There is still a great urge to build stone arch bridges. Not, perhaps, on the massive scale of old, but as attractive, durable footbridges over small streams. When done right, the results are quite rewarding, and the building process itself can be enjoyable. Furthermore, a stone arch bridge should need but little maintenance, especially compared to wooden structures. One of the main considerations to think about is: how thick should the arch ring be?
Calculating the Arch Ring
The arch ring must be a certain minimum thickness to ensure stability. A good rule for arch ring thickness is given by the following formula developed a long time ago based on empirical data:
Arch thickness (in feet) = (The square root of (½ span + radius of the arc of the arch))/ 4 + .2 ft.
Unless you are precisely cutting the stones to fit exactly, multiply that answer by 1.25.
So, for an example, say we are building a Roman arch with a span of 6 feet. One half the span is 3 feet, which number happens to also be the radius of the arc of the arch. Using the formula, we see the arch ring should be .81 foot thick. If we choose to avoid the precision-cut stones, we will have to build a “rubble” arch. This means we will need to multiply the arch thickness number by 1.25 and get the final answer: about 1 foot. The rise of the Roman arch is 3 feet; we added another foot, so that now our arch is 4 feet high total. Add a few inches of dirt or gravel on top, and we have a rather taller structure than, perhaps, we were planning on.
Arch Ring in Flatter Bridges
For small bridges with flatter arches, this problem becomes worse, for, although the rise is low, the arch thickness is very close to the rise of the arch and with the fill added it is easy to end up with a structure twice as tall as the height over the stream banks alone may dictate.
The end result is that an arch bridge will likely have a hump and long approaches, so at least a reasonably slight grade may be necessary.
If only people are going to be crossing the bridge, you might get by with steps to allow easy mounting of the structure. While you can skimp on the arch ring, you can only knock off a few inches for these small footbridge-type spans. The end result would not, for all practical purposes, be improved off the size dictated by the formula.
With a bigger arch, the arch ring is not so very significantly thick after all. Now assuming we have a Roman arch with a 60-foot span instead of six, the formula gives us a ring thickness of a little over 2 feet for a precision-cut arch, and about 2- 2/3 feet for a rubble arch. Notice that the increase in the thickness of the arch ring is not proportional to the increase in span.
The Problem With Roman Arches
The example above would probably not hold true in practical use, as a Roman arch has certain weaknesses peculiar to it which necessitates an increase of arch ring thickness. To sum up the difficulty, the lines of thrust of an arch tend to follow a more parabolic shape, which a Roman arch is not. To accommodate for this, the arch would have to be made significantly thicker.
The idea is that the lines of thrust of an arch should be kept in the middle third of the arch for maximum stability; if this is not the case, the stones can move, which is highly undesirable and simply asking for collapse.
In regards to the 60-foot-span Roman arch, one engineer tackled the problem with a Roman arch of that size and said for stability the arch should be 7 feet thick! In a bridge, these larger Roman arches are typically held in place by the weight of the fill material pushing down on the stones. So large Roman arch bridges with arches that are thinner than strictly required for a freestanding arch have successfully been made.
Flatter arches tend to follow the parabolic shape much more closely, with the result that the above formula tends to work well with such arches.