框架结构屈曲的精确有限元求解

EXACT FINITE ELEMENT SOLUTIONS OF BUCKLING ANALYSIS OF FRAME STRUCTURES

基于屈曲微分控制方程的一般解,构造了Euler梁在轴力作用下的精确形函数,建立了用于框架结构屈曲分析的精确有限单元,得到了单元刚度矩阵和几何刚度矩阵的显式表达,并提出了基于常规特征值计算的迭代算法以确定屈曲载荷及相应失稳模态的精确解。研究表明,对于线性稳定性分析而言,常规框架有限单元可视为精确有限单元的一种近似。若采用精确单元,无需进行网格细分就可以获得精确的屈曲载荷和失稳模态。数值算例证明了新单元和算法的效率和精度。

Based on general solution for the homogeneous governing equation for the linear buckling analysis of Euler beam, new shape functions are constructed and a new finite element is formulated. With the derived element stiffness matrix and geometric stiffness matrix, an iterative algorithm based on conventional eigenvalue calculation procedure is proposed for linear buckling analysis of frame structures. The conventional finite element is proved to be an approximation of the proposed element. By the application of the proposed element and algorithm, exact buckling solutions of frame structures can be obtained even with coarse meshes. Illustrative numerical examples are presented to show the effectiveness of the new element and algorithm.