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The theory proposed in this thesis is an attempt to bridge the gap that exists between the linear and nonlinear theories of Elasticity. The theory is applied to the solution of a hubbed, clamped, circular plate made of polyurethane whose hub is subjected to a "large" axi~symmetrioc twist. This particular problem is attacked in the conventional manner of the generalized plane stress problem of linear elasticity. However, the strain displacement relations are formulated in the Eulerian manner and the displacement gradients are not assumed to be small. In addition, a more general stress~strain relationship than the conventional Hookean form is assumed. The solution is checked by experiment; and in addition, three auxiliary problems; the uniaxial compression problem, the uniaxial tension problem, and a shear problem are checked experimentally to further cheek the validity of the proposed theory when applied to the finite strain response of polyurethane. / Ph. D.

From the results of the cases discussed, it can be concluded that the neglect of transverse-shear deformation and normal stress results in expressions for bending moments and deflection which may be seriously in error. / / Master of Science

Includes bibliographical references. / This investigation, primarily experimental, examines the failure of circular plates subjected to impulsive velocities. The experiments are conducted on fully clamped circular steel plates subjected to a uniformly distributed impulse. The strain-rate-sensitive mild steel plates fail with mode I (large ductile deformation), mode II (tensile-tearing and deformation) and mode III (transverse-shear) failure modes. The impulse is measured by means of a ballistic pendulum upon which the test plates are attached. During mode II and mode III failure the complete circumferential tearing of the test plate produces a circular disc. The velocity of this disc is recorded. An energy analysis is performed on the test results and an energy balance equation is formulated. Einput = Edeformation + Etearing + Edisc. The input and disc energies are obtained from the experimental measurements and the deformation energy is predicted by using the final deformed height and a shape function together with a rigid-plastic energy analysis adopted by Duffey. Etearing refers to the energy for tensile-tearing in mode II failure or the energy for transverse-shear in mode III failure. Good correlation is found and the experiments show good repeatability. The threshold velocities for the onset of failure modes II and III are given.

Problems which require a study of the static and dynamic response of plates can be approached by first considering the plate to be a portion of an infinite plate, the prescribed boundary conditions being temporarily ignored. Once the plate's boundary has been defined in the infinite plate, a numerical solution is initiated by dividing this boundary into N segments of arbitrary length. For the static case the desired loading can then be applied to the infinite plate, and its effect on the deflection and stresses at the midpoint of the N boundary segments computed. To satisfy the boundary conditions of elementary plate theory, a concentrated force and moment are applied at the midpoint of each boundary segment. The magnitudes of these N equivalent forces and moments are determined by specifying that their combined effects, together with the applied loading, satisfy the boundary conditions at the N boundary points. This yields a set of 2N simultaneous equations whose solution constitutes the solution to the problem. A similar approach can be utilized for the vibrating plate. For the dynamic case the applied loading is assumed as zero, and a harmonically varying force and moment placed at the midpoint of each of the N boundary segments. The magnitudes of the N harmonically varying forces and moments are determined by specifying that their combined effects satisfy the boundary conditions at the N boundary points. This, coupled with the assumption of homogeneous boundary conditions, yields a set of 2N homogeneous equations. The frequency equation follows by setting the determinant of the coefficients equal to zero. The above approach to the solution of boundary value problems is formally known as the Reflection Method. Application of the Reflection Method to the static plate was previously accomplished by placing the equivalent forces and moments in the infinite plate at a finite distance from the midpoint of each boundary segment. This finite distance was called a retracted distance, and the curve along which the equivalent forces and moments were applied, a retracted boundary. In this investigation, the magnitude of the retracted distance was found to influence the condition of the coefficient matrix, while the solution remained relatively independent. The static response of plates by the Reflection Method as presented here applies the equivalent forces and moments directly to the boundary of the plate. This was found to impressively improve the condition of the coefficient matrix and reduce the number of significant figures necessary to obtain a numerical solution. With no increase in the number of boundary points, results were obtained comparable to those utilizing a retracted distance. The equations enabling the forces and moments to be applied directly to the boundary are developed and several examples presented. Application of the Reflection Method to the problem of determining natural frequencies is first illustrated for beams and then extended to plates. In each case the necessary equations are developed and sample problems presented. / 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.

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

In the American Institute of Steel Construction (AISC) LRFD specification, the shear lag coefficient for plate members is dependent on only the ratio of the welding length to the plate width. To determine how this ratio influences the shear lag coefficient and whether there are other factors that should be considered, a finite element investigation was conducted. Experimental data from a previous study conducted at Virginia Tech was used for comparison with the analytical model. Upon completion of the verification study, several parameters were investigated to determine their influence on the shear lag coefficient. These included the effects of weld length, eccentric load and imperfection created by the welding process. Shear lag coefficients were determined from the finite element analyses and compared with the AISC specification values. / Master of Science

The failure behavior of composite laminates is modeled numerically using the Generalized Layerwise Plate Theory (GLPT) of Reddy and a progressive failure algorithm. The Layerwise Theory of Reddy assumes a piecewise continuous displacement field through the thickness of the laminate and therefore has the ability to capture the interlaminar stress fields near the free edges and cut outs more accurately. The progressive failure algorithm is based on the assumption that the material behaves like a stable progressively fracturing solid. A three-dimensional stiffness reduction scheme is developed and implemented to study progressive failures in composite laminates. The effect of various parameters such as out-of-plane material properties, boundary conditions, and stiffness reduction methods on the failure stresses and strains of a quasi-isotropic composite laminate with free edges subjected to tensile loading is studied. The ultimate stresses and strains predicted by the Generalized Layerwise Plate Theory (GLPT) and the more widely used First Order Shear Deformation Theory (FSDT) are compared with experimental results. The predictions of the GLPT are found to be in good agreement with the experimental results both qualitatively and quantitatively, while the predictions of FSDT are found to be different from experimental results both qualitatively and quantitatively. The predictive ability of various phenomenological failure criteria is evaluated with reference to the experimental results available in the literature. The effect of geometry of the test specimen and the displacement boundary conditions at the grips on the ultimate stresses and strains of a composite laminate under compressive loading is studied. The ultimate stresses and strains are found to be quite sensitive to the geometry of the test specimen and the displacement boundary conditions at the grips. The degree of sensitivity is observed to depend strongly on the lamination sequence. The predictions of the progressive failure algorithm are in agreement with the experimental trends. Finally, the effect of geometric nonlinearity on the first-ply and ultimate failure loads of a composite laminate subjected to bending load is studied. The geometric nonlinearity is taken in to account in the von Kármán sense. It is demonstrated that the nonlinear failure loads are quite different from the linear failure loads, depending on the lamination sequence, boundary conditions, and span-to-depth ratio of the test specimen. Further, it is shown that the First order Shear Deformation Theory (FSDT) and the Generalized Layerwise Plate Theory (GLPT) predict qualitatively different results. / Ph. D.