Column Buckling Calculator (Euler)

A steel tube post. A 60 mm round tube with a 4 mm wall has A = π(60² − 52²) ÷ 4 = 703.7 mm² and I = π(60⁴ − 52⁴) ÷ 64 = 277,264 mm⁴, so r = √(277,264 ÷ 703.7) = 19.85 mm. Pinned at both ends over 3 m, its slenderness is 3,000 ÷ 19.85 = 151.1, above the A36 transition of 125.7, so Euler applies: P = π² × 200,000 × 277,264 ÷ 3,000² = 60.81 kN (σ = 86.41 MPa). Carrying 25 kN, its safety factor against buckling is 2.43.

Changing the ends. The same tube set in concrete at the bottom and free at the top (K = 2) has an effective length of 6 m and buckles at a quarter of the load, 15.20 kN. Fixed at both ends (K = 0.5) it would take 243.2 kN in theory; with the AISC design value K = 0.65 the load is 1.69 times smaller than that.

The common mistake: using Euler on a short column. A 40 × 40 mm A36 bar 0.8 m long has r = 11.55 mm and KL/r = 69.3, well under 125.7. Euler gives 658.0 kN, a stress of 411 MPa that is far above the 250 MPa yield, so the bar would yield long before. The Johnson parabola gives σ = 250 − (250 × 69.3 ÷ 2π)² ÷ 200,000 = 212.0 MPa and a critical load of 339.2 kN, about half the Euler figure.

Leonhard Euler worked out the buckling of an elastic column in 1744, in an appendix on elastic curves to his book on the calculus of variations, Methodus inveniendi lineas curvas, and returned to the strength of columns in a paper of 1757. His result, that the load depends on stiffness and the square of the length but not on the strength of the material, went largely unused for a century, because the columns of the time were short and stocky and failed by crushing.

Iron and then steel made slender columns common, and tests showed that Euler’s formula badly overestimated shorter ones. Engineers fitted empirical curves to the test results; the parabola published by the American engineer J. B. Johnson in the 1890s is still taught because it meets Euler’s curve smoothly and caps the stress at the yield strength. Friedrich Engesser’s tangent-modulus theory of 1889 and F. R. Shanley’s paper of 1947 explained why columns that start to yield buckle at lower loads.

Modern codes such as AISC 360 in the United States and Eurocode 3 in Europe use column curves that also allow for initial crookedness and residual stresses from rolling and welding, so they give lower loads than either formula here.

This calculator works out the critical buckling load and stress of a straight, centrally loaded column, choosing Euler’s formula or the Johnson parabola from the slenderness. It can also give the longest column that carries a load with a chosen safety factor, or the second moment of area a load needs, with the solid round bar that would do. The section properties come from the same geometry engine as the Moment of Inertia Calculator, and buckling is checked about the weakest (minimum principal) axis, which matters for angles. I-beam and channel sections are modelled without their root fillets, so the area and I are a few percent lower than the steel tables.

The material values are typical: E = 200 GPa and a 250 MPa yield for A36, 210 GPa and 355 MPa for S355 (EN 10025-2), 68.9 GPa and 276 MPa for 6061-T6 aluminium. The K factors are the theoretical values, or the recommended design values from the AISC Commentary. This is an engineering estimate, not a code check: real columns are never perfectly straight or centrally loaded. Everything runs in your browser; nothing you enter is sent anywhere.

Related tools: Moment of Inertia & Centroid Calculator, Stress, Strain & Young’s Modulus Calculator, and Beam Calculator.