A hollow shaft is subjected to torsion. Which description correctly shows the shear stress variation along its radius?
Try it first: pick an option, then read the explanation.
Correct answer: C. Zero in the hole; across the wall, linear from a non-zero value at the inner surface to the maximum at the outer surface
Asked in SSC JE 03/03/2017 (Morning)
Detailed Explanation
The book shows four sketches (A)–(D); they are described in words here. Shear stress in a circular shaft is τ = Tρ/J, linear in the radius ρ wherever there is material.
\[ \tau(\rho) = \frac{T\rho}{J},\quad r_i \le \rho \le r_o \qquad \tau_{inner} = \tau_{max}\frac{r_i}{r_o} \]
Why “Zero in the hole; across the wall, linear from a non-zero value at the inner surface to the maximum at the outer surface” Is Correct
There is no material in the hole, so no stress there. In the wall, the stress starts at τmax·ri/ro at the inner surface and rises linearly to τmax — a trapezoidal diagram (book diagram C).
Why the Other Options Are Wrong
- A. Linear from zero at the centre to maximum at the outer surface, across the whole radius including the hole: There is no material, hence no stress, inside the hole.
- B. Uniform (constant) across the wall thickness: Uniform stress is only an approximation for very thin tubes; strictly it varies linearly.
- D. Zero in the hole; across the wall, linear from zero at the inner surface to the maximum at the outer surface: The inner surface is at radius ri > 0, so its stress is not zero.
Key Point
Hollow shaft: τ ∝ ρ within the wall — trapezoid from τmax·ri/ro to τmax.
Key Concepts for Students
- Same linear law as a solid shaft, minus the core.
- Thin tubes: nearly uniform τ (Bredt).
- Options reworded from the book's sketches.
Practice more: this is question 27 in Torsion MCQs – Part 2. Try the whole set and check every answer instantly.