Buckling is a stability failure
A compressed member can lose lateral stability before the material reaches a simple compressive strength limit. For an ideal slender elastic column, Euler gives Pcr = π²EI/(KL)².
Length is especially powerful
The critical load varies inversely with the square of effective length. Doubling effective length reduces the ideal Euler load by a factor of four. Bracing and end restraint can therefore change stability dramatically.
What K represents
The effective-length factor K represents idealized end restraint. Real joints are rarely perfectly pinned or perfectly fixed, so frame stiffness and connection behavior deserve attention.
Slender versus short columns
Euler theory is an elastic slender-column model. Short and intermediate columns may be governed by yielding or inelastic buckling instead. Local plate buckling can also govern thin-walled sections.
Imperfections matter
Initial crookedness, load eccentricity, residual stress and manufacturing tolerance reduce the usefulness of the perfect-column picture. Use a suitable code or validated nonlinear analysis when stability is safety-critical.
Machine-design verification
Check the full load spectrum, fatigue, lubrication, material data, environment, manufacturing variation and the relevant machine-element standard before release.