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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. arch bridges 27

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.

The thrusts on the ground add up to more than the weight of the arch, because it they are composed of vertical parts, which take the weight, and horizontal parts, which keep the arch from spreading.

Whatever the type of ground, we might ask what happens to the thrust as it enters the ground. It does something like this –

The pressure decreases as we look further from the springing of the arch. The purpose of abutments is to take the pressure until it has decreased to the point where the ground can be relied upon. The next picture hints at this.

Because the ground is not moving under the compressive forces (after everything has settled down), we know that some opposing forces must exist. The ground under the arch is in tension, but the stresses are very weak, because the forces are diffused through a large volume.

Making adequate foundations is not necessarily the end of the story. The foundations may be threatened by earthquake or flood, for example.

In fact, even the normal flow of river water past bridge piers can generate scouring which can bring down a bridge. The presence of the piers changes the flow, producing acceleration and turbulence. The lifting and carrying power of a fluid increase as a high power of the speed. The ancient Romans knew about this, and took precautions. Foundations need to penetrate to secure ground, and a pavement around piers can help to protect the bed. You can often see the results of scouring around a post or a boulder on a sandy beach, after the tide has gone out. The next diagrams, which are sections at right angles to the flow, show the general effect.

Furthermore, because of turbulence, the pressure on the bridge fluctuates with a wide frequency spectrum; people on a bridge that is nearly submerged report feeling strong vibrations.

These pictures show some bridges which are equipped with concrete or stone platforms to eliminate scouring. They all span the river Wye, which has carried away many spans in the past.

These platforms are normally under water, but have been revealed after a long drought. arch bridges 28

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. arch bridges 29

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. arch bridges 31

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 not absolute. What a transformation we see between a dull day and a sunny one. Bare, flat, dark concrete, especially, is particularly dull on a grey day.

Here is an arch that has been deformed by the weight above it. It is in the nave of Burford church, and carries a part of the weight of the tower and spire. The second picture has been compressed horizontally to clarify the distortion.

This 12th century arch has had plenty time to move, and it has developed an incipient hinge and a number of dislocations. The builders may have thought that a heavy arch would be strong, but a lighter one might have deformed less. Who knows?

Here are more arches that have yielded excessively to the loads on them. Perhaps the builder was over-ambitious in attempting these rather flat arches, with not enough buttressing at the left.

This arch was probably built in the form of three circular arcs. The adjusted picture suggests that there is a discontinuity on the left side in slope as well as curvature. Perhaps the arch was built like that because of an error in the centring, or it may have slumped slightly. The second picture is an ideal construction, for which the result is not very different. Perhaps the "problem" is an optical illusion.

Here is another way that an arch can be distorted – the building has cracked.

This skew bridge has a surprise for us. The first picture at left, squeezed horizontally, shows that the bridge is distorted. Remember that as a railway bridge, it should have a dead straight deck. The vertical yellow lines show two kinks in the masonry, and the two others show the misalignment in the courses. The second picture has circles added to show the distortion of the soffit from a single smooth curve. We have to be careful with this kind of thing to be sure that our lens has not caused some or all of the distortion. Using a wide-angle lens can also change the shapes of curves very significantly, as shown by the third picture, a poor example of a bridge picture.

The way we tried to show the distortion of the skew arch was very crude. A better way would be to use a suitable mathematical comparison with the correct curve. What if you don’t know the mathematical method? In the diagram below, we have taken a lot of points on the soffit of the arch, and we have moved them towards the centre by various fixed amounts. This gradually exaggerates any distortion, though in fact the distortion is great enough to be seen in the original curve, the outermost one, as a slight flattening, right of the crown. The advantage of mathematics is that we would be able to put a figure to the distortion, but if we only want to see that it is there, a pictorial method will do.

What is the first thing that you should do after doing a calculation? Distrust it. Try to check that you get the same result by a completely different method. Failing that, use the method on an example for which you know the answer. Let’s do it. Let’s begin with a perfect circular arc and see what happens.

OK – that’s not a mathematical proof, but it’s encouraging, so let’s put a small error in two points of the circle.

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. arch bridges 33

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. arch bridges 34

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. arch bridges 35