四角点支承四边自由矩形薄板屈曲问题的新解析解

角点支承矩形薄板的屈曲问题是板壳力学的一类重要课题,控制方程和边界条件的复杂性导致寻求该类问题的解析解十分困难。虽然各类近似/数值方法可用于解决此类难题,但作为基准的精确解析解在公开文献中鲜有报道。本文基于近年来提出的辛叠加方法,解析求解了四角点支承四边自由矩形薄板的屈曲问题。首先将问题拆分为两个子问题,接着利用分离变量与辛本征展开推导出子问题的解析解,最后通过叠加获得原问题的解。由于求解过程从基本控制方程出发,逐步严格推导,无需假定解的形式,因此本文解法是一种理性的解析方法。数值算例给出了不同长宽比和不同面内载荷比情况下,四角点支承四边自由矩形薄板的屈曲载荷和典型屈曲模态,并经有限元方法验证,确认了解析解的正确性。

A new analytic solution to the buckling problem of rectangular thin plates with four corners point-supported and four edges free

The buckling problem of rectangular thin plates supported by corner points is an important topic in mechanics of plates and shells.However,due to the complexity of the governing equations and boundary conditions,it was difficult to obtain the analytic solutions to such problems.Although various approximate/numerical methods have been developed to solve such problems,as benchmarks,accurate analytic solutions were rarely reported in the open literature.Based on the symplectic superposition method that was proposed in recent years,the buckling problem of rectangular thin plates with four corners point-supported and four edges free is analytically solved in this paper.The problem is firstly divided into two sub-problems;then the analytic solutions of the sub-problems are derived by separation of variables and symplectic eigen expansion.The solution of the original problem is finally obtained by superposition.Since the solution procedure starts from the basic governing equation and is derived rigorously,step by step,without assuming the forms of the solutions,the presented solution method is a rational analytic method.With different aspect ratios and different in-plane load ratios,the buckling loads and typical buckling mode shapes of the rectangular thin plates with four corners point-supported and four edges free are given in the numerical examples.The correctness of the analytic solution is validated by the finite element method.