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?
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AZ_p = (pi/32) * (D^4 - d^4) / (D/2)
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BZ_p = (pi/16) * (D^4 - d^4) / D
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CZ_p = (pi/32) * (D^4 + d^4) / D
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DZ_p = (pi/64) * (D^4 - d^4) / (D/2)
Answer
Correct Answer: Z_p = (pi/16) * (D^4 - d^4) / D
Explanation
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