We now know how to find the design load on a floor or roof. We also know how to take the load on a beam and calculate the reactions at each support. But there’s one step missing.

Floors are two-dimensional. Beams are one-dimensional. How much of the floor’s load goes into each beam?

The answer lies in a concept called tributary area, so named because each beam resembles a tributary of a river. Like water in a watershed, loads take the path of least resistance: they travel to the nearest beam. So if your beams are spaced 16 inches on center, each beam carries the 8 inches of floor on the left plus the 8 inches of floor on the right. Continuing the same example, if your floor has a uniform load of 60 pounds per square foot, then your beam carries a tributary load of (60 pounds/foot²) × (16 inches) × (1 foot/12 inches) = 80 pounds per foot. (Note that I converted 12 inches to 1 foot inside the equation… a nifty trick to ensure the units work out in the end.)

Tributary area of a typical floor joist.

Divide at the halfway line – that’s all there is to it. You probably could have guessed the solution on your own, even if you didn’t know it was called tributary area. If any part of your floor overhangs the outermost beam, then the entire overhang is included in the outermost beam’s tributary area. Not many building floors have overhangs, but roofs almost always do, and bridge decks as well.

You might also encounter a floor supported directly by columns, with no beams at all. This situation might arise when the floor is a concrete slab rather than plywood. In this scenario, the tributary area is a rectangular zone around each column. It’s exactly like a zone defense in football, where each linebacker is assigned to cover a section of the backfield and stands in the middle of that section. That zone is the tributary area; the player is the column. (Huh, I wonder why I have football on my mind today?)

Three linebackers cover their tributary areas.