Following up Monday’s discussion of philosophy[efn_note]Well, not really, but it’s hard to find a better word for stumbling around to explain how engineering design actually works.[/efn_note] and yesterday’s discussion of schematic analysis, here are some thoughts on general truss analysis. The photo above shows the not-so-soft underbelly of Edgefield County Bridge No. 3 in South Carolina, and accidentally makes a point important to today’s topic: the bridge deck, the thing that makes a bridge usable, isn’t very important to the basic structural action. The big beams running left to right are the deck beams, and they are how the deck is attached to the trusses[efn_note]Which are important to the basic structural action.[/efn_note]. The ends of the deck beams are attached to the verticals of the truss webs on each end, hung from the bottom chord connections. The deck beams then support the stringers[efn_note]The beams running parallel to the bridge span.[/efn_note] and the stringers support the deck proper. The diagonal rods are wind bracing and as such not part of today’s topic. As a reminder, here’s the bridge in a more traditional three-quarter profile:
In the pre-computer days, there were two methods commonly used for analysis of trusses like this – and I’ll get to what “like this” means in a minute. There was the method of joints, where you balanced[efn_note]I.e., calculated the vector sum using trigonometry.[/efn_note] the forces at each joint, typically starting at a support, and in doing so worked out the forces in one member after another. And there was the method of sections, where you cut an imaginary line through the portion of the truss that interested you and worked out the forces at that location, typically getting at once the top and bottom chords and one diagonal. If you wanted just the maximum forces, the method of sections was faster; if you wanted the forces in every member, the method of joints was faster. Both were worked out by mathematicians in (I think) the late 1700s; both made it to the US gradually in the mid-1800s. The first two American-published books that correctly showed you how to solve a truss were Squire Whipple’s A Work on Bridge Building: Consisting of Two Essays, the One Elementary and General, the Other Giving Original Plans and Paractical Details for Iron and Wooden Bridges in 1847 and Herman Haupt‘s later but much more readable General Theory of Bridge Construction in 1851.
Trusses “like this” are trusses that are statically determinate. Any formal definition of that phrase sounds like gibberish to non-engineers[efn_note]I guess mathematicians and physicists would have no problem with it, but what are they doing analyzing trusses?[/efn_note] but it boils down to whether or not there are multiple paths to support any given point. If there are, the structure is statically indeterminate and therefore more difficult to analyze; if there are not, the structure is statically determinate and can be solved using the two methods I mentioned above. There were pre-computer methods to solve statically indeterminate trusses, including graphic statics and, for masochists, energy methods, but they would be more work. Nearly all of the old trusses I’ve mentioned in the blog have been determinate, and that’s not by accident.
So you pick a determinate truss form, like a Pratt truss, use one of the easier methods, and get the forces in each member. It’s more accurate than using a schematic beam analogy but more work. I’m sure it will surprise no one that it’s still not quite right. The simple truss methods have…wait for it…hidden assumptions. The most important two are the idea that truss members are perfectly free to rotate at their joints and that the truss members only carry axial load in tension or compression.[efn_note]It can be argued that those are two ways of saying the same thing, but they’re stuck in my head as independent properties.[/efn_note] Neither is really true, but the implications of that will wait until Friday, because tomorrow I’d going to escape Flatland and discuss the implications of the bridge actually being a three-dimensional structure.
On an unrelated note, yes, I just installed an add-on that allows proper footnotes in the blog, and I’ll be amusing myself with that toy for some time.