Strain Energy in Torsion:

The element ds is now let subject to a torque T as illustrated in Figure generating an angle of twist dθ.

Figure

Strain energy = work done = (½) T d θ

But, from the theory of simple torsion, T/ J = G × (d θ / ds ) or d θ = T ds/ GJ

Here J refers to the polar moment of inertia and G is the shear modulus of the material.

Thus, total strain energy resulting from torsion,

For constant T, we get U = T ² L / 2 GJ