Chapter 4: Internal Forces in Bending¶
4.1 Concepts¶
Bending¶
外力/外力偶与轴线垂直，形成弯曲。
-
梁（beam）：以弯曲变形为主的构件。
-
纵向对称面
-
对称弯曲
- 外力都作用在对称面上
- 弯曲变形后仍对称，轴线在对称面内
4.2 Types of Supports & Loading for Beams¶
Supports¶
- Pin support：固定铰支端
- Roller support：可动铰支端
- Fixed support：固定支端
Loading¶
- Concentrated load：集中力
- Distributed load：分布力
- Couple of moment：弯矩
Basic types of beams¶
- Simple beam：简支梁
- Beam with overhang：外伸梁
- Cantilever beam：悬臂梁
Example 4-1
4.3 Shearing Force & Bending Moment¶
Internal force in bending¶
Example
- 求外力
\[ \begin{aligned} \sum F_x = 0 &\Rightarrow X_A = 0 \\ \sum M_A = 0 &\Rightarrow R_B = \frac{Pa}{l} \\ \sum F_y = 0 &\Rightarrow Y_A = \frac{P(l-a)}{l} \end{aligned} \]
- 求内力：截面法
\[ \begin{aligned} \sum F_y = 0 &\Rightarrow Q = Y_A = \frac{P(l-a)}{l} \\ \sum M_C = 0 &\Rightarrow M = Y_A \cdot x \end{aligned} \]
Bending moment \(M\)¶
构件受弯时，横截面上其作用面垂直于截面的内力偶矩。
Shearing force \(Q\)¶
构件受弯时，横截面上其作用线平行于截面的内力。
Sign convention¶
- Shearing force \(Q\)：
- Bending moment \(M\)：
4.4 Equations & Diagrams¶
4.5 Relations between \(q(x)\), \(Q(x)\), and \(M(x)\)¶
\[ \begin{aligned} Q(x) + \mathrm{d}Q &= q(x) \cdot \mathrm{d}x + Q(x) \\ \Rightarrow \frac{\mathrm{d}Q(x)}{\mathrm{d}x} &= q(x) \end{aligned} \]