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 beamDeflection 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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- 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
Hence
Solution
Strain Energy
Deflection