For a hollow (tubular) circular shaft under torsion, the same torsion formula relating shear stress, torque, radius, polar moment of inertia, shear modulus, and angle of twist per unit length applies as for a solid circular shaft, but shear strain and shear stress vary linearly with radial distance only across the annular material (zero at the (absent) center, minimum at the inner radius, maximum at the outer radius), so the polar moment of inertia must be computed as the difference between the polar moments of the outer and inner circular sections. For a bar of non-uniform cross-section or subjected to torques applied at multiple points along its length, the internal resisting torque varies by segment (found via free-body sections), and total angle of twist is obtained by summing each segment's twist (torque times length over shear modulus times polar moment of inertia) with appropriate sign, while the governing stress is the maximum among all segments — this belongs to the theory of torsion within mechanics of materials (strength of materials), extending the solid-shaft torsion theory to hollow and stepped/multi-loaded members.
For a hollow (tubular) circular shaft under torsion, the same torsion formula relating shear stress, torque, radius, polar moment of inertia, shear modulus, and angle of twist per unit length applies…