Load Combinations — EN 1990 Eurocode

EN 1990 ULS and SLS load combinations. Fundamental, accidental, and seismic combinations with partial factors gamma_G and gamma_Q for Eurocode design. Educational use only.

This page documents the scope, inputs, outputs, and approach of the EN 1990 Load Combinations tool on steelcalculator.app. The interactive tool runs in your browser; this documentation ensures the page is useful even without JavaScript.

What this tool is for

What this tool is not for

Key concepts this page covers

Inputs and outputs

Typical inputs: characteristic permanent actions Gk, variable actions Qk (imposed, wind, snow), accidental action Ad, seismic action AEd, and the consequence class.

Typical outputs: all applicable ULS and SLS combinations with computed factored values, the governing combination, and clear identification of the leading variable action.

Computation approach

The tool applies EN 1990 expression 6.10 (or the alternative 6.10a/6.10b twin expressions where permitted by the National Annex). Permanent actions are factored by gamma_G = 1.35 (unfavourable) or 1.0 (favourable). Variable actions are factored by gamma_Q = 1.5, with companion variable actions reduced by psi_0. The tool iterates through each variable action as the leading action to find the most onerous combination.

EN 1990 ULS Fundamental Combinations — Eq. 6.10

The fundamental ULS combination (EN 1990 Expression 6.10):

Ed = gamma_G,j × Gk,j "+" gamma_Q,1 × Qk,1 "+" gamma_Q,i × psi_0,i × Qk,i

Where:
  gamma_G,j = 1.35 (unfavourable permanent) or 1.0 (favourable permanent)
  gamma_Q,1 = 1.5 (leading variable action)
  gamma_Q,i = 1.5 (accompanying variable actions)
  psi_0     = combination factor for accompanying variable actions
  "+"       = "to be combined with" (not arithmetic addition)

Alternative twin expressions (6.10a / 6.10b) — where permitted by National Annex

6.10a: Ed = xi × gamma_G × Gk + gamma_Q × Qk,1 + gamma_Q × psi_0,i × Qk,i
        (xi = 0.85, reduced permanent factor with full variable)

6.10b: Ed = gamma_G × Gk + 0.95 × gamma_Q × Qk,1 + gamma_Q × psi_0,i × Qk,i
        (full permanent with slightly reduced variable)

Take max(6.10a, 6.10b) — always more economical than Eq. 6.10

Partial factors for buildings (EN 1990 Table A1.2(B))

Action Symbol Unfavourable Favourable
Permanent action (G) gamma_G 1.35 1.0
Variable action (leading) gamma_Q 1.5 0
Variable action (accomp.) gamma_Q 1.5 × psi_0 0

Psi factors per EN 1990 Table A1.1

Action Category psi_0 psi_1 psi_2
Category A: domestic/residential 0.7 0.5 0.3
Category B: offices 0.7 0.5 0.3
Category C: congregation 0.7 0.7 0.6
Category D: shopping 0.7 0.7 0.6
Category E: storage 1.0 0.9 0.8
Wind (altitude < 1000m) 0.6 0.2 0
Snow (altitude <= 1000m) 0.5 0.2 0
Snow (altitude > 1000m) 0.7 0.5 0.2

SLS Combinations per EN 1990

Three SLS combinations are defined:

Characteristic: Gk + Qk,1 + psi_0,i × Qk,i
  (used for irreversible limit states, e.g., cracking)

Frequent: Gk + psi_1,1 × Qk,1 + psi_2,i × Qk,i
  (used for reversible limit states, e.g., deflection)

Quasi-permanent: Gk + psi_2,i × Qk,i
  (used for long-term effects, creep, fatigue)

For a steel floor beam in an office: characteristic deflection uses Qk (full live), frequent uses 0.5 × Qk, and quasi-permanent uses 0.3 × Qk.

Accidental and Seismic Design Situations

Accidental combination (EN 1990 Eq. 6.11)

Ed = Gk "+" Ad "+" psi_1,1 × Qk,1 "+" psi_2,i × Qk,i

Where:
  Ad = design value of accidental action (fire, impact, explosion)
  psi_1 for frequent value of main variable action
  psi_2 for quasi-permanent values of other variable actions

Seismic combination (EN 1990 Eq. 6.12)

Ed = Gk "+" AEd "+" psi_2,i × Qk,i

Where:
  AEd = design seismic action from EN 1998
  Only quasi-permanent (psi_2) companion loads are included
  For offices: psi_2 = 0.3 (only 30% of live load present during earthquake)

Worked Example — EN 1990 Load Combinations for a Steel Beam

Problem: A simply-supported steel beam (IPE 300, S355) spans 6 m in an office building. Characteristic loads: permanent Gk = 15 kN/m, imposed load Qk = 20 kN/m (Category B office). Determine governing ULS and SLS factored loads.

Step 1 — ULS combination (Eq. 6.10)

Permanent (unfavourable): gamma_G = 1.35
Variable (leading): gamma_Q = 1.5

Ed = 1.35 × 15 + 1.5 × 20 = 20.25 + 30.0 = 50.25 kN/m

Check with counteracting permanent (if uplift case):
Ed = 1.0 × 15 + 1.5 × 20 = 15.0 + 30.0 = 45.0 kN/m (does not govern)

Step 2 — ULS factored moment

MEd = 50.25 × 6² / 8 = 50.25 × 4.5 = 226.1 kN.m

IPE 300, S355: Wpl,y = 628 cm³
MRd = Wpl,y × fy / gamma_M0 = 628 × 10³ × 355 / 1.0 / 10⁶ = 222.9 kN.m

MEd = 226.1 kN.m > MRd = 222.9 kN.m → MARGINAL (1.4% overstress)
Consider IPE 330 or check with twin expressions 6.10a/6.10b.

Step 3 — Alternative twin expressions

6.10a: xi × gamma_G × Gk + gamma_Q × Qk = 0.85 × 1.35 × 15 + 1.5 × 20
6.10a = 17.21 + 30.0 = 47.21 kN/m → MEd = 212.5 kN.m

6.10b: gamma_G × Gk + 0.95 × gamma_Q × Qk = 1.35 × 15 + 0.95 × 1.5 × 20
6.10b = 20.25 + 28.5 = 48.75 kN/m → MEd = 219.4 kN.m

max(6.10a, 6.10b) = 48.75 kN/m → MEd = 219.4 kN.m < MRd = 222.9 kN.m ✓

Using twin expressions, the IPE 300 works with 1.6% margin.

Step 4 — SLS deflection check (frequent combination)

Frequent SLS: Gk + psi_1 × Qk = 15 + 0.5 × 20 = 25.0 kN/m

IPE 300: Iy = 8,356 cm⁴, E = 210,000 MPa
Delta = 5 × 25.0 × 6000⁴ / (384 × 210,000 × 8,356 × 10⁴)
Delta = 5 × 25.0 × 1.296 × 10¹⁵ / 6.735 × 10¹² = 24.1 mm

Span/250 = 6000/250 = 24.0 mm → 24.1 ≈ 24.0 mm (borderline OK for frequent)

Partial Factor Safety System Comparison

Standard Permanent (unfav.) Variable (leading) Variable (companion) Favourable Perm
EN 1990 1.35 1.5 1.5 × psi_0 1.0
ASCE 7-16 1.2 1.6 Combination-fixed 0.9
NBCC 1.25 1.5 0.5 (companion) 0.9
AS/NZS 1170 1.2 1.5 psi_c × Qk 0.9

Eurocode partial factors are generally higher on permanent actions (1.35 vs 1.2) but lower on variable actions (1.5 vs 1.6) compared to ASCE 7. The net effect on design is typically within 5% for gravity-dominated structures.

Frequently Asked Questions

What is the difference between Eq. 6.10 and Eq. 6.10a/6.10b? EN 1990 Eq. 6.10 is the single expression: gamma_G Gk + gamma_Q Qk,1 + gamma_Q psi_0 Qk,i. Equations 6.10a and 6.10b are an alternative pair: 6.10a uses a reduced permanent factor (xi gamma_G) with full variable, while 6.10b uses the full permanent factor with psi_0 on the leading variable. Using the twin expressions can give more economical results, but requires checking both. Which option is allowed depends on the National Annex.

What are the psi factors and how are they used? Psi factors reduce variable actions for combination (psi_0), frequent (psi_1), and quasi-permanent (psi_2) values. Psi_0 is used at ULS to account for the low probability of two variable actions reaching their peaks simultaneously. Psi_1 represents the value exceeded for a small fraction of the reference period (used in SLS frequent combination). Psi_2 represents the long-term average value (used in SLS quasi-permanent combination and in accidental/seismic design situations).

How does EN 1990 differ from ASCE 7 load combinations? The most significant conceptual difference is EN 1990's explicit use of psi factors to distinguish leading and accompanying variable actions, while ASCE 7 uses fixed load factors per combination type. EN 1990 also provides separate partial factors for favourable and unfavourable permanent actions, whereas ASCE 7 uses a single factor (1.2 or 0.9) depending on the combination. The reliability targets differ: Eurocode calibrates to a 50-year reference period with beta = 3.8 for CC2, while ASCE 7 uses a different calibration basis.

What is the National Annex and why does it matter? Each country that adopts the Eurocode publishes a National Annex that specifies nationally determined parameters (NDPs), including partial factors, psi factors, and the choice between Eq. 6.10 and 6.10a/b. For example, the UK National Annex allows the twin expressions with xi = 0.925 (not 0.85), while Germany uses Eq. 6.10 directly with modified factors. Always check the applicable National Annex before applying EN 1990 load combinations.

What is the consequence class in EN 1990? Consequence classes (CC1, CC2, CC3) define the reliability requirement based on the consequence of failure. CC1 (low consequences, agricultural buildings) allows lower reliability. CC2 (medium consequences, normal buildings) is the default. CC3 (high consequences, grandstands, hospitals) requires higher reliability through higher partial factors or more rigorous analysis. Most building structures fall under CC2 with the standard partial factors in Table A1.2(B).

How are serviceability limit states handled differently between EN 1990 and other codes? EN 1990 defines three SLS combinations (characteristic, frequent, quasi-permanent) with specific psi factor reductions for each. Other codes typically define only two levels (total load and live load). The Eurocode approach is more nuanced: the frequent combination (psi_1) represents the load that is exceeded for about 5% of the time, while the quasi-permanent (psi_2) represents the sustained load. Different SLS criteria (irreversible damage, comfort, appearance) use different combinations.

What is the difference between EN 1990 and the Eurocode material-specific parts? EN 1990 (Basis of Structural Design) establishes the general principles, load combination rules, and partial factor framework. It does not contain material-specific design rules. The actual member design is done in the material-specific Eurocodes: EN 1992 (concrete), EN 1993 (steel), EN 1995 (timber), EN 1996 (masonry). EN 1990 tells you how to combine loads; the material Eurocode tells you how to compute resistance and apply material-specific partial factors (gamma_M1, gamma_M2, etc.). Both must be used together.

What is the difference between persistent, transient, and accidental design situations? EN 1990 defines three design situations based on duration and probability. Persistent situations cover normal service conditions (self-weight, occupancy loads, climatic loads) with a 50-year reference period. Transient situations cover temporary conditions during construction, maintenance, or repair (typically weeks to months). Accidental situations cover exceptional events (fire, impact, explosion) with a very low probability of occurrence. Each has different partial factors: accidental situations use factors closer to 1.0 because the event itself is already rare and additional safety margins would be disproportionate.

What are the gamma_M resistance partial factors in EN 1993-1-1? EN 1993-1-1 (steel design) defines resistance partial factors separate from the load factors in EN 1990. gamma_M0 = 1.0 for cross-section resistance (yielding), gamma_M1 = 1.0 for member stability (buckling), and gamma_M2 = 1.25 for net section fracture at bolt holes. Some National Annexes modify these values (e.g., UK NA uses gamma_M1 = 1.0, while Germany uses gamma_M1 = 1.1). These are applied to the characteristic resistance of the cross-section or member, not to the loads.

What is the difference between EN 1993-1-1 and the other parts of Eurocode 3? EN 1993-1-1 contains the general rules for steel building design: member resistance (tension, compression, bending, shear, combined), stability (buckling, lateral-torsional buckling), and connection design. Other parts cover specific topics: EN 1993-1-2 (fire design), EN 1993-1-3 (cold-formed steel), EN 1993-1-4 (stainless steel), EN 1993-1-5 (plated structures), EN 1993-1-8 (connection design with component method), EN 1993-1-9 (fatigue), EN 1993-1-10 (material toughness), and EN 1993-1-11 (tension components). For most building design, EN 1993-1-1 and EN 1993-1-8 are the primary references.

What is the Eurocode partial factor format and how does it differ from the global safety factor approach? The Eurocode uses a partial factor format where each variable (load type, material property, resistance model) has its own safety factor. This replaces the older global safety factor approach (e.g., working stress design with a single factor of safety of 1.5-2.0). The advantage of partial factors is that different sources of uncertainty are treated independently: loads with high variability get higher gamma_G/gamma_Q values, while materials with well-controlled production get lower gamma_M values. This produces a more uniform reliability across different design situations compared to a single global factor.

How does the Eurocode handle combination value for multiple variable actions? When multiple variable actions are present simultaneously, EN 1990 applies the full factor gamma_Q to only one variable action (the leading action) and reduced factors gamma_Q × psi_0 to all others (accompanying actions). The designer must check each variable action as the leading action to find the most onerous combination. For a roof beam with imposed load Q, snow S, and wind W: the designer checks Q leading (1.5Q + 1.5×0.5S + 1.5×0.6W), then S leading, then W leading. The governing combination is the one that produces the maximum demand on the member being designed.

Related pages

Disclaimer (educational use only)

This page is provided for general technical information and educational use only. It does not constitute professional engineering advice, a design service, or a substitute for an independent review by a qualified structural engineer. Any calculations, outputs, examples, and workflows discussed here are simplified descriptions intended to support understanding and preliminary estimation.

All real-world structural design depends on project-specific factors (loads, combinations, stability, detailing, fabrication, erection, tolerances, site conditions, and the governing standard and project specification). You are responsible for verifying inputs, validating results with an independent method, checking constructability and code compliance, and obtaining professional sign-off where required.

The site operator provides the content "as is" and "as available" without warranties of any kind. To the maximum extent permitted by law, the operator disclaims liability for any loss or damage arising from the use of, or reliance on, this page or any linked tools.