案例10 — 带单向受拉侧向拉杆的悬臂梁

描述

长1.0 m的竖向悬臂梁固定在上端。底部附近，两根长1.0 m的水平单向受拉单元将梁底节点与两侧固定锚点相连。在梁半高处（固支点以下0.5 m）施加1 kN水平集中荷载。

求：

• 每根单向受拉单元的轴力
• 梁底节点的水平位移

结构图示

验证模型的几何形状、边界条件和荷载施加情况。

模型参数

解析解

The system is statically indeterminate. The force in the tie is found from the compatibility condition of deformations:

$$\delta_{D,P} - T \cdot \delta_{11} = T \cdot \frac{L_h}{EA_h}$$

$$T = \frac{\delta_{D,P}}{\delta_{11} + \dfrac{L_h}{EA_h}}$$

where $\delta_{D,P} = \dfrac{5PL^3}{48EI}$ — deflection of point D due to the external load P, calculated on the released cantilever (without the tie);

$\delta_{11} = \dfrac{L^3}{3EI}$ — deflection of point D due to a unit force applied at D;

$L_h,\, A_h$ — length and cross-sectional area of the tie; $E$ — modulus of elasticity; $L$ — height of the cantilever segment BD.

We assume that there is no force in the opposite tie.

$$T = 0.3062\,\text{kN}$$ $$\Delta x(D) = 0.0186\,\text{mm}$$

数值结果

位移

轴力

RodX表格结果

对比

RodX获得的数值结果与解析解吻合。