I’ve written here about accuracy in engineering calculations three times – in 2016, in 2018, and last year – and it seems it’s time to do so again. As was true before, it’s encountering the accuracy-based limits on meaningful results that has triggered this train of thought, so you can expect me to keep returning to this topic every so often, for as long as I am doing analysis myself.
I was recently doing some crude, schematic-level analysis of the ability of masonry walls for 130-year-old high-rises to resist wind load. I generally hate* using spreadsheets for calculations, but I did so in this case because I was being lazy: the method I was using to get the stresses in the wall, a variation on the portal method, is conducive to working in a spreadsheet: the rows represent floors in the building as forces and moments add up from one floor to the next, and the various things that need to be calculated (for two examples, the wind shear and the shear area of the wall) make up the columns.
A spreadsheet, as modern software running on a modern computer, is going to give me a lot of digits in a calculation. One third is 0.3333333333333333, which is ridiculous. I use the formatting feature to hide most of the decimals while leaving them intact, since truncating or rounding makes the inaccuracy problem worse. The final results – the stresses I calculated in the masonry – showed two or three digits, which is wrong. Nothing about the calculation (see below) supported two significant figures, let alone three. The final result was 32 psi maximum tension and 224 maximum compression, but by tweaking the numbers I could easily have made the first number anything from 25 to 40 and the second anything from 200 to 250. At my first engineering job, I had a supervisor who used a phrase I truly hated – “let’s sharpen our pencils” – to mean “let’s tweak our assumptions to make the results look better.” One of my assumptions for this calculation concerned what wind pressure to use. I could use current code or I could use various code numbers that were in use when the building was constructed, but even that’s a morass. Weather in New York is demonstrably different in the last 150 years – the East River used to freeze at the surface enough so that people could walk across, but that doesn’t happen any more – so I don’t know if the data that is embodied in current code is accurate for the 1880s. Lower Manhattan used to be built up very solidly, with almost no open park or plaza space, so there’s a good argument to be made that the wind pressure on the lower five stories was quite light as a result. The old codes used to exclude low buildings from wind design for this reason and that has been empirically proven to be okay.
Some of the old prescriptive code language about wall geometry was optional. For example, non-load-bearing walls (the front facade on the building I was looking at) could be four inches thinner than similar bearing walls. Was it in the building I was looking at? The presence or absence of that four inches is good for an eight percent swing in my results.
The model assumes that the piers between windows are the same at the top and bottom of the window, but that’s not true: the top of the pier has the lintels for the windows embedded in it. Do the lintel ends act to weaken the pier at the top, by interrupting the brick continuity? Or to stiffen the piers, by preventing rotation? Any answer to this question, which has a spectrum of answers from full fixity to zero fixity, changes the assumptions.
But none of this mattered for what I was doing. It was a distraction. The tension and compression, taken as bell curves of possible values, were both high but not unreasonably so. Since that calculation was for (one possible value of) full wind load, that meant that the building should perform well under normal wind load. If I reduce my answers to one significant figure (tension of 30 psi, compression of 200 psi) I have a useful schematic answer that is no less accurate than 32 and 224 psi.
There’s an argument to be made that structural engineers would be better off going back to slide rules, which didn’t give all those meaningless extra digits in calculations.
* Why? I’ve almost certainly discussed it in this blog before, but (a) the equations are hidden from view and therefore mistakes are hard to find and (b) there’s no way to have units really properly attached to the numbers. Checking units is the fastest way I know to look for mistakes in equations.