# Strain Energy

Strain energy due to bending and deflection; calculated using Calculus.

## Introduction

Strain energy is a form of potential energy that is stored in a structural member as a result of an elastic deformation. The external work done on such a member when it is deformed from its unstressed state, is transformed into (and considered equal to) the strain energy stored in it. If, for instance, a beam that is supported at two ends is subjected to a bending moment by a load suspended in the center, then the beam is said to be deflected from its unstressed state, and a strain energy is stored in it.## Strain Energy Due To Bending.

Consider a short length of beam under the action of a Bending Moment M. If f is the Bending Stress on an element of the cross section of area at a distance y from the Neutral Axis, then the Strain energy of the length is given by:- For the whole beam: The product**EI**is called the

**flexural Rigidity**of the beam

## Deflection By Calculus

In "Bending Stress" equation (3),the general equation on bending was written. From this it can be seen that: And that in terms of the co-ordinates x and y:##### MISSING IMAGE!

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**The Elastic Line**. Notes on Application:-

- Take the X axis through the level of the supports.
- Take the origin at one end of the beam or at a point of zero slope.
- For built in or fixed end beams, or when the deflection is a maximum, the slope dy/dx=0
- For points on the X axis(usually the supports) the deflection y = 0

- E in lb./sq.in. ( or tons/sq.in.)
- I in
- y in in.
- M in lb.ft (or tons-ft.)
- x in ft.

Example:

##### Example - Strain Energy Due To Bending.

Problem

A simply supported beam of length

*l*carries a concentrated load*W*at distances of*a*and*b*from the two ends. Find expressions for the total strain energy of the beam and the deflection under load.Workings

The integration for strain energy can only be applied over a length of beam for which a continuous
expression for

*M*can be obtained. This usually implies a separate integration for each section between two concentrated loads or reactions. For the section AB. Similarly by taking a variable*X*measured from C Total But if is the deflection under the load, the strain energy must be equal to the work done by the load if it is gradually applied. For a Central Load HenceSolution

Strain Energy
Deflection