On Horseshoe Arches

Why use a horseshoe arch?

Historically many types of arch have been used to construct buildings and bridges. The underlying reasons for adopting each of the main types is normally relatively clear, however in the case of the horseshoe arch this is not so. The Romans predominately used the semi-circular arch. It has a geometrically simple form, which is straightforward to understand and straightforward to construct. It is perhaps the reference point for comparison with other forms.

The segmental arch is much flatter and is often used for constructing bridges. It is less efficient structurally than a semi-circular arch, in the sense that the lateral thrusts are greater. This means that larger abutments are required. The reason for choosing a segmental arch is to avoid creating steep inclines for the ramps onto a bridge. This is the reason heavy abutments are a price worth paying.

Pointed arches are characteristic of gothic architecture. There are very good reasons for adopting them. The first reason is geometrical. The rise of semi-circular arches varies with the span. Pointed arches can have the same rise for different spans. This is a useful trait when vaulting a gothic cathedral. A second benefit to pointed arches is that the lateral thrust at the abutments is less than for a semi-circular arch of the same span. This is also important for framing gothic cathedrals which are propped by delicate flying buttresses. Another reason, which is often sighted, though of little importance from a structural perspective, is that pointed arches can be used to create a higher ceiling. This enabled more light in the building and was viewed as being symbolic in a cathedral structure, because it was closer to God.

Now, returning to the archetype with which we began, the horseshoe or moorish arch. A google search will reveal that the purpose of this form is not at all clear. Even academic writings seem to be rather woolly on the topic. Explanations, particularly in architectural papers, seem to focus on somewhat subjective views about symbolic meaning and many of the structural explanations are far from compelling. 

Some point to the provision of a wider seating, which will reduce the bearing stress at the base. This is of course true, however since the compressive stresses in masonry structures are low to begin with its not really a material observation. 

Others say that horseshoe arches can be built without centring, as apposed to a semi-circular arch, which cannot. This argument makes no sense at all, at least to me. I do not see any property of a horseshoe arch relative to a semi-circular arch that would make that so.

There is a view, which seems to make sense if the horseshoe is seen as an intermediary between semi-circular and pointed arches, that the horseshoe provides a way of increasing the height of a space. It is self-evidently a taller structure than a semi-circular arch of the same span and, prior to the pointed arch, it would certainly be a good way to achieve a more spacious building interior with greater opportunity for light to penetrate.

One might also argue that the horseshoe is simply an aesthetic choice that was favoured by Moorish designers. This explanation is not terribly satisfactory; it would be disappointing if the traditional Roman semi-circle was replaced with a horseshoe just because it looked nice. I think there is more to it than that.

Analysis of a horseshoe arch with the same span as a semi-circular or segmental arch will show that it has a lower horizontal thrust at the base than either of the other two options. This is clearly an advantage, because the abutments can therefore be smaller. That said in order to re-directed thrust from horizontal to vertical the upper part of the arch must resist bending forces, which could cause buckling if the arch is too thin. 

An example of this form of failure occurred in 2004 when the concrete structure of the newly completed terminal 2E at Charles de Gaul airport collapsed killing 4 people.

The tendency to buckle can be resisted if the arch is confined on either side. Indeed, in every example I have seen of masonry horseshoes they are either confined by spandrels or are balanced by other arches pushing with an equal force in the opposite direction. Of course potential buckling forces were reduced by the subsequent development of the pointed arch. 

Nevertheless, if the upper part a horseshoe arch has sufficient bending capacity, in the case of modern materials, or is sufficiently confined, in the case of masonry, then the resulting thrusts at the base of the arch are somewhat reduced and this is a distinct advantage.

One of the reasons I think that Moorish designers knew what they were doing and were thinking about structural load paths is evident in the detail of their construction. It is noticeable that the portion of the arch below the semi-circle is invariably formed of a single piece of stone or from a series of specially shaped blocks that do not follow the standard format of the voussoirs above. This is essentially to ensure that the angle of the joints all point inwards thus making it easier to construct, because the stones cannot slip outwards before the arch has been completed. It likely also makes the arch less prone to fail due to lateral thrust in the permanent case.

In summary I rather suspect that the horseshoe arch was originally developed as a means of amending a semi-circular arch in order to create a taller space. It seems like a logical step to simply use a larger proportion of the circle. Through experimentation, probably with single arches to begin with, the behaviour of such structures started to be understood, which ultimately led to some of the rather impressive arcades and other structures that came to characterise Islamic Architecture to this day.

This is of course a speculation on my part, as I have not completed a detailed historical study of horseshoe arches. I trust however that being based on the structural properties of horseshoe arches, it has at least some interest and merit. I am certainly no less impressed by the clever use of horseshoe arches than I am the semi-circular Roman arches that preceded them or the pointed gothic arches that followed. I also have no doubt that the Moorish legacy of horseshoe arches in Spain would have been an important influence on medieval architecture in Europe thereafter.

On Gravity Glue

The importance of equilibrium

Michael Grab is an artist with a website called Gravity Glue. I think its a great title, which describes perfectly his stone balancing art work. It’s also a really good description for the concept of equilibrium, which is probably the most important principle in structural engineering. For unless there is equilibrium none of the other concepts much matter.

If you have not come across stone balancing before it is worth looking at Gravity Glue. The essential idea is to stack a pile of rocks one on top of the other without them toppling. The key is to find stones of different shapes and sizes and to join them in a way that intuitively seems unstable. There is nothing holding the stack of rocks together other than gravity acting on their weight and pushing them together; hence the term gravity glue. Finding precisely the right position and angle to stack each rock is tricky. It takes experience and patience to find the point of balance.

I don’t just like Michael Grab’s stone balancing, because of the engineering parallel, but also because the arrangements he creates are attractive in their own right. Some of this is attributable to the visual backdrops, but it is also more than that. His arrangements are clever and visually interesting. I too could balance some rocks on top of each other, but I don’t think the result would come close to what Grab achieves. He manages to surprise our innate sense of balance and to challenge our perception of what ought to be stable. I think it is these qualities, which provide the visual interest.

Equilibrium is simply the term that engineers use to describe a structure that is balanced. In order for balance to be achieved two criteria must be satisfied. Firstly, the magnitude and direction of all the forces acting on and within the structure must add up to zero i.e. for every force acting upwards there must be one of equal magnitude acting downwards. Similarly, for every force acting left there must be one of equal magnitude acting right. If this condition is not satisfied the structure will not be stable and will move in the direction of the unbalanced force. For example, if the force acting to the left is greater than that acting to the right then the structure will move to the left.

The second condition of equilibrium is that the moments resulting from the forces acting on a structure must also add up to zero. Moment is a turning action, which is the product of a force and its distance from the point of rotation. For this reason a small force acting at a large distance can generate the same moment, as a large force acting at a short distance. This abstract concept is easily illustrated by considering an adult and a child on a see-saw. To find the point of balance the adult must move closer to the fulcrum of the see-saw than the child. Conversely, if the adult sits too close to the end of the see-saw the child will be propelled upwards and the adult downwards. It follows that if the forces acting on a structure are balanced, but the moments are not, then the structure will topple in the direction of the unbalanced moment.

Applying these principles to stone balancing the forces acting on a stack of rocks are the self weights of the stones due to gravity. The first of the two equilibrium conditions is satisfied  by default. Since the stack is supported on the ground the ground will push back on the stones with an equal an opposite force. If it did not then the rocks would either sink into the ground or they would take off. The tricky bit is therefore balancing the moments.

Since the rocks are of different sizes and have unusual shapes their centre of gravity [the axis through which their weight acts] does not act through the point where the rock above is in contact with the rock below. An overturning moment is therefore generated. To balance the overturning moment the rock above must be rotated to move its centre of gravity towards the point of contact or the next rock up must be placed such that it generates a restoring moment in the other direction. As long as one of these two options are selected then the stack will be in equilibrium and will remain stable.

That said, if the weather were to turn and a strong wind were to blow then an additional external overturning moment would be generated. If this moment exceeds the effect of the stones self weight then the stones will topple. This thought introduces an interesting subtlety to the concept of equilibrium.

A structure can be in a state of equilibrium, however if that equilibrium is vulnerable to disturbance by an external action, particularly a small disturbance, then it is said to be in a state of unstable equilibrium. This is precisely the reason why stone balancing is difficult and requires such patience. The arrangements are invariably in a state of unstable equilibrium and in many cases the stability of the lower stones relies on the presence of the upper stones. I imagine that it must require octopus-like qualities to hold the lower stones in precisely the right place while the upper ones are added.

It is precisely this unstable form of equilibrium, which makes Michael Grab’s creations so visually interesting. Much like trying to stand a pencil on its end it does not seem possible to find the ‘Goldilocks point’ were there is neither too much over-turning moment in one direction nor in the other. Herein lies the patients and the skill.

Of course, while it creates an interesting work of art, unstable equilibrium is not at all desirable in buildings and bridges. It is self-evident that you want such structures to be resilient to disturbance so that they do not easily become unstable. Indeed this is a key structural design principle.

This does not stop an engineer from appreciating Michael Grab’s gravity glue. I would suggest it enhances your appreciation of his art.

On Gothic Cathedrals [yet again]

Flying Buttresses 

The flying buttress is synonymous with gothic cathedrals. It moves their structural skeleton out with the building envelope and exposes it to view. It is probably for this reason that it is readily identifiable as one of the defining features of gothic design.

The name flying buttress is also interesting, because if they were invented today we might simply have called them props. The term flying buttress is, I think, a reflection of the history and development of masonry structures.

As we have learned in prior blog posts, early barrel vaulted roofs required thick heavy walls to resist the lateral thrusts, which result from the vaults’ tendency to spread under the influence of their own weight. It was possible to obviate the need for heavy wall construction by concentrating these thrusts using ribbed vaults. This meant that the outer walls need only be reinforced with localised buttresses.

This was all well and good if the church had only a nave, however if there were aisles either side, or additional cloisters, then in order to avoid being in the way the buttresses had to be moved farther away from the nave. This led directly to a requirement for masonry props to ‘fly’ from the nave, over the aisles, and onto external buttresses. 

The flying buttress must resist three different types of loading. In the first instance it must resist its own self weight. It does this by forming a relatively flat arch, just like a segmental arched bridge. The vertical weight of the arch is supported on one side by the nave and on the other by the buttress. The lateral arch thrust is resisted by pushing back against the nave vaulting and against the external buttress. The thrust produced by the nave vaulting is much larger than that produced by the self-weight of the arch and therefore both the nave and the external buttresses can readily accept this load.

Of course, the thrust produced by the nave vaults is the second form of loading to be resisted. Unlike the curved load-path from the self-weight of the flying buttresses this load-path is essentially applied in a straight line, which is probably why the top surface of many flying buttresses is linear and not curved like their soffit.

There are two ways in which the nave thrusts may cause the external buttresses to fail. Firstly, they might rotate about their base due to the thrust being applied at their head. This would be an overturning or toppling failure. Providing there is sufficient mass in the buttress to provide a restoring force overturning will not occur. This is primarily a question of geometry.

The second potential mode of failure is a line of shear extending from the flying buttress to the outside face of the external buttress. In this scenario the top of the buttress is simply pushed laterally relative to the masonry below. To prevent this from happening most buttresses have a large pinnacle, whose weight squashes the shear surfaces together in order to prevent a crack plane from forming. It is in effect the application of a pre-stress, much like that which was encountered in a prior post about gothic window tracery.

The final type of loading to be resisted is wind load. Most cathedrals have a large wooden roof located above the masonry vaults. Without flying buttresses to transfer load from the base of the roof into the external buttresses the wind would generate an unacceptable thrust at the head of the nave walls. There is also a view that the weight of large timber roofs would be too great for timber ties to prevent them from spreading and consequently the flying buttresses must provide a load-path for restraint to roof spreading as well as a route for transferring wind load.

It is the requirement to provide restraint to the timber roof, which is responsible for the presence of high level flying buttresses located above those which prop the nave vaults. 

Something else which is interesting about flying buttresses is how load is actually transferred into them. This is not a trivial question. It is perhaps a statement of the obvious to say that flying buttresses are located outside the nave, while the vaults are located inside. What is perhaps less obvious is how load transfers from one into the other.

The flying buttresses are actually located just above the level at which the vaults are sprung on the inside. This is done to help facilitate lateral load transfer.

The masonry walls and piers in a cathedral are not normally solid, as you might suppose, and neither are the vault conoids. They are generally formed of dressed stone either side of a rubble-mortar infill. Medieval masons did not trust the rubble infill to transfer the vault thrusts from the solid ribs and therefore they would include full depth ‘through-stones’ known as ‘tas de charge’ just above the level at which the vault ribs spring from the internal piers. This is reflected in the external level of the flying buttresses.

The tas de charge was generally located at the top of the pier capitals and below the point at which the transverse and diagonal ribs [assuming a quadripartite vault] run together. This section of masonry was formed from several courses of single stones. There are three advantages of using single stone courses.

The first advantage is that they are able to bind the piers together and stop the dressed facing stones separating from the mortar-rubble infill. Secondly, they enable the tas de charge to transfer load into the flying buttresses efficiently. Finally, these courses can be placed without formwork before the ribs are constructed.

From the necessity for pinnacles, to pre-compress the external buttresses, to the positioning of a tas de charge to ensure that vault thrusts are transferred effectively, it is clear that medieval masons understood exactly what the load paths were in a system of flying buttresses and that they had thought about the details carefully. It is also clear that though the concept of a flying buttress is relatively simple there is actually some relatively complex thinking required to execute that concept.

On Gothic Cathedrals [again]

Quadripartite Vaulting

Another feature of Gothic Cathedrals, which makes their appearance distinctive, is the quadripartite vault. To full appreciate the impact of this innovation it is necessary understand what came before.

The first vaults are often attributed to the Romans, however both the Greeks and the Egyptians are known to have used them. Most early vaults were of the barrel variety, which is essentially a semi-circular arch extruded into the third dimension. The Chaldean version was formed of a series of arches inclined to one side. The inclined angle allowed them to be built without centring, assuming of course that the end was buttressed.

Masonry barrel vaults are useful for bridging long rectangular spaces, however they require heavy walls to support them due to the lateral thrusts imparted at their base. This is the same action that is found at the base of all arches and has been discussed in prior posts.

Since heavy walls were required to support barrel vault roofs it was difficult to fit large windows into the elevations. For this reason early churches constructed in this way were dimly lit. The Romans found an answer to this problem by interlocking barrel vaults in perpendicular directions. This concentrated loads at the junctions between vaults allowing them to be supported on large piers. The junctions between the vaults are known as groins leading to the eponymous name groin vaults. The obvious advantage being the ability to create windows in the elevations.

Groin vaults were structurally efficient, however they required very skilful masons to carve the groins’ complex geometry. For this reason they fell out of favour with the collapse of the Roman Empire.

Subsequently medieval masons introduced their own solution, which is known as the ribbed vault. Instead of constructing groins they bridged the building with a series of masonry ribs. Each bay had two diagonal ribs and three transverse ribs. This is known as the sexpartite ribbed vault, because the ribs divide the vault into six parts.

An advantage of this form is that load is concentrated at six piers, which means that, as with groin vaults, heavy walls were no longer required. Consequently, the space between the ribs can be used to form windows.

Since the ribs convey load to the supporting piers the infill between them need only span the short distance between the ribs. This was an improvement over the groin vault, which meant the infill could be both thinner and lighter. This in turn reduced the weight to be carried on the piers and also the magnitude of lateral thrust to be resisted.

One of the initial problems with sexpartite vaults was one of geometry. The span of the diagonal ribs is greater than the span of the transverse ribs. It follows that if the ribs are semi-circular in profile, like the barrel vaults from which they evolved, then the crown of the diagonals is necessarily at a different level to the transverse ribs. Masons solved this problem by using pointed arches for the transverse ribs. This meant that the level of their apex could be independent of their span. 

This innovation was also beneficial structurally. Since the inclination of a pointed arch is steeper than an equivalent semi-circular arch the magnitude of the resulting lateral thrusts is reduced, which also means the buttressing requirements are less.

Another problem with sexpartite vaults is that the intermediate piers carry less load than the diagonal ribs and this could lead to differential settlement. Medieval masons therefore began to omit the intermediate ribs. This had the advantage of concentrating both vertical loads and lateral thrusts at only four positions in each bay. It also limited the number of buttresses.

This new form of vaulting was known as the quadripartite vault, for the obvious reason that it was divided into four parts by the ribs. The effect it had on gothic architecture was profound. It led directly to the development of the flying buttress and also permitted spectacular windows to be constructed in the large spaces between ribs. Furthermore, the use of pointed arch ribs was instrumental in allowing cathedral roofs to be built ever higher. Not only did this lead to some astonishing spaces, but it was also of symbolic importance, as the resulting cathedrals were able to reach evermore heavenward.

 

The quadripartite vault was not the final stage in the long evolution of masonry vaulting, but it probably was the most definitive development. It is testament to the expertise of the Medieval Masons and the confidence they had to open up elevations to the extent that they did.

The legacy they have left remains with us to this day and will no doubt be with us for some time to come.

 

On Gothic Cathedrals

How tracery works

When it comes to gothic cathedrals most people tend to think of flying buttresses. There is of course no doubt that flying buttresses were a tremendous invention and are indeed an important part of gothic architecture.

Nevertheless, while flying buttresses are rather clever devices that provide a clear load-path, the thing that always leaves me astonished is how gothic tracery works. Even when you know its secret the slenderness of gothic tracery still appears to defy the laws of physics. Perhaps, one of the best examples is to be found at Gloucester Cathedral.

The stained glass window at the end of the Chancel is reported to be the size of a tennis court, and it fills almost the entire gable. The stonework seems impossibly thin, how on earth do the mullions manage to span from floor to roof? 

Before we answer that question we are first going to take a diversion into my childhood. Every year, after Christmas, one of the things that we liked to do as a family was settle down in from of the television to watch the World’s Strongest Man competition, which back then was televised on the BBC. We would cheer on Geoff Capes, who won the competition on several occasions, and was always in with a chance.

One of the delights of the competition, which has been lost in its modern incarnation, was the nature of the games in which the competitors were required to compete. While truck pulling and car lifting has been retained, but back then there was arm wrestling, bending iron bars behind the neck, balancing weights on your head and other such fun. Another of the lost games was brick lifting. If you have never seen brick lifting before its not as straightforward as it sounds. The bricks are laid out horizontally on table and are lifted by stretching out the arms and grabbing the stack at either end. It is self evident that simply holding the two end bricks and lifting would not work. There is nothing joining the bricks together and therefore gravity will keep them firmly planted on the table. To lift the bricks it is first necessary to apply axial pressure to the stack in order to generate friction between the bricks. The magnitude of the frictional force must exceed the force of gravity in order that the strongman can lift the stack without the bricks falling out. 

It is also important to maintain the axial force directly through the centre of the bricks to prevent an unstable hinge from forming. This is harder to do than it might seem because while the arms are clamping they are also required to be lifting. You can see in the image below Geoff Capes’ right arm has moved slightly forward and a hinge has already started to form. It will not be long before the bricks experience a rapid increase in their entropy!

 This may have seemed like an odd digression from gothic cathedrals, however it is nonetheless a relevant digression, because it illustrates the principle of prestress that is at work in window tracery. In the case of brick lifting the prestress, which is applied to the bricks before they can be lifted, is provided by the strongman. It is a far greater test of his strength than the actual weight of the bricks themselves. For as long as he maintains compressive force in the stack of bricks it will continue to bridge between his two hands. Since the dominant behaviour is compression the load-path he has created is actually that of a flat arch.

We can apply the same principle to window tracery by turning the problem on its side. But before we do this let us considered the case without prestress. If we imagine the stone mullions as a tall stacks of bricks spanning between the floor and roof it would be rather easy for the wind to simple blow them over. The reason for this is because masonry can resist compression, but not tension. When the wind blows on the window load is transferred from the glass into the stone. The stone begins to bend causing it to take up a curved profile. The windward side of the curve is squashed and is therefore in compression, while the leeward side is stretched and is in tension. Of course masonry cannot resist tension and therefore the masonry fails. 

It follows that the purpose of compressive prestress in window tracery is to overcome the tensile forces that wish to cause bending in the stonework. Providing the compressive stress in the mullions exceed the tensile stresses generated by the wind they will remain stable and can arch from top to bottom. The obvious next question is where the prestress comes from? 

The answer is both simple and clever. Medieval masons simply built some rather heavy masonry above tracery windows demonstrating that they understood exactly what the load-path was. In the example above at Gloucester Cathedral the head of the primary mullions have cleverly been bent to the form of pointed arches and are actually supporting part of the roof, as well as masonry above.

This load path does of course have an important implication that is not necessarily evident straight away. Arches produce lateral thrusts, which must be resisted or they will collapse.

In the example above there is one large arch, which spans the chancel and two subordinate arches that fall within the larger arch. It is also possible to divide the larger arch into three parts; a central arch bridging the two internal mullions with buttresses either side, which transfer load into the two outer mullions. In reality the complete load-path is a combination of the two descriptions and it ensures that there is sufficient compressive load in each mullion.

If we begin with the two subordinate arches it is clear that the thrusts, which push towards the middle of the window are balanced against each other. It is no accident that the heaviest transom in the window extends from the base of each arch and joins them together.

It is also not an accident that the outward thrust from the large arch and the subordinate arches occur in the same place, though the method of resistance is not obvious from inside the chancel.

The key is of course the use of heavy buttresses on the outside of the chancel and that brings us back nicely to where we started, the humble buttress and its more elaborate cousin the flying buttress, which we will tackle in a different post.

Notwithstanding all of the above the ability of gothic tracery to resist all that nature can throw at it still amazes me!