Column Buckling: Euler vs Johnson — Which Formula to Use

The slenderness ratio λ = Le/r captures everything about a column's susceptibility to buckling in one number. The effective length Le = K·L accounts for the end conditions: pinned-pinned K = 1.0, fixed-fixed K = 0.5, fixed-free K = 2.0, fixed-pinned K = 0.7. The radius of gyration r = √(I/A) comes from the cross-section geometry — a smaller r relative to the length means the column is more slender and more prone to buckling.

Three regimes follow directly from λ. Very stocky columns (λ ≤ 20) fail by simple compression yielding; the load capacity is just the yield strength times the area. Intermediate columns (20 < λ ≤ λ₁) follow the Johnson parabola. Long columns (λ > λ₁) fail elastically and the Euler formula governs. The tangent slenderness λ₁ = π·√(2E/Sy) marks the boundary where the Euler critical stress exactly equals half the yield strength — this is the cross-over point and is different for every material.

For λ > λ₁, the column buckles before the material yields. The Euler formula expresses the critical stress as a function of the elastic modulus and the slenderness squared. Because the stress varies as 1/λ², a doubling of column length reduces the critical load to one quarter. The critical load P_cr = σ_cr · A.

The Euler formula is purely a geometry and stiffness result — material strength does not appear. A long, slender steel column and a long, slender aluminium column of the same slenderness ratio have critical stresses in the ratio of their elastic moduli (~3:1), regardless of their yield strengths.

Between the crushing regime and the Euler zone, partial yielding and geometric imperfections lower the critical stress below the Euler prediction. The Johnson parabola fits this transition by starting at the yield strength (at λ = 0) and blending smoothly down to Sy/2 at the tangent slenderness λ₁, where it meets the Euler curve exactly. Using the Euler formula in this regime would overestimate the critical load and is unconservative.

In practice, most machine structural members — brackets, press frames, hydraulic cylinder rods and machine tool columns — fall in the Johnson regime. The boundary λ₁ for S355 steel (E = 210 000 MPa, Sy = 355 MPa) is about 108; for a softer S250 grade (Sy = 250 MPa) it rises to about 129. Always compute λ₁ for the actual material before deciding which formula applies.

The selection logic is: compute λ and λ₁, compare, then pick the formula. Always use the governing (lower) critical load — never apply Euler when Johnson gives a smaller result for the same slenderness. The MechanixCalc calculator runs this comparison in a fail-closed sequence so a column is never assigned the Euler load when the Johnson parabola would be more conservative.

For a column in service, a minimum safety factor SF = P_cr / P_applied ≥ 3 is commonly required (some codes and applications require higher — machinery frames often use SF ≥ 4). The safety factor must be applied to P_cr, not to the critical stress alone, because the cross-section area also carries uncertainty. Additional checks are needed for I-section columns (lateral-torsional buckling per EN 1993-1-1 §6.3.2), thin-wall sections (local plate buckling per EN 1993-1-5), and any column with an eccentric or combined axial-plus-bending load.