问题描述:
对于几何非线性中的大位移效应,能否列举简单的算例加以演示呢?
解答:
对于下图所示的结构,由于对铰 C 在竖直方向的位移缺少必要的约束,导致 C 点可沿竖直方向发生位移。然而一旦发生微小位移后,A、B、C 三个铰不再共线,C 点的竖直位移将受到相应的约束。这种原为几何可变,经微小位移后即转变为几何不变的体系,称为瞬变体系。工程结构中不能采用瞬变体系,而且接近于瞬变的体系也应该避免!
假设在跨中节点 C 处施加竖直向下的集中力 F
对于线性分析,由于该瞬变体系的初始刚度无法抵抗 C 点的竖向作用力(即相应刚度为零),且在整个计算过程中结构的刚度始终保持不变,故 C 点位移将为无穷大或分析无法进行(程序报错)!SAP2000 的计算结果如下图所示:
![](data:image/png;base64,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)
这种数量级巨大的位移结果通常表明结构存在刚度奇异,建议用户可以从以下几个方面检查模型错误:
a. 结构的约束是否足够限制其刚体位移;
b. 单元之间是否正确连接,是否存在重合的节点;
c. 结构的局部是否出现机构导致局部刚体位移;
d. 结构整体是否为瞬变体系(如本例所示);
e. 等等
事实上,该问题是存在理论解答的,只是需要同时考虑静力平衡方程、几何协调方程以及物理方程。如上图所示,具体求解如下:
a. 静力平衡方程:![](data:image/png;base64,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)
b. 几何协调方程:![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJwAAAAlCAYAAAC3fYDwAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAAJISURBVHhe7ZlbEoMgDEVZFwtyPazGzbgYWoK2EMERbGEg98z4YW37kTnmQZQFoCEQDjRFoHCrXZSy6n0t6/4RaIbMDLcZq5W2ZtvvQTNkCrcuVmlj4Vt7RAq3GW0V6mkXBAq3WaOV1ainXRA6NKB/64U84WhgWN7agR6IE476NwwM3RAn3Lqgf+uJMOH8wEAD6rpAvA6IzHBuy4BjkT7IGxpAV/4kHPaVdcwft2rhqDRdTXvYV9YxedzqhKOguDfxIjDYV9YxedyqhHNnWdoYSv+5SQ/7yjpmj1uFcN/VUL6s3tlX+u/QxHh1icqS8+95y4VzKf94A7P9BvaVdcwft0LhgoNTYs9SvAQ02Fcms2HjK+YHGbtB3HpTJpwLCAsY9RwsSPf2lSipnHtxG5si4ZJ7yH1iDZMc9pV1SIjbfeGy6Z6X1aDsYl9ZgIy43RbOl05W7qLrKyNNr+4z3tuBSyTErayHE8l33eQuvEPPgHCXeNk+5c0dCU0+Rf4bCHcBlbgwpVEfi/PFJ0C4HCm5soMTuMsEwvHzvFCIuP9KnXF9GnX2neSZGErqYwYXbhfqKHsuAx2SUDYKz7W8mNFxAxfI3dN/cYmDa/KD2X8ztHCUnTICpJ6le7LU5MmGBSIhLChmYOG8FGdZHAXCUJZz2YuX4lT/hoHhKeMKdyVA6hmJlRcmyoiJ35+yI6hiogznMtghCc9w/j4S5i1gmO3iISH+b79lwbDwC4bu4eJ1WyqjHc/OpZQyVvD8JFT0e8j2K4YWDoyGtS/C1jmt1axlHQAAAABJRU5ErkJggg==)
c. 物理方程:![](data:image/png;base64,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)
联立以上三个方程,则有:
这里,我们需要注意几点:
a. 静力平衡方程是根据结构变形后的几何形状建立的。
b. 外力 F 与 C 点的转角位移 θ 为非线性关系。
c. 结构最终的几何形状是与外荷载的大小相关的。
以上第一条就是大位移效应的本质:根据结构变形后的几何形状建立结构的平衡方程!
如果想要得到以上的理论解答,则必须进行非线性分析,并考虑几何非线性中的大位移效应。程序将根据结构的变形不断重新形成结构的刚度,通过迭代求解最终同时满足三套方程。正确的理论解答如下图所示:
a. 必须考虑大位移效应,只考虑 P-Delta 效应是无法得到正确解答的!
b. C 点的转角 θ (即上图中的 R2)与外力 F 完全满足:![](data:image/png;base64,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)
c. 对于一般的结构是不需要考虑大位移效应的,除非结构的变形显著不满足小变形假定(如Pushover分析)或结构包含特殊的柔性构件(如拉索)。