A slender member under compression can become unstable and deflect sideways suddenly even while the material remains elastic. This failure mode is buckling.

For an ideal straight column, Euler's critical load is

$$P_{cr}=\frac{\pi^2EI}{L_e^2},$$

where $E$ is Young's modulus, $I$ is the area moment of inertia about the buckling axis and $L_e$ is the effective length determined by the end restraints.

Instability rather than crushing

Buckling does not require the compressive stress to reach the material yield strength. A long slender column can lose stability at a much lower load.

Effective length

Different end conditions constrain rotation and translation differently. Their effect can be represented by

$$L_e=KL,$$

where $K$ is an effective-length factor.

Geometry matters strongly

Critical load grows linearly with $E$ and $I$ but falls with the square of effective length. Increasing the section's second moment of area or adding bracing can therefore greatly increase buckling resistance.