In a circular shaft under torsion, how does the shear stress at a point vary with its radial distance from the shaft axis?

Introduction / Context:
Torsion of circular shafts is a core topic in mechanics of materials. Designers must know how shear stress distributes from the center to the outer surface to ensure safe torque transmission.

Given Data / Assumptions:

• Solid or hollow circular shaft under pure torque T.
• Linearly elastic, homogeneous, isotropic material.
• No warping restraint beyond Saint-Venant torsion assumptions.

Concept / Approach:
The torsion formula gives shear stress at a radius r as tau(r) = T * r / J, where J is the polar moment of inertia of the cross-section. This indicates a linear variation of shear stress with radius—zero at the center and maximum at the outer surface.

Step-by-Step Solution:

Verification / Alternative check:
Mohr’s circle for pure shear at a point yields zero shear at the centerline because r = 0; experimental strain-gage data on shafts confirms linear radial variation.

Why Other Options Are Wrong:
Inverse or constant dependence contradicts tau = T r / J; saying maximum at center is incorrect as tau(0) = 0.

Common Pitfalls:
Confusing bending stress distribution (linear with y) with torsional shear; forgetting hollow shaft still has tau proportional to r within its wall.

Final Answer:

directly proportional to the distance from the axis