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Derive an expression for strain energy due to torsion.

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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 1/4th 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

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