UK Combined Loading Design — Axial Compression and Bending per EN 1993-1-1 Clause 6.3.3 + UK NA

Design of steel members subject to combined axial compression and bending moment (beam-columns) per EN 1993-1-1 Clause 6.3.3 with UK National Annex. Interaction formulae, kyy and kzy factors, Annex A (Method 1) vs Annex B (Method 2), and a worked example using a UK UC section in S355.

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Interaction Framework per Clause 6.3.3

For members subject to combined axial compression and bending, EN 1993-1-1 Clause 6.3.3 requires two interaction checks — one for each buckling plane:

y-y (major axis) buckling plane:

NEd / (χy NRk/γM1) + kyy × My,Ed / (My,Rk/γM1) + kyz × Mz,Ed / (Mz,Rk/γM1) ≤ 1.0

z-z (minor axis) buckling plane:

NEd / (χz NRk/γM1) + kzy × My,Ed / (My,Rk/γM1) + kzz × Mz,Ed / (Mz,Rk/γM1) ≤ 1.0

Where:

• NRk = A × fy (Class 1-3) or Aeff × fy (Class 4)
• My,Rk = Wpl,y × fy (Class 1-2), Wel,y × fy (Class 3), or Weff,y × fy (Class 4)
• Mz,Rk = corresponding minor-axis resistance
• kyy, kyz, kzy, kzz = interaction factors from Annex A or Annex B
• γM1 = 1.00 (UK NA value)

Annex B (Method 2) — Interaction Factors

Annex B is the simpler method suitable for manual calculation and is standard in UK design offices.

kyy (Major Axis Interaction)

kyy = Cmy × [1 + (λ̄y − 0.2) × NEd / (χy × NRk / γM1)]

but kyy ≤ Cmy × [1 + 0.8 × NEd / (χy × NRk / γM1)]

Equivalent Uniform Moment Factor Cmy

Cmy = 0.6 + 0.4ψ ≥ 0.4 (for end moments only, Table B.3)

kzy (Minor Axis from Major Axis Bending)

kzy = max(1 − [0.1λ̄z / (CmL,T − 0.25)] × NEd / (χz × NRk / γM1), 0.6 × kyy × [1 + (λ̄z − 0.2) × NEd / (χz × NRk / γM1)] )

For doubly symmetric sections without torsional susceptibility, kzy = 0.6 × kyy (conservative).

Worked Example — 254×254×89 UC, S355

Column:

• NEd = 1,200 kN, My,Ed = 80 kN·m (end moment, ψ = 0), Mz,Ed = 0
• L = 4.0 m, braced frame, Lcr,y = Lcr,z = 4.0 m
• Section: 254×254×89 UC, S355, A = 114 cm², Wpl,y = 1,220 cm³
• iy = 11.4 cm, iz = 6.59 cm

Section Resistance

NRk = 11,400 × 355 = 4,047 kN My,Rk = 1,220×10³ × 355 = 433 kN·m

Buckling Reduction Factors (from earlier example)

χy = 0.936 (curve a), χz = 0.729 (curve b) λ̄y = 0.459, λ̄z = 0.794

Check y-y Buckling Plane

NEd / (χy NRk/γM1) = 1,200 / (0.936 × 4,047) = 1,200 / 3,788 = 0.317

Cmy = 0.6 (ψ = 0, end moment, Table B.3)

kyy = 0.6 × [1 + (0.459 − 0.2) × 0.317 / 0.936] = 0.6 × [1 + 0.088] = 0.653

kyy ≤ 0.6 × [1 + 0.8 × 0.317 / 0.936] = 0.6 × 1.271 = 0.763 → kyy = 0.653 OK

Interaction: 0.317 + 0.653 × (80 / 433) = 0.317 + 0.121 = 0.438 ≤ 1.0 — OK

Check z-z Buckling Plane

NEd / (χz NRk/γM1) = 1,200 / (0.729 × 4,047) = 1,200 / 2,950 = 0.407

kzy = 0.6 × kyy = 0.6 × 0.653 = 0.392

Interaction: 0.407 + 0.392 × (80 / 433) = 0.407 + 0.072 = 0.479 ≤ 1.0 — OK

Verdict

The 254×254×89 UC in S355 is adequate for the combined loading (48 % utilisation in z-z plane governs).

Design Resources

• UK Steel Grades Reference — EN 10025-2 grade selection for UK projects
• UK Steel Mechanical Properties — fy, fu, and elongation tables
• UK Universal Beam and Column Sizes — UB/UC section dimensions and properties
• UK Bolt Capacity Tables — Class 8.8 and 10.9 bolt resistance
• UK Beam Design Guide — EN 1993-1-1 flexure, shear, and LTB
• UK Connection Design Guide — EN 1993-1-8 bolted and welded joints
• All UK Steel Design References — complete library

Frequently Asked Questions

What is the difference between Annex A and Annex B for combined loading?

Annex A (Method 1) uses refined interaction factors that account for the cross-section shape (through parameters Cyy, Czy, Cyz, Czz) and the degree of plasticity. Annex B (Method 2) is simpler and uses the Cmy equivalent moment factor approach. For doubly symmetric I-sections in braced frames with predominantly major-axis bending, both methods give similar results. Annex A is more accurate for asymmetric sections, biaxial bending, and slender members. The UK NA allows both methods.

When should I use the combined loading check instead of separate axial and bending checks?

The combined loading check per Clause 6.3.3 must be performed whenever both NEd > 0.25 Nb,Rd AND My,Ed > 0.04 Mpl,Rd (or Mz,Ed > 0.04 Mz,Rd). When either ratio is below these thresholds, the interaction effect is negligible and separate axial and bending checks suffice. For typical UK UC columns with moderate bending (NEd > 25 % of buckling resistance), the combined check is usually required.

What is the kyy factor for a column with uniform major-axis moment?

For a uniform moment diagram (ψ = 1.0), Cmy = 1.0. The kyy factor is: kyy = 1.0 × [1 + (λ̄y − 0.2) × NEd/(χy × NRk/γM1)]. For a typical UK UC column with λ̄y ≈ 0.5 and NEd/(χyNRk) ≈ 0.3, kyy ≈ 1.0 × (1 + 0.3 × 0.32) = 1.10. The factor increases the moment effect by approximately 10 % to account for second-order amplification in the beam-column.

Does the UK NA modify the Clause 6.3.3 interaction check?

No. The UK NA to BS EN 1993-1-1 adopts the Clause 6.3.3 interaction check without modification. Both Annex A and Annex B are permitted. The UK NA does not specify a preference, though UK design offices typically use Annex B (Method 2) for hand calculations and Annex A in design software.

Related Pages

• EN 1993 Steel Design Overview
• European Steel Properties
• EN 1993 Beam Design Guide
• EN 1993 Column Buckling
• EN 1990 Load Combinations
• UK Steel Chemical Composition
• UK Steel Charpy Values

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.