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Chapter 13: Energy Methods

13.1 Introduction

For solid deformable bodies under static loads,

\[ V_{\varepsilon} = W \]

Notes:

  1. 线弹性范围内，静载荷
  2. 用来求解静不定结构

13.2 Strain Energy

应变能

  1. 轴向拉压
\[ V_{\varepsilon} = \frac{F^2 l}{2E A} = \int_l \frac{F^2(x) \mathrm{d}x}{2E A} \]

应变能密度

\[ v_{\varepsilon} = \frac{1}{2} \sigma \varepsilon \]
  1. 纯剪切
\[ v_{\varepsilon} = \frac{\tau^2}{2G} = \frac{1}{2} \tau \gamma \]
  1. 扭转

扭转角 \(\varphi\) 与扭矩 \(M_{\text{e}}\) 之间呈线性关系：

\[ \varphi = \frac{M_{\text{e}} l}{G I_p} \]

扭转应变能：

\[ V_{\varepsilon} = \frac{M_{\text{e}}^2 l}{2 G I_p} = \int_l \frac{T^2(x) \mathrm{d}x}{2 G I_p} \]
  1. 弯曲

应变能

\[ V_{\varepsilon} = \frac{1}{2} M_{\text{e}} \theta = \int_l \frac{M^2(x) \mathrm{d}x}{2 E I} \]

综上，应变能可以统一写成 $$ V_{\varepsilon} = \frac{1}{2} F \delta $$

\(F\) 为广义力，\(\delta\) 为广义位移。

Castigliano's Theorem

Virtual Work Principle

  • Virtual displacement \(v^*(x)\)

  • B.C.

  • Continuous
  • Small deformation

Virtual work done by external forces:

\[ W^*_{\text{ext}} = F_1 v^*_1 + F_2 v^*_2 + \ldots + \int_l q(x) v^*(x) \mathrm{d}x + \ldots \]

Virtual work done by internal forces (Virtual strain energy):

\[ W^*_{\text{int}} = \int F_{\text{N}} \mathrm{d}(\Delta l)^* + \int M \mathrm{d}\theta^* + \int F_{\text{S}} \mathrm{d}\lambda^* + \int T \mathrm{d}\varphi^* \]
\[ W^*_{\text{ext}} = W^*_{\text{int}} \]

外力在虚位移上做的功等于内力在虚变形上做的功。

Example 9

A 点位移为 \(v\)，则三根杆的伸长量为

\[ \Delta l_1 = v, \quad \Delta l_2 = \Delta l_3 = v\cos\alpha \]

由 Hooke 定律：

\[ F_1 = \frac{E A}{l} \Delta l_1, \quad F_2 = F_3 = \frac{E A}{l} \Delta l_2 \]

假设 A 处的虚位移为 \(\delta v\)，则外力做的虚功为

\[ W^*_{\text{ext}} = P \delta v \]

虚应变能

\[ W^*_{\text{int}} = \int F_{\text{N}1} \mathrm{d}(\Delta l_1)^* + \int F_{\text{N}2} \mathrm{d}(\Delta l_2)^* + \int F_{\text{N}3} \mathrm{d}(\Delta l_3)^* \]

Mohr's Theorem

实际的变形视作虚拟的，单位载荷 \(1\) 和实际位移 \(\Delta\)