New analytic buckling solutions of side-cracked rectangular thin plates by the symplectic superposition method
New analytic buckling solutions of side-cracked rectangular thin plates by the symplectic superposition method
复制标题
DOI：
10.1016/j.ijmecsci.2020.106051
复制
发表时间：
2021-02-01
影响因子：
7.3
通讯作者：
Li, Rui
中科院分区：
文献类型：
--
作者：
Hu, Zhaoyang;Zheng, Xinran;Li, Rui
探索新的裂纹板屈曲解析解对于提供基准结果和基于显式参数分析和优化的初步结构设计具有重要意义。然而，高阶偏微分方程以及裂纹的几何不连续性本质上使问题复杂化，使得寻求解析解更加困难。本文首次尝试将一种新的辛叠加法推广到含侧裂纹矩形薄板的线性屈曲问题。将问题引入到哈密顿体系中，然后将含侧裂纹的板划分为若干个子板，用辛叠加法求解子板的解析解，在子板中建立了辛本征值问题，并进行了辛本征展开。在每个子板的边缘上施加了一些详细的力学量，其中多组常数由裂纹板的实际边界条件、沿着裂纹的自由边缘条件以及子板之间的界面连续性条件确定。通过子板解的积分，得到了边裂纹板的最终解析解。全面的基准结果制成表格，并绘制了典型板的屈曲载荷和模式的形状与侧裂纹从一个简单的支持的边缘，一个固定的边缘，或一个自由的边缘。裂纹长度效应也研究了所获得的解析解。由于该方法不需要预先确定解的形式，而是严格的数学推导，因此为寻求更多的解析解提供了一种合理的途径。
Exploring new analytic buckling solutions of cracked plates is of much importance for providing benchmark results and implementing preliminary structural designs based on explicit parametric analysis and optimization. However, the governing higher-order partial differential equations as well as the geometric discontinuity across a crack essentially complicate the problems, make it more intractable to seeking analytic solutions. This paper presents a first attempt to extend an up-to-date symplectic superposition method to linear buckling of side-cracked rectangular thin plates. The problems are introduced into the Hamiltonian system, and a side-cracked plate is then divided into several sub-plates that are analytically solved by the symplectic superposition method, where the symplectic eigenvalue problems are formulated, followed by the symplectic eigen expansion. Some elaborated mechanical quantities are imposed on the edges of each sub-plate, with multiple sets of constants determined by the actual boundary conditions of the cracked plate, free edge conditions along the crack, and interfacial continuity conditions among the sub-plates. The final analytic solution of a side-cracked plate is obtained by integration of the solutions of the sub-plates. Comprehensive benchmark results are tabulated and plotted for buckling loads and mode shapes of typical plates with a side crack from a simply supported edge, a clamped edge, or a free edge. The crack length effect is also investigated by the analytic solutions obtained. Due to the rigorous mathematical derivations without predetermination of solution forms, the present method provides a rational approach to exploring more analytic solutions.