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Euler buckling, explained

In Hebrew: קריסת אוילר

Euler’s formula gives the axial load at which a slender column suddenly bows sideways: P_cr = π²EI/(KL)². It depends on stiffness and length, not on strength.

Why columns buckle

Push on the ends of a long, thin rod and it stays straight, until a certain load. At that load a tiny sideways deflection no longer springs back: the bending caused by the axial load acting through the deflection beats the rod’s bending stiffness. The column bows out and carries no more load. This is an instability, and it can happen while the stress is far below yield.

That is why the formula has E, the elastic modulus, and not the yield strength. A high-strength steel buckles at the same load as a mild steel of the same shape.

End conditions and effective length

K turns the real length into an effective length: the length of the equivalent pinned-pinned column. Pinned at both ends, K = 1. Fixed at one end and free at the other, like a flagpole, K = 2, so it buckles at a quarter of the pinned load. Fixed and pinned, K ≈ 0.7. Fixed at both ends, K = 0.5 in theory. Real fixed ends are never perfectly rigid, so design guides recommend about 0.65 instead.

A column buckles about the axis with the smallest second moment of area I. For a rectangular section that is the thin direction, so check both axes.

When Euler applies

The slenderness ratio KL/r, with r = √(I/A), decides which regime you are in. For long, slender columns the critical stress is below the proportional limit and Euler holds. Short columns crush instead. In between, the Johnson parabola is the usual design curve. Real columns also have initial crookedness and off-centre loads, so they fail somewhat below the Euler load; codes cover this with their own column curves.

A worked number

A solid steel rod, 20 mm in diameter and 1 m long, pinned at both ends. E = 200 GPa and I = πd⁴/64 ≈ 7.85 × 10⁻⁹ m⁴. Then P_cr = π² × 200 × 10⁹ × 7.85 × 10⁻⁹ / 1² ≈ 15.5 kN.

Divide by the area, 3.14 × 10⁻⁴ m², and the stress at buckling is only about 49 MPa, roughly a fifth of a typical yield strength. The rod fails by buckling long before the material yields.

Common mistakes

  • Using the strong-axis I. The column picks the weak axis.
  • Assuming ideal fixed ends and taking K = 0.5 in a design.
  • Applying Euler to a short, stocky column, where it badly over-predicts the load.

Mechanical engineering, explained