Columns and Euler buckling

A short, stocky column fails by crushing. A long, slender one fails at a much lower load by bowing sideways: buckling. Buckling is a stability failure, so it depends on stiffness (E and I) and length rather than on material strength.

Euler's formula gives the elastic buckling load of an ideal straight pin-ended strut. The end conditions change the shape of the buckled curve, which is handled through an effective length: a column fixed at both ends buckles as if it were shorter, while a cantilever column (fixed base, free top) behaves as if it were twice as long.

Slenderness ratio, effective length divided by the least radius of gyration, is the single number that decides whether buckling governs. Real columns are never perfectly straight and loads are never perfectly central, so design codes use curves that blend crushing and buckling and fall below Euler's value.

Buckling always occurs about the weaker axis unless that axis is braced. Bracing a column at mid-height about its weak axis halves the effective length about that axis and raises its buckling load fourfold.