Arch

Imagine a grotesque animal with legs sprouting in all directions, but with no apparent head. It seems to be engaged in a futile struggle within itself, sometimes making convulsive movements in seemingly random directions. What is it? It’s a rugby union scrum. Sixteen people push with thirty-two legs, each half trying to push the other half back. If fifteen of the people could be instantly removed, the remaining one would fall to the ground. You don’t need to be a student of Zen to know the answer to half a scrum. One half cannot stand without the other.

Sometimes the scrum is almost perfectly balanced, at other times one side can push the other back. The scrum can even turn, which is not allowed, or it can be made to collapse, which is dangerous. What keeps it up? The heads and shoulders of the players push forward, and their weights pull down. These forces are resisted by the upward and forward push of the ground. The scrum is a living arch, and it demonstrates all the principles of that structure. It exerts a weight on the ground, and it exerts outward thrusts, resisted in this case by friction.

The cathedral of Seville, with its double row of buttresses on each side, is the structural equivalent of a scrum, with the two sides of the nave corresponding to the two front rows, and the two rows of buttresses on each side corresponding to the remaining two rows of players on each side.

Another living bridge is the case of a climber bridging across a wide chimney. In this example, her or his weight is resisted by the vertical friction that is generated by the outward force on the walls.

The problem

Here is the problem – how to get a road across a river.

The small river running from left to right in the picture is the river Windrush at Upper Slaughter, in the Cotswolds.

The solution

The simplest solution, shown here, is a ford, and as this river is normally only about 15 cm deep, a ford is adequate, if you drive slowly. On this occasion, however, the river was much deeper after heavy rain, almost covering the openings of the small two-arched footbridge on the right. Fords are only useful for small, shallow, slow-flowing rivers. The real answer is commonly a bridge, and rarely a tunnel, though ferries are still found.

Some Very Small Arches

The arches in the first picture, over the River Windrush at Barton, near Guiting Power, in the Cotswolds, must represent about the smallest size of arch that was worth building. In fact they are culverts rather than bridges, carrying the river under a minor road. Nowadays the bridge might be built another way, as in the examples in the panel below. Notice the deformation from the original semicircular shape. The other picture shows a similar construction in brick. Notice the voussoir (sector) that has moved out of position. We already learn something about arches. They seem to be quite stable when deformed.

Here is another small arch.

Here are more examples which have been constructed as pipes.

The small arches in these pictures, at Abbeydale, Gloucester, are based on tubes made of corrugated metal. Plenty of space has been provide for extra flow in times of spate. The ends are faced with reconstituted stone, which has not been arranged to look like arches. Whether structures should be made to conceal their structure is an interesting point. If one function of a structure is to look pleasant, then it may not be obligatory that its structure and technical workings should be obvious at the expense of appearance.

After all, stressing wires are seldom visible, so the observer cannot see true nature of a structure which uses them. And foundations are never visible.

A pleasant feature of both arches and suspension bridges is that their structure and function are fairly clear, in combination with an attractive form. The great importance of appearance is well covered by Fritz Leonhardt in his book "Bridges".

A similar construction takes Horsebere Brook (left) under a slip-road of the M5 motorway. You might think that this is a fairly simple thing to construct, but when you go into the tunnel, which is about fifty metres long, you find that large piles of stones have been placed along the stream on alternate sides, making artificial meanders which reduce the speed of flow in times of flood. The right hand bridge even has a road under it as well as over it: it allows access to a large car-park.

Some Larger Arches

Arches vary enormously in appearance, as the following pictures show. Some of the variables are –

Length of span

Width of deck

Height above ground or water

Height of piers

Ratio of span to width of pier

Ratio of span to rise

Number of spans

Materials

Segmented or curved

Type of construction

Arch at Niagara Falls

Bridges of Paris

The arches of this footbridge are very unusual. They appear to support the footway only at the half-span and quarter-span points, while requiring support themselves from the piers.

This is a part of the Musée d’Orsay in Paris. This wonderful building was once used to house a railway station.

This is a short tunnel, rather than a bridge, but it is a neat solution to a problem. Will the solution outlast the problem?

Some British Arches

Links to other arches in this web-site – Lune Arch Severn Arches Telford Arches

This is a very old bridge at Bradford-on-Avon, near Bath. This bridge, like the towns of Bradford and Bath, was built from Cotswold limestone, which is attractive, but susceptible to sulphur in the air.

Until Bath was cleaned in the 20th century, many of the buildings were almost black. The Pulteney bridge, built by Robert Adam between 176 and 1774, over the Avon at Bath has shops along both sides.

This bridge has two rows of very small voussoirs. There seems to be a concrete arch above them.

An elegant footbridge over the M42 motorway south of Birmingham. Is there a hint of Maillart?

Here is the bridge across the Sava Bohinjka at the eastern end of the beautiful Bohinjsko Jezero in Slovenija. The church is that of St John. It includes 14th century frescoes and is well worth seeing. The bridge has pedestrian refuges at the centre, and the wooden posts in the river bed are probably the remnants of defences against scouring in times of flood. After a heavy downpour in the Julian Alps, the river can rise rapidly, with an enormous increase in the rate of flow. Unfortunately these rather poor pictures don’t do justice to the scene. In the sixth picture, you can see, at the eastern end of Bohinjsko Jezero, the same bridge and church.

But the situation is reversed in the lower diagram – any movement, however small, lowers the centre of gravity of the system. So the symmetrical position is unstable. If the rods were to be released at the red position they would diverge even further from symmetry, until the right-hand rod would be resting on the line at the bottom.

We can use the same argument for more than three rods., and the limit of an large number of small rods, for a chain or cable. Any hinged polygon with more than three rods is not rigid, and is unstable if it is above the points of support.

Yet a stone or concrete arch looks very solid and robust. What is the secret?

One reason for the stability of many arches is that the volume between road and arch is filled in with masonry, which adds rigidity. In fact the masonry spreads a point load in such a way that its effects reach several voussoirs of the arch. The masonry holds the voussoirs together much as the hoops of a wooden barrel hold the staves.

A similar effect is seen in the brick walls of a small house. The opening for a small window needs no special treatment, even though there may be five metres of brickwork above. But the opening for French windows or patio doors needs a concrete or metal beam at the top, because the brickwork cannot spread the load across such a span.

We see from the bricks above the small opening that the bricks do not have to take the shape of an arch, as long as the line of thrust is within them, or there is enough support around them.

But here the opening is so wide that a reinforced concrete beam is needed to support the wall above.

Some houses have a shallow arch above windows and doors. The wall each side has to be able to absorb the side-thrust. A beam contains the tension and the compression within it, rather like an enclosed tied arch or self-anchored suspension bridge.

The other reason for the stability of an arch is that an arch has substantial thickness, so that even with variation in the load, the line of thrust passes through the voussoirs. In other words, the hinges mentioned above do not exist in a stone arch. But we also see that in principle an arch can be stable with up to three hinges, and such arches have been built. In some cases the voussoirs could in principle stand alone if the centring were removed. "Packhorse bridges" often consisted almost entirely of voussoirs.

For a semicircular arch this cannot be true, because the outward thrust cannot be made to vanish just by curving the arch. If you don’t believe this, imagine continuing an arch to more than a semicircle, in which case you are asking the thrust to turn inwards. In fact some Islamic buildings do contain arches that are greater than semicircular, but they always bear against something else, such as other arches.

Why build arches with hinges? If a system of parts, such as a truss, is assembled in such a way that it would be inherently rigid if hinged, and all the parts are rigidly fixed together, then if the fixing is not done exactly, there will be unwanted stresses in the parts. The system is in fact over-determined, or over-constrained, and at least some of the stresses cannot be calculated, so it is indeterminate.

For an arch, adding three hinges removes these effects completely, leaving two freely mobile halves propped against each other. Arches have actually been constructed with no hinges, one hinge, two hinges, or three hinges. Some indeterminate structures are provided with jacks, so that the stresses can be controlled. In some cases the foundations are then set in concrete, but in other there remains the possibility of later jacking to correct for subsidence.

The diagrams below show schematically the possibility of hinges or pins in arches. The one-pin cases are not useful in practice. It should be noted that the word "hinge" should not be taken too literally: if a part can be made narrow enough, the small angular movements it makes will not cause it to crack or crush. In such a case, no actual hinge mechanism need be constructed.

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Eiffel’s Garabit bridge is a magnificent two-pinned arch. The Hell Gate, Bayonne and Sydney Harbour trusses do not taper to the springings like that of Garabit, but nevertheless the thrust reaches the abutment through the lower chord only, the upper one being only for stiffening at these points. At the crown, however, by closing the truss with suitably dimensioned pieces, the forces can be shared between the upper and lower chords. Therefore, between springing and crown, thrust must migrate between the chords through the bracing struts and ties. See also the page called Arch or Beam.

The next diagram shows a set of designs for three pinned arches and another set for two pinned arches. Which ones do you think are bad designs? Which, if any, are potentially useful?

Transition from beam to arch

In the diagram below we see a transition from a simple beam at the bottom to something like an arch at the top, by adding more voussoirs and making them smaller, with thickness for stability.

Note that with an even number of segments, there is no keystone. The keystone of an arch has no significance in engineering (it is not even where the thrust is greatest), though it is sometimes quite prominent as in these photographs at right.

If we now replace all the straight parts by circular arcs, we get the second set of diagrams.

This wall has two openings for beams, below, and two for ventilation, above. The latter are primitive arches, such we saw in one of the diagrams above.

Voussoirs are not actually necessary – they are used in masonry arches for the obvious reason – it makes no sense to carve an arch from solid mass of stone, even if you could it to the site. But natural rock arches do exist, carved by natural forces, which will one day go on to reduce them to rubble.

The next picture shows a natural arch in the valley of the Mostnica river in Slovenia. This region of the Julian Alps is made of limestone, in which the fast flowing rivers carve all sorts of curves and holes, often by means of whirling stones in cavities. This arch may have been made where two cavities have become large enough to cut into each other. The hole would then have been enlarged by water flowing by and through.

Here are three snow bridges in the Alps. You might not think of them as arches, but their spans are so small compared with their thickness that some arch action may be occurring when they are loaded by a walker. In any case, if a snow bridge is so slender that it is definitely a beam, you definitely don’t cross it without great care. In the third picture, a large section of the bridge has collapsed into the crevasse. Snow bridges are most safe in the early morning before the sun has had time to start its destructive work. By the afternoon some of these bridges may become very dangerous. Even if you are roped to someone else, getting out if you fall in is not a simple business. One solution is to use jumars.

How can we suggest that a snow bridge acts as an arch? Nobody actually designed and built it, arch or no. Well, if it is to act as a beam, it has to take tension near the bottom surface. Snow is not well known for its tensile strength. In the same way, although children of all ages build structures from damp sand on beaches, any openings they make are in the form of arches or tunnels, and not in the form of beams: damp sand, like snow and like masonry, has little tensile strength.

In steel, as in stone, it is convenient to build the arch in sections, but a concrete arch may be poured complete, if required.

The way that the line of thrust behaves in a simple unstiffened arch can be seen by looking at this download. (Choose Run from Current Location.) It simulates loads with random weights moving with random speeds. In principle, for a masonry arch or concrete arch without reinforcement, the bridge will survive if the line of the thrust remains within the arch. To prevent disappearance of compression at any point, with the possibility of cracks, the line of thrust should stay within the middle third of the arch. With reinforcing or pre-stressing, or with steel arches, the line of thrust is not quite so critical.

The diagram below is a frame from the download, showing the case of a heavy load, indicated by the arrow. In such a case, the arch would be in tension around the line of the load. See also the page about the funicular. You could say that what distinguishes an arch from other structures is that if it is perfectly funicular, it will experience no bending moments. In practice, for economy of construction, there may be slight deviations from the funicular. And for reasons of practicality, such as headroom, there may be large deviations.

Note how the line of thrust moves towards the force. The bending moment thus created is proportional to the vertical distance of the thrust line from the centre line of the arch. It is also proportional to the horizontal thrust, which is the same throughout an arch. Deliberate moving of the line was used by medieval builders of cathedrals – they added weight to walls and to buttresses in order to change the shape of the line of thrust to better fit the shape of the structures.

If the line of thrust passes through the centre line, there is no bending moment. Therefore, if at some place an arch comes to a point, such as a hinge, there cannot be a bending moment there. That makes sense, as a hinge cannot resist such a moment.

The next picture shows an even more extreme case, with the line of thrust outside the arch. The position of a crack is shown. If the arch were to crack at two other points, thus producing three extra hinges, the risk of collapse would be very real.

At this point you will be asking why the line of thrust is so thin. In fact you should ask questions all the time. What you read, hear or see in any medium could just be wrong . . . . .

The answer in this case is that the line of thrust is not thin. It was drawn like that for simplicity. The force is really spread throughout the arch. The "line of thrust" is just the average line. Now we will do it more exactly, and we will understand a little more. Meanwhile, if you don’t believe in lines of thrust, stand up straight and get someone to push you in the back with steadily increasing force. You will find yourself rising on to your toes when the line of thrust goes too far forward. And if you carry an extremely heavy object in one hand, you automatically lean over.

Getting back to arches, look at the diagrams below, in which the compression is indicated by the brightness of the red hue. Going down the diagram, the point force at the crown of the arch is increasing in size.

In the lowest diagram the compressive stress just reaches zero at the top of the soffit (the underside of the arch). Any more weight on the arch, and tension would occur. The single line we saw before is actually the average of the red paths. The density of the red, that is, the compressive stress, varies linearly through the arch, and in the case where it runs from zero to a maximum, the mean line is one third of the distance from the top surface to the bottom.

This is an example of the rule of the middle third, which states that if the line of thrust goes outside the middle third of a section, tension will occur. The implications of this are striking. To get a sumo wrestler’s foot off the ground you only have to get his weight outside the middle third of his baseline, and that’s if you aren’t using his momentum. No wonder they place their feet so far apart. We see also the implication for the first Tay bridge and for Eiffel’s arches and tower.

And for a two dimensional section, such as that of a tower or a column, the middle third in two directions means the middle ninth in area. Now look at a picture of the leaning tower of Pisa. And think about the skills of medieval builders of religious buildings, which are described in another page of this web-site.

What is the correct shape for an arch? If the cross section is uniform, and no loads are present, apart from the self-weight, teh shape should be a catenary, the shape taken by a uniform flexible cable, hanging between two points.

This diagram shows an arch based on a catenary, the red curve.

To generate catenary arches with different ratios of height to span, click here to download program Brancat. By pressing the PrintScreen key, you can copy the picture into the clipboard to use it as the basis of a model.

Real arches, of course, carry loads, and they very often carry weight between the voussoirs and the road. The distribution of these loads means that many different shapes have been successfully used. And if the bridge is made of steel and not masonry, its rigidity enables it to deviate hugely from the funicular, the ideal line of thrust, for example to obtain headroom over a wide road.

Because all arches generate thrust at all points in the arc, they cannot stand up until they are complete. One solution is to build falsework, called centring. Another, less common, is to hold back the two halves using cables, until they are ready to meet. This is done with steel bridges, but not with concrete or masonry arches. The two URLs given here link to a picture by Canaletto, "London seen through an arch of Westminster bridge". The bridge had been built, but the centring had not yet been removed. A bucket hanging from the timber gives some idea of the scale. Canaletto 1 Canaletto 2

No law of physics forbids the line of thrust to be outside the structure. What is true is that in such a case, you cannot use materials that can only resist compression, such as masonry and unreinforced concrete. You need materials like reinforced concrete or steel, as in the two examples shown. Notice how the bridges are thickest where the deviation is greatest. This has two benefits. Firstly, the stiffness is increased where needed, and secondly, the extra weight moves the thrust line closer to the structure, which reduces the bending moment.

The line of thrust is always raised by the load, mainly near the centre. So we might think that the arch should be built with the dead-load thrust near the lower edge of the arch. We might also expect that some arches might taper almost to a point at the abutment, in view of the behaviour of the computer simulation mentioned above. This is not practical with masonry, but it is with iron and steel, as Gustav Eiffel showed. Maillart tapered a number of reinforced concrete arches early in the 20th century.

The next diagram shows a voussoir arch that is overloaded at the crown, making the line of thrust fall outside the arch. Three hinges have formed around this point, as well as two more at the abutments. Stability is impossible with more than three hinges. Is this diagram correct? Would hinges really form at the abutments, or would they form elsewhere? A similar effect could be produced by an overload at any point in the arch.

This diagram demonstrates two points – firstly that extra load does not necessarily increase the stress at all points in a structure, and secondly, that reduced stress is not always desirable. If you don’t balance your trailer correctly, you can reduce the load on the front wheels of your car, but this doesn’t do much for the steering.

If you look at the pages about beams and pre-stressing, you will see that beams made of concrete cannot withstand tension, a failing that is overcome by compressing the beams using pre-stressing wires. We could imagine that the arch is a type of beam which is pre-stressed by curving and letting its own weight do the compressing.

A Strange Arch

Here is a gateway with, apparently, an arch above it. But it the two ends of the arch, the bricks have been tapered the wrong way, so that the junctions with the walls are vertical. So the weight of the lintel has to be borne entirely by shear stress within the mortar. The flatness of the arch produces a large thrust into the mortar. Perhaps this helps to hold the thing together. But it is not an ideal way of doing things. In fact it is possible for such a structure to hold up even if the mortar loses its adhesion. If the angle from the horizontal is small enough, in relation to the coefficient of friction at the ends, the arch will stay up. This principle is used by climbers in the method known as bridging, in which they can climb a vertical chimney or a vertical crack by bridging across it, using the legs or the legs and body.

One advantage of the arch over the beam is that the ground is used to oppose the outward thrust. Near the abutments the ground is in compression, but under the arch it is in tension. Within a beam there are both compressive and tensile stresses, and of course shear stresses and bending moments. The arch avoids these, at least for the dead load. So it can be made lighter than a beam of the same span.

Therefore the longest arch is longer than the longest beam (Long spans). The beam does have three advantages; it can carry the deck directly, in principle it can be built as a whole and moved into position, and in a multiple span bridge the beams can be joined, and even stressed together to optimise bending moments. The idea of moving the whole span is possible in the case of a tied arch. An example is given later in this page.

The Romans built semicircular arches with very thick piers, so that any arch would remain standing if its neighbour was removed by flood or by enemy action. The thrust was meant to remain entirely within the piers. The Romans were not interested in record-breaking spans, only in utility and durability. That some of their bridges remain after about 2000 years of continuous scouring, in rivers which are subject to frequent heavy flooding, says it all. Military action has removed many that would otherwise have survived.

The diagram below, a vertical section, suggests the way that the ground transmits the tension below a two-pinned arch. It is not an exact calculation, only a rough sketch, and the lines would be distorted by variations in the ground. The actual force-field is continuous, and not really along narrow lines. Compare this with the stresses shown in the page on beams, and with Brunel’s Saltash bridge spans. These tensions in the ground are normally unimportant, because they are diffused over a large area. What matters is the stress, or force per unit area, which is large only at the abutments, where the compressive stress is largest.

The next diagrams are different ways of visualizing the stresses near the surface of the ground. The first one uses colours from yellow to red to show the intensity of the stress, along with contours at equal intervals of stress.

The next picture shows the same result by means of a grey scale.

Next we see two diagrams in three dimensions, with the stress represented by height. The diagrams differ in that the area of the abutment in the second case is much greater than in the first case, making the maximum stress much lower. These diagrams are intended only as a rough visual indication, having been derived from a very simplistic calculation.

The diagrams show that most of the ground experiences very low stress, and so its properties may be thought unimportant. But if the ground between the abutments contains geological faults, or is subject to slippage, it becomes very important indeed. In the case of a dam, the building of the dam, and filling with water, may have drastic effects, such as creating huge pressures, and lubricating fault planes. After the Fréjus dam was built, a gigantic earth-slide caused much of the water to be sent over the top of the dam in a few seconds, engulfing the towns downstream, with great loss of life.

In fact, in every type of bridge, except arches and suspension bridges, and beams with sloping struts, the horizontal forces are kept out of the ground, and carried in members that oppose the forces in the rest of the bridge. In those bridges, the forces on the ground are purely vertical.

The lines of force are spread sideways and vertically, as if they repel each other. Why do the lines of force not simply run straight along under the bridge? The energy density at a place is proportional to the square of the stress, for elastic material. Therefore the minimum energy state is found when the stress field is diffuse. Halving the stress at a place divides the energy density by four. The distribution is the one that minimises the total energy. Spreading it or shrinking it would increase the strain energy. These diagrams are not unlike the fields around a bar magnet or a pair of electric charges.

The stresses near the abutments are more complicated, because the arch induces compressions, which are present along with the tensions already described.

These diagrams make clear that the structure includes not only the visible part, but any part of any other object that is subject to significant stresses.

Stresses in the ground are perhaps most important in the construction of dams, where not only the dam, but a vast mass of water, creates great pressure on and in the ground, together with lubrication in cracks. See arch dams and gravity dams.

The diagrams below are outlines of some bridge types. Compare these with the previous diagrams.

At the bottom two beams are propped together. Because, together with the ground, the system forms a triangle, the beams can be hinged at all three joints. As explained in the page about beams, the variation of bending moment suggests that beams should be deeper in the middle, as in the next diagram up. Above that, a deck has been added, and in the top diagram it has been integrated into the arch. The beam would of course be supported at the ends. And so we see that an arch is not entirely unrelated to a beam.

The diagram at the top is an ugly version of a type of bridge that was beautifully designed by Maillart, and used many times since, though not always with the artistry that he possessed. Actually, Maillart arrived at his designs by a different line of thinking, starting from a normal arch, but the result was about the same. And here are some more ideas. In practice, the depth would probably be varied along the span to take advantage of the arch action when the supports are sloping inwards. Note how the deck is in compression when the struts are sloping.

In the structures above, there are straight struts and curved struts. Which are correct? If the struts are much lighter than the deck, they should be nearly straight, but if they are much heavier than the deck, they should be curved. Why? Some of the structures on the right resemble a gothic arch, and in fact they represent the true use of this shape. In gothic buildings, the point of the arch seldom corresponds to a load. The same is true of Sydney opera house.

Outward Thrust

An essential result of building an arch is that there will be an outward thrust at each end. This has to be resisted by the abutments. If you don’t believe this, try standing with one foot in a small boat and one foot on the river-bank, or better still, with your hands on the bank and your feet in the boat. You will very soon be in the water. Standing with your legs wide apart on ice will have a similar effect. The two pictures below show Telford’s bridge at Over, near Gloucester, which was completed in 1829. When the centring was removed, the crown sank about ten inches, because the thrust was not properly resisted, but the bridge was used until 1974, when a steel bridge was built nearby, to carry a much wider road, the A40.

The only way to avoid the thrust reaching the abutments is to tie the ends of the arch together, using the deck or some cables. This creates a tied arch, or bow-string arch. If the arch design is chosen to provide a passage for ships or traffic underneath then the tie method may not be acceptable, unless the whole thing can be built high enough. Then the road is very high, which creates problems with the approaches, unless the arch spans a deep narrow valley. But in such a case, the rocky sides will probably take the thrust in any case.

This arch, unsupported at the left, has sagged at the crown, as the green and white lines show. Most ceremonial arches and entrance arches would have more substantial supports, to make sure that the lines of thrust reach the ground well within the piers.

This type of entrance was a good solution in the days of horse-drawn coaches: with the advent of tall lorries the type of entrance shown below is more suitable.

This almost looks like a small building hanging between two bigger ones. Note the curved window to relieve the monotony of the rectangles. Comparing this picture with others in the web-site reminds us that the appearance of a building is is not absolute. What a transformation we see between a dull day and a sunny one. Bare, flat concrete, especially, is particularly dull on a grey day.

This picture shows another archway that has sagged.

Here is another apparent example. The masonry courses in the right hand arch suggest clearly that it has sagged. But look at the left hand one. What has happened there? Perhaps some rebuilding has been done.

Is there a slight hint of sagging around the top of this arch, or is it just slumping of the masonry around it. There is distortion of the courses below the arch as well. The cause is not pincushion distortion of the image by the lens, because the cropped outlines of the building were straight.

Apart from tied arches, you need a place with very good ground that can provide the reaction to the thrust. Somewhere under the arch, the ground is in tension, though of course it is in compression around the abutments.

A big advantage of a relatively small tied arch is that it can be built off-location and moved into place as a complete unit.

This was done in 1999/2000 on the M42 motorway, when a very elegant tied arch was built on land south of the road, and then translated across the road in the course of a single night.

The diagrams below show roughly the forces in an arch.

It is easy to calculate the horizontal thrust. The weight of the half-arch acts at the centre-of-gravity of that half. Its moment about the abutment must equal the moment of the horizontal component about that point.

The clockwise moment from the weight is W X D, and the anticlockwise moment from the thrust is T X H. These are equal, so T = W X D / H.

This makes it very clear that a flatter arch (smaller H) produces a greater horizontal thrust. A very flat arch such as the Pont Alexandre III in Paris exerts a huge thrust on its abutments. Whether this produces feelings of strain or unease in an observer is of course a subjective question, depending to some extent to whether it is seen as a geometrical shape or as a living structure. The Millennium Bridge in London is the tensile equivalent.

How about this for a flat arch? The building was put up by the Normans, but someone has added an amazing construction to the inner arch. Why? Perhaps they simply wanted a rectangular opening for the doors. How do we know that this flat soffit doesn’t make it a beam? In fact, suppose we could get a piece of stone of the size and shape of the complete set of blocks – would it be a beam or an arch?

One thing is clear from this and the many other alterations to old buildings – the owners did not let reverence for the past prevent them from adapting things to their current needs. Many a church and cathedral is a hotch-potch of added pieces, often with an astonishing mixture of styles. We have to remember that the purpose of a religious building is not to give satisfaction to aesthetes or engineers, whether present or future. It is built to enable people to worship the deity.

Note the slight sagging of the outer arch, shown by the line of stones above. This effect can be seen quite often in old buildings.

In the next diagram we see a crude three pin arch.

If the weight of the load is vastly greater than the weight of the arch, the graph represents the horizontal thrust as a function of the slope angle A. As A approaches zero, at the left, the thrust tends to infinity. So an arch cannot have zero rise. What about a beam? A beam is not an arch – it does not have a hinge. The beam is rigid, and in the page about Beams, we see that a solid beam contains within it both an arch and a suspension cable.

When you see a mathematical function, it is a good idea to ask what happens for all possible inputs. The next graph includes negative angles as well as positive ones.

Negative angles produce negative thrust: the structure is a crude suspension span. The jump from plus infinity to minus infinity would not happen in practice. No structure or supports could provide infinite force. What would happen is that at some very small angle, the compression produced by the thrust would be enough to let the arch fall through the gap and become a string.

If we keep the angle just above the critical point, the structure has two stable states, and we can cause a transition to the other state by adding a small extra force. Many latches work on this over-ride principle. Some electronic circuits are based on monostable or bistable systems. The lavatory cistern is a monostable system. If you operate it, the water pours out, leaving the cistern in an unstable temporary state. The water flows in, and eventually stability is reached when the valve stops the flow. The filling takes a considerable time, and in fact monostables are often used to generate timing periods in electronic systems. But for great precision, crystal clocks are more often used.