More About Arches and Domes
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On seeing these two buildings you probably wouldn't think of the Roman Colosseum of 80 AD, and you certainly wouldn't take the trouble to go and see them. Yet there is a connection between these buildings and the Colosseum.
These arches that are built on the round "corner" of each building have rather peculiar shapes, which are the result of placing an arc of a circle on a cylinder. Try drawing a circle on a piece of paper and then rolling the paper.
There are several consequences of building a corner arch. Firstly, the arch overhangs, and so there seems to be some cantilever action. Is this really the case? Secondly, the outward thrusts at the ends are not in line: in fact they are at right angles to each other. And throughout the arches, there is a continual change in the direction of the thrust, which is apparently not counteracted by any restraining force. If you look at the brightened up inset in the third picture, repeated here, you will see that the structure is not in fact a genuine brick arch: it's a concrete structure with some bricks at the front.
The Colosseum is a huge round building, with many arched openings, but these are small compared with the radii of curvature of the building, and as usual, the Romans provided supports between them which are thick in both dimensions. So the direction and position of thrusts was not critical. Even in their straight multi-arch bridges, the piers were so thick that the loss of one span would not bring down the entire bridge, a valuable property in time of war or flood. The diagram below shows a ring of thin parabolic arches based on a cylinder: the outward lean is visible at the edges.
Now we straighten the arches in vertical view to make a polygonal plan.
Now the arches do not lean outward, but now we see that where the arches meet, the planes that contain their lowest parts lean inwards. That means the thrusts of the arches have an outward component, so we must make another change - the piers must lean inwards.
If you don't believe this, look at the example shown here, a set of sharply angled windows. The angle is such that there is nowhere for the thrust of the jack arches to go, and in fact one has sagged, damaging the brickwork above. The second picture has been compressed sideways to clarify the sag. It is actually quite possible that these are not arches at all, and that the "voussoirs" have simply been stuck on to beams.
In this polygonal building, there is no outward lean, but the pillars are so thick, and the superstructure so heavy, that like the Arc de Triomphe, the structure contains within it all the required buttressing. The next diagram, however, shows a set of thin arches that do require tilted piers.
What would happen if we make everything lean sharply inwards, as in the next diagram?
It's obvious that the arches are going to fall inwards. But so far, we haven't shown the arches holding up a load. So let's add a load.
Now the compression in the ring can provide the outward forces needed to hold the arches in place. This isn't perhaps a very useful structure, but it can lead the way to an understanding of something very useful indeed, a dome.
Yes - the dome is closely related to the arch. A dome can be thought of as an arch that has been smeared out all around a vertical axis. But this is a geometrical thought, and geometrical thoughts, like this one, can obscure the physical reality. For example, the previous diagram hints at something you can do with a dome that you cannot do with an arch - you can build it without falsework, if you do it right, just as the Inuit used to build igloos. In the diagrams below, the arches have been replaced by struts.
Domes are often supported on cylindrical drums, with no visible means of containing the outward thrust. Buttresses are seldom used in this context. One solution is to surround the base of the dome with metal chains or cables, which are placed in tension by the thrust. Another solution, seen, for example, in Turkey and Bulgaria, is to surround the dome with smaller supporting domes.
Real domes often have a more complicated structure based on space frames or the tensegrity principle, allowing the creation structures which can range from small domes to complete spheres. Spheres are not strictly domes, because in a dome, almost everything is in compression, which cannot be the case in a sphere in a gravitational field. The "Millennium Dome" in London is not a dome at all: it is more like a tent, since the roof is held up by numerous cables which are attached to a ring of poles.
Domes of Islamic Buildings
You will have noticed that many domes on many Islamic buildings curve past the vertical and slope inwards at the base. This is incompatible with the existence of outward thrust. What is going on within these structures? In some cases the structure is not strictly a dome at all - it is made of metal, or in some modern examples, plastic or composite material. Another type of construction is explained in this page by the Traditional Structures Centre of the University of Isfahan.
The dome is so useful that it appears in many artefacts of many sizes. One example is the dome loudspeaker. The moving coil speaker is in principle very simple: a coil of wire lies in a cylindrical gap in a powerful magnet. Alternating currents cause the coil to oscillate. The motion is transmitted to a cone which is connected to the coil, and thence to the air. "Obviously" the larger the cone, the better the coupling to the air, but there are many problems.
Life is hard enough when you try to built something static, like a bridge, but when you want something to move in a controlled manner, things get very complicated indeed. A good example is the suspension on an automobile or cart, where we want the vehicle to ride smoothly across uneven ground.
The ideal material for the loudspeaker cone might seem to be something very light, so that it can be made to vibrate at up to 20 kHz, and very stiff, so that the motion at the centre is copied faithfully at the edge. Real cones are to some extent flexible, and may act like a series of cantilevers. The motion takes time to travel in the cone, so that the motion is like a wave, which at certain frequencies may become resonant. One source of the trouble is that the cone is a developable surface, meaning that if cut once, it can be flattened into a plane with no stretching or wrinkling.
The dome is one solution to this problem. The diaphragm is now within the circumference of the coil, rather than outside it. There are two aspects to the motion. Firstly, the mass of the diaphragm, and secondly the pressure of the air.
When the cone is accelerated forwards, the dome will experience compressive forces like those in the dome of a building. When the cone accelerates backwards, the forces are tensile. When the velocity is forwards, the pressure of the air on the front is increased, while when the velocity is backwards, the pressure is reduced. A well-designed speaker dome handles these effects cleanly. The example is a Morel W144 bass/mid-range unit. Many high-frequency units (tweeters) employ domes too.
This speaker has a very shallow cone around the dome, and around that a very compliant roll surround to allow large excursions from the mean position.
The speaker is almost black: consequently a photographic exposure which shows any detail at all also shows up even very tiny specks of dust. Pity the professional photographer who has to prepare photographs for brochures.
Most of the structures described in this web-site were built from highly processed materials, which have used a lot of energy in their preparation. We must mention the fact that a great many structures have been built from natural materials such as dried mud and snow. Any climber who ventures on to snow has to know how to make a snow hole in which to survive if caught out by bad conditions. A snow hole is not a built structure, but the act of removing material means that what remains is a simple structure, probably with some arch action.
In sub-Saharan west Africa, dried earth or mud is used to make buildings of many sizes, from small dwellings to large and impressive mosques. Like stone, dried mud cannot sustain tension, and it is not very resistant to shear, so the range of possible structures is very limited. Nevertheless, the material has the merits of cheapness, local availability, and low energy use in preparation. This page describes the Great Mosque at Djenné, the oldest known city in sub-Saharan Africa. Click here to see pictures showing some arches and vaults of the mosque at Gao.
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