Thesis (M.S.)--Rochester Institute of Technology, 1991. / Typescript. References: leaves 139-147.

Thesis (doctoral)--University of Hong Kong, 1982.

Thesis--Bonn. Extra t.p. with thesis statement inserted. / Includes bibliographical references (69-72).

Thesis (Ph. D.)--University of Wisconsin--Madison, 1965. / Typescript. Vita. Includes bibliographical references.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1970. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 144-148).

Thesis--University of Florida. / Description based on print version record. Typescript. Vita. Bibliography: leaves 126-130.

Various models are currently in existence for determining the deflection of lithospheric plates under an applied transverse load. The most popular models treat lithospheric plates as thin elastic or thin viscoelastic plates. The equations governing the deflection of such plates have been solved successfully in two dimensions using integral transform techniques. Three dimensional models have been solved using Fourier Series expansions assuming a sinusoidal variation for the load and deflection. In the engineering context, the finite element technique has also been employed. The current aim, however, is to develop an efficient solver for the three dimensional elastic and viscoelastic problems using finite difference techniques. A variety of loading functions may therefore be considered with minimum work involved in obtaining a solution for different forcing functions once the main program has been developed. The proposed method would therefore provide a valuable technique for assessing new models for the loading of lithospheric plates as well as a useful educational tool for use in geophysics laboratories. The multigrid method, which has proved to be a fast, efficient solver for elliptic partial differential equations, is examined as the basis for a solver of both the elastic and viscoelastic problems. The viscoelastic problem, being explicitly time-dependent, is the more challenging of the two and will receive particular attention. Multigrid proves to be a very effective method applicable to the solution of both the elastic and viscoelastic problems. / Science, Faculty of / Mathematics, Department of / Graduate

A knowledge of the natural frequencies of a vibrating plate is of great importance if an effective design is to be made which will prevent critical conditions of heavy vibration from occurring. Those frequencies which are associated with the symmetric modes are especially important. Many approximate methods have been devised to determine these natural frequencies. In this dissertation a method of frequency determination is suggested through an integral equation approach. The plate vibration problem is formulated as a problem in the solution of a homogeneous, linear Fredholm integral equation of the second kind in which the kernel is either symmetric or can be made so by a convenient transformation. The integral equation, as formulated, satisfies the boundary conditions in that it includes Green's function of the plate which is a solution of the isolated force problem. Three approximate methods for solving the integral equation are described mathematically and then applied to three elementary examples. The three methods used ares 1) method of successive approximation, 2) method of collocation and 3) the trace or the kernel. It is shown that using the trace of the kernel always gives a lower bound to the frequency and is particularly useful for the determination of the fundamental frequency. After solving the three elementary problems the integral equation approach is made to the uniform circular cantilever plate where the frequency is approximated both by collocation and by the use of the trace of the kernel. The first and second approximate mode shapes are then derived and shown graphically. The results are seen to compare favorably with results obtained from the Rayleigh-Ritz method. Finally, the fundamental frequency is determined for the circular, stepped cantilever plate and the clamped elliptical plate. For the stepped plate fundamental frequency curves are drawn for various positions and magnitudes of the step. The fundamental frequency curve of the clamped elliptical plate is drawn as a function of the eccentricity of the ellipse. A frequency obtained from experiment is reported along with a calculated value determined from the Rayleigh-Ritz method. It is seen that the integral equation approach is about 19% below the experimental value whereas the Rayleigh-Ritz method gives a fundamental frequency about 27% above the experimental value. / Ph. D.

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.