Hamiltonian Solution Approach For The Problems Of Rectangular Plates

As the important structural elements, elastic rectangular plates are widely used in various fields such as civil engineering, mechanical engineering, ocean engineering, aeronautics and astronautics. Solution of the plate problems (bending, vibration, etc.) have been one of the important research topics in engineering. However, it is hard to obtain the analytical solutions to most of these problems till now due to the mathematical challenge. In this dissertation, the bending and vibration problems of rectangular plates, including those based on the Kirchhoff theory (thin plate theory) and Reissner theory (moderately thick plate theory), are respectively introduced into the Hamiltonian system. Accordingly, the symplectic geometry is applied to solve the problems of rectangular plates with typical boundary conditions, some of which are known as the difficulties in elasticity.For the bending of thin plates, the Hamiltonian system is constructed from the governing equations, with the basic mechanical quantities as the symplectic variables. Then the symplectic approach is used to solve the canonical equation rationally. For the rectangular thin plates with two opposite edges simply supported, the Levy-type analytical solutions are derived; for those with two opposite edges clamped, the symplectic analytical solutions are obtained by determine the coefficients in the resultant series via the variational equations. In addition, the Hamiltonian system based solution method is extended to the bending of orthotropic plates and the sympletic analytical solutions of plates with two opposite edges simply supported or clamped are obtained. For the rectangular thin plates with complex boundary conditions such as cantilever plates, fully free plates and plates with two adjacent edges free, a symplectic superposition method is proposed, which is applicable to rectangular plates with any combinations of commonly used boundary conditions.For the bending of moderately thick plates, a concise form of the Hamiltonian system is constructed from the governing equations. Then the symplectic approach is used to solve the bending of moderately thick rectangular plates with two opposite edges simply supported as well as to explain the boundary effects in plate bending. Compared with the canonical equation based on basic mechanical quantities, the equation derived in this dissertation is simple in form and thus easy to solve.Free vibration problems of the thin and moderately thick plates are introduced into the Hamiltonian system and the plates with two opposite edges simply supported are rationally solved. The solution approach presented in this dissertation starts from the governing equations of plates. In the Hamiltonian system, using the symplectic geometry, the method of separation of variables as well as symplectic eigen expansion is adopted to obtain the analytical solutions of bending and vibration of thin and moderately thick rectangular plates. It is noted that the analytical solution procedure in this dissertation is completely rational without any predetermined trial functions such as the deflections, therefore, the solution methodology prevails over the conventional ones, as represented by the semi-inverse method. As a result, we conclude that the Hamiltonian system based solution approach enables one to obtain more analytical solutions which have not been obtained by other existing methods.