Eppur si muove

Continuing yesterday’s theme…

Different forms of analysis give very different forms of answers. (They also give different answers, period.) Back before I had a computer on my desk, I would occasionally use moment distribution for multi-span beams with complicated loading. The thing about moment distribution as a method is that it is explicitly iterative: there is no exact answer, there is only a rapidly-decreasing set of changes that allow you to see a convergence that you will never reach. But engineering isn’t math, and you don’t need an exact answer. When the next iteration of moment distribution changes the answer by a percent*, who cares? Your analysis wasn’t that accurate to begin with, so you’re done.**

Or for another example, closed-form equations – like the formula for the stress in a cantilever beam that Galileo was trying to derive in the essay excerpted above – have “exact” answers. Within the bounds of significant figures and assumptions, you run through the calculations and arrive at a single number answer. But those assumptions are important. One of them is that “plane sections remain plane” which is a fancy way of saying that the beam is slender enough relative to its length that it doesn’t warp internally, but rather simply bends. That’s fine for most beams but badly wrong if the beam is short and deep enough. So you do need to check those assumptions when doing something strange.

What got me thinking along these lines is the issue of frame analysis. The three methods used prior to computers – portal method, cantilever method, and moment distribution – don’t give answers about sidesway, the sideways drift of a building under lateral loads like wind. Those methods don’t give any deflection information at all. They give pretty good answers about moments and shears, and this is probably the place to point out that the first sixty years of skyscrapers were designed this way. Finite element analysis gives sidesway results because deflections are inherently part of the calculations in the analysis.

There was an interesting moment in structural analysis for tall buildings in the late 1960s. For architectural design and cost reasons, the heavy masonry walls of the past were gone, but not everyone had the computers, software, or knowledge to perform a modern-style analysis. The result was a series of buildings famous for excessive sidesway, including the John Hancock Building in Boston, and the Pan Am, Gulf & Western, and original World Trace Center buildings in New York. Problems ranged from facade damage to seasickness to elevators jamming in their tracks. The spread of better analysis and the explicit design of frames to limit sidesway eventually addressed the problem. More or less at the same time, the rules for providing expansion joints in facades were worked out.

If we look at 1920, we’ve got high-rises not designed for sidesway, with masonry walls without expansion joints and not designed at all, and no problems.*** If we look at 1990, we’ve got high-rises designed for sidesway, with all types of thin curtain wall with expansion joints and designed for wind load, and no problems. The problems occurred circa 1965-1975, when we had buildings not fully designed for sidesway, and curtain walls with inadequate or no expansion joints and only partially designed. There are people**** who look at that history and conclude that all buildings design before modern finite element analysis are suspect. Me, I look at it and say that the old style worked, the new style worked, but our profession fumbled the transition between the two.

* Or two percent, or half a percent. Everyone has their own personal threshold for accuracy. For the reason described in the next footnote, my threshold for some calculations is five or ten percent.

** One of the simplest measures of accuracy in calculations is the idea of significant figures. The live loads specified by building codes have one significant figure (e.g., the occupancy load for an office is 50 pounds per square foot, and only the 5 is significant) which means all of the calculations that follow – if we’re honest with ourselves – really only have one significant figure. I’d like to thank Professor Michael O’Rourke for pointing this out to me when I was 20, and thus saving me a lot of trouble over the years.

*** No problems within the scope of this discussion, that is.

**** Not a figure of speech: I’ve met some of these people.