Strength of materials – Polar modulus for a solid circular shaft\nFor a solid shaft of diameter D subjected to torsion, what is the polar section modulus (also called polar modulus) expressed in terms of D?

Introduction / Context:
The polar section modulus is a geometry property used in torsion of circular shafts. It links the applied torque to the maximum shear stress at the outer surface and is essential for sizing shafts to carry torque safely.

Given Data / Assumptions:

• Solid circular shaft with diameter D.
• Material is homogeneous and isotropic; elastic torsion theory applies.
• Saint-Venant torsion (no warping restraint).

Concept / Approach:
The torsion relation is T / J = τ / R = G * θ / L, where J is the polar moment of inertia and R is outer radius. The polar section modulus Zp is defined as J / R. For a solid circle, J depends on D^4, while Zp scales with D^3.

Step-by-Step Solution:
For a solid circular section: J = (π/32) * D^4.Outer radius: R = D / 2.Polar section modulus: Zp = J / R.Zp = ((π/32) * D^4) / (D/2) = (π/16) * D^3.Therefore, the correct expression is (π/16) * D^3.

Verification / Alternative check:
Dimensional check: J has units of length^4; dividing by R (length) gives length^3, which matches Zp. This is consistent with torsion formulas used in machine design textbooks.

Why Other Options Are Wrong:
(π/32) * D^3 and (π/64) * D^3 underestimate Zp by factors of 2 and 4 respectively; (π/8) * D^3 overestimates by 2. The option with D^4 is dimensionally incorrect for a modulus.

Common Pitfalls:
Confusing J (polar moment) with Zp (section modulus) or forgetting to divide by R. Another common mistake is using area moment of inertia I instead of polar J for torsion.

Final Answer:
(π/16) * D^3