Torsion of Shafts MCQ with answer and solution

Concept

Maximum torsional shear stress in solid shaft

Formula

τ_max = 16·T / (π × d³)

Step-by-step solution

1. Applied torque T = 1 kNm = 10⁶ Nmm; shaft diameter d = 40 mm. 2. Polar section modulus Z_p = π × d³ / 16 = π × (40)³ / 16 = 4,000 × π mm³. 3. Maximum shear stress τ_max = T / Z_p = 10⁶ / (4000 × π) = 250 / π. 4. Evaluating numerically: 250 / 3.14159 = 79.58 MPa approx 79.6 MPa. Answer: D

Explanation

Maximum shear stress in a circular shaft subjected to torque T is τ_max = 16·T / (π·d³). Substituting T = 1 kNm = 1,000,000 Nmm and diameter d = 40 mm yields τ_max = 16 × 10⁶ / (π × 64,000) = 250 / π = 79.58 MPa, which rounds to 79.6 MPa.

Why the other options are wrong

A) 39.8 MPa results from using 32 in the numerator instead of 16 (inverting Z_p formula). B) 49.7 MPa is an incorrect radius substitution. C) 59.7 MPa is an intermediate arithmetic error.

Shortcut

τ = 16 × 10⁶ / (π × 64000) = 250 / π ≈ 79.6 MPa.

Common mistake

Forgetting to convert torque from kNm to Nmm (10⁶ factor) or using 32 instead of 16.