Winkler地基上四边自由正交各向异性矩形薄板弯曲问题的辛叠加解

研究Winkler地基上正交各向异性矩形薄板弯曲方程所对应的Hamilton正则方程, 计算出其对边滑支条件下相应Hamilton算子的本征值和本征函数系, 证明该本征函数系的辛正交性以及在Cauchy主值意义下的完备性, 进而给出对边滑支边界条件下Hamilton正则方程的通解, 之后利用辛叠加方法求出Winkler地基上四边自由正交各向异性矩形薄板弯曲问题的解析解. 最后通过两个具体算例验证了所得解析解的正确性.

Symplectic Superposition Solutions of a Free Orthotropic Rectangular Thin Plates Bending Problem on Winkler Foundation

The orthotropic plates have widely applications in engineering problems. But due to the complexity of the orthotropic plate problems, it is generally difficult to obtain their analytical solutions, especially for fully free orthotropic rectangular plates on elastic foundations. At present, there are few methods to study this kind of problems. However, all these methods belong to traditional inverse or semi-inverse methods, with which it is rather difficult to seek a trial function satisfying corresponding boundary conditions. This shortcoming limits the scope of the application of the semi-inverse methods. Recently, a novel symplectic superposition method for elasticity has been rapidly developed in isotropic plate problems. Unlike the semi-inverse methods with predetermined trial functions, the symplectic superposition method is rigorously rational without any guess functions. However, because of the complexity of fully free orthotropic rectangular plates on elastic foundations, this kind of problems has not been solved by the symplectic superposition method. Based on the above, in this paper, the analytical bending solution of a fully free orthotropic rectangular thin plate, subjected to a concentrated load, resting on Winkler foundation is studied by the Symplectic superposition method. Firstly, the original bending equation is rewritten as a Hamiltonian canonical equations based on the known results. And the eigenvalues and eigenfunctions of the Hamiltonian operator matrix for the plate problem with two opposite edges slidingly supported are calculated. Then it is proved that the eigenfunctions are symplectic orthogonal and complete in the sense of Cauchy's principal value. Based on the completeness of the eigenfunctions, the general solution of the orthotropic rectangular thin plate with two opposite edges slidingly supported is derived. Secondly, the bending problem of fully free orthotropic rectangular thin plate, subjected to a concentrated load, resting on Winkler foundation is solved by superposing three sub-problems, which are all the bending problems of the orthotropic rectangular thin plates with two opposite edges slidingly supported. Finally, the deflection values of an isotropic rectangular thin plate and an orthotropic rectangular thin plate at some specific points are calculated by the obtained symplectic superposition solution respectively. Then we find that the numerical results obtained in this paper are in excellent agreement with the numerical results in the existing literature. Here, we only consider a concentrated load problem for the orthotropic rectangular thin plate, but the analytical solutions of the orthotropic rectangular thin plates under arbitrary loads can be calculated by using symplectic superposition method. Furthermore, the method can also be used to solve the problems of bending and vibration of orthotropic rectangular thin plates under more boundary conditions.