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 引用本文: 熊斯浚, 郑新然, 梁立, 周超, 赵岩, 李锐. 无拉力弹性地基上矩形板屈曲/后屈曲问题的辛求解方法. 力学学报, 2024, 56(2): 387-398.
Xiong Sijun, Zheng Xinran, Liang Li, Zhou Chao, Zhao Yan, Li Rui. The symplectic method for the buckling/post-buckling problems of rectangular plates on a tensionless elastic foundation. Chinese Journal of Theoretical and Applied Mechanics, 2024, 56(2): 387-398.
 Citation: Xiong Sijun, Zheng Xinran, Liang Li, Zhou Chao, Zhao Yan, Li Rui. The symplectic method for the buckling/post-buckling problems of rectangular plates on a tensionless elastic foundation. Chinese Journal of Theoretical and Applied Mechanics, 2024, 56(2): 387-398.

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

• 摘要: 无拉力弹性地基上矩形薄板的屈曲/后屈曲问题是板壳力学中一类重要课题, 在工程中有着大量应用. 因涉及接触非线性, 目前主要采用数值方法对该类问题进行求解, 发展具有重要基准价值的解析方法是当前面临的一项挑战. 针对上述问题, 本文将板划分为若干包含强制边界条件的板, 形成子问题, 在辛空间下利用分离变量与辛本征展开对子问题进行解析求解, 通过子问题边界处的连续条件确定板与地基的接触状态; 通过迭代求解上述过程, 获得子问题划分的收敛结果, 并得到最终屈曲载荷及模态. 结果表明, 无拉力弹性地基与Winkler地基上板的屈曲行为存在显著差异, 且无拉力弹性地基的刚度对板的屈曲载荷与屈曲模态均有重要影响. 在此基础上, 结合Koiter摄动法与辛方法, 对无拉力弹性地基上矩形板的后屈曲问题进行求解, 获得板的后屈曲平衡路径. 所得到的屈曲与后屈曲分析结果均与有限元计算结果吻合良好, 确认了本文结果的正确性. 由于本文方法数学推导严格, 求解效率高, 因此不仅为研究无拉力弹性地基上矩形薄板的屈曲/后屈曲行为提供了一种有价值的理论工具, 更有望拓展至更多复杂板壳力学问题的求解.

Abstract: 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.

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