Quick The strain energy is the increase in energy associated with the deformation of a member. The strain energy is equal to the work done by a slowly increasing load applied to a member.
Consider a rod BC of length L and uniform cross-sectional area A, which is attached at B to a fixed support, and subjected at C to a slowly increasing axial loadP, as shown:
As noted in the lesson Normal Strain, by plotting the magnitudeP of the load against the deformation x of the rod, a certain load-deformation diagram is obtained that is characteristic of the rod BC.
Now consider the workdU done by the loadP as the rod elongates by a small amount dx. This elementary work is equal to the product of the magnitude P of the load and of the small elongationdx:
dU = P dx
Note that the expression obtained is equal to the element of area of width dx located under the load-deformation diagram as shown:
The total work U done by the load as the rod undergoes a deformation x_{1} is then:
The work done by the loadP as it is slowly applied to the rod must result in the increase of some energy associated with the deformation of the rod. This energy is referred to as the strain energy of the rod. Then, by definition:
where P_{1} is the value of the load corresponding to the deformation x_{1}.
The concept of strain energy is particularly useful in the determination of the effects of impact loadings on structures or machine components. Consider, for example, a body of massm moving with a negative velocity v_{0} which strikes the end B of a rod AB as shown:
Neglecting the inertia of the elements of the rod, and assuming no dissipation of energy during the impact, it is found that the maximum strain energy U_{m} acquired by the rod is equal to the original kinetic energy:
T =
1
2
m_{1}v_{0}^{2}
of the moving body. The value P_{m} of the static load which would have produced the same strain energy in the rod is then determined, and the value σ_{m} of the largest stress occurring in the rod is obtained by dividing P_{m} by the cross-sectional area of the rod.