THE SYMPLECTIC METHOD FOR THE BUCKLING/POST-BUCKLING PROBLEMS OF RECTANGULAR PLATES ON A TENSIONLESS ELASTIC FOUNDATION

The buckling/post-buckling problems of rectangular thin plates on a tensionless elastic foundation constitute an important class of topics in mechanics of plates and shells, with extensive applications in engineering. Due to involving contact nonlinearity, this kind of problems have been primarily solved using numerical methods, while the development of analytical methods with significant benchmark value is currently a challenge. To address the aforementioned issue, a plate is divided into several subproblems in this paper, each containing enforced boundary conditions. The subproblems are solved analytically using the separation of variables and the symplectic eigen expansion in the symplectic space. The contact state between the plate and the foundation is determined by the continuity conditions at the boundaries of the subproblems. By iteratively solving the above process, the convergent division of the subproblems is obtained, along with the buckling load and buckling mode shape of the plate. The results indicate that there are significant differences in the buckling behavior between a plate on a tensionless elastic foundation and that on a Winkler foundation. The stiffness of the tensionless elastic foundation has a significant influence on both the buckling loads and buckling mode shapes. Based on this, the post-buckling problem of a rectangular plate on a tensionless elastic foundation is solved by combining the Koiter perturbation method with the symplectic method, yielding the post-buckling equilibrium path of the plate. The obtained buckling and post-buckling results both agree well with those by the finite element method, which confirms the correctness of the present results. Due to the rigorous mathematical derivation and high computational efficiency of the method proposed in this paper, it not only provides a valuable theoretical tool for the study of buckling/post-buckling behaviors of rectangular thin plates on a tensionless elastic foundation, but also can be extended to solve more complex mechanical problems of plates and shells.