---
title: "Short vs long column; limiting slenderness ratio"
subject: "Strength of Materials"
chapter: 8
key: "semester_3/strength_of_materials:short-long-column-slenderness"
source: https://www.bctengineer.com/q/semester_3/strength_of_materials/short-long-column-slenderness
site: BCT Engineering
---

# Short vs long column; limiting slenderness ratio

Strength of Materials · Chapter 8: Column Theory

- Asked in: 2081 Bhadra, 1+6 marks
- Video: [Difference between Short Column & Long Column](https://www.youtube.com/watch?v=RrII0bYwLL4) — Civil Engineering
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/strength_of_materials/short-long-column-slenderness
- Whole chapter: https://www.bctengineer.com/md/semester_3/strength_of_materials/chapter-8

## Question

Differentiate short column and long column based on failure mechanism. Calculate the maximum value of slenderness ratio for a column for which Euler's formula is valid. Take $E = 210\ \text{GPa}$ and $\sigma_{cr} = 330\ \text{MPa}$.

## Short answer

Short columns fail by crushing (material strength exceeded), long columns by elastic buckling. For E=210 GPa and σ_cr=330 MPa, the limiting slenderness ratio is π√(E/σ_cr) ≈ 79.3, above which Euler's formula is valid.

## Answer

#### Short column vs long column
| Point | Short column | Long column |
|---|---|---|
| Failure mechanism | Crushing (direct compression), when $P/A$ reaches the yield/crushing stress | Buckling (lateral deflection, elastic instability) at a load well below the crushing load |
| Slenderness ratio $\lambda = L_e/r$ | Small (below about 30 for steel) | Large ($\lambda \ge$ limiting value, about 80 to 100 for steel) |
| Governing formula | $\sigma = P/A$ | Euler's formula $P_{cr} = \pi^2EI/L_e^2$ |
| Dependence | On strength of material | On stiffness $E$, $I$ and length |
| Failure stress | $\sigma_y$ or $\sigma_{crushing}$ | $\sigma_{cr} = \pi^2E/\lambda^2 < \sigma_y$ |

#### Limiting slenderness ratio for Euler's formula
Euler's formula is valid as long as the critical stress does not exceed the given limiting (proportional) stress of 330 MPa. The critical stress is

$$
\sigma_{cr} = \frac{P_{cr}}{A} = \frac{\pi^2E}{\lambda^2}
$$

Setting this equal to the limiting stress of $330$ MPa:

$$
\lambda = \pi\sqrt{\frac{E}{\sigma_{cr}}} = \pi\sqrt{\frac{210\times10^3}{330}} = \pi\times25.22 = 79.25
$$

**Answer: the limiting slenderness ratio is $\lambda \approx 79.3$.** Euler's formula is valid for columns with $\lambda \ge 79.3$ (the critical stress then stays at or below 330 MPa). For smaller $\lambda$, it overestimates the load and must not be used.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. How does a short column fail compared with a long column?
   A. Short buckles; long crushes
   B. Both crush
   C. Both buckle at the same stress
   D. Short fails by crushing; long fails by buckling at a lower load
   Answer: D. Short columns reach crushing stress while long columns lose elastic stability. (Key point: Failure mechanism)

2. What is Euler's critical stress?
   A. pi^2E/lambda^2
   B. pi^2E lambda^2
   C. P/E
   D. pi E/lambda
   Answer: A. sigma_cr = P_cr/A = pi^2E/lambda^2. (Key point: Critical stress)

3. What is the limiting slenderness ratio?
   A. pi sqrt(sigma/E)
   B. pi sqrt(E/sigma) = 79.3
   C. pi^2 E/sigma = 6280
   D. 25.2
   Answer: B. Using 210000 and 330 gives 79.25. (Key point: Limit)

4. For which lambda is Euler's formula valid?
   A. lambda <= 79.3
   B. lambda < 30
   C. lambda >= 79.3
   D. any lambda
   Answer: C. Smaller lambda gives stress above the limit and Euler overestimates. (Key point: Validity)

## Related in Strength of Materials

- [Euler's formula: fixed-free column with limitations](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Aeuler-fixed-free-limitations) — Strength of Materials, chapter 8
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- [Euler's formula both ends fixed, with limitations](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Aeuler-both-ends-fixed-limitation) — Strength of Materials, chapter 8
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