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APPENDIX I APPENDIX IV NONLINEAR ANALYSIS OF DYNAMIC STABILITY OF ELASTIC SHELLS OF REVOLUTION By MARCUS GEORGE HENDRICKS 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 1974 TO MY WIFE, GENEVIEVE, AND MY DAUGHTER, CARMEN ACKNOWLEDGMENTS The author wishes to express his sincere appreci ation to Dr. S. Y. Lu, chairman of his supervisory com mittee, for his continuous guidance and encouragement throughout the entire period of this study. He also wishes to express his gratitude to Dr. I. K. Ebcioglu, Dr. U. H. Kurzweg, Dr. R. L. Sierakowski, and Dr. M. W. Self for their helpful discussions with the author and many valuable suggestions. TABLE OF CONTENTS Page ACKNOWLEDGMENTS.................................. iii LIST OF FIGURES................................. vi NOMENCLATURE..................................... vii ABSTRACT......................................... x CHAPTER 1 INTRODUCTION............................ 1 2 THEORETICAL DEVELOPMENT OF NONLINEAR SHELL EQUATIONS ................. 14 2.1 Shell Coordinates................... 14 2.2 Kinematic Relations................. 15 2.3 Constitutive Relations ............. 17 2.4 Kirchhoff's Hypothesis.............. 18 2.5 Variational Principle.............. 20 2.5.1 Hamilton's Principle........ 20 2.5.2 Equations of Motion ......... 21 2.5.3 Boundary conditions......... 26 2.6 Synthesis of Equations.............. 26 2.6.1 Shells of Revolution........ 27 2.6.2 Conical Shells.............. 27 3 DYNAMIC STABILITY EQUATIONS OF CONICAL SHELLS....................... 35 4 METHOD OF SOLUTION...................... 39 4.1 Circumferential Modal Analysis..... 39 4.2 Elimination of Meridional Derivatives ............. 42 4.2.1 Subdomain Method............ .45 4.2.2 Method of Satisfying Boundary Conditions ......... 47 TABLE OF CONTENTS (Continued) Page 5 NUMERICAL ANALYSIS FOR CONICAL SHELLS..... 52 5.1 Functional Description of the Computer Program....................... 52 5.2 Checking the Computer Program........ 54 5.3 Dynamic Stability Results............ 55 5.3.1 Dynamic Modal Preference...... 56 5.3.2 Linearized Dynamic Stability.. 58 5.3.3 Nonlinear Dynamic Stability... 60 6 RESULTS AND CONCLUSIONS................... 69 APPENDIX I SHELL SPECIMEN AND BOUNDARY CONDIT IONS ............................ 71 APPENDIX II CONVERGENCE.......................... 74 APPENDIX III FLOW DIAGRAM OF THE COMPUTER PROGRAM..................... 77 APPENDIX IV COMPUTER PROGRAM SOURCE LISTING................................ 81 BIBLIOGRAPHY........................................ 126 BIOGRAPHICAL SKETCH.................................. 131 LIST OF FIGURES Figure Page 1 Development Paths in Constructing Solution Methods for Problems of Dynamic Stability of Shells............. 4 2 Shell Element............................. 25 3 Conical Shell Coordinates................ 29 4 Fourier Wave Number Dependence.......... 57 5 Flexural Mode Preference During Dynamic Instability ..................... 59 6 Linearized Dynamic Stability for High Axial Load ......................... 61 7 Linearized Dynamic Stability for Intermediate Axial Load................. 62 8 Nonlinear Dynamic Stability for Heaviside Pressure Loading............... 65 9 Nonlinear Dynamic Stability for Intermediate Axial Load................. 66 10 Dynamic Stability Interaction Curves.... 68 11 Convergence Properties of Solution...... 76 12 Flow Diagram of the Computer Program.... 80 NOMENCLATURE A,B [B] D E G K L Ma MI, Ma8 N P P '\ P Q ,Q R R R ,RB T U,V,W ', W Coefficients of the first fundamental form Boundary condition matrix Flexural stiffness Young's modulus Shear modulus Membrane stiffness Lagrangian Function Stress couple components Stress couple components at edges Number of terms in power series Stress resultants Stress resultants at edges Axial load Potential energy Shear stress resultants Position Vector Radius of latitude circle Principal radii of curvature Kinetic energy Displacements of arbitrary point in the shell Displacement function for the buckling mode vii V an bnc n aa' a ,an ea e ,eB f,g,h h n j m n p ,p ,p PP pn q r s s t ,v,w ,v,w x3 z a,B Y E Region Temporal coefficients Coefficients of thermal expansion Strain components Differential operators Shell thickness Normal unit vector Number of essential boundary constraints Number of time dependent unknowns Circumferential wave number Surface loads Fourier component of normal pressure Transformed coordinate Radius vector Meridional conical coordinate Nondimensionalized meridional conical coordinate Time Middle surface displacements Nondimensionalized middle surface displacements Coordinate of axis of revolution Distance along i n Curvalinear coordinates or conical angles Strain tensor Error viii Circumferential conical coordinate Eigenvalue Poisson's ratio Density Stress tensor Nondimensional time Spatial coefficient Transformed circumferential conical coordinate Additional Notations [ ] [ ] ( ), ( ') ,( ) L J { I Number in brackets refers to reference numbers in the Bibliography Letter in brackets denotes matrix Comma denotes partial differentiation Denotes partial differentiations with respect to time Row vector Column vector Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy NONLINEAR ANALYSIS OF DYNAMIC STABILITY OF ELASTIC SHELLS OF REVOLUTION By Marcus George Hendricks December, 1974 Chairman: Dr. S. Y. Lu Major Department: Engineering Sciences It is the purpose of this dissertation to investi gate the dynamic stability of elastic shells of revolu tion. Two specific areas of this broad field are treated in detail. First, this analytical study generates pre viously unavailable interaction relations for combined dynamic loadings as they interact to cause passage from a dynamically stable state to other states. Secondly, the concept of linearized dynamic stability is extended to include geometrically nonlinear effects. One role of these effects is to allow the possibility of autoparametric ex citation of preferred flexural modes by the driven modes. The question of whether such excitation can occur during the primary dynamic instability motion sufficiently to af fect the magnitude of critical dynamic loads is studied. A modified version of the subdomain method in combination with circumferential modal analysis is developed for the solution method. A computer program is constructed to obtain numerical results and the dynamic stability charac teristics of a conical frustum are studied under a variety of combined dynamic loadings. Interaction curves for static stability, linearized dynamic stability, and non linear dynamic stability are generated. For the particular loading conditions studied, the results indicate that dynamic instability will occur at loads below critical static loads. This reduction in crit ical dynamic loads was shown to be the result of both the dynamic load factor effect and of autoparametric excitation. The interaction curves which are generated illustrate these effects quantitatively under conditions of combined dynamic loading. A criterion for dynamic buckling is established based on meridional mode shape changes. This ability to detect sudden jumps in the meridional profile helps to verify instability detected by divergence on displacement time history curves and provides additional information about the poststable state. U CHAPTER 1 INTRODUCTION A dynamic stability problem can be defined as a problem analyzed by Newton's equations of motion or any equivalent method [1]. Dynamic stability problems for continuous systems are nearly always governed by nonlinear partial differential equations [2]. The solution to a problem of dynamic stability in thinwalled structures may be stated as the determination of what loadtime combina tions cause the displacements of points in the material body to increase sufficiently to either interfere with operational specifications or cause a breakdown of the structure. Methods for obtaining such solutions may be categorized into three fundamental areas. The most obvi ous and usually most difficult method is the direct inte gration of the governing equations for sufficiently long periods to allow direct observation of the ensuing motions. A second technique is the investigation of the stability of equilibrium points. The question here is whether a slight perturbation of a dynamical system from an equilib rium state will produce motion confined to the neighborhood of the equilibrium point or a motion tending to leave that neighborhood [3]. This method is usually based on either perturbation or characteristic equation techniques. A third category of dynamic stability methods is on a higher level of abstraction than the previous two and relies on the existence and properties of variously de fined functionals. Energy functionals [4] and Liapunov functionals [5] are two examples. Definitions of dynamic stability may be categorized by methods of approach to such problems. In situations where the direct integration approach is employed, dynamic instability may be defined to exist when some character istic parameter, such as deflection, stress, or strain, becomes unbounded in one or more parts of the body. When the method of small oscillations about equilibrium points is used, instability may be defined as unbounded growth of perturbations with time or by the existence of one or more positive real parts of roots of a characteristic equation. In problems where the solution method is of the third above mentioned category a sufficient condition for stability is defined according to the definite form of the functional employed. In the case of energy func tionals, a boundary between stable and unstable regions is defined by the vanishing of the second variation of the energy functional. When Liapunov functionals are employed, stability is defined when the functional is positive definite and when its derivative is negative definite. The particular problem of interest in this study is the dynamic buckling of shell structures, which rep resents a specific application of the more general theory of dynamic stability principles. Just as the definitions of dynamic stability are related to the solution methods, the definitions of dynamic buckling of shells are related to solution methods as well as the specifics of the phys ical problem such as geometry, material properties, or loading rates. Solution methods, falling within the first of the three aforementioned categories, are direct temporal in tegration of equations of motion. They may be classified into five procedural divisions. These are finite differ ence, finite element, Hamilton's principle, energy methods, and methods of weighted residuals. Figure 1 illustrates the procedural flow and interrelations between these meth ods for the study of dynamic stability of elastic shells and is based on the most prominent articles dealing with this subject. First, let us consider recent significant contri butions falling within the finite difference method divi sion. Witmer et al. [6, 7, 8] have developed during the _ __ Numerical Integration * Shallow shell theory is often introduced here ** Bifurcation, if enforced, is usually introduced here Figure 1. Development Paths in Constructing Solution Methods for Problems of Dynamic Stability of Shells past decade a shell dynamics computer program known as PETROS which utilizes spatial finite difference methods in both the circumferential and meridional directions. Utilizing PETROS the authors have studied a number of explosive forming problems and have shown remarkable agreement with experimental results. However, consider ing that after a tenyear development period, computer run times are still measured in hours for moderately com plex problems, the practical usefulness of PETROS in study ing shell dynamic stability is somewhat limited. Using shallow shell theory, Longhitano [9] developed numerical solutions for the nonlinear dynamics of hemispherical shells also utilizing finite difference solutions in two directions. He showed that suddenly applied symmetic pressures may couple the breathing mode into more preferred axisymmetric or asymmetric vibratory modes causing instability at approx imately one half static buckling pressures. Again, however, the twodimensional discrete type of spatial treatment re sults in unusually long computer run times thus limiting the number of cases treated. Another class of finite difference solutions to shell dynamic stability problems have appeared in the lit erature which require only approximately three fourths of the computer run times as the previously cited twodimen sional discrete methods and appear to give comparable results. Such solutions based on elimination of the circumferential coordinate via modal analysis prior to finite differencing in the meridional direction have been presented by Cromer [10], Ball [11, 12], and Kim [13]. The results of Cromer and Ball also indicate that suddenly applied axisymmetric pressures may couple breath ing mode response into preferred vibratory modes to cause dynamic instabilities at pressure loadings lower than static critical loads. These studies indicate that cou pling between parametrically excited flexural modes is very weak and often may be neglected without compromising the results. Stricklin [14] and Wu [15] have made recent and significant contributions to the nonlinear dynamic analy sis of shells by employing the finite element method. Re sults published by both of these investigators indicate that accuracy comparable to methods emanating from the equations of motion is obtainable if sufficiently complex elements are used in large quantity. However, due to longer computing times required for finite element tech niques, their major advantage clearly lies in the treat ment of complex structures where equations of motion are not reasonably derivable. A seldom mentioned, yet almost universal, appealing characteristic of finite element techniques warrants a brief mention. This is that the unknown dependent variables in a finite element solution have a direct physical interpretation at all times as the solution proceeds. Thus, to some extent, the analyst never loses physical contact with his problem. This is in contrast to most other analysis techniques applicable to shell dynamics where the solution is carried forward in some transformed mathematical space. Klosner [16, 17] in a series of studies covering a fiveyear period studied the effect of suddenly applied axial loads on cylindrical shells. His solutions were ob tained by elimination of the circumferential coordinate from nonlinear shallow shell equations and applying the Ritz method or Hamilton's principle to the relations re sulting after introducing assumed solutions. His results indicate that suddenly applied axial loads do not exhibit the dynamic load factor effect found by others [9, 10, 11, 12] when suddenly applied normal pressures are considered. These investigations reveal that a valuable indication of the onset of dynamic instability can be found in the re versal in the trend of the time to first maximum deflec tion of the critical mode. Another subcategory of solution methods in studying dynamic shell stability by direct temporal integration is known as the energy method. This technique circumvents the derivation of the partial differential equations of motion. The method involves deriving internal energy expressions in conjunction with appropriate kinematic relations, substituting assumed solutions into these relations, and then applying Lagrange's equations to the result. This procedure leads to either a coupled system of nonlinear ordinary differential equations if the spatial integrals are evaluated analytically or a coupled system of nonlinear integrodifferential equa tions if the spatial integrals are to be evaluated numerically. In either case, the differentials are temporal and must be solved numerically. Goodier [18], Mclvor [19], and Lindberg [20] have presented solutions to a particular type of dynamic instability of cylin drical shells utilizing this solution procedure. Hubka [21] has presented alternate methods for numerical inte gration of the final equation sets for this problem. This problem is concerned with the circumferential modal coupling arising from the nonlinear membrane stiffness terms in the equation sets obtained by this energy method technique. The main results of these studies are that non linear coupling between circumferential modes may under conditions of minor excitation or initial imperfections in the preferred vibratory modes cause energy extraction from the driven breathing mode into one or more preferred modes such that the preferred mode becomes dynamically unstable. I Utilizing this same energy method derivation, Mente [22] has studied the dynamic nonlinear response of cylindrical shells subjected to asymmetric pressure loading. His computerized solution, known as DEPICS, is able to handle a variety of coupled modes during the temporal integration. The nonlinear stiffness is handled by solving a linear eigenvalueeigenvector problem at each time step then adjusting the succeeding stiffness value based on the nonlinear terms. Unfortunately this rather extensive computer program has to date been unable to satisfactorily define any regions of instability. Significant Russian contributions to the study of dynamic stability of shells have been made. Among these are works by Shumik [23, 24] who also employed solutions by direct temporal integration of expressions derived by the energy method. These works deal with suddenly applied loads using a linearized uncoupled dynamic stability theory. His results indicate that cones with large taper buckle dynamically at circumferential wave numbers higher than the critical wave number for static loading. Shiau [25] utilizing an energy solution studied the effects of imperfections on the dynamic stability of conical shells subject to axial impact. His results agree with Klosner's [16, 17] in so much that a reversal in the trend of the time to first maximum deflection of the critical mode may be viewed as an indication of the onset of dynamic buckling. The results, however, are so critically dependent upon the choice of the single term assumed buckling mode that the procedure appears to be of limited general value. A final division of solution methods in the cate gory of direct temporal integration is the method of weighted residuals. The details of the procedure will be covered in a later chapter. Nash [26] utilizing shallow shell theory in conjunction with the Galerkin method of weighted residuals investigated the response of thin conical shells to dynamically applied axial forces. He found that a rapid increase in end shorten ing may be used to denote the onset of dynamic instability. Fisher [27] employed a similar solution method to study the dynamic buckling of reinforced circular cylin ders subjected to suddenly applied axial compression load ings. He also found that a reversal of the trend of the time to maximum deflection corresponds to onset of dynamic buckling. Lakshmikantham [28] using shallow shell theory in conjunction with smeared analysis and the Galerken tech nique investigated the same problem as Fisher [27]. His solutions indicated that a jump in radial deflection was the correct criteria for indicating dynamic instability. An earlier work by Dietz [29] studying the same problem by shallow shell theory and the Galerkin method indi cated that loss of stability is best defined as a sudden drop in the axial load required to maintain a constant end shortening rate. By taking the results of the preceding literature survey in context, several comments may be made concerning the current stateofthe art in analytical studies of dynamic shell stability. First, it appears that nonlinear response problems which retain full circumferential modal coupling yield the most reliable indicators of the onset of dynamic buckling. Also the results appear to be highly sensitive to theoretical approximations and to the quality of the numerical solution procedures. Secondly, the study of passage through bifurcation points is well developed only for linearized dynamic stability techniques. The few articles proposing to deal with nonlinear dynamic stability unfortunately discarded the quadratic nonline arities, which are the most significant ones, retaining in stead only the cubic nonlinearities. This was done in order to circumvent having to deal with intermodal coupling. There fore, the results of these studies appear inconsistent. 1 It is the purpose of this dissertation to invest igate the dynamic stability of elastic shells of revolu tion. Two specific areas of this broad field are treated in detail. First, this analytical study generates previ ously unavailable interaction relations for combined dynamic loadings as they interact to cause passage from a dynamically stable state to other states. Secondly, the concept of linearized dynamic stability is extended to in clude geometrically nonlinear effects. One role of these effects is to allow the possibility of autoparametric excitation of preferred flexural modes by the driven modes. The question of whether such excitation can occur during the primary dynamic instability motion sufficiently to af fect the magnitude of critical dynamic loads is studied. The theory derived herein is suitable for the study of the dynamic stability of general shells of revolution. In order to obtain numerical results, however, a specific shape must be specified and the one used for numerical analysis in this study is a truncated circular conical shell structure. Conical frustum shells are in frequent use as structural elements. Moreover, a conical shell reduces to a cylindircal shell when the semivertex angle of the cone becomes zero. Similarly, it reduces to a flat circular plate when the semivertex angle of the cone ap proaches a right angle. Thus, the analysis and computer program developed for cones can be generalized to handle these two related problems. A modified version of the subdomain method in combination with circumferential modal analysis is de veloped for the solution method. A computer program is constructed to obtain numerical results and the dynamic stability characteristics of a conical frustum are stud ied under a variety of combined dynamic loadings. Inter action curves for static stability, linearized dynamic stability, and nonlinear dynamic stability are generated. A dynamic buckling criterion based on meridional mode shape changes is established and the quantitative impor tance of nonlinear coupling effects is illustrated. CHAPTER 2 THEORETICAL DEVELOPMENT OF NONLINEAR SHELL EQUATIONS 2.1 Shell Coordinates A set of normal curvilinear coordinates (0, B, z) is chosen in the shell space such that a and B lie on the undeformed middle surface, z is perpendicular to this middle surface, and z=O lies at the middle surface. The equation of the undeformed middle surface is given in terms of two independent parameters a and B by the radius vector r = r(a, ). (2.1) To describe the location of an arbitrary point in the space occupied by a shell, the position vector is defined as R(a,B,z) = r(a,B) + zin(a,8) (2.2) where z measures the distance of the point from the corres ponding point on the middle surface along in. The unit vector i is normal to the surface. 2.2 Kinematic Relations Acceptable nonlinear straindisplacement relations for shells may be obtained from one of two basically dif fering approaches. One approach is to utilize the finite strain tensor of the threedimensional theory of elasticity in conjunction with the GaussCodazzi relations for a sur face [30, 31, 32, 33, 34]. Another method is the two dimensional approach to shell theory that evades the ques tions of the approximations involved in the descent from three dimensions. This approach defines strain as one half of the difference between the material and spatial metric surface tensors. Sanders [35] presented the first consistent nonlinear kinematic relations derived by this method using a continuum called a Cosserat surface. Naghdi [36] discusses both of the above methods in detail. Nonlinear kinematic relations are seldom utilized in shell analysis in their complete form due to their complexity. Therefore, simplifications are routinely made. Throughout the following analysis it is assumed that the strains in the middle surface and that the rotations about coordinate axes are small. These assumptions imply that an element of area on the deformed middle surface is iden tical in size to an element of area on the undeformed middle surface and that the difference between the Christoffel symbols of the deformed and undeformed coordi nate systems is zero. These assumptions result in a con siderable simplification in the resulting equations with out unduly restricting the applicability of the solutions. The straindisplacement equations shown below were derived using the finite strain description of the three dimensional theory of elasticity in a manner similar to Ogibalov [32] and the nonlinear terms were regrouped into surface rotation expressions. The resulting kinematic relations are ea = (1/(l+z/Ra))[U, /A+A, V/AB+W/Ra+(l/(2(1+z/R))) x(W,a/A+U/R )2 e = (1/(l+z/R ))[V,'/B+B, U/AB+W/R +(1/(2(1+z/R ))) x(W, /B+V/R )2] (2.3) e = (1/(2(1+z/R )) (V, /AA, U/AB) +(1/(2(1+z/RB )) (U,g/BB,aV/AB) +(I/(2(l+z/R )(I+z/R )))(W,a/A+U/R )(W,/B+V/R ). In equations (2.3) the covariant components of the normal and shear strains are denoted by e e e U respectively. The components U, V, and W represent displacements of an arbitrary point in the shell. R and R are the a B principal radii of curvature and A and B are the coeffi cients of the first fundamental form. Comma denotes partial differentiation. 2.3 Constitutive Relations The relations between the components of strain and stress in an orthotropic, linearly elastic material are given by Hooke's law [37] as e= a/E v a /E a n/E n+a T a a e = V a /Ea+o aBn/E non/E +aT e = Vna0a/Ean aB6/E B+n/En+anT (2.4) 2eaB = oa6/GaB 2e = aB/G an n/Gan. Young's modulus in the a and B directions is de noted by E and E respectively. The Poisson Ratio vY designates the contraction in the a direction caused by a positive normal stress a in the B direction. G is aB the shear modulus in a plane which is tangent to the (c,B) coordinate surface and (a a ) are the coeffi cients of thermal expansion in the a and B directions respectively. These relationships are valid for materials subjected to stresses below the proportional limit. By introducing Kirchhoff's hypothesis and allowing for the symmetry of the elastic parameter matrix [38] the above equations may be written as oa = [1/(1 VBa)][Ea e +1BaE e(aa+ E a aE )T] B = [1/(1 aV B )][E e +ve E e a (a E +aa E )T] (2.5) aB = 2G aBe 2.4 Kirchhoff's Hypothesis The reduction of the threedimensional problem to a twodimensional one requires an assumption concerning the variation of strain, displacement, or stress across the shell thickness. To satisfy this requirement the fourth assumption in Love's first approximation [39], known as the Kirchhoff hypothesis, is introduced. This hypothesis entails the vanishing of transverse shearing and normal strains [33] and may be formulated as follows: U(a,B,z) = u( a, )+zea(a,B) V(a,B,z) = v(a,3)+zeg(6 ,8) W(a,B,z) = w(a,B) (2.6) e = u/R w,a/A = v/RBw, /B where u, v, and w are the components of displacement at the middle surface in the a,B, and normal directions, respectively, and 6 and 6 are the rotations of the normal to the middle surface during deformation about the B & a axes, respectively. The acceptance of this assumption is due to its great clarity [40]. Although Kirchhoff's hypothesis is a first approximation, its applicability to nonlinear shell theory is well known [33, 35]. The problem of determining the error intro duced by the hypothesis on the preservation of the normal has thus far not been solved exactly [41]. Novozhilov [42], Mushtari [43], and Koiter [44] have estimated that the errors introduced by Kirchhoff's hypothesis are of the order of h/R, which is small for thin shells. Kirchhoff's hypothesis, modified to account for transverse normal strain, has been employed in [31, 33, 34, 36, 45]. Effects of transverse normal strain were not included in the present analysis. 2.5 Variational Principle 2.5.1 Hamilton's Principle The Irish mathematician and physicist, Sir William Hamilton (18051865), formulated his celebrated principle in dynamics in which the governing equation depends ex plicitly on the energy of the system [46]. Hamilton's principle is stated as an integral equation in which the energy is integrated over an interval in time. In the language of the calculus of variations, Hamilton's prin ciple states that the first variation of the time integral of the difference between the kinetic energy T and the potential energy P of a dynamical system is zero, that is, t2 6 f (T P)dt = 0. (2.7) t1 The equation is assumed to hold for all dynamical systems whether they are conservative or nonconservative. The quantity (TP) is called the Lagrangian function and is denoted by L. With this designation equation (2.7) be comes t2 6 fLdt = 0 (2.8) tI and Hamilton's principle then asserts that the first vari ation of the timeintegral of the Lagrangian function is zero. Hamilton's principle is employed in the next sec tion to derive the equations of motion and the admissible boundary conditions for shells. 2.5.2 Equations of Motion Of the many procedures available for the derivation of the equations of motion for a differential shell element, Hamilton's principle is superior in nonlinear shell theory due to its efficient treatment of problems involving curv ilinear coordinates and because it gives the admissible boundary conditions, natural or forced, that are to be used with the theory [47, 48]. With the total potential energy P expressed as the difference between the internal strain energy and the potential energy of the external forces, we obtain from equation (2.8) t2 6f { v [(P /2)Vivi(1/2)siJYij]dv+/ si'Vids}dt = 0 (2.9) t1 O 0 Here the density of the undeformed body and the Lagrangian components of the displacement vector are p and V., re spectively. The symmetric Cauchy stress tensor sij is measured per unit area of the undeformed body and must be referred to the base vectors in the deformed body and the strain tensor, Yij, is defined by equation (2.3). The components of the surface force s are referred to the base vectors in the undeformed body and the volume and surface integrals in equation (2.9) must be extended over the volume and surface of the body in its undeformed state. Introducing the kinematic, constitutive, and Kirchhoff relations from previous sections, we eliminate the time derivatives of the variations by integrating by parts with respect to time and require the virtual dis placements to vanish at the end points of the arbitrary interval tl t t2, thusobtaining the equations of motion, stress resultants and couples, and boundary conditions. The equations of motion derived in this section reflect the assumptions of Kirchhoff's hypothesis. Small strains and moderately small rotations were also assumed. Thinness assumptions were delayed until after the equa tions were synthesized. The equations of motion are: (BN ), +(ANBa), +A, NaeB, N +ABQA/R AB(NQO +NB )/Ra +ABpp = ABphu(l+h2/12RR )+ABph3(1/R +1/R )5 12 (BNB'a),+(AN) +B, NBaA, Na+ABQS/R AB(NB e+N50 )/R +ABp = ABphV(I+h2/12R R )+ABph3(1/R +1/R ) /12 (BQ0) +(AQ), AB(N'/R +NO/R )(BO Na+BO ), 0 (2.10) (AeO +AO N ),+ABpn = ABphW(1+h2/12RaR ) (BM ) +(AMB), ,+A, Ma~B, MABQ0 = ABph3 [(1/+l/R )U+ a]/12 (BMVa) a+(AMB) ,+B, aMaA, M0ABQB = ABph3[(1/R +l/R )V+]/12 where (") denotes second partial differentiation with respect to time and p p p represent the applied surface loads. The stress resultants and stress couples are given by N0 h/2 a" {N = / { B }(l+z/R )dz N h/2 a NB h/2 ,B {N } /2{ I l+z/Ra)dz N h/2 u Mt h/2 a M } = / { (l+z/R )zdz MOB h/2 oa (2.11) {MB h/2 Ba(1+z/R )zdz Ma h/2 an6 Q } h/2 { l+z/RS}dz 0 h/2 an 1+z/R h/2 B = oadz h/2 h/2 M"B = aB zdz h/2 where h is the shell thickness. Figure 2 illustrates the relations between the stress resultants and couples to the generalized coordi nates and to the shell element. 1 Figure 2. Shell Element I 2.5.3 Boundary Conditions From equations (2.9), (2.10), and (2.11) we get the following natural boundary conditions that arise from the requirement of force balance. Along the edge of constant a the boundary conditions are N = N or u = u N B+MB/R = Qa+MBN N = p a pg B or v = v Q or w = w M0 = M or = a a . The boundary conditions from equation (2.12) by along constant B may be obtained interchanging a with B and u with v. 2.6 Synthesis of Equations The usual procedure followed is to reduce the number of equations and unknowns to a more manageable number by eliminating Qa and Q from the equations of motion (2.10) which are thus reduced from five to three. The force and moment resultant expressions (2.11) to gether with the constitutive relations (2.5) are then (2.12) substituted into the equations of motion. Finally, the straindisplacement equations (2.3) and (2.6) are substi tuted, to yield three differential equations of motion having u, v, and w as dependent variables and a, B, and t as independent variables. 2.6.1 Shells of Revolution In the preceding analysis R and R are arbitrary, but in all subsequent applications these principal radii will refer only to shells of revolution. It can be shown [47] that for a shell of revolution the principal radii may be expressed as R = [l+(aR o/ x 3)23/2/(D2R / x32 (2.13) R = R /1+(Ro0/3x3)2 where R is the radius of the latitude circle and x is the coordinate coincident with the axis of revolution. 2.6.2 Conical Shells The above described synthesis may be carried out using equation (2.13); however, the equations would be extremely unwieldy. Thus, the procedure will be followed only for a specific shell of revolution shape and the one used in the remainder of this study is a circular conical shell. A conical shell reduces to a cylindrical shell when the semivertex angle becomes zero. Similarly, it reduces to a flat circular plate when the semivertex angle approaches a right angle. Therefore, the following analysis can handle cones as well as the two related problems. The conical coordinate system adopted is shown in Figure 3. The coordinates, coefficients of the first fundamental form, and principal radii become a = s = 6 A = 1 B = s sin a (2.14) R = R = s tan a. In all subsequent relations, a will denote the semivertex angle. Assumptions relating to displacement magnitudes are now introduced into the analysis. The squares of inplane displacements are assumed to be small with respect to the squares of normal displacements and the shell thickness is assumed to be small compared to the radius. A L. Figure 3. Conical Shell Coordinates I Neglecting rotary inertia terms, the nonlinear equations of motion are s s 0 Os N +sNs N +N = sphu ctn a vN /s+ctna w, N /s+N +2NS+sN +ctna wsNso +ctna M, /s+2ctna MS/s+ctna M, = sphV W, S+sw, sNs+sw,sNssctna N ctna v, N6/s+w, N /s (2.15) ctna vN6 /s+w, N, /sctna v,s Ns+2w,s Nsectna vN0 +w, N, +w, sN,+2Ms,+sssM, +M /s+2M1,/s+2M +sp = sphw where i = 0 sina. I The nonlinear stress resultants and stress couples are Ns = [Eh/s(lv2)]{su, +s(w, )2/2+v[v, +u+ctna w+(w, )2/2s ctna vw, /s]} Nse = Nes = (Gh/s)(sv,s+u, v+w,sw,ctna vw,s) N = [Eh/s2(1v2)]{sv, +su+sctnc w+(w, )2/2ctna vw, +ctn2a v2/2+v[s2u,s+s2w,s)2/2]} (2.16) Ms = [Eh3/12s(1v2)]{sw,ss+ctna us+ctna(w,s)2/2 +v[w, /sw,s+ctna(w, )2/2s2] Ms = Ms = (Gh3/12s)(2w,s +2w, /s+ctna w, W,s/s) Me = [Eh3/12s2(lx2)](w, swsctna uctn2a wvs2w, s) Substituting equations (2.16) into equations (2.15) yield three equations of motion. Noting that Gh = K(1v)/2 we have K{[u/s+u, s+su, ss+ ,/2s3v, /2s+vs/2ctna w/s +(w,s/2s)(w, ctna v, )+(w,s/2s)(w, ctna v)+(w,) 2/2 (w,)2/2s2 +ctna vw, /s2ctn 2 v2/2s2+sw,sw,ss] (2.17) +v[u, //2s+v, s/2s+ /2+ctn w,sW, s/2(Ws+W,' /s ctna v, /s)+w, /s(w, s/2w, /2sctna v,s+ctna v/s) ctna vw, s/2s]} = phsu K{3u, /2s+u, s/2v/2s+v,s/2+sv,ss/2+v, /s+ctn w,/s +w, (w'ss/2+w, /s2)+w's('s +W/s)/2 +ctna[sv(w,ss/2+w, /s2)+w,(v+u, /2)+w, u/s]/s (2.18) +v[u, /2s+u, s/2+v/2sv,s/2sv,ss/2+w,5sW s/2w,s, /2s w,ssw, /2+ctna u,s/s(ctna v+w, )+ctna w, (vu, /2)/s +ctna vw,ss/2]} = phsV K{ctna (u+v, +ctna w)/s+ctna[(w, )2/2+ww, ]/s2+su,sw,ss +w,s(suss+V,/2+u, 4/2s)+w, v, /s2+w, (v,' /s2+v'ss/2 +u,'s2s)+w's ,(u, /s+v's)+u's5, s+uw, s2+u w ,/2s2 2 2 2 2 vw,so/SVs W,/2s+vw, /2s2 v, W,'s/2s+ctna(v, 2/s2 uv, /s2 vv, /s2u, v/2s2V,s2/2u,V,'s/2s+vV,s/svV,ss/2u'sv/2s v2/2s2)+ctn2a (v, w/s2)+v[ctna u,s+W, (v, +ctna w) +w, (u,s/2sv,ss/2)+us ,/s+Ws(v, /2+ctna w,s/2u, /2s) +uw,ss+us, svsW,'su, W,s/s+vws /s+uW, /2s2 (2.19) +V,s W/2svw, /2s2+v, W,s/2s+ctna (u,s V,/sU,s v/2s +v,2/2+u, v,s/2s+vv, s/2u, v/2s2vv,/s+v2/2s2)]} +D{sw, ssss2w,ss#/sw, /s3+2w',sW/s22w sss4w, /s3 +W,s/SW,/S +Ctn(Usss+Wss ssss ss +Ws 'sW, /s2W_'sW, /s3, 'ss'W,/s2 +(w, )2/s2 +v[(ctnm/s2)((7/2s)(w, w,s )+3(w, )2/s2+W, s W,/2s Wss W, W,'sWsOO)]}+spn = phsW 34 where the membrane stiffness is K = Eh/(lv2) (2.20) and the flexural stiffness is D = Eh3/[12(1v2)] (2.21) CHAPTER 3 NONLINEAR DYNAMIC STABILITY EQUATIONS OF CONICAL SHELLS The nonlinear dynamic stability equations are obtained from the nonlinear equations of motion by employing a perturbation analysis. Substituting the displacements u = I + uB v = v + vB (3.1) w = I + wB and the stress resultants and stress couple expressions Ns= Ns + N 1Ms = Ms + M I B I B N = N + N M = M6 + M (3.2) I B = M + M(B I N+NB I B and the load relation P = PI + PB (3.3) into equations (2.15) results in two coupled sets of equa tions. The quantities with subscript I are measured from the undeformed state and the quantities with subscript B are small but finite perturbations. This gives one equa tion set describing the motion in the stable space surround ing the undeformed state and another set, the stability equations, describing the motion during passage thru the nearest bifurcation point. The resulting set of nonlinear dynamic stability equations are as follows where the subscript B has been dropped for convenience: K[(u/s+u,s+su, ss+u,/2s3v,/2s+v,s /2ctna w/s) +v(u,0 /2s+v, /2s+vs/2+ctna w,S) (3.4) +(w, /2s+w2s+w ,/2s+w /2w. /2s 2+swsW, Ss/2w,/2s + sWss) +v(w,2/2w, w ./2s+w w /2sw 2/2s2)] = phsi 5 'ii 's) '= h K[(3u, /2s+u, s/2v/2s+vs/2+sv,ss/2+v, /s+ctn w,/s) +v(u, /2s+u, /2+v/2sv,s/2s,ss,/2) (3.5) +(w,W,'ss /2+w, w, /s 2+w, s s/2+Ws /2s+ctn 2 ww, /s2 +v(w,s W,s/2W, sW /2sw, ssW,/2)] = phsV K{ctna(u+v, +ctna w)/s+v(ctna u, ) +ctna [w,2/2s2+ww, /s2+v(w, /2+ww,s)]} 3 22 +Df(sw'ssss2w'ss"/sw, W /s 3 +2wsIK/S 2wsss 4w, /s3 +Wss/ss/ )+ctno[w, s2'ss W,/s2 +W,s2/S +W,sW,5s W', W's/s W,s,/s +w wss~/s2 (3.6) +W,sW, s/s 2+v(3w, W,'s/S +3w, /2s4 +WsW /s3 Wss"W,/s2 ,s s/s2)]} + spn ss +s'sss 's = phs +wN I+sw, NI+sw I s+W VN Is w, Nis = phsP           The single underlined quantities represent non linear membrane stiffness terms. The double underlined quantities represent nonlinear bending stiffness terms. The nonlinear stability terms are denoted by a dashed underline. These equations embody the assumptions that wI < wB and that uB < wB vB < wB In the present study, a thin truncated conical shell is loaded by a constant axial load and a nearly axisymmetric Heaviside pressure load. The membrane state stress resultants are given by Ns = P/(ms sin2a)pn(s tana)(1cosx t)/2 N = pn(stana)(lcosX t) where P is the applied axial load, pn is the magnitude of the Heaviside pressure load, and A is the natural frequency of the breathing mode of the shell. CHAPTER 4 METHOD OF SOLUTION 4.1 Circumferential Modal Analysis The nonlinear partial differential equations derived in previous chapters are spatially twodimen sional. The two directions are meridional and circum ferential. In this analysis, the twodimensional equa tions are reduced to onedimensional equations by utilizing circumferential modal analysis. The inde pendent variable e is eliminated by expanding all dependent variables into sine or cosine series in the circumferential direction. As a result of the trig nometic series expansions, there is one set of govern ing equations for each circumferential wave number considered. For a linearized analysis the sets are un coupled. For a nonlinear analysis the equation sets are modally coupled through the quadratic terms. This cou pling arises when use is made of the trignometric prod uct identities cos(ie)cos(je) = (1/2)[cos(ij)6+cos(i+j)e] (4.1) sin(ie)sin(je) = (1/2)[cos(ij)ecos(i+j)e] in order to facilitate the equating of like coefficients of Fourier expansion terms which is required to identify the equation sets. Substituting u(s,4,t) = Q v(s, ,t) = Z n=o Q w(sp,t) = Z n=o Q p(s,is ,t) = z n=o the expressions u n(s,t) cos(np/sina) v n(s,t) sin(np/sina) w (s,t) cos(nJ/sina) p (s,t) cos(n(/sina) into the nonlinear dynamic stability equations (3.4  3.6) and equating like coefficients of Fourier expan sion terms yield the following: (l/s){l+[n2(lv)/(2sin2 )]}u n+un's+sun'ss +(n/2s sina)(3+v)v +(n/2sina)(l+v)vns(ctna wn/s) (4.3) +vctna wn s = phs n/K (4.2) (n/2s sina)(3+v)u (n/2sina)(l+v)un's +{[(1v)/2](n2/sin2a)}(vn/s)+[( v)vn's/2] (4.4) +[(1v)svn,ss/2][n ctna wn/(s sina)] = phsVn/K ctna [(u /s)+vuns+(n/sina)(v n/s)+(ctn w n/s)] +(D/K){[(4n2/sin2 )(n4/sin4a)](wn/s3)+[(2n2/sin2 )+1] x[(Wn'ss/S )(wn,'s/s2)]2wn sss n ssss p n/K [P Wn'ss/(KFsin2a)]+(lcosX ot)[s tana Bcs 2 n / 2n 2n n/ s (s2tana ,ss/2)+(2n2 n/Sin2a)(2nBn/sina)]/K c 2 2 2 2 +Tc{ctna{(n2w2/4s2sin c)+v[(w ,2/2)+(w ,s/4) (4.5) +ww ss+(w w /2)]}+(h2ctna/12){w 2+w 2/2 o o'ss n n'ss o'ss n'ss [n2/(2s2sin2a)](wn wnsswn,2)}}+nc{ctna[(n2W Wo / (s2sin2a)) +v(w w +w w +w w ,s)]+(h2ctnx/12)[2w ,sW (Wo'sWnso'ssn onss onss n'ss 2 2 sin2= /K (n /s sin C)w w]} = 0hsvi/K where n = circumferential wave number c i n=o So n>o c Z w /2[ (i+n)+ P in c n o (4.6) = E wi/2[np(.i+n)+cPinli_nlP1 i=o n B = Wi/2[nsp (i+n) slinIp lin '], 1=0 s n=o ni n>o s 41 i n+l1 S=o i=n 1 isn1 4.2 Elimination of Meridional Derivatives The next phase of the analysis involves the re duction of the partial differential equations sets result ing from the circumferential modal analysis to ordinary temporal differential equation sets which are then amen able to numerical solution. Referring again to Figure 1 it is seen that the available solution methods for this task fall generally into four categories. These are finite element methods, finite difference methods, variational principles, and methods of weighted residuals. As noted by Finlayson [49], for certain types of linear problems these methods can be shown to be equivalent to each other. From the literature abounding on shell dynamics one can safely conclude that any of these methods when judiciously ap plied will give acceptable results. There are, however, marked differences between the methods in the amount and quality of manual mathematics and the quantity of machine computing time required to give equivalent results. The desired characteristics of the solution method for the particular problem of interest in this disserta tion are now discussed. The solution method utilized should yield an approximate direct solution where con vergence would be monotonic and assured. Spatial control of error residuals was desired. Because the taper in the conical shell results in nonsymmetric stiffness matrices, a solution technique applicable to nonselfadjoint prob lems was required. Finally, it was hoped that the chosen solution method would result in short computer compile times because experience has shown that the majority of computer time required for such a problem is in successive compile times during development and debugging. In matching these desired characteristics to the known characteristics of the classical solution methods an elimination process was begun which finally resulted in the algorithm used herein. I Finite element techniques were eliminated from consideration because this method is not competitive with other methods in problems with regular geometries and formulatable boundary conditions lying along coordinate lines. This method requires significantly more manual mathematics and computer execution time than equilibrium equation solution methods in order to give comparable results. Finite difference methods have been shown to be extremely powerful in solving nonlinear dynamic stability problems involving shells. This method has been used with particular success when coupled with circumferential mode analysis. It was shown, however, by Radosta [50] that convergence is slow when a meridional mesh is used with a shell under high axial load. In the particular problem at hand, i.e. a conical frustum subject to a high axial load combined with a time dependent pressure load, it was felt that a suitable selection of number and spacing of meridional mesh points would degrade into a highly time consuming trial and error effort. The final phase of solution method selection con sisted of choosing one of the four methods of weighted residuals: Galerkin, Least Squares, Collocation, and Sub domain. The least squares method was not seriously con sidered because it yields ordinary differential equations of the second degree in the temporal operator. The collocation technique was eliminated because it enhances the same point location selection drawbacks as does the finite difference method. Again, this is normally an insignificant problem but was made significant here by the expected buckled states resulting from the combined type loadings being considered. The final choice between the Galerkin and subdomain methods was based on the method of error residual control. In the Galerkin technique the error produced by each individual term in the assumed solu tion is forced to average to zero over the entire length of the shell. In the subdomain method the error in the total assumed solution is forced to average to zero over each subdomain. This type of spatial control for error residual was considered superior to the modal type of error control offered by the Galerkin method and there fore, the subdomain technique was selected for use in this investigation. 4.2.1 Subdomain Method A description of the theoretical basis of the sub domain method is now in order. Consider the equation L[u(x,t)] = 0 in V (4.7) where L is a nonlinear, nonselfadjoint, ordinary differential operator. If the approximation solution is represented in the form m u = Z a (t) i(x) (4.8) i= 1 1 and this is substituted into equation (4.7), we obtain an approximation L[(x,t)] = e (4.9) where E is the error. We now impose constraints on E by requiring it to average to zero over a subdomain v of the total domain V iL[u(x,t)]dv = 0 (4.10) v The number of subdomains v are chosen to equal m the number of time dependent unknowns in (4.8). The differen tial equation, integrated over the subdomain is then zero, hence the name subdomain method. As m increases, the dif ferential equation is satisfied on the average in smaller and smaller subdomains, and presumably approaches zero everywhere. This process yields a dynamically coupled second order set of nonlinear temporal differential equa tions which may in turn be solved by numerical methods. Equation (4.8) is called the trial function and may be selected in a variety of ways. It may be chosen to satisfy the boundary conditions, but not the differen tial equations or to satisfy the differential equations but not the boundary conditions. In order to simplify the required hand calculations, a set of complete power series were chosen as the trial functions for this prob lem. The trial functions were then forced to satisfy the boundary conditions by a modified application of the Lagrangian multiplier method which can be more properly thought of as a coordinate transformation to require an assumed power series solution to satisfy the essential boundary conditions. An approximate satisfaction of the differential equations was then obtained by the subdomain method. 4.2.2 Method of Satisfying Boundary Conditions A description of the technique utilized to satisfy boundary conditions will now be presented. Consider the trial function (4.8) in matrix form, u = L J {a} (4.11) (1xm)(mxl) and the j essential boundary constraints, [C.]{a} = 0. (4.12) (jxm)(mx1) Equation (4.12) may be partitioned as follows E[ C1 C2 [jx(mj)] ((jxj) r 1 (m 0. (mx1) Equation (4.13) may be rearranged as {q2} = [C2 ] [ C1 ] {q1 (jxl) (jxj) [jx(mj)][(mj)x1] and noting that {ql = [I]{ql} we obtain {a} = [ B ] {ql (mxl) [mx(mj)][(mj)x ] and (4.11) becomes u = L4J[B]{ql} where [B] may be considered as a projection operator. (4.13) (4.14) (4.15) (4.16) (4.17) After nondimensionalizing equations (4.3 4.5) according to s= s2s h = s2h Un = 2Un I = PcrI v = s2v n n P = 2TEh2cos2 //3(1v2) c r w = s cr n 2n( (4.18) S= /E/p(lv2) t/s2 P = Eh/(1v2)s2 Pn n = s2/p(1v2)/E A the tr The tr ial functions are 'ial functions are u a n n vn N1 b n n Wn =o Cn Pn Pn derivatives are likewise nondimensionalized. introduced into the equations. i i i. S. s . i i (4.19) After applying the transforms satisfying the boundary conditions, the equations are then integrated over (Nj) subdomains and the average residual error for I each subdomain equated to zero. During the original formulation it was noted that this approach leads to a singular mass matrix for equally spaced subdomains. To circumvent this problem the equations were mutliplied through by s. This is similar to a procedure known as the method of moments first employed by Polhausen [51]. Having completed the above sequence of operations results in Q sets of nonlinear second order temporal dif ferential equations. Each set contains 3 (Nj) equations and the sets are coupled in the nonlinear terms. In ma trix form a typical set may be expressed as [m][B]{q}(n)+[k][B]{q}(n)+[D][B]{q}(n)+{Nl.(n) = { n) (4.20) where [m] mass matrix [B] matrix requiring power series solution to satisfy the prescribed boundary conditions {q}(n) generalized coordinates of the nth cir cumferential mode [k] stiffness matrix [D] stability matrix NLi LqmJ[B T[C ][Bn]{qn} power series products {P} load vector I 51 As an example, for a problem retaining two circumferen tial modes with eight subdomains along the meridian, equations (4.20) would represent a total set of forty eight coupled nonlinear second order ordinary differen tial equations to solve. CHAPTER 5 NUMERICAL ANALYSIS FOR CONICAL SHELLS The sets of equations (4.20) were coded into computer language for numerical solution. A particular shell specimen was selected for study. The salient geometry and boundary conditions for the shell studied are given in Appendix I. Fortran IV, level H machine language was used and the program was constructed for double precision arithmetic. A flow diagram of the oper ational characteristics is given in Appendix III. A list ing of the program is given in Appendix IV. 5.1 Functional Description of the Computer Program If inplane inertia is considered ignorable with respect to normal inertia, then equations (4.20) separate into coupled sets of 2(Nj) nonlinear algebraic equations and 2(Nj) nonlinear first order differential equations. The flow of computations proceeds as follows. The linear and nonlinear stiffness matrices, mass matrices, boundary transform matrices, buckling matrices, and load ing vectors are generated by the program. At time equal to zero the problem solution begins with known external loading and initial conditions. The known normal deflec tions are substituted into the first two equations of equilibrium and the resulting u, v vectors are generated utilizing a Gauss elimination technique with complete pivoting. If nonlinear effects in the first two equa tions of equilibrium are considered, this mode is itera tive in nature. Having generated the u's and v's for time zero, they are substituted into the sets of 3rd dynamic equilibrium equations for temporal integration. The time integration is accomplished by a Hamming pre dictorcorrector integration technique. This generates the w's for the next time step. Normal velocities are also computed. This completes the sequence of advance to the first time increment beyond time zero and provides the required initial conditions in order to begin the next advance in time. The sequence then recycles and the temporal advance continues. At any time interval the an alyst chooses the solved for dependent variables are transformed from the solution space to real space and displayed as output. Each output consists of the com plete meridional profile for each mode in the solution. The build up of errors in the numerical solution is controlled by the following mechanisms. A loss of sig nificance control indicator is incorporated into the Gauss elimination routine which warns the analyst when solutions are being derived from a nearly singular coefficient matrix. An error bound control is available in the predictorcorrector integrator which limits this type of error by automatically adjusting the size of the time step. Round off errors are minimized by making all calculations accurate to sixteen significant figures. Such rigorous control of possible numerical errors is essential in dynamic stability analysis because of the possibility that numerical instability might be mistaken for actual structural instability. 5.2 Checking the Computer Program The control of clerical and conceptual errors in the construction of the computer program was done by con tinually checking back to simpler known cases. In some instances the known cases could be obtained by hand cal culations, in others, the results of other analysts were used. For the shell specimen studied, the computer pro gram has been verified by checking against known solutions for static stability, static deflection, free vibration modes and frequencies [39], linear dynamic response, and regions of parametric instability [52]. A convergence study was also conducted to determine the convergence properties of the solution for increasing numbers of subdomains. The results of this study are presented in Appendix II. 5.3 Dynamic Stability Results After the above verifications were completed, the program was used to study the linearized and the nonlinear dynamic stability of a short conical frustum with simply supported boundaries subject to various levels of constant axial loads and critical levels of nearly axisymmetric Heaviside pressure blasts. Eight subdomains along the meridional direction were used. The driven breathing mode and an arbitrary preferred flexural mode were retained in the analysis. Complete coupling between the modes was re tained through the quadratic terms for the nonlinear case. The critical static normal pressure for the speci men studied is 42.8 pounds per square inch and the critical circumferential wave number is eight. This represents a midspan deflection in the breathing mode of 9 percent of the thickness. The ordinate scale (W) of all figures depicting time history responses is normalized by this critical deflection. The critical static axial load is 40,400 pounds. The applied loads for time history graphs will be noted as (a, b, c) where a = P/P cr b = p/Pcr, c = P /Pcr. Additionally, note that P = P H(t) Pn = PnH(t) where H(t) denotes a Heaviside pressure pulse. 5.3.1 Dynamic Modal Preference The program was used to determine the critical flexural modes during dynamic instability. The results are shown in Figure 4 and the critical static buckling modes are indicated. Because only a few closely spaced critical modes dominate the flexural response and because 57 5 0  E 0 ~CO 4 z C III ( rl L1J o  C 0 0 0 Cci( ,aa) o0CO C 1 I U r S fO Sa I I II 5  0 ^0) ro "0  CCJ 0 0 0 CL~u r a a \\ I 011 I r \\ I a \\ O coupling between these modes is very weak [10], the param eter study that follows is reduced by considering the in stability of only a sequence of single flexural modes to find the mode indicating instability at the lowest load. 5.3.2 Linearized Dynamic Stability The linearized dynamic analysis was obtained by considering the vector {NL}(n) in equation (4.20) to be zero. This means that the internal loads from the pre buckling state may interact with postbuckling deflections to cause primary dynamic instability (i.e. snap through) but internal parametric instabilities due to the pulsat ing nature of the membrane state are prohibited. The critical circumferential wave number for dynamic instability was found utilizing this analysis and the results are shown in Figure 5. While conducting a series of linearized dynamic stability computer runs at various combinations of external loadings, a feature of the numerical model was discovered which represents an improvement over other computer programs used for this purpose. This feature is that the meridional profile undergoes a marked and rapid change of character during runs at or above the critical dynamic load. For loadings below critical, the meridional profile pulsates at 0 1. a) a ,E0 O (u C 'I) 4 C O E E O . SD  co 0 C one half wave, slightly skewed due to the conical taper. At or above the critical load the meridional profile begins to pulsate as a half wave but later snaps to two or more half waves. The time of snap, number and shape of the waves in the buckled meridional profile depend on the axial load to normal pressure ratio and their rela tion to static critical loads. Examples of this are shown in Figures 6 and 7. It is felt that this improved ability of detecting dynamic instabilities results from two areas of the analysis. One being the spatial error residual con trol offered by the modified subdomain method employed and the other being the strict control of error bound in the numerical work. Additionally, it should be noted that most published works on this type of problem utilize one term assumed solutions for preferred modes and clearly could not see this effect in their analysis. A large number of linearized dynamic stability runs were made at various combinations of static axial and Heaviside pressure loads. The results were collated into a dynamic stability interaction curve and are shown in Figure 10. 5.3.3 Nonlinear Dynamic Stability In the case of nonlinear dynamic stability, the possibility of autoparametric excitation of preferred 0 0 *r X x o 4, CC cu I 5 E OO o g 5 aO C *r LU' U u cO 03r 0 CD x E C: S +J ro C: 4 D ( +o 4E CA N S  It C 0 ' u LO O O flexural modes by the pulsating breathing mode is allowed, in addition to snap through or primary dynamic instability. The question here is whether such excitation will occur during the primary instability motion of the preferred mode sufficiently to affect the magnitude of the external loads required to induce dynamic instability. In contrast to the external parametric excitation problems, the total energy of the shell for the present problem is constant for all t>O. Thus, the unstable modes when growing must derive their energy from the breathing mode with a corresponding decrease of energy in that mode. To evaluate this inter play a full nonlinear analysis is necessary. Conceptually, the third equation of nonlinear dy namic stability may be expressed in an average sense as follows: + % 2 2 + 0 ow + k(uo) + f(w ) = oH(t) (5.1 a,b) n + [ g(o)w + (nVn) + (h(w)n = PnH(t) where uo, w represent breathing mode deflections measured from the undeformed state and u v n, w represent the poststable growth of the buckling mode. X and n denote the linear eigenvalues of the respective modes and f, g, h, k, and 1 are differential operators. The superscript a denotes an average over the shell length. Since w0 is excited by a Heaviside pulse, the wo response is initially periodic and equation (5.1 b) takes the character of an inhomogeneous Mathieu equation. The primary buckling terms denoted by a single underline cause an insignificant shift to the left on a parametric stability diagram. In the absence of imperfections in the preferred flexural mode, denoted by the double underlined term, equation (5.1 b) lies well within a stable zone of a Mathieu diagram and no significant parametric growth of the buckling mode would be expected. For finite deterministic imperfections, however, the location of stability boundaries becomes more difficult and for significantly large pn H(t), solutions to equation (5.1 b)may exhibit parametric growth at or below the dynamic loads required to produce immediate snap through and may lead to a delayed dynamic snap through. Whether or not this beating type of growth occurs with sufficient quick ness to have an effect on the primary dynamic instability can be determined only by direct temporal integration of two coupled sets of equation (4.20). A series of runs with the computer program was made to determine the significance of this effect. Typical re sults are shown in Figures 8 and 9. As in the case of C\j QJ E I 66 0 SC o _! *r O 0 0 no cr  ar 0 0 I' E E 00 g5 ( * \ E I~ lo 5 ___1 _______ 1 i linearized dynamic stability, a large number of computer runs were made at various combinations of static axial and Heaviside pressure loads and the results collated into a nonlinear dynamic stability interaction curve shown in Figure 10. The linearized dynamic stability curve lies below the static stability curve in Figure 10 due to the in creased severity of a suddenly applied load over that of a statically applied load. The peak membrane forces de veloped by the breathing mode which is excited by a Heavi side pulse are twice as large as that of an equivalent static case and contribute more to the developemnt of instability in preferred flexural modes than a static situation. This is known as the dynamic load factor effect. The nonlinear dynamic stability curve lies below the linearized dynamic stability curve because internal vibra tions interact between coupled modes so as to produce un stable beating resonances in the preferred buckling mode. Such unstable vibrations lead to a delayed dynamic snap through and this phenomenon is called autoparametric ex citation. Static Stability Linearized Dynamic  Stability SNonlinear Dynamic Stability n = 8 0 0.5 p H(t) / pcr Figure 10. Dynamic Stability Interaction Curves n 10 CHAPTER 6 RESULTS AND CONCLUSIONS In the present analysis, a nonlinear shell theory is derived and employed to study the nature of nonlinear dynamic instability of a truncated thin circular conical shell structure which is considered to be loaded by a constant axial load and a nearly axisymmetric Heaviside pressure load. A solution method is developed which sat isfies a boundary condition exactly and which converges toward the exact solution of the governing equations with increasing subdomains. The method avoids the necessity of assuming the shapes of prebuckling or postbuckling meridional profiles. The results of this study confirm the feasibility of the method of solution developed for this analysis. The method converges rapidly yet can be applied to nonlinear problems with minimum amounts of manual mathematics. The trial functions are simple to manipulate yet satisfy any formulatable boundary condition. The method is particu larly suitable for problems where poststable modes are not known in advance as is usually the case for combined dynamic loadings. For the particular loading conditions studied, the results indicate that dynamic instability will occur at loads below critical static loads. This reduction in critical dynamic loads was shown to be the result of both the dynamic load factor effect and of autoparametric ex citation. The interaction curves which are generated illustrate these effects quantitatively under conditions of combined dynamic loading. A criterion for dynamic buckling is established based on meridional mode shape changes. This ability to detect sudden jumps in the meridional profile aids to verify instability detected by divergence on displacement time history curves and provides additional information about the poststable state. APPENDIX I SHELL SPECIMEN AND BOUNDARY CONDITIONS The conical frustum utilized in this investigation has the following properties (see Figure 3): Material 1020 steel a = 200 s1 = 5.85 inches s2 = 14.22 inches RI = 2.0 inches R2 = 4.863 inches h = 0.02 inches v = 0.3 . The boundary conditions assumed for the analysis are: u = v = w = Ms = 0 at sI and s2 This results in eight constraint equations which when ex pressed in terms of power series solutions for eight sub domains along the meridian become 10 I ais = 0 at s = sI and s = s2 i=o 10 i Z b.s = 0 1=o 12 1 Z c.s = 0 1=o at s = sl and s = s2 at s = s1 and s = s2 12 z c.[(i)(i1) + v(n2/sin2a i=o + i)] i2 = 0 at s = s1 and s = s2 APPENDIX II APPENDIX II CONVERGENCE In the use of the subdomain method, convergence in the mean is desired. The main influence on convergence is the choice of trial functions. For assured convergence, the trial functions must be complete and linearly indepen dent [49]. The completeness property of a set of functions insures that we can represent the exact solution provided enough terms are used. The power series trial functions used in this study are complete, for example, so that any continuous function can be expanded in terms of them. To demonstrate the rate of convergence for the problem under study, a series of computer runs were con ducted to determine the dynamic perturbation response for different numbers of subdomains. Dynamic stability response for eight, ten, and sixteen subdomains was investigated. The results are shown in Figure 11. The ordinate designa tion implies percent differences with respect to the best approximation. I _ 76 8 6 I I o O  4 L\ U 2 0 I 5 10 Number of Subdomains Figure 11. Convergence Properties of Solution APPENDIX III APPENDIX III FLOW DIAGRAM OF THE COMPUTER PROGRAM For a better understanding of the computer program, a key which may be helpful in going from the theory to the program is given below: FORTRAN Name Theory Description N m Number of subdomains NW n Circumferential wave number Q Q Total number of n AP a Semivertex angle of cone RO p Material density E E Young's modulus V v Poisson's ratio F s2 Conical slant height BTT B Boundary condition matrix STT [k] Linear stiffness matrix DND [m] Mass matrix PV {p} Load vector PLUG {NL} Nonlinear stiffness vectors X T Nondimensional time FORTRAN Name Theory Descriptions U,V,Y u nv n, Nondimensional displacements P P Axial load PSD p Pressure level of Heaviside pulse PRMT(2) T2 Upper time limit PRMT(4) e Error bound ORAN(I)  Online core storage matrices DHPCG  Hamming's predictorcorrector integration subroutine DGELG  Gausselimination subroutine A flow diagram of the computer program is illustrated in Figure 12. APPENDIX IV FORTRAN IV SOURCE PROGRAM The computer program developed to provide the numerical solution to this dissertation problem is listed below. The program requires 182,000 bytes computer core storage space. A typical nonlinear dynamic stability run executes in approximately in seven minutes. The compile time is three seconds. 83 o0000000900000000000000000 o0000000000000000000000000 OOOOOOOOQOOOOOOOOOOQOQ00000OO0OO 000000000000000000000000000000000 N NO. 0 N 00 0 0  N .0 0 r 0 *4 0 o* o *c0 a m 0 NO N S NN0 * W If N J U U N *O N m Z (. 40 * a r \ * . W c N n *0 I0 0 > 0 N u< ** L co Co t* OD o O U O N N C 4 O o w )   E Nn N 4 N 3 COO* (M PN m V04 NNT . Z NN W O O .3 OC r* c t  0  4a C N M LJ Z C' I *Z' Z C O N N N Z N h zz N \ Z O C0O M 03 0 00 0  ON NC i* W *0 r N N 4 W*O0 N 03 W 0 m 0O  Q 0 N w u QQ) N . 4 rN O# O rO f NN2. 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( uo J l 0 c 0 LU 4 P uj w 4 0 4 > u_ 0 + _ x  Un 2 o U' LU Q   j r U 0 0 1 1 w *.j (/4 * i ut 3 ( _ Co  I EL Q a 1 *V) ^ )40 a 0 j UJ 0J .O'Un * 442 o+ II 3A II 0 J  LU CLJ , 00.t 20m 44 r ~ ~luLI^ > ! l l  i iii 1 i * m U 0 125 00000 0000000000 00000000000Q0000009 OOOOcOOOOOOOQOOQOOOOQOQOQUOOQOOOO 000000000000000Q000 00000000000000 vn oooQooo00o000o0000win0'orcouo, m f' en m m r (n ..7.7.4 .7 rr r r i7 o n. Lin n n toI LA 0in a0o a ' 3' T 'r .r y < 'T'L     < d  000 I 0 0 ( Z I 0 SI 4 UJ II IJ II (0 0 _I V) 27 Ln 0 l 0 , 0' 4 D a X 1  + f 2n + l e 2I S jI aE lZ 11 C n SV Vu v) 0 o A u Am juN j Lj*3 l co C1 C <) t II 9 I I I l N NI z a 9 U.; 011 a I 0 c w w z LU a L m (N UuUU SU U BIBLIOGRAPHY 1. Hoff, N. J., "Dynamic Stability of Structures," Proceedings of an International Conference on the Dynamic Stability of Structures, Northwest ern University, Evanston, Illinois, 1965, pp. 941. 2. Stoker, J. J., "Stability of Continuous Systems," ibid., pp. 4652. 3. Meirovitch, L., Methods of Analytical Dynamics, McGrawHill Book Co., New York, 1970. 4. Koiter, W. T., "General Equations of Elastic Stabil ity," Proceedings Symposium on the Theory of Thin Shells, University of Houston, Houston, Texas, 1967. 5. Meirovitch, L., op. cit. 6. Witmer, E. A., Leech, J. W., Pian, T. H. H., "Numer ical Calculation Technique for Large Elastic Plastic Transient Deformations of Thin Shells," AIAA Journal, Vol. 6, December 1968, pp. 2352 2359. 7. Witmer, E. A., Leech, J. W., Morino, L., "PETROS 2," BRL Contract Report 12, 1969. 8. Witmer, E. A., Leech, J. W., Morino, L., "An Improved Numerical Calculation Technique for Large Elastic Plastic Transient Deformations of Thin Shells," Journal of Applied Mechanics, Vol. 38, June, 1971, pp. 423436. 9. Longhitano, R., Klosner, J. M., "Nonlinear Dynamics of Hemispherical Shells," AIAA Journal, Vol. 11, August 1973, pp. 11171122. 10. Cromer, C. C., "An Investigation of the Nonlinear Dynamic Response of Cylindrical Shells Under Transient Pressure," Ph. D. Dissertation, Naval Postgraduate School, Monterey, California, 1969. 11. Ball, R. E., "A Computer Program for the Geometric ally Nonlinear Static and Dynamic Analysis of Arbitrarily Loaded Shells of Revolution," NASA CR1987, April 1972. 12. Ball, R. E., Burt, J. A., "Dynamic Buckling of Shallow Spherical Shells," Journal of Applied Mechanics,Vol. 40, June 1973, pp. 411416. 13. Kim, C. W., "Static and Dynamic Buckling of Nonhomo geneous Cylindrical Shells," Ph. D. Dissertation, University of Florida, Gainesville, Florida, 1972. 14. Stricklin, J. A., Martinez, J. E., Tillerson, J. R., Hong, J. H., Haisler, W. E., "Nonlinear Dynamic Analysis of Shells of Revolution by Matrix Dis placement Method," N 7030153, Texas A and M University, College Station, Texas, February, 1970. 15. Wu, R. W. H., Witmer, E. A., "Nonlinear Transient Responses of Structures by Spatial FiniteElement Method," AIAA Journal, Vol. 11, August 1973,pp. 11101116. 16. Klosner, J. M., Roth, R. S., "Nonlinear Response of Cylindrical Shells Subjected to Dynamic Axial Loads," AIAA Journal, Vol. 2, October 1964, pp. 17881794. 17. Klosner, J. M., Zweben, C., "Dynamic Instability of Circular Cylindrical Shells Having Viscoelastic Cores," AIAA Journal, Vol. 5, June 1967, pp. 11281134. 18. Goodier, J. N., Mclvor, I. K., "The Elastic Cylindri cal Shells Under Nearly Uniform Radial Impulse," Journal of Applied Mechanics, Vol. 86, June 1964, pp. 259266. 19. Mclvor, I. K., Lovell, E. G., "Dynamic Response of FiniteLength Cylindrical Shells to Nearly Uniform Radial Impulse," AIAA Journal, Vol. 6, December 1968, pp. 23462351. 20. Lindberg, H. E., "Stress Amplification in a Ring Caused by Dynamic Instability," Journal of Ap plied Mechanics, Vol. 41, June 1974, pp. 392400. 21. Hubka, W. F., "Dynamic Buckling of the Elastic Cylindri cal Shell Subjected to Impulsive Loading," Journal of Applied Mechanics, Vol. 41, June 1974, pp. 401406. 22. Mente, L. J., "Dynamic Nonlinear Response of Cylindrical Shells to Asymmetric Pressure Loading, AIAA Journal, Vol. 11, June 1973, pp. 793800. 23. Shumik, M. A., "Behavior of Conical Shells Under Dynamic Loading," Prikladnaia Mekhanika, Vol. 5, 1969, pp. 15 19 (in Russian). 24. Shumik, M. A., "On the Stability of Conical Shells Under Dynamical Longitudinal Compression," Priklednaia Mek hanika, Vol. 6, 1971, pp. 122126 (in Russian). 25. Shiau, A. C., "The Dynamic Buckling of a Truncated Coni cal Shell Structure," Ph. D. Dissertation, State Uni versity of New York at Buffalo, 1973. 26. Nash, W. A., Wilder, J. A., "Response of Thin Conical Shells to Dynamically Applied Axial Force," Interna tional Journal of NonLinear Mechanics, Vol. 7, 1972, pp. 6580. 27. Fisher, C. A., Bert, C. W., "Dynamic Buckling of an Axially Compressed Cylindrical Shell With Discrete Rings and Stringers," Journal of Applied Mechanics, Vol.40,September, 1973, pp. 736740. 28. Lakshmikantham, C., Tsui, TY., "Dynamic Stability of AxiallyStiffened Imperfect Cylindrical Shells Under Axial Step Loading," AIAA Journal, Vol. 12, February 1974, pp. 163169. 29. Dietz, W. K., "On the Dynamic Stability of Eccentrically Reinforced Circular Cylindrical Shells," Syracuse University Research Institute, Technical Report 1620, January 1967, pp. 12451260. 30. Novozhilov, V. V., Foundations of the Nonlinear Theory of Elasticity, Graylock Press, Rochester, New York, 1953. 31. Tsao, C. H., "Strain Displacement Relations in Large Displacement Theory of Shells," AIAA Journal, Vol. 2, 1964, pp. 20602062. 32. Ogibalov, P. M., "Dynamics and Strength of Shells," NASA TT F284, 1966. 33. Naghdi, P. M., Nordgren, R. P., "On the Nonlinear Theory of Elastic Shells Under the Kirchhoff Hypothe sis," Quarterly of Applied Mathematics, Vol. 21, No. 1, 1963, pp. 4960. 34. Biricikoglu, V., Kalnins, A., "Large Elastic Deformations of Shells with the Inclusion of Transverse Normal Strain," International Journal of Solids and Struc tures, Vol. 7, 1971, pp. 431444. 35. Sanders, J. L., Jr., "Nonlinear Theories for Thin Shells," Quarterly of Applied Mathematics, Vol. 21, No. 1, 1963, pp. 2136. 36. Naghdi, P. M., "The Theory of Shells and Plates," Report No. AM711, University of California, Berkeley, 1971. 37. Sokolnikoff, I. S., Mathematical Theory of Elasticity, McGrawHill Book Co., New York, 2nd edition, 1956. 38. Kalnins, Arturs, "Static, Free Vibration, and Stability Analysis of Thin, Elastic Shells of Revolution," Technical Report AFFDLTR68144, WrightPatterson Air Force Base, 1969. 39. Leissa, Arthur W., "Vibration of Shells," NASA SP288, 1973. 40. Novozhilov, V. V., The Theory of Thin Shells, P. Noord hoff Ltd., Groningen, 1959. 41. Gol Denveizer, A. L., Theory of Elastic Thin Shells, Pergamon Press, New York, 1961. 42. Novozhilov, V. V., Finkel'shtein, R., "On the Incorrect ness of Kirchhoff's Hypothesis in the Theory of Shells," P. M. M., Vol. 7, 1943, pp. 331340. 43. Mushtari, K. M., "NonLinear Theory of Thin Elastic Shells," N 62 72586, U. S. Department of Commerce, Washington, D. C., 1962. 44. Koiter, W. T., "A Consistent First Approximation in the General Theory of Thin Elastic Shells," I. U. T. A. M. Proceedings of the Symposium on the Theory of Thin Elastic Shells, Delft, NorthHolland Publishing Co., Amsterdam, pp. 1233. 45. Vlasov, V. S., "Basic Differential Equations in the General Theory of Elastic Shells," NACA TM 1241, 1951. 46. Hurty, W. C., Rubinstein, M. F., Dynamics of Structures, PrenticeHall, Inc., New Jersey, 1964. 47. Meirovitch, L., op. cit. 48. Kraus, H., Thin Elastic Shells, Wiley and Sons, Inc., New York, 1967. 49. Finlayson, B. A., The Method of Weighted Residuals and Variational Principles, Academic Press, New York and London, Vol. 87, 1972. 50. Radosta, R. J., "Stability of Nonhomogeneous Shells of Revolution," Ph. D. Dissertation, University of Florida, Gainesville, Florida, 1971. 51. Polhausen, K., "The Approximate Integration of the Dif ferential Equation for the Boundary Layer," Z. Angew, Mathematics and Mechanics, Vol. I, AD 64 5784 F. S. T. I., Springfield, Va., pp. 252268 (in English). 52. Tani, J., "Dynamic Instability of Truncated Conical Shells Under Periodic Axial Load," International Journal of Solids and Structures, Vol. 10, 1974, pp. 169176. BIOGRAPHICAL SKETCH Marcus George Hendricks was born May 9, 1940 at Pickens, South Carolina. He graduated from Pickens High School in May, 1958. He received a Bachelor of Science degree in Civil Engineering from Clemson University in June, 1962. Returning to Clemson University after serv ing two years in the U. S. Army as a commissioned officer, the author was awarded a Master of Science degree in Civil Engineering in August, 1966. In 1970 he enrolled in the Graduate School at the University of Florida in order to pursue work toward the degree of Doctor of Philosophy. He is married to the former Genevieve J. Boggs and has one daughter, Carmen. I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Sung Y. Lu, Chairman Associate Professor of Engineering Sciences I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. brahim K. Ebcioglu Professor of Engineering Sciences I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Ulrich H. Kurzweg Associate Professor of Engineering Sciences I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Robert L. Sierakowski Professor of Engineering Sciences I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Morris W. Self P ofessor of Civil/ Engineering This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate Council, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. December, 1974 Dei, College of Engineering e School ; Dean, Gradu te School " UNIVERSITY OF FLORIDA 3 1262 08666 450 4
