Consider a case when the torque applied to the circular shaft is within Hooke's law limit, so there is no permanent deformation.

Now, recall the expression for shearing strain. Multiplying it by the modulus of rigidity and using Hooke's Law for shearing stress and strain, an expression for shearing stress in a shaft can be determined.

Recall that the sum of the moments of the elementary forces exerted on any cross-section of the shaft must be equal to the magnitude of the torque exerted on the shaft.

Substituting for shearing stress and rearranging the terms gives an expression with an integral term, which represents the polar moment of inertia of the cross-section with respect to its center.

Further rearrangements and substitutions for maximum shearing stress give the elastic torsion formula for the shearing stress in a rigid uniform circular shaft.

However, for a hollow shaft with r1 and r2 as the inner and outer radii, the polar moment of inertia is expressed as a difference in the fourth power of two radii.