For a beam of uniform strength with constant depth:
Try it first: pick an option, then read the explanation.
Correct answer: A. width is directly proportional to the bending moment
Asked in SSC JE 22/01/2018 (Evening)
Detailed Explanation
Uniform strength means the same extreme-fibre stress σ = M/Z at every section.
\[ \sigma = \frac{6M}{b\,d^2} = \text{const},\ d = \text{const} \Rightarrow b \propto M \]
Why “width is directly proportional to the bending moment” Is Correct
With the depth fixed, the width is directly proportional to the bending moment.
| Cantilever of uniform strength (σ = 6M/bd² constant) | Varies as | UDL w, value at fixed end (x = l) |
|---|---|---|
| Depth d constant, width b varies | b ∝ M | b = 3wl²/(σd²) |
| Width b constant, depth d varies | d ∝ √M | d = l·√(3w/(bσ)) |
Why the Other Options Are Wrong
- B. width is directly proportional to the square root of the bending moment: b ∝ √M is wrong; it is the depth that goes as √M when the width is constant.
- C. width is directly proportional to three times the square root of the bending moment: No factor of three appears in the proportionality.
- D. width is inversely proportional to the bending moment: More moment needs more width, not less.
Key Point
Uniform strength: d constant ⇒ b ∝ M; b constant ⇒ d ∝ √M.
Key Concepts for Students
- Cantilever + point load, constant depth: triangular plan.
- Cantilever + UDL, constant depth: parabolic plan.
- Laminated (leaf) springs use this idea.
Practice more: this is question 58 in Bending and Shear Stress MCQs – Part 4. Try the whole set and check every answer instantly.