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.

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.

Because of its structural properties, simplicity of construction, economy, and the wide number of shapes that can be obtained from combinations of parabolas of different curvatures and from sections with principal axes rotated or translated, the Hyperbolic Paraboloid has become a popular form for roof shells. In this thesis a comprehensive method was presented for the analysis of a hyperbolic paraboloid surface with curved edges. Its reliability is verified by comparing the analytical and experimental results of displacements of a thin shell plaster model without edge beams and supported along two opposite edges. The model tested was 36" X 36" X 1/8" thick. The analysis is based on the Shallow Shell Theory and represents a combination of known techniques. In general good agreement was found between the analytical computations and the experimental measurements of displacements. / Master of Science

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

A model consisting of a circular plate fastened to the top of a circular cylinder was placed in a wind tunnel and the pressure caused by a wind load were read over the exposed surfaces. The pressure on the plate were approximated by four terms of a Fourier cosine series. By use of the governing equation or small deflection or plates, Fourier cosine series of four terms are developed for deflections of plates with a fixed boundary and with a simple support condition. Moments are developed from the deflections and results for both are presented in equation and tabular form. The results are left in a form that can be applied to any size thin circular plate under the action of a tangential wind load or the same flow regime as the one considered. / Master of Science

Thesis. 1978. M.S.--Massachusetts Institute of Technology. Dept. of Aeronautics and Astronautics. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND AERONAUTICS. / Includes bibliographical references. / M.S.

Thesis. 1978. Ph.D.--Massachusetts Institute of Technology. Dept. of Aeronautics and Astronautics. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND AERONAUTICS. / Vita. / Includes bibliographical references. / Ph.D.

Thesis. 1976. Ph.D.--Massachusetts Institute of Technology. Dept. of Aeronautics and Astronautics. / Microfiche copy available in Archives and Aero. / Vita. / Bibliography: p.194-204. / Ph.D.

Whereas analytic and experimental investigations of plates subject to lateral vibrations have been rather thorough, the present study is an analytic determination of the various critical frequencies of vertically cantilevered thin elastic rectangular plates vibrating freely within their own planes. Within the restrictions imposed by excluding any motion perpendicular to the face of the plate, the upright edges are free to move in the other two directions, as is the top horizontal edge. Three different base conditions are imposed: • A clamped lower edge; • A lower edge which is freely vibrating transversely in the plane of the wall where the vertical fibers of the wall are fixed at their roots; and • A horizontally freely pulsating lower edge where the vertical fibers of the wall are fixed at their roots. The first two conditions are considered in relation to plate vibrations which are essentially vertical while the first and third conditions are each employed with essentially horizontal plate vibrations. In every case the effect of a uniform load placed along the upper edge is studied. Critical frequencies and associated amplitude coefficients are obtained for various ratios of base length to wall height. The solution, which is presented in tabular and graphic forms, is obtained by using the method of iteration on the Rayleigh-Ritz energy procedure. It is concluded that, for a wall with a clamped base vibrating in accordance with the given stipulations, the fundamental period is proportional to the square root of the face area of the wall. When the base of the wall is vibrating there is only one critical period, and it varies with the height of the wall. The factor of proportionality should take into account the material of which the wall is composed. For designing unframed walls, subjected to dynamic loads in their plane, where the applied shear is to be taken as some constant times the dead load at the base of the wall, the recommended lateral force requirements of the Seismology Committee of the Structural Engineers Association of California, as set forth in 1959, seem adequate as modified above. / Ph. D.