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.

UK Combined Loading Design -- EN 1993-1-1 Clause 6.3.3 Axial-Bending Interaction

Members subject to simultaneous axial compression and bending -- beam-columns -- are ubiquitous in steel frames. Columns in sway frames, columns with eccentric connections, crane runway columns, and chord members in trusses all experience combined loading. EN 1993-1-1 Clause 6.3.3 provides the interaction formulae that govern these members, with two alternative methods in Annex A (Method 1) and Annex B (Method 2). The UK National Annex permits both methods without preference. This reference covers the interaction equations, the derivation of the k_ij interaction factors, equivalent uniform moment factors C_m, and worked examples for UK UC and UB sections acting as beam-columns.

The Interaction Equations -- Clause 6.3.3

For members not susceptible to torsional deformations, two interaction checks are required -- one for each buckling plane:

Major axis (y-y) buckling: N_Ed / (chi_y x N_Rk / gamma_M1) + k_yy x M_y,Ed / (M_y,Rk / gamma_M1) + k_yz x M_z,Ed / (M_z,Rk / gamma_M1) <= 1.0

Minor axis (z-z) buckling: N_Ed / (chi_z x N_Rk / gamma_M1) + k_zy x M_y,Ed / (M_y,Rk / gamma_M1) + k_zz x M_z,Ed / (M_z,Rk / gamma_M1) <= 1.0

Where:

N_Rk = A x fy for Class 1-3, A_eff x fy for Class 4
M_y,Rk = W_pl,y x fy for Class 1-2, W_el,y x fy for Class 3
gamma_M1 = 1.00 (UK NA)

The four interaction factors k_yy, k_yz, k_zy, k_zz account for the coupling between axial load and bending moment in the buckled configuration. The factor k_yz represents minor-axis bending induced by major-axis buckling, and k_zy represents major-axis bending induced by minor-axis buckling.

Annex B (Method 2) -- Standard UK Practice

Annex B is the simpler method, suitable for hand calculation, and is the standard approach in UK design offices for doubly symmetric I-sections in building frames.

k_yy -- Major-Axis Interaction Factor

k_yy = C_my x [1 + (lambda_bar_y - 0.2) x N_Ed / (chi_y x N_Rk / gamma_M1)] but k_yy <= C_my x [1 + 0.8 x N_Ed / (chi_y x N_Rk / gamma_M1)]

The equivalent uniform moment factor C_my accounts for the shape of the bending moment diagram about the y-y axis. For end moments only (no transverse loads):

C_my = 0.6 + 0.4 x psi >= 0.4

Where psi = M_2 / M_1, the ratio of end moments (|M_1| >= |M_2|), positive for single curvature, negative for double curvature.

For transverse loading: C_my values depend on the load case and end restraint conditions per Annex B Table B.3.

C_my Values for Common Cases

Loading and Support Condition	psi	C_my
End moments, uniform (psi = 1.0)	+1.0	1.00
End moments, triangular (psi = 0)	0.0	0.60
End moments, double curvature (psi = -1.0)	-1.0	0.40
UDL, simply supported	--	0.95
Point load at midspan, simply supported	--	0.90
UDL, fixed ends	--	0.85

k_zz and k_zy

k_zz = C_mz x [1 + (2 x lambda_bar_z - 0.6) x N_Ed / (chi_z x N_Rk / gamma_M1)] but k_zz <= C_mz x [1 + 0.8 x N_Ed / (chi_z x N_Rk / gamma_M1)]

For I-sections not susceptible to torsional deformation, Annex B provides: k_zy = 0.6 x k_yy k_yz = 0.6 x k_zz

These 0.6 factors are conservative simplifications. Annex A (Method 1) provides a more refined calculation that accounts for cross-section shape and the actual stress distribution.

Worked Example -- UC Column with Major-Axis Moment

Given:

Column: 254 x 254 x 89 UC, S355
L = 4.0 m, braced frame, Lcr,y = Lcr,z = 4.0 m
N_Ed = 1,200 kN
M_y,Ed = 80 kN.m (end moment from beam connection, psi = 0, triangular moment)
M_z,Ed = 0 (no minor-axis moment)

Section properties: A = 114 cm^2 = 11,400 mm^2 W_pl,y = 1,220 cm^3 = 1,220 x 10^3 mm^3 i_y = 114 mm, i_z = 65.9 mm tf = 17.3 mm <= 40 mm

Cross-section checks (Clause 6.2): N_pl,Rd = 11,400 x 355 / 1.0 = 4,047 kN M_pl,y,Rd = 1,220 x 10^3 x 355 / 1.0 = 433.1 kN.m

Check for shear: V_Ed is from beam end reaction. Assuming nominally pinned, V_Ed is negligible. Shear check passed.

Buckling check -- from previous buckling example: chi_y = 0.936 (lambda_bar_y = 0.459, curve a) chi_z = 0.729 (lambda_bar_z = 0.794, curve b)

Equivalent moment factor: psi = 0 (triangular moment, one end moment = 0) C_my = 0.6 + 0.4 x 0 = 0.60

Interaction factor k_yy: k_yy = 0.60 x [1 + (0.459 - 0.2) x (1,200/(0.936 x 4,047))] = 0.60 x [1 + 0.259 x 0.317] = 0.60 x [1 + 0.082] = 0.649

Upper bound check: k_yy <= 0.60 x [1 + 0.8 x 0.317] = 0.60 x 1.254 = 0.752. OK.

y-y buckling check: N_Ed/(chi_y x N_Rk/gamma_M1) + k_yy x M_y,Ed/(M_y,Rk/gamma_M1) = 1,200/(0.936 x 4,047) + 0.649 x 80/(433.1) = 0.317 + 0.649 x 0.185 = 0.317 + 0.120 = 0.437 <= 1.0. OK.

z-z buckling check: k_zy = 0.6 x k_yy = 0.6 x 0.649 = 0.389

N_Ed/(chi_z x N_Rk/gamma_M1) + k_zy x M_y,Ed/(M_y,Rk/gamma_M1) = 1,200/(0.729 x 4,047) + 0.389 x 80/(433.1) = 0.407 + 0.389 x 0.185 = 0.407 + 0.072 = 0.479 <= 1.0. OK.

The column is adequate with 48% utilisation. The z-z buckling plane governs because chi_z (0.729) is lower than chi_y (0.936), even though k_zy (0.389) is smaller than k_yy (0.649).

Worked Example -- UB Beam-Column with Minor-Axis Bending

Given:

356 x 171 x 51 UB used as a gable post in a portal frame building
N_Ed = 120 kN (modest axial from wall cladding + wind gable reaction)
M_y,Ed = 0 (no major-axis bending -- vertical load only)
M_z,Ed = 15 kN.m (wind load on gable post about minor axis)
L = 6.0 m, assume Lcr,z = 6.0 m

Section properties: A = 6,490 mm^2, W_el,z = 66.7 cm^3 = 66.7 x 10^3 mm^3 (Class 3, use elastic section modulus) i_z = 39.3 mm, i_y = 147 mm fy = 355 MPa

Buckling about z-z: lambda_bar_z = (6,000/39.3) / 76.4 = 152.7/76.4 = 1.998 approximately 2.0 chi_z = 0.215 (curve b, alpha = 0.34)

z-z buckling check: M_z,Rk = 66.7 x 10^3 x 355 = 23.7 kN.m

C_mz = 1.0 (uniform minor-axis moment from wind -- conservative)

k_zz = 1.0 x [1 + (2 x 2.0 - 0.6) x 120/(0.215 x 6,490 x 355/1.0)] = 1.0 x [1 + 3.4 x 120/(0.215 x 2,304,000)] = 1.0 x [1 + 3.4 x 0.242] = 1.0 x [1 + 0.823] = 1.823

N_Ed/(chi_z x N_Rk/gamma_M1) + k_zz x M_z,Ed/(M_z,Rk/gamma_M1) = 120/(0.215 x 2,304) + 1.823 x 15/(23.7) = 0.242 + 1.823 x 0.633 = 0.242 + 1.154 = 1.396 > 1.0. FAIL.

The UB 356 x 171 x 51 fails the combined loading check. The high k_zz factor amplifies the minor-axis bending effect because the axial load is near the buckling resistance. Options: (1) increase to 356 x 171 x 67 UB, (2) reduce effective length by providing intermediate lateral restraint, (3) use S460 steel to improve the buckling characteristics.

Annex A vs Annex B -- When to Use Which

Annex A (Method 1) is more accurate for cases where Annex B is conservative:

Sections with b/h approaching 1.0 (UC sections, where minor-axis bending is significant)
Members with significant minor-axis bending combined with major-axis bending
Asymmetric sections (channels, angles) where the Annex B simplification is not valid
High axial load ratios where Annex B is known to be conservative

In standard UK building design, Annex B is used for:

UB beam-columns with predominantly major-axis bending
UC columns with eccentric connection moments
Braced frame columns where the moment is secondary to the axial load

Design software typically implements Annex A for all cases, as the additional computational complexity is trivial for a computer.

UK National Annex Provisions

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

Adopts Clause 6.3.3 without modification.
Permits both Annex A and Annex B without expressing a preference.
Confirms gamma_M1 = 1.00.
For members susceptible to torsional deformations (Clause 6.3.4), the UK NA provides supplementary guidance referencing BS 5950-1 for asymmetric sections not covered adequately by the Eurocode method.

Design Resources

UK Column Buckling Reference -- Buckling curves and chi factors
UK Effective Length Factors -- Annex E k factor method
UK Steel Properties -- fy, fu tables
UK UC and UB Section Properties -- Section tables
All UK Steel Design References -- complete library

Frequently Asked Questions

What is the difference between Annex A and Annex B interaction factors?

Annex A (Method 1) uses refined interaction factors based on cross-section shape parameters (w_y, w_z, n_pl) derived from plastic section analysis. It provides separate factors for I-sections, hollow sections, and solid sections. Annex B (Method 2) uses simpler factors based on C_m equivalent moment factors and is valid for doubly symmetric sections. For UK UC columns with predominantly axial compression and secondary bending, Annex B gives results within 5-10% of Annex A. For sections with dominant bending and biaxial moments, Annex A is more accurate.

When must the combined loading check be performed?

Clause 6.3.3 requires the combined loading check when BOTH N_Ed > 0.25 N_b,Rd and M_Ed > 0.04 M_pl,Rd. When either ratio is below this threshold, the interaction is negligible and separate checks suffice. For a 254 x 254 x 89 UC with N_b,Rd,z = 2,950 kN: if N_Ed = 600 kN (0.20), the check is not required. If N_Ed = 1,200 kN (0.41) and M_Ed = 15 kN.m (0.035), the check is not required. If N_Ed = 1,200 kN and M_Ed = 80 kN.m (0.18), the check IS required.

What is the C_my factor for a beam-column with a UDL?

For a simply supported beam-column subject to a uniformly distributed load (no end moments), C_my = 0.95 per Annex B Table B.3. This accounts for the parabolic moment diagram being only slightly less severe than uniform moment for the purposes of beam-column interaction. For UDL with a fixed far end, C_my = 0.85, reflecting the additional benefit of end restraint.

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.