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How Do You Know When You’ve Gone Too Far?

That’s the second Stuckey’s Bridge over the Chunky River in Mississippi. The first bridge was built in the mid-1800s, this steel truss was constructed in 1901. The lurid gothic ghost story connected to the site concerns the first bridge and, being neither knowledgeable nor interested in ghosts, I couldn’t say if it makes sense to think that the ghost stuck around when the bridge was replaced. (The photo above was taken by Dudemanfellabra.)

The bridge is a modified Pratt truss, with additional web verticals and diagonals added after the original construction, converting it to a pseudo-Stearns truss. (Scroll down to the August 25, 2011 comment by Nathan Holth.) The Stearns truss form is itself an oddity, combining warren trusses of two different panels sizes to no obvious advantage. But what really jumps out at you is how slender every element of the bridge is.[efn_note]And thanks to Bill Harvey for bringing it to my attention.[/efn_note] The eyebars of the bottom chord, the latticed original verticals, the box top chord, the eyebar original diagonals, and the bar new diagonals and verticals are all incredibly light. Even the angles making up the portal bracing between the verticals are small. I’ve talked about some slender-built trusses before but this is light enough that it’s scarier than the purported ghosts are.

A question that comes up once in while about any structural form is “how far can I push this idea?” For example, starting maybe 110 years ago, engineers have looked at the question of the theoretical maximum height of a skyscraper. One answer is that tall buildings are constrained by the ability of bedrock to absorb column loads. In any case, Stuckey’s raises the question of how thin we can make a truss and still have it function. That question gets back to the issues I was talking about last week, because of the differences between how we model trusses in analysis and design and their actual physicality.

Some structural affects have linear relationships. Doubling the length of a uniformly-loaded beam doubles its end reactions. Some have relationships to the second, third, or fourth power: with the moment and maximum deflection in that beam as an example of the second and fourth power, respectively. Without trying to make it into a hard rule, I think that many engineers have an instinctive reaction that linear relationships are good and higher-power ones are bad. Using that beam as an example, you can pretty easily design a beam for twice as much shear, but designing it for four times as much moment means making it much bigger and designing it to prevent sixteen times as much deflection is a real pain.

There are a lot of small issues created by the differences between a truss model of idealized members with perfect joints and a real truss. We can ignore those issues when they’re small, but as the members get skinnier and skinnier, the effect of those issues increases non-linearly. For example, incidental moments in the truss verticals caused by connections more rigid than the theoretical pins become much more important as the verticals get thinner: the equation to review the effect of combined compression and bending has a way of blowing up for too-small members. Another example: any local bending caused by connections not meeting perfectly at a joint will also be resisted by a too-skinny piece of steel. And local failures can trigger a catastrophic collapse of the entire truss. In other words, the foundation pressures below a maximum-height skyscraper are part of the main thread of design for that theoretical building and increase in a predictable way as the we increase the building’s height, but the local bending stresses and interaction between compression and bending are not part of the initial truss design[efn_note]Although, to be fair, are a regular part of secondary design.[efn_note] and are difficult to predict because they are so badly non-linear.

In short: there’s no way to be absolutely sure that you’ve included all the nasty non-linear secondary effects as the truss gets thinner and thinner, and you don’t want to be there when it turns out that you missed one.

Memory is weird. I was going to title this post “Where am I to go now that I’ve gone too far?” but it occurred to me that a single line of lyrics from a song from 1982 might be a little too obscure, even though that song has, on occasion, been playing in my head since I first heard it on the radio.