Two beams, one of circular and the other of square cross-section, have equal areas of cross-section. When subjected to bending:
Try it first: pick an option, then read the explanation.
Asked in SSC JE 2012
Detailed Explanation
"Economical" here means more moment resisted for the same area (same weight of material). Compare the section moduli of the two shapes for equal areas.
\[ a^2 = \frac{\pi d^2}{4} \Rightarrow a = 0.886\,d \]
\[ \frac{Z_s}{Z_c} = \frac{a^3/6}{\pi d^3/32} = \frac{0.886^3\times 32}{6\pi} = 1.18 \]
Why “the square section is more economical” Is Correct
The square has an 18 % larger section modulus for the same area, so the square section is more economical. Its material lies farther from the neutral axis (the circle bulges at mid-depth, where bending stress is low).
| 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. the circular section is more economical: The circle has more material near the N.A., so it is less efficient.
- C. both sections are equally economical: Z differs by about 18 %.
- D. both sections are equally stiff: I also differs: I_s/I_c = (a⁴/12)/(πd⁴/64) ≈ 1.05, so they are not equally stiff.
Key Point
Equal areas: Z_square ≈ 1.18 Z_circle ⇒ square stronger and more economical.
Key Concepts for Students
- Efficient sections put material far from the N.A.
- I-section > square > circle for bending efficiency.
- Circles are preferred for torsion (shafts), not bending.
Practice more: this is question 35 in Bending and Shear Stress MCQs – Part 3. Try the whole set and check every answer instantly.