The ratio of the flexural rigidity of a beam of section b × d to that of another beam of section b × 2d of the same material is:
Try it first: pick an option, then read the explanation.
Correct answer: C. 1/8
Asked in SSC JE CWC & MES 2011
Detailed Explanation
Flexural rigidity is EI. The material (E) is the same, so the ratio is the ratio of I = bd³/12.
\[ \frac{EI_1}{EI_2} = \frac{bd^3/12}{b(2d)^3/12} = \frac{1}{8} \]
Why “1/8” Is Correct
Doubling the depth increases I eight times, so the ratio is 1/8.
| Section | I about centroidal axis | y max | Section modulus Z |
|---|---|---|---|
| Rectangle b × d (axis ∥ b) | bd³/12 | d/2 | bd²/6 |
| Square side a (sides horizontal) | a⁴/12 | a/2 | a³/6 |
| Square side a (diagonal horizontal) | a⁴/12 | a/√2 | a³/(6√2) |
| Solid circle, diameter d | πd⁴/64 | d/2 | πd³/32 |
| Triangle b × h (axis ∥ base) | bh³/36 | 2h/3 | bh²/24 |
| Hollow circle D, d | π(D⁴ − d⁴)/64 | D/2 | π(D⁴ − d⁴)/(32D) |
Why the Other Options Are Wrong
- A. 1/2: 1/2 would be true if I were proportional to d.
- B. 1/4: 1/4 is the ratio of Z (∝ d²), i.e. of strength, not rigidity.
- D. 1/16: 1/16 would need I ∝ d⁴.
Key Point
EI ∝ bd³: doubling depth ⇒ 8× stiffer, 4× stronger.
Key Concepts for Students
- Strength ∝ Z ∝ d²; stiffness ∝ I ∝ d³.
- Deflection ∝ 1/EI.
- Depth is the most effective way to stiffen a beam.
Practice more: this is question 57 in Bending and Shear Stress MCQs – Part 4. Try the whole set and check every answer instantly.