Investigation Of Analytical Solution Of Elastic Plane Beams Under Thermo-Mechanical Coupling

There are many problems on the area of Structural engineering, Geotechnical engineering, and Solid mechanics that can be reduced to the plane stress and strain of beam under thermo mechanical coupling, and the research on this subject has been active with the development of the technology. The problem was solved with the traditional method under Lagrange system, which involved higher orders of partial differential equations and faced the difficulty of handing the boundary conditions. So it is necessary to discuss a new effective way to solve the problem.The symplectic method has been successfully applied in the beam and plate elastic mechanics problem as a new system theory solution, which has specific advantage compared with the classic methods. So it is introduced to the isotropic, orthotropic and anisotropic plane elasticity problem under the thermo mechanical coupling in the present paper. With Hamiltonian variational principle and Legendre’s transformation, Hamiltonian canonical equations are obtained, together with side boundary conditions. Because of the influence of the thermo mechanical coupling reflected in inhomogeneous dual equations and the conditions, a special way is introduced to transform side loadings into inhomogeneous term of dual equations. So the original problem can be converted into determination of eigensolution with the help of separation of variables and symplectic expansion. The boundary conditions of three typical conditions such as simply supported, clamped and free are analyzed.The numerical examples give the results about the different material parameters, different boundary conditions and different depth-to-length ratio, which were compared with analytical solution and numerical solutions. From the results, we can know that the symplectic method has two obvious advantages:firstly, the analytic solutions have better convergence and precision. Secondly, the symplectic method is a rational method, and it not only can be applied to two opposite simply supported edges of isotropic beam, the edge clamped beam, free, and other boundary conditions, but also can be used to orthogonal anisotropy and anisotropic problem. The research will enlarge the scope of analytic solution. The work of this paper further extends the application field of Hamilton method, and it also will provide a theoretical basis of plane thermo elastic coupling problem.