The abutments of a stone arch bridge need to be adequately thick to resist the thrust of the arch. Just as the thrust of the arch needs to be contained in the middle third of the arch’s thickness for complete stability, so should the thrust of the arch as it passes through the abutment not pass into the outer third of the abutment on the side away from the arch.
Arches and Abutments
Not surprisingly, an abutment will need to be made thicker proportionately as the rise of the arch drops; this is because the lower the rise of the arch, the greater the horizontal component of the thrust. However, the inverse is not entirely true; in other words, a Roman arch will need an abutment thicker than just the width of the arch itself. This is true because the thrust of an arch is actually a catenary, so a Roman arch will exert a horizontal thrust; this thrust will exit the arch at an angle roughly around, say, 25 degrees. Thus, on a tall abutment, this small horizontal component of thrust will need to be considered.
A tall abutment does have one advantage: The thrust line is steered towards the ground by virtue of the weight of the masonry above. This means that the thrust line will start to angle more towards the ground, which in turn means the abutment will not have to be as thick as it otherwise would be to keep the thrust safely within it.
A Graphical Solution
So how thick should the abutment be? This is somewhat of a judgment call, and a question that has successfully been answered in many different ways over the centuries. However, as far as keeping the thrust from escaping the abutment or at least from entering the abutment’s outer third when supporting a segmental arch goes, you can draw a graph based on the angle of the arch’s skewback, making the thrust a straight line passing into the abutment from the end of the arch. Then make the abutment thick enough that the thrust line does not enter the end third of the abutment at any point. This would not take into account the fact that the thrust will actually start to angle towards the ground, which would suggest that this method is conservative. That is not a bad thing, as it is safest to have an excess of stability. One important thing to keep in mind is that the abutment must be solid to at least the top of the skewback.
As far as Roman arches are concerned, it helps to just treat the thrust line the same for a 130-degree segmental arch; the thrust exits the Roman arch at about a 25-degree angle, as would be the case for a 130-degree segmental arch.
Putting it Mathematically…
The conservative graphical method of abutment design can be expressed as a mathematical formula, making abutment design simple. This formula is as follows:
Abutment thickness = 1.35(y(tangent s’) + x(sine s’) + x(cosine s’))
Where y is the abutment height, x is the arch thickness, and s’ is the skewback angle.
The above formula applies to segmental arches 130 degrees down to 90 degrees. For Roman arches and segmental arches down to 130-degree segmental arches, the formula is:
Abutment thickness = 1.35(.47y + x(sine s’) + x(cosine s’))
Like all such “formulas” for arch building, it is important to understand the limitations:
- The formula is intended for stone arch bridges, as the weight of the fill above greatly increases stability.
- The abutment must be built solid to at least the top of the skewback.
- The formula assumes that the arch is thick enough for the thrust to stay within its middle third.
- While it probably is still applicable in most cases, it is not intended for arches below a 90-degree segmental arch due to the great horizontal thrust of these arches.
- This formula in no way takes into account factors such as the stability of the soil below; it is ideal for bridges founded on bedrock.
Stone Arch Bridges 