Column buckling and slenderness

Euler bucklingcritical loadslenderness ratioKL/reffective length factorradius of gyration

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

Euler Critical Buckling Load

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

Euler's critical buckling load for a slender column, using the end-condition factor K and the area moment of inertia in m⁴.

Area Moment of Inertia — Rectangle

I = \frac{b h^{3}}{12}

Area moment of inertia of a rectangle about its centroidal axis, I = bh³/12, returned in m⁴ and entered as a plain number.

Area Moment of Inertia — Solid Round Bar

I = \frac{\pi d^{4}}{64}

Area moment of inertia of a solid round bar about a diameter, I = πd⁴/64, returned in m⁴ and entered as a plain number.

Radius of Gyration (r = √(I/A))

r = \sqrt{\frac{I}{A}}

Radius of gyration of a cross-section, r = √(I/A), the shape property that decides how slender a column really is.

Slenderness Ratio (KL/r)

\lambda = \frac{K L}{r}

Slenderness ratio KL/r of a compression member, the single number that decides whether a column crushes or buckles.

How they fit together

A long column does not fail by running out of compressive strength; it fails by finding a sideways shape that costs less energy than staying straight. Euler's load says how much axial force that takes, and every term in it is geometric except E — the material's strength does not appear at all, which is why a mild-steel strut and a high-strength one of identical section buckle at the same load. Slenderness KL/r compresses the whole geometry into one number, with the effective-length factor K carrying what the end conditions do: 1.0 pinned, 0.5 fixed at both ends, 2.0 for a flagpole fixed at the base and free at the top.

The slenderness ratio is the decision rule, not an answer. Above roughly 100 for steel the column is slender and Euler governs; well below that it is a short column that will squash first, and Euler will hand you a critical load far above the material's yield strength — an answer that is arithmetically correct and physically nonsense. Always check the Euler stress against the yield stress before believing it. The other classic error is using the wrong I: a column buckles about its weakest axis, so a rectangular section takes the smaller second moment unless bracing prevents movement in that direction, and the radius of gyration r = √(I/A) must be taken about that same axis.