A slender column fails by suddenly bowing sideways — elastic buckling — governed by Euler's formula, Pcr = π²EI/(KL)², which depends only on stiffness (EI) and geometry, not material strength at all. A short, stocky column instead simply crushes at its yield stress before it ever has a chance to bow. Real columns of intermediate slenderness fail somewhere between those two extremes, which is exactly what Johnson's parabolic formula models — and the two formulas are constructed to meet exactly at the critical slenderness ratio, so this tool automatically picks whichever one actually applies to your column.
A 50 mm diameter, 3 m steel column, pinned at both ends, has a slenderness ratio of 240 — well into the Euler regime — giving a critical load of about 67 kN. Fix both ends instead of pinning them and the same column can carry exactly 4× that load, since fixed ends quarter the effective length in the formula.
Because elastic buckling happens before the material ever reaches its yield point — a slender column fails by suddenly deflecting sideways under a load its material could easily survive in pure compression, which is exactly why buckling is a stiffness problem, not a strength problem.
It appears squared in the denominator, so halving the effective length (say, by fixing rather than pinning an end) quadruples the critical load — end conditions can matter as much as the material or the cross-section itself.
A single number, r = √(I/A), that captures how efficiently a cross-section's material is spread away from its centroid — the same area arranged into a wide, thin shape (like a tube) has a much larger r, and therefore resists buckling far better, than a solid compact shape of equal area.
For a given amount of material (area), pushing that material further from the centreline dramatically increases I and therefore buckling resistance — a hollow tube can carry a far higher critical load than a solid rod of the same weight, which is exactly why so many structural columns are tubular.