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Load Combination Theory - EN 1990 (Eurocode 0)

The theory behind this load-combination generator: how permanent, variable and accidental actions are factored and combined to Eurocode (EN 1990), what the partial factors γ and combination factors ψ mean, and how the ultimate (STR/GEO) and serviceability combinations - including expressions 6.10, 6.10a and 6.10b - are formed.

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Eurocode EN 1990 sets out how the actions (loads) on a structure are factored and combined for design. Each combination represents a credible loading scenario; the structure must be verified for the most unfavourable of them. This page explains the partial factors, the combination factors and the standard ultimate and serviceability combinations the generator produces.

Actions and partial factors

Actions are grouped as permanent GG (self-weight, finishes), variable QQ (imposed, wind, snow) and accidental AdA_d. At the ultimate limit state each characteristic value is multiplied by a partial factor γ\gamma that covers the chance of an unfavourable deviation:

γG=1.35 (unfavourable permanent),γQ=1.50 (unfavourable variable)\gamma_G = 1.35\ (\text{unfavourable permanent}), \quad \gamma_Q = 1.50\ (\text{unfavourable variable})

Favourable permanent actions take γG,inf=1.0\gamma_{G,inf} = 1.0 (or less for the EQU check), and favourable variable actions are simply omitted - only loads that worsen the effect being checked are included.

Combination factors ψ

When several variable actions act together it is unrealistic to assume they all reach their characteristic peak at the same instant. The combination factors ψ\psi (EN 1990 Table A1.1) reduce the accompanying (non-leading) variable actions:

  • ψ0\psi_0 - combination value, for the accompanying variable actions in ULS and the characteristic SLS combination.
  • ψ1\psi_1 - frequent value, for the leading action in the frequent SLS combination.
  • ψ2\psi_2 - quasi-permanent value, for long-term effects (creep, settlement, appearance).

One variable action is taken as the leading action at full value; the others are accompanying and reduced by ψ0\psi_0. Because any variable action could lead, the generator cycles each one into the leading position and produces a combination for each.

Ultimate limit state (STR/GEO)

For strength (STR) and geotechnical (GEO) verifications, EN 1990 offers the single expression 6.10, or the less conservative pair 6.10a/6.10b (whichever the National Annex adopts):

∑γG,jGk,j  +  γQ,1Qk,1  +  ∑i>1γQ,i ψ0,i Qk,i(6.10)\sum \gamma_{G,j} G_{k,j} \;+\; \gamma_{Q,1} Q_{k,1} \;+\; \sum_{i>1} \gamma_{Q,i}\,\psi_{0,i}\,Q_{k,i} \qquad (6.10)
∑γG,jGk,j+γQ,1ψ0,1Qk,1+∑i>1γQ,iψ0,iQk,i(6.10a)\sum \gamma_{G,j} G_{k,j} + \gamma_{Q,1}\psi_{0,1} Q_{k,1} + \textstyle\sum_{i>1}\gamma_{Q,i}\psi_{0,i}Q_{k,i} \quad (6.10a)
∑ξj γG,jGk,j+γQ,1Qk,1+∑i>1γQ,iψ0,iQk,i(6.10b)\sum \xi_j\,\gamma_{G,j} G_{k,j} + \gamma_{Q,1} Q_{k,1} + \textstyle\sum_{i>1}\gamma_{Q,i}\psi_{0,i}Q_{k,i} \quad (6.10b)

In 6.10b the permanent action is reduced by a factor ξ≈0.85\xi \approx 0.85, which usually makes 6.10a/6.10b together less onerous than 6.10 for permanent-dominated structures. A separate set of factors (EQU) covers loss of static equilibrium (overturning, uplift).

Serviceability limit state

SLS combinations use unfactored permanent actions (γ=1.0\gamma = 1.0) and the ψ factors to scale the variable actions, in three forms for three purposes:

Characteristic:∑Gk,j+Qk,1+∑i>1ψ0,iQk,i\text{Characteristic:}\quad \sum G_{k,j} + Q_{k,1} + \textstyle\sum_{i>1}\psi_{0,i}Q_{k,i}
Frequent:∑Gk,j+ψ1,1Qk,1+∑i>1ψ2,iQk,i\text{Frequent:}\quad \sum G_{k,j} + \psi_{1,1}Q_{k,1} + \textstyle\sum_{i>1}\psi_{2,i}Q_{k,i}
Quasi-permanent:∑Gk,j+∑iψ2,iQk,i\text{Quasi-permanent:}\quad \sum G_{k,j} + \textstyle\sum_{i}\psi_{2,i}Q_{k,i}

The characteristic combination is used for irreversible limit states (e.g. first cracking), the frequent for reversible ones, and the quasi-permanent for long-term effects such as deflection and crack-width control.

The γ and ψ values and the choice between 6.10 and 6.10a/b are set by the National Annex - confirm them for your country. Accidental and seismic combinations follow expressions 6.11 and 6.12 with their own reduced factors.

Frequently asked questions

Partial factors multiply the characteristic values of actions at the ultimate limit state to allow for unfavourable deviations. For the STR/GEO checks the recommended values are γ_G = 1.35 for unfavourable permanent actions and γ_Q = 1.50 for unfavourable variable actions. Favourable permanent actions take γ_G,inf = 1.0, and favourable variable actions are omitted. The values may be adjusted by the National Annex, and separate factors apply to the EQU (equilibrium) and accidental situations.

The ψ factors (EN 1990 Table A1.1) reduce variable actions that accompany a leading action, because they are unlikely to peak simultaneously. ψ0 is the combination value, used for accompanying variable actions at ULS and in the characteristic SLS combination. ψ1 is the frequent value, used for the leading action in the frequent SLS combination. ψ2 is the quasi-permanent value, used for long-term effects (deflection, creep, crack width). Typical values for imposed office load are ψ0 = 0.7, ψ1 = 0.5, ψ2 = 0.3.

EN 1990 gives two ways to form the STR/GEO ultimate combination. Expression 6.10 applies the full permanent factor and full leading-variable factor in one equation. Alternatively, 6.10a and 6.10b are used together and the more onerous is taken: 6.10a reduces the leading variable by ψ0, while 6.10b reduces the permanent action by ξ ≈ 0.85. The 6.10a/6.10b pair is generally less conservative (more economical) for permanent-dominated structures. The National Annex states which approach applies.

When several variable actions act together, one is taken at its full characteristic value as the leading action and the rest are reduced by ψ0 as accompanying actions. Because it is not known in advance which action gives the worst effect, each variable action is taken as leading in turn, producing one combination per case. The design value of the effect is the largest result over all these combinations. This tool generates every leading-action permutation automatically.

All three use unfactored permanent actions (γ = 1.0) and differ in how the variable actions are scaled. The characteristic combination (ψ0 on accompanying actions) is for irreversible serviceability limit states, such as the onset of cracking. The frequent combination (ψ1 on the leading, ψ2 on the rest) is for reversible limit states. The quasi-permanent combination (ψ2 on all variable actions) is for long-term effects - deflection, creep and crack-width checks.

Ready to generate your own combinations? Enter the actions and get the full set of ULS and SLS load combinations with the governing case highlighted.

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