What it does
A slim column under compression does not crush first: it bows sideways and collapses at a load far below the crushing load. That is buckling. This calculator works out the load at which an ideal, straight column buckles:
- Section properties: area A, second moments of area Ix and Iy, and radii of gyration r = √(I/A), for an I or H section, a solid rectangle, a solid circle or a round tube.
- Effective length KL, where K depends on how the ends are held: 1 for pinned ends, 2 for a flagpole fixed at the base, 0.7 for fixed at one end and pinned at the other, 0.5 for both ends fixed.
- Slenderness λ = KL / r, using the smaller r, because the column buckles about its weakest axis.
- Euler’s critical load P = π²EI / (KL)², valid for slender columns.
- The Johnson parabola σ = fy − (fy² / 4π²E) λ² for intermediate columns, where Euler’s formula would give stresses near or above yield. The two curves meet at the transition slenderness Cc = √(2π²E / fy), where the stress is fy/2.
- An allowed load with a factor of safety you choose, and a check of an applied load if you give one.
For study and planning, not a substitute for professional judgement. Check your local code and a qualified engineer.
How to use it
- Choose the shape and type its sizes in millimetres.
- Type the length between supports, and pick the end conditions (or type your own K).
- Type the material’s Young’s modulus and yield strength in MPa (N/mm²). The starting values, 200,000 and 250 MPa, are example values for a mild steel.
- Choose a factor of safety, and type an applied load in kN if you want it checked.
The chart shows the column curve: the Johnson parabola up to Cc, the Euler curve beyond, and your column as a dot.
Worked example
A made-up I section 200 mm deep with 150 × 10 mm flanges and a 6 mm web, 4 m long with pinned ends, E = 200,000 MPa and fy = 250 MPa (the page opens with these):
- A = 2 × 150 × 10 + 180 × 6 = 4080 mm²; Ix = 3.002 × 10⁷ mm⁴; Iy = 5.628 × 10⁶ mm⁴, so ry = 37.14 mm. It buckles about the weak (y) axis.
- KL/r = 4000 / 37.14 = 107.7, below Cc = 125.7, so the Johnson parabola applies.
- σ = 250 − (250² / (4π² × 200,000)) × 107.7² = 158.2 MPa, so the critical load is 158.2 × 4080 = 645.4 kN. Euler alone would give 694.4 kN, which overestimates it.
- With a factor of safety of 2, the allowed load is 322.7 kN.
Fixing both ends (K = 0.5) halves the slenderness to 53.8 and raises the critical stress to 227 MPa.
FAQ
Why is my code’s answer lower? Real columns are never perfectly straight, have residual stresses from rolling or welding, and carry loads slightly off-centre. Design codes use their own column curves and factors to allow for this, and usually larger K values than the ideal ones. This calculator gives the ideal-column result for study and a first check.
Can a column buckle about a different axis? Yes, if it is braced differently in each direction. Work out each axis separately with its own effective length and use the lower load.
Does it cover local buckling of thin plates or tubes? No. Very thin flanges, webs or tube walls can buckle locally first; check their width-to-thickness ratios with your code.
Where the numbers come from
- Euler's buckling formula P = π²EI/(KL)², ideal effective length factors from elastic stability theory, and the Johnson parabola σ = fy − (fy/2π)² λ²/E for intermediate columns, as in mechanics of materials textbooks such as Timoshenko and Gere, Theory of Elastic Stability, and Gere and Goodno, Mechanics of Materials (formulas only).
- Section properties from the standard formulas for area and second moment of area.
- No design-code column curves or tables are used. E and fy are example values for a mild steel that the user replaces; the example column is made up.
- Checked against an independent Python calculation written for the tests: grid integration of the sections with NumPy and a finite-difference buckling eigenvalue solution (scipy.linalg.eigh) for pinned and clamped ends.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.