Torsion of Hollow Shafts — Polar Modulus What is the polar section modulus (Z_p) for a hollow circular shaft with outer diameter D and inner diameter d?

Introduction:
Polar section modulus relates torsional moment to maximum shear stress. For shafts, it is defined as Z_p = J / R, where J is polar moment of inertia and R is outer radius.

Given Data / Assumptions:

• Hollow circular shaft with outer diameter D and inner diameter d.
• Elastic torsion, circular shafts, Saint-Venant theory.
• Maximum shear occurs at outer surface (radius R = D/2).

Concept / Approach:
For a hollow circular section, J = (pi/32) * (D^4 - d^4). The polar section modulus is Z_p = J / R = J / (D/2) = (2J) / D.

Step-by-Step Solution:
Compute J: J = (pi/32) * (D^4 - d^4)Set R = D/2Z_p = J / R = ((pi/32) * (D^4 - d^4)) / (D/2)Simplify: Z_p = (pi/16) * (D^4 - d^4) / D

Verification / Alternative check:
Dimension check: numerator has length^4; dividing by D gives length^3, consistent with section modulus.

Why Other Options Are Wrong:
Option a: same as correct but unsimplified; both evaluate numerically equal, yet standard compact form is option b.Option c: uses D^4 + d^4, which is incorrect for hollow sections.Option d: mixes constants incorrectly and leads to a wrong multiplier.

Common Pitfalls:
Confusing J (pi/32)*(D^4 - d^4) with I (pi/64)*D^4 for bending.

Final Answer:
Z_p = (pi/16) * (D^4 - d^4) / D