Expression for strain energy:

Derive an expression for strain energy because of torsion.                                         

Sol.: The work done in straining shaft within elastic limit is called as strain energy. Consider a shaft having diameter D, and Length L, subjected to gradually applied torque T. Let θ be angle of twist. Energy is stored in shaft because of this angular distortion. This is called as torsional energy or Torsional resilience.

Torsional energy or strain energy = Work done by the torque = average torque X angular twist.

Thus; Strain energy per unit volume is 1/4th  ratio of square of shear stress to modulus of rigidity. For the hollow shaft: U = [ τ₂.(D₂  + d₂)/4.G.D₂].Volume of shaft

For the Solid shaft (d = : U = [ τ₂/4.G].

Volume of shaft

For very thin hollow shaft (d = 0): U = [τ₂/2.G].Volume of shaft