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Strain Energy

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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.


Equation

(Eq1)     U = ∫ x1   0 P dx

Strain energy

Nomenclature

P	load
dx	width of element under load-deformation diagram

Details

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 load P, as shown:



As noted in the lesson Normal Strain, by plotting the magnitude P 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 work dU done by the load P 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 elongation dx:

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 x₁ is then:

U =

∫

x1   0

P dx

and is equal to the area under the load-deformation diagram between x = 0 and x = x₁.

The work done by the load P 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:

(Eq1)

Strain energy = U = ∫ x1   0 P dx

In the case of a linear and elastic deformation, the portion of the load-deformation diagram involved can be represented by the following equation:

P = kx

as shown in the figure:



Substituting for P in Eq1, the following results:

U =

∫

x1   0

kx dx =

1 2

kx12

or:

(Eq2)

U =  1 2 P1x1

where P₁ is the value of the load corresponding to the deformation x₁.

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 mass m moving with a negative velocity v₀ 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

m1v02

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.