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Short vs long column; limiting slenderness ratio

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.

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 GPaE = 210\ \text{GPa} and σcr=330 MPa\sigma_{cr} = 330\ \text{MPa}.

Short column vs long column

PointShort columnLong column
Failure mechanismCrushing (direct compression), when P/AP/A reaches the yield/crushing stressBuckling (lateral deflection, elastic instability) at a load well below the crushing load
Slenderness ratio λ=Le/r\lambda = L_e/rSmall (below about 30 for steel)Large (λ≥\lambda \ge limiting value, about 80 to 100 for steel)
Governing formulaσ=P/A\sigma = P/AEuler's formula Pcr=π2EI/Le2P_{cr} = \pi^2EI/L_e^2
DependenceOn strength of materialOn stiffness EE, II and length
Failure stressσy\sigma_y or σcrushing\sigma_{crushing}σcr=π2E/λ2<σy\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

σcr=PcrA=π2Eλ2\sigma_{cr} = \frac{P_{cr}}{A} = \frac{\pi^2E}{\lambda^2}

Setting this equal to the limiting stress of 330330 MPa:

λ=πEσcr=π210×103330=π×25.22=79.25\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 λ≈79.3\lambda \approx 79.3. Euler's formula is valid for columns with λ≥79.3\lambda \ge 79.3 (the critical stress then stays at or below 330 MPa). For smaller λ\lambda, it overestimates the load and must not be used.

✓ Answer checked by Claude Opus 5.5 (high effort)

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