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.

An exact solution is presented for the large deformation response of a simply supported orthotropic cylindrical panel subjected to a uniform line load along a cylinder generator. The cross section of the cylinder is circular and deformations up to the fully snapped through position are investigated. The orthotropic axes are parallel to the generator and circumferential directions. The governing equations are ·derived using laminated plate theory, nonlinear strain-displacement relations, and applying variational principles. The response is investigated for the case of a panel loaded exactly at midspan and for a panel with the load offset from midspan. The mathematical formulation is one-dimensional in the circumferential coordinate. Solutions are obtained in closed-form. An experimental apparatus was designed to load the panels. Experimental results of displacement controlled tests performed on graphite-epoxy curved panels are compared with analytic predictions. This study demonstrates that panel shallowness, material orthotropy, and stacking sequence can influence the nonlinear static response. Initial geometric imperfections, observed during testing, were found to influence the response of the panels. However, the overall correlation of analytic and experimental results were good. / M.S.

A method is developed for analyzing the flexural-torsional and lateral-torsional buckling ("tripping") behavior of flanged stiffeners subjected to axial force, end moment, lateral pressure and any combination of these. The effects of cross-sectional distortion, postbuckling behavior of the plate (incorporated by considering the plate effective width), initial imperfections and plasticity are included. The method uses the Rayleigh-Ritz approach. Based on an assumed strain distribution, a displacement field is obtained for the tripping model, and the total potential energy functional is then derived. The strain distribution assumptions coincide with van der Neut's assumption. However, unlike the somewhat obscure differential equation approach given by van der Neut, this study provides a simple, clear, energy approach. Also the resulting method is applicable in the inelastic range, which is not possible with van der Neut's approach. Both the rigid web case and the flexible web case are studied. The effect of plate rotational restraint in the elastic range is accounted for. The method requires only four degrees of freedom and therefore the solution process is rapid. In order to verify the method in the elastic range, a number of sample stiffened panels are analyzed using the ABAQUS foote element program; the results are in quite good agreement. An inelastic tripping model is then developed based on the established elastic model, using deformation theory. Results obtained using the inelastic tripping method are shown to be in good agreement with experimental results, and to be more accurate than other methods. / Ph. D.

Active control of sound radiation due to subsonic wave scattering from an infinite or a finite fluid-loaded plate excited below the critical frequency is analytically studied. The disturbance is caused by a flexural wave in an infinite plate, or by a point force on a finite plate at subsonic frequencies. The wave scattering is caused by discontinuities on the plate or by the boundary conditions. A feed-forward control approach is applied by implementing either point/line forces or piezoelectric actuators on the plate. The amplitude and phase of control forces are determined by the optimal solution of a cost function which minimizes the far-field radiated acoustic power over a prescribed surface in the half space of the fluid field. The results show that for subsonic excitations, high global reduction in radiated pressure is possible with properly located active control forces. The number and location of control forces employed in order to obtain high control performance are related to the excitation frequency. The far-field sound radiation directivity pattern, the modal amplitudes of the plate vibration, the plate vibration autospectrum in the wave number domain, and the near-field intensity distribution are extensively studied in order to uncover the mechanisms of control. / Ph. D.

The in-plane vibration of a plate with an elliptic hole is studied. It is shown that standing waves whose wavelength is the same order of magnitude as the size of the hole are theoretically capable of causing microcracks which have been observed experimentally. Several approaches were used including reduction of the mixed boundary value problem to a Fredholm equation, and to a matrix eigenvalue problem. Contour curves of various stresses and displacements were obtained numerically. A new technique was developed for the solution of the wave equation appropriate for boundary conditions on an elliptical surface. / Ph. D.

The problem of determining the deflection and stress in a plate under transverse loading can be approached by first considering the plate to be a portion of an infinite plate, ignoring the prescribed boundary conditions. The known loads are then applied to the infinite plate and their effects are calculated at those points which correspond to the boundary of the original plate. A system of suitably chosen loads and moments is then applied on the infinite plate at points beyond the boundary of the original plate such that the prescribed boundary conditions are satisfied. For an exact solution, the number of external loads and moments would have to be infinite. However, in order to deal with the problem numerically, only a finite number of each are considered. Thus, solutions are obtained by satisfying the boundary conditions at only a finite number of points. The method is illustrated for beams and then extended to plates. Several problems with known solutions are solved and the results compared with the exact values. Also, plots of the deflection and moment along the centerline of a cantilevered triangular plate are presented. Discussions of the problem of plates with holes and the effect on the solution of various placements balancing loads are also presented. An IBM 1620 digital computer is used to facilitate calculations / 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 general method of analysis is developed for the solution of single or continuous span folded plate surfaces, with general end conditions, for any load. By considering the folded plate surface as a single, longitudinal unit throughout the analysis, the continuity of the surface is maintained over transverse supports, Within the limitations of the assumptions, this method yields an "exact" solution of the structural action based upon a difference technique in which the surface is divided into segments: compatibility of the surface is insured at the center of each finite segment along the span and over the supports. A computer program for the analysis of simply supported, single-span and continuous-span folded plate surfaces, with and without overhangs at each end, under three load conditions is described. A comparison is made between this general method of analysis and presently accepted techniques; and a final solution for a continuous folded plate surface is presented. The results of an experimental investigation to study the behavior of four different folded plate structures and to verify the general method of analysis are discussed. / Ph. D.

Unlike simple columns, rectangular plates which are supported on all edges may carry considerable load beyond their buckling load. Under some conditions it may be advantageous to utilize this additional load-carrying capacity. Von Karman has presented the basic nonlinear differential equations for a plate element undergoing large deflections. In this dissertation the nonlinear equations of von Karman are converted into a s~t of linear equations by expanding the displacements into a prn,er series in terms of an arbitrary parameter. The first few equations of the set can be identified as the usual (linear) small deflection equations. Solution of these and then some of the succeeding equations permits a study of the behavior of the plate at buckling and then beyond into the large deflection range. At present it seems that only plates without initial eccentricities subject to in-plane loading may be solved by the present method. The advantage of the present method is the simplicity of solution. The elastic postbuckling behavior of simply supported rectangular plates subjected to longitudinal compression and subjected to a unIform temperature rise is investigated in detail by solving the first few of the equations. Results are presented for these problems in the form of equations and curves. Load-shortening curves for the compression problem and similar curves for one of the temperature problems solved indicate that changes in buckle pattern will occur. Because of the incompleteness and the inconsistencies of the treatment of the phenomenon of change in buckle pattern in the literature, a study of this phenomenon is made. In order to analyze change in buckle pattern in a rigorous fashion the postbuckling behavior of a symmetric three element column on a nonlinear elastic foundation is determined. It is indicated how the principles learned from the column analysis may be applied qualitatively to plate problems. The results for the plate in compression are compared to previous theoretical results and to experiment. For a square plate the present results agree with previous exact results. For an infinitely long plate the present thesis gives more accurate (lower) loads than previous results. Experimental results which have not been reported previously are described in this thesis, and results from these and other experiment are compared with the present theory. Comparisons are made for total shortening and local strains and deflections which indicate good agreement between experimental results and theoretical results. / Doctor of Philosophy