The cantilever strip plate under torision, bending or flexure at infinity

Abstract

A homogeneous, isotropic plate occupies the region 0≤x1≤∞, |x2|≤a, |x3|≤h, where the ratio h/a is sufficiently small so that the classical theory of thin plate bending applies. The short end of the plate at x1=0 is clamped while the long sides are free. This cantilever plate is now loaded at x1=+∞ by an applied twisting moment, by a bending moment or by flexure. Despite the fundamental nature of these problems, and the long history of thin plate theory, no solutions are to be found in the existing literature that will determine (for instance) the important unknown resultants V1, M11 at the clamped end x1=0. The main reason for this is that this combination of boundary conditions leads to severe oscillating singularities of the field in the corners (0, ±a). The fact that such singularities must exist is widely known, but we present here for the first time a method of solution that takes these singularities fully into account.