The Rayleigh-Ritz method is used to determine both the buckling and postbuckling behavior of completely, enclosed prolate spheroidal shells under uniform external pressure. It is assumed that the shells are isotropic and elastic, and have a uniformly thin wall. It is further assumed that the prebuckling state can be described by membrane theory. In this analysis the buckling displacements are confined to a shallow cap located in the region of least curvature of the shell and the boundary of the buckled zone is considered to lie in a plane parallel to the axis of revolution of the shell. A coordinate transformation is performed so that one of the new coordinate curves coincides with the boundary or the buckled zone. The in-plane displacement component is then restricted to be normal to the family of curves which contain this boundary. In addition, both the in-plane and normal displacement components are considered to be functions of one variable only. Series expressions for the in-plane and normal displacement components, each involving M unknown parameters, are inserted into the total potential energy expression. The resulting functional is then minimized with respect to each displacement parameter and also the parameter characterizing the extent of the buckled zone to yield a system of 2M + 1 nonlinear algebraic equations. M + 1 of these equations are eliminated and the Newton-Raphson iterative procedure is employed to obtain the solutions to the remaining M equations. Results are presented for five shell geometries characterized by the ratio of the major diameter to the minor diameter: spherical shell is included as one of the cases. The numerical computations were performed on the Sperry-Rand LARC computer located at David Taylor Model Basin, Washington, D. C. Solutions for increasing values of M are compared in order to evaluate the convergence of the Rayleigh-Ritz method. For the determination of the buckling loads, the maximum value of M used is ten; the solutions to the postbuckling equations are limited to M=5. The numerical results represent a considerable improvement in scope and accuracy over previously published solutions to this problem. This is the first time that the postbuckling behavior associated with higher modes has been considered and it is demonstrated that there is a possibility of a mode shifting phenomenon occurring in the postbuckled state. A discussion is presented of a series of exploratory tests on spheroidal shell models made from an Epon-Versamid resin. Measurements and photographs of the buckled models give qualitative support to the theoretical work presented. / Ph. D.

A degenerated three-dimensional finite element based on the total Lagrangian, incremental, formulation of a three-dimensional layered anisotropic medium is developed, and its use in the geometrically nonlinear, static as well as dynamic, analysis of layered composite plates and shells is demonstrated via several example problems. For comparison purposes, a two-dimensional finite element based on the Sanders shell theory with the von Karman (nonlinear) strains is also presented. The elements have the following features: • Geometrically linear and nonlinear analysis • Static and transient analyses • Natural vibration (linear) analyses • Plates and shell elements • Arbitrary loading and boundary conditions • Arbitrary lamination scheme and lamina properties The element can be used, with minor changes, in any existing general purpose programs. The 3-D dimensional degenerated element has computational simplicity over a fully three-dimensional element, and the element accounts for full geometric nonlinearities in contrast to the 2-dimensional elements based on the Sanders shell theory. As demonstrated via numerical examples, the deflections obtained by the 2-D shell element deviate from those obtained by the 3-D element for deep shells. Further, the 3-D element can be used to model general shells that are not necessarily doubly-curved. For example, the twisted plates can not be modeled using the 2-D shell element. Of course, the 3-D degenerated element is computationally more demanding than the 2-D shell theory element for a given problem. In summary, the present 3-D element is an efficient element for the analysis of layered composite plates and shells undergoing large displacements and transient motion. / Ph. D.

Shells of double curvature are common structural elements in aerospace and related industries, but due to the complexity of their configurations and governing equations, little has been done to classify their general dynamic behavior. The subject of this dissertation is the determination of the effect of the meridional curvature on the natural vibrations of a class of axisymmetrically prestressed doubly curved shells of revolution. A set of linear equations governing the infinitesimal vibrations of axisymmetrically prestressed shells is developed from Sander's nonlinear shell theory and both the in-plane inertia and prestress deformation effects are retained in the development. The equations derived are consistent with first-order thin-shell theory and can be used to describe the behavior of shells with arbitrary meridional configuration having moderately small prestress rotations. A numerical procedure is given for solving the governing equations for the natural frequencies and associated mode shapes for a general shell of revolution with homogeneous boundary conditions. The numerical procedure uses matrix methods in finite-difference form coupled with a Gaussian elimination to solve the governing eigenvalue problem. An approximate set of governing equations of motion with constant coefficients which are based on shallowness of the meridian are developed as an alternate more rapid method of solution and are solved in an exact manner for all boundary conditions. The solutions of the exact system of shell equations determined from the numerical procedure are used to determine the accuracy of the approximate solutions and with its accuracy established, the approximate equations are used exclusively to generate results. The membrane and pure bending equations which correspond to the approximate set of equations are solved for a specific boundary condition. The effect of the meridional curvature on the fundamental frequencies of a class of cylindrical-like shells with shallow meridional curvature and freely supported edges are investigated. Results show that the positive Gaussian curvature shells have fundamental frequencies well above those of corresponding cylindrical shells. The fundamental frequencies of the negative Gaussian curvature shells generally are below those of the corresponding cylinders and evidence wide variations in value with large reductions in magnitude occuring at certain critical curvatures. Comparison of the membrane, pure bending and complete shell analyses shows that these critical curvatures represent configurations at which the fundamental mode of vibration of the shell is in a state close to pure bending. The membrane theory affords a simple method of determining the modal wavelength ratio at which the pure bending state exists for a given negative Gaussian curvature shell, while the pure bending theory gives a good estimate of the magnitude of the frequency for this wavelength ratio. Meridional edge restraints and internal lateral pressure reduce the wide variation of the natural frequencies in the negative curvature shells and in general raise the natural frequencies. External lateral pressure accentuates the reduction in natural frequencies of the negative curvature shells and causes instability at low compressive stress ratios. / Ph. D.

From the results and equations shown herein, several important conclusions are evident. The equations derived here considering bending deformations only are seen to be more general in form than existing solutions, and reduction to the existing cases is direct. For example if the loading is considered uniform in r and impulsive or step-wise uniform in time, the equations derived directly for such cases by Hopkins and Prager and Wang (refs. 2 and 5) appear exactly. Also, if the radial load distribution is considered uniform, and a general function of time is allowed (but assuming only inward hinge circle movement), the nonlinear equations of Perzyna (ref. 57) are found exactly. The conclusion of Perzyna that time variation is unimportant appears to be caused by an unfortunate choice of example time functions. He solves the specific non-linear equations for his example, and does not present any means for evaluation of his numerical method of solution. If the loading on the plate is considered to be a distributed Gaussian loading in r and impulsively applied, the equations derived directly for this case by Thomson (ref. 56) appear exactly herein. These two papers (by Perzyna and Thomson) are the only two papers available at present that allow variations of the loading, one in r and the other in t, and both sets of equations are included in the general expressions herein. In fact, the solutions currently available for bending theory are found to exist as special cases of these general equations. / Ph. D.

A probabilistic finite element analysis procedure for laminated composite shells has been developed. A total Lagrangian finite element formulation, employing a degenerated 3-D laminated composite shell element with the full Green-Lagrange strains and first-order shear deformable kinematics, forms the modeling foundation. The first-order second-moment technique for probabilistic finite element analysis of random fields is employed and results are presented in the form of mean and variance of the structural response. The effects of material nonlinearity are included through the use of a rate-independent anisotropic plasticity formulation with the macroscopic point of view. Both ply-level and micromechanics-level random variables can be selected, the latter by means of the Aboudi micromechanics model. A number of sample problems are solved to verify the accuracy of the procedures developed and to quantify the variability of certain material type/structure combinations. Experimental data is compared in many cases, and the Monte Carlo simulation method is used to check the probabilistic results. In general, the procedure is quite effective in modeling the mean and variance response of the linear and nonlinear behavior of laminated composite shells. / Ph. D.

The vibrational energy propagating in straight fluid-filled elastic pipes is carried by the structure as well as by the internal fluid. Part of the energy in the system may also transfer from one medium to the other as propagation occurs. For various types of harmonic disturbance, this study demonstrates that, whether the propagating energy is predominantly conveyed in the shell or in the fluid, large attenuations of the total power flow may be achieved by using an active control approach. As the shell and fluid motions are fully coupled, the implementation of intrusive sources/sensors in the acoustic field can be also avoided. The approach is based on using radial control forces applied to the outer shell wall and error sensors observing the structural motion. A broad analytical study gives insight into the control mechanisms. The cylindrical shell is assumed to be infinite, in vacuo or filled with water. The first disturbance source investigated is a propagating free wave of circumferential order n=0 or n= 1. The control forces are appropriate harmonic line forces radially applied to the structure. The radial displacement of the shell wall at discrete locations downstream of the control forces is minimized using linear quadratic optimal control theory. The attenuation of the total power flow in the system after control is used to study the impact of the fluid on the performance of the control approach. Results for the shell in vacuo are presented for comparison. Considering the breathing mode (n=O), the fluid decreases the control performance when the disturbance is a structural-type incident wave. Significant reductions of the transmitted power flow can be achieved when the disturbance is a fluid-type of wave. Regarding the beam mode (n=1), the fluid increases the control performance below the first acoustic cut-off frequency and decreases it above this frequency. / Ph. D.

The early stages of propagation of a water hammer disturbance are investigated, water hammer constituting a special case of axially symmetric elastic wave propagation in a fluid-filled cylindrical shell. Many of the objectionable features of the elementary (Joukowsky) water hammer theory are removed, and particular emphasis is placed upon consideration of the effects of radial inertia of the fluid and of the shell. The formulation is appropriate for consideration of any axially symmetric acoustic disturbance which originates in the fluid and any of the usual engineering boundary conditions which describe constraints on motion of the end, or ends, of the shell. Motion of the shell is described by a thin-shell theory, and motion of the fluid is described by the axially symmetric wave equation, nonhomogeneous boundary conditions providing coupling of the fluid and shell motions. Application of a finite Hankel transform to the axially symmetric wave equation yields an infinite system of one-dimensional wave equations representing motion of the fluid. Integration of a finite set of these wave equations in conjunction with equations governing motion of the shell is accomplished numerically after a straight-forward application of the method of characteristics. An analysis which includes bending, rotary inertia, and shear deformation in the shell is conducted for the case of sudden termination of uniform flow in a semi-infinite shell with a"built-in" end. For a relatively thick steel shell filled with water it is found that bending stresses and transverse shearing stresses at the end of the shell are significant, but that nowhere are there significant longitudinal membrane stresses. Maximum stresses and displacements are found to occur within the time required for an acoustic disturbance in unbounded fluid to traverse one diameter of the shell. The maximum radial displacement of the middle surface of the shell is found to exceed the value predicted by the elementary theory by about fifty percent. A solution based on the classical membrane theory of shells, neglecting longitudinal stresses, also is obtained by numerical integration of the ordinary differential equations arising from application of the method of characteristics to the governing partial differential equations. Considerable simplification of the numerical calculations results from the fact that only one pair of families of characteristic lines are involved in the membrane analysis as compared to three pairs of such families in the bending analysis. The membrane analysis is employed principally to show the adequacy of using the first five of the infinite set of one-dimensional wave equations governing motion of the fluid. The membrane formulation takes account of two important omissions of the elementary theory, namely radial inertia of the fluid and radial inertia of the shell. A representation of the fluid motion by a single one-dimensional wave equation is investigated. Radial inertia of the fluid is taken into account by attributing additional mass to the shell. This formulation is shown to produce the same results as the best available long-time asymptotic solution to a water hammer problem, but it is found, on the basis of an analysis employing the membrane theory of shells, to be inadequate for describing the early stages of a water hammer disturbance. / Doctor of Philosophy

The general nonlinear first approximation thin plate tensor equations in undeformed coordinates valid for large strains, rotations and displacements are developed based on the single assumption of plane stress. These equations are then reduced to the exact tensor and physical component equations for symmetrical circular plates. An order of magnitude analysis is performed on the resulting equations which shows that they reduce to the classical linear equations for very small deflections and to the von Karman equations for moderate deflections. However, the equations do not reduce to the Reissner equations for large deflections. The solution to the problem of a clamped circular plate loaded with a concentrated load on a central rigid inclusion was obtained and agreed with the solution of von Karman's equations for moderate deflections. Perhaps the most important result is that of finding the order of magnitude of the limiting value of deflection that would be allowed under the assumption of plane stress for this particular problem. It is shown that when the deflection approaches the order of magnitude of the radius, the boundary layer approaches the order of magnitude of the thickness and thus a first approximation theory is no longer valid. Two membrane problems are also solved. The first is that of a circular membrane deformed by a load which acts normal to the plane of a central rigid inclusion. A closed form solution is obtained for this problem when Poisson's ratio is equal to 1/3. An approximate solution is obtained for any value of Poisson's ratio for the case where the deflections are very large. The second problem is the same as the first with the addition of a small torque about a normal to the rigid inclusion. An approximate solution is obtained to this problem. / Ph. D.

The approach to the problem of a variable thickness plate used in this paper is different from the usual approach in that this paper starts with general stress-strain relations and a generalized form of the position vector as used by Green and Zerna in "Theoretical Elasticity". They use R̅=L[ r̅ (θ₁,θ₂)+ λθ₃a̅₃(θ₁,θ₂)] where θ₁,θ₂, and θ₃ are curvilinear coordinates with θ₁ and θ₂ being the coordinates of the middle surface and λ=t/L being a constant for a plate of constant thickness t. This paper takes λ = λ(θ₁,θ₂) as a function of θ₁ and θ₂ so that the variable thickness may be taken into account. General tensor notation is used so as to work independent of coordinate systems. Making simplifying assumptions only when necessary, the equations of equilibrium and stress-strain relations are derived in terms of tensors connected with the middle surface as was done by Green and Zerna for a constant thickness plate. The additional terms obtained in these equations due to the variation in λ help us to evaluate the effects of the varying thickness. Expressions for stress are developed and they include the effects of transverse shear deformation and normal stress as well as the variation in thickness. These expressions are very much like those used by Essenburg and Naghdi in a paper presented at the Third U.S. National Congress of Applied Mechanics, June, 1958. However, they assumed the form for the stresses while the present paper arrived at their assumed forms with some additional terms after starting with general stress-strain relations. Using the notation of Green and Zerna, a set of nine equations involving the nine unknowns, m ^αβ, w, n^αβ, and v^α is derived and under appropriate boundary conditions, this set will yield a solution to the problem which will be better than the classical solution. Two problems are solved and numerical results are obtained and compared with the classical solutions. One of the problems involves a rectangular plate clamped on one edge with a uniform shear load on the other. The other problem involves a circular ring plate clamped on the outer edge with a uniform shear load on the inner edge. A much better correlation for the deflection of the middle surface is obtained for the rectangular than for the circular ring plate. The deflection at the inner edge of the ring plate obtained by the theory of this paper is over twice that obtained in the classical solution of the same problem. In the previously mentioned set of nine fundamental equations, we have the stress resultants n^αβ and the deflections v^α. With appropriate boundary conditions, these equations could lead to a study of in-plane forces and buckling of variable thickness plates, a field in which not much progress has been made. This paper does not include any numerical work in this direction. It is felt, however, that one of the principal contributions of this paper to the literature is that the set of nine fundamental equations includes the stress resultants in n^αβ thus enabling us to study the effect of in-plane forces as well as that of transverse shear deformation, normal stress, and surface tractions. / Ph. D.

The Dugdale model is applied to an orthotropic plate. Stresses along the crack line and displacements along the crack and elastic plastic interface were found. The effect of orthotropy on several isotropic properties was found to be a multiplicative factor which is a function of the state of orthotropy. The yield stress is assumed to follow a von Mises' yield criterion which was adopted to the orthotropic case. A limit on the severity of orthotropy for a given external load was found as well as a limit on the external load for a given state of orthotropy in order that the material would still follow the Dugdale hypothesis. Finally, as long as the material satisfies the above mentioned limits, the plastic zone size was shown to be unaffected by orthotropy. / Master of Science