Solid circular shaft under torsion: The shear stress τ is not directly proportional to which quantity?

Introduction:
The question probes your understanding of torsion in circular shafts and which variables directly govern shear stress distribution. Knowing the exact proportionalities helps avoid design errors and separates material response from geometric and loading effects.

Given Data / Assumptions:

• Homogeneous, isotropic, circular shaft in pure torsion.
• Linear elastic behavior and Saint-Venant torsion theory apply.
• Polar moment of inertia J is for a solid circle: J = (π/32) * d^4.

Concept / Approach:
For a circular shaft under applied torque T, shear stress varies linearly with radius: τ(r) = T * r / J. This expression shows which parameters directly control τ for a given T: radius r (linearly), polar section property J (inversely), and torque T (linearly). Length L and modulus G do not appear in τ = Tr/J; they govern angle of twist, not stress, via θ = TL/(GJ).

Step-by-Step Solution:
Start from τ(r) = T * r / J (no L or G present).Therefore, τ is directly proportional to r and T and inversely to J.Angle of twist depends on L and G: θ = TL/(GJ), not τ.Hence τ is not directly proportional to shaft length L.

Verification / Alternative check:
Dimensional analysis of τ = Tr/J confirms independence from L and G for stress under given torque.

Why Other Options Are Wrong:
Radius of the shaft: τ increases linearly with radius.Angle of twist: Although related to T, τ is not directly proportional to θ unless L and G are fixed; the primary stress formula does not use θ.Modulus of rigidity: G influences θ, not τ = Tr/J.Applied torque: τ is directly proportional to T.

Common Pitfalls:
Mixing the stress formula (τ = Tr/J) with the twist formula (θ = TL/(GJ)); assuming that longer shafts have higher stress under the same torque—stress is geometry and load dependent, not length dependent.

Final Answer:
length of the shaft.