Torsion of a solid circular shaft A solid circular shaft of diameter d is subjected only to a torque T (pure torsion, no axial force or bending). What is the maximum normal (direct) stress induced anywhere in the shaft?

Introduction / Context:
This question tests fundamentals of torsion in solid circular shafts. Under pure torsion, the dominant stress is shear. Knowing which stresses appear (and which do not) is essential for safe shaft design.

Given Data / Assumptions:

• Solid circular shaft, diameter d.
• Applied loading is torque T only (no bending moment, no axial force, no internal pressure).
• Saint-Venant torsion; material is homogeneous and isotropic.

Concept / Approach:
In elementary torsion theory of a circular shaft, the shear stress varies linearly with radius: tau(r) = T * r / J, where J is the polar second moment of area. There is no accompanying normal (direct) stress field caused by torque alone. Normal stresses would arise from axial force (tension/compression), bending moment, or thermal/pressure effects—not from pure torsion.

Step-by-Step Solution:

Verification / Alternative check:
Mohr’s circle for pure shear shows principal normal stresses ±tau_max at 45° planes. However, on the physical cross-section (where normal stress is queried), the direct normal stress due to torsion is zero; the listed options refer to shear or unrelated expressions.

Why Other Options Are Wrong:

• T * r / J and 32 * T / (π * d^3) are shear-stress expressions, not normal stress.
• M / Z corresponds to bending normal stress, not torsion.
• E * θ / L refers to twist per unit length relationship, not a stress.

Common Pitfalls:
Confusing principal normal stresses on inclined planes with direct normal stress on the cross-section; the question asks for normal stress induced in the shaft material due to torsion alone.

Final Answer:
Zero (no normal stress under pure torsion)