The maximum value of the radius of gyration is obtained if the material ______.
Try it first: pick an option, then read the explanation.
Asked in SSC JE 11/10/2023 (9 am–11 am)
Detailed Explanation
Radius of gyration is the distance at which the whole area could be concentrated to give the same moment of inertia: I = A r².
\[ r = \sqrt{\frac{I}{A}}, \qquad I = \int y^2\, dA \]
Why “is placed farthest from the centroid” Is Correct
For a fixed area A, I increases as the material moves away from the centroidal axis (each element contributes y² dA). So r is maximum when the material is placed farthest from the centroid — which is why hollow tubes and built-up sections make efficient columns.
| Term | Meaning |
|---|---|
| Radius of gyration r | r = √(I/A); use the least value rmin = √(Imin/A) for buckling |
| Slenderness ratio λ | λ = le / rmin |
| Euler stress | σcr = P/A = π²E/λ² |
| Short column / pedestal | le / least lateral dimension ≤ 3 (fails by crushing) |
| Long column | Large λ; fails by buckling — Euler's formula applies |
Why the Other Options Are Wrong
- A. is placed nearest the centroid: Material near the centroid contributes little to I, so r is small.
- B. has the maximum area: More area alone does not raise r — r depends on how the area is distributed (I/A).
- C. is placed at the centroid: Material at the centroid adds area but no moment of inertia; r would be zero for a point.
Key Point
r = √(I/A): spread the material away from the centroid to raise r and lower the slenderness ratio.
Key Concepts for Students
- Hollow circular and box sections give a large r for little weight.
- Buckling uses the least radius of gyration.
- Higher r → lower λ → higher allowable stress.
Practice more: this is question 5 in Columns and Struts MCQs – Part 1. Try the whole set and check every answer instantly.