What this calculator does
Columns fail by buckling long before the steel reaches its yield strength. This calculator works out how slender the member is, applies Euler for long columns and the Johnson parabola for short ones, and tells you which mechanism controls.
How to use it
- Enter the effective length, the effective-length factor K, and the area and second moment of area of the section.
- K = 1.0 for a pinned-pinned column, 0.7 for fixed-pinned, 0.5 for fixed-fixed and 2.0 for a cantilever.
- The radius of gyration and slenderness are derived automatically.
- Apply the code safety factor afterwards — typically 1.67 for Eurocode buckling checks.
Governing formula
P_cr = π² E I / (K L)² ; P_johnson = A·(Fy − Fy²·λ² / 4π²E) Where:
-
λ= Slenderness = KL / r [–] -
r= Radius of gyration = √(I/A) [mm]
Derived Equations:
λ_c = √(2π²E / Fy) P_design = min(P_Euler, P_Johnson) Worked example — 4 m pinned column, S355
The figures below were produced by running this page's own engine with the example inputs (Column length (m), Effective length factor (K) (–), Second moment of area (I) (mm⁴), Cross-sectional area (A) (mm²), Yield strength (MPa), Modulus of elasticity (MPa)). If the calculator returns something different for the same numbers, that is a bug, not a rounding difference.
| Critical load | 1733 kN |
|---|---|
| Slenderness ratio | 30.86 – |
| Radius of gyration | 129.63 mm |
| Euler critical load | 10573 kN |
| Johnson critical load | 1733 kN |
| Governing formula | Johnson short-column |
| Assessment | Effective length 4 m, slenderness 30.9 — Johnson short-column; critical load 1733 kN. Apply the code safety factor (typically 1.67–2.0). |
- The radius of gyration is √(85.7 × 10⁶ / 5,100) ≈ 130 mm.
- With K = 1 the effective length is 4 m, giving a slenderness of about 31 — a stocky column.
- At this slenderness the Johnson formula gives a much lower load than Euler, so short columns fail by crushing rather than buckling.
Assumptions & limitations
Engineering calculation. Compares the Euler elastic critical load with the J.A. Johnson inelastic formula and reports the lower of the two. No residual-stress or code safety factor is applied.
- Idealised column: perfectly straight, no initial imperfections and no eccentric loading.
- The Johnson parabola is an empirical fit and does not replace the Eurocode buckling curves.
- Local buckling of thin-walled sections is not assessed; only global buckling is considered.
This tool supports professional engineering work; it does not replace a sealed design. Confirm every result against the applicable standard and a licensed engineer before it is built.
References & standards
- EN 1993-1-1 — buckling resistance of compression members.
- AISC 360 — column buckling and effective length.
- Timoshenko & Gere — Theory of Elastic Stability (Euler and Johnson derivations).