Eurocode's combination rules can generate a dizzying number of load cases. Every variable action takes a turn as the leading variable while the others are factored down by their ψ (psi) values, and the permutations multiply quickly. The honest question every engineer asks the first time they meet the combination matrix is: do all of these actually matter?
Why there are so many
EN 1990 builds ULS combinations by selecting one variable action as the leading action (taken at its full characteristic value) and treating the rest as accompanying actions, reduced by their combination factor ψ0. Cycle the leading role through dead, imposed, wind and snow, account for wind acting in different senses, and the count climbs into the dozens. Each combination is a legitimate question the code is asking: what if this action is the one at its peak while the others are merely present?
The ψ factors are the key idea. They encode the simple statistical fact that several independent variable actions are very unlikely to all peak at the same instant, so accompanying actions are taken at a reduced value rather than their full characteristic one.
A sensitivity study
To see how many combinations earn their place, take a concrete example: a 24 m single-bay portal frame with dead, imposed, wind and snow actions, and run every ULS combination under EN 1990.
Of fourteen combinations, just three ever produced a governing member force. The rest were always bracketed by those three.
That is a striking ratio, but it makes physical sense. A portal frame has a handful of critical actions and effects - the wind that pushes it sideways and uplifts the roof, and the gravity that loads the rafters and bases. Most of the combination matrix consists of cases where neither of those is at its worst.
What controlled, and why
In that frame:
- The leading-variable wind case governed the rafters and the eaves moment - wind uplift and side load are what a light portal is most sensitive to.
- The imposed-and-snow gravity case governed the column bases, where the downward load and its moment are largest.
- Everything else fell inside the envelope of those two.
The pattern - a few cases governing - is typical of simple, regular structures. More complex or irregular structures spread the governing cases more widely, which is exactly why you cannot assume the result.
The right lesson (and the wrong one)
The wrong lesson would be to hand-pick combinations and only run the three you think will govern. That is how you miss the case that governs on the next frame, or the member you did not anticipate.
This does not mean you should hand-pick combinations - automate them. It means a quick screen tells you where to focus the detailing effort.
The right lesson is twofold:
- Automate the full combination matrix. Modern analysis runs all of them cheaply, so there is no reason to prune the input. Completeness is free.
- Use the result to focus your attention. Once you see which two or three combinations govern, you know which member forces deserve careful detailing and which are comfortably bracketed - a screen for effort, not a licence to skip cases.
To build and check the combinations themselves to EN 1990, the load combination tool generates the full set with the correct ψ factors. The related wind-load workflow covers deriving the wind action that so often turns out to be the leading one.