The radius of gyration of a rectangular section (depth D, width B) about a centroidal axis parallel to the width is:
Try it first: pick an option, then read the explanation.
Correct answer: C. D/(2√3)
Asked in SSC JE 23/01/2018 (Evening)
Detailed Explanation
Radius of gyration r is the distance at which the whole area could be concentrated to give the same moment of inertia: I = A r².
\[ r = \sqrt{\frac{I}{A}} = \sqrt{\frac{BD^3/12}{BD}} = \sqrt{\frac{D^2}{12}} = \frac{D}{2\sqrt3} \approx 0.289D \]
Why “D/(2√3)” Is Correct
About the axis parallel to the width, the depth D is the dimension that matters, giving D/(2√3). The width B cancels out.
| 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. D/2: D/2 is the distance to the extreme fibre, not the radius of gyration.
- B. D/√3: D/√3 is the radius of gyration about the base (I = BD³/3).
- D. D/(4√3): D/(4√3) is half the correct value.
Key Point
Rectangle: r = D/(2√3) ≈ 0.289D about the centroidal axis parallel to B.
Key Concepts for Students
- Circle: r = d/4.
- Columns buckle about the axis of least r.
- Slenderness ratio = effective length / least radius of gyration.
Practice more: this is question 19 in Bending and Shear Stress MCQs – Part 2. Try the whole set and check every answer instantly.