Two shafts of the same length (one hollow, one solid) transmit equal torques and have the same maximum shear stress. Which property must be equal for the two shafts?

Introduction / Context:
Strength in torsion is governed by the relationship between applied torque and maximum shear stress. Distinguishing between stiffness quantities (like J and angle of twist) and strength quantities (like Zp) avoids design errors.

Given Data / Assumptions:

• Equal torque T on both shafts.
• Same maximum shear stress τ_max on both shafts.
• Same length; materials may be the same (for strength equality it is not required if τ_max is the same limit).

Concept / Approach:
The torsion formula for strength is τ_max = T / Zp, where Zp = J / R (R = outer radius). If T and τ_max are equal for both shafts, then Zp must be equal. J alone is insufficient because τ_max also depends on the outer radius.

Step-by-Step Solution:

Verification / Alternative check:
Angle of twist relates to stiffness: θ = T L / (G J). Equal θ would demand equal J, not required by equal τ and T; hence Zp is the correct equality for strength.

Why Other Options Are Wrong:
Polar moment J alone ignores the effect of R on stress; diameter equality is unnecessary; angle of twist pertains to stiffness, not strength.

Common Pitfalls:
Confusing Zp (strength) with J (stiffness) and designing for the wrong criterion.

Final Answer:

Polar section modulus (Zp = J / R)