Symplectic eigenfunction expansion theorem for elasticity of rectangular planes with two simply-supported opposite sides

The eigenvalue problem of the Hamiltonian operator associated with plane elasticity problems is investigated. The eigenfunctions of the operator are directly solved with mixed boundary conditions for the displacement and stress in a rectangular region. The completeness of the eigenfunctions is then proved, providing the feasibility of using separation of variables to solve the problems. A general solution is obtained with the symplectic eigenfunction expansion theorem.