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

The deflection u of a thin elastic plate is governed by the biharmonic equation 2u = 0, where is the two-dimensional Laplace operator. The problem of solving this equation in the domain D Occupied by the plate when u and A u are assigned on the boundary D is often called the supported plate boundaly value problem. Strictly speaking this terminology is not correct since A u should be replaced by the more complicated expression for the plate's moment M(u) on aD; however, when D consists only of rectilinear segments (or when the Poisson ratio is unity) M(u) reduces to 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 GoursatAlmansi method.