Compute the Euler critical buckling load Pcr = π²EI/(KL)² and critical stress of a slender column, with selectable end conditions and a yield-limit check.
Enter E, least moment of inertia I, length, area, and end condition to get the Euler critical load and stress.
About this tool
The Euler Column Buckling Calculator is a free, in-browser tool that finds the critical axial load at which a slender column becomes unstable and buckles. It applies Euler's formula Pcr = π²EI/(KL)², where E is the modulus of elasticity, I the least area moment of inertia, L the unbraced length, and K the effective-length factor set by the end conditions. All calculations run locally in your browser.
The end-condition factor K is selected from the classic cases: pinned-pinned K = 1.0, fixed-pinned K ≈ 0.699 (theoretical 0.7), fixed-fixed K = 0.5, and fixed-free (a flagpole cantilever) K = 2.0 — a lower K means a stiffer, stronger column. The tool also computes the radius of gyration r = √(I/A), the slenderness ratio λ = KL/r, and the critical (buckling) stress σcr = Pcr/A = π²E/λ². If you supply the material yield strength, it checks whether Euler's elastic result is valid: Euler applies only while σcr stays at or below yield, otherwise the column is short or intermediate and the Johnson parabola governs instead. Work in SI (GPa, mm⁴, mm, mm²) or imperial (psi, in⁴, in, in²).
As a worked example, a pinned-pinned steel column (E = 200 GPa, I = 500,000 mm⁴, A = 1,500 mm²) that is 3,000 mm long has Pcr = π² × 200×10⁹ × 5×10⁻⁷ / 3² ≈ 109.7 kN, a radius of gyration r = √(I/A) ≈ 18.3 mm, and a slenderness ratio λ ≈ 164 — well into the slender range where Euler buckling, not yielding, controls. Always apply a factor of safety to Pcr in design.
Frequently asked questions
What is Euler's buckling formula?
The Euler critical load is Pcr = π²EI/(KL)², where E is the elastic modulus, I the least moment of inertia, L the length, and K the effective-length factor. It gives the axial load at which an ideal slender column buckles elastically; real columns need a factor of safety applied to Pcr.
What effective-length factor K should I use?
K depends on the end restraints: pinned-pinned = 1.0, fixed-pinned ≈ 0.699 (theoretical 0.7), fixed-fixed = 0.5, and fixed-free (cantilever) = 2.0. A smaller K shortens the effective length KL and raises the critical load, so a fixed-fixed column is four times as strong as a fixed-free one of equal length.
When is the Euler result not valid?
Euler buckling assumes the column stays elastic, so it holds only while the critical stress σcr = Pcr/A is at or below the yield strength. If σcr exceeds yield, the column is short or intermediate and fails by yielding or inelastic buckling — the Johnson parabola formula applies instead. Enter a yield strength to get this flag.
What is the slenderness ratio?
The slenderness ratio is λ = KL/r, where r = √(I/A) is the radius of gyration. It combines effective length and section shape into one number: high λ means a slender, buckling-prone column governed by Euler, while low λ means a stocky column that fails by crushing or yielding.