How to use this calculator
Choose a material (or type E and yield strength), the unbraced length, the end conditions, and the section’s weakest second moment of area and its area. The calculator returns the Euler load and decides whether Euler’s formula applies by comparing the slenderness ratio KL/r with the transition value. When the column is too stocky for Euler, it also gives the Johnson parabola result.
Formulas
Effective length factor K
| End conditions | Theoretical K | Recommended design K |
|---|---|---|
| Fixed – fixed | 0.5 | 0.65 |
| Fixed – pinned | 0.7 | 0.80 |
| Fixed – fixed, one end free to sway | 1.0 | 1.2 |
| Pinned – pinned | 1.0 | 1.0 |
| Fixed – free (flagpole) | 2.0 | 2.1 |
| Pinned – fixed against rotation, free to sway | 2.0 | 2.0 |
The recommended values come from the AISC Specification commentary and allow for the fact that real “fixed” ends rotate a little. Select Custom K to use them. The calculator’s fixed–pinned option uses the exact theoretical value, 0.699.
Worked example
Pinned steel column, SI units (the calculator’s default values)
E = 200 GPa, σy = 250 MPa (A36), L = 4 m, K = 1, I = 220 cm⁴ = 2.2 × 10⁻⁶ m⁴, A = 3,000 mm².
- r = √(2.2 × 10⁻⁶ / 0.003) = 0.02708 m = 27.1 mm, so KL/r = 4 / 0.02708 = 147.7.
- Transition: (KL/r)c = √(2π² × 200,000 / 250) = 125.7. Since 147.7 > 125.7, Euler applies.
- Pcr = π² × 200 × 10⁹ × 2.2 × 10⁻⁶ / 4² = 271.4 kN, and σcr = 271,400 / 0.003 = 90.5 MPa, below half the yield strength.
- Shorten the column to 3 m and KL/r drops to 110.8, below the transition. Euler would give 482.5 kN, but the Johnson parabola gives σJ = 152.9 MPa and PJ = 458.6 kN.
Assumptions and limits
- The column is initially straight, prismatic and loaded exactly along its centroidal axis.
- Material is linear elastic up to buckling for the Euler result. The Johnson parabola is an empirical fit for inelastic buckling of intermediate columns.
- Only flexural buckling is considered. Local buckling of thin plates and torsional or flexural-torsional buckling are not checked.
- No safety, resistance or imperfection factors are applied. Real columns carry less than the theoretical load, which is why codes such as AISC 360 and Eurocode 3 use reduced column curves.
Frequently asked questions
Which I should I use?
The smallest one, about the axis the column can bend most easily, unless bracing prevents buckling in that direction. For a wide-flange column that is usually Iy. If the column is braced at different points in each direction, check both axes with their own unbraced lengths.
Why does the calculator say Euler doesn’t apply?
For short, stocky columns the Euler stress would exceed the material’s ability to stay elastic. The column yields or buckles inelastically first, at a lower load. Below the transition slenderness the Johnson parabola gives a more realistic estimate.
Is this the allowable load?
No. It is the theoretical load at which an ideal column buckles. Design capacities divide by a safety factor or multiply by a resistance factor and account for initial crookedness and residual stresses.
For education and preliminary checks only. Have a qualified engineer verify any design before it is built.