Radius of Gyration and Column Buckling per Eurocode 3

Area A measures how much steel resists squashing; the second moment of area I measures bending stiffness. The radius of gyration combines the two:

i = √(I/A)

Imagine concentrating the whole area of the section into a thin ring at distance i from the centroidal axis: that ring would have exactly the same bending stiffness as the real section. So i is a pure measure of shape efficiency: it tells you how far, on average, the material sits from the axis, independent of how much material there is. Two sections can share the same area yet differ enormously in i, and i is what decides how a compressed member behaves.

Because I is defined per axis, so is i. Every profile has i_y about the strong axis and i_z about the weak axis, and for angles also i_v about the inclined principal minor axis v–v. A hollow tube pushes all its material away from both axes at once, which is why tubes have the highest radius of gyration per kilogram of any common shape, and why they make such good columns.

A pin-ended column of buckling length L_cr buckles elastically at the Euler critical force N_cr = π²EI/L_cr², with E = 210 000 MPa for every structural steel grade. Dividing the buckling length by the radius of gyration gives the classical slenderness:

λ = L_cr/i

Slenderness is dimensionless and geometric: it compares how long the member is with how efficiently its cross-section resists bowing sideways. A stocky column (low λ) squashes; a slender one (high λ) buckles long before the steel yields.

Eurocode 3 goes one step further and normalises slenderness against the material, so that one set of design curves works for every steel grade:

λ̄ = √(A·f_y/N_cr) = (λ/λ₁), where λ₁ = π√(E/f_y)

The non-dimensional slenderness λ̄ compares the squash load A·f_y with the elastic critical load. At λ̄ = 1 the two are equal, the classic crossover point between yielding and elastic buckling. Below λ̄ = 0.2, Eurocode 3 lets you ignore buckling entirely and design for the full plastic resistance. A higher steel grade raises f_y and therefore λ̄ for the same geometry. For slender columns stronger steel buys surprisingly little, because buckling is governed by E, which never changes.

Real columns are never perfect: they carry residual stresses from rolling or welding, a slight initial bow, and small load eccentricities. EN 1993-1-1 folds them all into a single imperfection factor α and five buckling curves:

• curve a₀: α = 0.13 (near-perfect members, e.g. high-strength steel rolled sections)
• curve a: α = 0.21
• curve b: α = 0.34
• curve c: α = 0.49
• curve d: α = 0.76 (the most imperfection-sensitive members)

The curve turns λ̄ into a reduction factor χ ≤ 1 via Φ = 0.5·[1 + α·(λ̄ − 0.2) + λ̄²] and χ = 1/(Φ + √(Φ² − λ̄²)). The design buckling resistance is then simply

N_b,Rd = χ·A·f_y/γ_M1, with γ_M1 = 1.0 in most national annexes.

The curves diverge most around λ̄ ≈ 1, where residual stresses interact strongly with buckling: there the gap between curve a and curve d is a large fraction of the resistance, so choosing the wrong curve is not a rounding error.

The curve is assigned by Table 6.2 of EN 1993-1-1 according to how the section was made, its shape, and the axis of buckling:

• Hot-finished hollow sections (CHS, SHS, RHS): curve a for both axes: low residual stresses and a favourable stress pattern make tubes the best-treated family in the code.
• Cold-formed hollow sections: curve c for both axes. Cold forming locks in residual stresses around the whole perimeter, so the identical geometry is penalised relative to its hot-finished twin.
• Rolled I- and H-sections: the curve depends on the depth-to-width ratio h/b and on the axis. Deep, narrow sections with h/b > 1.2 (typical IPE) use curve a about y–y and curve b about z–z; squat wide-flange sections with h/b ≤ 1.2 (typical HEA/HEB) use curve b about y–y and curve c about z–z, with stricter curves for very thick flanges. Welded I-sections are generally one curve worse than their rolled counterparts.
• Angles (L-profiles): curve b, but always checked with the radius of gyration i_v about the inclined principal minor axis v–v, which is smaller than either i_y or i_z measured about the leg-parallel axes.

Every profile page on this site applies these assignments automatically: the built-in calculator picks the correct buckling curve per axis and returns N_b,Rd for your buckling length and steel grade.

A column does not get to choose its buckling axis: it fails about whichever axis gives the lowest χ·A·f_y, which under equal buckling lengths means the axis with the smaller radius of gyration. For I-sections the weak axis is dramatically softer: the web contributes almost nothing to I_z, so i_z is only a fraction of i_y. Wide-flange H-sections narrow that gap; square and circular tubes eliminate it entirely.

The design consequence: unless the weak axis is braced at closer spacing than the strong axis, weak-axis buckling governs and the strong-axis inertia you paid for is irrelevant. This is why unbraced columns are H-sections or tubes, and why a clever bracing layout (restraining z–z at mid-height, for instance) can balance the two slendernesses and unlock a lighter profile.

Angles deserve special care: their principal axes u–u and v–v are rotated relative to the legs, and the governing radius of gyration i_v is smaller than either leg-parallel value.

For a centrally loaded column, a simple decision logic emerges:

1. Both axes unbraced and equal lengths? Favour sections with i_y ≈ i_z: SHS/CHS (hot-finished, curve a on both axes) or wide-flange HEA/HEB. An IPE is a poor column here: its tiny i_z drives λ̄ up and χ down, wasting its excellent strong-axis properties.
2. Weak axis braced? With z–z restrained, an IPE's strong axis can be fully used and it may become competitive again.
3. Mind the fabrication route. A cold-formed SHS sits on curve c while a hot-finished one of the same size sits on curve a; at intermediate slenderness that difference alone can decide the utilisation check.
4. Angles and built-up members: check the v–v axis, and remember that connecting through one leg adds eccentricity on top of pure buckling.

Rules of thumb are a starting point, not a verification. Open any profile in the HEA, SHS or angle tables, enter your buckling lengths, and the calculator returns χ and N_b,Rd per axis with the correct Eurocode 3 buckling curve already applied.