Load combinations

What this page does

The generator produces every combination of actions required by EN 1990 (Eurocode 0) for a given set of actions: fundamental, accidental, seismic and the three serviceability combinations.

You enter the effect of each action — the moment, force, stress, any quantity linear in the loading — and the page applies the factors and enumerates the combinations, with the design value of each.

Why you cannot guess which combination governs

The most common mistake is assuming that the variable action with the largest effect is the leading one. It is not, because each accompanying action enters with its own \(\psi_0\).

The page's default example shows exactly this. With \(G = 100\), \(Q_1 = 50\) (category A, \(\psi_0 = 0.7\)) and \(Q_2 = 40\) (snow, \(\psi_0 = 0.5\)):

• \(Q_1\) leading: \(1.35 \cdot 100 + 1.50 \cdot 50 + 1.50 \cdot 0.5 \cdot 40 = 240\)
• \(Q_2\) leading: \(1.35 \cdot 100 + 1.50 \cdot 0.7 \cdot 50 + 1.50 \cdot 40 = 247.5\)

\(Q_2\) governs, even though its effect is smaller. Snow has a lower \(\psi_0\), so it is penalised less as an accompanying action — and that reverses the ranking. This is why the code requires enumeration, not reasoning.

Expression 6.10 and the pair 6.10a / 6.10b

For the ultimate limit state, EN 1990 offers two routes:

\(\text{6.10:} \quad \sum \gamma_{G,j} G_{k,j} + \gamma_{Q,1} Q_{k,1} + \sum \gamma_{Q,i} \psi_{0,i} Q_{k,i}\)

or the less favourable of:

\(\text{6.10a:} \quad \sum \gamma_{G,j} G_{k,j} + \gamma_{Q,1} \psi_{0,1} Q_{k,1} + \sum \gamma_{Q,i} \psi_{0,i} Q_{k,i}\)

\(\text{6.10b:} \quad \sum \xi_j \gamma_{G,j} G_{k,j} + \gamma_{Q,1} Q_{k,1} + \sum \gamma_{Q,i} \psi_{0,i} Q_{k,i}\)

The pair is more economical when permanent actions dominate: \(\xi = 0.85\) reduces them in 6.10b, while 6.10a compensates by reducing the leading variable to \(\psi_0\).

For a structure with \(G = 500\) and \(Q = 50\):

The pair gives 727.5 against 750 — 3% less. On heavy concrete structures the difference reaches 5-8%.

The two routes are alternatives, not cumulative. The national annex decides which applies. Do not take the envelope of both — the page warns if you request them together.

The partial factor sets

Table A1.2 gives three sets, for three different failure modes:

EQU deserves a note. There the structure is treated as a rigid body, and the check is against overturning or flotation. The gap between 1.10 and 0.90 on the same self-weight is small in absolute terms but decisive: it is exactly what separates "stands" from "overturns". That is why in EQU the favourable and unfavourable actions must be modelled separately, not netted off.

Favourable actions

A permanent action that relieves the structure receives \(\gamma_{G,inf}\); a favourable variable action is omitted entirely (\(\gamma_Q = 0\)).

There is a trap here that clause 6.4.3.1(4) closes explicitly: a permanent action from a single source must be treated consistently across the whole structure. You cannot take a slab's self-weight as unfavourable in one span and favourable in another — it is one action, with one factor everywhere.

Accidental and seismic combinations

\(\text{6.11b:} \quad \sum G_{k,j} + A_d + (\psi_{1,1} \text{ or } \psi_{2,1}) Q_{k,1} + \sum \psi_{2,i} Q_{k,i}\)

\(\text{6.12b:} \quad \sum G_{k,j} + A_{Ed} + \sum \psi_{2,i} Q_{k,i}\)

All partial factors are 1.0: the event is already extreme, so it is not amplified further. In the accidental combination, clause 6.4.3.3(2) leaves the choice between \(\psi_1\) and \(\psi_2\) on the leading action to the national annex — the page has a switch.

In the seismic case there is no leading action: every variable enters with \(\psi_2\).

Serviceability combinations

\(\text{characteristic:} \quad \sum G_{k,j} + Q_{k,1} + \sum \psi_{0,i} Q_{k,i}\) \(\text{frequent:} \quad \sum G_{k,j} + \psi_{1,1} Q_{k,1} + \sum \psi_{2,i} Q_{k,i}\) \(\text{quasi-permanent:} \quad \sum G_{k,j} + \sum \psi_{2,i} Q_{k,i}\)

Each has its purpose: the characteristic for irreversible effects (cracking), the frequent for reversible ones, the quasi-permanent for long-term effects — creep and consolidation settlement.

Input data

• Factor set — STR, EQU or GEO.
• Permanent actions — name, effect, favourable or not.
• Variable actions — name, effect, EN 1990 category (which gives \(\psi_0\), \(\psi_1\), \(\psi_2\)) or manual values.
• Combination families requested.
• \(A_d\) and \(A_{Ed}\) — only if those families are requested.

Assumptions and limitations

• Action effects are superposed, which is valid only for linear behaviour. For second-order or nonlinear analysis, effects must be computed on the combined loading; superposition no longer applies.
• The factors are those recommended by EN 1990. The national annex may change both the partial factors and the \(\psi\) values — all are editable.
• Fatigue cannot be computed here, and should not be. §6.4.3 is titled “Combinations of actions (fatigue verifications excluded)”, and the note to §6.4.1(1)d refers fatigue combinations to EN 1992 … EN 1999. EN 1990 gives no expression for them.
• Combinations for several simultaneous directions of action are not generated — if wind can act along two directions, run each separately. The 100/30 rule for seismic action is in EN 1998-1 §4.3.3.5, not here.

What was added

Prestress, with its own \(\gamma_P\)

The term \(P\) appears in every expression, from 6.10 through 6.16b. But EN 1990 does not tabulate it: Table A1.2 has no column for prestress. The value comes from the material code — EN 1992-1-1 §2.4.2.2 gives 1.0 in the ordinary case and 1.3 for local effects or for stability with external prestress. Hence \(\gamma_P\) is an editable field, with a note on every run.

One subtlety, with a test: \(\xi\) does not apply to \(P\). In expression 6.10b the reduction factor accompanies only \(G_{kj,sup}\); the term \(\gamma_P P\) is untouched. At accidental, seismic and serviceability, \(\gamma_P = 1.0\).

The combined EQU/STR check

Table A1.2(A), NOTE 2, offers an alternative to the two separate checks, for the case where the static equilibrium check also involves the resistance of structural members: \(\gamma_{G,sup} = 1.35\), \(\gamma_{G,inf} = 1.15\), \(\gamma_Q = 1.50\).

It is the only set in the code where \(\gamma_{G,inf}\) exceeds 1. The note explains why: applying 1.00 to both parts of the permanent actions does not produce an increased unfavourable effect. On a favourable action of 100, EQU gives 90 while the combined set gives 115.

The fire situation

The same expression 6.11b, with all factors 1.0. It was separated from the other accidental actions for a concrete reason: whether the leading variable takes \(\psi_1\) or \(\psi_2\) is a distinct National Annex decision (EN 1991-1-2), and the two choices must be able to coexist in one calculation.

The \(A_{d,fi}\) field is for indirect actions — restrained thermal expansion. The fire itself is not a term in the combination: it enters through the reduced material strengths.

UPL and HYD — but from EN 1997, not EN 1990

Here the premise I had started from was wrong. There is no “Table A1.2(D)”. §6.4.1(1) of EN 1990 lists exactly four ultimate limit states — EQU, STR, GEO and FAT — and A1.3.1(7) refers hydraulic failure and uplift explicitly to EN 1997.

The factors come from there: Table A.15 for UPL (1.00 / 0.90 / 1.50) and Table A.17 for HYD (1.35 / 0.90 / 1.50). The difference is not cosmetic — the same destabilising permanent action is charged 35% more for hydraulic failure than for uplift.

They also have a different form: the other combinations produce one number to compare against a resistance, whereas here two sums of actions are compared, destabilising against stabilising, each with its own factor.

Related calculators

• Wind action — where the wind action effect comes from
• Snow load — and the snow one
• Timber deflections — uses the SLS combinations and the \(\psi\) factors directly