Chapter 6: Deformations of Beams¶
6.1 Introduction¶
绝大部分情况下，不希望变形太大
6.2 Deflection and Angle of Rotation¶
- The deflection curve（挠曲线）
- The deflection of beam（梁的挠度）：竖直方向上的位移
- The angle of rotation（转角）：梁绕中性轴转过的角度
The equation of the deflection curve
\[ \left \{ \begin{aligned} &y = f(x) \\ &\theta \approx \tan\theta = \frac{dy}{dx} \end{aligned} \right. \]
Discussion
- 小变形：\(y_{\max} < \frac{l}{1000} \sim \frac{l}{250}\)
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Convention of sign:
-
\(x\) 轴正方向：右
- \(y\) 轴正方向：上
- \(\theta\)：逆时针
6.3 Approximate Differential Equations of the Deflection Curve¶
\[ \underset{\text{Pure bending}}{\frac{1}{\rho} = \frac{M}{E I}} \overset{l \gg h}{\longrightarrow} \text{Shearing bending} \]
小变形，曲率很小
\[ \kappa = \frac{1}{\rho} = \pm \frac{\frac{d^2y}{dx^2}}{\left[1 + \left(\frac{dy}{dx}\right)^2\right]^{\frac{3}{2}}} \overset{\frac{dy}{dx} \ll 1}{=} \pm \frac{d^2y}{dx^2} \]
得到
\[ \frac{d^2y}{dx^2} = \frac{M(x)}{E I} \]