Keystones, Windows and More About Arches
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is a keystone? A keystone is the stone at the top of an arch. Here is
an arch seen from the viewpoint of a rather self-important keystone.
One thing is sure, the keystone is not in the position of maximum stress. But do we need a keystone at all? What can we deduce without getting out our finite element analysis program, or using the calculus of variations? Yes, we should always try to use the simplest means to discover answers, though it is a good idea to use at least two methods of obtaining an answer. Don't forget - if you calculate something twice, using the same initial assumptions, and you get two different answers, at least one of them must be wrong, and both may be wrong. The same is true for measurements. Even if two different results agree, they are not thereby proved correct. To be more accurate, the results should not differ by an amount which is incompatible with the accuracy of your method of working.
Let's ask a question about the stress in the keystone. In which direction does it operate? Imagine a vertical line through the middle of the arch. By symmetry, the forces in the two halves must be mirror images. In the keystone, if the forces had a vertical component at the centre line, then these would not cancel out, though the horizontal forces would. Therefore the stone would move, in translation or in rotation, until the vertical force vanished or the arch collapsed, and in particular, no shear would remain. So the forces along the centre line must be horizontal. So if we cut the keystone in two halves down its middle, no sliding would occur, because no shear stress is present. Now we don't have a keystone - we have two equal partners at the top. So an arch can work perfectly well with an even number of voussoirs, and no keystone.
Since this page is rather negative about keystones, we could also mention an article in Arab Construction World Nov 2005 / Vol XXII - Issue 7, in which Hakan Sandbirg points out that the value of the keystone is that it reminds us that "until the structure is complete, we have to keep thinking" - an important point.
You will have noticed that we have stated, without proof, that the stresses are least at the top and greatest at the bottom. Can you prove this without doing any mathematics?
The porch of St Brelade's church in Jersey has a most peculiar crown of the arch, and there is a concrete beam across the doorway. The circular window above has no keystone. The vault of the nave is composed mainly of thousands of small irregular stones, though the crossing vault uses blocks.
Keystones are not needed. So we don't need a page about keystones. Ah well, let's just fill up this page with miscellaneous observations about beams and arches. Actually, you will probably have already realised from the existence of three-pin arches that the keystone is inessential.
By the way, the arch in the diagram at the top of the page will not stand unless the lowest voussoirs are provided with inward forces to stop them spreading, unless the mortar (and the blocks) are very resistant to shear and bending. Can you see why?
What do clocks, lamps and mirrors and dinner plates have in common? An invitation to designers to produce not only work of distinction, but work with just about every other characteristic, not always tasteful, as you can see in shops and advertisements. You can even buy plates and mirrors that are clocks. Architecture is a much more restrained discipline, if only because you have to build something that stands up for a reasonable time. Even the architecture of dictators and totalitarian states usually errs in the direction of big, empty, and pretentious, rather than over ornate, at least on the outside. The pyramids of Giza are extreme examples. If large structures are dominated by the laws of physics, what about small details, such as windows?
The person who designed this window may not have known whether he intended to make a beam or an arch, nor how to make either of them. Notice the crack in the left hand half of the lintel, shown in the second picture. The top edge is clearly in tension. That is definitely out of order for an arch. Someone has tried to fill the crack with mortar. This will at least keep out water and frost, but it won't cure the problem. It might have been better to make the lintel in one piece.
The picture at left has been exaggerated vertically to show the angular change in the left hand block, which has cracked, and the slippage of the right hand block, resulting in the cracks at the top right hand of the picture.
These keystones dominate so completely that the arches are barely noticeable. The brickwork over the left hand window has distorted and cracked, because the lintel has sagged. This type of brickwork is called a jack arch.
The example shown here is 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. he 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 a beam.
These arches go to the other extreme. There is little possibility that they will give problems. The designers have tried to integrate the voussoirs into the buildings. The middle picture shows a part of a bank. Banks were often built in this very solid looking style.
Here is another arch with large voussoirs. One possible problem with an arch between buildings is that you may wonder what happens inside the buildings, since the walls cannot be expected to take the thrust. If the soffit reaches the ground, the question does not arise, but such a design is unusual, because it prevents people from walking under the bridge near the buildings. So - what does happen inside the buildings. This example in Birmingham connects two parts of the art gallery and museum, and is wide enough to serve as a display room in its own right.
This neat arch has two rows of voussoirs, and the designer has chosen the proportions so that 15 voussoirs in the upper row match 14 in the lower row. A massive block at each end carries the vertical and horizontal components of the thrust into the wall. Although keystones are strictly unnecessary, this one lifts what might have been a rather ordinary entrance.
Here is another keystone. There is definitely no arch. The lintel probably refers to classical Greek ideas, via various later European practices. The "ancient Greeks" were not interested in arches, and they did not use keystones, so this design is rather a hotch-potch of styles. The windows, not shown, follow yet another style. On the other hand, this keystone might be an architectural joke.
And here is another non-arch, in fact a beam, with a massive non-keystone. The keystone, beginning as an unnecessarily large voussoir, has evolved into a decorative block that still acts as a voussoir in arches, and finally here to an obvious piece of pure decoration. These beams may even have been carved from single blocks of stone.
The pictures below show openings in buildings, arranged roughly in order of the ratio of rise to span. Can you classify each one as either arch or beam?
These two lintels are subtly different. The lower is a pointed arch with a large wedge shaped keystone. The upper is very unusual. It has a beam below, deeper in the middle, reflecting the greater bending moment there. Above it is something that looks like a brick arch. But does it act as an arch or not? Perhaps it just rests on the lintel as if it were the first course of bricks above the window. If it does act as an arch, how is the load shared with the beam? What if the stone, brick and mortar all have different coefficients of thermal expansion? What about the effects of differing elastic moduli? In fact, it is not uncommon to see flat arches in brickwork, with some horizontal bricks underneath, and sometimes a lintel beneath, of stone or wood. In such cases, the window frame or lintel has only to support the bricks that lie between it and the arch. The bulk of the wall is supported by the arch.
Our last keystone example is truly weird. On the left, the bricks above the window can just about be taken for a jack arch, and on the right? We see two halves of arches, with a supporting column between, and on top of that, a keystone looking like a giant exclamation mark.
Voussoirs are the wedge-shaped blocks that make up a masonry arch. But are they actually acting as wedges? Look at the diagram below, showing how a wedge is used to raise a heavy object, or perhaps to stop a door moving sideways.
The red arrows represent the main forces acting on the wedge. They are not equal and opposite. In the absence of any other forces, they would force the wedge out of the gap. But there are, in fact, two other forces, shown by the small yellow arrows. These are the forces of friction. The angle of the wedge in radians must be less than the coefficient of friction, that so-called constant which is so popular with the authors of elementary physics books.
As voussoirs are wedge-shaped, we might wonder whether they might act as wedges. The answer, it turns out, is no, not if the arch is funicular. The page about funiculars shows how a non-funicular arch requires internal forces to resist shears and bending moments. These may be taken to correspond loosely to the friction in the wedge.
The distinction made here between wedges and voussoirs is probably not completely clear-cut. An essential difference is that a wedge is usually a small object that is used to separate two large objects, whereas a voussoir is surrounded by other voussoirs of similar size. We seldom have to consider the weight of a wedge, whereas that of a voussoir is highly significant.
Let us consider a fragment from a semicircular arch, because it is easy to draw. And if a semicircular arch was good enough for the Romans, it will be good enough for us.
There is a fundamental difference between the voussoir and the wedge. The weight of the voussoir is the force that turns the line of thrust through the arch: no friction is needed in order to balance the parts. If you were to replace the marked block by a much lighter one, then in the absence of any friction, it would be squeezed out like a frictionless wedge, or at least the arch might be distorted in some other way. And if the one voussoir were very heavy, and the rest of the arch very light, that one voussoir could slide inward.
The next question is this - Is the position as built, the most stable? If you were to displace one or more blocks radially, inwards or outwards, would the total energy go up, go down, or stay the same? If you think about a hanging chain or cable, any disturbance raises the energy. Presumably any disturbance of the inverted case, an arch, lowers the energy. So the arch is not inherently stable, unless it is given rigidity by virtue of its thickness. Do you think that the minimum thickness for a stable arch is related to the criteria for a stable strut, such as the Euler criterion?
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