Hollow circular shaft – torque capacity expression\n\nFor a hollow circular shaft of outer diameter D and inner diameter d subjected to torsion, which expression gives the transmitted torque T in terms of maximum shear stress τ_max?

Introduction / Context:
Design of shafts under torsion requires the correct relationship among torque, section geometry, and shear stress. For hollow shafts, capacity depends on the polar section modulus rather than simple area.

Given Data / Assumptions:

• Hollow circular shaft with outer diameter D and inner diameter d.
• Maximum shear stress in the material is τ_max.
• Elastic torsion theory applies (linear elastic, Saint-Venant assumptions).

Concept / Approach:
The torsion formula is T / J = τ / R = G θ / L. The torque–stress relation uses the polar second moment J and radius to the outer fiber R = D/2. The polar section modulus Z_p = J / R directly gives T = τ_max * Z_p.

Step-by-Step Solution:
Polar second moment: J = (π/32) * (D^4 - d^4).Outer radius: R = D / 2.Polar section modulus: Z_p = J / R = [(π/32) * (D^4 - d^4)] / (D/2) = (π/16) * (D^4 - d^4) / D.Torque capacity at τ_max: T = τ_max * Z_p = (π/16) * τ_max * (D^4 - d^4) / D.

Verification / Alternative check:
For a solid shaft (d = 0), the formula reduces to T = (π/16) * τ_max * D^3, which is the standard solid-shaft result (since J = π D^4 / 32 and Z_p = J/R = π D^3 / 16).

Why Other Options Are Wrong:
(a) omits division by R and thus gives J scaled by τ rather than T; (c) uses area terms D^2 - d^2 which are not appropriate for torsion; (d) has an incorrect constant; (e) is dimensionally inconsistent.

Common Pitfalls:
Confusing polar second moment with area; forgetting to divide by the outer radius; mixing diameter and radius.

Final Answer:
T = (π/16) * τ_max * (D^4 - d^4) / D