I read “What Is the Difference Between Accuracy and Precision?” by Anne Marie Helmenstine yesterday (steered there by Bill Harvey), and of course I immediately started thinking about structural design in that light. I remember one of my professors saying, circa 1985, that our calculations simply weren’t accurate so we shouldn’t be worrying about every decimal of our results.
The cartoon above, from XKCD, says most of what needs to be said about accuracy. If all of the numbers you’re dealing with are accurate, then you’ve got some reason to believe that your results will be accurate; otherwise, you’ve got garbage. Garbage numbers – inaccurate or imprecise ones – infect everything around them.
What does that mean for structural design? If we start with the simplest case, looking at a simple floor beam, the results don’t look so good. The dimensional properties of steel shapes are tabulated to at least 3 and often more significant figures, but, like all manufactured items, rolled steel shapes have some variability in their sizes. The variations are enough to kill the third significant figure, so the values in the tables are fine as averages but are less accurate than they seem. The accuracy drops quickly when we look at loads. We think we know the dead load – the weight of the building itself – accurately, but do we? Is the concrete slab that we think is 4 inches thick (and therefore weighs 50 pounds per square foot) actually 4 inches thick? Maybe, but again that number is an average, and the actual slab may be thicker or thinner. More importantly, “50 psf” has only one significant figure, so we can throw away all of the accuracy we thought we had for the steel beam. Then there’s the live (occupancy) load. Say the building is an apartment house, so the live load is 40 psf. Again, that’s only one significant figure, but the numbers are actually worse. We allow probabilistic reduction of live loads on the basis of the unlikelihood of large areas of floor being loaded to their maximum expected load simultaneously. So that 40 psf might be 40 psf for a regular floor beam, and 32 psf for a girder, and 27 psf for the column supporting the girder. That’s looking less like a precise number and more like the blur I see when I take my glasses off.
None of this invalidates our calculations. Engineering, as has been said elsewhere, is about making the best decisions we can with the information we have. But it throws light on a comment that my classmates and I heard from so many professors that it became a joke. When working through a problem in class, if the result was less than five percent over the allowable, the professor would often shrug and say “It’s within an engineering approximation.” That sentence became a punchline for me and a lot of my classmates; I remember another engineer saying that the 1986 Boston Red Sox won the World Series within an engineering approximation. But if you only have one significant figure of accuracy, which another way of saying that the number you’re using is in the center of a ten-percent range of possibilities, then a five percent error can be considered literally meaningless. So we perform our calculations as accurately as we can, but when we get to a result, we don’t let minor discrepancies there drive our non-numeric logic and conclusions.