 1 for the definitions of each quantity in this equation. 20 21 If the shell is sufficiently shallow so that the middle surface of the shell can be approximated by a paraboloid, thus Z ~ Z0 ( r / r0 ) 2 and the curvature of the middle surface has to satisfy the condition: dr2 Ro ' These last two equations give us the useful relation: Zo ~ (2.2) By virtue of equation (2.2), a dimensionless form of (2.1) can be written as: yÂ¡ + (76 + 3.025 0 ) r) 5-11 rÂ¡2 + 2-1075 r/3 where the nondimensional quantities: f *1 (2.3) v = i=\frTp */r0 , 2 Q h I Z0 %(?)= jT^z $ (t'> (2.4 have been used, and p is the mass density of the shell material; the dot () now represents d/dt. In that case the external pressure is an ideal rectangular impulse as shown in Figure 6. For sufficiently small ^T, as 22 discussed in reference (28), this problem can be formulated in the following fashion: the oscillation of the system is essentially a free oscillation started at the time t = AT (i.e., at- O ), with certain initial conditions derived from the external disturbance. It is conventional to take the displacement at the new initial time, i.e., t = AT, still being the same as the one at t = 0. For an initially undisturbed system, the new initial deflection will remain zero because of the smallness of AT. The velocity at t = At, however, will not be the same as that at t = 0, because of the sudden and instantaneous external disturbance. If the system is originally undisturbed, i.e., velocity is zero at t = 0, then the velocity at the new initial time can be easily found by use of "the principle of linear impulse and linear momen tum, i.e., It is noted that equation (2.3) is equivalent to a nonlinear spring-mass system with a unit mass (m = 1) and a constant force in its nondimensional form, Therefore, the above equa tion reads: e j ax = i (*t) o where AX is the dimensionless form of the time duration AT, by the definition in (2.4). 23 From the above discussion, we have our new initial condi tions, thus at Z = dt V(*z) = 0 I (AX) = l) (4-c) = || 0 I* (2.5) where I* = q.(Atr) is the dimensionless impulse. Because of the smallness of the time duration of the impulse ( AZ ta O ), it is conventional to solve this problem by taking (2.5) to be the initial conditions at X : 0, thus 3? = Tb 7] (0) 0 (2.6) and a free oscillatory system with a dynamic equation obtained from (2.3) by dropping the terra on the right-hand side, which is involved with the external load q. Thus 7] + [i 2 76 t 3 025 0*)r) $-31 + 2 /o7- 0 The oscillatory motion of an autonomous system characterized by equation (2.7) and initial conditions (2.5) can be discussed quali tatively by a phase plane method. We shall follow reference (23) and write (2.7) in the following form: 1 = l (2.8) 24 Here it is apparent that Â£ is the dimensionless velocity of the motion. Equation (28) can be integrated once and becomes 7 4 -3. Â£4 rf + ( 2-76 + 3-025 >2) V1 (2.9) where C is an integration constant. Equation (2.9) is the equa tion of the trajectories in the phase plane. It describes all possible motions of a system characterized by (2.8). The stability of the motion will be analyzed in the following. The singular points (or critical points) of (2.8) are found from the equation (2.8) by putting 7] and % equal to zero simul taneously, i.e. \ 0 and 7 [ 2-lo8 7)1 5-31 7 + (2-7 + 3 o2$ e2) ] = 0. It is clear that the critical points are located on the "7 -axis. One of them is the origin of the phase plane and the other two points are defined by the equation: 77 = 'o 4-2/6 $-31 5-0$ Vo-'9103 0X (2.11) The stability nature of these singular points can be determined 1 from the following characteristic equation, (9), i.e., Refer to p. 317, reference (9). 25 Det (A) -A f -[(2-76 + 3-o25 02) lo (>2 \ + \Z ] -A (2.12) It is given by the stability theorem^ that corresponding to a pair of real eigen values of (2.12), an unstable singular point will satisfy the following condition: [ 6-324 *)02 10-62 \ +(276+3-025 62)] <0. (2 It is also apparent from equation (2.12) that there are only two types of singular points for this system, namely, the stable critical points of "center" type and the unstable points of the type of "saddle points." Let us return to equation (2.11). It is clear that there exist real positive nontrival values of r]o (singular points) provided, 01 4 o 19 303 or @ 4 0- 44 (2.14) Physical significance of this condition is that if a shell is sufficiently shallow such that its geometrical parameter 6 is larger than a certain limit, i.e. ^This is the characteristic equation of the linear approx imation. The discussion of using this approximation is referred to in Appendix Io 26 0 > 0-44, 1 (2.15) There will be no "snap buckling" under the action of an impulse. For this reason, we shall be interested only in those shells with geometrical parameter Q 0-IU+. By virtue of (2.13), we know the root: 1 4.2I& $31 o 19303 9 (2.16) of (2.10) is the unstable saddle point. The trajectory passing through this point is called a separatrix. The motion on the separatrix is essentially unstable, and the motion described by a trajectory inside of the separatrix, in general, has different character than the one described by a trajectory outside of the separatrix. Therefore, the problem of determining the critical condition of the system reduces to one of find the motion whose trajectory is the separatrix. Because the separatrix passes through the singular point defined in (2.16), then by using equation (2.9) we can find its equation. For the purpose of emphasis, let us replace the symbol by I** from now on, T)'** in equation (2.16) is read This condition is equivalent to defined in equation (4). \< J37 where A is 27 From (2.9), for the separatrix, we have - 1-0$38 I* 3-& Vj + (2 76 + 3 02$ e2) 712 /cr ~ C ] ~ (2.17) It must be remembered that from (210), "?7cralso satisfies the condition:^ ( 276 + 3 02$ 02) = $-3lVtr- 2 1o8 Vet - Using this equality and (2.17), we can determine the constant C = Cs, which will yield equation (2,9) as the equation of the separatrix. Thus, Cs = 1-77 r,l 1.0538 V* (2.18) where, from (2.16), \ = 1-2$94$ 119181 V0-19303 Q2 (2.19) Let us return to (2.5), the initial conditions which define the motion of the system under the action of an impulse. From the first condition in (2.5), i.e., '*)(o) = 0 . It should be noted that ^cr is subjected to the condition that o i.e., vÂ¡Lt is nontrivial. 28 It is apparent from this condition that the motion always starts at the point where its trajectory intersects the Â£ axis. In case the motion is on the separatrix, i.e., at the critical condition, from equation (2.9), (2.18), and with condition (2.5), we have the following result: or (2.20) where 1?tr is defined in (2.19).. These two equations will give the critical impulse for any shallow spherical shell whose geometrical parameter 0 is known and satisfies the condition in (2.14). An example is given by taking 6 = 0.26, (A 5). We shall see, particularly in this numerical example, that the result obtained by using a phase plane method will be the same as obtained by using the buckling criterion proposed in the last chapter. Furthermore, the same result may be obtained if Budiansky-Roth's criterion and techniques in (8) are employed. It should also be mentioned that the result reported in reference Q8) is numerically more accurate than that given by equation (2.20) because of higher order approxi mations used by those authors.^ However, the problem solved by 1 They used a five-degree-of-freedom system. 29 this simple but precise method will permit certain qualitative conclusions which could not be obtained, or would cause much labor in calculations if other methods are employed. When 0 r: 0-26 i (2.5), and (2.6) read: # V = - 2-/075 V( V- o.S3$o3)(rj- 1-68453) T)Co) = O f yj (o) 0-536 2$ IA'. (2.21) Three singular points on the T) -axis are: >? = O a center V = 0-Â£3503 a saddle point T) = 1 6^4 53 a center. Let 'r)Cr = 0.83503; from (2.20), we found lcr = 1-34233 (2 o 22) The equation of the separatrix is found as follows: Â£2 = 1-0^2 Tj1* + 3 $4 rf 2-9645 i)1 + 0-SI83C. This equation and other phase plane trajectories have been plotted and are shown in Figure 7. In Figure 7, when I < I cy given in (2.22), trajectories of the motion are closed curves around the center point at the 30 origin, e.g., curve 1; when 1 '= Icr the motion is unstable and * -r * on the separatrix 2; when I > 1 cr the motion is on a trajectory such as 3. It is also easy to explain the occurrence of the "buckling" from this figure. The points where trajectories intersect the ^ -axis correspond to the situation that the response curve reaches its amplitude. Therefore, the coordinate of these points (e.g., 0A) is the measure of the maximum inward central deflections of the shell. It is clear that the maximum inward central deflection increases with I in a continuous fashion when 1 < lcr As soon as 1 is slightly larger than I tr the maximum central deflection undergoes a severe change, from some value less than 0.84 (e.g., 0A) to some value greater than 2 (e.g., OC). Because of this severe change of deflection, snap- buckling of the shell occurs. 2. A Study of the Dynamic Response It is noted that the differential equation has the form of (2.7), and with initial conditions (26) can be integrated. The solution of such an equation, in general, is involved with Jacobian Elliptical Functions, It is still impossible to give a nontrivial expression for the solution of the equation, which is of the same form as equation (27) However, when 0 is taken to be a definite value, the solution of (2.7) can always be obtained. For an example, 31 in solving such type equations, we shall take the system defined in equation (2.21); i.e., 0 is taken to be 0.26 in equation (2.7) and (2.6). All numerical work involved will be presented in detail. We feel that the result of this section will clarify certain impor tant points in both the last section and the following section on the justification of the buckling criterion. Let 0 = 0.26 in equation (2.9); we have the equation of the trajectories for the system (2.21) in the following form: } = J C 2-964$ Y i 3-54 T]3 1-0538 V* , (2.23) where C is an integration constant. By using the second initial condition in (2.21), C has the following expression: C = O-S3625 I*)2 (2.24) Our problem is to find the response zj (?) corresponding to each disturbance I*. It is still impossible to obtain the general expression and only particular cases will be given. We shall study the responses corresponding to two individual disturbances: 1 = 1.2, and 1 = 1.5. We shall see, in contrast to the small increment in the disturbance I the corresponding responses will undergo characteristic changes. When 1 = 1.2, the positive branch of (2.23) reads: d V dz h -I-03S ( T}~ 2-0049 )(>}-1 loStld-o + 0-3/47) . (2.25) 32 This can be transformed into an elliptical integral of the Legendre's standard form.^ Let us, first, formulate the quadratic equation: Z-Sb \bb V2 4-7766 V + 1-fo2$3 = 0 (2.26) Its coefficients are related to the zeros of the algebraic equa tion under the radical sign in (2.25) in definite ways which can be found in almost any textbook treating on elliptical functions and 2 will not be given here. The two roots of (2.26) are: P = 0 21585 % = 7- 397 /S' (2-27> Now, we use the following transformation: = f* U 1 + z 0-27585 + 1 397 18- Z 1 + Z (2.28) and dl) = [ 1 ,2,3V(i +z)2 ] z , and then equation (2.25) can be transformed into the following form: 1 Refer to reference (41). 2 ecg., reference (43) 33 I.12133 oU (1 + 2)2 dt J 0-tz)2 J \ 0*55% (O y7721 Z2- / 4342)p 4272 Z2- 0-16983) or dz dr 0-21196 fZ2- 8 o9$59)(22' 0-11899) or dz 0 46039 dr = /"Z2-8 ?SÂ£9)( 2J- 0-//899) . 1 (2.29) Referring to reference (41)/ the function z can be written in the form: Z = 0-34495 Sn (u | m) , (2.30) where u = 1-3099J (r Co) * m = o- o J469 (2.31) Therefore, as we substitute (2.30) into equation (2.28), the solution of the problem can be formally expressed as: The expressions are on p. 26, reference (41). 34 V = o- 2 7 fe % 5 + o 4S96 Snfu-I'fli) l + o 3449fe Sn ("u ) m) (2 o 32) where U and m have their definitions in (2.31)o The value X o in (2.31) is determined by the initial condition: when X = 0, xj = 0. In this fashion, we have XQ satisfying the following equation Sn (-1-30995 Zo | 0 01469 ) = 0-fe72 3* . (2.33) As an approximation, Zo 0 469^ (2.34) Solution (2.32) has the following general properties: A. It is periodic because it involves the double periodic function S'n(u|irn.) The real period of Sn(ulm) is 4K = 4(1.57658). The period, P of *] then, is equal to 4.81419 according to equation (2.31). B. The 7} values are bounded in the interval -0.31464 4 V i 0.56344, because the value of Sti (u Im)varies between -1 and +1. Therefore, the maximum amplitude of the dynamic response V) corresponding to the disturbance I* = 1.2 is Y) = 0.56344. ' max The next example; when I* = 1.5, the positive branch of (2.23) reads: ~ = Â£ = J 0-64702 7964* + 3 *4 rf l-o$3? . 35 Different from the last case, the rational function in the radical sign in the above equation has complex roots. As a counterpart of equation (2.25), we can express the above equation in the following form; thus dr) f- = y-i o538 crj-ot) C7)-fi)( rÂ¡-r)( o) (2.35) where d 2. 0 = 0-38067 , = o-d- 0 28-537 t 6 = o-8M6 o-28*J7 i. The counterpart of equations (2.26), (2.27), and (228) are respectively 2 O 0170 3 If + 3 13426 V 2-6321$ O. p- -I84-879 ft -3*87 (2.36) and ^ -184 #79 + 0-83$&l Z 1 + 2 d7) = [ 17/u8] / (I-b Z)2 J dz (2.37) The transformation in (2.37) brings equation (2.35) into the simpli fied form as in the following: dz A / f 22 - i 0-999j + Mi O O2IUH ) y where A Ji U9993 ) o 081 uu M, = 34,49) *6o$9 , M2 = 34, 429-9792 3 . Therefore, we have (2.38) Z = I5l- 64249 Nc (aim) (2.39) where U. = 1 29092 (T- r0) , m = 0-94g,49. (2.40) Substituting (2.39) into (2.37), we have the solution -I U2l9 Cn(u|m) + 126 -7534 I C-n(uim) + 151-64249 (2.41) where the definition: Nc = < / Cn , has been used, and u, m are defined in (2.40). 37 In a similar way, To in (2.40) can be determined by requiring 1](Z=o)= O thus lQ -0.70755 (2.42) It is obvious that the solution in equations (2.41), (2.40), and (2.42) is characteristically different from the solution represented by equations (2.32), (3.31), and (2.34). Solution in (2.41) has the following characteristics: A It is periodic, with period P = 9.03 Bo 7) values are in the interval -3.8079 Â£ 2,06868. The maximum amplitude of the dynamic response, ^max 2,06868 Notice the characteristic change in the form of the dynamic response and the severe increment in the amplitude (from ^ = 0563 to 2 .068) as I* value changed from 1,2 to 1.5. We can max conclude that the critical load Icr must be some value in between the two values. One gets a satisfactory justification by referring back to equation (2.22), where the critical impulse was found to be 1.34233. Response curves corresponding to I* = 1.2 and I* = 1.5 are presented in Figure 8 38 3o A Justification of the Buckling Criterion In the last section, we have seen that the dynamic response for a system defined by (2.21) can be found by integrating the differential equation directly, and the solution in terms of Jacobian Elliptic Functions. The dynamic responses corresponding to other external impulses than those given in the last section may also be obtained in a similar manner, yet the procedure is laborious. If merely the amplitudes of the response curves are desired, then for a one-degree-of-freedom system, as equation (2.21), the difficulty of integrating the differential equations can be removed by use of the information obtained from the previously discussed topological method. We have mentioned that the intersection points of the trajectories and the V -axis in the phase plane are the points where the response curves reach their amplitudes, because at those points, the velocity Â£ is equal to zero. This fact suggests that we obtain the amplitude-impulse relation for the system (2.6) and (2.7), or their special case (2.21), from the equation of trajectories, i.e., equation (2.9). Let us restrict ourselves to dealing with the special case defined in equation (2.21); i.e., 0 is taken to be the value 0.26. Substituting (2.24) into equation (2.9) and using 0 = 0.26, we have the following equation for the trajectories of this system: 39 % 2 |.o(54 r)l + %uS rf 02%1$G I*2 J (2.43) By virtue of the above discussion, the amplitudes of the response curves, rj' s, are found by setting the velocities t,'s equal to zero By doing so, from (2.-43) the following relation is obtained: _J o-2$7$6 - 3 54 7m + /%4$ (2.44) For the solution to be physically meaningful, the positive branch of the last expression should be used; thus, - /3M32 7)* [ rj* 3.3*8-63 Vm + 2-WG\\ (2>45) This relation is shown in Figure 9. Good agreement between the result presented in Figure 9 and the result for the two cases worked out in the last section indicates the correctness of this technique. In Chapter II we have proposed that the shell will buckle when the characteristic load-deformation curve reaches the first unstable state of static equilibrium There are three particular static equilibrium states, i.e=, the positions of rest of the system (2.21) which can be found from the first equation of (2.21) by setting the inertia force, i,e,, y term, equal to zero. Thus, the three positions of rest (where ij = = E, = 0) are obtained: iO il) 3> , rÂ¡ 0 r) o-fa 7} l-GS . (2.46) 40 In T)max I* plane, these curves will be straight lines parallel to the I* axis. According to the criterion, the first unstable equilibrium position ( ^ = 0.84) defines the critical condition. As shown in Figure 9, the I*cr thus found is identical with that found previously by the phase plane method. It is also of interest to see the comparison between the present criterion and that proposed by Budiansky and Roth in (^). According to Figure 1 and equation (2.0), we have: t o Therefore, certain definitions in reference (8) assume the following expressions: Z and By definition we have A = C-/zc = V . (2.47) According to (2.47), the measure of buckling used in (8>) , /\ =1, corresponds to 7) = 1 in the notation of this paper, max max max 41 It is apparent from Figure 9 that the same critical impulse will be obtained if the criterion proposed in reference (8) is employed. Furthermore, it is also indicated in the same figure that the measure used by Budiansky and Roth falls into the unstable branch of the load-deformation curve and is close to the point of instability. Therefore, the criterion proposed by these two authors is proved to give satisfactory accuracy for this specific problem. Critical impulses for other values of 8 based upon equa tions (2.19) and (2.20) have also been calculated. Results are presented in Figure 10. It is shown in the figure, for 0.44 > 0 > 0.32 (or 3 = 87 < X c 453) that equation (2.20) agrees well with the result given in (8) and appears almost the same as the result of reference (13) when 0< 0.15 (or A > 6,6). It is believed that this analysis is parallel to the presentation of (15), yet with a cut-off point at a larger 0 value, i.e., the present analysis admitted a shallower shell to buckle under the applied impulse. This tendency seems to be correct as compared with the result of using a higher degree of approximation given in reference (8). 4. A Note on the Effect of Initial Geometrical Imperfections It is rather interesting that we may conclude, on the basis of Figure 7, that any axially symmetric geometrical imperfection will give a deduction of the critical impulse for the shell. It is the nature of the equation of the separatrix of having a relative max imum when "*7=0, i.e., on the axis. Any initial imperfection 42 (deflection) of the shell is equivalent to set the motion starting at C 1 instead of at C as shown in Figure 7. It is seen from S 3 this figure, that Cs' has a smaller ordinate than Cg Therefore, the critical impulse based upon the former will have a lower value. For an example, let us assume that the initial deflection of the shell is axially symmetrical and has the same form as the deflection of the shell, i.e, can be described by equation (2=0); further more, it has a central deflection = 0-0$ z0 J t or 7Ji 0-0$ (2.48) Based upon this value, the critical impulse will be 6 per cent less than that directly given by equation (2.20). CHAPTER III BUCKLING OF A CLAMPED SHALLOW SPHERICAL SHELL UNDER A SUDDENLY APPLIED CONSTANT PRESSURE OF INFINITE TIME DURATION 1. A Qualitative Study of the Instability of the Structure We have seen, from the last section, the solution of the shell buckling under a pure impulse with infinitesimal time dura tion did not answer the buckling problem of the shell under the impulsive pressures with sufficiently long time durations. In this section we shall solve the problem, which is another limiting case of the above-mentioned problem; the time duration of the applied impulsive loading is infinite. Let us assume, without loss of generality, that before the time t = 0 the system is at rest, and at time t = 0 a uniformly distributed constant pressure q is suddenly applied on the surface of the shell. The history of such a load is shown in Part B of Figure 6. The dynamic equation of this problem can be obtained directly from (2.3) by taking q as a constant. Thus, V = [(7 + J O25 02) 7] ~ 1-3 I 1JZ + 2 lo? I/3 1 l > 0 (3.1) 43 44 We have mentioned that the system is initially at rest, i.e., ^(o) = 0 Â£ (0) = o (3.2) are the conditions at V = 0. Note that the autonomous system in (3.1) can be integrated once and results in the form: * 2 A + (u-125 9 (3.3) where A is the integration constant. Equation (3.3) is obviously the equation of the phase plane trajectories. The singular points (critical points) of the system in equation (3.1) are found from: V = \ = o i.e., on the *) axis where l 6f [ (2-76 + 3 02$ ?) 5.3! Tj1 (3. + 2 108- TJ? J Q _ Stability conditions in the neighborhood of these singular points are determined by the following characteristic equation, thus - /0 62 7) + (276 + 3 025 02)] = 0, 1 + f 6 7255 7)z (3.5) 45 Corresponding to the real eigen values yielded by (3,5), the saddle points satisfy the following condition: 3225 ^ Â¡o 62 7jo + (21G 4 3 025 Q1) < 0 , (3.6) where is the root of equation (3.4), The problem of finding the critical load q which will make the motion of the system described by (3d) and (3.2) lose its stability is equivalent to the problem of finding the q value which will put the motion of the system on a trajectory passing through the saddle point in the phase plane, i.e., on the separatrix. By virtue of its initial conditions, the trajectory of the motion of the system (3.1) and (3.2) passes through the phase plane origin ( = E, = 0). Therefore, the equation of this trajectory is obtained from (3.3) by taking A = 0, thus 2 (3.7) If, furthermore, this trajectory also passes through the saddle point ( V) 0), it is obvious that the following condition has to be satisfied: or equivalently 46 for "70 1 0. Attention is invited to the fact that this problem reduces to finding common roots between equations (3.4) and (3.8).^ A solution to a similar problem for a complete spherical shell is given in Chapter V, where the technique is discussed in detail. Equations (3.4) and (3.8) will have common roots provided _ t 11 I O5U % = 0 2.062$ 9 Substituting (3.9) into (3.4), we have 3 162 rj* JOS' 7)2 + ( 2 76 + 3 02$ 92 ) ^ (3.9) 0 , (3.10) which has three roots. The two nontrivial roots are solved from the following equation: rll f. 11954 / ( 13 2 1792 3*26020 Q2) V 39 99291 (3.11) It should be noted that it is not true that both solutions in (3.11) are the singular points of the system, because they are not solved directly from (3.4). It merely says, at the present 1Let us recall that V0 in (3.8) is one of the roots of equation (3.4); it satisfies the condition (3.6). 47 time, that both ">7, and rj1 in (3.11) are the possible common roots between (3.4) and (3.8) when the corresponding q value in (3.9) is taken. In other words, if 1) is substituted into (3.9) to yield a particular value of q and this value is used to replace the parameter q in both (3 4) and (3=8), then these two equations will possess a common root, V = r)l (Note that '*]2 may not be a root of either of these two equations.) Since equation (3.10) gives the common root of (3=4) and (3.8) which specifies the buckling of the shell, we may deduce one of the important con clusions i = e., 0 0-630 7 1, (3.12) has to be satisfied; otherwise, no buckling will occur, because equation (3.10) has no real positive nontrivial roots, as indicated in (3.11)= We shall see that only one of the two nontrivial roots of (3.11) will satisfy condition (3.6), i.e., be a singularity in the form of a saddle point. Discarding the trivial solution T) n 0 from (3.10), we have the following condition: ( ?-76 +3 02$ Q1) 7 .o% rj 3 162 (3 = 13) ^It corresponds to A > 3.2123, where A is defined in equation (4). 48 which is satisfied by both Tjt and ^ in (3=11)= Substituting (3.13) into (3.6), after combining terms in the same order of ?) , the following condition is obtained: 3 162 17 2 3 4 v) < 0 . As we mentioned, both ?)]_ and ^ are real positive values, because we have put the restriction (3.12) into our problem. If this is the case, and it is noted that 77 is solved from 'o equation (3.10), condition (3.6) is reduced to the following form: V} 4 f. 11954 (3.14) It is apparent that only the smaller root in (3.11) will satisfy the condition (3.14) and be a possibly unstable saddle point for the dynamic system (3 = 1) More definitely, let us put the critical deflection as follows: I I1954 - ( I 21792 38 26o2o Q ) 39 99297 (3.13) Therefore, without even going back to the phase plane, we can write a general solution for the critical pressure by substituting (3=13) into (3.9), i=e., 2 ' 77 Vtr 1-0*4. ^Icr 2.0625 e (3=16) 49 where 7J has its definition in (3.15). It may be apparent that certain techniques and their logical foundations have not been made clear in the above discussions, and they are very difficult, if not impossible, to be discussed on a general basis. Therefore, we shall give an example using a specific shell with 6 > 0,26, which has been used in the previous problem. Let us first find the solution, i.e., the critical deflec tion and load, and then go back to verify that these critical quantities do put the motion of the system on its separatrix and cause the shell to reach the threshold of instability. From (3,15), when 0 = 0.26, we have 7jir = 754 (3*17) and from (3.16), we obtain immediately, rcr 0 6*5 3? (3.18) Our question has been whether the value of q in (3.18) does cause the instability of the system (3=1) when Q = 0.26, in other words, to make the motion of the system on its separatrix, or equivalently, does the value of Y) in (3.17) define the saddle point in the phase 50 plane when q takes the value in (3.18)? To answer this question, let us write down equation (3.4) and (3.8) by taking 9 = 026 and q = qcr defined in (3.18); thus, equation (3.4) is in the form:* 2.lo? rjZ $-31 r]2 + 2 96449 V ~ 0 ?S1$Z 0 (3.19) which has three real positive roots: t] o nSoS rj2>- 0 "7 = 17863?. (3.20) These are the singular points in the phase plane, while equation 2 (3.8) has the following form: [ i o$u 7}3 3$4 72 +2-964$ r) o 73507 ] O, (3.21) which has three roots: Tj ?.2d3fe$ , and a double root at Tj o 567^4 (3.22) Comparing (3.20) and (3.22), it is clear that 7 : 0.55754, i.e., the critical value defined in (3.17) is the common root between We have multiplied the value (-1) through the original equation (3.4). 2 Similarly, a quantity (-1) has been multiplied through the original equation. 51 (3.19) and (3.21). Furthermore, tested by condition (3.6), this value ( 7? = 0.55754) defines a saddle point in the phase plane. A further remark on the equations (3.21) or (3.8) will completely answer the above question. Let us replace the 7J in the first equation of (3.8) by f) and comparing the final equation with equation (3.7), we find (3.8) is merely the condition of the inter section of a phase plane trajectory with the "^-axis ( Â£> = 0) Because of the preceding discussion, we conclude that the phase plane trajectory for the motion of this particular shell ( 9 = 0.26) will pass through the saddle point if the parameter q is defined in (3.18); therefore, the value of q in (3.18), or in general in (3.16), is the critical value for the external pressure. The phase plane trajectory of the motion of the shell at the threshold of the instability is shown in Figure 11 Based upon equation (3.16) and (3.15), relations between the critical load and the geometrical parameter Q are found and given in Figure 12. Curves indicating the variation of critical deflections with 0 values are given in Figure 13. Comparison between this problem and its counterpart in the static case has been made and is also shown in the figures mentioned. 52 2. Another Justification of the Buckling Criterion In Section 3, Chapter II, we have discussed a method of obtaining the load-deformation curve. For a different problem in this chapter, the same method may still be used to obtain the relation between the characteristic load and the characteristic deformation. The critical condition of this system, according to the discussion in Chapter I, can be determined, and the result thus found will be compared with that obtained in the last section as a justification of the proposed criterion. If the characteristic deformation is chosen as the ampli tude of the central deflection following Section 3, Chapter II, the amplitude-pressure relation for system (3.1) can be obtained from the equation of trajectories, i.e., equation (3.7). By equating 1, to zero, we obtain the following relation between the nontrivial amplitude of the response curves and the corresponding dynamic loadings. ( 2 76 + 3 o2^ 02 ) ^ 3 rf + I-te rf 4-2S 9 (3.23) For the purpose of further discussions, let us take a specific shell with B = CL26. From (3.23), we obtain: 1 ol2 5 (3.24) 53 The static load-deflection relation can be easily found from equation (3.1) by taking the inertia term, i.e., the ^ term equal to zero. For 6 = 0=26, we have 296U49 f) $-11 V2 + 2.108 V3 0-53625 (3 o 25) Equations (3.24) and (3.25) are plotted in Figure 14. The inter section point of these two curves is at Vm = 0.55754, where the dynamic curve has a relative maximum. It is obvious that the inter section point falls into the unstable branch of the static curve. According to Case 1 of the buckling criterion in Figure 4, the corresponding pressure at the point of inteisection is the critical dynamic load, thus qcr = 0.68539. These results are identical with those given by (3.15) and (3.16) based upon the stability theorem. The identical results obtained via two different approaches have established the following facts: A. In certain cases the dynamic criterion proposed in Chapter I is identical with the stability theorem. B. Without given explanations, we have taken the specific expression in equation (3.9), which led to finding the solution of the problem. This turns out to be correct, since the solu tion in (3.23) is entirely independent of (3.9), yet the same result was yielded. A comparison between the present theorem and reference (30) is presented in both Figure 12 and Figure 14. It is interesting 54 that the critical dynamic load given by the present theorem falls in between the two values for the critical dynamic load obtained by use of different methods given in reference (30) The critical deflections given by (30) are seen to be larger than those given by equation (3.15). This is partially due to the fact that a different deformation form was chosen in reference (30). However, a similar relative relation between static and dynamic curves, as indicated in Figure 14 was also seen in reference (30). From the resulting curves presented in (30), by utilizing the buckling criterion in Chapter I, the critical dynamic pressure can be obtained with negligible errors as compared with the solution. From this point of view, the result of the analysis in (30) may be used as another justification of the proposed criterion, 3. A Discussion of the Results In both Chapters I and II, a qualitative method has been used to discuss the motion of the system and to determine the critical condition of the dynamically loaded structure. It is also evident in these two chapters that good agreement exists between the result obtained in this way and the result by the use of the buckling criterion outlined in the first chapter. This gives, at least qualitatively, a justification of the proposed criterion. From the point of view of the applications, the proposed criterion is subjected to no restrictions of any sort, while the topological discussion would meet certain difficulties when the system is nonautonomous or of higher degrees of freedom. However, it is 55 rather convincing that the phase plane method is suitable for use in discussing the dynamic buckling problems. The accuracy of this method may be restricted by the fact of using a single degree of freedom system. The general behavior of the motion, however, is much clearer as being plotted in trajectories. Furthermore, through the examples given in the last two chapters, one can see the direct connection between the energy method and the method of response curves, which were usually employed in solving the dynamic buckling problems. This fact was clearly shown in Figures 7, 8, and 9. The motion on the separatrix, which passes a saddle point corresponding to a certain extreme of the energy level is the threshold of the substantial change in the amplitude of the dynamic response. In the application of the proposed criterion to dynamic buckling problems considering higher axisymmetrical modes or unsyrametrical forms of deformations, a suitable characteristic deformation has to be chosen. One of the examples is to take the mean deflection of the shall as the authors of reference (8) did. A more general problem is that of rectangular loadings characterized by the application of pressure q at time t = 0, which is held constant for a time duration A t and then suddenly removed. This would require the simultaneous solution of two nonlinear differential equations of the following form: 56 rÂ¡ r 7^ Q% [(2-76 -h?2S92)V *T 3 I rf + 2jo7^^?]? O < T < T , rÂ¡(o) = ij(o) O (3.26) [(276 + 5 02$ e1)^ fr-3/ rf + 2 /07T 7? ] , t < r < 00 , (3.27) where Â£ is the dimensionless forra of the quantity d t, according to the definition in (2.4). The stability problem of this system may not be solved without having a general solution of (3.26). It should be noted that it is not possible to give a general expression for the response of the load q in (3.26) in a nontrivial form, This point has also been mentioned in Section 2, Chapter 2. It is for this reason that only discussion of obtaining the results will be given in the following. It is also understood that the shell buckling occurs after the load is removed, i.e., the time duration of the applied loading is sufficiently small. Therefore, the loss of stability of the system is largely due to equation (3.27)o The present problem, by virtue of the above discussion, has the same characteristics as the problem that has been con sidered in Chapter II; in fact, the latter is merely a limiting case of the former. Much as we have done in the previous sections, this problem also can be phrased in the language of the topological 57 method. After doing this, the condition of instability can be formulated in a straightforward manner. The motion of the shell under the action of the said dynamic loading is described by both equation (3.26) and (3.27). Specified clearly in these two equa tions, the motion will be on the trajectories of (3.26) until Z = Z o Immediately after Z = Z the motion of the shell is on the trajectories characterized by equation (3.27). It is obvious that the displacement ^ at t = I is the common solution of both (3.26) and (3.27). The critical condition of the struc ture will correspond to the following situation in the phase plane: the response of q and its time derivative in equation (3,26) at time Z = Z, i.e., ^(t) and Â£( z ) which gives the initial conditions for (3.27), will put the motion of the structure on the separatrix of system (3.27). We shall discuss this matter as follows: when Z Z > the equation of the trajectory^- is Jb2 = (U 12$ 9f ) r) ~ ( 2-76 + 3 2$ 62 ) r) + 3 T4 V l-ob , (3.28) ~ 2 when 2 > Z > the equation of the trajectory of the motion reads: (3.29) 1 Refer to equation (3.7). 2 Refer to equation (2.9). 58 The continuation property of the system at l = I requires: % a) = yr(z) ^ (z) = y(z) . Therefore, we have C = 4-l2i> 0 l ) 7J(i) , (3.30) by comparing equation (3.28) and (3.29). The critical condition of the system under load q now depends entirely upon the value of C. Reference is made to equation (2.18), which defines the equation of the separatrix; the condition of instability of the system characterized by equations (3.26) and (3.27), or alternatively, equations (3.28) and (3.29) then turns out to be (3.31) where C has been defined in (2*18) and (2.19). As an example, when 0 = 0.26, according to the previous analysis in Chapter II, Cc = 0.5183. Therefore, O cr = 8326 (3.32) It should be noted that in equation (3.32), Zj (z ) is also a func tion of q. This is obvious as shown in equation (3.26). Since no analytical form of the solution of (3.26) can be given, further discussions would require a great number of calculations. A procedure for determining the critical pressure, qcr, is suggested as follows: 59 A A time duration z was preassigned, based on the external impulsive loading, B. Assign also a series of numerical values for the loading q in (3.26); these values are arranged in an ascending order of magnitude and with sufficiently small increment. By taking Q as a certain value, e.g., 0.26, corresponding to each q, every equation in the form of (3.26) can be integrated either analytically (in terms of Jacobian Elliptic Functions), or numerically. Therefore, the response of each q at time Z = Z , i.e., 'T) (r ) can be found. C. The critical pressure qcr is the one which satisfies the condition (3.31), or when 0 = 0.26, the condition (3.32) is satisfied. It is tedious, yet straightforward. CHAPTER IV BUCKLING OF A CLAMPED SHALLOW SPHERICAL SHELL UNDER A UNIFORMLY DISTRIBUTED PRESSURE LINEARLY INCREASED WITH TIME1 1. The Solution of the Problem Equation (2.1) in Chapter II can be written in the following form: (4.1) 2 If the following definitions are employed, 7 = e = */z. V (4.2) ^Refer to Figure 1 for the geometry of the shell. 2 Refer to equations (2.2) and (2.4). 60 61 Then (41) can be written as 2 dl + [2H + 3-o2* 02) rÂ¡ $ 31 + 2 log rÂ¡ If the load q is a function which is linearly increasing with time t, i.e., the form as shown in Part C of Figure 6, then it can be represented by the expression: ^ Q t (4.4) where Q is the pressure increasing rate and has the dimension psi. Following reference (1_), we shall use a "transformed" time variable -j *, (4.5) where pQ is the nondimensional critical load for a complete shell given by classic linear theory and has the following form: - 2 f t/jo-v1; for = 0.3, pQ = 1.21; Q is defined in the following: q r; e h Q 2 9 62 which has a dimension: 1/sec. By utilizing (4.4) and (4.5), equation (4.3) reads: + 2 I off r)3 (4.6) where Y (4.7) , a nondimensional quantity. It is clear that the character of the solution of (4.6) depends entirely upon the two parameters 'I/ and 0 We also like to mention here that the transformed time variable is the "dynamic overload factor"; i.e., the ratio of the critical dynamic load to the corresponding critical static load of a complete sphere, which has been defined in equation (4.5). Therefore, the response curve obtained by integration (46) is actually the load- deflection relation of this problem. Let us assume the initial conditions as follows: 7} ( Z* = 0 ) = 0 0 (4.8) which imply an initially undistrubed shell. 63 The nonautonomous system (4,6) with initial conditions (4,8) is best solved by a numerical method. Different values of the rates of dynamic loading and geometrical shape of shells have been selected to substitute the parameters ^ and 0 in (4.6), and response curves were obtained by integrating the equations numerically on the University's IBM-7090 computer.^ Dynamic buckling loads were determined by use of the criterion proposed in Chapter L The static load response curves were found by n 2 dropping the inertia term, i.e., d^7? /d-Â£* from (4.6), and the typical form of those curves was shown in Figure 15. In Figure 15, the form and the nature of the response curves are very similar to their counterparts in reference (1^). For a rapidly applied load with larger Q(e.g., V = 0.3), deflec tion increases slowly at the beginning and has a vigorous change at the time of buckling. For a certain shell, i.e., 0 E, etc., are fixed, the faster the rate of increasing the dynamic load, the higher the dynamic overload pressure will be. It is also seen in the same figure, for a very slow rate of load (e g., \f/ = 100), the buckling of the shell approaches the static case as it should be and the "creep pheonomenon" strongly indi cates that loss of stability is of the "classic type." Followed by several cycles of oscillation, the dynamic curves for '4/ = 100 converge to the static curve. "Runge-Kutta method" was employed. The technique of this method is found in (22). 64 Another set of curves was presented in Figure 16, based upon various geometrical parameters As a different feature from the static case, it is found that the critical load decreases monotonically with the increasing Q values (or decreasing in shallowness) As shown in Figure 18, no relative minimum corre sponding to a certain 0 value seems to exist as it usually does in the static case (Refer to Figure 5.) Also as indicated in Figure 16, the curves move toward the left as the Q value increases, corresponding to a decrease in critical pressure. However, for a very shallow shell, i.e., a sufficiently large value of Q e.g., 0 = 10 ( X as 0.806), as indicated in Figure 16, the curve does not follow the above argument and falls to the right of the curves with Q values smaller than 10. Because this curve remains at very small deflections at a very large pressure, it is clear that the failure of this structure will not be by buckling. As is well known in hydrostatic loading problems, buckling will not occur for a very shallow shell which has a geometry close to a circular plate. This is also observed in this dynamic loading problem. The limiting value of Q i.e., the largest value of 9 for buckling to occur, has not been found. It is the feeling of the author that the limiting value of 8 depends also upon the rate of the loading, i.e, the parameter i// Therefore, the general answer may not be found without considerable costly computations. A further discussion on this matter will be given later. 65 Figure 19 shows the relation between the critical dynamic overload factor and the critical central deflection of the shell, i.e., the deflection at the time of the buckling In this figure we observe the rate of change of the upper ^cr is decreasing as Z *cr increases. This phenomenon can be explained as due to the development of membrane stresses which usually play an important role in the large deflection theory of plates and shells. Further more, these curves seem to approach asymptotically to different limits. These limits depend solely upon the geometry of the shell; the shallower the shell is the higher the limit of the upper Jcr values will be. However, there exists a particular shape of the shallow shell, which corresponds to 0 0.35. For any other shape of the shell shallower than this value (i.e., 0 > 0,35), the critical deflection decreases. This is to say, there exists a maximum for the upper rJCY 9 curves as shown in Figure 20. We do not know whether this value ( 0 s 0.35) is the common maximum point for all values of V ; it would require much computation to answer this question. We are rather interested in the significance of the existence of such a point. Intuitively, it is reasonable to believe that a shallow shell would permit more severe deflection, as compared to the height of its raise, than a nonshallow one would. This is noted as being true for all cases where 0 < 0.35 in Figure 20. However, a contradiction arises for Q > 0.35 (or, A < 4.31). Two possible hypotheses may be provided: first, for all 0 > 0,35 there is no dynamic buckling 66 occurring because of the shallowness of the shell geometry; second, for 0 ~ 0.35 the deflection of the shell has the highest sensi tivity, or the weakest shape with respect to the dynamic load. Therefore, one may thinkthat this value of Q may have something to do with the size of the buckling region of a complete sphere. Both Figures 19 and 20 indicate that if we want to select a certain value of characteristic deflection as the measure of the buckling of the shell, then it may be necessary to take a different value of this quantity for different geometrical shapes of the shell as well as for various dynamic loads. Comparing Figures 17 and 21, we are able to conclude that the effect of the load-rate parameter y on the critical dynamic load is much more significant than that due to the geometrical parameter Q We mentioned that the critical dynamic overload factor !cr* is a function of two parameters (variables) Y and 0 i.e., it is a surface in the ( Y, 0 Z *) space. Based upon the curves in Figures 17 to 21, we have the approximate equations for the surface: (For, l) = 0.3) / -o8\ ( 1 + V ) -7 3 9 7 Ur] u l.frl 4 2.0 Si e 132. (27 +W -99 I 31 1 o.79 6 (4.9) 67 We found that in the range of 0.5 Â£ V 3.5 and 0,15 Â£ 9 035, results obtained from equation (A.9) agree well with that solved from equation (A.6) and initial condition (A.8). The accuracy of (A.9) can be proved only by experiment. We believe, however, that a design formula for different values of Poisson's ratio can also be established in the following form -bw ^ K 9 n Zcr = (a + V )( C t Â£ e ) (A.10) with a, b, c, g, and k determined by experimental tests The projection of the trajectories on the ^ V plane are shown in Figures 23 and 2A. According to the criterion in Chapter I, the critical state is bounded by the two points indicated by L and U, ice, the lower and the upper bounds. It is seen from these trajectories that the deformation 7) increases very slightly after passing the point U and oscillates about different equilibrium points on the "^-axis with an increase in amplitude. 2 A Conclusive Discussion of the Problem The nature of the response of shallow spherical shells to a high-speed dynamic load with linearly increasing pressure intensity were found and represented in Figures 15 to 2A. We found the functional relation between the critical load and the geometry of the shall has a characteristic difference from the 68 static case and this is shown by comparing the curves in Figure 17 with their counterpart in Figure 5. Differing from the result obtained by Agamirov and Vol'mir, we found the critical dynamic load (or, D.O.L.F.^) depends upon the ratio of the speed of elastic waves in the shell material and the product of the shell radius with the increasing rate of the intensity of the external pressure, instead of solely upon the increasing rate of the load as presented in (_1) . A functional relation between the critical dynamic overload pres sure and the two parameters ^ and G was formulated in equa tion (49); it gives the critical dynamic load from the given geometric shape, the material properties, and the increasing rate of the dynamic loading. Another suggestion was also offered by formula (4.9). Let us return to the definition of 1//" in (4.7), i.e, which is a dimensionless quantity. The requirement of similitude is very conveniently furnished by the quantity For an example, if we choose the same geometric parameter Q for the model and the prototype, then one can determine the nature of Dynamic overload factor. 69 dynamic buckling of the prototype under a very high rate of dynamic loading, i.e, a very large Q, from the test of a model with lower Young's Modulus and a relatively lower rate of the dynamic load, provided they have the same value of ^. Discussion of the buckling criterion was also made. It is seen from the results that both the increasing rate of the dynamic load and the geometrical shape of the shell have influ ences on the critical deflection of the structure. Therefore, the criterion proposed in Chapter I, which permitted "the measure of the critical state, to change along with different dynamic loadings and geometric shapes of the shall, has definite advantages. CHAPTER V BUCKLING OF A COMPLETE SPHERICAL SHELL UNDER SUDDENLY APPLIED UNIFORMLY DISTRIBUTED DYNAMIC LOADINGS Introduction The nonlinear problem of a complete spherical shell has been discussed by numerous people since the article by von Karman and Tsien first appeared in 1941 (17) The progress toward a substantiated explanation of the discrepancy between the classic linear theory of Zolley and experiments has been rather slow. This is partially due to the lack of reliable experimental data on the buckling load for a complete spherical shell. For many years Tsien's (31) energy criterion and the "lower buckling load" (32) have been used to determine the load-carrying capacity of a thin elastic spherical shell. In the year 1962, two inde pendent experimental tests contributing to the problem were reported by Thompson (33) and Krenzke (19). Both of these two experiments show that the critical load of a complete spherical shell can be much higher than the lower buckling load offered by Tsien; it ranges from 45 per cent to 70 per cent of the critical pressure predicted by classic linear theory. A fairly precise theoretical analysis based upon a large-deflection strain energy theory was also given by Thompson in reference (33). Thompson 70 71 ended his research with the conclusions: A. The initial buckling (pre-buckling) was seen to be classical in nature, i.e., the load deflection process is con tinuous. B. The buckling load was about 75 per cent of Zolley's "upper buckling load." C. The stable post-buckling states are observed to be rotationally symmetrical Both Thompson and Krenzke's tests suggested further theoretical investigations. It seems that more precise work on the numerical investigation of the governing differential equa tions as has been done for shallow shells is very much to be desired.^ Among all the works on the stability problem of complete spherical shells, Vol'mir's first approximation (35) remains unknown to most of the western scholars, mainly because it was not a successful one. He attempted to investigate the post-buckling behavior of a complete spherical shell by using a very simple form of deformation, i.e., the form in equation (2.0) and integrated the equation approximately by a Galerkin method. His assumption on the buckling process was the same as the one assumed by von lE.g., (16), (7), (27), (38), (39), and (40). 72 Kantian and Tsien, ie., the sphere is contracted to a sinaller sphere and forms a single dimple after buckling. By virtue of this buckling mechanism, Vol'mir allowed a uniform membrane stress: - V-At. to be distributed in the buckling region before the loss of stability and considered the buckling region as a shallow spherical cap In other words, the differences between Vol'mir1s method and Karman-Tsien's is that Vol'mir used a Galcrkin's method based upon a variation of the governing differential equations and also considered the variation only of a single parameter, which is the central deflection, while Karman- Tsien based their method upon the variation of the total energy with two parameters. In a limiting case of small deflection theory, Vol'mir found a buckling load twice as large in magnitude as predicted by the classic linear theory and failed to obtain a lower buckling load. In Vol'mir's approximation method, we find certain ques tions which may have led to his failure in obtaining an approx imate solution. First, he considered that buckling occurs at the transition of the membrane state and the bending state of the shell; therefore, his effort was concentrated on finding a post- buckling load. Second, the assumption of a uniform membrane 73 stress in the buckling region before the loss of stability requires an abrupt change of stress state, which seems impossible under a continuous loading process. If the structure is perfect in geometrical shape, this change of stress distribution would require different equations to describe the equilibrium of the shell and result in mathematical difficulties of integrating the differential equations. It seems to us that Vol'mir's method may be restudied using a different contemplation of the buckling mechanism of a complete spherical shell. 1o On a New Mechanism of the Buckling of a Complete Spherical Shell We shall assume that the buckling of the shell follows a possible mechanism which permits the transition from a membrane state of stress to a bending stress state occurring in a con tinuous fashion and the transition occurs before the loss of stability of the shell. This is described in the following paragraphs A. When the external pressure q is much less than the critical value, the shell contracts to a slightly smaller sphere; 74 as shown in Figure 25, Part A, the original shell contracts to a sphere with radius equal to Rq R, As q increases, AR increases and a significant change in curvature occurs because of the change in radius. B. Let us take another assumption that the structure has a resistant nature against the higher pressure and has a tendency to resume its original curvature. Based on experimental evidence that spheres form a single dent after buckling, we think that the resumption of the curvature starts at a small region, or we might say at a point. The effect of the resumption of the curvature from a larger one to a smaller one has introduced a pure moment, which will be in the same direction as that caused by the external pressure.^ In other words, we consider that the initiation of bending stress in the shell is due to the imperfect nature of the structure; however, the external pressure will certainly help to increase the magnitude of the bending stress and build up the inward deflections- 1 Refer to Figure 25, Part C. 75 Co The existence of such a single point for the first resumption of the original curvature may be explained as due to the "imperfections," Let us assume a spherical shell with perfect geometrical shape all around except a very small hole at point 0 (The advantage of the assumption of a small hole is that we do not have to make any other assumptions on the form of the imperfections) When the original shell contracts to a smaller one (refer to Figure 25) so that A moves to A', and 0 to O', the small hole is contracted to an infinitesimal one. Under such a condition, the membrane stress at O' is certainly zero. Do If we allow the existence of such an infinitesimal hole at the point 0 in Figure 25, then the bending state is inherent in the problem itself. As shown in Figure 25, Part C, in the immediate vicinity of O', the situation is very similar to a clamped circular plate with a central hole The idea of "boundary layer" may be best fitted into this particular circular region; outside of this region, a pure membrane state remains. When the external pressure q increases, this circular 76 region dilates (or the thickness of the boundary layer increases) and forms the buckling region after the loss of stability of the shello By virtue of the a bove described buckling mechanism, we arrive at the conclusion of the existence of a boundary layer at the vicinity of a point 0' In this region, both membrane and bending stresses exist at the time of the stabil ity of the shell. We are interested in the distribution of the membrane stress in the boundary layer region during the load ing process. As we have mentioned, the stress at O' is zero and outside of this region the shell maintains a momentless state with a membrane stress (Jr = 0/2h. By referring to the stress distribution in a bent clamped circular plate,^ a reasonable assumption in the boundary layer region will be a parabolic variation, i,e., (5.1) ^Refer to (34), pp 54-63. ^Refer to Figure 25, Part B. 77 We shall analyze a nonlinear problem of the loss of stability of a complete sphere by taking the buckling region as a clamped shallow spherical segment with a nonuniform membrane stress in the form of (5.1) distributed in the middle surface before its loss of stability. 2 Buckling of the Shell under a Static Load We shall take the buckling region of a complete spherical shell as a shallow spherical segment clamped along a circular boundary, as shown in Figure 1, Part B. From the discussion in Appendix II for shallow spherical shells, we have the governing differential equations for such a shell under a uniformly distributed static load q (p.s.i.) in the following form:^ The equation of equilibrium, D(V) i iz r <*r J 2 (5.2) and the compatibility equation, d_ dr ( V24>) - Â£ \ J- ( )2 + i d w j [ 2r ( dr J Ko dr J ) where VZ = di dr1 + Ld. r dr (5.3) 1 Refer to equations (A.29) and (A.30) in Appendix II. 78 Since we have restricted ourselves in the problem of axi- syrametrical deformations, then equation (2.0) can be used again as a first approximation of the deflections in the buckling region; thus, we have w = Â£ f(r/re2)J (5.4) Substituting (5.4) into (5.3) and integrating, we have which is the condition that the strains or stresses in the middle surface due to large deflections have to satisfy. However, it should be remembered that there is a membrane stress already in the middle surface due to the contraction effect of the rest of the spherical shell outside of the boundary layer region. By using the relation between the stress function (fi and radial stress (Tr and after including equation (5.1), the above equa tion of the compatibility condition of deformations in the middle surface of the shell reads (5.5) 79 Substituting (5.5) into (5.2), the equation of equilibrium yields the following form a (5-3W i v Â£ 4 rc (7> - ' 0 6Ra i~v 2(2-V) ,r_, ( rj Hkf 0 (5.6) We shall use Galerkin's method. This method allows equation (5.A) and (55) to be the approximate solutions of equations (5.2) and (5.3), provided ff G['-(r/r.)2J dA = 0 JJ a (5.7) is satisfied. In (5.7) "A" represents the area of the circular 1 region with radius r It should be noted that r is not a o o constant; it is the thickness of the boundary layer, which varies with the external pressure. After performing the above integration, we find that the central deflection 4 has to satisfy the following equation, which describes the equilibrium conditions. 1 Refer to Figure 25. 80 r \ i E hZ o-2) C 1 PF?o2 1 24 ) i-if ) -}] E [ tn 75 if R<>r02 4? (i-v) (>219 2704 E r ff-jjv b(l-v) C il 8 h 2 i ^ re2y .3, equation (5=8) reads f ' + ' Y 11 '0 ? ) c Rl ) t to-62 2 Roro ^ Â£ 45 T = 'n + 8 3 Ro / 2 r2 S 'o J (5.8) (5.9) As in the case of shallow shells, we shall employ the approx imations and dimensionless quantities described in equations (2.2) and (2.4), i.e., 20 T/zR0 0 = VZo . It should be noted that among these quantities, z is neither a o fixed constant nor a given value as in the case of shallow shells It depends upon the size of the buckling region as indicated in (5.10)o For the same reason, Q should be treated as a parameter 7 = V7 , f = W/Eh1 . 81 in the sequel, and it will be used as a measure of the size of the buckling region, or the thickness of the boundary layer, to replace the variable rQ By utilizing (5.10), equation (59) takes the following form -[(li + 3 025 e2)7? 5 31 + 2.1 off = 0 ; (5U) this is the load-deflection relation when the shell is under static equilibrium Let us take this opportunity to discuss the nature of the load-deflection relation and its associated stability proper ties. A typical curve of equation (5.11) is in the form of curve 1 as shown in the figure on the following page. Curve 1 constitutes three branches: the unbuckled stable branch 0A, the unstable branch AB, and the buckled stable branch BA'. Instead of calling point A and B the bifurcation points or branch points, we shall directly call them the critical points. The feature of the loss of stability is such that, during the loading process, the equilibrium position of the structure moves from 0 to B' and then to A in a continuous and monotonously increasing fashion; any slight increment of the pressure at the equilibrium position at A would cause a sudden and large increase in deflection, which 82 83 brings the equilibrium position from A to some point above A' on the buckled stable branch Therefore, the equilibrium condi tion at A is certainly a "critical" situation, and the load corresponding to the equilibrium condition at A on curve 1 deserves the name of "critical load." In the static loading analysis, we shall permit only one type of buckling of the shell that is due to the loss of stability after passing the point A; therefore, the critical load at point A is also the buckling load of the structure. The significance of the equilibrium situation at the critical point B defines the equilibrium condition where the "outward snap" of the shell occurs. For the purpose of emphasis, we shall repeat the argument that equilibrium condition at point B has no significance to the instability of the shells, if a classic buckling criterion*' is used. We would like also to point out that it is incorrect when we have a P- S curve in the form of curve 1; this then implies the necessity of using an "energy criterion," The main difference between these two criterions is the method of determining the buckling load. The classic criterion defines the buckling load by having a hori zontal tangent at the critical point, i.e., the buckling of the We use the definition given by Kaplan and Fung in reference (16) 84 shell is solely due to the loss of stability and the load-deflection curve usually is in the form of curve II in the figure referred to. On the other hand, energy criterion permits a "jump" from the state at B' to B during the inward deformation process; thus, a lower buckling load corresponding to the pressure at the equilibrium state at B is defined. Curves I and II indeed represent two different types of instability. According to Biezeno and Grammel (6), the instability represented by curve I is called "transitional instability" and the other is called "complete instability," which does not have the monotonously increasing branch beyond the range of instability. An example based upon a classic criterion was given in reference ^6),^ where the load deflection relation was in the form of curve I. We shall investigate the instability of the shell, basing the investigation upon a classic criterion. The condition of equili brium corresponding to point A in the above-mentioned figure will be referred to as the "unstable equilibrium position" or the "critical position" because it defines the loss of stability of the system as well as the buckling of the shell. Let us return to equation (5.11) Geometrically, it repre sents a one parameter family of curves in the ^-q plane; 6 is the parameter. For each 0 value, equation (5.11) shows a possible ^Refer to reference (6), pp. 484-496. 85 load-deflection relation during the buckling process. Let us say that the true P- Â£ relation during the buckling of the shell will be the one with 6 = Cr anc* a ^orm similar to curve I, which has been discussed above. We shall define the 0cr in the following fashion: 0cr will make the system reach its "unstable equilibrium position" with the smallest value of q0 By the fact that the state of unstable equilibrium corresponds to a relative maximum position on the qC7]) curve mathematically, the problem of finding Q is equivalent to seeking a least maximum for the family of curves in equation (5.11). After calculations, the locus of the "unstable equilibrium position" was found to be as shown in Figure 26. It has a relative minimum when Q- 0.548. Therefore, we have 0cr = 0$48\ (5.12) As Q = @cr = 0.548, equation (5.11) takes the following form: | 3668,71 V ~ $31 rf + 2108' 0-US ( I-37& + 07 rj) (5.13) This equation describes the curve shown in Figure 26, which repre sents the "best possible" relation between the load and deflection in the sense of yielding a smallest critical load. It posses a maximum at "7 = 0.418 corresponding to the critical position and a minimum at *1) = 1.22. Corresponding to 7= 0.418, we have the critical load from equation (5.13): 86 = 0-82 1 (5.14) I) cr i which is about 68 per cent of Zolley's result based on a classic linear theory and matches very well with both Thompson and Krenzke's experimental results. It also should be noted that corresponding to the minimum position on the ^ curve there is a i = 0-32 (5 o 15) oa which is about 26 per cent of the result of the classic linear theory. As we have mentioned, this load corresponds to the outward snapping load in the unloading process. The experimental test by Thompson gave an outward snapping load for shells with (RQ/h) ~ 20 of the magnitude about 22 per cent of the linear classic result. The change of volume during the loading process can also be obtained from the analysis and has the following form: TT y = 2itU-v) + j rc Â£ . The first term in the above equation was due to the membrane contrac tion, while the second term was the volume developed by inward deformation in the buckling region and was found by the following equation: Krenzke found experimentally, qcr = 0.84; refer to reference (19). 87 AV', = w r dr de Let us define a dimensionless change of volume in the form: AV = ( AV fio lo then, from the above equations, we have A? = 2 TT ( I V) el | + T ^ , where 0cr = 0.548, defined in (6.12), and q, t) were also given previously. We shall use the subscript "cl" to indicate quantities corresponding to the result obtained from the classic linear theory, i.e., 1rCL ~ V, jK'-ll2) Wei ~ 2n O-v) (Ro/h)(oW)2 Therefore, for V = 0.3,(R/h) = 20, we obtain the following expression: r c l A V AV, 26-UI62 l + -o9k4 ^ "cl 31 96? Together with equation (5.13), the relation between ( /a\/c\) anc^ (q/qcl) can fund> it as given in Figure 27. As far as the critical loads are concerned, the theoretical result is qualitatively good as compared with experiment. The rate of increasing of pressure in the post-buckling region was seen to be faster than the experi mental results given by (33) 88 3o Buckling of the Shell under a Suddenly Applied Constant Pressure'*' Let us assume that Reissner1s simplified theory on transverse vibration of a thin shallow elastic shell, i.e., the inertial forces in the middle surface are neglected as being small compared to the transverse inertial force does also hold true in the case of a complete spherical shell. We, therefore, obtain the equation of motion for such a shell by adding one term: (5.16) which is due to the transverse inertial effects, to the right-hand side of the equation of equilibrium (5.2). Equations of compati bility are kept in the same form as equation (5.3). By taking the same form of equation (5.4) for the axisymmetrical dynamic deforma tions of the shell and considering the central deflection varying along with time, after performing a similar integration of the Galerkin's functional in the form of (5.7), we obtain the counter part of equation (5.9), i.e., the dynamic equation of a metallic ( "0 = 0.3) complete spherical shell in the following form: 1 Refer to Figure 6, Part B 89 If the following dimensionless quantities, which have been employed in the case of a shallow spherical shell are used^ (5.18) then equation (5=17) takes the following nondimensional form: jj + [( 2-76 + 3 02$ 92)y] Â£ 31 rf + 2 lo? rf ] - +ze 7i]. (5.19) If a shell is under such condition that it is initially undisturbed, then at time X = X 0 = 0, we have the following initial conditions 7] (o) = 0 and rÂ¡ (O) = 0 . (5.20) 2 Because of the 7] term appearing in the nonlinear part, equation (5.19) does not belong to any well-known class of equations 1 Refer to equation (2.4). 90 whose behavior has been systematically discussed. In the case of a given load q, the dynamic response can be obtained by integrating (5.19) and the solution, in general, in terms of elliptical functions. Now, in this problem, q is taken as an unknown parameter. We are looking for the critical value of q which will result in the motion of the system being unstable. Since the system in (5,19) appears to be autonomous, then a qualitative discussion of the motion is possible by use of a topological method. Equation (5.19) can be integrated once when q is taken as a constant, and the following result is obtained: (if = A + (-Â£ fe) 17 (?H + 302? e2-1 %e) T)2 + 3-$u rf i $U 74 } r>o. (5.21) When q, 9 and the arbitrary constant A are assigned certain values, equation (5.21) is the equation of the trajectories in the phase plane of the system (5.19), This equation, in general, posses three (3) singular points at the 7 values solved from the following equations: = v = 0 (5.22) or on the 7 axis where |

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