Strain Energy in Shear:

Assume now an element of length ds of the bar subjected to a shear load Q at one end causing deformation by an angle γ (the shear strain) & a shear deflection δ, as illustrated in Figure.

Figure

Strain energy, dU, of the element = work done = (½) Q δ = (½)Q γ ds

But, shear strain, γ = Shear Stress, τ/ Shear Modulus, G        =  Q /GA

∴          dU =(1/2) Q ×( Q/ GA) ds=   Q²ds/2GA

∴          Overall strain energy in the bar, resulting from shear is as:

=          π² /2 G × Volume of the bar