EN 1993 Combined Loading — Axial + Bending Interaction per Clause 6.3.3

Complete guide to combined axial compression and bending (beam-column) design per EN 1993-1-1:2005 Clause 6.3.3. Interaction formulas for in-plane and out-of-plane buckling, Cm equivalent moment factors, k_yy and k_zz interaction factors. Two full worked examples: a standard IPE 300 beam-column and a slender HEA 200 column with biaxial bending.

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Design Philosophy — Clause 6.3.3

Beam-columns are checked for two buckling modes:

1. In-plane buckling — Buckling in the plane of the bending moment (interaction of N and M about that axis)
2. Out-of-plane buckling — Buckling perpendicular to the bending plane (includes LTB effects)

EN 1993-1-1 Clause 6.3.3 provides two alternative methods:

• Method 1 (Annex A): Based on accurate theoretical derivation — more accurate but computationally intensive
• Method 2 (Annex B): Simplified interaction formulas — more common in practice

This guide presents Method 2 (Annex B), which is used in most European design offices.

Interaction Formulas — Method 2 (Annex B)

In-Plane Buckling Check (Clause 6.3.3(4))

N_Ed / (χ_y × N_Rk / γ_M1) + k_yy × M_y,Ed / (χ_LT × M_y,Rk / γ_M1) + k_yz × M_z,Ed / (M_z,Rk / γ_M1) ≤ 1.0

Out-of-Plane Buckling Check

N_Ed / (χ_z × N_Rk / γ_M1) + k_zy × M_y,Ed / (χ_LT × M_y,Rk / γ_M1) + k_zz × M_z,Ed / (M_z,Rk / γ_M1) ≤ 1.0

Interaction Factors k_yy, k_yz, k_zy, k_zz

For Class 1 and 2 Sections (Annex B, Table B.1)

Key insight: The interaction factors amplify the moment term to account for second-order (P-δ) effects. When axial load is low, k_yy and k_zz approach C_my — the equivalent moment factor. When axial load approaches the buckling resistance (N_Ed / N_b,Rd → 1.0), the interaction factors can exceed 1.5, amplifying the moment contribution significantly.

Equivalent Moment Factors C_m — Table B.3

The C_m factors convert the actual moment diagram to an equivalent uniform moment:

Where ψ = M_end,min / M_end,max (ratio of smaller to larger end moments, maintaining sign convention for double curvature).

Worked Example 1 — IPE 300 Beam-Column in S355

Interaction Check

Parameter calculations:

k_yy = 0.95 × (1 + (0.62 - 0.2) × 0.230) = 0.95 × 1.097 = 1.042 k_zy = 0.6 × 1.042 = 0.625

In-Plane Check

0.230 + 1.042 × 80 / (0.62 × 223.1) = 0.230 + 1.042 × 0.578 = 0.230 + 0.602 = 0.832 ≤ 1.0 OK

Out-of-Plane Check

0.349 + 0.625 × 80 / (0.62 × 223.1) = 0.349 + 0.625 × 0.578 = 0.349 + 0.361 = 0.710 ≤ 1.0 OK

Both checks satisfied. The in-plane check governs at 83% utilisation.

Worked Example 2 — HEA 200 Column with Biaxial Bending

Problem: A corner column HEA 200 (S355) in a braced frame carries N_Ed = 280 kN, M_y,Ed = 45 kN·m and M_z,Ed = 15 kN·m. Column height = 3.5 m, pinned-pinned both axes. Verify the biaxial interaction.

Parameters:

• A = 5380 mm², W_pl,y = 429.5 × 10³ mm³, W_pl,z = 203.8 × 10³ mm³
• N_Rk = 5380 × 355 = 1910 kN
• M_y,Rk = 429.5 × 10³ × 355 / 10⁶ = 152.5 kN·m
• M_z,Rk = 203.8 × 10³ × 355 / 10⁶ = 72.3 kN·m
• i_y = 82.0 mm, λ̄_y = (3500/82.0) / 76.4 = 0.559
• i_z = 49.2 mm, λ̄_z = (3500/49.2) / 76.4 = 0.931
• Buckling curve b (major), c (minor): χ_y = 0.87, χ_z = 0.63
• N_Ed/χ_y N_Rk = 280/(0.87×1910) = 0.169
• N_Ed/χ_z N_Rk = 280/(0.63×1910) = 0.233

Biaxial interaction (in-plane about y-y, with M_z contribution):

C_my = 0.95 (UDL), C_mz = 0.90 (point load at midspan — conservative). k_yy = 0.95 × (1 + (0.559 - 0.2) × 0.169) = 0.95 × 1.061 = 1.008. k_zz = 0.90 × (1 + (2 × 0.931 - 0.6) × 0.233) = 0.90 × (1 + 1.262 × 0.233) = 0.90 × 1.294 = 1.165. k_yz = 0.6 × 1.165 = 0.699. k_zy = 0.6 × 1.008 = 0.605.

In-plane check:

0.169 + 1.008 × 45/152.5 + 0.699 × 15/72.3 = 0.169 + 0.297 + 0.145 = 0.611 ≤ 1.0 OK.

Out-of-plane check:

0.233 + 0.605 × 45/152.5 + 1.165 × 15/72.3 = 0.233 + 0.179 + 0.242 = 0.654 ≤ 1.0 OK.

Result: HEA 200 is adequate at 65% utilisation. The biaxial bending contribution (M_z) accounts for roughly 37% of the total utilisation — neglecting minor axis bending would be unconservative by a factor of 1.6.

Simplified Method — Clause 6.3.3(6)

For sections where N_Ed / N_Rk ≤ 0.25 and λ̄_max ≤ 0.5, a simplified interaction check may be used:

N_Ed / N_Rd + M_y,Ed / M_y,Rd + M_z,Ed / M_z,Rd ≤ 1.0

This simplified check is conservative and should not be used when the interaction factors k_yy, k_zz can be reliably calculated. Use for preliminary sizing only.

Practical Design Tips

1. Check both in-plane and out-of-plane. It is not always obvious which governs. For weak-axis buckling (small i_z), the out-of-plane check often governs even when the major-axis moment is larger.

2. C_m factors are powerful. A double-curvature moment diagram (ψ = -1.0, C_m = 0.2) reduces the equivalent moment to 20% of the maximum, dramatically improving the interaction check. Designers should detail connections to achieve double curvature where practical.

3. Steel grade matters. For S460, the non-dimensional slenderness λ̄ increases by sqrt(460/235) = 1.40, moving the column into a more slender buckling regime where the interaction effects are stronger.

4. Torsional buckling of open sections. For IPE sections with short weak-axis buckling lengths, torsional-flexural buckling (Clause 6.3.1.4) may produce a lower χ than flexural buckling. Always check both.

Frequently Asked Questions

When should I check combined loading per EN 1993-1-1? Combined loading must be checked whenever a member is subject to both axial compression and bending moment. This applies to beam-columns, frame members with axial load and frame moments, columns with eccentric loads, and any member where M_Ed ≥ 0.05 × M_Rd and N_Ed ≥ 0.05 × N_Rd simultaneously.

What is the difference between Method 1 (Annex A) and Method 2 (Annex B) for combined loading? Method 1 (Annex A) is based on a more rigorous theoretical approach and provides more accurate interaction factors for non-standard cases. Method 2 (Annex B) uses simplified formulas that are easier to apply manually and are generally more conservative. Most design offices use Method 2 for routine design and Method 1 for complex cases or where economy is critical. The interaction factors from Method 1 are typically 5-10% lower than Method 2 for standard beam-column configurations.

How do I determine the equivalent moment factor C_m for a continuous beam-column? For a beam-column in a continuous frame, the moment diagram is typically a combination of end moments and span loads. Per Table B.3, C_my = 0.6 + 0.4ψ ≥ 0.4 for end moments, where ψ is the ratio of end moments. For the span loading contribution, use C_my = 0.95 (UDL) or 0.90 (point load). The combined C_my = C_my,end × (1 - α_s) + C_my,span × ÃƒÂŽÃ‚±_s where α_s = M_span / (M_span + |M_end,max|).

Can I use the simplified method for all beam-column checks? No. The simplified method (Clause 6.3.3(6)) has strict limits: N_Ed/N_Rk ≤ 0.25 AND λ̄_max ≤ 0.5. Outside these limits, the Method 2 Annex B interaction factors must be used. Using the simplified method outside its valid range can be unconservative by up to 40% because it neglects the amplification of moments due to axial load (P-δ effect).

Related Pages

• Column Design Guide — Compression per EN 1993-1-1
• EN 1993 Beam Design — Flexural design guide
• Column K-Factor — Effective length per Annex E
• Compact Section Limits — Class 1-4 per Table 5.2
• EN 1993 Lateral-Torsional Buckling — LTB reduction factor χ_LT
• All European References
• Beam Capacity Calculator — Free Tool
• Column Capacity Calculator — Free Tool

Educational reference only. Design per EN 1993-1-1:2005 + A1:2014 Clause 6.3.3 and Annex B. Interaction factors per Table B.1. Verify actual moment diagrams and axial forces. Results are PRELIMINARY — NOT FOR CONSTRUCTION without independent verification.