The Green’s Function Method for the Supported Plate Boundary Value Problem

Abstract

The deflection $u$ of a thin elastic plate is governed by the biharmonic equation ${\Delta }^{2}u=0$, where $\Delta$ is the two-dimensional Laplace operator. The problem of solving this equation in the domain $D$ occupied by the plate when $u$ and $\Delta u$ are assigned on the boundary $\delta D$ is often called the supported plate boundaly value problem. Strictly speaking this terminology is not correct since $\Delta u$ should be replaced by the more complicated expression for the plate’s moment $M(u)$ on $\delta D$; however, when $\delta D$ consists only of rectilinear segments (or when the Poisson ratio is unity) $M(u)$ reduces to $\Delta u$. Here, the supported plate problem is solved by a Green’s function method, closed form solutions are obtained for the disk and the half-plane, and the supported plate Green’s functions for these domains are computed explicitly. As a check, the solutions of these boundary value problems are also derived using a modification of the Goursat-Almansi method.