Check a compression member against buckling — critical load by Euler for slender columns and the Johnson parabola for stocky ones, the slenderness ratio that decides which governs, and how dramatically the end conditions change everything.
Typical: 2–3 steel, 3–4 aluminum, 4–5 timber.
Critical stress versus slenderness. Stocky columns (left) crush near yield along the Johnson parabola; slender ones (right) buckle elastically along the Euler hyperbola. The two meet at Cc. Your column's operating point sits on whichever branch governs.
Slenderness. λ = K·L / r_min, where r_min = √(I_min/A) is the radius of gyration about the weakest axis — buckling always happens about the axis with the smallest I. The effective-length factor K encodes the end restraint: 1.0 pinned–pinned, 0.5 fixed–fixed, 0.7 fixed–pinned, 2.0 fixed–free.
Which formula. The transition slenderness Cc = √(2π²E/Sy) divides the two regimes. For λ ≥ Cc (long, slender) the column buckles elastically — Euler: σcr = π²E/λ², i.e. Pcr = π²EI/(KL)². For λ < Cc (short, stocky) yielding intervenes first — Johnson: σcr = Sy − (Sy²/4π²E)·λ².
End conditions dominate. Because Pcr ∝ 1/(KL)², going from pinned–pinned (K=1) to fixed–fixed (K=0.5) quadruples the capacity, while a cantilever (K=2) drops it to one-quarter — a 16× swing across the range. Fixing column ends is often the cheapest way to add buckling capacity.
Design load. Allowable load is Pcr/FoS. Buckling is unforgiving and imperfection-sensitive, so factors of safety are higher than for pure strength (2–3 steel, 3–4 aluminum).
Ideal, concentrically-loaded, initially-straight columns. Real columns have eccentricity and out-of-straightness — use the secant formula or a code curve (AISC E3, Eurocode 3 curves a–d) for those. Local buckling of thin walls and flanges is a separate check.