ArchTwo

Arches – Part Two

August 2002

Beam Box Girder Cable Stayed Cantilever Pre-Stressed Suspension Truss

Arches in architecture Arches in religious buildings

This page continues from Arches – Part One

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 at the start, 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.

These three pictures show an arch which is bowing outwards. In the third picture, two cracks are indicated by red arrows, and a surveying target by a yellow arrow. Evidently this arch is causing concern. The buttress looks very substantial, and there is little weight on the arch, so perhaps the foundation is not adequate.

Another arch nearby is in an equally bad way. This arch is actually double: someone in the distant past has built a new arch against the older one, with complete disregard for style. But we must remember that old abbeys were life support systems, not works of art. They were perhaps like space stations, supporting a small number of people with some supplies of food from the outside world, though they had their own gardens as well.

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.

This row of arches illustrates very nicely some of the problems with arches. At the two ends, the thrust has pushed the piers outwards, and on the left, two incipient hinges have formed. The other piers remain upright, because the forces are more or less balanced.

To look at a possible consequence of the formation of hinges, we go back many years to the invention of an early electronic device, the triode valve or tube. The curve below shows roughly how the current might vary with the potential difference across the valve.

A great problem with the triode was the capacitance between the anode (output) and the grid (input), which had a severe effect on the gain at high frequencies. A solution was found by the use of a second grid, between the first grid and the anode, to act as an electrical shield. This solution to the capacitance problem introduced a new problem, shown by the curve below.

There is a region in which the curve slopes downwards instead of upwards. The consequence of this "negative resistance" is that the device can oscillate, which may or may not be desirable. The first solution to this problem was to insert a third grid. A second solution was the "beam grid" tetrode. This type of curve is found in some more modern electronic devices such as Gunn diodes, IMPATT diodes, and tunnel diodes. In all cases they can be used in oscillator circuits.

Strangely enough, such a curve can arise in the case of an arch which does not fit together correctly, and the arch, too, can oscillate.

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.

In the picture below, we can see the way the curvature of a spider web is related to the weight per unit length.

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 in a flying buttress 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.

Enough of this uncertainty: here is a structure that is definitely not an arch. It comprises a set of planks, some of which act as beams, others partly as cantilevers as well. If there is no friction between the planks, there can be no horizontal force. The rule is very simple – no thrust: no arch. In a real construction, there would be friction, and some lateral forces would be created. But in the frictionless case, what carries the weight of the unsupported parts? See beams and cantilevers.

You could probably say that this is an arch. At any rate, it generates lateral thrust. Like a gothic arch, its apex does not correspond to extra load: quite the reverse in fact. As we saw earlier, the curvature of an arch or a cable is closely related to the load that pushes or pulls it. So a sharp corner ought to be resisting a localised force.

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 necessary, 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 Millennium 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.

The next example, in the spectacular Vintgar gorge in Slovenija, is a high arch spanning the top of the deep gorge. A narrow, deep valley presents a superb opportunity for the bridge builder to produce a design that will complement the landscape. This picture exemplifies the difficulty and cost of providing transportation in mountainous terrain.

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 different functions into different components. 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. Nor has the animal world, apart from gliding snakes.

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 and velocity, causing vibrations in immersed structures. 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 of the water. The next pictures show the bridge in time of flood, though nowhere near the highest that can occur.

The first picture shows the turbulence induced by the nose of a cutwater of the Loyn bridge, in water that is already swirling at high speed. The second picture shows the flow of the water as it enters the central arch. Since the piers of the arches reduce the width available for the flow, you might expect that the water would pile up. But no, it slopes down along the nose of the cutwater, and it drops again as it enters the arch. Hydrodynamics, like aerodynamics, is not a subject that you can work out using "common sense". The third picture follows the flow right under a span, and out the other side, while the fourth picture shows the view from the downstream side. Note the turbulence downstream of the bridge.

Note also the double arrays of voussoirs and the slight angle at the crown. This may have been produced by spreading of the central arch, which is off the picture to the right. This bridge is thought to be of late 17th century construction. Like many old bridges, this one has refuges which are continuations of the cutwaters. These allow pedestrians to avoid vehicles on the bridge, which is very narrow.

Natural events affect not only people and their works, but every form of life. Events can occur on time-scales from milliseconds (an exploding meteorite) to hundreds of millions of years (motion of continents). Slow changes can stimulate evolution, medium ones can cause discomfort, fast ones can cause catastrophe. These particular floods can have a disastrous effect on the breeding of sand martins, oystercatchers, and some other birds that nest near the river.

The pictures here show one of four sand martins – Riparia riparia – which had climbed on to the river bank to escape their flooded tunnel. They were coaxed into a tube to allow them to dry out, and when the rain had stopped, and the water level had started to fall, the tube was buried in the bank, to make an imitation tunnel.

Not far from the Loyn bridge stands a private bridge of the early 19th century, also over the Lune. Its two main arches are ribbed, and a nice is the use of a slightly different colour for the stone above the voussoirs. In a small niche in one wall of the arch – last picture – a grey wagtail – Motacilla cineria – had built a nest. Judging by the quantity of material, the niche had probably been used before. The niche harbours at least one growing plant, which helps to disguise the nest.

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 Caille Bridges.

Firenze

The Ponte Vecchio needs no comment. Brunelleschi’s dome is a work of genius. A dome is like an arch rotated about a vertical axis. Unlike an arch, it can in principle be built without centring, like an igloo. The reason is that whereas an arch is made of parallel sections, a dome comprises tapered sectors. In a globe of the earth these would be called gores. Brunelleschi’s dome, like some others, actually comprises an inner dome and an outer shell. The thickness of the inner dome is such that at every level the octagon contains a complete circle.

Roma

The "ancient Romans" were skilled, though conservative, in the construction of arches. Their piers were so wide that most of their bridges could survive the loss of a span or two by flood, scouring, or act of war. To have made great bridges and aqueducts that have survived for up to 2000 years is a tremendous achievement.

Here is a picture of the Ponte Sant’Angelo, spanning the Tiber after more than 1800 years. The balustrade and the statues were added in the 17th century by Bernini. The original name, when the bridge was built in about 135 AD, was Pons Aelius.

Note the large platforms for the piers. The Roman builders were well aware of the dangers of scouring, and took their piling and piers down to good ground. They had a form of concrete that could set under water.

Another fine Roman bridge.

Karluv Most – Praha – Charles Bridge – Prague

This beautiful bridge was begun in 1357 under Charles IV, who had founded the new town in 1348. Statues were added from 1706 to 1714.

Bayonne Bridge

Completed in 1931 by Othmar Amman, the Bayonne bridge is, at 50 feet shorter than the New River Gorge bridge, the second longest arch span. The span is slightly longer than that of Sydney harbour bridge. It connects New Jersey with Staten Island. This is an example of truss construction, in which most of the volume is empty space, the forces being channelled along struts and ties.

To avoid the construction of massive and expensive falsework, which would obstruct the channel, such arches are often built as cantilevers, the halves being pulled back by temporary cables. The Sydney Harbour bridge was built in this way. The diagrams below show another style of trussed arch.

The next diagram shows the same design with redundant members removed. See also Indeterminacy.

All these main spans with through connection to the side spans can in principle be constructed by cantilevering, keeping the navigation channel clear at all times. The next diagram shows the two sides at different stages of construction.

Now let’s colour some of the members in red for compression and blue for tension.

But what happens when the two halves meet, and we complete the top and bottom chords? We could arrange over-size rivet holes and join the two parts, retaining the existing forces, but that is not the usual way. By jacking the bottom chords apart at the crown, the span is turned into an arch, relieving the tension at the two outer piers, and creating outward thrust at the main abutments. But there is still the top chord. That, too, can be jacked if required, until it is in compression and not tension.

Can we then confidently assign blue and red to the vertical and sloping members? What we learn from this thought experiment is that we cannot always, just by looking at a structure, know even qualitatively what all the parts are doing. In fact, it is quite possible to construct in such a way that we cannot fully understand even by calculation.

This type of bridge can be constructed as a cantilever bridge, so that there is no arch action at the main supports. Can you see advantages or disadvantages for the arch construction or the cantilever construction?

An Unusual Arch at Ross-on-Wye

This bridge over the river Wye at Ross-on-Wye is built of red sandstone. It has a very unusual feature. Instead of the voussoirs having the normal slightly tapered trapezium shape, every one of them has a zig-zag shape on both sides. One of them has been outlined in white to show this more clearly, as the photograph is poor.

An enlargement is inset at the lower left of the picture at left . It is as if the designer thought that there could be shear between each voussoir and its neighbours. This should not, of course, happen in a well designed arch.

Here is a picture of one arch. Note the flood relief arches in the distance, under the Ross-on-Wye bypass A40. The river Wye and the river Severn are very prone to flooding, which has caused great damage in several recent years.

Perhaps the designer had experienced a problem with a previous construction, and was trying to make sure that nothing could go wrong. This bridge has very large cutwaters, perhaps because the Wye in spate has a very fast flow.

_______________________________________________

Ribblehead Viaduct

The magnificent Ribblehead viaduct seen (just about) from near the summit of Ingleborough. This how not to do it – on a dull, windy day, without a filter, without a tripod, from too far away. This viaduct is a fitting construction for the locality. If you look up the height of Ingleborough you will probably not be impressed. Don’t be fooled, it can be a very wild place. The wind can whip across the summit plateau so fast that you cannot walk against it: in those conditions, to go upwind, you have to crawl. These pictures were taken from the edge of the plateau.

Newcastle arch under construction This link is to a photograph showing how an arch can be constructed in two halves, each held back by cables. This avoids expensive falsework and keeps the navigation channel open during construction.

Gothic Arches

The points of Gothic arches, like the pointed tops of Sydney opera house, do not seem to correspond to anything in the loading. Like the pointed arches found in many other beautiful buildings, they remind us that functionality includes the function of pleasing the eye and the mind.

What is the correct shape of an arch? The Romans always used semicircles, but this cannot be the correct shape for a bare arch of voussoirs, because at the two ends the curve is vertical, providing no provision for containing the outward thrust of the arch.

The theoretical shape for a set of uniform voussoirs would be a catenary, the same shape as a hanging cable. But when masonry or other loads are added above the voussoirs, the weight distribution and the added stiffness can allow many different shapes. In fact the masonry spreads the load, so the effect of any live load will probably not move the thrust line as much as if it were applied direct to one voussoir.

The pointed arch was used in a great number of medieval bridges as well as buildings.

So the shapes of arches vary considerably. Brunel built two very flat elliptical brick arches at Maidenhead, for the Great Western Railway. They are still in use. The illustrated bridge carries a road.

Three Hinged Arch

As mentioned earlier, arches may include up to three hinges. if an arch departs from the optimal shape it will require some form of stiffening to transmit the bending moments. The arch shown here is a three-hinged reinforced concrete arch over the M6 at New Hall in Lancashire. Effectively the ground and the two halves of the bridge form a triangle, which is a stable shape. A fourth hinge would render the structure unstable.

The span of this reinforced concrete arch is about 150 feet. Arches can be built with three hinges, two hinges, one hinge, or none at all, as in traditional masonry arches.

What about bridges like this? Are they too far from the usual arch shapes to merit the name? This is a three-pinned footbridge with helical ramps.

These ramps are a visually interesting solution to the difficult problem of integrating ramps with footbridges. In a town there would not often be room, and straight or zig-zag ramps would be used.

Here are departures from the funicular in the opposite direction.

Bridge at Auxerre

This picture shows a wonderfully slim footbridge over the beautiful river Yonne at Auxerre in France, comprising two concrete arches. We see a very narrow central pier, from which we deduce that thrust from a load is carried over into the other arch and into the abutments.

A Devil’s Bridge

Several bridges in Britain are called "Devil’s Bridge". This one is at Kirby Lonsdale. It is hardly surprising that people could be astonished by the building of a bridge like this, and, as a result, attribute its construction to a supernatural being. If you don’t know about centring it certainly looks difficult. This is a splendid structure; two ribbed arches cross the fast flowing river Lune at a place where it is quite narrow. Ribbed arches were not uncommon in medieval bridges. Ribs reduce the weight, and improve the appearance. The outline of the walls is distinctive without being overdone.

The bridge is very narrow, and it now carries pedestrians only: a rather undistinguished – though not unpleasant – concrete bridge carries the road nearby.

Not so Fast

We are used to the idea of bridges as a means of getting people and goods across an obstacle as quickly as possible, often avoiding a long detour. But many of us need sometimes to go slowly. Parks and gardens are found in many countries for this purpose. Streams and lakes, sometimes artificial, may need to be crossed by paths, and so the most naturalistic garden may include an engineering construction.

Many Japanese gardens look very natural, but like many other gardens, result from careful planning and continual maintenance. Some consist only of gravel with a few rocks. Others abound with plants and water. All exhibit the ability to make the planned object look unaffected and accidental. The bridge here is in Korakuen in Tokyo, near a huge amusement park. The old Korakuen is an amusement park from a different age, when people would stroll through a garden, stopping at strategic places to contemplate the view. A bridge might be more than a crossing point: it would probably be one of the viewing places.

Use of Local materials

Using the local stone makes it more likely that a bridge, or indeed any structure, will fit in well with the landscape. In Derbyshire, Lancashire and Yorkshire, and of course many other places, there are many old bridges that achieve this. A good design may be a personal creation, or it may be a team creation, but it will have character, whether it be in stone, pre-stressed concrete or steel. If the design is right, the material used need not be a bar to integrating structure into a site. This is not to be confused with adding decoration or unnecessary features to an uninspired piece of work.

Some good examples of well-attuned bridges have already been shown.

Here are some other bridges, built in local stone, which work well in their surroundings.

Note, in the third picture, the cutwater on dry land. In fact the upper Thames floods quite frequently, so they may have some function. In any case they fit in with the general design of the bridge, and prevent it ending tamely at the abutment.

A bridge doesn’t have to be the biggest to be successful. It doesn’t have to have a feature that nobody else has used. All it needs is to be a good answer to the problem in hand, and to be inoffensive to the people who have to see it.

__________________________

Although a fine bridge in local stone is a pleasant sight, the ability to transport materials and to subject them to processes which change their appearance or characteristics, as in brick making, has been of great importance in the progress of technology. As far back in time as the building of parts of Stonehenge, some people thought it was worth moving large stones long distances.

Some of the examples have shown that the distinction between bridges and buildings is not clear-cut. The Ponte Vecchio in Florence and the Pulteney Bridge in Bath have shops on them. This was not uncommon in older times. An early London Bridge had houses along its entire length.

As many bridges were built on the boundaries of counties or other areas, they sometimes include gate-houses, as in the Monnow bridge at Monmouth.

And of course many buildings include arched doorways, windows and gateways.

See also Arches in architecture

Destroying arches

Arches are surprisingly resilient. In many parts of England you can see arched ruins left from the dissolution of the monasteries. Here are some arches left after bombing during the 20th century.

The vault of the banqueting hall at Ctesiphon, Iraq, looks fairly close to a catenary. The walls are thickened towards the ground, to make sure that the funicular remains inside them. This is shown clearly in Figure 20 of "The Story of Architecture" by Patrick Nuttgens, Phaidon, ISBN 0-7148-3616-8. Click here for a picture which does not show the side walls quite so well. Built in about 550 AD, this building survived until 1987, when most of it was destroyed by a flood. ……………………………

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.

Here are pictures of the beautiful Slap Savica in the Triglav National Park, Slovenija. The dam has only slight curvature, and probably acts mainly as a gravity dam.

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.

The arch dam is commonly seen in the form of the paired gates of dry docks and locks, which in effect form three-hinged arches. You might wonder why very large lock gates, such as those in the Panama canal, are not made as sections of cylinders. Perhaps the extra expense is not worthwhile, and also there would be some awkward forces at the hinges. Even the deepest lock doesn’t compare with the highest dams. Here is an idealised plan view of a pair of lock gates.

These pictures show a dry dock and its gates. When the dock is to be filled, it is not possible to open the gates against the pressure of the water: instead the dock is filled by opening valves in pipes that connect the main basin with the dry dock. …………………………………

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.

.===

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 it 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. The stresses in an arch are purely compressive, and the horizontal component is constant throughout, but the vertical component varies, because each part has to support the higher parts, but not the lower parts.

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, and hence enables arches of many different shapes to be built. The ancient Romans did not seem to know this, or if they did, they did not care. Their designs were very conservative by today’s standards.

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.

If you got this far, try a superb game about bridge building – http://firingsquad.gamers.com/games/pontifex/default.asp .

===

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