Some mixed and associated boundary value problems in the theory of thin plates

Abstract

The bending of thin flat plates has occupied the interests of mechanicians and applied mathematicians since J. L. Lagrange discovered the differential equation characterizing the behavior of such structural members.

One particular phase of investigation in this field concerns itself with the solution of the differential equation subject to given boundary conditions. Indeed, it may be safely stated that the bulk of the literature on the subject of flat plates is concerned with the solution of problems involving the specification of the transverse loading on the plate and the conditions at the boundary of the plate. Various mathematical techniques are available for the solution of such problems. Among these, the most prominent are, a.) the method of series, b) the method of singularities, and c) the complex variable techniques.

A survey of the literature in this area has revealed a paucity of solutions of certain types of problems; notably, those problems in which boundary conditions are mixed a.long a portion of the edge of the plate which ha.s a continuously turning tangent. By mixed boundary conditions, we mean a. change in condition from prescription of bending moment and vertical shear to assignment of slope and deflection along a portion of the edge which has a continuously turning tangent.

In the first section of this thesis, a number of problems are considered for the half-plane. The attendant boundary conditions considered a.re combinations of clamping and simple support.

The second portion consists of a number of problems associated with the quarter-plane. Solutions for these problems are obtained by utilizing the method of images in conjunction with the solutions presented in the first section.

After this, we examine some problems connected with the circular plate. In particular, a numerical solution is given for a uniformly loaded circular plate simply-supported over half of its boundary and clamped over the remaining portion.

The last chapter is a brief discussion of plates in the form of rectangles. Here, a closed solution is presented for the bending moments in terms of Weierstrassian elliptic functions. Another numerical example is included for a uniformly loaded plate clamped over a portion of one edge and simply- supported over the remainder of its boundary.