Strain energy due to bending and deflection; calculated using Calculus.
IntroductionStrain 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:-
Deflection By CalculusIn "Bending Stress" equation (3),the general equation on bending was written. From this it can be seen that:
- 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 - Strain Energy Due To Bending.
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.
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.