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：\(y_{\max} < \frac{l}{1000} \sim \frac{l}{250}\)
-
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
\[ \pm \frac{d^2y}{dx^2} = \frac{M(x)}{E I} \]
根据 \(M(x)\) 的符号定义：
- \(M(x) > 0\)：凸向下，\(y'' > 0\)
- \(M(x) < 0\)：凸向上，\(y'' < 0\)
可知 \(M(x)\) 与 \(y''\) 同号。故
\[ \begin{equation} \tag{1} \frac{d^2y}{dx^2} = \frac{M(x)}{E I} \end{equation} \]
6.4 Beam Deformations by Integration¶
对式 \eqref{1} 积分一次，
\[ EI \frac{dy}{dx} = \int M(x) \, \mathrm{d}x + C \]
再积分一次，
\[ EI y = \iint M(x) \, \mathrm{d}x\mathrm{d}x + C_1 x + C_2 \]
Boundary conditions¶
- Rigidly fixed support（刚性固定支撑）
\[ \begin{aligned} \left. y \right|_{x = 0} &= 0 \\ \left. \theta \right|_{x = 0} &= 0 \end{aligned} \quad \text{or} \quad \begin{aligned} \left. y \right|_{x = 0} &= 0 \\ \left. y' \right|_{x = 0} &= 0 \end{aligned} \]
- Movable (or fixed) hinged support（活动或固定铰支撑）
\[ \begin{aligned} \left. y \right|_{x = 0} &= 0 \\ \left. y \right|_{x = l} &= 0 \end{aligned} \]
铰链不提供转矩！
Continuity conditions¶
- Deflection
\[ \left. y_1 \right|_{x = a} = \left. y_2 \right|_{x = a} \]
- Angle of rotation
\[ \left. \theta_1 \right|_{x = b} = \left. \theta_2 \right|_{x = b} \quad \text{or} \quad \left. y_1' \right|_{x = b} = \left. y_2' \right|_{x = b} \]
6.5 Beam Deformations by Superposition¶
在小变形假设下，\(y\) 和 \(\theta\) 都是外力的线性函数！
结构形式叠加原理（逐段刚化法）
外伸梁 = 简支梁 + 悬臂梁