Hamiltonian System In Anisotropic Plate Bending Problems

There is analogy relationship between plane elasticity and plate bending. Their governing equations are both bi-harrnonic equations. Strain, stress function- stress relationship and stress-strain relationship in plane elasticity correspond to bending moment, deflection-curvature relationship and bending moment- curvature relationship in plate bending respectively. Thus the Simplectic solution system can be applied in plate bending as in plane elasticity. Dual equations and Hamiltonian operator matrix can be derived from the H-R variational principle. After separation of variables it becomes the transverse eigenfunction vector problem of Hamiltonian matrix. There is adjoint Simplectic orthogonality relationship between the eigenfunction vectors, so any solution can be expanded by the eigenfunction vectors. The general form of non-zero eigenfunction vector can be written and be substituted into lateral boundary conditions to get the transcendental equation for non-zero eigenvalues. Thus we find the non-zero eigenvalues and eigenvectors. According to the adjoint Simplectic orthogonality relationship and the expansion law we can write down the representative which satisfies the in-domain equation and two lateral boundary conditions. After substituting it into two end boundary conditions and determining the constants in it we get the solution for the original problem. Several solutions are given in this paper and the results show that they reach a relatively high accuracy with only several eigenvalues used. The new method presents the analytical solutions in rectangular plate via the methods of separation of variables and eigenfunction- vector expansion; it breaks through the limitation of traditional semi-inverse solution. The results show that the new method has vast application vista.