Old-Style Analysis

This is going to be a really geeky post. All of my posts are geeky, but usually in an architecture, history, or preservation way; today I’m geeking structural analysis.

There are structures ranging from very simple (individual beams, for example) to medium complicated (a truss roof) to very complicated (a large-building frame). That has always been true. What is true now that was not true, say, 100 years ago, is that we have a lot of different analytical tools that have different strengths and therefore different levels of applicability to the different structures. In the last thirty years I’ve used (in no particular order):

• closed-form equations like the ones illustrated above (mostly for individual beams or other members),
• moment distribution (for individual beams or for frames),
• graphic statics (for individual beams, small frames, or trusses),
• finite-element method analysis (for pretty much anything),
• and guesswork (for pretty much anything).

(There are two other methods that were common 100 years ago that I have not used but would love to try: the portal method and the cantilever method, both good for large frames.)

All have their strengths and weaknesses. More importantly, they are all constrained in accuracy and usefulness by the assumptions that are used to create their models of reality. If the building doesn’t match the assumptions (or, more realistically, vice versa) then the answer you get will be garbage regardless of the beauty of your analytic effort. A second overarching theme is that finite element analysis can’t realistically be performed without a computer because of the sheer number of calculations it requires, while the others can all be done, with varying degrees of difficulty, using pencil and paper. The last and most important theme is that only closed-form equations are exact (within the constraint of the accuracy of the assumptions) but they are difficult to derive for any but the simplest structures.

This came up recently in the discussion of how to best analyze a beam that supports gravity load but is curved in plan. A straight beam supporting gravity load has a simple closed-form solution and will have moment and shear in the vertical plane; the curve introduces both horizontal-plane moments and torsion. The standard solution in current-day structural engineering would be to create a small finite element model of that beam, breaking it into 10 or 20 straight-line segments to approximate the curve, and let the computer deal with all the math.

I suggested a different approach that I’ve used before. There is an exceedingly weird and useful book called Roark’s Formulas for Stress and Strain that consists of nothing but closed-form equations for non-standard structural elements. Since those equations are derived – with more effort than I care to even imagine – using partial differential equations for specific conditions, they are timeless. We have the second edition and the sixth edition in the office and while the sixth is longer and has more cases, the parts that are common to both editions are identical. One of the chapters concerns curved beams, and then there’s a table for beams loaded at right angles to the plane of curvature, and once you select your end conditions you have a bunch of equations that give you the moments, shears, deflections, and so on. Here’s the latest version, if you have a McGraw-Hill account.

Why use the closed-form solution instead of finite elements? The finite element method is faster to set up if the structure is complicated, and closed-form equations are simply not practical if the structure to be analyzed is too complex. In this single-beam case, I could certainly use the equations faster than I could set up the model. But beyond that, monoculture is as bad an idea in engineering as it is in ecosystems. Using more than one tool makes you think about what the different tools are doing and why. Coming at a problem in a completely different way, once in a while, makes you think about what the physical nature of the problem actually is, since you’re using a tool you’re less familiar with.

There’s an old joke that engineers repeat often: if the only tool you have is a hammer, everything looks like a nail. That kind of tight focus is, in my experience, not conducive to good design.