Euler Buckling Stress⚠ unverified
Mechanical / Columns · Critical buckling stress of a column
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| E | E | Pa | 200000000000.0 | Young's modulus |
| I | I | m^4 | 1e-05 | Second moment of area |
| A | A | m^2 | 0.01 | Cross-sectional area |
| L | L | m | 3.0 | Length |
| K | K | — | 1.0 | Effective-length factor |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| sigma_cr | σcr | Pa | Critical stress |
The science & history
Understanding the Parameters
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Young's modulus $E$ — appears linearly: buckling stress is proportional to stiffness, and again independent of material strength. Two steels of equal $E$ buckle at the same stress even if one yields at twice the other.
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Section properties $I$ and $A$ — combine into the radius of gyration $r=\sqrt{I/A}$ and hence the slenderness $\lambda = KL/r$. It is the ratio $I/A$, not their absolute sizes, that matters — spread the same area farther out (larger $r$) and the buckling stress rises.
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Length $L$ and factor $K$ — enter through $\lambda$. Because $\sigma_{cr}\propto 1/\lambda^2$, the buckling stress falls off steeply with length: a column twice as slender buckles at a quarter of the stress.
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The result $\sigma_{cr}$ — compare it with $\sigma_y$. If $\sigma_{cr} < \sigma_y$ the column is slender and buckling governs; if $\sigma_{cr} > \sigma_y$ Euler theory is invalid (the column would yield first) and an inelastic formula must be used.
Derivation (Approaching a Proof)
Begin with the Euler critical load (Euler Buckling Load column):
$$P_{cr} = \frac{\pi^2 E I}{(KL)^2}.$$
The buckling stress is the load divided by the area:
$$\sigma_{cr} = \frac{P_{cr}}{A} = \frac{\pi^2 E I}{(KL)^2 A}.$$
Substitute $I = A r^2$ (definition of radius of gyration) to eliminate the absolute section size:
$$\sigma_{cr} = \frac{\pi^2 E (A r^2)}{(KL)^2 A} = \frac{\pi^2 E}{(KL/r)^2} = \frac{\pi^2 E}{\lambda^2}.$$
The area cancels, leaving buckling stress as a function of material stiffness and slenderness only. This is the key insight: for a slender column, allowable stress is set by geometry (slenderness), not by material strength. The curve $\sigma_{cr}(\lambda) = \pi^2 E/\lambda^2$ is truncated at $\sigma_{cr}=\sigma_y$, defining the transition slenderness (Critical Slenderness).
Dimensional check. $[\sigma_{cr}] = \dfrac{\text{Pa}}{(\text{dimensionless})^2} = \text{Pa}$. ✓
History and Development
Recasting Euler's 1744 load as a stress-vs-slenderness curve was the 19th-century engineering contribution that made buckling usable in design. Plotting $\sigma_{cr}$ against $\lambda$ exposed the gap at low slenderness where real columns failed early, motivating the Rankine–Gordon, Johnson parabola (Intermediate Column Load), and tangent-modulus theories, and ultimately the smooth column curves of AISC and Eurocode 3.
Related Concepts: Euler Buckling Load column, Slenderness Ratio, Critical Slenderness, Intermediate Column Load, Factor of Safety, Radius of Gyration
Notes: Valid only where $\sigma_{cr} \le \sigma_y$ (slender columns). For short/intermediate columns use $\min(\sigma_y, \sigma_{cr})$ or the Johnson parabola. Real columns with initial crookedness and eccentricity buckle below the ideal $\sigma_{cr}$ — see Secant Formula Stress.