# Where do the tributary angles/ratios for tributary areas come from?

When preparing load calculations for a slab, the quick method is to define the tributary areas of a slab for the beams/columns below. These are generally taken as lines at mid-span of the slab panels and, depending on the support conditions, at the edges at angles of 30 & 60 or 45 & 45 degrees.

Where do the 60/30 and 45/45 angle ratios come from? Example distributions are shown in the image below. I know that in reality the ratios depend on the stiffness and end conditions of the various structural members, I also take it that the tributary areas are somewhat affected by the load they are carrying. Would excessive loading on one panel, compared to the adjacent panel affect the areas?

I wouldn't be surprised if these tributary angles are merely a rule-of-thumb obtained over time. (Correct me if I'm wrong!). While it makes obvious sense that, in a slab with neighboring fixed and pinned edges, the fixed edge will support a larger proportion of the load, I can't point to any derivation of the 30/60° division. It might have something to do with plastic hinges, which take a very similar shape as the tributary boundaries.

Also, different loads in different panels shouldn't affect the tributary areas due to the superposition principle. Assuming this principle is valid, the result of loading slab $A$ with $Q_A$ and slab $B$ with $Q_B$ should be equal to loading only slab $A$ and then adding the results due to loading only slab $B$.

This is obviously different from saying that loading slab $B$ won't affect slab $A$: any load applied to one slab will always generate stresses in neighboring, joined slabs (a fact usually ignored). It merely means that loading one slab won't affect the calculated tributary areas for another.

The angles of those lines are determined by yield line theory. They are the axes of bending, should the slab decide to fail, which will require the least internal work to yield.

Here is a very simple, watered down explanation, but if you Google it, you'll find some white papers (or whatever colour you decide to print on :P) with more technical explanations and all the math.

• Moment coefficients are derived from yield line theory, yes. But my question was particular to the 45/60 rule of thumb for tributary areas when doing load estimates, which I think is a further simplification of yield line theory. Jan 16, 2018 at 10:52
• Yes. I don't have a specific source I could cite on that, but I remember one of my professors told me that a bunch of old guys with thick glasses and lab coats had a meeting and said that this is a good idea. As for the 45/60 rule, that would only be perfect for a perfectly fixed edge of the slab and even loads and geometry of adjacent slabs will affect that. Coming to think of that now, I wonder if that would affect the tributary area in reality.
– ChP
Jan 16, 2018 at 11:51