An Analysis Method Of Symplectic Dual System For Beams /Plates Structures

There are many problems that can be reduced to beam and plate structural system.The research on this subject has been ongoing for many years. The traditional method solves this problem in Euclidean space under Lagrange system, which involves solving higher orders of partial differential equations and faces the difficulty of handing the boundary conditions.So it is necessary to investigate a new and efficient solving method. Symplectic dual system is introduced to theory of elasticity mechanics by Professor Zhong Wanxie.And then symplectic system of elasticity is established and developed so that many effective mathematics and physics methods such as separation of variables, conjugated symplectic orthogonal and sumplectic eigenfunction expansion method etc. are made to be implemented. So the symplectic solution system of elastic mechanics is introduced to the beam and plate structures in the paper.The main study works and conclusions are as follows:(1) The symplectic method was applied to analyze the bending of composite laminated beam. Based on the equations of elasticity, the analytic solutions were obtained for all kinds of span to depth ratio and boundary conditions, which avoid the lacking of imprecision effect on the structures coming from shearing deformation and transverse normal stress based on simplified theory. The parameters'influence on mechanical property are discussed, such as span to depth ratio, ply angle, Arrangement of plies and anisotropic degree. Some laws are derived.(2) For symmetric angle-ply laminated plates, the solutions are not limited to simple support boundary condition and geometrical shape, which are further applied to analyze the raft foundation.The results of cross-ply and angle-ply laminated graphite–epoxy composite plate are also shown. The parameters'influence on mechanical property are discussed, such as ply angle, Arrangement of plies and anisotropic degree.(3) The closed polynomial expressions for shear lag coefficient and effective width are given at the condition of Saint-Venant analytical solutions to full-span uniform load on flange slab of box girder. Availability of formula in this paper is demonstrated by comparing results with international norm and measured values.(4) The paper proposes the application of the symplectic elasticity solution system in solving of electromechanical problem exists in functionally graded piezoelectric material for the first time.After introducing the material nonhomogeneous along axial direction and constructing the new stress, the electromechanical problem are formulated in the form of symplectic system. The shift Hamiltonian matrix is raised and the conjugated symplectic orthogonal between the eigenvectors are re-established, which provide a new method for solving piezoelectric problems and promote the use of symplectic elasticity solution in solving intellectual material problems at the same time.(5) Based on the convergence being guaranteed by the completeness of symplectic expansion, the stress singularities problem of anisotropic beam basing on two-dimensional elasticity is investigated, which is reduced to solve the eigenvalue problem of the operator matrix. Especially, the nonzero eigen-solutions that have fading characteristic usually ignored by Saint Venant's principle can explain the local phenomenon. It is especially good in describing the localized effects and their reduction, which are seldom mentioned in former researches.(6) A new method of structural dynamic equation is put forward by introducing Pade series expansion.The algorithm is based on the coupled precise time integration and Newmark method,which avoid the deficiency of traditional method and overcome the difficulties for time integration reducing orders and maintain the fine precision.The dynamic response of prestressed Timoshenko beams fully and partially resting on a two-parameter elastic foundation under moving harmonic load is investaged.A beam element with shear deformation is formulated.The effectd of prestress,foundation support ,moving velocity and excitation frequency on dynamic characteristics of the beams are studied and described in detail.The research results show that the symplectic method fits the beam/plate problems well and have complete space of eigensolutions, which is simple, straight, effective and have more advantages than the traditional method, which make a breakthrough of higher orders of partial differential equations and the difficulty of handing the boundary conditions.Based on the results of herein, one may conclude that the symplectic method for beam/plate structure is usefule in engineering application and has good development and applied prospects for structural engineering.