Stress and Strain - Structural and Stress Analysis - page 167

Substituting for in Eq. (7.27) gives

P 2 L 0
2 AE

U = (7.29)

It is often convenient to express strain energy in terms of the direct stress σ . Rewriting
Eq. (7.29) in the form

P 2
A 2

1
2

AL 0
E

U =

we obtain

σ 2
2 E ×

U = AL 0 (7.30)

in which we see that AL 0 is the volume of the member. The strain energy per unit
volume of the member is then

σ 2
2 E
The greatest amount of strain energy per unit volume that can be stored in a member
without exceeding the limit of proportionality is known as the modulus of resilience
and is reached when the direct stress in the member is equal to the direct stress
corresponding to the elastic limit of the material of the member.

The strain energy, U , may also be expressed in terms of the elongation, , or the direct
strain, ε . Thus, substituting for P in Eq. (7.29)

EA 2
2 L 0

U = (7.31)

or, substituting for σ in Eq. (7.30)

1
2 E ε 2

U = × AL 0 (7.32)

The above expressions for strain energy also apply to structural members subjected
to compressive loads since the work done by P in Fig. 7.14(a) is independent of the
direction of movement of P . It follows that strain energy is always a positive quantity.

The concept of strain energy has numerous and wide ranging applications in structural
analysis particularly in the solution of statically indeterminate structures. We shall
examine in detail some of the uses of strain energy later but here we shall illustrate its
use by applying it to some relatively simple structural problems.

DEFLECTION OF A SIMPLE TRUSS

The truss shown in Fig. 7.16 carries a gradually applied load W at the joint A.
Considering the vertical equilibrium of joint A

P AB cos 45 ◦ − W = 0