Arch thickness is an important design criteria. Obviously, the thickness of the arch largely determines the strength of a stone arch bridge, much the same as the thickness of a wooden beam determines the strength of the said beam. And while there are many empirical rules for arch thickness, it is helpful to understand exactly why we are doing what we are doing.
The Proportions
The arch thickness is, of course, related to the span. The longer the span the thicker the arch should be. However, as it happens, this is a nonlinear relationship; this is why many empirical formulas for arch thickness have a square-root function in them. In other words, the relationship between span and arch thickness is such that smaller arch spans will need to be proportionately thicker than larger arches to achieve the same strength.
Small Arches
Why do small arches need to be proportionately thicker than large arches? This has to do largely with the weight of the arch. For example, in a large stone arch bridge of, say, 50-foot span, the arches can easily contain about 100 tons of masonry, depending, of course, on the arch’s width and thickness. The average car weighs around 2 tons, give or take. It will be seen that the arch weighs about 50 times more than the load expected to cross it. Now, if we take a small stone arch bridge of about 5′ span, the arch weighs considerably less, while, of course, the load it is expected to carry weighs the same. The more the arch weighs than the load, the more stable it will be; the moving load will have less of an effect on the arch, as the much greater weight of the arch essentially “swamps” the weight of the load. Thus, when it comes down to it, the smaller, lighter arch needs to be made proportionately thicker to accommodate the load.
Forces
This brings us to another consideration. In the above example we showed that larger arches are less affected by the loads and that we can thicken a short arch to allow it to accommodate proportionately heavier loads. However, despite making a small arch thicker, we will not in most cases be able to practically make it weigh nearly as much as the longer span arch. And yet these small arches are still stable, despite not being able to “swamp” the weight of the vehicle as effectively. So why does thickening the arch work so effectively? The answer is the thicker arch gives the line of thrust more room to work in.
Keeping the Forces Within the Arch
The thrust line is used to describe how the force of gravity working on the arch travels through all the stones to the ground, holding the structure in compression and stability. A moving vehicle will obviously cause a change in the thrust line of the arch as compared to a freestanding bridge. The key to arch stability is for this thrust to remain within the arch; some authorities would also point out that when the thrust line nears the outer edges of the arch stability is lost. Regardless of how you look at it the idea of thickening the arch remains the same: Thicker arches have more room within themselves to accommodate the changing thrust caused by traffic on the bridge. Thus, the thicker arch can accommodate changes in loads better simply because it has more area in which the forces can play without compromising the stability of the structure.
A Final Practical Consideration
For very small arches, many empirical formulas for arch thickness give answers less than 1′ thick, yet most authorities agree that the arch should never be thinner than 1′. This is not necessarily because there is anything wrong with the empirical rules (it is entirely possible to build stable arches much thinner than 1′), but simply because arches less than 1′ are not terribly practical to build. Unless everything is very well fitted, there is some error, so a little extra arch thickness makes the build much more forgiving by allowing for sufficient effective arch thickness; in other words we guarantee sufficient thickness within the arch despite some errors in how the stones fit. This is helpful, as small arches tend to be harder to turn than larger arches.
Stone Arch Bridges 