Internal and external diameters - solid shaft:
Two shafts are of the similar material, length & weight. One is solid & of 100 mm diameter, the outer is hollow. If hollow shaft is to be store 25% more energy than the solid shaft while transmitting torque, what should be its internal & external diameters? Consider the similar maximum shear stress in both the shafts.
Solution
Consider A is the solid shaft of diameter D_{a} & B is the hollow shaft with external diameter D_{b} & internal diameter d_{b}. As both of the shafts are to be the similar material and of equal weight, their volumes must be equal.
Therefore,
or
As the maximum shear stress is the similar for both the shafts, we have,
τ_{a} = τ_{b}
T_{a} D_{a} /2J a = T_{b} D_{b}/2J B ( since τ = T × r/ J = T × D/2J )
Therefore,
Ta/ Tb = D_{b}/ D_{a } × J _{a}/ J_{b}
But it is also given that the strain energy of B = 1.25 × strain energy of A.
Thus,
From the relationship, this reduces to
After that, ∴
Therefore, the internal & external diameters of the hollow tube are 57.74 mm & 115.47 mm, respectively.