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, and step-by-step worked example.
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Design Philosophy — Clause 6.3.3
Beam-columns are checked for two buckling modes:
- In-plane buckling — Buckling in the plane of the bending moment (interaction of N and M about that axis)
- 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
- 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)
| Factor | Formula |
|---|---|
| k_yy | C_my × (1 + (λ̄_y - 0.2) × N_Ed / (χ_y × N_Rk / γ_M1)), but ≤ C_my × (1 + 0.8 × N_Ed / (χ_y × N_Rk / γ_M1)) |
| k_yz | 0.6 × k_zz (conservative) |
| k_zy | 0.6 × k_yy (conservative) |
| k_zz | C_mz × (1 + (2 × λ̄_z - 0.6) × N_Ed / (χ_z × N_Rk / γ_M1)), but ≤ C_mz × (1 + 1.4 × N_Ed / (χ_z × N_Rk / γ_M1)) |
Equivalent Moment Factors C_m — Table B.3
The C_m factors convert the actual moment diagram to an equivalent uniform moment:
| Moment Diagram | C_my, C_mz | C_mLT |
|---|---|---|
| Uniform moment (ψ = 1.0) | 0.6 + 0.4ψ ≥ 0.4 | 0.6 + 0.4ψ ≥ 0.4 |
| Triangular (ψ = 0) | 0.6 | 0.6 |
| Reverse curvature (ψ = -1.0) | 0.2 | 0.2 |
| UDL on simply supported | 0.95 | 0.95 |
| Point load at midspan | 0.90 | 0.90 |
Where ψ = M_end,min / M_end,max (ratio of smaller to larger end moments).
Worked Example — IPE 300 Beam-Column in S355
| Parameter | Value |
|---|---|
| Section | IPE 300 (Class 1) |
| Steel | S355 (fy = 355 MPa) |
| A | 5380 mm² |
| W_pl,y | 628.4 × 10³ mm³ |
| L | 4.0 m |
| N_Ed | 400 kN (compression) |
| M_y,Ed | 80 kN·m (UDL) |
| χ_y | 0.91 (from column design) |
| χ_z | 0.60 (from column design) |
| χ_LT | 0.62 (from LTB check) |
Interaction Check
Parameter calculations:
| Parameter | Value |
|---|---|
| N_Rk | 5380 × 355 = 1910 kN |
| M_y,Rk | 628.4×10³ × 355 = 223.1 kN·m |
| C_my (UDL, no end moments) | 0.95 |
| C_mLT | 0.95 |
| λ̄_y | 0.62 |
| λ̄_z | 1.08 |
| N_Ed / (χ_y × N_Rk / γ_M1) | 400 / (0.91 × 1910) = 0.230 |
| N_Ed / (χ_z × N_Rk / γ_M1) | 400 / (0.60 × 1910) = 0.349 |
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 ✓
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 ✓
Both checks satisfied. The in-plane check governs at 83% utilization.
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
With the addition for Class 1/2 sections: N_Ed / N_Rd + M_y,Ed / M_pl,y,Rd + M_z,Ed / M_pl,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.
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.
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
- All European References
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.