An Arch Puzzle

Imagine that you wish to build a wall with an opening for a door or a gate. The opening could be rectangular, or it could have a shaped top, for example, a semicircle.

This appears to be a simple enterprise, and it is, but the purpose of this page is to ask some questions about stresses, and the wall will serve as a simple example.

One way to build the wall would be to make some shuttering from wood, and then to pour concrete in between the wooden surfaces. The wood is removed when the concrete has set.

Another way, using rather less wood, is to make the shuttering horizontal, and to pour the concrete in as a pool. The whole thing can eventually be rotated into a vertical position, and the shuttering removed. In fact, prefabricated concrete walls can be made at a distance from the site, and transported when required. This is very useful when several identical or similar buildings are being built.

There is a difference between the horizontally and the vertically poured sections, in that the final stresses will be different. Why is that? It does not matter, because the material in small walls is seldom at a stress that is likely to break it. So why mention this subject at all? In order to introduce the pictures below.

ArchBeam2669.JPG (119987 bytes) LintelBeamArch2.jpg (89350 bytes)

In both pictures there appears to be an arch above a beam, a not uncommon sight in England. The purpose of the arch in many cases is probably to reduce the weight on the beam, especially when the span is large. Whether the beam is of stone or wood, the weight of a heavy wall would be difficult to hold unless the beam were deep. The problem with this arch idea is how to make it work. We saw in the examples above that the stresses in a structure can depend on the method of building. Let us go back to the idea of horizontal building, and imagine building the wall with the arch and beam system lying on a horizontal platform. Imagine too, that the properties of the mortar between the bricks, when set, exactly matches the properties of the bricks - same elastic moduli, same thermal expansion coefficient, etc. When the wall is finally rotated into a vertical plane, does it now matter that we put in an arch-like course of bricks? How can this fact affect the stresses, if the materials have the same properties throughout? Does it help if we build the wall vertically in the usual manner of bricklaying? If so, how?

We can also imagine the much more likely case in which the mortar has different properties from the bricks. Does this help the arch action? What differences should there be between bricks and mortar to get the arch to relieve the beam? Would it in fact help to have no mortar underneath the arch?

This may seem academic, but building something based on an idea is no guarantee that the structure will behave as hoped, unless the theory is correct and the design is carefully done. In older times, it is quite possible that many structures were stable, not because they worked as the builders imagined, but because they were built in a traditional manner that was known to work, or because they were over-engineered. The value of theoretical knowledge is that structures and machines can be built much more efficiently and economically if accurate calculations can be done. Of course, the builders are then obliged to know the sizes of likely loads on the structures, because if you are not going to over-engineer the thing, you must not run a significant risk of failure.

During the 19th century, a few cast iron beams had pre-stressed wrought iron bars beneath them, to relieve the tension which is the bugbear of cast iron. Over the years, unless the designs were very carefully calculated and built, the pre-stressing forces would gradually fade away as the structure moved, or the metal crept.