关于结构刚度方程病态问题的疑难分析

Puzzle Analysis of Ill-Conditioned Structure Stiffness Equation

本文围绕矩阵位移法求解结构位移的误差,讨论了结构刚度矩阵的病态问题,指出了计算上的病态常常是结构刚度分布和结构拓扑的不合理,而不一定是计算本身的问题。首先,文章依据误差界的概念和范数的性质推导了衡量结构刚度矩阵病态程度的条件数的公式。然后,阐明了基于不同范数的条件数在衡量矩阵病态程度上的等价性。最后,通过两个具有代表性的算例,分别从结构构件的刚度差异和结构几何拓扑两方面,计算和分析了结构刚度矩阵的病态程度随结构本身性质的改变而改变的规律。在特定的算例中,还有使结构刚度矩阵的病态程度降到最低的最佳刚度分布。

Around the possible error in solving displacements of structures using matrix displacement methods, the issue of ill-conditioned stiffness matrix of structures was discusses, and it clarified that the ill-conditioning problem in calculation is not always occured due to the calculation itself, but usually related to the irrationality of the stiffness distribution and the structure topology. In terms of the concept of error bonds and the features of norm＇s, the formula of the condition number used for measuring the degree of stiffness matrix＇s ill-condition was derived and clarified that the condition numbers based on different norms is equivalent in measuring the degree of stiffness matrix＇s ill-condition. With two characteristic examples, representing members＇ stiffness difference and structures＇ geometric conformation separately, the change rule of the degree of ill-condition with the changing of structure＇ s own properties was analyzed Furthermore, there exists the best stiffness distribution, which will minimum the error bound in a certain structural problem.