问题描述:
如下所示的两端铰支柱和悬臂柱,分别以欧拉公式和屈曲分析计算临界力。为何 SAP2000 给出的结果(屈曲分析)无法与理论解(欧拉公式)完全一致呢?
解答:
上述 SAP2000 计算模型并不满足欧拉公式的两个基本假定,即:仅考虑弯曲变形,三角(正弦或余弦)函数的挠曲线。默认情况下,SAP2000 中的框架单元均为铁木辛柯梁,根据剪切刚度计算剪切变形;欧拉公式则假定剪切变形为零。为了实现欧拉公式“仅考虑弯曲变形”的假定,我们可以对框架对象或框架截面指定属性修正系数。如下所示,根据需要将沿 2 轴或 3 轴的剪切面积的修正系数设置为零即可。
另外,SAP2000 中框架单元的插值函数为三次函数,欧拉公式则采用三角(正弦/余弦)函数的挠曲线。因此,单个框架单元的变形曲线无法精确拟合欧拉公式采用的挠曲线。不过,我们只需要对框架对象进行适当的剖分,即可无限逼近理论上的精确解。
针对以上两点修改计算模型,并将框架对象分别剖分为 1 个单元、2 个单元、4 个单元。如下所示,随着单元数量的增加,屈曲分析的结果快速逼近理论解。
![](data:image/png;base64,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)