The torsion of a thm circular tube is a relatively simple problem as the shearing stress may be assumed constant throughout the wall thickness.
The case of a solid circular shaft is more complex because the shearing stresses are variable over the cross-section of the shaft. The solid circular shaft of Figure has a length L and radius a in the cross-section.
Figure : Torsion of a solid circular shaft.
When equal and opposite torques Tare applied at each end about a longitudinal axis we assume
(1) the twisting is uniform along the shaft, that is, all normal cross-sections the same distance apart suffer equal relative rotation;
(ii) cross-sections remain plane during twisting; and
(iii) rahi remain straight during twisting.
If the material is elastic, and has a shearing (or rigidity) modulus G, then the circumferential shearing stress on this elemental tube is
The thickness of the elemental tube is 6r, so the total torque on this tube is
This is the polar second moment of area of the cross-section about an axis through the centre, and is usually denoted by J.
We may combine equations