Torsion of circular shafts – identify the false proportionality for shear stress In a solid circular shaft under torque T, the shear stress τ at a radius r obeys which of the following proportionalities? Choose the statement that is NOT correct.

Introduction / Context:For circular shafts in pure torsion, the shear stress distribution is linear with radius. Recognizing the correct dependencies helps in sizing shafts and checking maximum shear at the outer surface.

Given Data / Assumptions:

• Circular shaft, linear elastic, Saint-Venant torsion.
• No stress concentrations or warping considered.
• Polar moment of inertia J is constant for a given cross-section.

Concept / Approach:The torsion formula for circular shafts is τ(r) = T * r / J where T is applied torque, r is radial position, and J is the polar moment of inertia. From this, τ ∝ r, τ ∝ T, and τ ∝ 1/J. It is not inversely proportional to r.

Step-by-Step Solution:Write torsion relation: τ = T r / J.Inspect proportionalities: increases linearly with r; doubles if T doubles; halves if J doubles.Therefore the statement “τ inversely proportional to r” is false.

Verification / Alternative check:At the center (r = 0), τ = 0; at the surface (r = R), τ = T R / J (maximum). This confirms τ grows with r.

Why Other Options Are Wrong:

• (a) and (d) are consistent with τ = T r / J.
• (c) is also correct since larger J reduces stress for a given T.
• (e) is wrong because one incorrect statement exists, namely (b).

Common Pitfalls:Confusing stress variation in torsion (linear) with bending (also linear for normal stress) but with different governing equations.

Final Answer:τ is inversely proportional to the distance r from the axis