Euler Critical Buckling Load

Also known as column buckling load · critical load

P_{cr} = \frac{\pi^{2} E I}{(K L)^{2}}

Worked example: pinned 4 m steel column, I = 1e-5 m^4 → 1234 kN — press Try an example to run it live, then adjust anything.

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Euler Critical Buckling Load explained

A slender column does not fail by crushing — it goes sideways, suddenly and without warning, at a load that has nothing to do with the material's strength. Leonhard Euler solved the problem in 1744 as an appendix on elastic curves, and the result is startling: only E, I and length appear. A high-strength steel column and a mild steel column of identical shape buckle at exactly the same load. A 4 m pinned steel column with I = 10⁻⁵ m⁴ carries $P_{\mathrm{cr}}$ = π² × 200 × 10⁹ × 10⁻⁵ ÷ 4² = 1.234 × 10⁶ N, about 1234 kN.

K captures the end restraint by converting the real length into the effective length between inflection points: 1.0 pinned-pinned, 0.5 fixed-fixed, 0.7 fixed-pinned, and 2.0 for a flagpole fixed at the base and free at the top — which means a cantilevered column buckles at a sixteenth of the pinned-pinned load. Two traps. Use the least I of the section, since the column buckles about its weak axis, and check slenderness: below KL/r of roughly 100–120 for steel the column yields before Euler's elastic curve is reached, and the formula badly overpredicts. The 1907 Quebec Bridge collapse, which killed 75 workers, came down to compression chords whose buckling capacity had been overestimated on exactly that kind of error.

Euler Critical Buckling Load formula

P_{cr} = \frac{\pi^{2} E I}{(K L)^{2}}

Where

$P_{cr}$ = Critical buckling load (N)
$E$ = Young's modulus (kPa)
$I$ = Least area moment of inertia (mm⁴)
$K$ = Effective length factor
$L$ = Unbraced length (m)

Missing one of these? Work it out first, then come back

Critical buckling load — Pulley System Effort Force, Max Bending Moment — Centre Point Load
Young's modulus — Young's Modulus (E = σ/ε), Axial Deformation (δ = PL/AE)
Least area moment of inertia — Bending Stress (σ = Mc/I), Beam Deflection — Simply Supported, Centre Load
Effective length factor — Slenderness Ratio (KL/r), Universal Soil Loss Equation (USLE)
Unbraced length — Slenderness Ratio (KL/r), Normal Strain (ε = δ/L)

Part of this formula set

Column buckling and slenderness

5 formulasThe Euler critical load, radius of gyration and slenderness ratio — how a column fails by going sideways rather than crushing.