column_model.py. End-fixity factors are the idealised textbook values; real end conditions are rarely that clean, and material constants here are representative.
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.
Releases in which this page changed, newest last. Derived from the archived copy of every release, not from notes written afterwards — so it reflects what actually shipped. Site-wide passes are left out; they are in the full changelog.