弹性地基上四边自由矩形板问题的一种新型挠度函数

A New Deflection Function for the Problem of a Rectangular Plate on Elastic Foundation with Four Free Edges

许多工程实际问题都可以抽象为搁置在弹性地基上的四边自由的矩形板问题 ,而弹性地基上四边自由矩形板问题的解析解 ,长期以来被认为是一个比较困难的问题。根据对矩形板解析解的研究 ,本文构造了一个包含三角函数和多项式组成的新的挠度函数 ,它满足四个自由边上全部边界条件和自由角点条件。基于这个新的挠度函数表达式 ,利用迦辽金方法本文导出了弹性地基上四边自由的矩形板弯曲、自由振动和稳定问题的解析解。通过文献中找到的两个算例 ,与有限元方法、有限差分法和其它文献中方法给出的计算结果进行了比较 ,结果吻合良好 ,证明了本文所给出的新型挠度函数是可以求解弹性地基上的四边自由矩形板问题的。这一新型挠度函数形式具有较广阔的工程应用前景。

Many engineering problems can be modeled by a rectangular plate on elastic foundation with four free edges.However,the analytical solution to this problem is considered to be difficult for a long time.Based on a comprehensive study on the existing analytical solutions for rectangular plates,a new deflection function is proposed in this paper which consists of trigonometrical functions and polynomial expressions.This function is orthogonal and satisfies all the boundary conditions on four free edges and the conditions at four free corner points.Solutions for bending,free vibrations and stability of a rectangular plate on elastic foundation with four free edges are derived by using Galerkin method.Two examples found from references are analyzed and the results are compared with those from the literature which used other methods such as finite element method,finite difference method.The agreement is found to be satisfactory which proves the validity of the new deflection function in solving the problem of a rectangular plate on elastic foundation with four free edges.This new deflection function will be of important practical applications.