# 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: The sign depends upon the convention for axes. For beams in normal engineering practice, the slope is very small everywhere, and may be neglected in comparison to 1 in the denominator. Taking y as positive upwards, under the action of a positive Bending Moment, the curvature of the beam is shown in the diagram. It can be seen that dy/dx is increasing as x increases. Hence, or, Thus, provided that M can be expressed as a function of x, equation(43) can be integrated to give the slope dy/dx and the deflection y can be found for any value of x. Two constants of integration will be involved and these can be found by substituting known values of slope or deflection at particular points. A mathematical expression is thus obtained for the form of the deflected beam (also known as**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