Euler's formula is valid for:
Try it first: pick an option, then read the explanation.
Asked in SSC JE 2010
Detailed Explanation
Euler's derivation assumes the column fails purely by elastic buckling, with the stress still below the proportional limit and direct compressive stress negligible.
\[ \sigma_{cr} = \frac{\pi^2E}{\lambda^2} \le \sigma_{p} \;\Rightarrow\; \lambda \ge \pi\sqrt{E/\sigma_p} \approx 90\text{–}100 \text{ for mild steel} \]
Why “long columns only” Is Correct
For short columns Euler's formula would predict enormous loads, far above the crushing load. It is valid for long columns only; Rankine's formula is used for the intermediate range.
| 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. short columns only: Short columns fail by crushing; Euler's formula overestimates their strength.
- C. both short and long columns: It is unsafe for short columns.
- D. none of the above: It is valid for long columns.
Key Point
Euler → long columns (buckling). Short columns → crushing. Rankine–Gordon → covers the full range empirically.
Key Concepts for Students
- Rankine: 1/P = 1/Pc + 1/Pe.
- Euler stress must stay below the proportional limit.
- Limiting slenderness for mild steel ≈ 80–100.
Practice more: this is question 26 in Columns and Struts MCQs – Part 2. Try the whole set and check every answer instantly.