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

1. Simple beam：简支梁
2. Beam with overhang：外伸梁
3. Cantilever beam：悬臂梁

Example 4-1

4.3 Shearing Force & Bending Moment

Internal force in bending

Example

1. 求外力

$$\begin{array}{rl}\sum F_x=0& \Rightarrow X_A=0\\ \sum M_A=0& \Rightarrow R_B=\frac{Pa}{l}\\ \sum F_y=0& \Rightarrow Y_A=\frac{P(l-a)}{l}\end{array}$$

1. 求内力：截面法

$$\begin{array}{rl}\sum F_y=0& \Rightarrow Q=Y_A=\frac{P(l-a)}{l}\\ \sum M_C=0& \Rightarrow M=Y_A\cdot x\end{array}$$

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 $M(x)$

$$\begin{array}{rl}Q(x)+\mathrm{d}Q& =q(x)\cdot \mathrm{d}x+Q(x)\\ \Rightarrow \frac{\mathrm{d}Q(x)}{\mathrm{d}x}& =q(x)\end{array}$$