四边固支矩形弹性薄板的精确解析解

Exact Analytic Solution of Rectangular Thin Plate with Four Edges Clamped

利用辛几何方法本文推导出了四边固支矩形弹性薄板弯曲问题的精确解析解。由于在求解过程中不需要事先人为的选取挠度函数,而是从弹性薄板的基本方程出发,首先将矩形薄板弯曲问题表示成Hamilton正则方程,然后利用分离变量和本征函数展开的方法求出可以完全满足四边固支边界条件的精确解析解。本文中所采用的方法突破了传统的半逆法的限制,使得问题的求解更加合理化。文中还给出了计算实例来证明推导结果的正确性。

The exact solution of rectangular thin plate with four edges clamped was derived by symplectic geometry method. Firstly, the basic equations for elastic thin plate were transferred into Hamilton canonical equations. And then the whole state variables were separated. Finally, according to the method of eigen function expansion in the symplectic geometry, the exact solution of fully clamped rectangular thin plate was obtained. Since only the basic elasticity equations of thin plate are used and there is no need to select the deformation function arbitrarily, the approach goes beyond the limits of classical semi-inverse method thus more reasonable and theoretical. To verify the accuracy and validity of the formulations, the numerical results were presented to compare with those of other references.