# Why are truss bridges the way they are?

Just by taking a train ride across my home city I can see truss bridges like the one in above picture everywhere. There are numerous variations, but the most common design seems to be this. But why are they built specifically this way?

I can intuitively kind-of see why such a design probably is strong, but is there any kind of in-depth reason? I would be interested to know the answer as much from the physics side of things as possible. Googling didn't help much; I could find information on the different variations and many examples, but none really covered what is it about this design specifically that makes it so popular.

• Amount of material required cf. Strength provided also skill required for construction, then time necessary to construct are all factors... Commented Jul 23, 2018 at 16:51
• Try playing some World of Goo and see how you end up building things (roughly) like this yourself! Commented Jul 24, 2018 at 14:24

That looks like a Pratt truss.

These trusses have diagonals which go from the outer-top nodes to the inner-bottom nodes (i.e. they connect to the top chord on the node furthest from the center of the span, and to the bottom chord on the node closest to the center). This design means that the diagonals are under tension and the verticals are under compression.

Another famous design is the Allan truss, which is the exact opposite: the diagonals go from the inner-top nodes to the outer-bottom nodes, which means that the diagonals are under compression and the verticals are under tension.

The reason the Pratt truss is so common in steel bridges is because it tends to be more economical. This is because steel works better under tension than under compression.

Under tension, steel can theoretically operate very close to its yield stress. Under compression, however, there's the risk of buckling.

Buckling is a behavior of slender elements under compression to effectively collapse at loads far below their yield stress (think of the classic "compress a ruler from both ends" experiment). "Slender" here means beams which are very long and with relatively small cross-sections (see Wikipedia's slenderness ratio page). Steel beams are frequently slender and therefore buckle under compression (as opposed to simply being crushed). The longer the element, the smaller the buckling stress and, therefore, the larger the beam's cross-section must be to resist buckling.

So, with the Pratt truss, the verticals are under compression and the diagonals are under tension. As can be clearly seen in the image (or derived from geometry), the diagonals are longer than the verticals. Therefore, the diagonals' buckling load is smaller than the verticals'.

So in an Allan truss, the longer diagonals will have a larger cross-section and the shorter verticals will have a smaller cross-section.*

With a Pratt truss, however, the longer diagonals can have a smaller cross-section while the verticals will have a larger cross-section.*

So the advantage of the Pratt truss is that material tends to be used more efficiently: the longer elements have as small (and therefore light and cheap) a cross-section as possible by "sacrificing" the shorter elements. This works because those shorter elements require a smaller "upgrade" to resist buckling than the longer ones.

* Note that when I say above that, for example, "the verticals will have a larger cross-section", I don't mean that the verticals' cross-section will be larger than the diagonals'. I mean only that it will be larger than if buckling weren't an issue.

• Will either truss design work just as well if turned upside down (space below bridge permitting)?
– Bent
Commented Jul 24, 2018 at 8:12
• @Bent: If you literally flip a Pratt truss upside down, you basically end up with an Allan truss, and vice versa. So an upside-down Pratt truss has diagonals under compression and verticals under tension. So everything I said above is still valid in this case, only its inverted.
– Wasabi
Commented Jul 24, 2018 at 11:10

That is a steel truss design from the recent era when good engineering analysis of simple designs was possible. The accurate analysis allowed the use of longer, thinner elements, as you see here. But only for simple designs, and even then only with considerable effort.

Originally, truss bridges were almost always built to a standard design. An individual Engineer would have a standard design, and an individual Rail Company would have a standard design, and an individual agency would have a standard design.

Your state might have paid for all of the important road bridges in your state: your rail company might have put up all the original rail bridges in your state. When looking at the history of a collection of bridges, you find that they were all built in a 20 year period, all built by the same agency, all with the same chief engineer. Then he retired, and a different design was adopted.

When better analysis became possible, it was used to build cheaper bridges, but the effort involved still meant that nobody wanted to use a different truss structure for every cheap bridge. They just choose one truss design, and worked with that.

For some of the truss designs, there are clear advantages: you see that in the posted picture, both sides of the truss are identical, there are common profiles, verticals are interchangeable, and vertical-plan diagonals are interchangeable. Truss designs where that is not true can save on weight and material cost -- but at the cost of construction, design, and manufacture cost. The advantage of standard elements continues when more bridges are built -- all the bridges you saw were probably built to the same height and length, with repeating sections where required.

Generally, the reason you see repeated bridges with identical truss design is institutional costs, not because that design is clearly superior to other similar truss designs.