Hidden But Present

I wrote about the Arch Bridge at Bellows Falls, Vermont, briefly last year, but I want to discuss one specific detail that I mentioned then but did not really describe. Descriptions of the bridge always call it a three-hinged arch, which is not a minor point as it was the longest arch bridge in the country when it was completed 1905, but I can’t find the mid-span hinge in the photo above, taken shortly after completion. The HAER profile photo from 1979 shows something going on at the top of the arch, but it certainly doesn’t look like a hinge:

If you showed me that photo without telling me there was a hinge at the top, I’d say it was a two-hinged arch (the hinges at the abutments are obvious) and that the Pratt-truss bracing was doubled at the center two panels. And I’d be wrong. The third hinge is hidden but it exists in the form of a compression strut on each side of the center, forming the only connection between the two halves, and disguised by bracing that follows the line of the top and bottom chords. In other words, rather than forming the hinge by bending the chords together to meet, as happens at the abutment hinges, the line of force goes from the chords to the diagonals to the strut. A picture is worth at least the 79 words I’ve used trying to describe it, and the HAER documentation included two: a line drawing and a detail. Here’s the full explanation:

Here’s the line drawing and explanation:

And here’s the detail:

The HAER documentation also included some of the original design drawings which, while less clear, also show this detail. Here’s the detail itself, which clearly shows that the only connection that runs past the center of the truss is at that extra horizontal strut:

But the true engineering proof is in the force diagram for the arch:

Here’s a blow-up, showing the forces at the center:

The legend for the sheet says that forces are in thousands of pounds, with compressions positive and tensions negative. The chord forces – the general compression in the arch – increase towards the ends because of the cumulative effect of the roadbed hangers, so the progression from 306,000 pounds (upper chord) and 330,000 (lower chord) at the second panel from the center to 321,000 and 375,000 at the third panel makes sense, but the chords loads are suddenly halved at the center panel. Even better, what’s going on with the diagonals in the center panel? They switch from compression to tension at the midpoint of the panel, which is not normal truss behavior. It’s easily explained: if you take the 615,000 pounds of compression in the horizontal strut (that runs from the diagonals crossing at the center of the panel to the right edge), divided it by four (for the four half-diagonals at its left end, with the force evenly distributed among them because all are of equal stiffness) and then multiply that by the square root of 2 (I’ll get to why in a moment) you have 217,000 pounds. The square root of 2 is based on the panel being a square (it’s not, but it’s close) and that’s converting the horizontal force to its 45-degree equivalent. So the strut is pushing to the left at 615,000 pounds, and that force is taken by the four half-diagonals equally; the two to the left in compression and the two to the right in tension, since the direction of the force remains to the left. The two diagonals on the right convert their forces into the compression in the top and bottom chords (154,000 is just about 219,000 divided by the square root of 2). The reason that things don’t perfectly add up is that there is a roadbed hanger at the center joint adding its load.

In other less-geeky terms, you can always figure out the design intent from a force diagram even if you don’t know exactly what the details look like. Whether the structure functions as intended is a different story.