Strain energy due to bending and deflection calculated using Calculus
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:-
- 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.
- Compare the strain energy of a beam, simply supported at its ends and loaded with a uniformly distributed load, with that of the same beam centrally loaded and having the same value of maximum bending moment. (U.L.) If l is the span and EI the Flexural Rigidity, then for a uniformly distributed load w, the end reactions are and at a distance x from one end:-
- A concentrated load W is gradually applied to a horizontal beam simply supported at its ends, produces a deflection y at the load point. If this falls through a distance h onto the beam find an expression for the maximum deflection produced. In a given beam, for a load W, y = 0.2 in. and the maximum stress is 4 tons/sq.in.. Find the greatest height from which a load of 0.1 W can be dropped without exceeding the elastic limit of 18 tons/sq.in. (U.L.) The loss of Potential Energy by the load = The gain in Strain Energy by the beam i.e.
Deflection By CalculusIn "Bending Stress" equation (3) it 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.
- Obtain expressions for the maximum slope and deflection of a cantilever of length l carrying (a) a concentrate4d load W at its free end and (b) a uniformly distributed load w along its whole length. (a) If the origin is taken through the free end and the X axis through the fixed end then at a distance x from the origin:-