对合变换和薄板弯曲问题的多变量变分原理

  目的  矩形薄板的屈曲研究具有重要的理论和实际意义。针对工程中常见的矩形薄板结构，为了研究其在任意弹性边界条件下受轴向压力的屈曲特性，给出一种基于系统最小势能原理计算弹性失稳时屈曲载荷的方法。  方法  首先，在板结构模型的四条边界上分别设置旋转约束弹簧和横向约束弹簧，并设定两类弹性弹簧的刚度值大小以模拟任意弹性边界条件。由于经典傅里叶级数形式的位移函数在边界上的导数可能存在不连续问题，因此引入辅助函数，并以三角级数形式建立位移函数的几何表达式。然后，建立矩形板系统的势能表达式，结合最小势能原理，对未知傅里叶系数求偏导建立线性方程组。最后，求解得到矩形板临界屈曲载荷等参数，给出不同边界条件下弹簧刚度的合理取值，并将本研究所提方法得到的屈曲载荷与文献中的计算结果进行对比。  结果  结果显示，采用本研究方法所得屈曲载荷与文献中的计算结果吻合良好，验证了本文研究方法的正确性和收敛性。  结论  研究成果可为船舶相关结构的分析提供参考。

Involutory transformations and variational principles with multi-variables in thin plate bending problems

  Objectives  A rectangular plate buckling study is of great theoretical and practical importance. To study the axial compression buckling behavior of the rectangular plate structure commonly used in engineering under arbitrary elastic boundary conditions, we present a method for calculating buckling loads caused by elastic instability based on the minimum potential principle.   Methods  Rotating restrained springs and lateral restrained springs are respectively arranged on the four boundaries of the plate model. The geometric expression of the displacement function is established based on the improved Fourier series method（IFSM）. The derivative function of the displacement function on the plate boundaries may have discontinuous issues, which can be solved by introducing auxiliary functions on the basis of the classical Fourier series. The potential energy function of the rectangular plate system is established, and by using the minimum potential energy method, the linear equations of the unknown Fourier coefficient are obtained, then the buckling characteristic parameters such as the critical load of the rectangular plate are solved.  Results  The reasonable values of spring stiffness under different boundary conditions are given, and the obtained results are consistent with the exact solution and finite element software. The method is proved to be correct and convergent.   Conclusions  The contents of this paper have a certain reference value for the related structural analysis of ships.