Torsion of Circular Shafts – Effect of scaling torque and diameter For a solid circular shaft (diameter d) with maximum shear stress τ under torque T, if torque becomes 4T and diameter becomes 2d, what is the new maximum shear stress?

Introduction / Context:
The torsional shear stress in a solid circular shaft depends on applied torque and the polar section modulus. Understanding how stress scales with changes in torque and size is key to quick design checks and optimization.

Given Data / Assumptions:

• Original shaft: solid, diameter d, torque T → maximum shear τ.
• New condition: torque 4T, diameter 2d.
• Elastic torsion theory applies (no yielding).

Concept / Approach:

For a solid circular shaft: τ_max = 16 * T / (π * d^3). The stress is directly proportional to T and inversely proportional to d^3. Scale T and d accordingly to find the new stress τ′.

Step-by-Step Solution:

Verification / Alternative check:

Dimensional check confirms stress halves because diameter cubed increases by 8 while torque increases by 4 (net factor 4/8 = 1/2).

Why Other Options Are Wrong:

2τ and τ overestimate stress by ignoring the strong d^3 effect; τ/4 would imply torque unchanged or diameter growth even larger.

Common Pitfalls:

Forgetting the cubic dependence on diameter or confusing polar moment with second moment of area.

Final Answer:

τ/2