Different Structural Forms Combined

by Don Friedman on July 17, 2017

Now that the weekend’s over, back to the Pearl Street bridge that is part of the approach to the Brooklyn Bridge. First, a minor correction to the first part of my street analysis of this structure: the original truss is not a double-diagonal warren truss. It’s a subdivided pratt truss, as some of the compression diagonals are missing. I’m sure that everyone is relieved that I’ve corrected this.

The picture above (click to enlarge) is unfortunately about the best that can be done in terms of getting the side elevation of the bridge. There are plate-girder approach ramps on either side of the main bridge that block the view from both the south and the north. In any case, the picture above shows the basic form of the bridge as it exists today: the original truss bridge is propped on steel arches, with each of the four original parallel trusses getting its own arch.

That arrangement seems backwards, historically. The use of first wrought iron and then steel freed bridge builders from arches, allowing them to build girder and truss bridges that had greater vertical clearance and did not develop lateral thrust. Why would a nineteenth-century truss be reinforced with a twentieth-century arch? The answer has to do with load transfer.

When you join two parallel structures and try to share load between them, you don’t get to decide how much load goes in each one. They decide for you. Not because they know more than you do, because they don’t: they’re dumb pieces of metal, or wood, or masonry. The load will split between the two structures based on the ratio of their stiffnesses. The simplest example is to picture two parallel and joined beams spanning, say, ten feet: one a wood 2×6 and the other immediately next to it (and fastened to it) a piece of steel with the same cross-section as a 2×6.* The elastic modulus for steel is about twenty times that of wood, depending on the wood species. To keep the math simple and since this is just an example, we’ll say that the ratio is exactly twenty to one. In that case, when we load the beams** together, 1/21 of the load will be carried by the wood beam and 20/21 by the steel beam. The reason is simple: if the beams are constrained to move together, the load will equalize in the ratio of their stiffnesses. If either beam were to take more than its ratio share, it would deflect more than the other beam, which isn’t possible if they’re joined.

If you look at the composite arch/truss bridge above and the approach ramp blocking the view, you get a sense of the geometry. The ramp is just about as low as it can be without blocking truck traffic on Pearl Street. The vertical distance from the ramp to the underside of the truss is the space that was available to put in reinforcing, and it’s significantly less than the depth of the original truss. For beam-type structures, including trusses, stiffness more or less correlates with depth, so if a beam or truss had been used below the original truss, it would likely have been more flexible*** than the original and therefore would carry less than half the load. On the other hand, arches with a decent rise (the vertical distance between the spring point and apex) are a lot stiffer than beams of the same span, so the arch will carry most of the load.

In short, if the purpose of the reinforcing was to relieve the existing structure of most of its load, using arches of significantly greater stiffness than the trusses was a good way to do it. If it weren’t the famous and landmarked Brooklyn Bridge, it would have been even easier to remove the old trusses and substitute modern structure, but that wasn’t an option here.

* I.e., 1-1/2 inches wide by 5-1/2 inches high, solid.

** The easiest way to load them equally is to have a floor that runs over both of them and is supported by another beam off to the side, so the floor spans perpendicularly to our composite wood/steel beam.

*** I.e., less stiff.

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