Subject : Structural Analysis 1

Topic : General Principles

Difficulty : Low

Types of Energy Theorems:

1. Castigliano's first Theorem
2. Castigliano's Second Theorem
3. Castigliano's Theorem of least work
4. Maxwell - Betti Reciprocal Theorem
5. Principle of Superposition
6. Principle of Virtual work (Unit Load Method)

Castigliano's Second Theorem states that:

The displacement at the point in a body is equal to the first partial derivative of the strain energy in the structure with respect to a force acting at the point, in the direction of the displacement.

$\triangle point i = \frac{\partial U}{\partial P_{i}}$ of the system

$\theta point i = \frac{\partial U}{\partial M_{i}}$ of the system

For example, we can find the deflexion at the end of the axially loaded bar using Castiliano's theorem as follows.

$\nu =\frac{1}{2}\frac{P^{2}L}{AE}$

$\triangle_{i}=\frac{\partial v}{\partial p}=\frac{\partial }{\partial b}\left[\frac{1}{2}\frac{P^{2}L}{AE}\right]=\frac{PL}{AE}$

$\vee=v\left(P_{1},P_{2}.....P_{n},M_{1},M_{2}....M_{n}\right)$

$\therefore \partial v=\frac{\partial u}{\partial P_{1}}dP_{1}+\frac{\partial u}{\partial P_{2}}dP_{2}+...+\frac{\partial u}{\partial P_{k}}dP_{k}+..+\frac{\partial u}{\partial m}dm$

$\therefore \partial v=\frac{\partial u}{\partial P_{k}}dP_{k}$

Work of the external force:

$\therefore W_{e}+dW_{e}$ $\therefore W_{e}+(dP_{k})\triangle_{k}$ (Defination of complementary work)

$\therefore W_{e}+(dP_{k})\triangle_{k}=\nu+\frac{\partial u}{\partial P_{k}}dP_{k}$

$\therefore (dP_{k})\triangle_{k}=\frac{\partial u}{\partial P_{k}}dP_{k}$

$\therefore \triangle_{k}=\frac{\partial u}{\partial P_{k}}$

Similarlly, $ \triangle_{k}=\frac{\partial u}{\partial M_{k}}$