Arch

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. The other picture shows a similar construction in brick.

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.

Further upstream, the brook is crossed by the link road from Gloucester Business Park to the Brockworth bypass and southbound M5 motorway, over an arch based on curved concrete slabs. The use of a slightly non-circular profile increases the headroom over the farm track and the footpath. Although this bridge is seen by a relatively small number of people, the designers have achieved a very pleasant appearance. At the other end of the tunnel, a wooden beam footbridge crosses the brook.

This picture shows the bridge carrying the B4425 road over the river Coln at Bibury in Gloucestershire. This is definitely a real bridge, even having small cutwaters.

There are many such bridges, of varying sizes in the Cotswold region, usually made, like this one, from Cotswold limestone. Other regions of Britain have bridges made from their own local stone.

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?

Elegance in Toscana

How Arches Work

The arch is one of the older forms of bridge. It is rather like an inverted suspension bridge, with all the tensions replaced by compressions, and vice versa. The other great difference is in the stability of the system. You can hang a rope across a gap, and it will return to its original position, after some oscillation, if disturbed.

But you cannot hang it in the shape of an arch. Even if it could be positioned correctly, the slightest disturbance would send it flying. The diagram below shows a very simple system of hinged rods, to explain this fact, obvious though it seems.

The blue lines in the upper diagram represent three rods, hinged at their ends, hanging from a fourth rod, drawn in black. The red lines represent a different position of the rods. We see that the centre of gravity of the rod on the right has moved up, while that on the left has moved down, but by a smaller amount. The central rod has moved upwards, so the net result is a higher centre of gravity of the whole system.

Therefore the symmetrical position was more stable than the unsymmetrical one, because a system is most stable in the position of lowest energy. If the rods were to be released at the red position they would swing back to the symmetrical position, and oscillate about that position until friction would have removed all the energy.

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.

The next diagram shows a set of designs for three pinned arches and 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.

The same transition with circular arcs

Voussoirs are not actually necessary – they are used in masonry arches in order to keep the sizes of the blocks manageable.

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.

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.

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 over loaded 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 form at the abutments or elsewhere?

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 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.

In fact, in every type of bridge, except arches and suspension bridges, these horizontal forces are removed from the ground and carried in members that oppose the forces in the rest of the bridge.

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. This diagram is 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.

This diagram makes 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 diagram.

Propped beams become Maillart arch

The next diagram develops one of the shapes seen in an earlier diagram..

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.

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. 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 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.

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.

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.

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.

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 at a time of low traffic, during a single night.

The diagrams below show roughly the forces in an arch.

These drawings were made by hand. This is not the way to do it for accurate results. The drawing below was made and drawn by computer calculation, for a simple deck-stiffened arch.

The horizontal component of the thrust (pale blue) is the same throughout the arch. It must be so, because the spandrel walls exert only vertical forces. The vertical component (green) increases towards the abutment as it is the sum of all the weight from the centre to a given point. The total force (red) of course acts along the arch.

In the next picture the height of the structure has been reduced by a half. Look at the effect on the horizontal component of the thrust.

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 diagrams above represent two concrete bridges, one flatter than the other. The lines at the right abutment represent the force provided by it. he vertical line represents the weight, and the horizontal line represents the thrust. The flatter the arch, the greater the horizontal thrust.

A simple way to see that there must be outward thrust is to imagine a simple arch of two loosely hinged rods, standing on ice. Obviously it will collapse, because the ice cannot oppose the outward thrust.

You can try this by standing with your legs wide apart on an ice rink. On second thoughts, don’t try it. When you put a ladder against a wall, the wall provides a force as if there were a second ladder leaning against the first, like a tall narrow arch. The ladder must not be too far from the vertical, otherwise friction at the ground cannot balance the outward thrust. If someone stands on the bottom of the ladder, this adds weight, and increases the available friction.

If we look at the two pictures above we can see why a circular set of voussoirs cannot be the correct shape. We assume for this purpose that the joints between the voussoirs cannot sustain a shear force.

In the upper diagram of a semi-circle, the thrust at the bottom can only be vertical, and so it cannot counteract the outward thrust of the upper voussoirs.

In the lower picture the situation is even worse. The thrust into the bottom of the ring is outwards, when it needs to be inwards. Finally, as a set of voussoirs must be like an inverted cable, it is immediately obvious that these shapes cannot work.

The inverse example of a hanging cable works in the same way. The two ends of a hanging cable are never vertical, unless they are at the same place. Perhaps the purest analogue of the cable is the Gateway Arch in St Louis, which is 192 metres wide and 192 metres high. From the point of view of resisting transverse wind forces, this arch is a huge cantilever.

In a real masonry arch, the volume above the voussoirs is generally filled with massive material, which changes the line of thrust to a steeper line. This material also help to rigidise the bridge.

The builder of this magnificent barn evidently did not believe that the line of thrust could remain in an arch which was almost a semicircle, even with a heavy wall above it, for he has provided a buttress that pushes well above the arch. At any rate, if he did believe it, he was forced to think again, and added the buttresses afterwards. The lines in the second picture hint at the reason for the building of the buttresses. See buttresses for more on this subject.

Here is another Cotswold building, with the usual roof of Cotswold limestone tiles. This stone does not split neatly into thin layers like slate, and these roofs are very heavy. This barn has substantial buttresses to resist the thrust of the roof, which can be thought of as a three pinned arch resting on the walls.

The bridge shown below is a propped beam, but it could be looked at as a three segment arch, and indeed the sagging span and the reflections in the water show that the thrust has slowly pushed the piers outwards.

The diagrams below this panel suggest how an arch can behave as a five-pinned structure, which is unstable. Two suitably placed and well-founded buttresses can reduce this to a stable three-pinned arch. Click on the diagram to see a picture of a foam plastic model. This material exaggerates the strains to make them more obvious. In a real structural material they are so small that strain gauges and amplifiers are usually needed to measure them. Measuring stress within a material is usually very difficult, so it has to be inferred from strain.

We can look at this a little more mathematically. The diagram at right shows three forces in equilibrium, acting at a point. This is possible only if the vectors can be joined up to form a triangle. This is only a calculational tool. Three forces acting like that would actually tend to rotate an object, although it would not be translated to another position.

An example would be the forces at some piece an arch. We have the thrusts from the rest of the arch, and the weight of the piece. These must balance at all points, unless the structure has enough stiffness to cope with the induce bending moment.

The picture shows the changes of angle where larger dew-drops hang on a thread made by a spider. At every point where a drop hangs, the tensions in the visible thread and the invisible weight of the drop are in balance, leading to a change in angle.

From the rule given above, we can deduce the transverse force on a curved structure such as an arch or a cable sustaining a force F, and having a radius of curvature R. If the transverse force per unit length is f, then

f = F / R

As an arch becomes steeper towards the ends, the transverse component of the weight becomes less and less, and is zero if the arch becomes vertical. If f is decreasing, either R must increase, that is the curve becomes straighter, or F must increase, in other words the arch must become thicker.

For a masonry arch, a part of f is provided by the masonry between the voussoirs and the deck, which provide both weight and sideways force.

In medieval cathedrals, buttresses, often spectacular flying buttresses, were deployed in order to control the forces generated by large and heavy roofs and vaults. The piers of the buttresses were often higher than the buttresses themselves in order to add weight, thus getting the forces into the right alignments. The towers at the left are provided for this purpose.

The transverse force is equivalent to the pressure on an airliner fuselage. A three-dimensional analogue is the excess pressure inside a water-drop.

Notice, in the buttresses above, that the builders did not just build the arch part – they, like almost everyone else, used the arch to support a straight row of blocks. Are these blocks there to add weight, helping to steepen the line of thrust, or do they take some thrust themselves?

In practice, the actual forces might well be distributed throughout the buttress, but these structures illustrate the idea that the line of thrust does not necessarily remain within an arch which is a circular arc. If the thrust lies within the straight row of blocks, the buttress has merely pushed the problem further out, forcing the pier to do what the wall of the cathedral cannot do, that is, resist the transverse force. To find out exactly what goes on, we would need to calculate everything from the dimensions of the structure. In practice, the builders probably built up empirical knowledge from years of experience.

The upper line of blocks looks a bit like a sloping beam. But what is a beam? It is usually a more-or-less horizontal, more-or-less straight object that rests on supports and does not take any thrust. And it is an object that does not fall apart when moved. So these blocks do not form a beam. What is a pier? A more-or-less vertical rigid object that takes thrust. And what about an arch? A more-or-less curved thing that takes thrust. Our straight line of blocks illustrates the fact that you can always find things that cannot be placed into simple categories. Perhaps we should call it a sloping pier, prevented from bending by the arch below.

We cannot manage without categories: the time taken to analyze everything from first principles would make life impossible. The use of categories appears to be almost instinctive, but when we categorise people or things wrongly, or we fail to take account of the vagueness of the boundaries, we can create problems. Ancient people correctly placed the planets and the stars into two distinct categories, based on apparent movement, but few realised that the earth was a moving planet, that the sun was a star, that neither was special or at the centre of the universe.

In England, but not in France, people have separate words for butterflies and moths, and jam and marmalade. In both countries there are separate words for frog and toad, but are these really biological categories above the species level? In some species of fish, individuals can even change sex. The simplest example of the problem in biology expresses itself in the distinction between "lumpers" and splitters". No doubt there is a vague boundary between these categories as well. The supreme example of splitting is to deny that evolution took place, and to deny that species are connected at all.

In engineering, these ideas may not matter as much as they do in other fields, because the important thing is to find out what works. Nevertheless, generalisation is a powerful tool, which enables the same calculational techniques to be used again and again. But if we extrapolate further than the regime in which a technique has been tested, we can find problems. For example, a short stout pillar works well, and we can build taller and taller examples, until we suddenly find that buckling occurs, at a width-to-height ratio calculated by Euler. An important part of a code of practice is its range of validity.

Arch and Portal Frame

The diagram below takes an "arch" from a previous diagram, with three voussoirs, and adds a beam on top, to make a type of bridge with sloping piers which is quite often seen over motorways.

Some examples of this type of bridge can be seen in the page about beams. Again there is some relationship between beam and arch. Not every bridge can be unambiguously placed into one of the basic categories.

The next picture shows a five-pinned arch being stabilised by a stiff deck. The central tie is not strictly necesssary, but enables a thinner deck to be used.

Developing a Beam into a Truss and an Arch

At the bottom of the picture the diagram represents a simple plate girder. In the next diagram some attempt has been made to shape it to suit the bending moments. In the third diagram this is taken further, and in the fourth picture the structure is greatly lightened by changing it into a truss. Finally, at the top, we see a tied arch or bowstring arch.

The point is to get the material as far from the neutral axis as possible in order to oppose the bending moment. Material near the neutral axis isn’t doing anything useful in this context. For a tension member, of course, you might as well use a wire as a tube, unless the member is very long and in danger of vibrating.

A Very Peculiar Bridge

What is wrong with this design? It is a three-pinned arch with a haunched beam on top.

A Millenium Arch

This beautifully simple tied arch bridge was completed in the year 2000. It takes an A road over the M42 motorway south of Birmingham. The span was built on land to the left of the picture, and was moved across to its final position during one night, minimising disruption to traffic. This possibility, together with the absence of outward thrust at the abutments, makes the tied arch very attractive for this type of location. This is one of the most elegant bridges depicted in this web-site.

So here is a much larger copy of the picture, unfortunately taken on a dull wet day. Note the great disparity between the thin hangers and the stout arch, as is usual for tension and compression members. This is a true millennium bridge.

Here is another tied arch, which you cross as you enter Evesham from the Cheltenham road. The Abbey bridge, across the Avon, is made of reinforced concrete. The soffits curve down sharply at the abutments. Do you think this is a good idea? Does the curve correspond to any forces?

Many bridges in this region have approach viaducts across the flood plains. The design of these approach viaducts neatly echoes that of the main deck, though the effect is not easily apparent because of the many trees. But the pillars at the ends of the arch suggest an uneasy compromise with a type of design which would once have been used for a masonry arch.

The pictures below are of Evesham’s other Avon bridge, a fine structure dating from 1856.

This bridge does not at first sight look especially noteworthy, being like many others in England. But look at the decoration and the balustrade – just enough to add interest without over-doing anything. And look at the piers – how narrow they are compared with the Roman ones. By this time, engineers had realised that all the horizontal thrust from any arch in a multi-span bridge need not be carried by its piers – it could be carried over through the other arches to the abutments. Of course, an excessive load on any one arch could in principle collapse the bridge.

The carrying through of loads is an elegant solution to many problems. Another type of through load is seen in mutlt-span beam bridges that rest on narrow piers under the centre-line. Torsional forces are carried through to the abutments by means of the stiffness of the deck. An excellent curved example is seen where the M42 motorway joins the M5 motorway south west of Birmingham.

This little bridge in Evesham is noteworthy for the interesting treatment of the railings. Is it an arch, or is it a beam?

Hollow Spandrels

This bridge is one of two at junction 18 of the M4, which connects that motorway with the A46.

It is not always necessary for the space between arch and deck to be filled in. One of Robert Maillart’s early reinforced concrete bridges developed cracks. Instead trying to make stronger designs he realised that the cracks were indicating that some of the material wasn’t doing what he thought it was doing, and in fact, wasn’t needed.

Like many other apparently lucky discoveries, this one came to a prepared mind. As a result he was able to design some beautiful bridges with a lot less concrete. Although the load is carried to the arch only around the centre of the arch, this is not reflected in the shape, which is a continuous curve. His designs have been very influential, and even today you can see bridges being built which owe much to him.

Stiffening Arches

The deck of a bridge like this must be stiff enough to carry the live loads between the spandrel walls. In fact, precisely because the deck is connected to the arch by rigid spandrel walls, we could imagine a bridge with a deck so stiff that the arch could be made quite thin. In such a deck-stiffened arch, the arch need only be thick enough to take the thrust without crumbling or buckling. So a bridge like this one could have a thick deck and a thin arch. The arch can then be quite light. It can also be made in straight sections.

The deck can also be light, because it can be made in cellular form. It is obviously easier to design and build ribs or boxes for a straight deck than for a curved arch. The deck can be lighter than a simple beam, because its weight is taken by the arch. This is one of the many examples of separating the material of a structure and putting it where it is most useful. (Lune Arch)

We also need to look at the appearance of an arch. If we make either the deck or the arch much thicker than the other part, it becomes obvious which is doing what. But if both look similar, there is confusion. The overall effect may also be rather uninteresting.

The way that the line of thrust behaves in a simple deck-stiffened 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 contrast with the normal self-stiffened arch the arch has no ability to withstand the bending moment: this is absorbed by the beam at the top.

To avoid any hint of tension in an arch, the line of thrust should lie within the middle third of the section at all times.

We also see why the suspension bridge is so difficult to build for railways. The cable has no stiffness, and so the deck must provide it. To ask that of a 1000 metre deck is asking a great deal; it would have to be very heavy.

The deck-stiffened arch is a beautiful example of the benefits that can accrue from separating functions. The opposite is often true, as in the wing box used as a fuel tank. But nobody has made much progress with aircraft which are all wing and no fuselage.

If we compare the deck-stiffened arch with the beam we can see how the benefits arise, (considering only static loads).

Straight beam

No horizontal thrust at supports

Very strong horizontal tension and compression within beam

Strong bending moment within beam

Simple arch

Strong horizontal thrust at supports

No tension within arch

No bending moment within arch

Arch must be thick enough to resist buckling

Deck-stiffened arch

Strong horizontal thrust at supports

No tension within arch

No bending moment within arch

Arch need only be thick enough to withstand compression

Only small forces in short deck sections

No overall bending moment in deck

Loyn bridge over the river Lune

This fine bridge spans the river Lune in three arches. The simple design gives a monumental effect. The piers are extended by pointed cutwaters, to steer the water around them. The line of the cutwaters reaches the top of the bridge, where they provide refuges for pedestrians when vehicles cross the bridge.

Like many bridges in this area, the Loyn bridge has a heavy stone pavement around the piers, to counteract the possibility of scouring in time of spate, when the flow of water down from the fells is very powerful. Remember that a cubic metre of water weighs a tonne, and at even 15 km per hour, it can exert strong forces. The effect of turbulence is to cause local variations in pressure, causing vibrations in the structure. These variations, added to the aerofoil effect over curved stones, and the effect of Archimedes principle, enables rushing water to move and lift large objects. The lifting power increases as a large power of the speed.

Bruges

Bruges has many bridges and arches in buildings.

Les Ponts de la Caille (Quail)

At Cruseille, in Haute-Savoie, between Annecy and Geneva, two bridges cross a deep limestone gorge in which flows the River Usse. The arch bridge was built between 1925 to 1932. It has a span of 140 metres.

See also Arches in architecture

Developing the Arch

Since the semi-circular Roman arch has served so well, some examples still standing after nearly 2000 years, it must have some strong points. Let’s see what we can do by messing around with the semicircle.

If we reflect it about a diameter we have a ring. Add some spokes and we have a wheel, such as a bicycle wheel. We can learn a lot from the bicycle. The Wright brothers did, and of course they were brilliant and patient reseachers.

A bicycle wheel is very light, but very strong. It is rather like a bowstring arch in which the straight part has been squashed to a point, and the curved part wrapped right round it. Although the rim and the spokes are not very rigid, the assembly is very rigid indeed.

Since the spokes cannot take compression without bending, the weight of the bike must be transmitted from the hubs to the top of the wheel through the upper spokes.

Other spokes hold the rim in shape as the force is taken down an around the rim. At the bottom of the wheel, the weight is tending to push the rim towards the hub. The vertical spokes cannot resist this action, but for one part of the rim to move inward, another part must move outward. So the spokes as a whole are keeping the wheel in shape.

The spokes in a bicycle wheel are not along radii, because they have to transmit rotational torque from hub to rim. So they are tangential to the hub. The hub is wider than the rim in an axial direction to keep the hub rigidly in the mid-plane of the rim.

A pulley wheel is more like an arch with a distributed load.

The rim of a bicycle wheel is not very stiff, so the wheel is like a deck-stiffened bowstring arch, where the deck has become a point. But some motor-cycle wheels have stiff rims, with as few as three spokes. They are more like self-stiff arches.

Strangely enough, some wheels are the exact opposite: they have rims in tension around a compressed wheel. In such cases, a metal rim or tyre is heated, fitted to the wheel, and allowed to cool and contract on to the wheel.

Related to the bicycle wheel are the rackets for games like badminton, squash and tennis, shown diagrammatically below. These have an oval frame with a grid of taut strings. The array must be two-dimensional, because one parallel set of strings alone would easily distort the frame. Striking a ball would be ineffective, because the distortion would increase, reducing the ability of the tension to send the ball on its way.

If we think of the frame as two opposing arches, pulled together by parallel strings (the load), the other set of strings can be thought of as resisting the outward thrust, (the abutments of an arch). In effect, these rackets are like double tied arches.

If we rotate a ring about its centre we get a ball. Balls are ubiquitous. A ball flies straight if not spun, and it rolls straight and bounces true – hence its use in games. It can take the tension if inflated, and the light table-tennis ball can take surprising amounts of compression if applied uniformly. One make of gardeners’ barrow uses a ball instead of a wheel, to spread the load on soft ground.

Many balls, such as footballs, are under internal pressure, and so, although they look like domes, they are actually in tension. In fact, vessels to contain gases at high pressure have been made by winding steel wires around spheres or cylinders. Golf balls have been made by winding rubber threads around and around. Although the bands are in longitudinal tension, they are compressed laterally by the threads outside them.

Then, of course, there are eggs and skulls. Nature did it first, as usual. When a chick tries to break out of a shell, does it need a greater force than a predator trying to break in?

Stringed musical instruments are of two main types; those like the harp, the harpsichord, and the pianoforte, in which a strong frame surrounds the strings, and those like the violin family and the guitar family, in which the strings are stretched across a box, and spaced from it by a bridge. In all cases, the rigid part has to be strong. The total force on the frame of a large piano is enormous. There may be over eighty keys, many working two or even three strings.

The diagrams below represent the outline of a zither or cembalom. If the frame were made of the same material as the strings, we might expect that the cross-sectional area of the two compression legs need be no bigger than the total area of the strings. On this basis the frame on the left is too thick. In fact it is too thin. Why? The reason is that long before failing under compression, the frame will fail by buckling. Euler showed that a strut under compression will buckle in response to a tiny deflection, if its thickness is less than a critical value.

Because the top and bottom of the frame do not follow the funicular, they have to be very thick as well. In a sense, the frame contains both an arch and a suspension cable; this is discussed in beams. The shape is determined by musical, and not structural requirements: it has to suit the lengths of the strings that produce the right wavelengths and therefore the right frequencies. By arranging the strings in a different order and at varying angles, it might be possible to use a funicular frame, but the striking mechanism of a piano would be very complex if the strings were all at different angles, and the non-musical order of the notes would be most unhelpful to the player.

In fact, by varying the thickness and tension of the strings, the musical requirements can be adapted to fairly simple shapes of frames. Since the time of Stradivari and Guarneri, violins have been dismantled and rebuilt with the fingerboard at a different angle, to allow for the greater tensions demanded by modern music and modern methods of playing in large halls as opposed to the small rooms for which chamber music was intended.

The strings of a racket, like the string of an archery bow, must impart maximum energy and momentum to the projectile, and retain as little vibrational energy as possible. A piano string, on the contrary, must absorb most of the hammer’s energy, while bouncing it quickly away, so that it cannot damp the vibrations. The mechanism of a piano is very ingenious: the parts have to be very light, yet they must be rigid enough to transmit the force of the finger exactly. The piano frame has to transmit the vibration of the string to the air, like the diaphragm of a loudspeaker. In this, the instrument is the opposite of a bridge, which is not supposed to oscillate, either wholly or in part.

The frame or body of a musical instrument must transmit vibration well, yet have no strong narrow resonances. It must give tone, but not colour the sound unduly. In electric guitars and electric violins, the body has no sonic function, and is purely a mechanical platform.

The archery bow differs in that the string is purely a means of transmitting the strain energy of the bow to the arrow. If a bow is shot without an arrow, where does the energy go? Don’t try it.

When lightness is paramount, as in the masts of yachts and other small craft, and in communications masts, the structure is thin, and is braced externally by wires. It is effectively divided into sections which are individually rigid.

From 1937 to 1940, Henry Moore made a number of sculptures which included strings or wires, but he stated that these were derived from organic forms, and indeed they do not convey a strong impression of tension against compression. Barbara Hepworth and Naum Gabo also made sculptures which included wires. One example by Gabo is shown here. Do you know of any works of art which really convey the ideas of structural forces?

This discussion could as well have been included in the pages about beams, cable-stayed bridges or suspension bridges, because many structures, including some bridges, cannot be assigned into a simple category.

Half a ball, or an arch rotated about its apex, makes a dome, again a strong shape, whether a human skull or the dome of St Paul’s cathedral. Distort a ball, and you have a rugby ball, an American football, or an egg. An egg is very strong, unless you poke it with something fairly sharp. In the same way, the curved case of a shell-fish is strong, until, for example, the sharp bill of an oystercatcher stabs it or prises it open.

If we rotate the ring about a line that is outside it, we get a torus, for example the inner tube of a tyre that fits the wheel already mentioned.

If we rotate a narrow arch about one of its abutments we get a circular shape, well seen in the beautiful chapter house of Salisbury cathedral. This building has a vault supported on a circular wall and a narrow central pillar.

Extended arches, intersecting in various ways, form the vaults of medieval cathedrals.

If we translate a ring we get a tube, described in another page of this web-site. (Tubes) The tube is of course a ubiquitous device in both the animate world and the inanimate world.

If we turn the tube on to its end we have the shape of a factory chimney or a lighthouse. An early example was the Eddystone lighthouse by Smeaton.

Some buildings, such as oast-houses or pottery kilns, have been made in conical shapes, with curved or straight sides.

Finally, if we turn an arch on its side, we have the shape of an arch dam. Of course the arch needs to be thicker at the bottom than at the top, to withstand the greater pressure. In fact the shape of a dam has to be such that it is stable for all levels of the water.

Some dams (cupola dams) are even curved in both directions, like a section of a dome. They can be quite thin, and even overhanging, near the top. Unlike a bridge, a dam does not experience rapidly changing live loads. The only change is in the water level, principally while the reservoir fills for the first time.

The cathode ray tube of a television set has a thick curved faceplate, and in fact the entire surface of the CRT is curved. This makes it possible to withstand the pressure of the atmosphere, with no pressure from inside the evacuated tube. It is analogous to a cupola dam, except that the pressure is the same all over.

The ideal shape for a pressure vessel is a sphere, a shape is often used for deep sea exploration. But a sphere would be impractical for a domestic device, and so some compromises have to be made . The faceplate must be thick enough to make sure that the surface of thrust lies entirely within the glass. Large CRTs are therefore very heavy.

What do you think is a typical total force on the front of a TV tube?

Let’s consider a TV screen of 40 cm X 30 cm. The total area is 1200 cm², or 0.12 m². The pressure of the atmosphere is about 0.1 MPa.

So the total force is 0.12 X 100000 N, which is 12000 N. This is equivalent to a weight of about 1200 kg, or 1.2 tonnes. So the face-plate has to be very thick and very tough. The largest TV tubes are extremely heavy. What is the energy released if such a tube were to implode?

Let’s double the total force to allow very roughly for the back of the tube, making 24000 N. The energy needed to push against this force is roughly obtained by using the mean radius of the tube. We can take 20 cm, or 0.2 m, as a rough value. The energy is 0.2 X 24000 J = 4800 J. This is a lot of energy, and so these tubes are dangerous if handled wrongly.

These pictures show a small part of a long reservoir that is held by an arch dam at the left. The pressure at any point on a dam is dependent on the depth and the density of water. The width and length of the reservoir have absolutely no effect on the pressure.

To see this, imagine a vertical plate in the dam. It has the same pressure on both sides, otherwise it would move. If it is enlarged to divide the dam into two parts, we could imagine filling one part with earth, without changing the pressure in the other part.

Because the arch dam and cupola dam rely on the rock to sustain the thrust, the quality of the rock is of the utmost importance. Grouting of the rock around the dam, both laterally and below, is usual. The water also exerts great pressure on the rock, which may create significant stresses. The lubricating and uplifting effects of water that has been forced into cracks may have serious consequences, as in the case of the Vajont dam. This was a magnificent feat of engineering, but in 1963 a fall of rock into the reservoir displaced a gigantic mass of water over the dam, killing about 2000 people.

Although arch dams can be quite thin, their weight can be used to aid stability if they are made thick. The dam is then a gravity-arch dam. See Gravity dams and Top Ten dam sites.

Continuing the arch dam to make a complete free-standing circle, We get a cylindrical caisson or cofferdam, which keeps water out while a bridge pier is built. Turning the cylinder on its side, making it long, and adding end caps, we have a submarine. Some changes in shape are required to obtain minimum drag and to accommodate the necessary equipment. For a submersible which does not need to go fast, a simple cylinder with hemispherical end caps is adequate. In fact, some early submersibles were spherical.

Robert Maillart

The name of Swiss engineer Robert Maillart will always be associated with arch bridges, but he was in fact a brilliant and versatile creator, contributing to the development of efficient and elegant reinforced concrete buildings as well. It has been said by an aircraft designer – "If it looks right, it is right." That just about sums up the work of Maillart.

No building was too humble to benefit from Maillart’s best efforts. He was a true innovator, looking at problems and finding good solutions, looking at requirements, assessing known designs, and finding economic answers.

In his reinforced concrete buildings, he used columns which spread at the top, merging into the deck above. This looked good, better than the right-angle that we so often see. And it allowed the forces to flow from the deck into the column, at the same time reducing the spans between the columns, with a consequent saving of material by reducing the thickness of the deck. In fact, in these buildings, there is more than a hint of the medieval fan vault.

Nature seldom joins things together in a crude manner. Look at the way that a tree grows its branches – if you cut through the wood, you see the lines of force well inside the main branch, showing where the subsidiary branch grew out. The first two pictures below show how palm leaves grow.

The other pictures show pieces of wood cut from a place where two branches grew out, together with a computer simulation. If you look at an old fallen tree you can often see clearly the flow of the stresses to which its growth was a response. Given the enormous time-scale of evolution, we can assume that natural structures represent good compromises between all the requirements for survival and reproduction.

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Definition of an Arch

After looking at this page, what do you think an arch is? This web-site is not intended as a text-book, and is not arranged in the logical fashion of a text-book, and doesn’t include many definitions. Although structures can be classified broadly into different basic types, in practice, few structures are pure examples.

Let’s look at arches. How’s this for a description of a "pure" arch?

An arch is a structure in compression, which follows the funicular, and is only thick enough to contain the live loads at all times. It will usually include extra members to support a more or less horizontal deck, though some older Chinese and Japanese bridges used the arch itself as a deck, sometimes with steps. It cannot exist without abutments that can react against the horizontal thrust, except in the case of a tied arch, which could be regarded as a beam in which the tension and compression have been separated.

In practice, arches often deviate from the funicular, sometimes to obtain clearance over a greater width, or for structural reasons. If an arch deviates too far from the funicular, it will require stiffness, and so it will to some extent have to behave like a beam.

Some arches are in the form of trusses, so that page should be looked at for further information. Masonry arches are often solid, and the the distribution of the weight of course affects the shape of the funicular.

Many real structures are far from being "pure", but the ideas like "arch", "beam" and "truss" are useful in learning to understand. Conversely, many elegant structures have been made by combining features of different types. Look at some structures and work out what is going on in them.

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Links about Robert Maillart

Web-site including pictures

Tavanasa bridge – pictures

Salginatobel – Schwandbach

Bridges of Paris

Book in German

Photographs

Scientific American – July 2000

Niagara Falls bridges

Beam Box Girder Cable Stayed Cantilever Pre-Stressed Suspension Truss

Severn Arches Musical Arches

Arches in architecture Arches in religious buildings

Back to Home Page Back to Bridges

Arch simulator download Deck stiffened arch simulator download

Links to other web-sites about arches

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