问题描述:
下图所示的外伸梁在均布线荷载作用下,其自由端的竖向位移与外伸长度 X 之间并非单调的递增或递减关系。也就是说,随着 X 的增加,自由端的竖向位移既可能增加也可能减少。请问,如何从理论上解释这个现象呢?
![](data:image/png;base64,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)
解答:
对于上图所示的外伸梁,在只考虑弯曲变形及 EI 为常数的情况下,利用单位荷载法(图乘法)计算自由端的竖向位移 Δ,如下所示。具体计算过程详见结构力学的相关教材,此处不再赘述。
![](data:image/png;base64,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)
其中,Δ 以竖直向下为正,竖直向上为负。当 X 取零时,Δ 等于零,即:外伸梁退化为简支梁。同理,当 X = l 时,Δ =
ql^4/8EI,即:外伸梁退化为悬臂梁。基于上述公式,外伸长度 X 与自由端竖向位移 Δ 之间的函数关系如下图所示:
![](data:image/png;base64,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)
可以看出,随着外伸长度的增加,自由端竖向位移的数值先减后增。如果考虑竖向位移的方向,可将以上曲线划分为三段,从左向右依次为:向上增加段 → 向上减少段 → 向下增加段。
因此,当线荷载的作用方向(竖直向下)与自由端的位移方向相反(负值 = 竖直向上)时,随着外伸长度 X 的增加,自由端的竖向位移 Δ 并非单调递增或单调递减,这与外伸长度的变化范围密切相关。