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