Question

Derive an expression for strain energy due to torsion.

Answer 1

The work done in straining the shaft with in the elastic limit is called strain energy. consider a shaft of diameter D, and Length L, subjected to a gradually applied torque T. Let θ be the angle of twist. Energy is stored in the shaft due to this angular distortion. This is called torsional energy or the Torsional resilience. 

Torsional energy or strain energy = W.D. by the torque = average torque X angular twist. 

= (T/2).θ 

= ½.T.θ 

= ½. (τ.J/R) (τ.L/R.G) 

= ½. (τ²/G).(J/R²).L 

= ½ .(τ²/G).[(πD⁴/32)/(D/2)²].L 

= ½ .(τ²/G).(πD²/8).L = ½. (τ²/G).(π.4R²/8).L 

= ½. (τ²/G).(π. R².L/2) = ¼. (τ²/G).Volume 

= ¼. (τ²/G).Vor; 

U/V = ¼. (τ²/G) 

So; Strain energy per unit volume is 1/4^th ratio of square of shear stress to modulus of rigidity. 

For a hollow shaft : U = [τ².(D² + d²)/4.G.D²].Volume of shaft 

For a Solid shaft (d = : U = [τ²/4.G].Volume of shaft 

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