Euler's elastic buckling model works best for sufficiently slender columns. Shorter columns can instead yield or crush before an elastic instability is reached.

Radius of gyration

A cross-section's radius of gyration is

$$r=\sqrt{\frac{I}{A}},$$

where $I$ is the area moment of inertia about the relevant buckling axis and $A$ is cross-sectional area.

Slenderness ratio

The dimensionless slenderness ratio is

$$\lambda=\frac{L_e}{r}.$$

Large $\lambda$ describes a long slender member; small $\lambda$ describes a stockier member.

Using Euler's formula, the elastic critical stress can be written

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

Elastic and inelastic regions

At high slenderness, elastic buckling can govern. At low slenderness, material yielding or crushing becomes more important. Between them lies an inelastic buckling region in which both material nonlinearity and instability matter.

Real design rules therefore use column curves or code-specific formulas rather than applying Euler's ideal formula to every compressed member.