Symplectic system based analytical solution for bending of rectangular plates on Winkler foundation

Abstract

Based on an appropriate definition of symplectic inner product, we establish a symplectic space formed by generalised displacements and their dual variables for the bending of rectangular plates on Winkler foundation. By using the Hellinger–Reissner variational principle, dual equations in the symplectic space are obtained and its operator matrix proved to be a Hamilton operator matrix. So the original plate bending problem is converted into the symplectic system. By using separation of variables and symplectic eigenfunction expansion, a novel analytical method is presented for solving the aforementioned plate bending problem. The symplectic eigenvalue problems for rectangular plates with two opposite sides simply supported and the other two opposite sides clamped are discussed. The transcendental equation of its eigenvalues and symplectic eigenvectors is derived in analytical forms. Analytical solutions of two examples are presented by using this method. The solution for a simply supported plate under un...