Eurocode combination principles — ULS equations 6.10 and 6.10a/6.10b, ψ combination factors, seismic combination, all three SLS combinations, and a complete worked example for an office building floor beam.
EN 1990 §6.4 governs load combinations for all limit states. The fundamental principle: when multiple variable actions are present, only one is taken at its characteristic value (the leading variable action Qk,1); all others are reduced by combination factors ψ0,i.
The logic behind ψ0: it is statistically unlikely that all variable loads simultaneously reach their characteristic (98th-percentile) values. For an office floor, full live load at the same instant as peak wind is improbable — ψ0 = 0.7 for offices acknowledges that wind-accompanying imposed load is at most 70% of its peak value.
EN 1990 requires checking all permutations — each variable action must be treated as the leading action in turn, with the rest as accompanying actions. In practice, the combination producing the highest effect governs. Common cases:
The base ULS fundamental combination (EN 1990 Eq. 6.10) applies a single partial factor set to all actions:
EN 1990 §6.4.3.2 offers an alternative to Eq. 6.10: use whichever is more unfavourable of two separate equations. This approach is commonly adopted by National Annexes (UK, Sweden, etc.) because it produces less conservative results for permanent-action-dominated cases.
For a typical office slab (Gk = 5 kN/m², Qk = 3 kN/m²):
EN 1990 Table A1.1 provides ψ0 (combination), ψ1 (frequent), and ψ2 (quasi-permanent) for all variable action categories.
| Action Category | ψ0 | ψ1 | ψ2 |
|---|---|---|---|
| Imposed — Residential (A) | 0.7 | 0.5 | 0.3 |
| Imposed — Offices (B) | 0.7 | 0.5 | 0.3 |
| Imposed — Public assembly (C) | 0.7 | 0.7 | 0.6 |
| Imposed — Retail (D) | 0.7 | 0.7 | 0.6 |
| Imposed — Storage (E) | 1.0 | 0.9 | 0.8 |
| Snow — altitude ≤ 1000m | 0.5 | 0.2 | 0.0 |
| Snow — altitude > 1000m | 0.7 | 0.5 | 0.2 |
| Wind | 0.6 | 0.2 | 0.0 |
| Temperature (non-fire) | 0.6 | 0.5 | 0.0 |
ψ2 = 0 for wind and low-altitude snow means these loads make no contribution in the quasi-permanent SLS combination — they are absent on an "average" day. This is important: SLS deflection checks for concrete beams use Gk + ψ2·Qk (imposed only), with wind excluded.
The seismic design situation uses a separate combination. Permanent actions are unfactored (γ = 1.0), and variable actions are reduced to their quasi-permanent level:
Real earthquakes shake a structure in all directions simultaneously. EN 1998-1 §4.3.3.5.1 specifies how to combine the horizontal seismic components. Because the worst-direction angle of attack is unknown at design stage, the code uses the following permutations (where "+" means combined with, not arithmetic addition):
Both combinations must be checked, and the envelope governs. This means every member must be designed for the worst of (full X + 30% Y) and (30% X + full Y).
EN 1998-1 §4.3.3.5.2 requires the vertical seismic component EEdz to be combined when any of the following apply: the design vertical ground acceleration avg > 0.25g, horizontal cantilevers or horizontal prestressed members span > 20 m, base-isolated structures, or near-fault sites. In those cases:
EN 1990 defines three SLS combinations, used for different serviceability checks:
| Combination | Governs for | EC2 Application |
|---|---|---|
| Characteristic | Irreversible limit states (first crack, first yield) | Crack width (prestress zones); stress limitation w/ prestress |
| Frequent | Reversible limit states, short-term loads | Crack width in decompression check; stress limitation in RC |
| Quasi-permanent | Long-term effects (creep, shrinkage) | Deflection calculation; long-term crack widths in RC |
| Item | Eurocode (EN 1990) | ASCE 7 + ACI 318 |
|---|---|---|
| Gravity ULS | 1.35G + 1.5Q = 43.2 kN/m | 1.2D + 1.6L = 1.2×22 + 1.6×9 = 40.8 kN/m |
| Wind ULS (leading) | 1.35G + 1.5W + 1.05Q | 1.2D + 1.0W + 1.0L |
| SLS (deflection) | G + ψ2·Q = G + 0.3Q | D + 0.5L (sustained portion) |
| Resistance factor | φ = 1.0 (use γC, γS) | φ = 0.65–0.90 (no γM) |
| Governing combination | 43.2 kN/m (Eq. 6.10) | 40.8 kN/m (6% less) |
The minor difference in total design load reflects similar reliability targets — the Eurocode and ASCE 7 formats are calibrated to achieve approximately the same probability of failure. Neither system is systematically more conservative than the other across all cases.
Select your building's occupancy category and tick the load types present. All required EN 1990 ULS and SLS combinations are listed below.
ψ values per EN 1991-1-1 Table A1.1 (recommended; National Annexes may differ). Snow ≤1000 m a.s.l. assumed. Ed = design seismic action per EN 1998-1 — determined by site-specific hazard analysis; X and Y components combined via 100%+30% rule (EN 1998-1 §4.3.3.5.1). * = likely governing combination.