THE SYMPLECTIC METHOD FOR THE BUCKLING/POST-BUCKLING PROBLEMS OF RECTANGULAR PLATES ON A TENSIONLESS ELASTIC FOUNDATION
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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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Key words:
- symplectic method/
- buckling/
- post-buckling/
- tensionless elastic foundation/
- plate
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图 1无拉力弹性地基板示意图
Figure 1.Schematic diagram of a plate on a tensionless elastic foundation
图 2无拉力弹性地基板屈曲问题求解流程图
Figure 2.Solution flowchart for the buckling of a rectangular plate on a tensionless elastic foundation
图 4屈曲模态迭代的收敛性验证(kw= 2000,a/b= 2)
Figure 4.Iterative convergence verification of buckling mode shapes (kw= 2000,a/b= 2)
图 5无拉力弹性地基上矩形板的临界屈曲模态(kw= 2000,a/b= 2)
Figure 5.Critical buckling mode shape of a rectangular plate on a tensionless elastic foundation (kw= 2000,a/b= 2)
图 6不同长宽比矩形板的临界屈曲模态
Figure 6.Critical buckling mode shape of rectangular plate with different aspect ratio
图 7无拉力弹性地基的力−位移曲线示意图
Figure 7.Schematic diagram of the force–displacement relation of a tensionless elastic foundation
图 8无缺陷和含缺陷简支板的后屈曲平衡路径 (续)
Figure 8.The post-buckling equilibrium paths of perfect and imperfect simply supported plates (continued)
表 1不同长宽比及弹性地基刚度下矩形板的临界屈曲载荷$ - {{{N_{{\text{cr}}}}{b^2}} \mathord{\left/ {\vphantom {{{N_{{\text{cr}}}}{b^2}} D}} \right. } D}$
Table 1.Critical buckling load factors, $ - {{{N_{{\text{cr}}}}{b^2}} \mathord{\left/ {\vphantom {{{N_{{\text{cr}}}}{b^2}} D}} \right. } D}$, of rectangular plates with different aspect ratios and elastic foundation stiffnesses
${k_w}$ Method a/b =1.5 a/b =2 a/b =2.5 a/b =3 1 FEM 44.228 39.502 44.046 44.105 present 44.258 39.529 44.075 44.027 10 FEM 43.038 39.923 42.683 42.265 present 43.067 39.949 42.711 42.292 100 FEM 43.687 42.172 42.119 41.436 present 43.716 42.199 42.146 41.462 200 FEM 43.038 42.998 40.998 39.770 present 43.068 43.026 41.023 39.794 2000 FEM 42.835 44.134 40.804 39.488 present 42.865 44.163 40.829 39.512 
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