There are three broad classes of steel columns that are defined by the slenderness ratio of the column. The slenderness ratio is the ratio of the length of the column (L) to its radius of gyration (r). We will assume loading at the centroid of the column and no material imperfections. For steel columns:
a) Short column when L/r <40. The failure mode is by yielding (like a compression specimen).
b) Intermediate column when 40 L/r <120. The failure mode is a combination of yielding and inelastic buckling. The pre-buckled deflections are small but some stresses are beyond the linear range. A nonlinear buckling analysis is required for intermediate columns.
c) Slender (Euler) column when L/r > 120. The failure mode is by elastic buckling. The pre-buckled deflections are small and critical load is reached before the material yields. A linear buckling analysis is required for this Euler column. The long slender member subjected to an axial force will respond by deforming laterally. The actual value of the force required to make the structure buckle can vary by up to a factor of four depending on how the two ends of the column are restrained.