Slenderness Ratio (KL/r)

Slenderness is the ratio of a column's effective length to the spread of its own cross-section, and it sorts compression members into three families. Below about KL/r = 40 a steel member is "short" and simply squashes at its yield stress. Above roughly 120 it is genuinely slender and Euler's elastic buckling governs. In between sits the awkward inelastic range where partial yielding and residual stresses from rolling and welding drag the real capacity below both curves — the reason design codes use empirical transition formulas rather than Euler alone. A 3 m pinned column with r = 0.02887 m has λ = 1.0 × 3 ÷ 0.02887 = 104, right in that transition band.

Codes cap slenderness for practical reasons too: KL/r ≤ 200 for compression members and ≤ 300 for tension members in most steel practice, mostly so that pieces are not so whippy they get damaged in shipping and erection. Two traps. Use the least radius of gyration, and use the unbraced length for that axis — a column braced at mid-height about its weak axis has two different lengths to check, and the governing λ may belong to either. K itself is where judgement lives: the theoretical 0.5 for fixed-fixed is almost never achieved in real construction, so 0.65 is the recommended design value.