# Displacements by energy principles

## Displacementsby energy principles beam of length L under pure bending

The total strain energy stored in a bending beam is then

and if the moment and other quantities vary along the beam,

### Castigliano’s Second Theorem

The work-energy method is the simplest of energy methods. A more powerful method is that based on Castigliano’s second theorem2 , which can be used to solve problems involving linear elastic materials. As an introduction to Castigliano’s second theorem, consider the case of uniaxial tension, where

. The displacement through which the force moves can be obtained by a differentiation of this expression with respect to that force,

These are examples of Castigliano’s theorem, which states that, provided the body is in equilibrium, the derivative of the strain energy with respect to the force gives the displacement corresponding to that force, in the direction of that force. When there is more than one force applied, then one takes the partial derivative. For example, if n independent forces P1, P2, … … , Pn act on a body, the displacement corresponding to the ith force is

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