ON THE ANALOGUES RELATING FLEXURE AND EXTENSION OF FLAT PLATES

Abstract

The displacement of a flat plate bent by transverse loading, and the extensional (Airy) stress-function in ‘plane strain’ or in ‘plane stress’, are governed by equations of identical form; and the boundary conditions have identical form when edge-displacements are specified in the flexural, edge-tractions in the extensional problem, so mathematically, in these circumstances, only a single problem is presented. This, the ‘first analogue’ relating flexure and extension, is well known. A ‘second analogue’, relating the flexural problem when edge-tractions with the extensional problem when edge-displacements are specified, is believed to have been first propounded in 1941. By introducing two quantities U and V, analogous with the components μ and ν of extensional displacement, it permits a treatment of the flexural problem by any method—e.g. ‘two-diagram (relaxational) technique’— which yields extensional solutions of this second type. In this paper both analogues are combined in an inclusive statement covering the perforated (multiply connected) plates which were discussed in 1948. Reasons are stated for believing that ‘two-diagram technique’ is preferable in problems governed by ‘mixed’ boundary conditions.