GlosCathArch

If you walk into Gloucester from the River Severn at Over you can find the cathedral, which includes semicircular Romanesque arches built by the Normans between 1089 and 1100, and much later ones in the perpendicular style, with more or less pointed tops. The official web-site of the cathedral is at http://gloucestercathedral.uk.com. It contains a wealth of information about the cathedral, including a history and a plan which you can use in conjunction with this page, and many excellent photographs. The best thing to do, of course, as with all the topics in this web-site, is to go and see for yourself. See also The abbey mason.

Although the empire Romans had collapsed hundreds of years earlier, some of their ideas were more robust, and the style brought to England by the Normans continued to use the massive piers and arches that the Romans had favoured. Many Roman buildings, bridges and aqueducts are still standing, and in good condition. Some would say that this proves that they over-engineered. Yes, but their bridges could withstand the loss of an arch or two from enemy action or from flooding. And it depends on your definition of good engineering. In modern times we would of course include the cost, not only financial, but also often human and environmental, in our deliberations. For the Romans, these constraints may not always have been felt so strongly.

When you see this porch outside Gloucester cathedral, you are not prepared for the sight of the massive Norman piers and arches that support the nave. Some of these are shown in the next two pictures. The Normans brought much else besides a style of architecture, in terms of organisation and culture. Inside this cathedral, you can find the tomb of Robert of Normandy, or more correctly, Robert de Normandie, as French was the official language for a long time after the Norman invasion. French was just one of the numerous linguistic influences on the language that came to be called "English".

One of the pleasant features of Gloucester Cathedral is that you don’t have to pay to go in, though the maintenance of such a building is tremendously expensive. There is only a box for voluntary collections. You are allowed to take photographs, and this is free as well.

Romanesque Style

Some of the piers in Gloucester cathedral are reddish at the bottom because the thatched roof caught fire and collapsed into the nave. The intense heat from the burning debris affected the surface of the limestone.

Unlike these 11th century piers in Gloucester cathedral, those in the cathedral of Durham, the oldest Norman building in England, are carved with decorative patterns. Durham cathedral is a magnificent edifice, on a commanding site. If you go there, don’t forget to see the ingenious Kingsgate foot-bridge over the river Wear. This bridge was built in two halves, each parallel to the river. The two parts were then rotated about a vertical axis and then joined to form a single beam. A splendid construction. Tewkesbury abbey is also well worth a visit.

The crossing of Gloucester cathedral has two of these flimsy looking east-west arches. What do they do? What is not apparent from these photographs is that they carry the springing of one part of the 14th century vault above the choir. Evidently the builders did not think that they could carry the forces from the vault into the existing walls and arches. From details like this we can see that this type of building was not created from a single plan, as would be done today. Cathedrals often grew over several centuries, and sometimes parts fell down, even, in some cases, soon after completion. The cathedral at Beauvais would have been one of the wonders of the world, if it had been completed, and if it had survived. In fact it is but a fragment of the intended structure.

The gothic cathedral gives the lie to the idea that design grows out of engineering, and that engineering grows out of science. These ancient people did not understand what they were doing, in the technical sense that builders do today, but almost a thousand years later, people still come to marvel at what they created. In fact, people are still making things of which not all the details are fully understood, though finite element analysis does enable stresses to be calculated throughout an object, allowing designers to pare away the least stressed parts. Using an evolutionary or iterative technique, it is possible to design parts with reduced mass of material, valuable in structures, aircraft, ships and spacecraft, and the engines that power them.

Much later, the industrial revolution began long before anyone understood much about energy – even the law of conservation of energy was unknown when the first steam engines were built. Thermodynamics was far in the future. So the first steam engines were almost incredibly inefficient. What did it matter? Trees and coal abounded in England.

That is not to denigrate science: the world has changed, and we now realise the immense benefits that come from knowing the science. People in Britain, France, Germany and USA, for example, made great contributions to thermodynamics and other branches of science during the nineteenth century, revealing that a perfectly efficient heat engine was a myth, but at the same time enabling engineers to approach closer to the limits of the possible. From that work sprang the internal combustion engine.

Later, the discovery of quantum mechanics made possible the discovery and design of semiconductor components upon which our communications, computing and entertainment technologies are based. The world has changed incalculably during the last hundred years: many of the changes would probably not have happened had people not been prepared to spend money on pure research. Perhaps that is too strong a statement – what do you think?

We got off the subject here. But then, there is no subject: everything is created from stuff, so everything has structure, and everything feels forces, so the subject has no boundaries.

Keystones

In engineering, the keystone has no structural significance, but it has developed into a decorative element in many buildings. Sometimes, looking at a small window or porch, there is an unworthy suspicion that the whole arch, including the "keystone", is cast as one piece of concrete. This building is a Unitarian chapel.

21st century tubular scaffolding on the south transept of Gloucester cathedral. Note the outward lean of the south wall at left. It is being pushed by buttresses that are connected to the tower. They are referred to later in this page. This is not the subject of the repair. These arches are more recent than the Norman ones.

Ruins in Gloucester, near the library. Well built arches are very resilient. These have been here for a very long time, but are in quite good condition. A modern building has been built on to these ruins, rather incongruously.

Part of the North side of Gloucester cathedral, with newer pointed arches at left close to older Norman ones at right. Many English cathedrals and churches contain a mixture of styles like this. Evidently people then were not prevented from adding new styles to old buildings by considerations of incompatibility.

Domes

One of the lovely features of architectures, for example, Islamic and Christian, is the way that the idea of the arch has been extended far beyond the basic spanning and supporting element. Rotating the arch about the centre produces the dome. The dome of the Taj Mahal is a well-known example.

Another is Brunelleschi’s dome in Firenze. The other dome below is in Isfahan. The picture was downloaded from Welcome to Isfahan, which offers an informative guide to Islamic Iranian architecture.

The dome idea can be extended to the cone and polygonal pyramid, as in church spires, and of course the igloo. The oast houses of Kent, and many pottery kilns in the Midlands, also used cones.

Domes are surfaces with positive curvature. Surfaces of negative curvature are seen in the cooling towers of power stations, in the form of hyperboloids of revolution. These surfaces have the theoretical advantage of being developable from straight lines, like the once fashionable hyperbolic paraboloidal roof.

Here are some views of the sky-line of a typical small English city. We see the elegantly simple domes and minaret of the modern mosque, the intricately constructed tower of the medieval cathedral, surrounded by 20th century rectangle-esque, or trabeate, offices, the modern but ageing hospital, and the rusting gas-holder – nearly 900 years of history in building. The gas-holder has been painted pale blue since this picture was taken. The hospital is going to be replaced.

Note that the gas-holder is cylindrical because the pressure is equal all around, and that the top is slightly domed for rigidity. The circular shape allows the use of helical guides for expansion. With vertical guides, each cylinder could wobble inside the one below. Each cylinder has a helical inspection stairway which is high enough to reach the next level when the gas-holder is full.

The three pictures below are very large JPEGs showing general views of parts of the same city, the last being a part of a typical English suburb, with its numerous variations on the themes of semi-detached houses, detached houses, and bungalows. Here and there are older buildings that preceded the development from a scattered village, based around the route of a Roman road, to a large suburb. From the third picture we see that the cathedral is almost at the far side of the city. The town was originally a small religious community near the river. Because the other side of the river is a low-lying flood plain, the town developed almost entirely on one side. Even so, the streets nearest are frequently flooded, to the despair of the inhabitants, and the foundations of the cathedral are said to be permanently waterlogged. What would happen if they dried out?

The Brunelleschi dome in Florence clearly demonstrates one big difference between domes and arches, the distribution of forces. This particular dome comprises eight pointed sectors (on a globe map of the earth they would be called gores). Actually, the inner dome is so thick that at all levels it contains a horizontal circle. An arch is generally made of parallel sections, with the same width throughout. So the distribution of forces and weights is subtly different, leading to the other big difference between a dome and an arch.

The second important difference is that a dome can in principle be erected without centring, as the Inuit do with igloos. An arch needs centring, unless the halves are treated as cantilevers and held back during construction. This was done in the cases of the Eads bridge and in the Sydney harbour bridge. The dome can be built with falsework supported on the built sections.

On the other hand, arches are generally built on the ground, which reacts against the thrust. Domes, except igloos, seldom spring from the ground. They often include chains or cables around the base to contain the outward forces. In this they resemble bow-string arches or tied arches, except that in the case of a dome, the constraint does not interfere with the space within. In that sense, a dome can include its own abutments.

A plan view of a dome shows that if we imagine it divided into sectors, the width of the sectors is proportional to the distance from the centre. For a simple cable or arch with a catenary shape, the mass per length along the curve is constant.

From this information can you guess what the ideal shape for a uniform shell would be as compared with a catenary for a uniform arch?

The shape of this sea-urchin shell perhaps owes more to surface tension than to gravity. Many kinds of curves exist in nature and artefact, such as those of liquid bubbles and drops, in which the tension is constant throughout the surface, and the pressure difference across the surface is uniform. The pressure difference across the skin of an airliner or a space-craft is also constant all over, which is why cylinders and spheres are so prevalent.

Here are some eggs, one of which has hatched. Egg-shells are sometimes used as a symbol of fragility, and certainly a small piece of a small egg is not very strong. But opening a complete egg is not quite as easy as cracking a flat sheet, and some birds drop eggs to open them. The dispute between the big-endians and the little-endians in Gulliver’s Travels is well known as a symbol of factional behaviour. A sphere, an ellipsoid and a cylinder are all good shapes for withstanding pressure differences in either direction.

Salisbury Cathedral

Here are three diagrams, showing an arch, a dome, and a peculiar spidery object. To find out what it is, visit Salisbury cathedral and go to the beautiful chapter house. There you can forget about stresses, strains and bending moments, and just enjoy the view.

Salisbury’s chapter house is made by rotating an arch about one pier, producing a kind of arch with a narrow central pier and an octagonal wall around the outside. It dates from the middle of the 13th century. The central pillar at first looks amazingly slim, but it is only supporting about a quarter of the weight of the roof, and it feels virtually no lateral forces because of the symmetry. Its function is analogous to that of the neutral wire in the three phase electrical supply that would be invented hundreds of years later.

If the thrust of the ith arch of N is Ti, the total thrust is given by

which is zero in the symmetrical case where all the T_i are equal. For the case of an N-phase power supply, current in the neutral wire is

which is zero in the case where all the currents Ii are equal.

In the arch formula, the value N = 2 corresponds to a pair of arches in a normal multi-arch bridge.

This chapter house includes a medieval frieze of sculptures of early bible stories. You can also see there a Magna Carta from 1215. Only three other originals exist. A similar construction, but in a rectangular space, is found in Marienburg castle. These two links are about the roof of the Great Court of the British Museum and Great Court – B M.

This shows the interior of a fort on the lovely island of Jersey. Like the chapter house of Salisbury cathedral, it has a toriodal arch, but is continuous instead of being vaulted or segmented.

All these ideas are based more or less on the funicular, but if you use material which has strength in both tension and compression, you can depart considerably from funicular shapes. The further you go from the ideal shape, the greater the bending moment you have to support. A tent shows its departure from the arched funicular by sagging, and indeed forms a funicular in tension rather than compression. Roofs based on curved sheets in tension are now quite common. In the picture below you see only one strut per section, but in analysing any structure, we must not forget the surrounding supports, including the ground. You cannot make a structure comprising only ties, you need at least one strut per spatial dimension.

But to hold a rigid shape you need beams. The Roman Catholic Metropolitan cathedral in Liverpool is a cone, and the struts create far more outward thrust at the ground than a dome would make. Further still from the funicular is the Metropolitan Cathedral – Nossa Senhora Aparecida in Brasilia.

In Salisbury cathedral you can see the four main piers, visibly bowed by the weight of the tower, surmounted by a spire for which the columns were not designed. The columns are stabilised by inverted arches.

The spire is visible for many miles around, even though Salisbury is in a valley. Salisbury cathedral without the spire is unthinkable, yet that is how it was designed. The transmission of the weight of the spire from the polygonal cone to the four great pillars must entail some interesting engineering, especially as the tower was not designed to receive these forces. Some churches have a narrow spire on a wide tower. We should not imagine that the spire has been simply plonked onto a flat roof – inside the tower there will be the necessary structures to carry the weight of the spire down into the walls of the tower. This is a case where visible form does not follow function, but is none the worse for that.

We are so used to this building that we sometimes do not see that the tower and the spire are quite different in style. In a more purist age, the original building might have been "listed", and the building of the spire might not have been allowed. In fact, some other English cathedrals lost their spires, either by collapse or by deliberate removal.

In Salisbury cathedral you can also see the oldest clock in England, dating from the late 14th century.

In any case, this is surely one of the most beautiful buildings in England. Painting by Constable

Wells Cathedral has a most peculiar construction which looks like an inverted arch above a normal one. This was built as an attempt to strengthen the structure after cracks appeared in the tower in 1338.

A picture taken in the cloister garden in Gloucester cathedral, showing the large windows, and the buttresses that take the thrust of the fan vaulting that forms the ceiling inside the cloisters.

Vaults

Here is a picture of the north aisle of Gloucester cathedral. Diagonal arches spring from the walls and the piers. Between these, vaults, parallel and perpendicular to the aisle, complete the roof. The transverse parts open the way for the pointed windows. This is a very simple vault. The later styles produced some very complicated designs, including the skewed lierne vaults in Lincoln cathedral. Here the apices of the bays of the north and south windows do not meet: they are offset by a fraction of a bay that looks, from a photograph, like about the golden ratio cubed, 0.236, but this could be purely coincidence. If you did not find the golden section elsewhere in this building, and preferably in other cathedrals of the time, you would not have evidence of deliberate use of this ratio.

Here in the west vaults of Gloucester cathedral we see the juxtaposition of two completely different styles, in the first two pictures. This would probably not be done today, though we cannot know what will happen to current buildings in a hundred years time. The plain vault dates from the 13th century, while the more complicated western part dates from the 15th century, having been built after the western towers fell down. The third picture shows a part of the south transept.

Looking up at these vaults we are reminded of the triangulation of a truss. The force of gravity acts in the plane of a truss, but here it is at right angles to the structure, which is why the arches are needed.

Fan Vaulting

Here are pictures of a part of the cloister of Gloucester cathedral. From the narrow piers in the walls, arches grow out like the branches of a tree, forming fan vaulting. At the top they support pieces of stone that are flat, apart from the carving. These are held in place by the inward thrust of the arches. Whereas a dome has positive curvature in both directions, these surfaces have negative curvature in one direction and positive in the other. In the second picture the architect has suggested the flow of the forces from the fan down into the arch. Is this realistic? Wouldn’t it have been sufficient to have the three forces meeting at a point? Sometimes you see an ogee arch like this in a wall. In such a case the shape is purely artistic, because the wall pushes on the arch right across its width.

The fifth picture shows the external buttresses that make all this possible: in a sense the interior is a little fraudulent in that it looks as though the forces flow down into the wall. What does it matter – it looks good – one purpose of a religious building is to create an atmosphere inside that is conducive to worship. From this point of view, all else is subsidiary, including the outside. Question – Why does a tent need guy ropes while a church needs buttresses?

If you walk right round the cloisters, and look carefully at the vaults, you will spot some variations. On the south side, nearest the cathedral, the builders have made circular arcs that join along the centre line in a symmetrical manner. But on the west side of the cloister, the arcs on the west half of the vault are complete, while those on the east half are interrupted – in other words the west half was built first, and the east half was fitted to it. The effect is not obvious unless you are looking for it.

To the east of the cathedral, in Kings Walk, you can see a covered shopping area which echoes the fan vaulting. Unfortunately, this roof is a complicated array of steel tubes, which although easily understood and completely logical, looks a mess. Furthermore, the struts come down to no obvious supports above the shops. Under the coverings they must rest on cantilevered beam ends. The eye is given no guidance at all – it just does not know where to look.

Nearby is Kings Square.

Here are two pictures taken in Cheltenham, showing much neater solutions to the problem of supporting a glazed roof.

Returning to the subject of fan vaulting, not far away, in Cirencester, you can see a nice example of fan vaulting in a side chapel of the splendid church. From the top of the tower, on a fine day, you can view the town and countryside.

Between Gloucester and Cirencester, if you leave the main A417 road, you can view styles from pre-Norman onwards, in the villages of Brimpsfield, Syde, Winstone, Duntisbourne Abbots, Duntisbourne Rouse, Daglingworth, North Cerney, Bagendon, Rendcomb, Elkstone, Colesbourne and Coberley. Not one village lies on the A417, which follows the line of the Roman road, later called Ermin street, or at least it did, until recently built bypasses changed the route at several places, cutting fifteen to twenty minutes off journey times, and greatly benefitting not only drivers, but the people of Cirencester and Latton. Another bypass takes the route around Brockworth, again with great improvement to life in the village.

Then there are the wool churches at Cirencester, Fairford, Lechlade and Northleach, built with money from the wool trade. The above pictures of Northleach church show that the construction is much lighter than the Norman work at Gloucester. The concave octagonal pillars are unusual.

Here are details from a church in Evesham. The builders make the stone seem to float like feathers in the air. Yet it is the same material that was used to build heavy looking medieval castles.

These three pictures were taken at Birmingham airport, Geneva airport, and Jersey airport. A similar style can be seen at Stansted airport and inside some parts of Heathrow airport. In a sense, these supports represent fan vaulting simplified and taken to an extreme.

The other extreme is the simple pillar and flat roof so often seen at petrol garages, such as the one in this picture. The flat roof is usually made of I-beams, covered with some form of sheeting. Between the fan vaulting and the petrol garage we can place the mushroom headed columns of Maillart, which spread the load smoothly into the roof.

Here is a part of a footbridge between Balexert, a shopping centre, and Les Avanchets, a large apartment complex, in Geneva. The fanning out of the piers was more or less forced by the truss design, the alternative being four separate piers at each node of the bridge.

In this design the fan-out is taken to the extreme: there is no vertical part at all. Why do think this was done instead using a simple pier? Hint: look at the middle of the span.

Here are some example diagrams. They show how a trabeate or beam building and a fan vaulted building are two extremes. In between are many variations, including Maillart’s mushroom pillars.