Abstract 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.

中文翻译：

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侧裂矩形薄板的辛叠加法新解析屈曲解

摘要 探索裂纹板的新解析屈曲解对于提供基准结果和实施基于显式参数分析和优化的初步结构设计非常重要。然而，控制高阶偏微分方程以及裂缝上的几何不连续性本质上使问题复杂化，使得寻找解析解更加困难。本文首次尝试将最新的辛叠加方法扩展到侧裂矩形薄板的线性屈曲。将问题引入哈密顿系统，然后将侧裂板分成几个子板，通过辛叠加方法解析求解，其中辛特征值问题被公式化，其次是辛本征展开。在每个子板的边缘上施加了一些详细的机械量，多组常数由裂纹板的实际边界条件、沿裂纹的自由边缘条件和子板之间的界面连续性条件确定。侧裂板的最终解析解是通过对子板解的积分获得的。综合基准​​结果被制表和绘制，用于典型板的屈曲载荷和模式形状，从简支边、夹紧边或自由边产生侧裂纹。裂纹长度效应也通过获得的解析解来研究。由于没有预先确定解形式的严格数学推导，