-- | ----- | -------------------------------------------------------------------- | | a0 | 0.13 | Hot-finished hollow sections (CHS, RHS), S460 up to 40 mm | | a | 0.21 | UC and UB about y-y axis (tf <= 40 mm), hot-finished RHS S355 | | b | 0.34 | UC and UB about z-z axis (tf <= 40 mm) -- most common UK column case | | c | 0.49 | UC and UB z-z (tf > 40 mm), channels, angles, cold-formed RHS | | d | 0.76 | Cold-formed RHS S420-S460, welded box sections with tf > 40 mm |

The selection depends on the cross-section type, the axis of buckling, the steel grade, the flange thickness, and the manufacturing process (hot-rolled vs cold-formed vs welded). Table 6.2 provides the definitive mapping.

Buckling Curve for UK Sections

For the vast majority of UK building columns (UC sections, S275 or S355, tf <= 40 mm), buckling about the z-z (weak) axis with curve b (alpha = 0.34) governs.

Lambda_1 Values for UK Steel Grades

For S355, lambda_1 = 76.4, meaning that a column with slenderness ratio Lcr/i = 76.4 has lambda_bar = 1.0 -- the point at which buckling and yielding effects are equal. Below this, yielding dominates; above, buckling dominates.

Tabulated Chi Values

Worked Example -- 254UC in S355

Given:

• Section: 254 x 254 x 89 UC, S355J2
• L = 4.0 m, pinned ends, braced frame: Lcr,y = Lcr,z = 4.0 m
• N_Ed = 1,800 kN

Section properties (from SCI Blue Book): h = 260.4 mm, b = 256.3 mm, tw = 10.5 mm, tf = 17.3 mm A = 114 cm^2 = 11,400 mm^2 i_y = 11.4 cm = 114 mm, i_z = 6.59 cm = 65.9 mm tf = 17.3 mm <= 40 mm, so curve a for y-y, curve b for z-z.

Slenderness: lambda_1 = 93.9 x sqrt(235/355) = 76.4 lambda_bar_y = (4,000 / 114) / 76.4 = 35.1 / 76.4 = 0.459 lambda_bar_z = (4,000 / 65.9) / 76.4 = 60.7 / 76.4 = 0.794

Reduction factors: y-y, curve a (alpha = 0.21): Phi_y = 0.5 x [1 + 0.21 x (0.459 - 0.2) + 0.459^2] = 0.5 x [1 + 0.054 + 0.211] = 0.633 chi_y = 1 / [0.633 + sqrt(0.633^2 - 0.459^2)] = 1 / [0.633 + 0.435] = 0.936

z-z, curve b (alpha = 0.34): Phi_z = 0.5 x [1 + 0.34 x (0.794 - 0.2) + 0.794^2] = 0.5 x [1 + 0.202 + 0.630] = 0.916 chi_z = 1 / [0.916 + sqrt(0.916^2 - 0.794^2)] = 1 / [0.916 + 0.456] = 0.729

Buckling resistance: N_b,Rd,y = 0.936 x 11,400 x 355 / 1.0 = 3,788 kN N_b,Rd,z = 0.729 x 11,400 x 355 / 1.0 = 2,950 kN (governs)

Utilisation: N_Ed / N_b,Rd,z = 1,800 / 2,950 = 0.61. OK.

Worked Example -- UB Section Used as Column

Given:

• Section: 356 x 171 x 51 UB, S355
• L = 3.5 m, pinned ends
• N_Ed = 850 kN

This UB section is being used as a lightly loaded column, perhaps in a low-rise frame. Note that UB sections used as columns are more efficient about the y-y axis but the z-z axis buckling is more critical.

Section properties: h = 355.0 mm, b = 171.5 mm, tw = 7.3 mm, tf = 11.5 mm A = 64.9 cm^2 = 6,490 mm^2 i_y = 14.7 cm = 147 mm, i_z = 3.93 cm = 39.3 mm tf = 11.5 mm <= 40 mm.

Slenderness: lambda_bar_y = (3,500 / 147) / 76.4 = 23.8 / 76.4 = 0.312 lambda_bar_z = (3,500 / 39.3) / 76.4 = 89.1 / 76.4 = 1.166

The weak-axis slenderness of 1.166 means the column is firmly in the buckling range.

Reduction factor z-z (curve b): Phi_z = 0.5 x [1 + 0.34 x (1.166 - 0.2) + 1.166^2] = 0.5 x [1 + 0.328 + 1.360] = 1.344 chi_z = 1 / [1.344 + sqrt(1.344^2 - 1.166^2)] = 1 / [1.344 + 0.667] = 0.497

Buckling resistance: N_b,Rd,z = 0.497 x 6,490 x 355 / 1.0 = 1,145 kN > 850 kN. OK at 74% utilisation.

This is a typical efficiency for UB columns: chi_z approximately 0.5 at moderate slenderness.

UK National Annex Provisions

The UK NA to BS EN 1993-1-1 confirms:

1. gamma_M1 = 1.00 for buckling resistance of members, unchanged from the recommended value.
2. The buckling curves a0-d and their associated imperfection factors are adopted without modification.
3. For S460 steel, the UK NA confirms that the curve selection for hot-rolled sections is the same as for S235-S355, with curve a0 applicable only to hot-finished hollow sections.
4. For flexural-torsional buckling (Clause 6.3.1.4), the UK NA adopts the recommended provisions without change, noting that single-symmetric sections (channels, tees) require explicit consideration of the interaction between flexural and torsional buckling modes.

Design Resources

• UK Steel Grades Reference -- EN 10025-2 grade selection
• UK Steel Properties -- fy, fu tables
• UK UC and UB Section Properties -- Dimensions and section properties
• UK Effective Length Factors -- Annex E k factors
• UK Combined Loading Design -- Beam-column interaction
• All UK Steel Design References -- complete library

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Frequently Asked Questions

Which buckling curve applies to a UK UC column?

For a Universal Column (UC) in a UK building, with tf <= 40 mm: buckling about the y-y (major) axis uses curve a (alpha = 0.21), and buckling about the z-z (minor) axis uses curve b (alpha = 0.34). Since columns typically buckle about the weaker z-z axis, curve b is the most common for UK UC sections. For heavy UC sections with tf > 40 mm, the z-z axis curve upgrades to c (alpha = 0.49).

How do I calculate the non-dimensional slenderness lambda_bar?

lambda_bar = (Lcr / i) / lambda_1, where lambda_1 = 93.9 x sqrt(235/fy). For S355: lambda_1 = 76.4. For S275: lambda_1 = 86.8. The effective length Lcr = k x L, where k is the effective length factor from EN 1993-1-1 Annex E. For a braced frame column with simple connections: k = 1.0, giving Lcr = L.

What effective length should I use for a braced frame column?

For a braced frame column with nominally pinned connections, Lcr = 1.0 x L (storey height). This is the standard assumption for UK simple construction. With partial rotational restraint (e.g., flush end plate connections), k may reduce to 0.85-1.0 per Annex E. The UK NA confirms that simple connections (fin plates, partial-depth end plates) do not provide significant rotational restraint, supporting the conservative use of k = 1.0.

At what lambda_bar does a column transition from cross-section yielding to buckling?

The transition region is lambda_bar = 0.2 to 0.4. Below 0.2, chi >= 0.98 and buckling effects are negligible -- the column can be designed for cross-section resistance only (N_Ed <= N_pl,Rd). Above 0.4, chi drops significantly and buckling governs. At lambda_bar = 1.0 (the Euler reference slenderness), chi = 0.58 for curve b -- the column achieves 58% of its squash load. At lambda_bar = 2.0, chi = 0.22 -- only 22% of the squash load is available, and a more efficient section should be selected.

Educational reference only. All design values are per BS EN 1993-1-1:2005 + UK National Annex and BS EN 10025-2:2019. Verify all values against the current editions of the standards and the applicable National Annex for your project jurisdiction. Designs must be independently verified by a Chartered Structural Engineer registered with the Institution of Structural Engineers (IStructE) or the Institution of Civil Engineers (ICE). Results are PRELIMINARY -- NOT FOR CONSTRUCTION without independent professional verification.