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Euler Buckling Load Calculator

Enter the column's E, I, length, and end-fixity factor K to find the critical buckling load.

How it is calculated

The tool evaluates Euler's formula Pcr = π² · E · I ÷ (K · L)². The units are fixed: E (Young's modulus) in GPa, I (second moment of area of the cross-section, about the axis it buckles around) in mm⁴, L (column length) in metres, and K the dimensionless effective-length factor. Internally E is converted to pascals and I to m⁴, and the load comes out in newtons and in kilonewtons.

If K is left empty it is taken as 1 (pinned at both ends). Common values are 0.5 fixed-fixed, 0.7 fixed-pinned, 1 pinned-pinned and 2 fixed-free. E, I, L and K must be above zero. The result is the elastic critical load only: it does not include a safety factor, crushing of short columns, imperfections or code reductions.

Worked example

A steel column with E = 200 GPa, a square 100 mm section (I = 100⁴ ÷ 12 ≈ 8,333,333 mm⁴), length 3 m and pinned ends (K = 1).

E·I = 200×10⁹ Pa × 8.333×10⁻⁶ m⁴ ≈ 1.667×10⁶ N·m². Effective length K·L = 3 m, so (K·L)² = 9. Pcr = 9.8696 × 1.667×10⁶ ÷ 9 ≈ 1.8277×10⁶ N, which the tool shows as 1,827.70 kN (1,827,704.45 N). Fixing both ends (K = 0.5) would quadruple this to 7,310.82 kN.

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Common mistakes

  • Entering E in MPa or Pa. The field is in GPa, so steel is 200, not 200000.
  • Using the wrong I. For a rectangle I = b·h³ ÷ 12 about the axis the column bends around, which is the weaker axis (the smaller dimension), and the answer must be in mm⁴.
  • Entering the length in millimetres. The length is in metres, and because it is squared, an error of a factor 1000 makes the load a million times off.
  • Choosing K by guesswork. Doubling K cuts the capacity to a quarter, and real end connections are rarely perfectly pinned or fixed.
  • Treating Pcr as a safe design load. It has no safety factor and does not apply to short, stocky columns that crush before they buckle.

About this calculator

Slender columns don't fail by crushing — they fail by buckling, suddenly bowing sideways under a compressive load well below what the material could otherwise withstand. Leonhard Euler derived the critical load at which this happens in 1757: Pcr = π²·E·I / (K·L)², where E is the material's Young's modulus, I is the cross-section's moment of inertia, L is the column length, and K is the end-fixity factor that accounts for how the ends are restrained (K=1 for pinned-pinned, K=0.5 for fixed-fixed, K=2 for fixed-free).

This calculator finds that critical load from the column's properties, a standard early check for structural and mechanical engineers sizing columns, struts, and slender compression members before running a full code-based design check — buckling is a stability failure, not a strength failure, so it can occur even in columns that would easily survive the same load in pure compression.

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