On Lunes & Cracks

Conserving St Peter’s Basilica

The dome of St Peter’s Basilica in Rome is one of the most recognised structures in the world. It was completed around 1590 and was conceived by the genius Michelangelo. There are many reasons why it is special structure, but perhaps the most important is amongst the least well known.

By around 1680 cracks were being reported in the dome, which unsurprisingly caused some to question its safety. Concerns were exacerbated following an earthquake in 1730. Meridional cracking, associated with a dome’s tendency to spread at the base, was well known in the sixteenth century and therefore, following a detailed investigation, a recommendation was made to supplement the existing wrought iron hoops, which were intended to prevent spreading, with three or four more.

It would seem that the Vatican had travelled a distance since Galileo’s heresy trial and the incumbent Pope Benedict XIV, unlike many designers and practitioners of the day, was impressed by the progress made by scientists and mathematicians of the day. He therefore commissioned three of them; Thomas Seur, Francois Jacquier & Roggiero Boscovich to examine the subject. The publication of their findings in 1743 was a seminal moment for Structural Engineering, because they had based their conclusions on a mathematical analysis of the dome. It was the first known occasion when this was done in any meaningful way. Their approach included a model of the dome’s weight, its materials and two different behavioural scenario’s. The method they adopted for combining this information would today be called ‘virtual work’.

They concluded that the existing iron rings embedded within the dome were insufficient to prevent spreading and that the dome would collapse. They therefore proceeded to calculate the number and proportions of additional rings. This was of course a safe recommendation, however it overlooked the rather important fact that the dome had not in fact collapsed and was very much still standing.

Though they were three of the smartest mathematicians of their day they had made the same basic error that almost every graduate engineer makes at some point. They had placed their confidence in their model over what they could see with their eyes. One of the most important truisms of structural engineering is that a structure will remain in place until it has exhausted every possible means of standing. Consequently, if a mathematical model says a structure will collapse, but it stubbornly refuses to do so, then one is obliged to conclude that the model is wrong and not reality.

The presiding committee responsible for the church’s upkeep did what committees often do. They ignored the expert report and continued to monitor the structure. Benedict was also dissatisfied, but wished to persevere with a scientific approach. He commissioned a new study by a different expert Giovanni Poleni. 

Poleni criticised aspects of the first report and tackled the problem with a different approach. While he conceived his own mathematical model, Poleni was also aware of Robert Hooke’s work on the stability of arches, which is described in an earlier post [On Balloons, Chains and Arches]. He therefore imagined the dome to be split into a series of lunes each of which rested against an opposing lune on the other side of the dome. He then used a physical model to demonstrate that the line of thrust for a pair of lunes lay within the depth of their cross section and would therefore meet Hooke’s criteria for a stable arch. By this reasoning the whole dome, being a series of balanced lunes, would be stable in spite of its meridional cracks.

Nevertheless, Poleni also recommended four additional wrought iron hoops, which were installed in 1744. A further hoop was added in 1747, when it was discovered that one of the originals had in fact fractured.

While the application of mathematics to structural engineering problems, which was pioneered by Seur, Jacquier and Roggiero, would prove successful in the long run it is not difficult to see why it was unsuccessful to begin with. 

While mathematicians and scientists had been publishing treatise on engineering subjects from the early eighteenth century, architects and engineers of the day were unacquainted with mathematical argument and treatise were therefore largely ignored.

A second, and perhaps more significant issue, was the relative maturity of the respective disciplines. Eighteenth century mathematical models, though brilliant in their conception, were no match for 1,000 years of engineering experience, which had refined and optimised known structural forms about as much as it was possible. The only conceivable  advantage for science would be for the conception of structures for which there was no precedent.

This dichotomy caused a divergence in engineering practise between Britain and its European neighbours. While France and Germany forged ahead with academic schools of engineering Britain largely adopted an empirical approach. Unsurprisingly the continental Europeans produced more impressive academic works, however Britain prospered with its empirical approach, which produced closer alignment with the real world and therefore greater efficiency.

Britain’s engineers did not dismiss theoretical works, because they were less intelligent or less capable. They simply knew that the best academic theories of the day could not get close to matching the empirical approach they were pioneering.

There is a lesson in this for the modern engineer. Modern codes of practise are becoming increasingly academic and less practical. It is not clear to this engineer that the additional effort required to use them yields a justifiable benefit. I am also quite certain that as in the eighteenth century some modern methods are less efficient than the empirical methods they have replaced.

In some circles our profession needs to rediscover the once obvious truth that a theory, which does not match past empirical experience, is not a good theory.....particularly when it is more complex to use than its predecessor.

On Pyramids & Ziggurats

Smarter engineering than you might think

It is self evident that ancient structures were not conceived using modern codes of practice and that their designs were based on rules of thumb evolved from a process of trial and error, however there is perhaps some misunderstanding as to what this means in practise. 

It was not, as you might suppose, a process of edging successive designs slowly towards failure with fingers crossed; hoping to stop before you get there.

A rule of thumb, by its mere existence, presupposes the existence of mathematical relationships between objects. How else might proportions and limits be implemented. Furthermore, evidence suggests that trial and error was purposeful and based on underlying principles of logic. As we shall see ancient structures are more sophisticated than you might think.

Ziggurats were built in Mesopotamia on the great plain that lies between the Tigris and Euphrates rivers in what is today part of Iraq. These two great rivers carry vast amounts of silt, often depositing it along the way in times of flood. This created thick deposits of alluvial soil, which are ideal for agriculture, but much less so for constructing large, heavy buildings.

This was not the only geological issue to be overcome. With the Mesopotamian plain being covered to depth with alluvial soil, there is little building stone from which to construct monumental structures.

For this reason Ziggurats were constructed of bricks, made of locally available mud baked in the sun. Sadly, without a protective stone skin, most surviving examples are heavily eroded. That said, on account of their exposed condition, those which remain provide us with clues about how they were constructed.

 It would have quickly become apparent to the Mesopotamians that soft alluvial soils would undergo significant settlement when subjected to the weight of a ziggurat and their steep sided construction would have a tendency to spread at the base. 

Successive layers of construction would suggest that they paused and restarted the works on many occasions until the settlements and spreading eventually ceased. In this way the lower layers became layers of fill below a wide plinth or temenos, on which a great temple could be constructed.

This process is not unlike the modern technique of preloading soft ground with great berms of earth. The same technique has been used to improve sites adjacent to the River Clyde in Glasgow.

The Mesopotamians were not, however, satisfied with the pace of construction that this method afforded and they soon conceived another ingenious plan, which engineers today might consider modern.

After every eight or nine courses of brickwork they began to add a thin layer of sand containing matts of woven reeds and cables made of plant tissue. Together these innovations allowed the Mesopotamians to create a primitive form of reinforced earth not unlike that which is achieved today using geosynthetic grids and textiles.

The great weight of construction generates friction between the mud bricks and the reinforcement; clamping them together so that they cannot move relative to each other. This allows the tensile capacity of the reinforcing matts and cables, which is not possessed by the brickwork itself, to be mobilised such that the steep walls of the ziggurat are prevented from spreading laterally. If this were not clever enough the layers of sand in which the reinforcement was laid had two ingenious roles. Firstly, it would have helped to bed the bricks evenly onto the reed matts helping to ensure an even distribution of load and to prevent sharp or uneven edges from causing unwanted damage. Secondly, the sand would suck moisture from the mud-bricks and provide a route for it to escape. This leads to consolidation, increased density and greater strength.

The evidence is clear; Mesopotamian ziggurat builders were not simply stacking bricks until failure was reached. These innovations demonstrate a knowledge of complex engineering principles. 

The Pyramid’s of Egypt are built between the Libyan dessert and the western bank of the river Nile, as it flows towards its Mediterranean delta. On the face of it they appear to have much in common with Ziggurats. They are both large, heavy structures, with steep sides, constructed from masonry. They both impose massive loads at their base and are subject to lateral spreading forces.

This, however, is where the similarities end, because the great pyramid designer Imhotep came up with some rather different solutions. Perhaps the most obvious difference is that Imhotep, and those who followed him, adopted locally available limestone en lieu of mud bricks. It is a much stronger material, which requires a different treatment.

Perhaps the first thing to note is that a pyramid’s weight is not evenly distributed. The maximum pressure is exerted below its centre, reducing towards the edges. This means that a pyramid’s core and perimeter will settle differentially. 

We know that Imhotep understood this because he devised a clever method of preventing the rigid stone blocks from being fractured by said settlement. 

Pyramids are not solid structures. Examples investigated at Saqqarah, Meidum and Dahshur consist of a solid stone core laid at a steep angle, which is surrounded by independent concentric squares of masonry. The inner portion of each square, roughly 4/5, is of roughly cut stone laid in mortar while the final 1/5 is of dressed stone with smooth contact surfaces. The central core also has an outer facing of dressed zone.

Each independent square can slip relative to its neighbour, thus accommodating differential settlement. The efficacy of this process is enhanced by the smooth surface of the facing stones. 

Nevertheless, cutting and dressing smooth stone surfaces is difficult, time-consuming, expensive work, particularly using bronze age tools. It therefore made sense to minimise this type of work by using rough cut stone as the backing, although this does have consequences. While the dressed facing stones have good contact surfaces that distribute load evenly and provide a solid stable base, the rough cut stones have poor contact surfaces resulting in greater potential for consolidation and outward movement.

Imhotep would have known that the inclination of the dressed facing masonry had to be optimised so that it leans into the rough stone and contains its tendency to spread. It has been found that the angle adopted corresponds to the prime numbers 2, 7 & 11. 

These observations demonstrate that Imhotep, and those who followed, had a clear understanding of structural load paths and of building materials. Furthermore, what evidence we have for design by trial and error falls within this rational framework.

The Stepped Pyramid at Medium and the ‘Bent’ Pyramid at Dahshur are good examples.

The former has a strange shape, which archaeologist originally presumed to be the result of stolen facing stones. It is not clear why one would steal from the top and not the base;  engineering appears to provide a better explanation. While the construction follows Imhotep’s settlement mitigation strategy some of the stone has been found to be of poor quality. It’s friable nature caused a local collapse by creating the conditions for a slip plane to develop thus causing the loss of several structural layers due to spreading. 

Similarly, the so called bent pyramid clearly shows that the designer realised part way through the build that the angle of inclination was too steep and had to be reduced to maintain equilibrium and thereby prevent spreading. This demonstrates that he understood something was going wrong and then knew what to do about it.  

It follows that for both the ziggurat’s of Mesopotamia and the pyramid’s of Egypt there is clear evidence of structural principles being understood and refined by purposeful trial and error and captured in rules of thumb with a basis in Maths. 

One might argue that they are good examples of qualitative design. They are certainly a reminder for modern engineers that complex sums are secondary to clear thinking about underlying load-paths and a practical knowledge of material.

On Ice Arches

Understanding ice flows in the Nares Strait

Today an article on the BBC website caught my interest, but perhaps not for reason the author intended. The article describe the premature disintegration of an ice arch that bridges between Greenland and Ellesmere Island. The arch had blocked the so called Nares Strait, much like the damming of a river, thus preventing the southern migration of the ice flow. Viewed from space the arch is spectacular. The images below show before and after disintegration.

The point of the BBC article was to highlight the effect of a changing climate and what might result from increased ice flow. I have no interest in discussing this; it doesn’t remotely fall within my expertise and I don’t have anything new or novel to bring to the topic. 

Instead, what initially crossed my mind was whether this, and similar structures, I assume other examples must exist, might be the largest arches on the planet. I have no idea whether this is the case or not, but one has to think that they must be contenders.

My mind then turned to the question of why such arches form and whether they were in fact true arches at all. As it turns out I think the two questions could be linked. Not that I really know anything about this subject either, but I do at least feel more comfortable to speculate on the basis of structural principals.

The ‘before’ photo shows the ice flow firmly interlocked with the coast of each land mass. The interlocking extends over a distance, which exceeds the width of the strait and that seems significant to me. The reason I view this as important is that a member whose depth exceeds half its width fits the definition of a deep beam. A deep beam is one that is of sufficient depth that its behaviour is no longer governed by bending effects. I therefore wondered whether what we actually had was a deep beam rather than an arch. That gave me cause to think about how a deep beam actually works. It was this that lead me to postulate how, and I suppose why, an ice arch might form.

The effective depth of a deep beam is roughly equal to its span. It follows that there is little load resisting contribution from the ice flow beyond that point. Normally within a deep beam’s effective depth the stresses behave a bit like an imaginary tied arch with a compression zone at the top, which pushes outwards towards the sides. There must also be a corresponding tension zone at the bottom of the beam, the tie, which prevents the internal ‘arch’ from spreading and tension cracks from forming.

The trouble is that this isn’t what we see in the satellite image. The tension zone is completely missing, leaving the observed arch profile at the base of the ice flow. The obvious reason for this would be that ice is an anisotropic material; it is strong in compression, but has little or no strength on tension. Self-evidently, since there is is an absence of tension capacity, the section of ice exposed to tension has, presumably, cracked and floated away prior to the picture being taken, thus leaving behind an arch profile.

That said, while this may explain the formation of an arch profile it can’t be the whole story. If there is no tie stopping the ice arch from spreading why hasn’t the arch itself collapsed? The answer must be that the land mass on either side of the ice flow provide strong buttresses, which contain and resist the outward thrusts. 

This, however, still doesn’t entirely explain what is going on, because if the buttresses are secure then no tension can be present in the ice flow and if that is so why did the bottom of the ice flow fall out.

Assuming the buttress theory to be correct I can think of several potential mechanisms, but I am not sure which, if any, are contributory.

My first thought would be that perhaps the ice formation takes time to interlock with the land mass and the arch is able to spread while it is still forming. Perhaps during the formation process ice at the land interface cracks and breaks, as the ice flow moves south only becoming solid and immovable as it is slowly pushed and squashed into all the available gaps. Perhaps there is also undulation in the ground and some of the ice rides up over the shallow flat parts until it is is resisted by projections.

Maybe, despite the appearance of static resistance, even the arch is moving slowly southwards and the buttresses gradually shift and adjust. Not enough for the arch to fail, but enough for tension cracks to form at the base of the ice flow.

These would seem, at least to me, plausible explanations for the formation of an arched profile. The question is whether what we are actually seeing is in fact a true arch. We had been considering the possibility of deep beam behaviour, but have, without noticing, slipped back into describing the structure as an arch.

I think it would be just as correct to view the structure as a buttressed deep beam with the effect of rigid coastlines replacing the beam’s tension zone. Though this is perhaps an unusual description I happen to think it is a better description than a true arch. I have three reasons for this.

Firstly, I don’t think the ‘arch profile’ forms without the ice flow first trying to behave as a deep beam. Secondly, due to the re-distribution of forces within the depth of the structure, caused by the buttresses adjusting I don’t think you can avoid the conclusion that stresses are set up across the full width and effective depth of the ice flow. Thirdly, using classical arch theory, which has been discussed in prior posts about masonry arches, I don’t think you could stop the thrust line leaking out of the optimal arch shape due to the depth of the structure and the behaviour of the abutments...actually I’m not sure that’s not just another way of stating reason two.

So, that would be my answer. I think that the base of the ice flow has the appearance of an arch, but is in fact a buttressed deep beam, or maybe that’s just a speculative folly on my part. 

One notable point is that I have offered no comment on the ‘after’ picture. I guess that’s because I wanted to talk about the apparent arch itself, it seems obvious that it would start to fail if the ice melts. Maybe the ‘after’ scenario shows what the ice flow looks like as the structure is forming and the ice is being squashed together. I have no idea if that is true, but aesthetically I am drawn to the idea of a circular process. Perhaps if the temperature is lower at higher latitudes a new wider ‘arch’ will form further north.

On Devorgilla Bridge

Some Characteristics of Stone Bridges 

Last summer I took a camping trip with friends and family to the Scottish borders. We happened to visit the town of Dumfries, which straddles the river Nith and has a number of interesting bridges. One of these is known as the Devorigilla Bridge, although in reality the name applies to a succession of bridges dating to circa 1270. The first of these was built by, or at least funded by, Lady Dergovilla, the mother of John Balliol, who became king of Scotland in 1292. Dergovilla was also responsible for founding Balliol college in Oxford. 

I new of John Balliol, he was king at the outset of Scotland’s War of independence, and I knew of Balliol college, though I had not until this point linked them together. I did not know anything of Devorigilla Bridge, however by observation it was clear to me that it was certainly old and had undergone several periods of change …. and that was interesting.

The bridge spans the river with six masonry arches, which appear to be of equal size. The masonry is generally a red sandstone, which is squared and coursed, although the parapets are grey rubble, which is random in some locations and coursed in others.

There are notable features at either end of the bridge, which appear to provide some clues about its past. The first arch, which springs from the western river bank, is pointed while the five remaining arches are semi-circular. On the eastern bank there is a staircase, which descends from the bridge to current street level.

Although semi-circular arches were used by the Romans, and therefore predate later Gothic [pointed] arches, the change from romanesque to gothic happened in the 13th century prior to most stone bridges in Scotland being built. Romanesque designs returned around the 16th century during the renaissance period. 

This implied that the single gothic arch was much older than the five romanesque ones. Anecdotally this would seem correct, as we might expect a bridge to be re-built after being overwhelmed and destroyed by flood waters. It seems more plausible that the single arch located next to the river bank would survive intact rather than those located in the middle of the river where the flow is greatest.

On the opposite bank a staircase to exit the bridge seemed like an odd feature. While today the bridge carries only foot traffic, and vehicles pass over more modern bridges, in its heyday Devorgilla bridge would surely have carried carts and wagons. It must be likely that tolls on goods would have been collected to help fund the bridge. Why then would the bridge have stairs at one end? 

The conclusion I drew is that the bridge was not constructed this way. There must have been at least one additional span, which was for some reason truncated or removed at a later date. It seemed to me that the likely period when this might have happened would be the 17th century. My logic for this was the distinct parapet stonework, which I have already mentioned in passing. 

Bridges before the 17th century tended to have coursed, well-squared stones, whereas for a 100 years or so afterwards courser rubble masonry was often used. If then the end of the bridge had to be re-modelled, due to the end span being truncated, the works would necessarily need to include the parapet. The opportunity may then have been taken to look at the parapet in general. This work would then be reflected in the courser stonework that can be seen in the masonry today.

Having made what seemed to me some reasonable engineering observations I wanted to know more about the history of the bridge and I was therefore delighted to find a small plaque located near to the bridge on the western bank of the river. The plaque confirmed some of my suspicions.

The bridge was built in 1431 to replace the first wooden structure erected by Lady Devorgilla, but was largely destroyed in the 1620’s resulting in a nine arch bridge being built to replace the 1431 version. 

It also noted that in 1790 the Buccleuch Street bridge was constructed, which currently carries road traffic to the north, and has itself undergone alterations. In order to build the Buccleuch Street bridge land on the eastern bank was built up to meet it. This action would have required the Devorgilla bridge to be truncated.

1790 is probably a bit late for the general remodelling of the parapets, although some works would have been required where the steps were created. It certainly looks like more work has been done to the parapets above the final two arches.

It follows that while some of my conclusions remain speculative, their broad outline seem to be borne out by the official history. My curiosity has therefore been sufficiently satisfied to move on to something new, as my interest lies in observation of the structure rather than pure historical research. I shall leave it to others with greater historical interest to work out the details and show where I have gone wrong….not too much I hope.

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.