Symplectic System Based Analytic Solutions For Reissner Plate Bending

The traditional method of elasticity is mainly the semi-inverse method,which is always confined to the solution of higher-order partial differential equations by eliminating the various unknown variables.Hence,the effective methods in mathematical physics such as variable separation and expansion of eigenfunctions become inapplicable.In contrast to the semi-inverse approach based on the Euclidean space with one kind of variables,the symplectic dual methodology is consist of two kinds of variables based on the symplectic space.It forms a rational solution method by using the symplectic eigenfunction-vector expansion and the method of separation of variables,and expands the scope of analytic solutions.Based the Hellinger-Reissner variational principle for the Reissner plate bending,presented problem can be derived to the symplectic dual system,and the dual equations can be presented.Therefore this forms the rational solution method for the Reissner plate bending problem by the schemes of effective methods of separation of variables and the symplectic eigenfunction-vector expansion.In symplectic eigenvalue problem for Reissner plate bending with two opposite sides free,the zero eigenvalue is a very special one,which eigensolutions contain particular significance in elasticity and are basic solutions in the Saint-Venant principle.On the contrary,the eigensolutions for nonzero-eigenvalues are one with local effect,and decay drastically with respect to distance and are covered by the Saint-Venant principle.However,these eigensolutions for nonzero-eigenvalues must be considered for the plate with small ratio of length-width or other boundary conditions.Substitution the general solution of non-zero eigensolutions for two opposite boundary conditions can lead to the transcendental equation on non-zero eigenvalues and analytic expressions of their eigensolutions.From the eigenvalues and eigensolutions obtained and based on the adjoint symplectic orthogonality property,the analytic solution can be established by using the expansion theorem.Based the symplectic system for Reissner plate bending,Reissner plate bending problems with various boundary conditions are discussed in this paper,and more analytical solutions for Reissner bending plate are provided.Firstly,the analytical solutions of bending plate with two opposite sides clamped and the other first simply supported and one with three clamped and the other first simply supported are presented by applying the established symplectic eigensolutions for Reissner plate with two opposite sides clamped.Subsequently, eigensolutions for nonzero-eigenvalue for Reissner plate with two opposite sides free and second simply supported are discussed,so the general solutions in symplectic expansional eigen-solution are formed.Lastly,the analytical solutions of bending plate with two opposite sides free and the other clamped,fully second simply supported plate,and plate with two opposite sides second simply supported and the other free,are presented in detail.Comparing results with the literature show that,the new method has higher precision and convergence rate,and is a very effective analytic solution.